FREQUENCY ANALYSIS

Transcription

FREQUENCY ANALYSIS

OFFICE OF PUBLIC WORKS
FLOOD STUDIES UPDATE PROGRAMME
WORK-PACKAGE WP-2.2
“FREQUENCY ANALYSIS”
Final Report
Department of Engineering Hydrology
&
The Environmental Change Institute
National University of Ireland, Galway
September 2009
Table of Contents
List of Figures.............................................................................................................. iv
List of Tables .............................................................................................................. vii
List of Equations........................................................................................................ viii
List of Symbols and Abbreviations ............................................................................... xi
Abstract ..................................................................................................................... xiv
1
2
3
Introduction and data............................................................................................. 1
Trend and Randomness ......................................................................................... 5
Descriptive Statistics and Inferences from them .................................................... 8
3.1
Descriptive Statistics ..................................................................................... 8
3.2
Preliminary distribution choice with help of Skewness versus Record Length
Plots .................................................................................................................... 25
3.3
Preliminary distribution choice with the help of L-Moment Ratio diagrams. 29
3.4
Largest recorded values and outliers ............................................................ 34
4
Probability Plots and Inferences from them ......................................................... 36
4.1
Probability Plots .......................................................................................... 36
4.2
Linear Patterns............................................................................................. 40
4.3
Non Linear Patterns ..................................................................................... 41
4.4
Variation of shape of probability plot with Skew coefficient........................ 45
4.5
Flood volumes associated with largest peaks on convex probability plots .... 49
5
Flood Seasonality................................................................................................ 57
6
m-year maxima ................................................................................................... 63
7
Flood statistics on some rivers with multiple gauges............................................ 66
8
Determining the T-year Flood Magnitude QT by the Index Flood Method ........... 69
8.1
Introduction................................................................................................. 69
8.2
Subject, Donor and Analog Sites ................................................................. 71
9
Determining Qmed ................................................................................................ 72
9.1
Subject site is Gauged.................................................................................. 72
9.2
Subject size is Ungauged ............................................................................. 73
9.2.1
Use of Regression equation.................................................................. 74
9.2.1.1 Regression equation on its own ........................................................ 74
9.2.1.2 Regression equation used with donor site ......................................... 75
9.2.1.3 Regression equation used with analog site........................................ 76
9.2.2
Region of Influence Method................................................................. 76
9.3
Assessment of Qmed estimation using donor or analog sites .......................... 78
9.3.1
Tests on Qmed estimates via donor and analog catchments .................. 81
9.3.2
Results................................................................................................. 83
9.3.2.1 Results at individual subject sites ..................................................... 83
9.3.2.2 Results from all donor sites collectively ........................................... 86
9.3.3
Conclusions: ........................................................................................ 92
9.4
Summary of Qmed estimation and associated error........................................ 92
10 Determining the Growth Factor XT...................................................................... 94
10.1 Introduction................................................................................................. 94
10.2 At-site and Pooling Group Estimates of XT .................................................. 94
10.3 Pooling Groups............................................................................................ 95
10.4 Growth curve estimation.............................................................................. 97
11 Effect of catchment type and period of record on XT and QT ............................. 101
11.1 Data and Data Screening............................................................................ 101
ii
11.2 Effect of catchment type on pooled growth curve estimates. ...................... 104
11.3 Temporal effect on pooled growth curve estimates .................................... 111
11.4 Arterial drainage effect on pooled growth curve estimates ......................... 118
11.5 Implications for Flood Frequency Estimation ............................................ 121
12 Selection of distance measure variables............................................................. 123
12.1 Selecting variables for pooling................................................................... 123
12.2 Pooled uncertainty measure, PUM ............................................................. 124
12.3 Selecting pooling variables for Irish data ................................................... 125
13 Standard errors of Qmed, QT and XT.................................................................... 130
13.1 Standard Error of Qmed............................................................................. 131
13.2 Standard Error of At-site estimate of QT .................................................... 131
13.3 Standard Error of pooled estimate of XT and of QT .................................... 135
13.4 Standard Error of QT based on ungauged catchment estimate of Qmed and
pooled estimate of XT ............................................................................................ 140
14 Guidelines for determining QT........................................................................... 142
14.1 Introduction............................................................................................... 142
14.2 Determining QT from at-site data ............................................................... 143
14.3 Determining QT from Pooled data.............................................................. 145
14.3.1
Data available at the subject site......................................................... 145
14.3.1.1
Determining Qmed..................................................................... 145
14.3.1.2
Determining XT .......................................................................... 146
14.3.2
Subject site is ungauged ..................................................................... 146
14.3.2.1
Determining Qmed..................................................................... 146
14.3.2.2
Determing XT ............................................................................. 147
14.4 Some examples.......................................................................................... 147
14.5 Flood growth curves with an upper bound or without an upper bound?...... 156
14.6 Choice between at-site and pooled estimates and between 2 and 3 parameter
pooled estimates.................................................................................................... 157
15 Summary and Conclusions ................................................................................ 159
15.1 Data........................................................................................................... 159
15.2 Descriptive Statistics ................................................................................. 159
15.3 Seasonal Analysis...................................................................................... 159
15.4 m-year Maxima ......................................................................................... 160
15.5 Estimation of Design Flood ....................................................................... 160
References ................................................................................................................ 163
Appendix 1................................................................................................................ 165
Appendix 2................................................................................................................ 166
Appendix 3................................................................................................................ 177
Appendix 4................................................................................................................ 194
iii
List of Figures
Figure 3. 1 A map showing Specific Qmed (m3/s/km2) for A1 , A2 and B category
stations........................................................................................................................ 19
Figure 3.2 A map showing estimated Cv at 110 station locations (A1 & A2 category
stations) ...................................................................................................................... 20
Figure 3.3 A map showing estimated Hazen skewness at 110 station locations (A1 & A2
category stations) ........................................................................................................ 21
Figure 3.4 Histograms of Cv and L-Cv values............................................................. 22
Figure 3.5 Histograms of Hazen skewness and L-skewness values ............................. 23
Figure 3.6 L-Cv Vs Cv for A1 and A2 stations. Note if intercept is included the
relevant trendlines are y = 0.4426x + 0.0268. .............................................................. 24
Figure 3.7 L-Skewness Vs Hazen Skewness for A1 and A2 stations. Note if intercept is
included the relevant trendlines are y = 0.1225x + 0.0145. .......................................... 24
Figure 3.8 Observed Skewness calculated at 110 no. A1 and A2 gauging stations plotted
versus record length. Also shown are 67% and 95% Confidence Intervals for skewness
calculated from (a) Normal distribution samples and (b) EV1 distribution samples. .... 26
Figure 3.9 Observed L- Skewness calculated at 110 no. A1 and A2 gauging stations
plotted versus record length. Also shown are 67% and 95% Confidence Intervals for LSkewness calculated from (a) Normal distribution samples and (b) EV1 distribution
samples. ...................................................................................................................... 28
Figure 3.10 L-moment ratio diagram for annual maximum flow for 110 stations of
Ireland ........................................................................................................................ 30
Figure 3.11 L-moment ratio diagram of a) observed data b) data simulated from EV1
distribution c) data simulated from LN2 distribution and d) data simulated from LO
distribution.................................................................................................................. 30
Figure 3.12 First set of two realizations of four realizations of L-moment ratio diagrams
of data simulated from EV1, LN2 and LO distribution ................................................ 32
Figure 3.13 Second set of two realizations of four realizations of L-moment ratio
diagrams of data simulated from EV1, LN2 and LO distribution ................................. 33
Figure 4.1 (a, b) Random Samples of (a) size 25 (b) size 50 from an EV1(100,30)
population. (Cv = 0.33) ............................................................................................... 39
Figure 4.2 Hydrograph volumes of different durations during the 4 largest annual
maximum flood peaks at 24082 River Maigue at Islandmore. ..................................... 52
maximum flood peaks at 7006 River Moynalty at Fyanstown...................................... 53
maximum flood peaks at 15003 River Dinan at Dinan Bridge. .................................... 55
Figure 4.5 A plot between FAI vs Hazen Skewness..................................................... 56
Figure 5.1 Seasonal occurrence of annual maximum flood values in 202 stations........ 57
Figure 5.2 Flood seasonality showing actual magnitude in 202 Irish stations............... 58
Figure 5.3 Flood seasonality with specific flood magnitude in 201 Irish stations ......... 59
Figure 5.4 Month corresponding to maximum flow in each of 202 AM series ............. 60
Figure 6.1 Probability plots of 1, 2 and 3 year maxima showing contrasting behaviours.
................................................................................................................................... 65
iv
Figure 7.1 (a) Probability plot summary for stations on Barrow River (b) a scatter plot
of Qmed vs Catchment area for these stations. Stations are ordered from upstream. .... 66
Figure 7.2 (a) Probability plot summary for stations on Suir River (b) a scatter plot of
Qmed vs Catchment area for these stations. Stations are ordered from upstream. ........ 67
Figure 7.3 (a) Probability plot summary for stations on Suck River (b) a scatter plot of
Qmed vs Catchment area for these stations. Stations are ordered from upstream. ........ 68
Figure 9.1Increase in R2 value with number of catchment descriptors included in
regression equation in log domain, (i) as in FSR (1975) and (ii) in NUI Maynooth Draft
WP2.3 Report (2008). ................................................................................................. 75
Figure 9.2 Relationships between different methods of estimating Qmed or of enhancing
a Qmed estimate for an ungauged catchment................................................................. 77
Figure 9.3 Location of 38 potential subject sites.......................................................... 80
Figure 9.4 predicted Qmed by regression vs observed Qmed . Each point represents a
catchment , 35 in total as all catchment descriptors were unavailable for 3 of the 38 sites
................................................................................................................................... 83
Figure 9.5 Performance of data transfer method of estimating Qmed at Station 14006,
Barrow at Pass Bridge, using regression based adjustment. The figures above the boxes
are catchment areas. The final figure labelled Regression refers to estimate obtained by
eq (10). ....................................................................................................................... 85
Barrow at Pass Bridge, using area based adjustment.................................................... 86
Figure 9.7 Adjusted Qmed (Regression method) vs observed Qmed. Each point
represents a catchment and panels relate to different transfer method .......................... 90
Figure 9.8 Adjusted Qmed (Area method) vs observed Qmed (m3/s). Each point
represents a catchment and each plot relate to different area based data transfer methods
................................................................................................................................... 91
Figure 10.1 Simplified qualitative outline of simulation results about flood quantile
estimates. (a) and (b) show the effect of choice of wrong distribution having fixed but
incorrect skewness. (c) and (d) show the effect of using a flexible distribution i.e. low
bias and large standard error when used in at-site mode but with much reduced se as
well as low bias when us in at-site +pooling group mode. (Adapted from Figure 5.1 in
WMO ,1989)............................................................................................................. 100
Figure 11.1EV1 probability plot for the discordant stations....................................... 104
Figure 11.2 Location of the 85 stations and the four regions...................................... 105
Figure 11.3 Box plots of X100 values from region of influence pooling estimates of
X100 obtained by (a) GEV, (b) GLO estimating equations showing effect of %peat and
FARL. The numbers in brackets (85, 44, 30…) indicate the number of X100 values
contributing to each box plot..................................................................................... 106
Figure 11.4 Values of (a) H1 (b) H2 for the analyses summarised in Figure 11.3 ...... 107
Figure 11.5 Box plots of (a)X100 (GEV) (b) X100 (GLO) showing effect of geographical
area and catchment size............................................................................................. 109
Figure 11.6 Values of (a) H1, (b) H2 for the analyses summarised in Figure 11.5...... 110
Figure 11.7 Variation of X100 with period of record (first half versus second half of
each record) and variation of X50 with decades, as estimated by (a) GEV, (b) GLO
from each individual stations pooling group using the 5T rule................................... 112
Figure 11.8 Values of (a) H1, (b) H2 for the analyses summarised in Figure 11.7...... 113
v
Figure 11.9 (a)Column chart of X50 (GEV) values (b) Column chart of Qmed values (c)
Column chart of Q50 values for the decadal data (‘70, ‘80 and ‘90) of 50 stations...... 115
Figure 11.10(a)Column chart of X50 (GEV) values (b)Column chart of Qmed values (c)
Column chart of Q50 values for the decadal data (’50, ’60 ‘70, ‘80 and ‘90) of 26
stations...................................................................................................................... 116
Figure 11.11 (a) Column chart of X50 (GEV) values (b) Column chart of Qmed values (c)
Column chart of Q50 values for the decadal data (’57-66, ’67-76 ’77-86, ’87-96 and ’9706) of 34 stations ...................................................................................................... 117
Figure 11.12 (a) Column chart of X50 (GEV) values (b) Column chart of Qmed values
(c)Column chart of Q50 values for the pre- and post-drainage data of 16 stations ....... 120
Figure 12.1 Box-plot of 100-year PUM values for different sets of catchment
descriptors used in defining the distance measure dij. (a) subject site is not included in
its own pooling-group when PUM is evaluated (b) subject site is included in its own
pooling-group when PUM is evaluated. .................................................................... 128
Figure 12.2 Box-plot of (a) H1 values, (b) H2 values for different sets of catchment
descriptors used in defining the distance measure dij. ................................................ 129
Figure 13.1 (a) Box-plot of percentage standard error of flood quantile estimates for
EV1 distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100, 500year return period. (b) The theoretical EV1 percentage standard errors for the average
sample size of 37 years and for a range of L-Cv values ............................................. 133
Figure 13.2 (a) Box-plot of percentage standard error of flood quantile estimates for
GEV distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,
500-year return period. (b)The theoretical GEV (k=-0.1) percentage standard errors for
the average sample size of 37 years and for a range of L-Cv values........................... 134
Figure 13.3 Box-plot of percentage standard error of growth factors estimates for EV1
distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year
return period. ............................................................................................................ 138
Figure 13.4 Box-plot of percentage standard error of quantile estimates for EV1
return period. ............................................................................................................ 138
Figure 13.5 Box-plot of percentage standard error of growth factors estimates for GEV
return period. ............................................................................................................ 139
Figure 13.6 Box-plot of percentage standard error of quantile estimates for GEV
return period. ............................................................................................................ 139
Figure 14.1 At-site and pooled quantiles for station 25016 ........................................ 147
Figure 14.2 At-site and pooled quantiles for station 9001 .......................................... 149
vi
List of Tables
Table 2.1 Number of cases in which the null hypothesis of zero trend or randomness is
rejected at stated level out of 91 no. AM cases .............................................................. 6
rejected at stated level out of 117 no. 5-year median cases ............................................ 7
Table 3.1 Calculation of moments and their dimensionless ratios ................................ 10
Table 3.2 Summary statistics for 45 No. A1 stations ................................................... 11
Table 3.3 Summary statistics for 70 No. A2 stations ................................................... 12
Table 3.4 Summary statistics for 67 No. B stations...................................................... 14
Table 3.5 Average values of some statistics at 43 No. A1 and 67 No. A2 gauging
stations........................................................................................................................ 17
Table 3.6 High outliers from 110 Irish stations............................................................ 35
Table 3.7 Low outliers from 110 Irish stations ............................................................ 35
Table 4.1 Linear pattern statistics for A1 and A2 stations combined. Scores at individual
stations are tabulated in Table 4.3 and Table 4.4. ........................................................ 40
Table 4.2 Non- linear pattern statistics for A1 and A2 stations. Scores at individual
stations are tabulated in Table 4.3 and Table 4.4. ........................................................ 42
Table 4.3 Probability plot linear and non-linear scores for A1 category Stations.......... 43
Table 4.4 Probability plot linear and non-linear scores forA2 category Stations .......... 44
Table 4.5 A1 stations ranked by skewness.................................................................. 46
Table 4.6 A2 stations ranked by skewness................................................................... 47
Table 4.7 Sample of 8 stations showing convex upward appearance on probability plots.
................................................................................................................................... 50
Table 5.1 Station Number with percent of peak floods in the October-March
(percentage) ................................................................................................................ 61
Table 5.2 Station number with month corresponding to maximum flow in an AM data
series........................................................................................................................... 62
Table 6.1Examples of m-year maxima ........................................................................ 64
Table 9.1 List of Rivers that have 3 or more gauging stations on them: ....................... 78
Table 9.2 38 Gauging Stations used in the Qmed Assessment....................................... 79
Table 9.3 Estimates from donor catchments – summary of minimum, maximum and
average percentage difference from observed Qmed.................................................... 84
Table 9.4 Qmed estimates from donor catchments – minimum, maximum and average
% difference form observed Qmed.............................................................................. 87
Table 11.1 Stations with site characteristics, L-moment ratios and discordancy measure
................................................................................................................................. 102
Table 11.2 Four stations out of 94 showing large discordancy values ........................ 104
Table 11.3 List of 16 no. Pre and post- drainage stations ........................................... 119
Table 12.1Summary of AMF data sets used in dij study ............................................ 126
Table 12.2 Variation in the mean PUM100, H1 and H2 values for different sets of
pooling variables....................................................................................................... 127
Table13.1 Ranges of percentage se for growth factors, XT, and quantile estimates, QT.
................................................................................................................................. 137
vii
List of Equations
QT = F-1 (1-1/T)
QT = QI XT
--- (8. 1) ......................................................................... 70
--- (8. 2) .................................................................................... 70
0.36
se(Qmed ) =
--- (9. 1) ....................................................................... 72
Qmed
N
 s

s
d  Qmed by regression 
--- (9. 2)............................................. 76
Qmed = Qmed
 Q d by regression 
med


s
d
Qmed
= Qmed
As
Ad
--- (9. 3).................................................................................. 76
s
 Qmed
by regression 
s
a
 a

Qmed
= Qmed
--- (9. 4) .............................................. 76
Q
by
regression
 med

 1 m  Q i 
s
--- (9. 5) ................................................................... 76
Qmed
= As  ∑  med 
 m i =1 Ai 
Q med = 0.000302 × AREA 0.829 × SAAR 0.898 × BFI -1.539
2
--- (9. 6) ........... 82
2
2
1  ln AREAi − ln AREAj   ln SAARi − ln SAARj   BFIHOST
i − BFIHOST
j 

 +
 +
 --- (10. 1)
dij =






2
σ ln AREA
σ ln SAAR
σ BFIHOST
 
 

................................................................................................................................... 96
2
2
2
 ln AREAi − ln AREAj   ln SAARi − ln SAARj   ln BFI i − ln BFI j 
 + 
 + 
 --- (10. 2)
dij = 
σ ln AREA
σ ln SAAR
σ ln BFI

 
 

................................................................................................................................... 96
t2
m
m
i =1
i =1
∑ wi t 2(i ) / ∑ wi
=
m
t3 =
xT = 1 +
--- (10. 3)........................................................................ 97
m
∑ wi t3(i) /
∑ wi
i =1
i =1
--- (10. 4)....................................................................... 97
β
T k
k
) ,
 (ln 2) − (ln
k
T −1 
k ≠0
--- (10. 5).............................................. 98
ln 2
2
3 + t 3 ln 3
k = 7.8590c + 2.9554c2 where c =
--- (10. 6) ................................. 98
kt 2
--- (10. 7)........................................... 98
t 2 Γ(1 + k ) − (ln 2) k + Γ(1 + k )(1 − 2 − k )
--- (10. 8) .................................................... 99
xT = 1 + β (ln(ln 2) − ln(− ln(1 − 1 / T )))
t2
--- (10. 9) ..................................................................... 99
β=
ln 2 − t 2 [γ + ln(ln 2)]
β =
(
xT = 1 +
)
β
(1 − {(1 − F ) / F } )
k
k
= 1+
β
(1 − {T − 1} ),
k
−k
viii
k ≠0
--- (10. 10) .................. 99
kt 2 sin(πk )
--- (10. 11)................................................ 99
kπ (k + t 2 ) − t 2 sin(πk )
xT = 1 + β ln(T − 1)
--- (10. 12)............................................................................ 99
--- (10. 13) ............................................................................................ 99
β = t2
k = - t3 and β =
Di =
1
T
N (u i − u ) A −1 (u i − u )
3
n
d ij =
∑ W (X
k
--- (11. 1) ............................................................. 101
− X k, j )
2
k ,i
--- (12. 1)................................................................ 123
k =1
2
2
2
 ln Ai − ln Aj 
 ln SAARi − ln SAARj 
 FARLi − FARLj 
 FPEXT
i − FPEXT
j  --- (12. 2)
 + 0.5
 + 0.1
 + 0.2

dij = 3.2
0.37
0.05
0.04
 1.28 






................................................................................................................................. 124
2
M long
∑ n (ln x
i
PUM T =
Ti
− ln x
p
Ti
)
i =1
--- (12. 3) ..................................................... 124
M long
∑n
i
i =1
dij =
 ln AREAi − ln AREAj
1.7 
σ ln AREA

2
  ln SAARi − ln SAARj
 + 
σ SAAR
 
2

 BFIi − BFI j
 + 0.2 
σ BFI


2

 --- (12. 4)

................................................................................................................................. 127
EV1:
GEV:
α
SE Qˆ T =
n
[ ]
[1.1128 + 0.4574 y + 0.8046 y ] --- (13. 1) ......................... 131
α
[exp{a (T ) + a (T )exp(− k ) + a (T )k + a (T )k }] --- (13. 2) 131
se[Qˆ ] =
n
2
2
0
T
N
1
SE ( X T )[%] =
N
∑
1
SE (QT )[%] =
N
N
Qˆ T = Qˆ med × xˆT
i =1
∑
i =1
1
1
M
 Xˆ T − Xˆ T
i
 i, m
∑
T

X
m =1 
i

1
M
 Qˆ T − Qˆ T
i
 i, m
∑
T

Qi
m =1 

M
M
2




3
1/ 2
3
2
× 100
--- (13. 3)............................... 135
2


 × 100

--- (13. 4).................................. 135
--- (13. 5)............................................................................. 140
) + E (Qˆ ) 2 Var ( xˆ ) + 2 E (Qˆ ) ⋅ E ( xˆ ) ⋅ cov(Qˆ ⋅ xˆ ) --- (13. 6)140
≈ E ( xˆT ) 2 Var (Qˆ med
med
T
med
T
med
T
2
2
ˆ
ˆ
Var (QT ) ≈ ( xT ) | m Var (Qmed ) + (Qmed ) | m Var ( xˆT )
--- (13. 7)............................ 140
0.36
--- (13. 8).......................................................................... 140
se(QT ) QT ≈
N
y i = − ln(− ln( Fi )) where Fi =
i − 0.44
, i = 1,2,........N
N + 0.12
ix
--- (14. 1)........................ 144
y i = Φ −1 ( Fi )) where Fi =
i − 3/8
, i = 1,2,........N
N + 1/ 4
--- (14. 2) .............................. 144
2 M 110 − M 100
L2
=
--- (14. 3).............................................................. 144
ln(2)
ln(2)
u = M 100 − 0.5772 α = L1 − 0.5772α
--- (14. 4) ................................................. 145
QT = u + αyT
--- (14. 5)................................................................................. 145
n
1
--- (14. 6) ..................................................... 145
µ z = ∑ z i where zi = log10(Qi)
n i =1
α =
1/ 2
 1 n

σ z =
( zi − z )2 
--- (14. 7) .............................................................. 145
∑
 n − 1 i =1

QT =10 ZT
--- (14. 8) .................................................................................... 145
Qmed rural = 1.083 × 10 −5 AREA 0.938 BFIsoils −0.88 SAAR 1.326 FARL2.233 DRAIND 0.334
(14.
× S1085 0.188 (1 + ARTDRAIN 2) 0.051
9) .............................................................................................................................. 146
x
List of Symbols and Abbreviations
Symbols
T
return period in years
N, n
record length
Q
flood peak discharge
QI
index flood
Qmed
median of AM flood series
Qmean,, Q , Qbar
mean of AM flood series
Qmax
maximum flood on record
Qmin
minimum flood on record
QT
the T year flood
XT
the T year growth factor
dij
similarity distance
D
discordancy measure
Cv
coefficient of variation
H-skew
Hazen corrected skewness
L-Cv
coefficient of L-Cv
L-Skew
coefficient of L-skewness
L-kur
coefficient of L-kurtosis
t2
sample L-Cv value
t3
sample L-Skewness value
t4
sample L-kurtosis value
ln
natural logarithm
log10
10 base logarithm
E
expected value
Var
variance
F(Q), F
cumulative distribution function
f(Q)
probability density function
w
weighting term
R2
coefficient of determination
A1
gauging station grade A1
xi
A2
gauging station grade A2
B
gauging station grade B
H1
heterogeneity (using L-Cv)
H2
heterogeneity (using L-Cv and L-skewness)
s
Qmed
Qmed at the subject site
d
Qmed
Qmed at the donor site
a
Qmed
Qmed at the analog site
As
catchment area of the subject site
Ad
catchment area of the donor site
u
location parameter in extreme value distributions
α
scale parameter in extreme value distributions
β
scale parameter (growth curve)
k
shape parameter of Generalized Extreme Value distribution
k
shape parameter of Generalized Logistic distribution
γ
Euler’s constant
Γ
Gamma function
M100
1st PWM
M110
2nd PWM
M120
3rd PWM
λ1
1st L-moment
λ2
2nd L-moment
λ3
3rd L-moment
µ
mean
σ
standard deviation
µ3
3rd central moment
g
coefficient of skewness
y
reduced variate
z
variate used in statistical exposition, z = ln X
σz
standard deviation of z = ln X
Φ
Standardised Normal Distribution Function
xii
Abbreviations
A, AREA
catchment area (km2)
AEP
annual exceedance probability
AM
annual maximum
ARTDRAIN2 % of the catchment river network included in the Drainage Schemes
BFI
baseflow index
BFIHOST
baseflow index derived from HOST soils data (U.K)
BFIsoils
baseflow index derived from soils data
DRAIND
drainage density
EPA
Environment Protection Agency
ESB
Electricity Supply Board
EV1
Extreme Value Type 1 (distribution), (= Gumbel distribution)
FAI
flow attenuation index
FARL
flood attenuation by reservoir and lake
FEH
Flood Studies Handbook
fse
factorial standard error
FSR
Flood Studies Report
FSU
Flood Studies Update
GEV
Generalized Extreme Value (distribution)
GLO
Generalized Logistic (distribution)
iid
independently identically distributed
LN2, LN
2 parameter Log Normal (distribution)
LN3
3 parameter Log Normal (distribution)
LO
2 parameter Logistic (distribution)
OPW
Office of Public Works
PEAT
% of land area covered by peat bogs
PUM
Pooled Uncertainty Measure (as defined in FEH)
PWM
Probability Weighted Moment
S1085
10-85% stream slope (m/km)
SAAR
standard annual average rainfall (mm)
se
standard error
xiii
Abstract
In 2005 the Office of Public Works announced a Flood Studies Update Programme
arising out of the report of the Flood Policy Review Group in 2004.
The work
described in this report deals with the topic of flood frequency analysis and estimation.
Flood frequency analysis addresses the issue of the rarity of large and medium floods
for flood risk assessment and the specification of flood flow values for the design flood
of alleviation and control works and for flood zoning and spatial planning.
The report provides a statistical summary of flood flow records in Ireland which have
been taken from the data archives of the OPW, EPA and ESB. The summary is based
on analysis of annual maximum flow records, ranging from 10 to 55 years long, at
approximately 200 river flow gauging sites.
More detailed research has been carried
out on a subset of approximately 85 gauging sites, at which the quality of the gauged
flows is the most reliable.
Most floods occur during the winter half of the year with some notable floods occurring
during summer, especially in August, also.
Because of Ireland’s humid climate the
range of variation of flow values from year to year, as measured by coefficient of
variation, is quite small by international standards. Likewise the skewness of flow series
is modest. While no single statistical distribution can be considered to be “best” at all
locations it has been found that both the Extreme Value Type 1 (Gumbel) and the
lognormal distributions provide a reasonable model for the majority of stations.
Guidance is provided on the estimation of the design flood of required annual
exceedance probability at both gauged and ungauged locations and on how to express
the uncertainty in the resulting estimates. The use of growth curves or growth factors
based on regional pooling of data with the use of suitable 3 parameter distributions is
recommended for most applications. A number of examples are presented in Chapter
14 which discusses a range of difficult cases that can arise in practice. In some of these
cases the user may be recommended to consider using an at-site based estimate in
preference to the generally recommended regional pooling based method.
What this
indicates is that flood estimation cannot be reduced to a single strict formula based
procedure but that individual analysts must make choices which depend on the problem
circumstances and which take into account their own knowledge and experience.
xiv
1 Introduction and data
Flood frequency analysis is concerned with the assessment of flood magnitudes of
stated frequency or stated degree of rarity for use as input into the process of flood risk
assessment and management. Flood risk assessment is needed in the design of flood
relief and protection works and in the assessment of the safety of existing and planned
elements of infrastructure whether they be domestic houses, commercial or industrial
buildings, bridges and roads, railways or other vital pieces of infrastructure such as
hospitals, electrical stations, gas stations or water and waste-water works. No project
can be guaranteed to be immune from flooding during its projected life. Flood proofing
every project would be prohibitively costly.
As a result projects for which the
consequence and cost of occasional flooding is modest may be required to tolerate
occasional flooding. Calculation of the flood risk associated with such projects requires
that the flood magnitude which needs to be exceeded to cause flooding has to have its
associated probability of occurrence calculated.
It is these probabilities that are
calculated with the help of flood frequency analysis.
This report arises in the context of the Government's multifaceted approach to flood risk
assessment and management which was first outlined in the Report of the Flood Policy
Review Group (2004) prepared by an expert group appointed by the Minister of State at
the office of Public Works. This approach ranges from hard engineering solutions such
as constructed flood relief schemes to improved flood awareness, education and
preparedness and the implementation of planning and development control. One key
element identified was the need for improved hydrological information for input into
decision making processes and a proposal was included for a Flood Studies Update
(FSU) programme to achieve this. This programme has many components relating to
rainfall frequency analysis, flood frequency analysis, hydrograph analysis, flood plain
effects on floods as well as the application of Geographic Information Systems to
automate the measurement of catchment descriptors from digital mapping data and the
use of Information Technology techniques to deliver much of the FSU's output to
potential users via the world wide web. These components were divided into Work
Packages, one of which is WP2.2 Flood Frequency Analysis. This report describes the
work carried out in this Work Package at NUI Galway between 2006 and 2009.
1
It also makes recommendations about guidelines for future practice in the area of flood
frequency analysis and design flood determination. The work programme undertaken is
as proposed in the NUI Galway Department of Engineering Hydrology’s Tender
Document of October 2005.
The Work Package 2.2 study is based on the annual maximum flow data for 203
hydrometric stations which were provided by OPW in stages between June 2006 and
February 2007. A final set of 3 data sets from ESB stations were provided in April
2008. Attention was focused on those stations whose quality grading, based on rating
curve and water level measurement reliability as indicated by OPW, fall into the
categories A1, A2 and B. There are
•
45 Grade A1 stations
•
70 Grade A2 stations
•
67 Grade B stations and a further
•
17 stations that have pre- and post-drainage records
giving a total of 199 gauging stations.
The main recommendations of the study are based solely on the results obtained from
A1 and A2 graded stations. The B grade stations are analysed and displayed but no
general deductions are made from them. However the median values obtained from
both A and B grade stations have been used in determining relationships between Qmed
and catchment descriptors, both in part of this study (i.e. equation (9. 6)) and in the
separate work Package WP2.3.
Some of the studies described in this report are based on subsets of the entire A1 and A2
dataset. This is due to a number of reasons – some studies needed data that did not have
gaps in the record, some studies needed multiple gauges on each river to be included,
some needed Baseflow Index (BFI) which was not available for all catchments, some
studies were conducted before some of the data series became available. In each part of
the report a note is included in the relevant introduction about the selection of data used
in that part of the study.
2
A number of remarkable floods which have occurred in Ireland do not appear in these
systematically measured data series because gauging stations did not exist on some such
rivers. The River Dargle at Bray is an example where Hurricane Charlie caused a
notable flood in August 1986. Earlier ungauged large floods also are known to have
occurred in the Dargle in 1904, 1932 and 1967.
Another such example is the
Newcastlewest flood of 1st August 2008. Some catchments which are known to have
experienced significant floods, eg Deel at Crossmolina, have gauging stations but data
have not been supplied for them because the gauging stations are not up to the required
standard for inclusion in category A1, A2 or B. Further examination of these floods is
merited because some of the floods occurred on small, relatively steep catchments
which are not well represented in the main database of A1 and A2 grade stations
although this further examination is not reported on here.
This report first describes the results of statistical analyses of the annual maximum
series under the headings;
•
Trend and randomness
•
Descriptive statistics
•
Probability plots
•
Flood seasonality
•
m-year maxima
•
Statistics of some rivers with multiple gauges.
It then considers methods of determining the T year return period flood magnitude,
including
•
Introduction to index flood method
•
Determination of the index Flood Qmed in both
gauged and
ungauged catchments using regression and/or donor or analog
catchments
•
Assessment of donor and analog catchment techniques
•
Determining the Growth Factor from at-site and from pooling group data
•
Effect of catchment type and period of record on pooled estimates of QT
•
Choice of catchment descriptors for pooling group selection
3
•
Standard errors of Qmed, XT and QT.
Finally summary guidelines for determining the design flood QT are given, followed by
some examples where the unchallenged application of a single prescribed strategy may
provide QT estimates that are counter intuitive and this leads on to a discussion of such
topics as
•
Flood growth curves with or without an upper bound
•
Choice between at-site and pooled estimates of QT and
•
Choice between 2 and 3 parameter distributions.
This discussion shows that there can be no blind application of a single rule based flood
frequency estimation method but that a number of factors such as existing at-site flood
experience, the trade-off between at-site and regional estimates and the plausibility or
otherwise of an upper bounded distribution must be considered in any determination of
QT.
The most suitable measure of rarity of any flood magnitude for general communication
with the public is the annual exceedance probability, AEP, of that flood. For example a
large flood may have an AEP of 0.01 or 1% meaning that it has only one chance out of
100 of being exceeded in any year. However as in most technical reports the measure of
rarity of any flood magnitude used in this report is return period, T, which is defined as
the average number of years elapsing between successive exceedances of that flood
magnitude. The word average in the definition needs to be stressed as any magnitude
can be exceeded at unequal intervals of time. The two measures are consistent with one
another since AEP = 1/T.
4
2 Trend and Randomness
It is assumed in flood frequency analysis that successive values in a flood series have all
emerged randomly and independently from a common population of flood values. In
particular this means that there should be no upward or downward trend in the series of
data nor should there be any clustering together of groups of larger than average or
smaller than average values during the passage of time. To determine how well these
assumptions are met recognised tests of trend and randomness are conducted.
This testing work is based on analysis of data from 117 Grade A1 and A2 gauging
stations, including some post-drainage records of catchments which had earlier
experienced arterial drainage works. A number of stations from among those currently
available were not available when the trend and randomness analyses were carried out
in 2007, hence the number 117 rather than 120.
A total of 6 tests were applied. These took into account the tests which were reported in
UK Flood Studies Report (NERC, 1975), WMO Report on Trend (Kundzewicz, 2004)
and also software called TREND developed by University of Melbourne for
hydrological purposes. These tests are
A)
B)
Tests for serial persistence or trend:
i)
Mann-Kendall (non-parametric test for trend)
ii)
Spearman’s Rho (non-parametric test for trend)
iii)
Mean-weighted Linear Regression test (parametric test for trend)
Tests for progressive change in the mean and median with time:
iv)
Mann-Whitney U test (non-parametric test for step jump in
median)
C)
Tests for serial dependency of time series
v)
Turning Points (Kendall non-parametric test for randomness)
vi)
Rank Difference (Meachem non-parametric test for randomness)
5
The results are summarised below and are presented in a separate 50 page report by Mr.
U. Mandal, which is attached at the end of this volume.
Initial exploratory data analysis (EDA) found that 11 stations showed, on a visual basis
on a chronological plot, an upward or downward trend with time. In this section also
EDA of pre- and post-drainage records was conducted. Surprisingly three of 14 such
stations showed a decrease in flood magnitudes during the post-drainage period.
Statistical trend analyses were carried out on
91
No. annual maximum series
117 No. 5-year median series.
The median series, from non-overlapping 5 year segments of each record, are examined
in order to give a macro view of the data which is unaffected by occasional larger than
usual or smaller than usual values in the annual maximum series. Only 91 stations
contributed to tests carried out on annual maximum series since 26 of the 117 data
series available had some gaps that prevented application of the tests.
When 6 tests are applied to data of approximately 100 no. stations it is to be expected
that the null hypothesis (of zero trend or of independence etc.) would be rejected on a
small number of occasions even if no trend were present or if all data sets were truly
random. The results are summarised below in Table 2.1 and Table 2.2 (which are Tables
7 and 8 of Mr. Mandal's report, Appendix 1).
rejected at stated level out of 91 no. AM cases
Sing.
Level
Split
Trend and persistence
test
sample
Randomness test
Mann-
Spearman Mean-weighted
Mann Whitney Turning
Rank
Kendall
Rho
Linear Regressn.
U test
point
difference
11
12
7
22
4
8
25
25
14
36
8
14
1%
5%
6
rejected at stated level out of 117 no. 5-year median cases
Sing.
Level
1%
5%
Split
Trend and persistence
sample
test
Randomness test
Mann-
Spearman Mean-weighted
Mann Whitney Turning
Rank
Kendall
Rho
Linear Regressn.
U test
point
difference
9
14
7
47
2
10
22
35
10
73
4
18
These results show an unexpected degree of non-randomness where, across all tests on
annual maximum series, approximately 10% of stations reject the null hypothesis at the
1% level whereas in a truly random situation only 1 station from approximately 100
stations would be expected to reject the null hypothesis. Similar remarks can be made
about the 5 year median tests. Thus a big question mark hangs over the traditional
"independently identically distributed" (iid) assumption on which all flood frequency
analysis is based. However use of the iid assumption still allows progress to be made in
estimating flood probabilities and the strengths of the method, which has been widely
used and developed throughout the world, mean that it is the most suitable form of
assumption on which to base a national flood frequency methodology. The iid
assumption also allows data series which are incomplete due to gaps in the record, due
to malfunctioning recorders or subsequent loss of data, to be treated in the same way as
continuous records. For instance if the annual maximum values from 5 years of a 40
year record are unavailable the record is considered to be equivalent to a record of 35
years regardless of whether the 5 unavailable years are contiguous or not.
7
3 Descriptive Statistics and Inferences from them
3.1 Descriptive Statistics
The steps carried out here involved calculating basic statistics and corresponding
dimensionless statistics for each station, the preparation of probability plots on Extreme
Value Type I (EV1) and logistic bases as well as log-normal probability plots and
studies which attempt to infer from the dimensionless statistics collectively what the
most appropriate form of 2 parameter distribution might be best suited as a universal
choice for annual maximum flood data.
While data are available for 115 stations and descriptive statistics are calculated for
them (Tables 3.2 and 3.3) and their data are displayed on probability plots (Chapter 4
and Appendix 4) the results for eight of these stations are not included in the statistical
summary in Table 3.5 nor in the inferences about distribution suitability based on
skewness and L-Moment ratios in Sections 3.2 and 3.3 below. These so omitted include
•
27070 River L. Inchiquinn at Baunkyle because of suspected extreme lake
effects
•
36037 River Woodford (Ballyconnell Canal East) at Ballyheady because of
suspected man made effects
•
Stations 25001, 25002, 25003, 25004, 25005 and 25158 in the Mulkear
catchment because of embankments which contain 4 out of 5 annual maximum
flows but which are overtopped by the larger flows causing damping of the
larger flow magnitudes which leads to extremely concave downwards
probability plots.
The annual maximum flood values used in the present study provide the basic
information on which the entire study depends. The entire data sets can be viewed as
histograms or as probability plots but for the purposes of applying statistical models,
such as probability distributions based on the identically independently distributed
assumption, the data need to be summarised by a number of statistical measures for use
as input into the calibration of such models. Such summary statistics, especially those
expressed in dimensionless form can also be used for comparison of flood frequency
behaviour on different catchments and for drawing inferences about the suitability of
8
probability distributions for describing flood data. This Chapter of the report describes
these measures and the results of calculating them from the available data series.
In flood hydrology the most useful statistics relate to
1. overall magnitude of the flood data as conveyed by their mean µ or median
Qmed
2. the scale or spread of the data as measured by standard deviation σ or 2nd Lmoment λ2
3. the skewness or the lack of symmetry of the data, as measured by the 3rd central
moment µ3 or by the 3rd L-moment, λ3.
Dimensionless forms of (ii) and (iii), expressed as
•
Coefficient of variation, Cv
•
L-Cv
•
Skewness, g
•
L-Skewness
are defined in Table 3.1 in terms of the basic population quantities. Formulae for
calculation of these from sample data are also provided in Table 3.1.
For comparative studies these dimensionless measures of scale and asymmetry are
useful for detection of common behaviour among the several series in the data set.
Dimensionless measures of magnitude are not feasible although specific flow value
Q/(catchment area) is sometimes used for comparison purposes. Other dimensionless
quantities such as Qmax/Qmed, Qmax/Qmean or Qmean/Qmed can also be compared between
catchments although it has to be realised that Qmax/Qmean and Qmax/Qmed are very much
dependent on the period of record available and have high sampling variability.
The basic statistical descriptors mean, median, Qmax, Cv, Hazen Skew, L-Cv, L-Skew
and L-Kurtosis as well as ratios Qmax/mean, Qmax/median, Qmax/Area and
Qmean/Qmedian are shown below in tabular form in Table 3.2, Table 3.3 and Table 3.4
for A1, A2 and B grade stations respectively. The quantity specific median flood,
Qmed/Area, for A1, A2 and B stations is mapped in Figure 3. 1.
9
Table 3.1 Calculation of moments and their dimensionless ratios
Definitions based on ordinary moments
Population quantities
µ = Mean
Sample estimates
µ3 = 3rd Central Moment
g=
N
∑Q
i
i =1
(Qi − Q) 2
∑i=1 N − 1
N
N
µ̂ 3 =
∑ (Qi − Q ) 3
( N − 1)( N − 2) i=1
σˆ
Cˆ v =
Q
µˆ
gˆ = 33
σˆ
 8.5 
gˆ H = gˆ 1 +
 = Hazen's Unbiassed
N 

Skewness
σˆ =
σ = Standard deviation
Cv =
1
N
Q=
σ
= Coefficient of variation
µ
µ3
= Coefficient of Skewness
σ3
N
Definitions based on Probability Weighted Moments (PWMs)
Population quantities
Sample estimates
Mˆ 100 = Q
M100 = 1st PWM
1
(i − 1)
Mˆ 110 = ∑
Q(i )
N i =1 ( N − 1)
1 N (i − 1)(i − 2)
Mˆ 120 = ∑
Q(i )
N i =1 ( N − 1)( N − 2)
where Q(i) is the ith smallest value
M110 = 2nd PWM
M120 = 3rd PWM
λ1
λ2
λ3
= sample mean
N
= M100 = 1st L-Moment
= 2M110 - M100 = 2nd L-Moment
= 6M120 - 6M110 + M100 = 3rd L-Moment
λ2
= L-Cv
λ1
λ
τ 2 = 3 = L-Skewness
λ2
τ =
Sample values of M100, M110 and M120 are inserted in these expressions to give the
samples estimates of L-Cv and L-Skewness.
10
Table 3.2 Summary statistics for 45 No. A1 stations
Station No., River & Station Name
Area
N
Qmean
Qmed
Qmax
H.Skew
CV
L-CV
LSkew
L-Kur
Qmax/
Qmean
Qmax/
Qmed
Qmax/
Area
Qmed/
Qmean
06011 RIVER FANE @ MOYLES MILL
234
48
15.86
15.39
26.36
0.81
0.20
0.11
0.09
0.07
1.66
1.71
0.11
0.97
06013 RIVER DEE @ CHARLEVILLE WEIR
307
30
27.81
27.37
41.84
0.10
0.27
0.16
0.02
0.01
1.50
1.53
0.14
0.98
06014 RIVER GLYDE @ TALLANSTOWN
270
30
22.56
21.46
39.40
1.31
0.27
0.15
0.22
0.12
1.75
1.84
0.15
0.95
06025 RIVER DEE @ BURLEY BRIDGE
176
30
18.32
18.69
23.57
-0.33
0.16
0.09
-0.07
0.19
1.29
1.26
0.13
1.02
06026 RIVER GLYDE @ ACLINT BRIDGE
144
46
13.87
12.30
24.12
1.05
0.32
0.18
0.24
0.09
1.74
1.96
0.17
0.89
06070 RIVER MUCKNO L. @ MUCKNO
153.5
24
13.32
13.19
20.93
0.78
0.25
0.14
0.14
0.14
1.57
1.59
0.14
0.99
07009 RIVER BOYNE @ NAVAN WEIR
1610
29
162.64
134.80
297.60
0.93
0.38
0.21
0.21
0.11
1.83
2.21
0.18
0.83
37
20
5.62
5.32
8.96
1.98
0.21
0.11
0.27
0.17
1.59
1.68
0.24
0.95
08002 RIVER DELVIN @ NAUL
08009 RIVER WARD @ BALHEARY
09001 RIVER RYEWATER @ LEIXLIP
62
11
10.38
6.59
53.60
5.43
1.41
0.56
0.68
0.65
5.17
8.13
0.86
0.64
215
48
38.71
35.46
91.50
1.17
0.44
0.24
0.19
0.15
2.36
2.58
0.43
0.92
09002 RIVER GRIFFEEN @ LUCAN
09010 RIVER DODDER @ WALDRON'S
BRIDGE
10021 RIVER SHANGANAGH @ COMMON'S
ROAD
37
24
7.24
5.40
23.70
2.40
0.83
0.42
0.39
0.25
3.28
4.39
0.64
0.75
95.2
19
70.15
48.00
269.00
3.24
0.86
0.42
0.42
0.30
3.83
5.60
2.83
0.68
30.9
24
7.87
7.36
14.30
0.95
0.37
0.21
0.19
0.07
1.82
1.94
0.46
0.94
10022 RIVER CABINTEELY @ CARRICKMINES
10.4
18
3.84
3.85
6.89
0.36
0.40
0.23
0.06
0.06
1.79
1.79
0.66
1.00
14006 RIVER BARROW @ PASS BRIDGE
1096
51
83.76
80.52
137.38
1.46
0.20
0.11
0.24
0.21
1.64
1.71
0.13
0.96
14007 RIVER STRADBALLY @ DERRYBROCK
115
25
16.94
16.20
29.30
1.12
0.30
0.17
0.22
0.07
1.73
1.81
0.25
0.96
14011 RIVER SLATE @ RATHANGAN
163
26
12.07
12.30
18.70
0.19
0.25
0.14
0.02
0.16
1.55
1.52
0.11
1.02
2415
51
141.83
147.98
216.07
0.22
0.24
0.14
0.04
0.06
1.52
1.46
0.09
1.04
0.99
14018 RIVER BARROW @ ROYAL OAK
14019 RIVER BARROW @ LEVITSTOWN
1660
51
103.46
102.41
162.86
0.61
0.24
0.14
0.09
0.13
1.57
1.59
0.10
16011 RIVER SUIR @ CLONMEL
2173
52
234.52
223
389
0.42
0.3
0.17
0.09
0.05
1.66
1.74
0.18
0.95
397
51
69.45
68.98
92.79
0.05
0.15
0.08
0.00
0.11
1.34
1.35
0.23
0.99
25003 RIVER MULKEAR @ ABINGTON
25006 RIVER BROSNA @ FERBANE
25014 RIVER SILVER @ MILLBROOK BRIDGE
25017 RIVER SHANNON @ BANAGHER
1207
52
86.77
81.91
147.21
0.71
0.25
0.14
0.14
0.18
1.70
1.80
0.12
0.94
165
54
17.67
17.25
27.03
0.57
0.23
0.13
0.10
0.13
1.53
1.57
0.16
0.98
7980
55
413.25
407.68
596.51
0.18
0.20
0.12
0.04
0.08
1.44
1.46
0.07
0.99
25023 RIVER LITTLE BROSNA @ MILLTOWN
25025 RIVER BALLYFINBOY @
BALLYHOONEY
25027 RIVER OLLATRIM @ GOURDEEN
BRIDGE
116
52
12.14
11.22
20.05
0.57
0.29
0.16
0.14
0.06
1.65
1.79
0.17
0.92
160
31
10.15
10.18
17.40
0.57
0.29
0.17
0.08
0.15
1.71
1.71
0.11
1.00
118
43
23.32
22.10
40.46
0.27
0.28
0.16
0.05
0.13
1.73
1.83
0.34
0.95
25030 RIVER GRANEY @ SCARRIFF BRIDGE
279
48
43.80
40.64
87.04
1.01
0.32
0.18
0.18
0.13
1.99
2.14
0.31
0.93
11
25158 RIVER BILBOA @ CAPPAMORE
116
18
47.66
43.88
75.09
0.17
0.29
0.17
0.05
0.13
1.58
1.71
0.65
26006 RIVER SUCK @ WILLSBROOK
182
53
26.57
24.23
70.06
3.87
0.37
0.15
0.40
0.41
2.64
2.89
0.38
0.91
1184
53
91.75
88.15
147.84
1.05
0.19
0.11
0.16
0.16
1.61
1.68
0.12
0.96
26008 RIVER RINN @ JOHNSTON'S BRIDGE
292
49
23.68
22.94
41.02
1.49
0.19
0.10
0.19
0.19
1.73
1.79
0.14
0.97
26019 RIVER CAMLIN @ MULLAGH
260
51
22.34
21.18
37.03
1.17
0.25
0.14
0.23
0.12
1.66
1.75
0.14
0.95
26020 RIVER CAMLIN @ ARGAR BRIDGE
128
32
11.21
11.27
15.59
0.16
0.19
0.11
0.03
0.07
1.39
1.38
0.12
1.01
26059 RIVER INNY @ FINNEA BRIDGE
249
17
12.98
12.20
16.70
0.31
0.18
0.10
0.11
0.17
1.29
1.37
0.07
0.94
27002 RIVER FERGUS @ BALLYCOREY
562
51
34.22
32.60
59.76
1.37
0.23
0.12
0.18
0.22
1.75
1.83
0.11
0.95
29001 RIVER RAFORD @ RATHGORGIN
29011 RIVER DUNKELLIN @ KILCOLGAN
BRIDGE
119
40
14.17
13.46
19.66
0.34
0.19
0.11
0.09
0.10
1.39
1.46
0.17
0.95
26007 RIVER SUCK @ BELLAGILL BRIDGE
31002 RIVER CASHLA @ CASHLA
33070 RIVER CARROWMORE L. @
CARROWMORE
34018 RIVER CASTLEBAR @ TURLOUGH
0.92
354
22
31.94
28.89
66.52
3.37
0.30
0.14
0.41
0.32
2.08
2.30
0.19
0.90
72
26
12.89
12.16
21.10
1.92
0.24
0.13
0.31
0.18
1.64
1.74
0.29
0.94
90
28
7.90
7.67
11.97
1.19
0.16
0.09
0.10
0.18
1.52
1.56
0.13
0.97
93
27
11.50
11.28
17.33
0.93
0.20
0.11
0.18
0.03
1.51
1.54
0.19
0.98
36010 RIVER ANNALEE @ BUTLERS BRIDGE
774
50
66.56
66.80
106.62
1.05
0.22
0.12
0.15
0.22
1.60
1.60
0.14
1.00
36012 RIVER ERNE @ SALLAGHAN
263
47
14.22
14.12
21.63
0.21
0.22
0.13
0.03
0.10
1.52
1.53
0.08
0.99
36015 RIVER FINN @ ANLORE
36018 RIVER DRONMORE @ ASHFIELD
BRIDGE
175
33
23.14
22.08
49.99
2.62
0.32
0.16
0.32
0.30
2.16
2.26
0.29
0.95
233
50
15.84
16.25
24.43
0.43
0.18
0.10
0.04
0.10
1.54
1.50
0.10
1.03
Table 3.3 Summary statistics for 70 No. A2 stations
Qmed
Qmax
H.Skew
CV
LCV
LSkew
LKur
Qmax/
Qmean
Qmax/
Qmed
Qmax/
Area
Qmed/
Qmean
Area
N
Qmean
06031 RIVER FLURRY @ CURRALHIR
45.3
18
13.58
11.70
35.80
3.12
0.52
0.26
0.39
0.32
2.64
3.06
0.79
0.86
07006 RIVER MOYNALTY @ FYANSTOWN
176
19
26.73
27.93
34.05
-1.09
0.21
0.12
-0.20
0.07
1.27
1.22
0.19
1.05
0.98
07033 RIVER BLACKWATER @ VIRGINIA HATCHERY
129
25
14.93
14.62
26.58
1.74
0.24
0.13
0.16
0.27
1.78
1.82
0.21
08005 RIVER SLUICE @ KINSALEY HALL
10.1
18
3.04
2.50
7.81
1.54
0.68
0.38
0.23
0.19
2.57
3.13
0.77
0.82
08008 RIVER BROADMEADOW @ BROADMEADOW
110
25
44.55
40.90
123.69
1.74
0.63
0.34
0.28
0.16
2.78
3.02
1.12
0.92
12001 RIVER SLANEY @ SCARAWALSH
1036
50
169.50
157.00
399.00
1.59
0.36
0.19
0.18
0.17
2.35
2.54
0.39
0.93
14005 RIVER BARROW @ PORTARLINGTON
398
48
40.81
38.27
80.42
2.01
0.29
0.15
0.29
0.22
1.97
2.10
0.20
0.94
14009 RIVER CUSHINA @ CUSHINA
68
25
6.69
6.79
11.19
1.42
0.23
0.13
0.14
0.24
1.67
1.65
0.16
1.02
14013 RIVER BURRIN @ BALLINACARRIG
154
50
16.54
16.05
26.32
0.37
0.26
0.15
0.07
0.06
1.59
1.64
0.17
0.97
14029 RIVER BARROW @ GRAIGUENAMANAGH
2762
47
162.54
160.74
206.21
0.18
0.14
0.08
0.06
0.07
1.27
1.28
0.07
0.99
14034 RIVER BARROW @ BESTFIELD
2060
14
137.30
125.00
247.00
2.23
0.32
0.17
0.33
0.17
1.80
1.98
0.12
0.91
12
15001 RIVER KINGS @ ANNAMULT
443
42
89.39
88.75
151.00
0.14
0.28
0.16
0.00
0.08
1.69
1.70
0.34
0.99
15002 RIVER NORE @ JOHN'S BRIDGE
1605
35
211.98
197.00
393.00
0.59
0.31
0.18
0.08
0.09
1.85
1.99
0.24
0.93
1.05
15003 RIVER DINAN @ DINAN BRIDGE
298
50
143.58
150.76
187.52
-0.78
0.20
0.11
-0.15
0.09
1.31
1.24
0.63
15004 RIVER NORE @ McMAHONS BRIDGE
491
51
38.96
37.28
74.96
0.96
0.31
0.17
0.13
0.15
1.92
2.01
0.15
0.96
16001 RIVER DRISH @ ATHLUMMON
140
33
15.65
15.66
24.49
0.30
0.22
0.12
0.01
0.12
1.56
1.56
0.17
1.00
16002 RIVER SUIR @ BEAKSTOWN
512
51
55.40
52.66
123.88
1.79
0.30
0.16
0.17
0.19
2.24
2.35
0.24
0.95
16003 RIVER CLODIAGH @ RATHKENNAN
246
51
31.17
29.98
45.72
0.85
0.18
0.10
0.21
0.05
1.47
1.53
0.19
0.96
16004 RIVER SUIR @ THURLES
236
48
22.17
21.37
34.89
0.53
0.20
0.11
0.07
0.11
1.57
1.63
0.15
0.96
16005 RIVER MULTEEN @ AUGHNAGROSS
87
30
23.11
21.79
34.31
1.33
0.18
0.10
0.20
0.13
1.48
1.57
0.39
0.94
16008 RIVER SUIR @ NEW BRIDGE
1120
51
90.66
92.32
110.91
-0.32
0.13
0.07
-0.06
0.05
1.22
1.20
0.10
1.02
16009 RIVER SUIR @ CAHIR PARK
1602
52
159.29
162.21
206.00
-0.41
0.17
0.10
-0.10
0.05
1.29
1.27
0.13
1.02
18004 RIVER BALLYNAMONA @ AWBEG
324
46
30.96
31.20
52.70
1.25
0.17
0.09
0.04
0.33
1.70
1.69
0.16
1.01
18005 RIVER FUNSHION @ DOWNING BRIDGE
363
50
56.69
53.05
109.73
1.63
0.27
0.14
0.22
0.18
1.94
2.07
0.30
0.94
19001 RIVER OWENBOY @ BALLEA UPPER
106
48
15.87
15.42
22.03
0.48
0.17
0.09
0.09
0.12
1.39
1.43
0.21
0.97
19020 RIVER OWENNACURRA @ BALLYEDMOND
75
28
24.63
22.40
38.70
0.12
0.34
0.20
0.03
0.04
1.57
1.73
0.52
0.91
23001 RIVER GALEY @ INCH BRIDGE
196
45
97.39
99.05
210.07
1.22
0.33
0.18
0.13
0.18
2.16
2.12
1.07
1.02
23012 RIVER LEE @ BALLYMULLEN
60
18
16.87
15.66
31.74
2.98
0.29
0.15
0.39
0.33
1.88
2.03
0.53
0.93
24002 RIVER CAMOGUE @ GRAY'S BRIDGE
231
27
24.06
23.49
35.21
0.32
0.19
0.11
0.06
0.17
1.46
1.50
0.15
0.98
0.98
24008 RIVER MAIGUE @ CASTLEROBERTS
805
30
120.96
119.13
194.86
0.28
0.26
0.15
0.05
0.09
1.61
1.64
0.24
24022 RIVER MAHORE @ HOSPITAL
39.7
20
9.83
9.80
20.50
1.12
0.41
0.23
0.12
0.16
2.09
2.09
0.52
1.00
24082 RIVER MAIGUE @ ISLANDMORE
764
28
135.47
140.01
206.35
-0.22
0.26
0.15
-0.04
0.13
1.52
1.47
0.27
1.03
25001 RIVER MULKEAR @ ANNACOTTY
646
49
133.95
132.88
178.58
-0.19
0.16
0.09
-0.02
0.16
1.33
1.34
0.28
0.99
25002 RIVER NEWPORT @ BARRINGTONS BRIDGE
223
51
61.15
62.64
74.64
-0.65
0.16
0.09
-0.14
0.06
1.22
1.19
0.33
1.02
25005 RIVER DEAD @ SUNVILLE
190
46
28.73
29.63
33.42
-0.98
0.11
0.06
-0.19
0.10
1.16
1.13
0.18
1.03
25016 RIVER CLODIAGH @ RAHAN
274
42
23.04
22.57
36.14
0.62
0.22
0.13
0.08
0.16
1.57
1.60
0.13
0.98
25021 RIVER LITTLE BROSNA @ CROGHAN
493
44
28.03
28.58
35.80
-0.13
0.14
0.08
-0.03
1.28
1.25
0.07
1.02
25029 RIVER NENAGH @ CLARIANNA
301
33
54.12
56.48
74.06
-0.09
0.24
0.14
-0.02
0.07
0.02
1.37
1.31
0.25
1.04
25034 RIVER L. ENNELL TRIB @ ROCHFORT
12
24
1.50
1.48
2.21
-0.48
0.29
0.17
-0.08
0.12
1.47
1.49
0.18
0.99
25040 RIVER BUNOW @ ROSCREA
30
20
3.78
3.59
6.29
1.32
0.27
0.15
0.20
0.18
1.67
1.75
0.21
0.95
25044 RIVER KILMASTULLA @ COOLE
98.9
33
25.38
22.70
45.75
1.23
0.34
0.19
0.25
0.15
1.80
2.02
0.46
0.89
25124 RIVER BROSNA @ BALLYNAGORE
254
18
12.79
13.65
22.50
-0.31
0.36
0.20
-0.07
0.23
1.76
1.65
0.09
1.07
26002 RIVER SUCK @ ROOKWOOD
626
53
56.98
56.56
104.87
2.29
0.22
0.11
0.22
0.29
1.84
1.85
0.17
0.99
26005 RIVER SUCK @ DERRYCAHILL
1050
51
92.80
93.21
135.94
0.27
0.18
0.10
0.03
0.14
1.46
1.46
0.13
1.00
97
35
13.66
13.22
18.76
0.93
0.16
0.09
0.20
0.11
1.37
1.42
0.19
0.97
26009 RIVER BLACK @ BELLANTRA BRIDGE
13
26018 RIVER OWENURE @ BELLAVAHAN
118
49
9.19
8.95
13.98
0.77
0.20
0.11
0.13
0.11
1.52
1.56
0.12
0.97
26021 RIVER INNY @ BALLYMAHON
1071
30
65.88
66.34
92.51
-0.86
0.25
0.14
-0.11
0.19
1.40
1.39
0.09
1.01
26022 RIVER FALLAN @ KILMORE
950
33
6.64
6.49
11.06
0.47
0.30
0.17
0.09
0.05
1.67
1.71
0.01
0.98
27001 RIVER CLAUREEN @ INCH BRIDGE
48
30
20.65
20.10
31.70
1.55
0.20
0.11
0.19
0.25
1.53
1.58
0.66
0.97
27003 RIVER FERGUS @ COROFIN
168
48
24.01
22.92
40.50
0.71
0.24
0.13
0.09
0.22
1.69
1.77
0.24
0.95
29004 RIVER CLARINBRIDGE @ CLARINBRIDGE
123
32
11.39
11.30
14.77
0.77
0.15
0.08
0.14
0.08
1.30
1.31
0.12
0.99
123.5
26
16.00
15.70
24.30
0.74
0.23
0.13
0.11
0.19
1.52
1.55
0.20
0.98
458
31
61.93
62.98
95.98
1.30
0.20
0.11
0.11
0.20
1.55
1.52
0.21
1.02
29071 RIVER L.CUTRA @ CUTRA
30007 RIVER CLARE @ BALLYGADDY
30061 RIVER CORRIB ESTUARY @ WOLFE TONE
BRIDGE
3111
33
274.97
247.97
601.59
3.04
0.32
0.15
0.41
0.38
2.19
2.43
0.19
0.90
32012 RIVER NEWPORT @ NEWPORT WEIR
138.3
24
30.06
29.85
36.60
-0.12
0.12
0.07
0.00
0.18
1.22
1.23
0.26
0.99
34001 RIVER MOY @ RAHANS
1911
36
174.76
174.61
286.56
1.08
0.19
0.10
0.08
0.21
1.64
1.64
0.15
1.00
34003 RIVER MOY @ FOXFORD
1737
29
180.42
178.00
282.00
1.26
0.17
0.09
0.11
0.25
1.56
1.58
0.16
0.99
34009 RIVER OWENGARVE @ CURRAGHBONAUN
113
33
28.37
27.48
38.58
0.43
0.17
0.10
0.08
0.16
1.36
1.40
0.34
0.97
34011 RIVER MANULLA @ GNEEVE BRIDGE
144
30
18.80
18.73
26.05
0.78
0.16
0.09
0.10
0.23
1.39
1.39
0.18
1.00
34024 RIVER POLLAGH @ KILTIMAGH
128
28
20.70
20.80
24.70
-0.39
0.12
0.07
-0.05
0.14
1.19
1.19
0.19
1.00
35001 RIVER OWENMORE @ BALLYNACARROW
299
29
30.52
31.16
46.04
0.10
0.21
0.12
-0.01
0.19
1.51
1.48
0.15
1.02
35002 RIVER OWENBEG @ BILLA BRIDGE
90
34
51.78
50.48
69.37
0.07
0.17
0.10
0.03
0.08
1.34
1.37
0.77
0.97
35005 RIVER BALLYSADARE @ BALLYSADARE
642
55
77.78
75.42
132.71
1.04
0.26
0.14
0.20
0.14
1.71
1.76
0.21
0.97
35071 RIVER L.MELVIN @ LAREEN
247.2
30
26.95
26.29
37.91
0.76
0.18
0.10
0.12
0.20
1.41
1.44
0.15
0.98
35073 RIVER L.GILL @ LOUGH GILL
384
30
54.81
54.05
78.40
0.34
0.22
0.12
0.08
0.14
1.43
1.45
0.20
0.99
36019 RIVER ERNE @ BELTURBET
1501
47
89.60
89.95
119.43
-0.16
0.18
0.10
-0.03
0.06
1.33
1.33
0.08
1.00
0.94
36021 RIVER YELLOW @ KILTYBARDEN
23
27
24.96
23.37
43.57
1.85
0.22
0.12
0.20
0.22
1.75
1.86
1.89
36031 RIVER CAVAN @ LISDARN
52
30
6.85
6.45
13.70
4.19
0.22
0.10
0.37
0.39
2.00
2.12
0.26
0.94
39008 RIVER LEANNAN @ GARTAN BRIDGE
78
33
28.34
28.18
43.88
0.73
0.26
0.15
0.13
0.11
1.55
1.56
0.56
0.99
39009 RIVER FERN O/L @ AGHAWONEY
207
33
45.91
45.72
76.67
1.03
0.26
0.14
0.16
0.15
1.67
1.68
0.37
1.00
Area
N
Qmean
Qmed
Qmax
H.Skew
CV
LCV
LSkew
L-Kur
Qmax/
Qmean
Qmax/
Qmed
Qmax/
Area
Qmed/
Qmean
1041 RIVER DEELE @ SANDY MILLS
45.3
32
85.08
82.61
147.16
0.22
0.29
0.17
0.02
0.13
1.73
1.78
3.25
0.97
1055 RIVER MOURNE BEG @ MOURNE BEG WEIR
10.8
9
2.92
2.70
4.72
1.03
0.38
0.23
0.16
0.02
1.61
1.75
0.44
0.92
6021 RIVER Glyde @ Mansfieldstown
321
50
21.54
21.50
33.00
0.45
0.24
0.13
0.09
0.09
1.53
1.53
0.10
1.00
6030 RIVER BIG @ BALLYGOLY
10.2
30
20.58
10.03
122.00
3.42
1.61
0.61
0.68
0.48
5.93
12.17
11.96
0.49
6033 RIVER WHITE(DEE) @ CONEYBURROW BR.
57.4
25
27.88
18.60
92.30
2.49
0.84
0.41
0.46
0.27
3.31
4.96
1.61
0.67
Table 3.4 Summary statistics for 67 No. B stations
14
8003 RIVER BROADMEADOW @ FIELDSTOWN
76.2
18
26.88
22.55
110.00
4.21
0.87
0.39
0.36
0.34
4.09
4.88
1.44
0.84
8007 RIVER BROADMEADOW @ ASHBOURNE
34
15
9.88
8.24
18.79
0.89
0.49
0.29
0.18
-0.01
1.90
2.28
0.55
0.83
8011 RIVER NANNY @ DULEEK ROAD BRIDGE
181
23
31.00
32.22
45.41
-0.93
0.25
0.14
-0.15
0.20
1.46
1.41
0.25
1.04
8012 RIVER STREAM @ BALLYBOGHIL
22.1
19
4.21
4.35
8.15
-0.54
0.58
0.33
-0.09
0.11
1.93
1.87
0.37
1.03
9035 RIVER CAMMOCK @ KILLEEN ROAD
54.7
9
12.04
11.70
27.80
2.75
0.60
0.33
0.33
0.23
2.31
2.38
0.51
0.97
10002 RIVER AVONMORE @ RATHDRUM
233
47
88.19
83.49
266.64
2.85
0.43
0.21
0.27
0.31
3.02
3.19
1.14
0.95
10028 RIVER AUGHRIM @ KNOCKNAMOHILL
204.1
16
56.69
46.95
102.00
1.57
0.38
0.21
0.30
0.08
1.80
2.17
0.50
0.83
11001 RIVER OWENAVORRAGH @ BOLEANY
148
33
49.85
47.17
120.7
3.03
0.33
0.16
0.24
0.3
12013 RIVER SLANEY @ RATHVILLY
185
30
45.16
43.55
72.30
-0.01
0.27
0.15
0.03
0.12
1.60
1.66
0.39
14033 RIVER OWENASS @ MOUNTMELLICK
185
22
22.59
19.50
33.00
0.53
0.28
0.16
0.14
-0.07
1.46
1.69
0.18
0.86
15005 RIVER ERKINA @ DURROW FOOT BRIDGE
387
50
28.47
27.44
61.85
1.78
0.34
0.18
0.19
0.21
2.17
2.25
0.16
0.96
15012 RIVER NORE @ BALLYRAGGET
945
16
77.16
77.11
133.00
0.90
0.30
0.17
0.08
0.21
1.72
1.72
0.14
1.00
16006 RIVER MULTEEN @ BALLINCLOGH BRIDGE
75
33
30.37
27.87
58.07
0.34
0.39
0.23
0.06
0.03
1.91
2.08
0.77
0.92
16007 RIVER AHERLOW @ KILLARDRY
273
51
79.18
75.84
138.03
0.46
0.32
0.18
0.09
0.06
1.74
1.82
0.51
0.96
16011 RIVER SUIR @ CLONMEL
2173
52
234.52
223.00
389.00
0.42
0.30
0.17
0.09
0.05
1.66
1.74
0.18
0.95
16012 RIVER TAR @ TAR BRIDGE
228
36
55.20
54.57
92.20
0.39
0.28
0.16
0.06
0.08
1.67
1.69
0.40
0.99
2.42
2.56
0.82
0.95
0.96
91
33
101.69
93.21
207.02
0.86
0.42
0.24
0.16
0.08
2.04
2.22
2.27
0.92
16051 RIVER ROSSESTOWN @ CLOBANNA
34.18
13
2.95
2.85
5.67
2.53
0.36
0.19
0.33
0.26
1.92
1.99
0.17
0.96
18001 RIVER BRIDE @ MOGEELY BRIDGE
18002 RIVER BALLYDUFF @
BALCKWATER(MUNSTER)
335
48
71.07
71.49
96.93
-0.16
0.19
0.11
-0.03
0.10
1.36
1.36
0.29
1.01
2338
49
353.65
344.00
479.00
0.24
0.16
0.09
0.06
0.10
1.35
1.39
0.20
0.97
18003 RIVER BLACKWATER @ KILLAVULLEN
1258
49
282.76
266.15
502.74
1.10
0.23
0.13
0.16
0.10
1.78
1.89
0.40
0.94
18006 RIVER BLACKWATER @ CSET MALLOW
1058
27
291.30
286.00
397.00
1.10
0.14
0.08
0.19
0.12
1.36
1.39
0.38
0.98
18016 RIVER BLACKWATER @ DUNCANNON
113
24
80.99
79.65
114.82
0.49
0.23
0.14
0.12
0.00
1.42
1.44
1.02
0.98
18048 RIVER BLACKWATER @ DROMCUMMER
881
23
222.77
220.00
269.00
1.11
0.08
0.05
0.15
0.22
1.21
1.22
0.31
0.99
18050 RIVER BLACKWATER @ DURRIGLE
244.6
24
121.96
124.50
175.00
0.12
0.20
0.11
-0.01
0.09
1.43
1.41
0.72
1.02
19014 RIVER LEE @ DROMCARRA (ESB)
N/A
20
79.69
71.89
157.21
1.67
0.38
0.21
0.30
0.12
1.97
1.40
0.69
0.90
19016 RIVER BRIDE @ OVENS BRIDGE (ESB)
N/A
8
28.74
29.58
34.87
-1.35
0.16
0.09
-0.14
0.22
1.21
1.65
0.60
1.03
16013 RIVER NIRE @ FOURMILEWATER
19031 RIVER SULLANE @ MACROOM (ESB)
N/A
9
131.09
135.90
201.18
1.46
0.27
0.16
0.16
0.17
1.53
2.27
0.67
1.04
19046 RIVER MARTIN @ STATION ROAD
60.4
9
31.09
29.95
41.95
-0.34
0.26
0.16
-0.04
0.06
1.35
2.01
0.70
0.96
20002 RIVER BANDON @ CURRANURE
431
31
140.60
126.28
287.11
2.13
0.37
0.19
0.37
0.28
2.04
1.43
0.44
0.90
20006 RIVER ARGIDEEN @ CLONAKILTY W.W.
79.3
25
30.25
27.70
55.60
1.67
0.30
0.16
0.27
0.18
1.84
1.22
0.25
0.92
22006 RIVER FLESK @ FLESK BRIDGE
325
51
165.89
169.09
282.83
0.66
0.25
0.14
0.07
0.12
1.70
2.07
0.43
1.02
22009 RIVER DREENAGH @ WHITE BRIDGE
37
24
11.91
11.47
16.35
1.17
0.16
0.09
0.18
0.19
1.37
1.64
0.63
0.96
559.65
14
112.81
116.40
141.48
-0.86
0.20
0.12
-0.15
0.03
1.25
1.29
0.40
1.03
22035 RIVER LAUNE @ LAUNE BRIDGE
15
24004 RIVER MAIGUE @ BRUREE
246
52
54.86
50.63
104.55
0.91
0.39
0.22
0.18
0.10
1.91
1.39
0.29
0.92
24011 RIVER DEEL @ DEEL BRIDGE
273
33
103.01
104.55
171.74
0.42
0.22
0.12
-0.02
0.24
1.67
1.41
0.48
1.01
24012 RIVER DEEL @ GRANGE BRIDGE
359
41
110.45
109.99
141.95
-0.02
0.16
0.09
0.01
0.13
1.29
2.37
0.16
1.00
24030 RIVER DEEL @ DANGANBEG
248
25
52.89
52.00
72.20
0.88
0.15
0.08
0.11
0.10
1.37
2.05
0.45
0.98
25004 RIVER BILBOA @ NEWBRIDGE
125
30
41.70
42.30
59.50
-0.16
0.23
0.13
-0.04
0.12
1.43
1.73
0.49
1.01
25011 RIVER BROSNA @ MOYSTOWN
1227
51
85.64
82.02
194.25
1.31
0.34
0.18
0.14
0.23
2.27
2.38
0.41
0.96
25020 RIVER KILLIMOR @ KILLEEN
197
35
46.60
43.65
89.55
0.93
0.33
0.19
0.16
0.11
1.92
1.66
0.32
0.94
25038 RIVER NENAGH @ TYONE
139
17
42.08
39.30
67.90
1.30
0.27
0.15
0.17
0.22
1.61
2.28
0.21
0.93
26010 RIVER CLOONE @ RIVERSTOWN
100
35
20.03
17.17
40.80
1.64
0.41
0.22
0.35
0.18
2.04
1.27
0.14
0.86
26014 RIVER LUNG @ BANADA BRIDGE
222
16
44.10
42.82
70.90
1.80
0.23
0.12
0.19
0.26
1.61
2.72
0.77
0.97
26058 RIVER INNY UPPER @ BALLINRINK BRIDGE
59
24
5.98
5.35
12.20
1.60
0.38
0.21
0.28
0.18
2.04
1.62
0.15
0.89
26108 RIVER OWENURE @ BOYLE ABBEY BRIDGE
533
15
56.29
57.32
73.07
0.28
0.18
0.11
0.06
-0.06
1.30
1.23
0.14
1.02
28001 RIVER INAGH @ ENNISTIMON
168
17
52.69
47.58
129.28
4.89
0.40
0.16
0.47
0.61
2.45
2.23
0.44
0.90
29007 RIVER L.CULLAUN @ CRAUGHWELL
278
22
27.83
26.49
42.93
1.03
0.22
0.12
0.15
0.20
1.54
1.30
0.14
0.95
30012 RIVER CLARE @ CLAREGALWAY
1075.4
9
126.89
126.00
155.00
1.00
0.12
0.07
0.13
0.15
1.22
1.79
0.02
0.99
30021 RIVER ROBE @ CHRISTINA'S BR.
138
26
28.17
27.20
60.70
2.17
0.34
0.18
0.21
0.29
2.15
2.38
0.12
0.97
30031 RIVER CONG @ CONG WEIR
891
24
94.35
93.88
122.28
-0.62
0.18
0.10
-0.02
0.13
1.30
1.93
1.82
0.99
30037 RIVER ROBE @ CLOONCORMICK
210
21
1.80
1.79
3.19
-0.42
0.37
0.21
-0.09
0.18
1.77
1.96
1.59
0.99
31072 RIVER CONG @ CONG WEIR
891
26
49.08
43.20
103.00
2.04
0.38
0.20
0.31
0.20
2.10
1.94
0.57
0.88
32011 RIVER BUNOWEN @ LOUISBERG WEIR
68.6
26
74.88
64.87
125.00
0.75
0.30
0.17
0.17
0.06
1.67
2.35
1.28
0.87
33001 RIVER GLENAMOY @ GLENAMOY
73
25
62.11
59.30
116.00
1.65
0.28
0.15
0.20
0.19
1.87
1.07
0.53
0.95
34007 RIVER AT DEEL @ BALLYCARROON
156
53
90.37
84.48
198.91
0.96
0.36
0.20
0.14
0.10
2.20
1.25
0.12
0.93
34010 RIVER MOY @ CLOONACANNANA
471
12
123.29
113.72
193.07
1.31
0.3
0.17
0.21
0.11
35011 RIVER BONET @ DROMAHAIR
294
36
116.02
115.36
188.01
0.04
0.30
0.17
0.01
0.08
36011 RIVER ERNE @ BELLAHILLAN
318
49
17.91
18.23
23.6
-0.51
0.18
0.1
-0.1
0.11
36071 RIVER L.SCUR @ GOWLY
66
20
6.36
6.49
8.13
-0.06
0.15
0.09
-0.04
0.01
1.28
1.78
3.25
1.02
38001 RIVER OWENEA @ CLONCONWAL
109
33
70.02
70.63
113.38
1.78
0.16
0.09
0.08
0.26
1.62
1.75
0.44
1.01
39001 RIVER NEW MILLS @ SWILLY
49
30
44.88
44.25
61.50
0.06
0.20
0.12
0.01
0.08
1.37
1.53
0.10
0.99
16
1.57
1.62
1.32
1.7
1.39
1.29
0.41
1.26
0.07
0.92
0.99
1.02
Hazen skewness g h is ordinary skewness g multiplied by the factor {1 + 8.5/N}. This
correction factor described by Hazen (1932) was shown by Wallis et al. (1974) to
provide an almost unbiased estimator of skewness for lognormal and EV1 distributions
and for [mildly] skewed distributions in general.
The values relating to Cv, Skewness and Qmax are of particular interest because they
convey information about the flood regime of a region which may be used for intercomparison with other regions. The average values of these for A1 and A2 stations are
shown in Table 3.5 below. The estimated Cv and Skewness values are shown in map
format in Figure 3.2 and Figure 3.3.
The average values of Qmed/Qmean are 0.94 for A1 stations and 0.98 for A2 stations.
These are just below the values expected in data from normal distributions whereas if
data were from an EV1 distribution this ratio would have a value of 0.95 for population
Cv values in the range 0.2 to 04, values which are relevant for Ireland. The inverse of
0.95 ≈ 1.06 which is similar to the average value of Qmean/Qmed report in NERC (1975),
Volume 1, p-132.
Table 3.5 Average values of some statistics at 43 No. A1 and 67 No. A2 gauging stations.
Station
Category
Qmed/Qmean
Cv
L-Cv
H.Skew
L-skew
Qmax/Qmean
Qmax/Area
A1
A2
A1& A2
0.94
0.98
0.96
0.31
0.25
0.28
0.16
0.14
0.15
1.15
0.84
1.00
0.17
0.11
0.14
1.85
1.65
1.75
0.28
0.31
0.29
The table shows that average Cv lies between 0.25 and 0.31 which is a little lower than
the average value reported in the FSR for Ireland (0.30 based on 63 stations of record
lengths > 15 years). The average Hazen corrected skewness lies between 0.84 and 1.15
which is considerably less that the average value of 1.63 reported in FSR, without
Hazen's correction, which was also based on 63 stations with record lengths > 15 years.
However the average record length of these 63 records used in FSR was < 20 years
whereas the average used in this part of the present study is 36 years. Thus Ireland has
a low Cv and low Skewness flood hydrology regime, typical of very humid conditions
where between year variation is relatively small. Furthermore the degree of "lowness"
17
has decreased from the 1960s, whether due to real hydrological change or due to change
in available record length.
The Cv and skewness statistics are shown in graphical form also, in Figure 3.4 and
Figure 3.5 while Figure 3.6 and Figure 3.7 show the relationships between Cv and LCV and between L-Skewness and Hazen Skewness.
While Cv and skewness vary between gauging stations some of this variation is due to
sampling variability while the remaining may be due to true differences between the
gauging stations themselves.
From Table 3.5 it can be seen, using the combined average A1 and A2 values, that the
ratio L-CV / Cv = 0.535 and the ratio L-skewness / Skewness = 0.14. These compare
very well with the corresponding theoretical values of 0.540 and 0.150 respectively in
an EV1 distribution. Alternative values of these ratios, namely 0.52 and 0.13, can be
obtained from the slopes of the lines on Figure 3.6 and Figure 3.7, where individual
values of L-Cv are plotted against individual values of Cv and likewise for L-Skewness
and H-Skewness.
18
Legend
Sp.Qmed
(
!
0.006829 - 0.151544
(
!
0.151545 - 0.270321
(
!
0.270322 - 0.418750
(
!
0.418751 - 0.812329
!
(
0.812330 - 1.823686
(
!
!
(
(
!
(
!
(
!
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
!
(
!
(
(
!
(
!
(
!
!
((
!
(
!
(
!
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(
!
(
!
(
!
(
( !
!
!!
(
(
(
!
(
!
(
!
(
!
!
(
(
!
(!(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!!
(
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(!(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
! !
(
(!( !
!
(
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(!(
!
!!
(
(
(
!
(!(!(
!
(
!
!!
(
(
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(
!
(
!
(
!
(
!
(!
!
(
(
!!
(
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
!!
(
(
´
Figure 3. 1 A map showing Specific Qmed (m3/s/km2) for A1 , A2 and B category stations.
19
´
Legend
CV
(
!
0.110000 - 0.190000
(
!
0.190001 - 0.270000
(
!
0.270001 - 0.440000
(
!
0.440001 - 0.860000
!
(
0.860001 - 1.410000
39009
(
!
39008
(
!
35071
(
!
35073
35005
(
!
(
!
35002
(
!
33070
(
!
36015
34001
(
!
36019
3601036021
35001 34010
(
!
(34003
!
(
!
34018
(!(
!
34024
(
!
34011
(
!
36031
(
!
36011
(
26018 !(26009 !
(26008
!
26017
(!(
!
26006
2602026022
( !(
!
(
!
6025 6013
7033
!
26059 (
(
!
(
!
(
!
7006
(
!
(
!
7009
(
!
26021
(
!
8002
(
!
(
!
26005
80088009
8005
25124
(
!
(
!
30061
25006 25016
(
!
( 25014
!
14011
14009
25017
(
!
(
!
14005
( 14006
!
(
!
( !(
!
25021
(
!
14007
29071
25025
(
!
(
!
(
!
2504025044
27003
14019
(25023
!
25030
(
!
(
!
(
!
(
!
25029
15004
14034
27002
(
!
(
!
25027
(
!
27001
14013
(!
!
(
(2900429011
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
25002
(
!
25003
25158
(
!
(
!
(!(25005
!
24008
( 24082 24022
!
( !(24002
!
(
!
16004
(16001
(!
!
16003
(16002
!
16005
(
!
(
!
16008
!
(
!
(!
(
9010
10021
( !(10022
!
90019002
(
!
26007
23001
(
!
6026
!!(26019
(
26002
6031
6011
(
!
(
!
30007
31002
(
!
(
!
36012
(
!
(
!
6070
36018
(
!
(
!!
(
(
!
32012
(
!
(!(
!
15003
(
!
14018
(
!
15001 14029
(
!
(
!
11001
12001
(
!
(
!
(
!
25034
(
!
16009
(
!
23012
(
!
18004
(
!
18005
(
!
19020
(
!
19001
(
!
Figure 3.2 A map showing estimated Cv at 110 station locations (A1 & A2 category
stations)
20
Legend
´
H_SKEW
(
!
-1.090000 - -0.090000
(
!
-0.089999 - 0.620000
(
!
0.620001 - 1.460000
(
!
1.460001 - 2.620000
!
(
2.620001 - 5.430000
39009
39008
(
!
(
!
35071
(
!
35073
35005
(
!
35002
(
!
33070
(
!
(
!
35001
(
!
(
!
36015
(
!
(
!
(
!
26009 (
26018 26008 !
!!(
(
26020
(
!
26006
!
(
30007
26005
26007
(
!
(
!
7009
(
!
25124
(
!
(
!
30061
2900429011
(
!
( 29001
!
2500625016
(
!
14011
( 25014
!
14009
25017
(
!
(
!
14005
(
!
( 14006
!
(
!
25021
(
!
14007
29071
25025
(
!
(
!
(
!
2504025044
27003
14019
(25023
!
25030
(
!
( 14034
!
( 27002
!
15004
( 25027!(25029
!
27001
14013
(
!
(
!
(
!
!
(
!
(
(
!
!
( !(!(
10021
(!(
!
16004
(
(!
!
16003
( 16002
!
16005
(
!
24008
( 24082 24022
!
( !
!
(24002
(16008
!
(
!
15003
14018
(
15002 !
(
!
(
!
15001 14029
(
!
(
!
12001
(
!
(
!
25034
16009 16011
(
!
(
!
23012
!
(
8002
!80088009
(
(
!
!
((8005
90019002 !
(!
!
( 9010
10022
(
!
(
!
23001
(
!
(
!
7006
(
!
(
!
31002
6014
26021
(
!
(
!
(
!
6031
!
(
6011
(6026
!
!!
(
(
6025 6013
7033
(
!
26059
26022
(
!
(
!!
(
26019
26002
(
!
!
(
36012
(
!
6070
36019
36018
(
!
(
!
36031
36021
36010
34009
320123401134018
(
!
( 34024
!
(
!
(
!
(
!
34001
(
!
18004
(
!
18005
(
!
19020
(
!
19001
(
!
Figure 3.3 A map showing estimated Hazen skewness at 110 station locations (A1 & A2
category stations)
21
Cv Range
0
0
Figure 3.4 Histograms of Cv and L-Cv values
22
L-Cv Range
>0
.3
10
0.
3
10
L-Cv Range
to
20
0.
2
>0
.3
0.
3
0.
25
to
to
0.
25
0.
2
to
0.
1
>0
.3
0.
3
0.
25
to
to
0.
25
0.
2
0.
2
0.
15
to
to
0.
1
0.
05
to
to
0.
15
L-Cv Range
0.
25
30
0.
25
Cv of 110 A1 and A2 Stations
to
50
0.
2
60
0.
15
0
0.
2
Cv of 67 A2 Stations
to
0
0.
05
0.
0
No. of Occurances
No. of Occurances
Cv of 43 A1 Stations
0.
15
10
0.
15
10
to
20
0.
15
30
to
50
0.
1
40
No. of Occurances
60
0.
1
0.
05
0
to
to
6
0.
6
0.
5
0.
4
0.
3
0.
2
0.
1
>0
.
to
to
to
to
to
to
10
0.
05
0.
0
0.
5
0.
4
0.
3
0.
2
0.
1
0.
0
20
0.
1
40
No. of Occurances
No. of Occurances
30
0.
1
0.
05
0.
6
0.
5
0.
4
0.
3
0.
2
0.
1
>0
.6
to
to
to
to
to
to
40
to
to
6
0.
5
0.
4
0.
3
0.
2
0.
1
0.
0
50
0.
05
0.
0
>0
.
0.
6
0.
5
0.
4
0.
3
0.
2
0.
1
No. of Occurances
Cv Range
to
to
to
to
to
to
Cv Range
0.
5
0.
4
0.
3
0.
2
0.
1
0.
0
60
60
50
L-Cv of 43 A1 Stations
40
30
20
10
0
60
50
L-Cv of 67 A2 Stations
40
30
20
60
50
L-Cv of 110 A1 and A2 Stations
40
30
20
o1
.5
o2
.5
20
15
10
10
5
5
0
0
Skewness Range
23
L-Skewness Range
Figure 3.5 Histograms of Hazen skewness and L-skewness values
>
30
0.4
30
0.4
35
to
H.Skewness of 110 A1 & A2 Stations
0.
0.
L-Skewness Range
0.3
35
0.3
40
to
Skewness Range
0.2
3
2
4>
0.
4
0.
3
0.
to
to
0
0.
2
5
0
to
10
5
1
10
0.
H.Skewness of 67 A2 Stations
0.2
Skewness Range
to
15
0.
3
0.
2
0.
1
0.
4
0.
3
0.
2
0.
1
0
0.
4>
to
to
to
to
to
0
0.1
20
0.
1
30
to
30
0
35
0.1
35
to
40
.1
1
10
0.
0
-0
<0.
No. of Occurances
15
0
3.5
o3
2.5
o2
1.5
1
3.5
>
3 to
2.5
t
2 to
1.5
t
1 to
0.5
to
0.5
o0
20
to
5t
30
1
25
No. of Occurances
<-1
-0.5
0 to
-0.
to
25
0.
-0
.
-0
.1
>
.5
3
.5
2
.5
1
.5
3.5
o3
<
3t
to
o2
2.5
2t
to
o1
1.5
1t
to
o0
0.5
0t
to
0
-1
30
to
0
1
No. of Occurances
35
-0.1
.1
25
No. of Occurances
<to
-0.
5
-0 .
5
-1
No. of Occurances
H.Skewness of 43 A1 Stations
35
0.0
.5
3.5
>
o3
<-0
3t
2.5
to
3
2t
1.5
to
2
1t
.5
0
to
1
o0
0.5
0t
to
o0.5
-0 .
5
-1
t
<1
No. of Occurances
40
40
L-Skewness of 43 A1 Stations
25
20
15
10
5
5
0
L-Skewness Range
40
L-Skewness of 67 A2 Stations
25
20
15
40
L-Skewness of 110 A1 & A2 Stations
25
20
15
0.80
y = 0.5152x
0.70
0.60
L-Cv
0.50
0.40
0.30
0.20
0.10
0.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
Cv
Figure 3.6 L-Cv Vs Cv for A1 and A2 stations. Note if intercept is included the relevant
trendlines are y = 0.4426x + 0.0268.
0.80
y = 0.1293x
0.70
0.60
0.50
L-skew
0.40
0.30
0.20
0.10
0.00
-0.10
-0.20
-0.30
-2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
H-skew
Figure 3.7 L-Skewness Vs Hazen Skewness for A1 and A2 stations. Note if intercept is
included the relevant trendlines are y = 0.1225x + 0.0145.
24
3.2 Preliminary distribution choice with help of Skewness versus
Record Length Plots
Another way of viewing skewness values is shown on Figure 3.8 and Figure 3.9 which
show skewness or L-Skewness plotted versus length of record along with confidence
intervals based on Normal distribution and EV1 distribution assumptions.
In the interest of model parsimony it is always desirable to consider the adequacy of 2 –
parameter models and consequently only Normal and EV1 distributions are considered
in this sub-section. The normal distribution confidence intervals for skewness are
calculated using the standard expression
se( skewness) =
6 N ( N − 1)
( N − 1)( N + 2)( N + 3)
while all other confidence intervals are obtained by simulation. At the lower end of
skewness range it is seen that there is a sizeable number, 14, of stations which have
negative skewness which fall below the EV1 confidence interval band. At the upper end
of the skewness range there are 46 stations which fall above the upper confidence band
for the Normal distribution. There is a group of 49 stations which fall within both sets
of confidence intervals.
Because of the relatively low values of observed skewness at many gauging stations it is
appropriate to ask whether the data as a whole could be considered to have come from a
Normal distribution. All but 1 of the lower values fall above the 95% lower Normal CI
while 46 of the larger values lie above the upper 95% Normal CI. If the data as a whole
were Normal then only 2.5% of 110 or say 3 values would lie above the upper 95%
Normal CI, so that as a whole the Normal hypothesis cannot be accepted. The same
question is considered in Figure 3.8(b) in relation to the EV1 distribution. Only 5
values lie above the upper 95% EV1 CI ( ≈ 3) while 14 values ( >> 3) lie below the
lower 95% EV1 CI. Thus the hypothesis that all the samples have come from the EV1
family cannot be accepted as a whole.
25
Skewness vs RL (Normal Dist.)
4
3
Skewness
2
1
0
-1
-2
0
10
Observed
20
30
Record Length
40
Theoretical Skew. Value for N.D.
50
95% C.I.
60
67% C.I.
Skewness vs RL (EV1)
4
3
Skewness
2
1
0
-1
-2
0
10
Observed
20
30
Record_Length
Theoretical Skew. value for EV1
40
50
67% C.I.
60
95% C.I.
Figure 3.8 Observed Skewness calculated at 110 no. A1 and A2 gauging stations plotted
versus record length. Also shown are 67% and 95% Confidence Intervals for skewness
calculated from (a) Normal distribution samples and (b) EV1 distribution samples.
26
Very similar findings are obtained when analogous plots of L-Skewness are examined,
Figure 3.9 below.
The overall result is that, on the basis of skewness values, the stations with the larger
values of skewness could be modelled by an EV1 distribution but with the reservation
that there is a serious number (19 stations or 17% of all stations) which are not in
keeping with this choice. Thus while 83 % of station skewness values are consistent
with EV1 there is a large proportion, 53%, which is consistent with the normal
distribution and 45% of all stations are consistent with both Normal and EV1
assumptions.
Thus it would seem that the choice of a single distribution for all stations in Ireland is of
doubtful validity. However skewness in the usual sample sizes found in hydrology is
not renowned for its precision. L-Skewness offers some advantages but the same
conclusions can be drawn from Figure 3.9 as from Figure 3.8. An alternative to the
Skewness versus Record Length analysis is the use of L-Moment Ratio diagrams and
these are considered in the next section.
27
L-Skewness vs RL (Normal Dist.)
0.8
0.6
L-Skewness
0.4
0.2
0.0
-0.2
-0.4
0
10
20
Observed
30
Record_Length
Theoretical L-Skew.value for N.D.
40
50
95% C.I.
60
67% C.I.
L-Skewness vs RL (EV1)
0.8
0.6
L-Skewness
0.4
0.2
0.0
-0.2
-0.4
0
10
Observed
20
30
Record_Length
Theoretical L-Skew value for EV1
40
50
67% C.I.
60
95% C.I.
Figure 3.9 Observed L- Skewness calculated at 110 no. A1 and A2 gauging stations plotted
versus record length. Also shown are 67% and 95% Confidence Intervals for L-Skewness
calculated from (a) Normal distribution samples and (b) EV1 distribution samples.
28
3.3 Preliminary distribution choice with the help of L-Moment Ratio
diagrams
Another way to explore the suitable of different probability distributions is to use
moment ratio diagrams. Moment ratio diagrams based on conventional moment based
skewness and kurtosis are regarded as having very poor discriminating capability
(WMO, 1989, Appendix 3) but L-moment ratio diagrams can be more reliable as a
diagnostic tool (Hosking and Wallis, 1997 p-40). Nevertheless even L-moment ratio
diagrams cannot be regarded as entirely reliable in the context of the available sample
sizes.
The L-moment ratio diagram is a graph between L-kurtosis and L-skewness. Usually a
two-parameter distribution with a location and a scale parameter plots as a single point
while a three parameter distribution with location, scale and shape plots as a line or
curve on the diagram. Generally the distribution selection process involves plotting the
sample L-moment ratios as a scatter plot and comparing them with theoretical Lmoment ratio points or curves of candidate distributions.
Plots of dimensionless L moment ratios are shown in Figure 3.10. It compares the
observed and the theoretical relations between L-skewness and L-kurtosis for the annual
maximum flood flows at the 110 A1 and A2 sites.
In the context of 2-parameter distributions alone the data are on average closer to the
population L-moments of an EV1 distribution rather than to those of the LO and LN
distributions. Figure 3.11 shows sample L-moments of data simulated from EV1, LO
and LN distributions each with the same record lengths and parameters as the actual
records. In the case of EV1 simulated data the scatter of the points is narrower than in
Figure 3.10 especially on the left hand side. There are many more negative historical
values of L-skewness than there are among the simulated data. The LO and LN
simulated L-moments cover the range of observed values in the negative L-skewness
domain more adequately than the EV1 values but do not extend as far to the right as the
EV1 simulated and observed L-skewness values. Four further realizations of this kind
of simulation are shown in Figure 3.12 and Figure 3.13.
29
0.70
L-Moment Ratio Diagram for 110 A1 & A2 Stations
0.60
0.50
L-Kurtosis
0.40
0.30
0.20
sample_data
sample_avg
0.10
EV1
LO
0.00
LN
GEV
LN3
-0.10
GLO
Normal
-0.20
-0.30
-0.20
-0.10
0.00
0.10
0.20
L-Ske w ness
0.30
0.40
0.50
0.60
0.70
Figure 3.10 L-moment ratio diagram for annual maximum flow for 110 stations of Ireland
0.7
0.7
L-Moment Ratio( Observed data)
L-Moment Ratio (EV1 simulated data)
0.6
0.6
0.5
0.5
L-Kurtosis
L-Kurtosis
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.3
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
-0.2
-0.3
0.7
-0.1
0.0
0.1
0.2
0.3
L-Skewness
0.4
0.5
0.6
0.7
L-Moment Ratio (LO simulated data)
L-Moment Ratio( LN simulated data)
0.7
0.6
0.6
0.5
0.5
0.4
L-Kurtosis
0.4
0.3
0.2
0.3
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.2
-0.3
-0.2
0.7
0.8
L-Kurtosis
0.3
-0.2
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
-0.3
0.7
-0.2
-0.1
0.0
0.1
0.2
0.3
L-Skewness
0.4
0.5
0.6
Figure 3.11 L-moment ratio diagram of a) observed data b) data simulated from EV1
distribution c) data simulated from LN2 distribution and d) data simulated from LO
distribution
30
0.7
While on the basis of average values EV1 looks a more suitable candidate than either of
LO or LN the occurrence of so many negative values of L-skewness among the
observed data throws doubt on the suitability of EV1 as a universal 2 parameter model
for Ireland. Note also that the average of the data points falls roughly half way between
the GEV and GLO curves.
31
0.7
0.7
0.6
0.6
0.5
0.5
L-Kurtosis
L-Kurtosis
0.4
0.4
0.3
0.2
0.3
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.3
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
-0.2
-0.3
0.7
-0.2
-0.1
0.2
0.3
L-Skewness
0.4
0.5
0.6
0.7
0.6
0.6
0.5
0.5
0.4
0.4
L-Kurtosis
L-Kurtosis
0.1
0.7
0.7
0.3
0.2
0.3
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.2
-0.3
0.0
-0.2
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
0.7
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
L-Skewness
0.4
0.5
0.6
0.7
0.7
0.7
0.6
0.6
0.5
0.5
L-Kurtosis
L-Kurtosis
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.3
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
-0.2
-0.3
0.7
-0.1
0.0
0.1
0.2
0.3
L-Skewness
0.4
0.5
0.6
0.7
0.6
0.6
0.5
0.4
0.4
L-Kurtosis
0.5
0.3
0.2
0.3
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.2
-0.3
-0.2
0.7
0.7
L-Kurtosis
0.3
-0.2
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
-0.3
0.7
-0.2
-0.1
0.0
0.1
0.2
0.3
L-Skewness
0.4
0.5
0.6
0.7
Figure 3.12 First set of two realizations of four realizations of L-moment ratio diagrams of
data simulated from EV1, LN2 and LO distribution
32
0.7
0.7
0.6
0.6
0.5
0.5
L-Kurtosis
L-Kurtosis
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.1
-0.3
0.3
-0.1
-0.2
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
-0.3
0.7
-0.2
-0.1
0.2
0.3
L-Skewness
0.4
0.5
0.6
0.7
0.5
0.6
0.7
0.5
0.6
0.7
0.5
0.6
0.6
0.6
0.5
0.5
0.4
0.4
L-Kurtosis
L-Kurtosis
0.1
0.7
0.7
0.3
0.2
0.1
0.3
0.2
0.1
0.0
0.0
-0.1
-0.1
-0.2
-0.3
0.0
-0.2
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
0.7
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
L-Skewness
0.4
0.7
0.7
0.6
0.6
0.5
0.5
L-Kurtosis
L-Kurtosis
0.4
0.4
0.3
0.2
0.3
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.3
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
-0.2
-0.3
0.7
-0.2
-0.1
0.7
0.6
0.5
0.5
0.4
0.4
L-Kurtosis
L-Kurtosis
0.1
0.2
0.3
L-Skewness
0.4
0.6
0.3
0.2
0.3
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.2
-0.3
0.0
0.7
-0.2
-0.2
-0.1
0.0
0.1
0.2
L-Skewness
0.3
0.4
0.5
0.6
-0.3
0.7
-0.2
-0.1
0.0
0.1
0.2
0.3
L-Skewness
0.4
0.7
Figure 3.13 Second set of two realizations of four realizations of L-moment ratio diagrams
of data simulated from EV1, LN2 and LO distribution
33
3.4 Largest recorded values and outliers
The largest flow values, Qmax , in each record are also listed in Tables 3.2 to 3.4 as are
dimensionless values Qmax/Qbar and Qmax/Qmed and the specific value Qmax/A.
The largest values of Qmax/Qbar and Qmax/Qmed among the 45 Grade A1 stations are
6.1 and 10.7 respectively, at station 08009 on the River Ward at Balheary. These
resemble those found in UK data but there are only two other stations where these
statistics exceed 3 and 4 respectively, namely the 09002 River Griffeen at Lucan and the
09010 River Dodder at Waldron’s Bridge.
At Balheary the record is 17 years long but there are 7 incomplete years, leaving only 10
valid annual maxima which are
10.30, 6.59,
6.77, 3.62,
6.39, 2.72,
6.95, 2.08,
3.31, 53.6
Regardless of whether the record is short or long the 1993 (04/02/1994) value of 53.6,
even allowing for difficulties of measurement and extrapolation of the rating curve, is
truly a large outlier. This station's data are not included in the inferences and summaries
of the preceding sub-sections.
The large value at Waldron's Bridge occurred in August 1986 as a result of Hurricane
Charlie and of all the exceptional flows which occurred in South Dublin and Wicklow at
that time it is the only flow to have been measured. Again, even allowing for difficulties
of measurement and extrapolation of the rating curve, it is truly a very large flood in the
Irish context.
At all other 42 A1 stations the values of these ratios are less than 3 which are extremely
low by European standards. It should be noted that these quantities depend on record
length and not all records in the data set are of equal length, varying from 15 to 55
years. The corresponding largest values of Qmax/Qbar and Qmax/Qmed among 70
Grade A2 stations are 2.8 and 3.1 respectively.
The largest specific values Qmax/A in cumec/km2 are also listed as are the quantities
Qmax/A . The last quantity is of interest because as is well known flood values do not
increase linearly with A but rather with Aa where the power a varies from 0.75 to 0.85
depending on circumstances.
34
An outlier is an observation that lies an abnormal distance from other values in a
random sample from a population. In other words an outlier lies significantly away from
the trend of the remaining data. The presence of outliers causes problem with linearity
of pattern on probability plots, see next Chapter. The 17 stations with the most obvious
outliers have been selected from the probability plots and are listed in Table 3.6. This
shows only three stations with a Qmax/Qmed value in excess of 3 and only 8 stations
with values in excess of 2. These numbers are extremely low by comparison with UK
data for instance. There are other instances of notably large floods which have occurred
on ungauged catchments or at Grade B stations and these are not included in Table 3.6
eg Dargle at Bray in 1986 (ungauged) and Deel at Crossmolina ( no grade).
Table 3.7 lists two stations with notable low outliers. Low outliers can have a negative
effect on skewness values and on the appearance of probability plots.
Table 3.6 High outliers from 110 Irish stations
Station
No
Station
Class
Record
Length
largest Q
or Qmax
2nd
Largest Q
3rd
Largest Q
Qmed
Qmax/
L.Q/
2ndL.Q/
Qmed
2ndL.Q
3rdL.Q
6011
A1
48
26.36
21.07
19.99
6031
A2
18
35.8
24.4
18.1
15.39
1.71
1.25
1.05
11.7
3.06
1.47
1.35
7033
A2
25
26.58
20.56
8009
A1
11
18.71
14.62
1.82
1.29
1.1
11.8
156
10.3
112
6.59
8.13
4.54
1.15
9010
A1
19
53.6
269
48
5.60
1.72
1.39
14005
A2
48
15004
A2
51
80.42
80.42
59.99
38.27
2.10
1
1.34
74.96
73.61
56.88
37.28
2.01
1.02
1.29
16002
A2
23012
A2
51
123.88
84.42
81.86
52.66
2.35
1.47
1.03
18
31.74
26.27
19.12
15.66
2.03
1.21
24022
1.37
A2
20
20.5
15
14.2
23.49
0.87
1.37
1.06
26006
A1
53
70.06
68.66
41.81
29.23
2.40
1.02
1.64
29011
A1
22
66.52
44.34
40.61
28.89
2.30
1.5
1.09
33070
A1
28
11.97
9.44
9.35
7.67
1.56
1.27
1.01
34001
A2
36
286.56
224.44
219.12
174.61
1.64
1.28
1.02
36015
A1
33
49.99
43.3
31.77
22.08
2.26
1.15
1.36
36021
A2
27
43.57
32.16
31.79
23.37
1.86
1.35
1.01
36031
A2
30
13.7
8.87
8.32
6.45
2.12
1.54
1.07
Table 3.7 Low outliers from 110 Irish stations
Lowest
Q
2nd
Lowest
3rd Lowest
Station
No
Station
Class
Record
Length
or Qmin
Q
Q
15003
A2
50
61.13
96.72
98.02
26059
A1
17
8.23
10.7
11.4
35
Qmed
2nd L.Q
3rd L.Q
/Qmin
/L.Q
/2nd L.Q
150.76
2.47
1.58
1.01
12.2
1.48
1.3
1.07
Qmed
4 Probability Plots and Inferences from them
4.1 Probability Plots
Probability plots are useful techniques for the graphical display and analysis of flood
data, particularly to determine whether or not a particular sample is consistent with from
a particular population distribution. Probability plots employ an inverse distribution
scale so that a simple, usually 2 parameter, cumulative distribution function (CDF) plots
as a straight line, i.e the points (x(i), yi); i = 1, 2, ..., n expect to be close to line y =a +
b*x, where a and b are location and scale parameters respectively; conversely, strong
deviation from this line is evidence that the distribution did not produce the data. It is
acknowledged that what constitutes "strong deviation" from such a line is usually
judged subjectively. Nevertheless they are widely used in flood hydrology for data
display if not for determining a final distribution choice.
A small selection of probability plots of samples, of size 25 and 50, drawn randomly
from an EV1 population are shown in Figure 4.1(a) and (b). These show, in some cases,
departures from linear patterns that could cause an EV1 hypothesis to be rejected, on the
basis of visual inspection, in an isolated case that might occur in practice. Hence
deductions from probability plots have to be considered carefully.
Three probability plots namely Gumbel (EV1), logistic (2 parameter or LO) and
Lognormal (2 parameter or LN) are plotted for the annual maximum discharge data of
187 Irish stations. The annual flood data from these 186 stations vary in record length
from 8 to 55 years and the associated catchment areas range from 10.1 to 2780 km2.
Data from an additional 17 stations which have pre- and post-drainage records are also
plotted. The probability plots are shown in Appendix 4 which is subdivided into A1,
A2, B gauging station categories and which contains two further sections for ESB
stations and pre- and post-drainage stations.
Plotted data for 110 A1 and A2 stations were investigated from the standpoint of
linearity and non-linearity. In the linearity investigation a best-fit least squares straight
line is drawn through the data and is given a score from 1 to 5 where, 1 stands for a very
poor fit and 5 stands for a very good fit. On the other hand in the non-linearity
36
investigation an eye-guided assessment of the plotted data is given a curve pattern like
linear, concave, convex or S-curve by visual judgment. It is acknowledged that
deductions drawn from probability plots have low statistical power i.e. there is a high
probability that a conclusion could be adopted even though it is not true. Nevertheless
probability plots have played a prominent role in hydrological frequency analysis in the
past and it is considered worthwhile to use them in this study as a way of viewing the
available data.
37
300
250
300
250
200
250
200
150
150
Q
Q
Q
200
100
50
50
0
0
-2
-1
0
1
2
3
4
5
50
0
-2
-1
0
1
EV1 y
2
3
4
5
-2
0
1
4
5
2
3
4
5
2
3
4
5
300
250
200
Q
Q
150
200
150
100
100
3
350
200
150
2
EV1 y
250
250
Q
-1
EV1 y
300
100
50
50
0
0
-2
-1
0
1
2
3
4
5
50
0
-2
-1
0
1
EV1 y
2
3
4
-2
5
350
300
300
200
250
250
200
200
Q
150
100
50
0
Q
350
150
150
100
100
50
50
0
-1
0
1
2
EV1 y
0
1
EV1 y
250
-2
-1
EV1 y
300
Q
150
100
100
3
4
5
0
-2
-1
0
1
2
3
4
EV1 y
5
-2
-1
0
1
EV1 y
Figure 4.1(a)
38
350
350
250
300
300
250
250
200
200
150
Q
Q
200
100
50
Q
300
150
150
100
100
50
0
50
0
-2
-1
0
1
2
3
4
5
0
-2
-1
0
1
450
400
350
300
250
200
150
100
50
0
4
5
-2
2
3
4
Q
100
50
50
-1
0
1
2
3
4
-2
5
200
200
150
150
150
Q
250
200
Q
300
250
100
100
100
50
50
50
0
0
2
EV1 y
0
1
3
4
5
2
3
4
5
2
3
4
5
EV1 y
300
1
-1
EV1 y
250
0
5
0
-2
300
-1
4
150
EV1 y
-2
3
200
100
5
2
250
150
0
1
1
300
200
0
0
350
250
-1
-1
EV1 y
300
-2
Q
3
EV1 y
Q
Q
EV1 y
2
0
-2
-1
0
1
2
3
4
5
-2
-1
0
1
EV1 y
EV1 y
Figure 4.1 (a, b) Random Samples of (a) size 25 (b) size 50 from an EV1(100,30) population. (Cv = 0.33)
39
4.2 Linear Patterns
Each data series was examined from the point of view of linearity and subjectively
scored from 1 to 5 by visual judgment,
where 1= very poor;
2= poor;
3= medium;
4= good; and
5= very good fit.
These scores are listed individually for each station in Table 4.3 and Table 4.4 for A1
and A2 stations respectively and a summary of these scores is given in Table 4.1, as
follows:
Table 4.1 Linear pattern statistics for A1 and A2 stations combined. Scores at individual
stations are tabulated in Table 4.3 and Table 4.4.
Linear pattern counts for A1 stations
Type of plot
EV1
LO
LN
Modal Score
4
2
3
Mean Score
3.3
2.6
3.3
Linear pattern counts for A2 stations
Type of plot
EV1
LO
LN
Modal Score
4
3
4
Mean Score
3.4
3.1
3.4
Linear pattern counts for A1& A2 stations
Type of plot
EV1
LO
LN
Modal Score
4
3
4
Mean Score
3.3
2.9
3.3
The above statistics show that EV1 and LN give almost identical results and they
provide a better fit than LO. It can be deduced from the linear patterns that Irish flood
data are more likely to be distributed as EV1 or LN rather than LO among 2-parameter
distributions.
40
4.3 Non Linear Patterns
Each data series was investigated from the point of view of non-linearity and was
assigned a curve pattern by visual judgment. Four curve patterns linear, concave,
convex and elongated S are introduced and, again each has been classified as degree one
and degree two for greater discrimination. However, there is also an ‘X’ introduced if
the patterns of the end points of the plot do not match the main body pattern well. In
summary, non-linear patterns are classified as:
L1= perfectly straight line
L2= little deviation from straight line
L2X= body pattern is quite straight but end points disturb the linearity
U1= mild concave upwards
U2= severe concave upwards
U1X= mild concave with end disturbance
U2X= severe concave with end disturbance
D1= mild convex upwards
D2= severe convex upwards
D1X= mild convex upwards with end disturbance
D2X= severe convex upwards with end disturbance
S1= mild S-curve
S2= severe S-curve
S1X = mild S-curve with end disturbance
S2X= severe S-curve with end disturbance
These scores are listed individually for each station in Table 4.3 and Table 4.4 for A1
and A2 stations respectively and a summary of these scores is given in Table 4.2 as
follows:
In EV1 probability plots for A1 stations, 22 out of 43 show a linear trend and a convex
trend comes out as second dominant pattern with 8 stations; while for A2 which
contains 67 stations, the corresponding numbers are 29 and 16 respectively.
Interestingly very few stations exhibit concave pattern on EV1 probability plots. This is
41
because in many cases the largest floods on record are close together in magnitude and
in very few cases there are notably large floods with very few cases of large outliers.
Table 4.2 Non- linear pattern statistics for A1 and A2 stations. Scores at individual
stations are tabulated in Table 4.3 and Table 4.4.
Curve
Patterns
L1
L2
D1
D2
U1
U2
S1
S2
D1x
D2x
U1x
U2x
S1x
S2x
L2x
Non linear score counts
: A1
EV1
5
17
8
0
2
3
1
1
0
0
0
1
2
2
1
Type of plot
LO
0
9
1
0
11
4
8
3
0
1
1
1
1
1
2
LN
5
17
1
0
5
0
8
1
0
1
0
1
2
1
1
: A2
EV1
5
24
9
7
0
3
9
3
0
1
0
0
0
1
5
Type of plot
LO
0
19
2
1
17
5
13
5
0
1
1
0
0
1
2
LN
7
21
7
2
3
2
11
5
1
0
0
1
1
0
6
: A1& A2
EV1
10
41
17
7
2
6
10
4
0
1
0
1
2
3
6
Type of plot
LO
0
28
3
1
28
9
21
8
0
2
2
1
1
2
4
In Logistic probability plots concave and linear patterns emerge as the dominant
patterns and a significant number of S shapes was also found. Hardly any convex
pattern was found in LO plots. There was no station found with a perfect straight line.
In the case of lognormal plots a linear trend was found as the dominant one. A
considerable number of S and convex patterns was also observed. In Table 4.3 and
Table 4.4 it was seen that in many cases EV1 and Lognormal complement one another.
In many cases it was observed that those stations which fit quite straight on EV1 plots
show a concave fit on Logistic paper. The reason might be due to the range of the
probability axis, because the probability axis of LO is relatively compressed compared
to EV1. That is why it was also noticed that those which fit as convex on EV1 plots are
found to be quite linear on LO.
42
LN
12
38
8
2
8
2
19
6
1
1
0
2
3
1
7
Table 4.3 Probability plot linear and non-linear scores for A1 category Stations
Station No, Location
6011 FANE
6013 DEE
6014 GLYDE
6025 DEE
6026 GLYDE
6070 MUCKNO
7009 BOYNE
8002 DELVIN
8009 BALHEARY
9001 RYEWATER
9002 GRIFFEEN
9010 DODDER
10021 SHANGANAGH
10022 CARRICKMINES
14006 BARROW
14007 DERRYBROCK
14011RATHANGAN
14018 BARROW
14019 BARROW
16011 SUIR
25006 BROSNA
25014 SILVER
25017 SHANNON
25023 LITTLE BROSNA
25025 BALLYFINBOY
25027 OLLATRIM
25030 GRANEY
26006 SUCK
26007 SUCK
26008 RINN
26019 CAMLIN
26020 CAMLIN
26059 INNY
27002 FERGUS
29001 RAFORD
29011 DUNKELLIN
31002 CASHLA
33070 CARROWMORE
34018 CASTLEBAR
36010 ANNALEE
36012 ERNE
36015 FINN
36018 DRONMORE
Record
Length
48
30
30
30
46
24
29
20
11
48
24
19
24
18
51
25
26
51
51
52
52
54
55
52
31
43
48
53
53
49
51
32
17
51
40
22
26
28
27
50
47
33
50
EV1
4
2
4
2
3
4
4
4
1
5
1
2
4
4
5
4
3
3
3
3
4
4
3
2
5
4
4
1
5
4
4
3
2
4
3
2
2
4
4
4
3
2
4
43
Linear Score
LO
2
3
2
4
1
4
1
1
1
2
1
2
2
3
2
2
3
3
3
2
3
3
4
2
4
5
2
1
3
3
3
3
2
2
4
2
1
3
3
3
4
2
4
LN
3
4
3
3
2
5
3
2
2
5
4
4
3
4
3
3
3
4
4
3
4
4
5
2
5
3
4
1
5
3
3
2
1
4
4
2
1
4
3
4
3
2
4
Non linear Score
EV1
LO
LN
L2
U1
L2
D1
S1
L2
L1
U1
L2
D1
L2
S1
S1
S2
S2
L2
L2
L1
L2
D2X
L2
L2
U1
U1
U2
U2
U1
L1
U1
L1
S2X
U2
L1
U2
U2
L2
L2
S1
L2
L2
L2X
L2
L1
U1
U1
L2
U1
S1
L2
L2
L2
L2
S1
S1
D1
S2
S1
U1
S1
S1
L2
L2
S1
L2
U1X
L2
D1
L2
L1
S2
S2
S1X
L1
L2
L2
D1
L2
D1
S1X
U1
L2
S2X
S2X
S2X
L1
U1
L1
L2
U1
L2
L2
S1
L2
L2
S1
L2
D1
D1
D2X
L2
U1
L2
D1
S1
S1
U1
S1
S1
U2X
U2X
U2X
L2X
L2X
L2X
L2
U1
U1
S1X
S1X
S1X
D1
L2
L2
U2
U2
U1
L2
L2
L2
Table 4.4 Probability plot linear and non-linear scores forA2 category Stations
Station No,Location
6031 FLURRY
7006 MOYNALTY
7033 BLACKWATER
8005 SLUICE
8008 BROADMEADOW
12001 RIVER SLANEY
14005 BARROW
14009 CUSHINA
14013 BURRIN
14029 BARROW
14034 BARROW
15001 RIVER KINGS
15002 NORE
15003 DINAN
15004 NORE
16001 DRISH
16002 SUIR
16003 CLODIAGH
16004 SUIR
16005 MULTEEN
16008 SUIR
16009 SUIR
18004 BALLYNAMONA
18005 FUNSHION
19001 OWENBOY
19020 OWENNACURRA
23001 GALEY
23012 LEE
24002 CAMOGUE
24008 MAIGUE
24022 MAHORE
24082 MAIGUE
25016 CLODIAGH
25021 BROSNA
25029 NENAGH
25034 L. ENNELLTRIB
25040 BUNOW
25044 KILMASTULLA
25124 BROSNA
26002 SUCK
26005 SUCK
26009 BLACK
26018 OWENURE
26021 INNY
26022 FALLAN
27001 CLAUREEN
27003 FERGUS
29004 CLARINBRIDGE
Record
Length
18
19
25
18
25
50
48
25
50
47
14
42
35
50
51
33
51
51
48
30
51
52
46
50
48
28
45
18
27
30
20
28
42
44
33
24
20
33
18
53
51
35
49
30
33
30
48
32
Ev1
2
2
4
4
4
4
3
4
4
3
4
2
4
2
5
5
3
4
5
5
2
2
2
4
4
3
4
2
4
3
5
2
4
3
3
2
4
3
3
2
4
4
4
1
4
4
4
4
44
Linear Score
LO
2
3
3
3
2
2
2
3
3
4
2
2
3
3
3
4
2
3
4
2
4
3
2
2
4
4
2
1
5
5
3
5
4
4
3
4
3
3
4
1
4
3
3
2
3
3
4
3
LN
3
2
3
5
4
3
2
4
4
5
2
3
4
2
5
5
4
3
5
2
3
3
2
3
4
3
3
1
4
4
5
3
5
5
4
3
4
3
2
2
5
2
4
1
4
3
4
3
Non Linear Score
EV1
LO
LN
U2
U2
U2
D2
D1
D2
S1
U1
S1
L2
U1
L1
L2
U1
L2
L2
U1
L2
L2
U1
U1
S1
U1
S1
S1
S2
S1
D1
S1
L1
L2X
U2X
U2X
D2
S2
S1
L2
U1
L2
D2
S1
D2
L2
U1
L2
L1
U1
L1
L2X
U1X
L2X
L2
S1
S1
L1
U1
L1
L2
U1
U1
D2
L2
D1
D2X
D2X
D1X
S2
S2
S2
L2X
U2
L2X
L2
L2
S1X
D2
S1
D1
L2
U1
L2
U2
U2
U2
L2
L2
L2X
L2
L2
L2X
L1
U1
L2
D1
L2
D1
L2
L2
L2
D1
L2
L2
S2
S2
S1
D2
L2
D1
L2
U1
L2
S1
S1
S2
D1
L2
D1
S2X
S2X
L2X
L1
L2
L1
S1
S1
S2
L2
S1
L2
D2
D1
D1
L2
S1
S1
L2
U1
L2
L2
L2
L2
L2
L2
S1
29071 L.CUTRA
30007 CLARE
30061 CORRIB ESTUARY
32012 NEWPORT
34001 MOY
34003 MOY
34009 OWENGARVE
34011 MANULLA
34024 POLLAGH
35001 OWENMORE
35002 OWENBEG
35005 BALLYSADARE
35071 L.MELVIN
35073 L.GILL
36019 ERNE
36021 YELLOW
36031 CAVAN
39008 GARTAN
39009 FERN O/L
26
31
33
24
36
29
33
30
28
29
34
55
30
30
47
27
30
33
33
4
4
1
3
4
3
4
4
3
4
3
4
4
3
2
3
2
3
5
3
3
1
5
3
3
4
4
4
5
5
2
3
4
3
3
1
2
3
4
4
2
3
4
3
5
4
3
4
4
4
4
4
3
3
1
3
5
L2
L2X
U2
D1
L2X
S1
L2
S1
D1
L2
D1
S1
L2
D1
D1
L2
S2
S1
L1
S1
L2X
U2
L2
L2X
S1
L2
L2
L2
L2
L2
S1
U1
L2
S1
U2
S2
S1
U1
4.4 Variation of shape of probability plot with Skew coefficient
Having noted that skewness is negative or a small positive value in many AM series the
estimated skewness values have been ranked in increasing order and the corresponding
curve pattern of the data series was noted alongside for the case of EV1 probability
plots. This summary is presented in Table 4.5 and Table 4.6 for A1 stations and A2
stations respectively.
These tables show that:
1. There are 1 A1 station and 13 A2 stations with negative skewness and a further 22 A1
stations and 27 A2 stations with skewness between 0 and 1. The average skewness of
110 Irish stations is 1.0.
2. Stations with negative or left skewness predominantly have convex upwards curve
patterns.
3. Stations with higher or right skewness mostly have concave upward shape.
4. In addition, most of the high outliers are found in the high right skewness zone and
conversely most of the low outliers are found in left skewness zone.
45
L2
L2X
U1
L2
L2
S2
L1
L2
L2
L2
L2
L2
L2
D1
S1
S1
S2
S1
L1
Table 4.5 A1 stations ranked by skewness
Hazenskew
-0.33
0.10
0.16
0.18
0.19
0.21
0.22
0.27
0.31
0.34
0.36
0.43
0.42
0.57
0.57
0.57
0.61
0.71
0.78
0.81
0.93
0.93
0.95
1.01
1.05
1.05
1.05
1.12
1.17
1.17
1.19
1.31
1.37
1.46
1.49
1.92
1.98
2.40
2.62
3.24
3.37
3.87
5.43
6025 DEE
6013 DEE
26020 CAMLIN
25017 SHANNON
14011RATHANGAN
36012 ERNE
14018 BARROW
25027 OLLATRIM
26059 INNY
29001 RAFORD
10022 CARRICKMINES
36018 DRONMORE
16011 SUIR
25014 SILVER
25023 LITTLE
BROSNA
25025 BALLYFINBOY
14019 BARROW
25006 BROSNA
6070 MUCKNO
6011 FANE
34018 CASTLEBAR
7009 BOYNE
10021 SHANGANAGH
25030 GRANEY
26007 SUCK
6026 GLYDE
36010 ANNALEE
14007 DERRYBROCK
26019 CAMLIN
9001 RYEWATER
33070 CARROWMORE
6014 GLYDE
27002 FERGUS
14006 BARROW
26008 RINN
31002 CASHLA
8002 DELVIN
9002 GRIFFEEN
36015 FINN
9010 DODDER
29011 DUNKELLIN
26006 SUCK
8009 BALHEARY
Record
length
30
30
32
55
26
47
51
43
17
40
18
50
52
54
Curve patterns on
EV1
convex
convex
linear
convex
linear
convex
linear
convex
convex
convex
linear
linear
convex
linear
52
31
51
52
24
48
27
29
24
48
53
46
50
25
51
48
28
30
51
51
49
26
20
24
33
19
22
53
11
linear
linear
convex
linear
linear
linear
linear
linear
linear
s-curve
linear
s-curve
s-curve
linear
linear
linear
linear
linear
linear
linear
linear
concave
linear
s-curve
concave
concave
concave
s-curve
concave
46
Remarks
Low outlier
High Outlier
High Outlier
High Outlier
High Outlier
High Outlier
High Outlier
High Outlier
Table 4.6 A2 stations ranked by skewness
Hazenskew
-1.09
-0.86
-0.78
-0.48
-0.41
-0.39
-0.32
-0.31
-0.22
-0.16
-0.13
-0.12
-0.09
0.07
0.10
0.12
0.14
0.18
0.27
0.28
0.30
0.32
0.34
0.37
0.43
0.47
0.48
0.53
0.59
0.62
0.71
0.73
0.74
0.76
0.77
0.77
0.78
0.85
0.93
0.96
1.03
1.04
1.08
1.12
1.22
1.23
1.25
1.26
1.30
7006 MOYNALTY
26021 INNY
15003 DINAN
25034 L. ENNELLTRIB
16009 SUIR
34024 POLLAGH
16008 SUIR
25124 BROSNA
24082 MAIGUE
36019 ERNE
25021 BROSNA
32012 NEWPORT
25029 NENAGH
35002 OWENBEG
35001 OWENMORE
19020 OWENNACURRA
15001 RIVER KINGS
14029 BARROW
26005 SUCK
24008 MAIGUE
16001 DRISH
24002 CAMOGUE
35073 L.GILL
14013 BURRIN
34009 OWENGARVE
26022 FALLAN
19001 OWENBOY
16004 SUIR
15002 NORE
25016 CLODIAGH
27003 FERGUS
39008 GARTAN
29071 L.CUTRA
35071 L.MELVIN
26018 OWENURE
29004 CLARINBRIDGE
34011 MANULLA
16003 CLODIAGH
26009 BLACK
15004 NORE
39009 FERN O/L
35005 BALLYSADARE
34001 MOY
24022 MAHORE
23001 GALEY
25044 KILMASTULLA
18004 BALLYNAMONA
34003 MOY
30007 CLARE
Record
length
19
30
50
24
52
28
51
18
28
47
44
24
33
34
29
28
42
47
51
30
33
27
30
50
33
33
48
48
35
42
48
33
26
30
49
32
30
51
35
51
33
55
36
20
45
33
46
29
31
47
Curve patterns on
EV1
convex
convex
convex
convex
convex
convex
convex
convex
convex
convex
convex
convex
s-curve
convex
linear
convex
convex
convex
linear
linear
linear
linear
convex
s-curve
linear
linear
linear
linear
linear
linear
linear
s-curve
linear
linear
linear
linear
s-curve
linear
s-curve
linear
linear
s-curve
linear
linear
linear
s-curve
s-curve
s-curve
linear
Remarks
Low outlier
High outlier
High outlier
High outlier
1.32
1.33
1.42
1.54
1.55
1.59
1.63
1.74
1.74
1.79
1.85
2.01
2.23
2.29
2.98
3.04
3.12
4.19
25040 BUNOW
16005 MULTEEN
14009 CUSHINA
8005 SLUICE
27001 CLAUREEN
12001 RIVER SLANEY
18005 FUNSHION
8008 BROADMEADOW
7033 BLACKWATER
16002 SUIR
36021 YELLOW
14005 BARROW
14034 BARROW
26002 SUCK
23012 LEE
30061 CORRIB
ESTUARY
6031 FLURRY
36031 CAVAN
20
30
25
18
30
50
50
25
25
51
27
48
14
53
18
linear
linear
s-curve
linear
linear
linear
linear
linear
s-curve
linear
linear
linear
linear
s-curve
concave
33
18
30
concave
concave
s-curve
48
High outlier
High outlier
High outlier
High outlier
High outlier
High outlier
High outlier
4.5 Flood volumes associated with largest peaks on convex probability
plots
For many of the stations which display a convex upwards appearance on probability
plots it is noted that the largest and the 2nd largest floods are not appreciably larger than
the 3rd largest flood. This contributes to the flattening out of the probability plot and
the calculated skewness tends to be small or negative. Such stations are identified as
D1, D2 or DX in Table 4.3 and Table 4.4 where there are 8 and 21 such occurrences
respectively. These stations are identified as "convex" in Table 4.5 and Table 4.6.
There are other stations, not classified as D1, D2 or D2X in the subjective classification,
which also display such behaviour to a lesser degree but such stations are not considered
further here.
It could be asked if this behaviour can be attributed to catchment slope or tendency to
show over-bank flow at high flow values. In other words is there any storage available
which would dampen out the flood peaks while the flood volumes are different between
the largest and second/third larges floods i.e. could volume be increasing when peak
discharge is stagnant?
To investigate this question hydrograph volumes have been calculated for each of the
three or four largest flood peak values in a selection of 8 series from the 29 series
identified as having convex upwards series.
Having examined the results, see
discussion below, it was decided not to proceed with assembly of hydrograph data for
the remaining 21 stations. The stations examined are listed in Table 4.7.
49
Table 4.7 Sample of 8 stations showing convex upward appearance on probability plots.
Station
Hazen
Duration
Grade
Skewness
hydrograph
Station No.
River and Station Name
7006
R. Moynalty at Fyanstown
A2
-1.088
1956-2005*
15003
R. Dinan at Dinan Bridge
A2
-0.996
1972-2005
16008
R. Suir at New Bridge
A2
-0.324
1954-2005
16009
R. Suir at Cahir Park
A2
-0.409
1940-2005
24013
R. Deel at Rathkeale
A1
-0.908
1972-2003
24082
R. Maigue at Islandmore
A2
-0.216
1975-2001
25017
R. Shannon at Banagher
A1
0.018
1989-2003
25021
Little Brosna at Croghan
A2
-0.134
1961-2003
of
data
available
* (1987-2006 post-drainage period alone was used in this part of the study)
Volume calculation:
On examination of hydrographs it is seen that some of the larger annual maximum flows
occur as a result of a single rainfall event but many of them occur as a result of longer
rainfalls of varying intensities which lead to multi peak hydrographs around the time of
the annual maximum flood. As a result no single time duration for calculation of
hydrograph volume can be specified. Hence volumes were calculated for a number of
time durations, or windows, symmetrically located around the time of the peak. The
windows corresponded to ±12hours, ± 24 hours, ±84 hours, ±1week and ±2 weeks
leading to window sizes of 1, 2, 7,14 and 28 days respectively.
The results for stations 24082 River Maigue at Islandmore and 7006 River Moynalty at
Fyanstown are summarised in Figure 4.2 and Figure 4.3 below. The former example is
based on 46 years observations, 1977 – 2004 while the later one is based on 19 years of
post drainage observations, 1986 – 2004.
50
250
EV 1
AMF(m3/s)
200
'89
'98 '88
'00
150
w inter peak
summer peak
100
50
2
5
10
25
50
100
500
0
-2
-1
0
250
1
2 EV 1 y 3
4
Hydrogra ph during Ma x im um Flood in the ye a r 1989
5
6
7
Red zone= Vol. of 1day
Yellow zone= Vol. of 2day
Blue zone= Vol. of 1week
Discharge (m3/s)
200
150
100
50
0
01/02/1990 00:00
03/02/1990 00:00
200
05/02/1990 00:00
07/02/1990 00:00
Time
09/02/1990 00:00
180
11/02/1990 00:00
160
Discharge (m3/s)
140
120
100
80
60
40
20
0
31/10/2000 00:00
02/11/2000 00:00
04/11/2000 00:00
06/11/2000 00:00
Time
51
08/11/2000 00:00
10/11/2000 00:00
12/11/2000 00:00
200
180
160
Discharge (m3/s)
140
120
100
80
60
40
20
0
14/10/1988
00:00
16/10/1988
00:00
200
18/10/1988
00:00
20/10/1988
00:00
22/10/1988
00:00
Time
24/10/1988
00:00
180
26/10/1988
00:00
28/10/1988
00:00
160
Discharge (m3/s)
140
120
100
80
60
40
20
0
23/12/1998 00:00
25/12/1998 00:00 27/12/1998 00:00
160
140
29/12/1998 00:00 31/12/1998 00:00
Time
02/01/1999 00:00 04/01/1999 00:00
V olum e of hydrographs of diffe re nt ye ar during m ax pe ak
1998
1988
Mill cu. m eter
120
100
2000
1989
80
60
40
20
0
1
2
Days
7
14
30
maximum flood peaks at 24082 River Maigue at Islandmore.
In Figure 4.2 the flood hydrographs are shown in decreasing order of peak magnitudes
from 1989 (largest) to 1998 (smallest) while in the final part of Figure 4.2 the flood
volumes are shown in increasing order of magnitude 1998, corresponding to the
52
smallest of the 4 floods, 1988 (2nd smallest), 2000 (2nd largest) and 1989 (largest of the
record).
One day and two day flood volumes are noticeably similar for these four events. For 7
and 14 and 30 days the 1989 event, the largest peak, gives the largest volume among the
four.
40
7006 RIVER MOYNALTY @ FYANSTOW N
35
'00 '04
AMF(m3/s)
30
'93
'95
w inter peak
summer peak
25
20
15
10
5
2
5
10
25
50
100
500
0
-2
-1
25
0
1
2 EV 1 y 3
4
5
6
7
Volume of hydrographs of different year during max peak
2000
20
1995
Mill. cu. m eter
1993
15
10
5
0
1
2
7
Days
14
30
maximum flood peaks at 7006 River Moynalty at Fyanstown.
In the Figure 4.3 the individual hydrographs are not shown and the flood volumes are
shown in the order 2000 (corresponding to the smallest of the 3 floods), 1995, and 1993
(largest flow on record).
The flood peak magnitudes increase slightly in the order shown, i.e. 2000 peak is the
smallest while 1993 peak is the largest. One day flood volumes are noticeably similar
for these three events. For 2, 7 and 14 and 30 days the 1995 event shows a slightly
53
higher volume than the 2000 event but the 1993 event, that with the largest peak gives
the least volume among the three; in fact it is 50% of the 1995 event volume for 7, 14
and 30-day. The 2000 and 1995 flood events display multiple peaks which probably
explain why they have greater volumes than the 1993 event.
Of the remaining 6 stations so considered, 5 of them show the same behaviour as that
shown above, namely 1-day and 2-day volumes which are almost identical across the 3
or 4 flood events analysed. As a result it may be concluded that flood volumes do not
reveal any noticeable growth with rank among the highest ranking floods in those series
which display convex upwards patterns on probability plots. That is, no clue about the
flattening out of the plots can be found among the flood volumes which would seem to
rule out local attenuation effects, meaning that attenuation effects if any are attributable
to the effects of the upstream catchment.
The exception is 15003 River Dinan at Dinan Bridge where there is noticeable growth
of volume with flood peak magnitude, at 1 and 2 day durations and slight growth of
volume at 7 days see Figure 4.4 below. However this is reversed for the larger durations
where the largest flood peak (1985) is associated with the smallest volume.
54
200
'98 '68
'85
'97
AMF(m3/s)
150
w inter peak
summer peak
100
50
2
5
10
25
50
100
500
0
-2
-1
50
Mill. cu. meter
1
2 EV 1 y 3
4
5
6
7
Volume of hydrographs of different years during max peak
45
1998
40
1997
35
0
1985
30
25
20
15
10
5
0
1
2
Days
7
14
30
maximum flood peaks at 15003 River Dinan at Dinan Bridge.
The entire sample of 8 stations is presented in Appendix 3. The conclusion remains
that, except in the case of the Dinan, where the largest flood peaks show very little
variation the flood volumes also remain more or less constant so that no explanation for
the flattening out of the probability plots can be found among the flood volumes.
As part of Work Package 5.3, a catchment descriptor called Floodplain Attenuation
Index, FAI, was developed for each gauged catchment in the study. The Index falls in
the range 0 to 1 and it might be hoped that it could explain some of the behaviours
described above. A plot of skewness , as an indicator of convex upwards behavour ,
against FAI for those catchments whose skewness is less than 1 does not reveal any
repentance between these variables (Figure 4.5) leading to the conclusion that the
concave downwards behaviour cannot be explained by the Flood Attenuation Index.
55
FAI vs H.skewness
Data point: 60 ( stations whose skewness are less than 1.0)
0.35
0.30
0.25
FAI
0.20
0.15
0.10
0.05
0.00
-1.50
-1.00
-0.50
0.00
Hazen Skewness
Figure 4.5 A plot between FAI vs Hazen Skewness
56
0.50
1.00
1.50
5 Flood Seasonality
Annual maximum floods were examined, with respect to seasonality, on 202 stations
(A1, A2 and B categories). After the seasonality study had been completed some of the
202 stations were considered unsuitable for further statistical analysis because data
quality issues. These issues did not affect the seasonality studies. A total of 6969 station
years of data were involved. Data for 203 stations are available for categories A1, A2
and B but one of these does not have date of occurrence listed.
1. Floods are found to occur at all times of year, but most rivers register in between 65%
and 100% in the winter time i.e. October-March half-year. In all, 6094 of the 6969
annual maxima occurred in winter. Table 5.1 lists all stations with the corresponding
percentage of peak floods in the October-March period. There are 11 stations where no
peak floods were observed in the summer time i.e. April-September.
2. Figure 5.1 shows the frequency of flood peaks in each of the 12 months. The months
December and January are associated with the greatest number of AM flood events
followed by November and February. In the summer time considerable numbers of
flood peaks were observed in August while July has the least number of AM floods.
Seasonal Flood Frequency
1800
Total no. of station
years: 6969
1600
No. of Occurances
1400
1200
1000
800
600
400
200
0
Aug
Sep
Oct
Nov
Dec
Jan
Feb
M onth
Mar
Apr
May
Figure 5.1 Seasonal occurrence of annual maximum flood values in 202 stations
57
Jun
Jul
3. Figure 5.2 illustrates the seasonal distribution of AM flood peaks for the 202 gauging
stations. The radial position indicates season and the distance from the center shows
magnitude of the flood peak. As stated above most of the big floods were found in the
months between October and February. There were six occasions when more than 600
cumec discharges were observed, one of which occurred in August. However, all these
discharges occurred at a single station, 23002 Feale at Listowel with a drainage area 646
km2.
Apr
800
600
5 8 0
400
200
1 2 0
Jul
-8 00
-3 4 0
120
580
Jan
-3 4 0
-8 00
Oct
Figure 5.2 Flood seasonality showing actual magnitude in 202 Irish stations
4. As drainage area exercises a substantial control over flood peaks, specific AM (flood
magnitude divided by catchment area) values are plotted on Figure 5.3, which is similar
in appearance to Figure 5.2. A number of large specific floods were observed in August
58
and September. In fact among the biggest five, four were from a single station Sandy
Mills (1041) on the river Deele with drainage area 45.3 km2, while the other one was
found in August during Hurricane Charlie in August 1986 at the station Waldron's
Bridge (9010) of the river Dodder. It is also known that very large floods occurred in
other rivers along the east coast as a result of Hurricane Charlie in August 1986, the
river Dargle being a notable one where Q/A may have exceeded 3 cumec/ km2.
However this and others are not included in Figures 13 and 14 as no gauged record
exists.
Apr
3.25
3 .2 5
2
1
0.5
Jul
0
-3 .2 5
0
-3 .2 5
Oct
Figure 5.3 Flood seasonality with specific flood magnitude in 201 Irish stations
59
32
. 5
Jan
5. Finally the months corresponding to occurrence of maximum flow in each AMF data
series are listed in Table 5.2 and are shown graphically in Figure 5.4. Among the 202
stations examined, Figure 5.4 shows that the largest value in the series occurred during
the summer months at 20 of them. Of those 20, 8 occurred in August. The month of
December provided the series maximum in 91 stations (45%).
Month corresponding to Max. flow in an AM series
100
90
No. of occurences
80
70
60
50
40
30
20
10
0
Jan
Feb
Mar
Apr
May
Jun
Jul
Month
Aug
Sep
Figure 5.4 Month corresponding to maximum flow in each of 202 AM series
60
Oct
Nov
Dec
Table 5.1 Station Number with percent of peak floods in the October-March (percentage)
Station
No
%of
events(octmar)
Station
No
%of
events(octmar)
Station
No
1041
91
14006
88
20006
84
26009
89
34018
1055
89
14007
76
22006
86
26010
91
34024
90
3051
100
14009
80
22009
79
26012
98
34029
86
6011
100
14011
81
22035
100
26014
100
35001
90
6013
90
14013
82
23001
87
26017
94
35005
93
6014
97
14018
96
23002
80
26018
96
35011
89
6021
94
14019
92
23012
83
26019
88
35071
90
6025
80
14029
98
24001
88
26020
88
35073
87
6026
91
14033
82
24002
81
26021
90
36010
96
6030
63
14034
82
24004
87
26022
88
36011
96
6031
94
15001
88
24008
87
26058
83
36012
98
6033
80
15003
84
24011
85
26059
96
36015
94
6070
85
15004
92
24012
88
26108
100
36018
92
7002
96
15005
90
24013
87
27001
73
36019
96
7003
83
15012
75
24022
100
27002
96
36021
73
7004
98
16001
88
24030
88
27003
88
36027
93
7005
91
16002
90
24082
82
27070
79
36031
90
7006
84
16003
82
25001
73
28001
82
36071
75
7007
91
16004
90
25002
76
29001
85
38001
73
7009
90
16005
87
25003
78
29004
81
39001
83
7010
93
16006
82
25004
87
29007
82
39008
82
7011
98
16007
84
25005
87
29011
86
39009
88
7012
94
16008
90
25006
92
29071
83
7033
84
16009
90
25011
78
30001
94
7041
100
16011
90
25014
81
30004
97
8002
76
16012
92
25016
93
30005
86
8003
78
16013
88
25017
98
30007
94
8005
67
16051
85
25020
91
30012
100
8007
73
18001
90
25021
91
30021
92
8008
84
18002
94
25023
71
30031
100
8009
67
18003
96
25025
90
30037
57
8011
87
18004
85
25027
88
30061
91
8012
68
18005
84
25029
85
31002
81
9001
77
18006
93
25030
92
31072
69
9002
68
18016
83
25034
83
32011
65
9010
63
18048
83
25038
82
32012
88
9035
44
18050
92
25040
75
33001
68
10002
10021
83
63
19001
19014
92
90
25044
25124
83
94
33070
34001
89
97
10022
60
19016
100
25158
72
34003
97
10028
69
19020
89
26002
91
34004
92
11001
91
19031
89
26005
94
34007
85
12001
12013
90
73
19046
20001
67
93
26006
26007
94
94
34009
34010
91
58
14005
90
20002
94
26008
92
34011
97
61
%of
events(octmar)
Station
No
%of
events(octmar)
Station
No
%of
events(octmar)
100
Table 5.2 Station number with month corresponding to maximum flow in an AM data
series
Station
No.
1041
1055
3051
6011
6013
6014
6021
6025
6026
6030
6031
6033
7002
7003
7004
7005
7006
7007
7009
7010
7011
7012
7033
7041
8002
8003
8005
8007
8008
8009
8011
8012
9001
9002
9010
9035
10002
10021
10022
10028
11001
Month
corresponding
max flow
in a series
12
12
12
12
12
12
12
11
12
11
12
12
11
12
11
12
10
12
12
11
11
12
1
1
11
8
11
8
11
6
12
8
12
12
12
10
11
5
5
10
8
Station
No.
12001
12013
14005
14006
14007
14009
14011
14013
14018
14019
14029
14033
14034
15001
15003
15004
15005
15012
16001
16002
16003
16004
16005
16006
16007
16008
16009
16011
16012
16013
16051
18001
18002
18003
18004
18005
18006
18016
18048
18050
19001
Month
corresponding
max flow
in a series
11
11
12
12
2
2
2
12
2
2
12
12
2
12
8
2
12
2
12
12
12
12
9
12
9
12
12
11
11
11
1
10
10
12
1
12
11
1
12
12
1
Station
No.
19014
19016
19020
19031
19046
20001
20002
20006
22006
22009
22035
23001
23002
23012
24001
24002
24004
24008
24011
24012
24013
24022
24030
24082
25001
25002
25003
25004
25005
25006
25011
25014
25016
25017
25020
25021
25023
25025
25027
25029
25030
62
Month
corresponding
max flow
in a series
12
1
12
2
3
10
12
12
12
12
1
12
8
8
10
2
12
12
12
12
12
12
8
12
12
10
12
12
11
12
12
12
12
12
1
12
11
1
12
2
12
Station
No.
25034
25038
25038
25040
25044
25124
25158
26002
26005
26006
26007
26008
26009
26010
26012
26014
26017
26018
26019
26020
26021
26022
26058
26059
26108
27001
27002
27003
27070
28001
29001
29004
29007
29011
29071
30001
30004
30005
30007
30012
30021
Month
corresponding
max flow
in a series
12
12
2
10
12
2
1
10
12
12
11
11
10
11
2
11
12
12
10
12
12
12
1
1
1
1
12
12
12
12
1
1
1
1
12
12
11
12
2
12
1
Station
No.
30031
30037
30061
31002
31072
32011
32012
33001
33070
34001
34003
34004
34007
34009
34010
34011
34018
34024
34029
35001
35005
35011
35071
35073
36010
36011
36012
36015
36018
36019
36021
36027
36031
36071
38001
39001
39008
39009
Month
corresponding max
flow in a
series
12
11
1
10
7
9
12
9
10
10
10
10
10
11
6
10
12
11
1
10
10
10
1
12
12
12
12
10
12
12
10
1
10
4
9
9
12
12
6 m-year maxima
The theory of extreme values states that if Z is the maximum of a number of other
independent random variables X1, X2, ……XN then the distribution of Z converges
towards one of the 3 types of Extreme Value distribution as N becomes infinitely large.
In practice it is assumed that the limiting case is reached or almost reached for quite
modest values of N, c.f. the analogous case of the Central Limit Theorem, although it is
unlikely that the CL findings would easily transfer to the EV case. In the context of
extreme value theory Gumbel (1940) suggested that annual maximum flows ought to
have an extreme value distribution because each one is the maximum of a sufficiently
large number of individual flows during the year and he used some examples where the
EV1 distribution looked to be a good descriptor of AM flow values. However what
constitutes "large" in the context of N was never really discussed. Some writers felt that
N=365 based on the number of days in the year but this is hardly realistic since
successive daily flows are not independent. In most Irish rivers there are no more that
10 or 12 separate flood peaks in any one year (leaving aside consideration of what
constitutes independent peaks) and this number may not be sufficient to support the
large N assumption.
If N in individual years is insufficient to produce EV behaviour then one would hope
that decade maxima might display such behaviour and analyses of such values might
lead to identification of which type of EV distribution would be appropriate. However
with the length of record available a realistic assessment cannot be made on the basis of
decade maxima. Instead a more modest approach base on m-year maxima where m=1,2
and 3 is explored here in order to see what information might be unearthed.
Probability plots of 1-year , 2-year and 3-year maxima have been prepared thus far for
the 41 No. A1 grade stations. The results obtained are not uniform. In some cases the
three lines are parallel, in some case the three lines diverge at high return periods and in
some cases they converge at high return periods. Examples of each type of case are
shown overleaf inTable 6.1 and Figure 6.1. It is disappointing that no clear single
behaviour exists.
The behaviour is further compared with the behaviours observed in previous plots and
tables. The parallel line case, Station 9001, shows the classical EV1 behaviour with a
maximum score of 5 for linearity in Table 4.3 and a skewness almost identical to the
theoretical EV1 skewness of 1.14. The diverging lines case, Station 36015, scores very
63
low in linearity on any type of plot and is classified a severe concave upwards in Table
4.5, with a large skewness and presence of outliers. In the converging case, Station
6013, while Table 4.3 shows medium scores for linearity Table 4.5 shows convexity and
accompanying low skewness.
The pattern seems to be that the behaviour of m-year maxima is very much predictable
from the annual maximum behaviour. It appears that here is no relief to be gained, from
m-year maxima analysis, from the uncertainty of identifying a single underlying parent
distribution for Irish AM floods.
Table 6.1Examples of m-year maxima
Station
and
AM Prob. Plot Behaviour
Skewness
pattern of
Table 8
Table 10
Tables 3 and
m-year maxima
EV1,LO,LN
EV1,LO, LN
12
9001 parallel
5, 2, 5
L1, U1, L1
1.17
No
36015 diverging
2, 2, 2
U2,U2, U1
2.616
Yes
6013 converging
3, 3, 4
D2, S1, L2
0.10
No, but convex
64
Outliers?
m-year maxima, cumec
Station 9001
100
y = 14.032x + 40.423
80
y = 13.351x + 47.107
60
y = 13.641x + 30.97
40
20
0
-2
-1
0
1
2
3
4
5
4
5
EV1 Reduced Variate
Station 36015
60
50
y = 7.1632x + 22.326
40
30
y = 8.285x + 24.198
20
y = 5.7818x + 19.878
10
0
-2
-1
0
1
2
3
EV1 Reduced Variate
Station 6013
60
50
y = 4.798x + 29.494
40
y = 4.4222x + 31.09
30
y = 5.8484x + 24.516
20
10
0
-2
-1
0
1
2
3
4
5
EV1 Reduced Variate
Figure 6.1 Probability plots of 1, 2 and 3 year maxima showing contrasting behaviours.
65
7 Flood statistics on some rivers with multiple gauges
As part of exploratory data analysis it is interesting to view flow data from several
gauging stations on the same river, where such multiple gauging stations exist. This
gives an opportunity to see how flood magnitude increases in the downstream direction.
The data for catchments (Barrow, Suir, and Suck) are illustrated below in the form of
probability plots and plots of median annual flood vs catchment area. Catchment area
acts as a surrogate for distance along main channel. It can be seen that median annual
flood increases in a regular, mostly linear, manner with area.
250
Probability Plot Summary for stations on Barrow River
200
14005
AMF(m3/s)
150
14006
14018
14019
100
14029
50
2
0
-2
-1
0
5
1
10
2
EV1 y
25
3
50
4
100
500
5
6
7
Qmed vs Catchment Area relationship for stations on Barrow River
180
160
Qmed (m3/s)
140
120
100
80
60
40
20
0
0
500
1000
1500
2000
2500
3000
Catchment Area (km2)
Figure 7.1 (a) Probability plot summary for stations on Barrow River (b) a scatter plot of
Qmed vs Catchment area for these stations. Stations are ordered from upstream.
66
500
Probability plot summary for stations on Suir River
450
2
5
10
25
50
100
500
AMF(m3/s)
400
350
16004
300
16002
250
16008
200
16009
150
16011
100
50
0
-2
-1
0
1
2
3
4
5
6
7
EV1 y
Qmed vs Catchment Area Relationship for stations on Suir River
250
Qmed (m3/s)
200
150
100
50
0
0
500
1000
1500
2000
2500
Figure 7.2 (a) Probability plot summary for stations on Suir River (b) a scatter plot of
67
160
Probability Plot Summary for stations on Suck River
140
AMF(m3/s)
120
26006
100
26002
80
26005
26007
60
40
20
2
5
10
25
50
100
500
0
-2
-1
0
1
2
3
4
5
6
7
EV1 y
Qmed vs Catchment Area relationship for stations on Suck River
100
90
80
Qmed (m3/s)
70
60
50
40
30
20
10
0
0
200
400
600
800
1000
1200
1400
Figure 7.3 (a) Probability plot summary for stations on Suck River (b) a scatter plot of
68
8 Determining the T-year Flood Magnitude QT by the Index
Flood Method
8.1 Introduction
Since the time of Fuller (1914) other developments were made by Foster (1924) and
Hazen (1932). Gumbel (1940) brought the basis of analysis to a new level by applying
to extreme value theory and the findings of Fisher and Tippet (1928) Gumbel
introduced the Extreme Value Type I distribution to flood frequency analysis. This
distribution is often known as the Gumbel distribution but is denoted EV1 in this report.
Powell (1943) introduced extreme value probability paper , which in one form or
another (ruled in terms of probability or reduced variate) is still widely in use today.
Then Langbein (1949) provided the link between the T-year flood estimated by both
annual maximum and partial duration (or peaks over a threshold) models.
The concept of standard error and bias of statistical estimates of QT was introduced by
Kimball (1948) for the EV1 distribution and by Kaczmarek(1957) for EV1 and other
distributions although this information filtered only slowly into hydrological practice.
The first simulation based studies of standard errors were reported by Nash and
Amorocho(1965) and by Lowery and Nash (1969) while by the mid-1970s extensive
simulation studies on bias and standard error were reported by Mathalas and
wallis(197); Mathalas, Wallis and Slack (197) and other analogous studies up to
including that by Lu and Stedinger(1992) were reported in the intervening years.
Since the time of Fuller (1914) flood magnitudes have been estimated by fitting
distributions, either graphically or numerically, to series of annual maximum floods
treated as if they are random samples from known distributions.
The T-year return period flood is the flood magnitude which is exceeded on average
once in every T years and which has a probability of being exceeded in any one year of
p =1/T. Let F(Q) be the distribution function of annual maximum flood magnitudes,
then
69
F(Q) = Non-exceedance probability of magnitude Q in any one year = 1 – 1/T
and
QT = F-1 (1-1/T)
--- (8. 1)
as illustrated in Figure 8. 1
Distribution Function
N on Ex ceedance Probability, F(q)
Probability Density F unction
Probability Density Function
0.4
0.3
area =
1 - 1/T
0.2
area = 1/T
0.1
0
QT
1
1 - F(q) = 1/T
0.8
0.6
F(q) = 1 - 1/T
0.4
0.2
0
QT
Figure 8. 1 probability density function and distribution function
Regional or geographical pooling of dimensionless flow values was introduced by
Dalrymple (1960) so as to provide regional growth factors XT for use at ungauged sites,
in the form
QT = QI XT
where QI
XT
--- (8. 2)
=
Index Flood = Mean or Median annual flood
=
QT / QI = Growth factor.
FSR (1975) introduced the idea that equation (8. 2) also be used for QT estimation at
gauged sites if T > 2N where N = period of observed or gauged record while FEH
(1999), more stringently, recommended that equation (8. 2) be used at gauged sites if T
> N/2. In between publication of FSR and FEH considerable research was carried out
on the efficiency of the index flood approach (Greis & Woods (1981), Hosking, Wallis
& Woods (1985), Lettenmaier, Wallis and Wood (1987), WMO (1989), Hosking and
Wallis (1997)) and the index flood approach is now widely accepted as a valid and well
understood method (both in its implementation and its properties). While Dalrymple
(1960) and FSR (1975) used the Index Flood a method with geographically defined
regions the Region of Influence method of selecting catchments for regionalisation or
pooling developed by Burn (1990) was favoured and found to be more effective by FEH
(1999).
70
The index flood approach involves selection of a suitable index flood such as Q or Qmed
or possibly Q5. FSR choose Q , FEH choose Qmed while FSR Volume II (Rainfall
Studies) choose M5 the 5 year return period rainfall. In the present study Qmed is chosen
in line with FEH.
Use of the index flood approach involves
(i)
estimation of the index flood for the site in question – the subject site.
(ii)
estimation of the growth factor XT or the growth curve, the X-T relation.
These two steps are more or less independent and will be dealt with separately in the
following sections. First some terminology relating to subject, donor and analog sites
is explained.
8.2 Subject, Donor and Analog Sites
Any river location at which flood estimate is required for any purpose or project is
called a subject site or alternatively a project location.
In some cases a subject site
coincides with or nearly coincides with a gauging station location but the majority of
subject sites do not. If not, the subject site is then an ungauged site.
In keeping with the terminology adopted by FEH (1999) a donor site is considered to be
a gauging station on the same river as a subject site and either upstream or downstream
from it.
Likewise in keeping with FEH (1999) an analog site is considered to be a gauging
station which is not on the same river as the subject site but may be on another tributary
within the same parent catchment as the subject site or it may be on a neighbouring
catchment area. It is assumed that an analog site is chosen so that its catchment’s main
characteristics such as area, soil, slope etc. are similar to those of the subject site’s
catchment area. While traditionally an analog site was presumed to be "close" to the
subject site that restriction may be relaxed if an objective measure of similarity of
catchments, such as Euclidean distance measure in catchment descriptor space, e.g.
equation (10. 2) below, is used.
71
9 Determining Qmed
9.1 Subject site is Gauged
At a gauged site Qmed is obtained as the median of the annual maximum flow series
Qmed = Median (Q1, Q2 ….. QN)
The Qi values are ranked from smallest to largest
Q(1) < Q(2) <
……. Q(N)
If N is odd there is a single mid value which can be taken as Qmed. If N is even the two
middle most values are averaged to give Qmed.
The standard error of the median in a random sample from a normal distribution is
se(Qmed) ≈ 1.253 σ /
N.
Since Irish flood data are slightly more skewed than the normal distribution then
se(Qmed) for flood data will be slightly greater, the multiplier 1.25 being increased to
approximately 1.28 to 1.30 when N = 5, a result obtained by simulation. Adopting the
_
larger value of 1.30 and taking average values of Cv and of the ratio Qmed/ Q for Irish
A1 and A2 stations into account an approximate value of the multiplier of Qmed is
obtained as follows.
se(Qmed ) =
0.36
N
Qmed
--- (9. 1)
se(Qmed) ≈ 0.161 Qmed
( = 0.22 Qmed in FEH,3, Table 2.2)
If N = 10, se(Qmed) ≈ 0.114 Qmed
If N = 5,
If N = 50, se(Qmed) ≈ 0.051Qmed ( not included in FEH,3,Table 2.2)
The multipliers given for FEH are slightly larger possibly because of the large Cv and
skewness of UK data.
In FEH it is recommended that if N < 14 then Qmed can be estimated with slightly
smaller standard error from a peak over a threshold series (POT).
72
The reduction in
standard error in the UK case is from 22% Qmed to 18% Qmed when N = 5, with similar
kinds of reduction when N < 5 and with lesser reductions when 5 < N < 14 (FEH, 3,
Table 2.2).
The standard error of the median is 25% large than that of the mean but this is partially
balanced by the fact that Qmed is not affected by low or high outliers in the data sample
and by the fact that its return period of 2 is easier to interpret.
Another method of reducing the standard error of estimate is by “record extension”
whereby a short at-site record at the subject site is “extended” by exploiting the
correlation between the AM series at the subject site and the AM series of one or more
neighbouring sites (Fiering, 1963). This has not been considered further here. However
another form of data transfer is described in the following section.
9.2 Subject size is Ungauged
The standard method is to use an equation, calibrated by multiple regression analysis,
which gives Qmed in terms of catchment descriptors.
Qmed,cds = cAαSsRr…………..
This can be used for a quick preliminary estimate for a pre-feasibility study but such an
estimate should not be relied on its own, because of its low inherent precision, until all
other possibilities have been investigated. Every effort should be made to find suitable
donor or analog sites that may be able to enhance the precision of the Qmed estimate.
This enhancement results from what is called "data transfer" from one or more donor or
analog sites where such sites are available or have been identified. Selection of analog
sites could be made by human judgement as to similarity of catchments or more
formally by using a region of influence method where a Euclidean distance measure in
catchment descriptor space is used to express similarity.
When data transfer is employed either the regression based estimate is adjusted so that it
conforms with reality at a donor station on the same river as the subject site or at an
analog site if there is evidence to support confidence in the use of a particular analog
site.
73
Use of a regression equation on its own could also be considered to be a "data transfer"
method so that these procedures are not entirely independent. Their dependencies will
be considered further below.
The alternative to use of the regression method as the starting point is to calculate the
average specific median annual flood at a group of analog sites, selected by a region of
influence method, and then to transfer this to the subject site and scale it up by A s , the
catchment area at the subject site. This is a less tried and tested method of estimation
although it has been used in the area of low flow estimation.
The following sections give a more formal description of the possible methods as
follows
(i)
Use of Regression model.
a. Regression on its own
b. Regression equation used with donor site or with analog site
(ii)
Region of Influence Method as an alternative to Regression model.
A comparison of the similarities and differences between these approaches can be made
with the help of Figure 9.2 below.
9.2.1 Use of Regression equation
The regression model can be used on its own, section 9.2.1.1, or it can be used in
conjunction with a donor station, section 9.2.1.2 below.
9.2.1.1 Regression equation on its own
In this case a regression based estimate is used namely
Log10 Qmed = c + a log10 AREA+ r log10 SAAR + s log10 S1085 + …..
or Qmed = 10c AREAa SAARr S1085s ………..
where AREA, SAAR, S1085 …. are numerically expressed catchment descriptors and
c, a, r, s, … are parameters of the linear model estimated by regression analysis in log
space.
74
In such equations AREA plays a dominant role and on its own explains
R 2 = 55% - 60% of the variance of log10 Qmed. When the 3 most significant variables
are included about 80% of the variance is explained. Generally it is useful to include
no more than a further 4 variables in which case R2 may increase to c. 90%. The
relationship between R2 and number of variables included is demonstrated in Figure 9.1.
The data from FSR (1975) is largely based on UK data, with some Irish data included.
It is included here to show that the nature of the relationship is almost the same for the
two eras and two countries.
FSR(1975)
NUI Mayn'th(2008)
R Squared
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
Number of Variables
Figure 9.1Increase in R2 value with number of catchment descriptors included in
regression equation in log domain, (i) as in FSR (1975) and (ii) in NUI Maynooth Draft
WP2.3 Report (2008).
As pointed out by Nash & Shaw (1966), the use of such a regression equation is only as
good as having an at-site record length of 1 i.e. Qmed ( Q in their case) estimated from a
single year of gauged record would have the same precision, as expressed by standard
error, as the regression estimate.
This was confirmed in a simulation experiment by
Hebson & Cunnane (1988). However, other authors claim that the regression estimate
is as useful, from a standard error point of view, as up to 5 years of flow data.
9.2.1.2 Regression equation used with donor site
Because of the large standard error associated with the regression method it is
advantageous to use it in conjunction with the donor site approach.
The regression
d
estimate at the subject site can be modified by the ratio of observed Qmed
to the
regression estimate of Qmed at the donor site e.g.
75
 Q s by regression 

Qmed = Qmed  med
 Q d by regression 
 med

s
d
--- (9. 2)
This method of adjustment is recommended in FEH 3, Chapter 4. It was also hinted at
in FSR I, pp 342-3. If the regression model were based on the single dominant variable
Area (A) alone then equation (9. 2) would be reduce to a specific flood form of
adjustment
s
d
Qmed
= Qmed
As
Ad
--- (9. 3)
9.2.1.3 Regression equation used with analog site
In the absence of any suitable donor site and if a suitable analog site can be identified
then the regression estimate can be adjusted by
s
 Qmed
by regression 
s
a
 a

Qmed
= Qmed
 Qmed by regression 
--- (9. 4)
a
Where Qmed
and Qmed by regression are the observed and calculated values of Qmed at
the analog site.
9.2.2 Region of Influence Method
Another method of obtaining Qmed at ungauged site is based on a region of influence
approach in which the similarity of all possible analog sites to the subject site is
expressed by a Euclidean distance measure in catchment descriptor space and the 1, 2,
3, 4, 5, … closest catchments chosen on this basis as is done in regional frequency
analysis.
The average of the specific median flood values is then transferred to the
subject site and scaled up by its catchment area As,
 1 m  Qi
s
Qmed
= As  ∑  med
 m i =1  Ai



--- (9. 5)
where m is the number of sites included.
The relationships between the ungauged catchment estimators can be seen in Figure 9.2
below.
76
Regression Equation on its own
Qmed,cds = cAαSsRr…………..
Use of Regression Equation
Use of Regression Equation
with donor site
Q
s
med
=Q
d
med
s
 Qmed
,cds

 Qd
 med ,cds
with analog site




Q
s
med
=Q
a
med
s
 Qmed
,cds

 Qa
 med ,cds




Special
case
Special
case
Use of 1 variable regression
Use of 1 variable regression
Equation with donor site
Equation with analog site
Q
s
med
=Q
d
med
 As
 d
A
If α = 1
and m = 1
Donor case



α
Q
1 m Qj
s
Qmed
= A s  ∑  medj
 m j =1  A
s
med



=Q
a
med
 As
 d
A



α
If α = 1 and
m=1
Analog case
Transfer of pooled average specific
median annual flood
Figure 9.2 Relationships between different methods of estimating Qmed or of enhancing a
Qmed estimate for an ungauged catchment
77
9.3 Assessment of Qmed estimation using donor or analog sites
A total of 184 hydrometric gauging stations were identified as being potentially suitable
candidates for this assessment.
It was decided to limit the assessment to subject sites on rivers that have at least three
gauging stations on the same river and to stations having at least 20 years of data so that
se (Qmed) < 10% Qmed approximately. Therefore on each river so selected there is a
choice of 3 subject sites, each of which have 2 possible donor sites. There were 38 such
gauging stations available located in 10 different catchments. These stations are
displayed in Figure 9.3, and the stations are identified further in Table 9.1and Table 9.2.
Table 9.1 List of Rivers that have 3 or more gauging stations on them:
River Name
Basin Name
Hydrometric Code
No. of Gauging Stations
Suir
Suir
16
5
Barrow
Barrow
14
6
Blackwater
Blackwater
18
5
Erne
Erne
36
3
Suck
Upper Shannon
26
4
Inny
Upper Shannon
26
3
Brosna
Lower Shannon
25
3
Deel
Shannon Estuary
24
3
Maigue
Shannon Estuary
24
3
Glyde
Glyde
6
3
78
Table 9.2 38 Gauging Stations used in the Qmed Assessment
Station
No
6014
6021
6026
14005
14006
14018
14019
14029
14034
16002
16004
16008
16009
16011
18003
18006
18016
18048
18050
24004
24008
24011
24012
24030
24082
25006
25011
25124
26002
26005
26006
26007
26021
26058
26059
36011
36012
36019
Station Name
Tallanstown Weir
Mansfieldstown
Aclint
Portarlington
Pass Bridge
Royal Oak
Levitstown
Graiguenamanagh
Bestfield
Beakstown
Thurles
New Bridge
Caher Park
Clonmel
Killavullen
Cset mallow
Duncannon
Dromcummer
Duarrigle
Bruree
Castleroberts
Deel Bridge
Grange Bridge
Danganbeg
Islandmore Weir
Ferbane
Moystown
Ballynagore
Rookwood
Derrycahill
Willsbrook
Bellagill
Ballymahon
Ballinrink br.
Finnea br.
Bellahillan
Sallaghan
Belturbet
River_Name
Glyde
Glyde
Lagan (Glyde)
Barrow
Barrow
Barrow
Barrow
Barrow
Barrow
Suir
Suir
Suir
Suir
Suir
Blackwater
Blackwater
Blackwater
Blackwater
Blackwater
Maigue
Maigue
Deel
Deel
Deel
Maigue
Brosna
Brosna
Brosna
Suck
Suck
Suck
Suck
Inny
Inny upper
Inny
Erne
Erne
Erne
79
Area, km2
270
321
144
398
1096
2415
1660
2762
2060
512
236
1120
1602
2173
1258
1058
113
881
245
246
805
273
359
248
764
1207
1227
254
626
1050
182
1184
1071
59
249
318
263
1501
Record
length
30
50
46
48
51
51
51
47
17
51
48
51
52
52
49
27
24
23
24
52
30
33
41
25
28
52
51
18
53
51
53
53
30
24
23
49
47
47
Obs Qmed
21.46
21.50
12.30
38.27
80.52
147.98
102.41
160.74
117.00
52.66
21.37
92.32
162.21
223.00
266.15
286.00
79.65
220.00
124.50
50.63
119.13
104.55
109.99
52.00
140.01
81.91
82.02
13.65
56.56
93.21
24.23
88.15
66.34
5.35
12.20
18.23
14.12
89.95
´
36019
!
(
36011
06026
!
(36012
!
(
26059
!
( (
!26058
26006
!
(
26002
!
(!
(!
(06021
06014
26021
!
(
!
(
26005
25124
!
( 26007
!
(
!
(
25011
(25006
!
(!
1400514006
!
(!
(
14019
!
( 14034
!
(
16004
16002
24008
24082
!
(!
24012
(
24030 24004
!
(
!
(24011
!
(
!
(
(
!
(!
16008
14018
!
(
14029
!
(
!
(
1600916011
!
( !
(
18006
1801618050
!
(18003
(18048
!
( !
(
!
(!
Legend
!
(
subjectsite
areas
Figure 9.3 Location of 38 potential subject sites
80
9.3.1 Tests on Qmed estimates via donor and analog catchments
The approach used here to estimate Qmed at ungauged sites consists of three steps:
•
selection of appropriate donors and analogues for each subject site,
•
determination of Qmed and catchment descriptors at the donor or analog site
•
calculation of adjusted Qmed for the subject site using data of each donor and
analog site. Simultaneous use of multiple donor sites has not been considered.
The donor site selection technique uses the nearest gauge on the same stream as the
subject site. The 2nd nearest, 3rd nearest etc. in the upstream and downstream directions
are also examined where such sites exist. The donor site selection methods examined
are the
•
up-stream site(s),
•
down-stream site(s) and
Analog sites, analogs are selected according to the similarity of physiographic
catchment characteristics. For this component of the study Catchment area, SAAR and
BFI have been used to find the similarity between subject sites and possible analog
sites. For the Area based adjustment catchment area alone is used for identifying an
analog station.
For each subject site and donor/analog site combination two methods for determining
Qmed are examined here.
The first method, adopted from FEH, adjusts the Qmed at the donor / analog site by the
ratio of the regression estimates of Qmed at the subject site and donor/analog sites that
is use of equation (9. 2).
In order to apply this method a regression model has been developed between log Qmed
and log catchment descriptor values, independently of any work being carried out in
Project WP2.3, as WP2.3 Work was being carried out in parallel when this work was
being done.
81
Data from 164 sites are used for the calibration of the regression model. Three
catchment descriptors namely Catchment Area, SAAR and BFI were used in this model
to find the regression equation for use in this part of the study and these descriptors are
also used in the pooling analysis to pool the data when selecting catchments for the
analog with regression based adjustment.
The resulting regression model is found to be:
Q med = 0.000302 × AREA 0.829 × SAAR 0.898 × BFI -1.539
--- (9. 6)
which explains 82% of the total variance in lnQ med with RMSE = 0.4725 in the ln
domain and RMSE = 0.205 in the log10 domain and hence FSE = 10
0.205
= 1.60. (In
WP2.3 the same 3 variables are reported to account for 83% of the variance), the
difference being due to the different basic data set used in each case.
Figure 9.4 shows the scatter plot between predicted Qmed using this regression equation
vs observed Qmed for the 35 stations used in the current tests (the other 3 stations are
not used due to unavailability of BFI catchment descriptors) with a coefficient of
determination of 80% (which, as expected is similar to the value of 82% found when
the equation was derived).
The second method adjusts the Qmed obtained at the donor/analog site by scaling it by
the ratio of catchment areas of the subject site and the donor/analogue site that is use of
equation (9. 3).
The second method is a subset of first method and can be expected to be less effective.
82
300
y = 0.8169x
Regression
R 2 = 0.8033
250
Qmed_pred
200
150
100
50
0
0
50
100
150
200
250
300
350
400
Qm e d_obs
Figure 9.4 predicted Qmed by regression vs observed Qmed . Each point represents a
catchment , 35 in total as all catchment descriptors were unavailable for 3 of the 38 sites
9.3.2 Results
The results are considered in two subsections, one deal with details at individual subject
sites and one which amalgamates the results and examines them collectively.
9.3.2.1 Results at individual subject sites
The results are provided in
Appendix
2
and the details for Site 14006, River Barrow at Pass Bridge are
demonstrated here. This station has one donor station located upstream of it, namely
14005 River Barrow at Portarlington, and 4 potential donor stations downstream of it,
namely 14019 (Levitstown), 14018 (Royal Oak), 14034 (Bestfield) and 14029
(Graiguenamanagh) in order of increasing catchment area. Station 14034 is not used in
this case – as all of the catchment descriptors for it are not available.
Figure 9.5 and Figure 9.6 show in tabular and graphical form the estimates of Qmed at
station 14006 for both area based adjustment and regression based adjustment, where
Donor_up means station 14005, Donor_down_1 means Station 14019, Donor_down_2
83
means Station 14018 and so on. The estimated Qmed at 14006 for each case is shown
in the 2nd last row of the each table and this value expressed as a percentage of observed
Qmed at 14006 is shown in the last row.
These latter percentages are shown in
graphical form also where the red line represents 100%.
Summaries of these
percentages for the 4 donor catchment based estimates are shown in Table 9.3. These
show that the regression based adjustment provides more reliable estimates than the area
based adjustment although the average performance is comparable in magnitude
between the two, 21% and 18%. Both donor methods are better, in this case, than the
estimate obtained from the catchment descriptor based regression equation alone (9. 6).
Table 9.3 Estimates from donor catchments – summary of minimum, maximum and
average percentage difference from observed Qmed.
Minimum difference
Maximum difference
Average abs. difference
Difference using catchment descriptors
Regression only
Regression based
adjust'nt
3%
23%
18%
+31% (Absolute
value)
Area based adjustment
-16%
31%
21%
+31% (Algebraic
value)
Use of analog catchments is much more subjective and the choice of catchment greatly
affects the performance measure of the Qmed estimate. For the Area based adjustment
analog sites have been chosen by a Euclidean distance which is based on Area alone. On
the other hand for the Regression based adjustment analog sites have been chosen by a
Euclidean distance which is based on Area, SAAR and BFI.
84
St.No.
Obs Qmed
Area,km2
Adj.Qmed
%obs Qmed
Subject_site
14006
80.5
1096
80.5
100
Donor_up
14005
38.3
398
62.4
78
Minimum difference via donors
Maximum difference via donors
Average abs. difference via donors
Difference from regression estimate
Regression Adjustment
Donor_down1 Donor_down2
14019
14018
102.4
148
1660
2415
82.7
98.7
103
123
Donor_down3
14029
160.7
2762
98.7
123
Analog
24008
119.1
806.04
130.3
162
+3%
+23%
18%
+31%
14006 Barrow @ Pa ss Bridge, Area =1096 km2
(Regression Adjustment)
180
806.04
160
140
2415
%obs Qmed
120
2762
1660
100
80
398
60
40
20
0
Donor_up
Donor_down1 Donor_down2 Donor_down3
Analog
Regres s ion
Data Trans fer Category
Barrow at Pass Bridge, using regression based adjustment. The figures above the boxes
are catchment areas. The final figure labelled Regression refers to estimate obtained by eq
(10).
85
Regression
105.2
131
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs Qmed
Subject_site
14006
80.5
1096
80.5
100
Donor_up
14005
38.3
398
105.4
131
Average abs. difference via donors
Area Adjustment
14019
14018
102.4
148
1660
2415
67.6
67.2
84
83
Donor_down3
14029
160.7
2762
63.8
79
Analog
30012
126
1075.4
128.4
159
-16%
+31%
21%
+31%
14006 Barrow @ Pass Bridge, Area=1096 km2
(Area Adjustment)
180
1075.4 km2
160
140
398 km2
%obs Qmed
120
100
1660 km2
2415 km2
2762 km2
80
60
40
20
0
Donor_up
Analog
Regression
Data Transfer Category
Barrow at Pass Bridge, using area based adjustment.
9.3.2.2 Results from all donor sites collectively
Results are assessed in 2 ways, one being as in Table 9.3 above, see Table 9.4 below,
and one in which Qmed estimated by each method is plotted against observed Qmed,
see Figure 9.7 and Figure 9.8 below.
86
Regression
105.2
131
Table 9.4 Qmed estimates from donor catchments – minimum, maximum and average %
difference form observed Qmed
Subject
Site
ST14005
ST14006
ST14019
ST14034
ST14018
ST14029
ST18016
ST18050
ST18048
ST18006
ST18002
ST25124
ST25006
ST25011
ST24030
ST24011
ST24012
ST36012
ST36011
ST36019
ST6026
ST6014
ST6021
ST26058
ST26059
ST26021
ST24004
ST24008
ST24082
ST26006
ST26002
ST26005
ST26007
ST16004
ST16002
ST16008
ST16009
ST16011
Average
St dev'n
Min diff%
Reg
Area
29
-24
3
-16
-3
-1
2
0
1
0
-2
2
-28
0
38
20
8
-4
-8
0
69
10
24
-9
-1
-10
2
12
46
-19
-20
-11
25
7
39
5
-28
-4
2
-7
0
7
19
-12
-32
14
26
-21
-21
5
-11
-5
24
-14
12
-1
-32
-3
-2
1
2
5
21
26
-9
1
0
-21
10
-1
1
-11
0
0
14
3
21
Max diff%
Reg
Area
58
-41
23
31
-25
56
69
-37
57
-37
65
-17
-79
-19
-71
29
182
-22
161
-19
378
23
26
12
-21
-19
-20
38
83
-28
-45
23
-32
12
39
-6
-28
-10
2
-22
0
-16
27
27
-46
45
85
-31
46
23
-28
17
39
-19
-19
9
-44
5
47
11
50
10
79
74
14
-35
-20
38
25
-35
-19
-42
-20
3
31
26
82
Avr. abs. diff%
Reg
Area
44
35
18
21
16
18
24
18
18
18
21
7
58
7
52
24
84
10
76
7
194
17
25
11
11
15
11
25
64
23
33
17
28
9
39
5
28
7
2
14
0
12
23
20
39
29
56
26
34
14
20
11
31
17
16
5
37
4
22
6
23
8
40
52
12
17
8
26
20
17
8
23
8
18
12
32
33
Diff. from
Regression
Actual
Abs
+69
69
+31
31
+27
27
N/A
+7
7
+7
7
-32
32
-33
33
-18
18
-36
36
-33
33
+13
13
+3
3
-8
8
-18
18
-41
41
-27
27
N/A
+41
41
+1
1
+40
40
+38
38
N/A
+22
22
+39
39
-4
4
-15
15
-19
19
-31
31
+35
35
+30
30
+37
37
+24
24
+41
41
-8
8
+12
12
-9
9
-19
19
5
29
25
15
The values in Table 9.4 show that the regression based adjustment is more reliable.
This is to be expected since the 3 variable regression equation accounts for 82% of the
87
variance of lnQmed while area accounts for only 55%-57% of that variance. While the
average minimum differences of 3 and 0 are quite similar the standard deviations of 21
and 14 respectively show that the regression based method is more reliable in terms of
the minimum differences.
In terms of maximum differences the regression based
method is clearly superior both in average value and standard deviation and the same
applies to average absolute differences. These performances can be judged relative to
the result of using an estimate based on regression on catchment descriptors alone,
equation (9. 6), where the mean absolute difference is 25(15) in comparison to 32(33)
and 18(11) for area based and regression based adjustment respectively, where the
figures in brackets are the relevant standard deviations.
This one measure alone confirms that the catchment descriptor based regression formula
estimate is better than the area based adjustment donor estimate. This happens probably
because the variance in Qmed is better explained by the 3 catchment descriptors
(Catchment Area, SAAR and BFI) rather than the by Catchment Area alone, see Figure
9.1. Area based adjustment can be considered as a special case of regression based
adjustment (see Figure 9.2) where only Catchment area has been used in the regression
equation and hence would be expected to be less effective than the regression on
catchment descriptors.
Qmed values obtained by each method have also been compared to the observed Qmed
values in Figure 9.7 which shows the scatter plot between predicted Qmed vs observed
Qmed when the regression based method has been used for the adjustment of Qmed at
the subject site. Each point represents a catchment and the 3 plots in Figure 9.7 relate to
different data transfer methods. The downstream donor selection method performs best
with a very good coefficient of determination of 95% followed by upstream donor
selection method (91%) and analogue selection method (79%).
Figure 9.8 shows the corresponding results as Figure 9.7 for the case where the Area
based method is used in the data transfer procedure. These results show that the
downstream donor selection method performs best with the coefficient of determination
of 78% while in case of analogue selection method it performs worst with a low
coefficient of determination of 3%. The other method, upstream donor selection also
88
performed well with coefficients of determination of 77%. It is quite noticeable that
with this adjustment technique the coefficient of determination for analogue selection
method is smaller compared to the case of regression based transfer.
It should be noted that fewer data points i.e. subject sites are used in Figure 9.7 than in
Figure 9.8 due to unavailability of catchment descriptor values for some subject sites at
the time that this part of the study was conducted. If the same donor sites were used in
Figure 9.8 as in Figure 9.7 the corresponding coefficients of determination would then
be 76%, 77% and 3% for upstream selection, downstream selection and analogue
selection method respectively.
89
700
Regression based adjustment
Upstream site
600
y = 0.9551x
R2 = 0.9169
Qmed_pred
500
400
300
200
100
0
0
100
200
300
400
Qmed_obs
500
600
700
500
600
700
600
700
700
Downstream site
600
500
Qmed_pred
y = 1.0608x
R2 = 0.9515
400
300
200
100
0
0
100
200
300
400
Qmed_obs
700
Analog site
600
y = 0.957x
R2 = 0.7908
Qmed_pred
500
400
300
200
100
0
0
100
200
300
400
Qmed_obs
500
Figure 9.7 Adjusted Qmed (Regression method) vs observed Qmed. Each point represents
a catchment and panels relate to different transfer method
90
700
Area based adjustment
Upstream site
600
500
Qmed_pred
y = 1.2819x
R2 = 0.7719
400
300
200
100
0
0
100
200
300
400
Qmed_obs
500
600
700
500
600
700
500
600
700
700
Downstream site
600
500
Qmed_pred
y = 0.8568x
R2 = 0.7816
400
300
200
100
0
0
100
200
300
400
Qmed_obs
700
Analog site
600
500
Qmed_pred
y = 0.8098x
R2 = 0.0291
400
300
200
100
0
0
100
200
300
400
Qmed_obs
Figure 9.8 Adjusted Qmed (Area method) vs observed Qmed (m3/s). Each point represents
a catchment and each plot relate to different area based data transfer methods
91
9.3.3 Conclusions:
1. The comparisons show that the downstream selection method performs best in
both adjustment techniques.
2. Upstream selection method also performs well and is better than analogue site
selection method.
3. Table 9.4 shows a substantial improvement in using regression based adjustment
over using area based adjustment in terms of average minimum difference
between observed Qmed and adjusted Qmed. Likewise in terms of average
maximum difference and in terms of average of the average difference measured
at the 38 sites. These differences are also apparent in Figures 8 and 9. In the
case of analogue sites the regression adjustment transfer method performs much
better than the Area adjustment method. This happens probably because the
variance in Qmed is better explained by the 3 catchment descriptors (Catchment
Area, SAAR and BFI) rather than the Catchment Area alone, see Figure 9.1.
Area based adjustment can be considered as a special case of regression based
adjustment (see Figure 9.2 and equation (9. 2)) where only catchment area has
been used with exponent α =1 in the regression equation and hence would be
expected to be no more effective than the regression on catchment descriptors
alone.
9.4 Summary of Qmed estimation and associated error
•
At site data available:
•
Qmed can be estimated from the at-site data and the corresponding
standard error is approximately 0.363 Qmed/√N.
•
At site data not available (ungauged catchment case):
•
Qmed estimated from catchment descriptor based formula.
i. If this estimate is as good as only one year of data, as described
by Nash & Shaw (1965), then se(Qmed) = σQ, the standard
deviation of the flood population ≈ 0.28µ ≈ 0.29 Qmed.
92
ii. If this estimate is as good as, say, 4 year's data then
se(Qmed) = σQ/√4 = 0.5 σQ = 0.5 x 0.28µ = 0.14 µ ≈ 0.145
Qmed.
iii. The factorial standard error, FSE, of such estimates varies from
1.47 (in FSR) to 1.55 (in FEH) with WP2.3 giving 1.36. These
give lower 67% confidence limits of between 65% Qmed and
70%Qmed and upper 67% confidence limits of between 1.44
Qmed and 1.55 Qmed.
(i) and (iii) should be compatible since they are derived from the same basic
information, namely the rmse (see) of the regression residuals. If se(Qmed) = 0.29Qmed
then Qmed – 1 se(Qmed) = 0.71Qmed which could taken as equalling the 70% Qmed in
(iii).
•
If a donor catchment is used the analysis described above gives some
indication of the average estimation error as follows from Table 9.4:
i. Using regression based data transfer: Abs(Av. abs diff) = 18%
ii. Using area based data transfer:
Abs(Av. abs diff) = 32%
iii. The corresponding values using regression formula alone = 25%
iv. The corresponding values for transfer from a single analog
catchment are 30 % and 69% respectively using regression based
and area based data transfer.
Note that these are average values of differences from a finite number of cases and are
not directly comparable to standard error values which are expected root mean square
values rather than average values.
93
10 Determining the Growth Factor XT
10.1 Introduction
The growth factor XT is the factor which when multiplied by the index flood QI gives
the flood magnitude of return period T, QT, as in equation (8. 2) : QT = QI. XT
The relationship between XT and T is often referred to as the growth curve.
In this
study Qmed is used as the index flood.
In this Section two methods of determining the growth factor XT and the resulting QT
are outlined. The effects of catchment type (Area, Storage amount and Peat content),
Geographical Location and Period of Record on XT and on QT are examined. Also the
effectiveness of different combinations of catchment descriptors in defining the distance
measure on which region of influence pooling groups are formed is examined.
10.2 At-site and Pooling Group Estimates of XT
If a subject site coincides with a gauging station and if a sufficiently long record of AM
floods exists at the station then a suitable distribution can be fitted to the data of that site
for direct estimation of QT.
Here the word “direct” means that QT can be obtained
without going through the intermediate steps of determining Qmed and XT.
What constitutes a “sufficiently long record” is arguable. In FSR (1975) the criterion N
> 0.5T was adopted, where N is available record length and T is required return period
whereas FEH (1999) adopted the much more stringent criterion of N > 2T.
If the
record is not sufficiently long the standard error of estimate of QT, se (QT) is large.
Adopting a pooling group estimate of XT, and hence of QT, reduces the value of se(QT).
Obviously the choice of using a single site estimate or pooling group estimate involves a
trade-off between the convenience of a single site estimate and the extra work involved
in assembling and analysing data for a pooling group estimate as well as adoption of the
assumption of pooling group homogeneity.
Another question arises as to what constitutes a suitable distribution for at-site
estimation. Generally it is better to adopt a 2-parameter distribution than a 3-parameter
94
distribution because se(QT) is much larger in the latter case even though bias may be
reduced.
Some guidance on choice of 2-parameter distributions for Irish data is
available from Chapter 2,
where EVI and lognormal distributions are seen overall to
be better descriptors of gauged AM data than the 2 parameter logistic distribution.
However there are many samples of data which seem not to be able to be described by
any 2 parameter distribution either through extreme concave or convex upward
curvature or an S-shape on a probability plot. In the case of these latter samples perhaps
this is a genuine permanent departure from 2 parameter behaviour or it may merely be a
period of record effect i.e. the next N years may not exhibit this behaviour.
One approach might be to avoid using an at-site estimate at a station where the sample
shows a strong departure from 2 parameter behaviour and to adopt a pooling group
estimate instead, on the basis that this reflects an average type of Q-T behaviour for the
type of catchment in question.
Note also that probability plots of random samples
drawn from a 2-parameter distribution can display non-linear behaviour in a small
proportion of cases (see Figure 4.1).
10.3 Pooling Groups
While XT can be estimated at a gauged site from the at-site record by conventional flood
frequency analysis methods it is generally accepted that an advantage can be gained, in
terms of reduction of standard error of estimate of QT, if the assumption is made that XT
does not vary between gauging stations that belong to a homogeneous pooling group.
A pooling group of gauging stations is said to be homogeneous if the same growth
curve applies to all stations in the group (or even more strictly if the same value of XT,
for stated T, can be applied to all stations in the group).
With this assumption a space
– time trade-off is achieved e.g. n years of record at each of m stations is assumed to be
equivalent to nm years of record at a single site. Hence if XT is estimated from the nm
values (after they have been suitably standardised e.g. by division by their respective
Qmed values) it will have smaller standard error of estimate than an XT value obtained
from an at-site estimate, the reduction being of the order of
95
m.
Pooling groups can be formed by using geographical regions but FEH (1999, III, Fig.
16.5) found that such pooling groups were less homogeneous than those formed by a
region of influence approach of the type proposed by Burn (1990).
The region of
influence approach selects stations, which are nearest to the subject site in catchment
descriptor space, to form the pooling group for that subject site. For instance in FEH
(1999) the distance between subject site i and any other site j is taken to be
2
dij =
2
2
1  ln AREAi − ln AREAj   ln SAARi − ln SAARj   BFIHOST
i − BFIHOST
j 
 +
 +
 --- (10. 1)







2
σ ln AREA
σ ln SAAR
σ BFIHOST
 
 

In choosing a pooling group and distance measure dij a decision has to be made about
how many and which catchment descriptors are to be included in the distance measure
(3 in the above case) and what weightings are to be applied to them and whether
logarithms or other transformations are to be used. In addition a decision has to be
taken about how many stations are to be included in the pooling group.
FEH (1999)
investigated a range of pooling group sizes and decided on adoption of the 5T rule,
namely that the total number of station years of data to be included when estimating the
T year flood should be at least 5T. Adoption of such a rule is a compromise. If too
few stations are included precision of the QT estimate is sacrificed whereas if far too
many stations are included the assumption of homogeneity may be compromised.
Hosking & Wallis (1997) however show that a small departure from homogeneity can
be tolerated so that having too few stations included may be less desirable than having
slightly too many.
In this study the distance measure
dij =

σ ln AREA

has been adopted.
2
  ln SAARi − ln SAAR j
 + 
σ ln SAAR
 
2
  ln BFI i − ln BFI j
 + 
σ ln BFI
 
2

 --- (10. 2)

Choice of AREA and SAAR follows from the FEH case, combined
with the knowledge that both Area and SAAR play a major role in regression studies of
relations between Qmed and catchment descriptors.
Measured BFI is used as the third
variable because it has been found to be useful in regression studies of relations between
Qmed and catchment descriptors and also between some hydrograph shape parameters
and catchment descriptors.
96
In this study also the 5T rule, for determining which stations to include in the XT
estimation, has been used. It is recognised that in some practical situations the 5T rule
cannot always be implemented.
10.4 Growth curve estimation
Growth curves are expressed as algebraic equations for XT such as equations (10. 5) and
(10. 8) below. Suppose there are m gauging stations in a particular group.
Let Qi , i =
1, 2 … ni be the recorded annual maximum floods at the ith of m sites.
Probability weighted moments M1jo are calculated at each site and the dimensionless Lmoment ratios L-CV = t2 and L-Skewness = t3 are calculated for each site.
The regional weighted average value of each of these is calculated as
m
t2 =
∑w t
(i )
i 2
i =1
m
t3 =
∑
i =1
m
/ ∑ wi
--- (10. 3)
i =1
wi t 3(i ) /
m
∑ wi
--- (10. 4)
i =1
where t 2(i ) and t 3(i ) are the values of t2 and t3 at the ith site and wi is a weighting factor.
If equal weights are given to all stations then wi = 1 for all i and Σ wi = m. If t values
are weighted according to length of record then wi = ni and Σ wi = total number of
station years of data in the pooling group.
Note that the dimensionless 1st L-Moment is unity.
A distribution's parameters can be estimated from the dimensionless L –Moments,
from 1 and L-Cv
in the case of a 2 parameter distribution, such as
EV1 or LO
97
from 1, L – Cv and
L-skewness
GEV or GLO
from 1, L-CV, L-Skewness
the
& L-Kurtosis
kappa.
In pooled regional flood frequency estimation the values t 2 , t 3 , and t 4 if required,
are equated to expressions for these quantities written in terms of the distribution's
unknown parameters (when the distributions growth curve is expressed in dimensionless
form) and the resulting equations are solved for the unknown parameters.
A three parameter distribution can be used for a pooling group because the extra data
involved in the estimation ensures that the resulting standard error is smaller than in the
at-site or single sample case.
Further the three parameter distribution avoids any
possible bias resulting from using a fixed shape 2 parameter distribution when in fact a
3 parameter distribution is more appropriate --- see (d) in Figure 10.1.
In this study the GEV and its special case EV1 and GLO and its special case LO forms
of growth curve are used.
The dimensionless GEV growth curve is defined by a shape factor k and a scale
parameter β as follows:
xT = 1 +
β
k
T k

k
) ,
 (ln 2) − (ln
T −1 

k ≠0
--- (10. 5)
The parameters k and β are estimated from sample t2 = L-CV and sample t3 = Lskewness, as follows Hosking & Wallis (1997, p 196)
k = 7.8590c + 2.9554c2 where c =
β =
2
ln 2
3 + t 3 ln 3
kt 2
t 2 (Γ(1 + k ) − (ln 2) k )+ Γ(1 + k )(1 − 2 − k )
98
--- (10. 6)
--- (10. 7)
The EV1 growth curve is defined as:
xT = 1 + β (ln(ln 2) − ln(− ln(1 − 1 / T )) )
β=
--- (10. 8)
t2
ln 2 − t 2 [γ + ln(ln 2)]
--- (10. 9)
where, t 2 is the L-coefficient of variation (L-CV) and γ is Euler’s constant = 0.5772.
The dimensionless GLO growth curve is defined by a shape factor k and a scale
parameter β as follows:
xT = 1 +
β
k
(1 − {(1 − F ) / F } )
k
= 1+
β
k
(1 − {T − 1} ),
−k
k ≠ 0 --- (10. 10)
The parameters k and β are estimated from sample t2 = L-CV and sample t3 = Lskewness, as follows Hosking & Wallis (1997, p 197)
k = - t3 and β =
kt 2 sin(πk )
kπ (k + t 2 ) − t 2 sin(πk )
--- (10. 11)
The LO growth curve is defined as:
xT = 1 + β ln(T − 1)
--- (10. 12)
β = t2
--- (10. 13)
where, t 2 is the L-coefficient of variation (L-CV)
99
FIXED SKEWNESS MODELS (usually with two parameters)
(a) Model Skewness < Parent
Skewness Hence negative bias i.e.
underestimation at large T
(b) Model Skewness > Parent Skewness
Hence positive bias i.e. overestimation at large
T
1
1
0
T
1
0
T
1
VARIABLE SKEWNESS MODELS (usually with > two parameters)
(c) At-site use
(d) Pooled XT estimation + At-site Qmed
Small bias
and small se
Small bias,
large se
1
1
0
T
0
T
1
1
T
10
100
10
100
Bias/QT%
[ Bias ± 1.96 se ]/QT %
Figure 10.1 Simplified qualitative outline of simulation results about flood quantile
estimates. (a) and (b) show the effect of choice of wrong distribution having fixed but
incorrect skewness. (c) and (d) show the effect of using a flexible distribution i.e. low bias
and large standard error when used in at-site mode but with much reduced se as well as
low bias when us in at-site +pooling group mode. (Adapted from Figure 5.1 in WMO
,1989).
100
T
11 Effect of catchment type and period of record on XT and
QT
11.1 Data and Data Screening
Data of 88 A1 and A2 category stations which have BFI values available were used in
this part of the study.
Very unusual or discordant data sets have been excluded using the Hosking Wallis
discordancy measure:
Di =
1
T
N (u i − u ) A −1 (u i − u )
3
--- (11. 1)
where u i is a vector containing the L-moment ratios for station i, namely the L-Cv (t),
L-Skewness (t2), and L-Kurtosis (t3), u is the unweighted regional average and A is
the matrix of sums of square and cross products.
The D values are shown in Table 11.1 where 3 values are seen to be greater than the
discordant cut-off value of 3. The data of these stations have been examined to and
possible reasons for their discordancy are listed in Table 11.2. The EV1 probability
plots of these data are shown in Figure 11.1. These stations were excluded from further
analysis for this part of the study even though it can be argued that this reduces the
natural variability within the data set being studied.
101
Table 11.1 Stations with site characteristics, L-moment ratios and discordancy measure
StationNo
6011
6013
6014
6026
6031
6070
7006
7009
7033
8002
8005
9001
9002
9010
10021
10022
12001
14005
14006
14007
14009
14011
14018
14019
14029
15001
15003
16001
16002
16003
16004
16005
16008
16009
16011
18004
18005
19001
19020
23001
23012
24008
24022
24082
25006
25014
25016
25023
RL
48
30
30
46
18
24
19
29
25
20
18
48
24
19
24
18
50
48
51
25
25
26
51
51
47
42
50
33
51
51
48
30
51
52
52
46
50
48
28
45
18
30
20
28
52
54
42
52
AREA
229.19
309.15
270.38
148.48
46.17
162.02
177.45
1658.19
124.94
33.43
9.17
209.63
34.95
94.26
32.51
12.94
1030.75
405.48
1063.59
118.59
68.35
162.30
2419.40
1697.28
2778.15
444.35
299.17
135.06
485.70
243.20
228.74
84.00
1090.25
1582.69
2143.67
310.30
378.47
103.28
73.95
191.74
61.63
806.04
41.21
762.84
1162.76
164.42
275.17
113.86
SAAR
1028.98
873.08
927.45
940.87
930.66
1046.28
936.67
868.55
1032.22
791.12
710.76
783.26
754.75
955.04
799.07
821.92
1167.31
1014.90
899.07
814.07
831.24
806.97
857.46
861.46
876.50
935.24
933.86
916.42
932.15
1192.01
941.36
1153.57
1029.63
1078.57
1124.95
985.41
1190.37
1175.68
1179.07
1084.01
1264.00
939.47
942.31
941.70
931.99
1007.65
947.09
922.49
BFI
0.71
0.62
0.63
0.66
0.56
0.73
0.55
0.71
0.44
0.60
0.52
0.51
0.67
0.56
0.65
0.60
0.72
0.50
0.57
0.64
0.67
0.60
0.67
0.62
0.69
0.51
0.38
0.61
0.63
0.55
0.58
0.56
0.64
0.63
0.67
0.68
0.71
0.64
0.66
0.32
0.46
0.54
0.53
0.52
0.71
0.67
0.61
0.65
FARL
0.87
0.97
0.93
0.92
1.00
0.83
0.99
0.99
0.89
1.00
1.00
1.00
1.00
0.96
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.96
1.00
1.00
1.00
102
L-Cv
0.113
0.157
0.149
0.177
0.258
0.145
0.118
0.211
0.129
0.112
0.378
0.243
0.417
0.423
0.213
0.233
0.194
0.149
0.106
0.170
0.125
0.143
0.138
0.136
0.078
0.159
0.111
0.124
0.156
0.099
0.111
0.098
0.072
0.099
0.173
0.087
0.142
0.095
0.196
0.180
0.146
0.151
0.233
0.152
0.137
0.128
0.125
0.163
L-Skew
0.090
0.019
0.224
0.240
0.392
0.135
-0.205
0.206
0.162
0.271
0.228
0.191
0.390
0.416
0.186
0.057
0.176
0.291
0.238
0.219
0.136
0.020
0.036
0.092
0.058
0.003
-0.146
0.012
0.172
0.207
0.069
0.204
-0.062
-0.096
0.089
0.043
0.224
0.095
0.029
0.128
0.386
0.055
0.123
-0.036
0.140
0.103
0.082
0.142
L-Kur
0.074
0.012
0.121
0.092
0.317
0.143
0.074
0.106
0.272
0.175
0.189
0.146
0.253
0.305
0.068
0.061
0.171
0.223
0.212
0.072
0.242
0.157
0.064
0.126
0.067
0.075
0.087
0.125
0.191
0.054
0.108
0.133
0.055
0.052
0.053
0.333
0.182
0.119
0.037
0.181
0.326
0.092
0.164
0.130
0.183
0.125
0.155
0.062
Discordancy
0.438
0.971
0.572
1.019
1.978
0.019
2.462
0.738
0.721
0.859
4.715
0.826
6.261
6.855
1.100
1.275
0.193
0.535
0.549
1.125
0.473
0.423
0.389
0.052
0.733
0.488
1.618
0.294
0.050
1.851
0.190
0.731
0.939
1.014
0.618
2.891
0.213
0.284
0.987
0.181
1.650
0.223
0.809
0.804
0.040
0.071
0.080
0.679
25025
25027
25029
25030
25034
25040
25044
25124
26002
26005
26006
26007
26008
26018
26019
26021
26022
26059
27001
27002
29004
29011
30007
30061
31002
32012
34001
34003
34009
34018
34024
35001
35002
35005
35071
36015
36018
36019
36021
36031
31
43
33
48
24
20
33
18
53
51
53
53
49
49
51
30
33
17
30
51
32
22
31
33
26
24
36
29
33
27
28
29
34
55
30
33
50
47
27
30
161.20
118.87
292.67
280.02
10.77
28.02
92.55
215.45
641.45
1085.38
184.76
1207.22
280.31
119.48
252.96
1098.78
61.88
256.64
46.70
564.27
121.44
354.14
469.90
3136.08
71.35
146.16
1974.76
1802.38
117.11
95.40
127.23
299.45
88.82
639.66
247.21
153.06
234.40
1491.76
23.41
63.77
904.54
1021.15
1108.68
1183.81
968.60
989.64
1186.86
954.84
1067.03
1054.40
1120.64
1045.62
1035.47
1043.90
979.62
945.25
915.82
976.44
1476.89
1336.35
1107.47
1079.37
1115.06
1422.43
1530.25
1784.36
1322.66
1339.66
1256.71
1554.59
1177.46
1172.84
1380.56
1198.32
1364.46
1090.72
950.12
971.21
1569.64
910.43
0.73
0.65
0.58
0.54
0.76
0.64
0.58
0.87
0.61
0.56
0.54
0.65
0.61
0.72
0.54
0.83
0.58
0.91
0.28
0.70
0.52
0.63
0.65
0.78
0.53
0.59
0.78
0.80
0.40
0.66
0.52
0.60
0.42
0.61
0.77
0.42
0.69
0.79
0.27
0.48
103
1.00
1.00
1.00
0.85
1.00
1.00
1.00
0.78
0.98
0.98
0.97
0.98
0.86
0.76
0.99
0.81
1.00
0.73
0.99
0.84
0.99
0.98
0.99
0.66
0.63
0.84
0.83
0.82
1.00
0.73
0.92
0.92
0.99
0.90
1.00
0.96
0.85
0.76
1.00
0.96
0.167
0.158
0.139
0.178
0.166
0.154
0.188
0.200
0.111
0.101
0.152
0.107
0.104
0.114
0.139
0.140
0.173
0.099
0.105
0.122
0.083
0.145
0.107
0.147
0.129
0.071
0.104
0.093
0.097
0.114
0.066
0.119
0.098
0.144
0.100
0.159
0.104
0.104
0.119
0.098
0.083
0.046
-0.020
0.185
-0.080
0.204
0.248
-0.069
0.224
0.035
0.397
0.164
0.193
0.126
0.234
-0.112
0.091
0.111
0.193
0.181
0.144
0.410
0.112
0.410
0.308
0.001
0.080
0.114
0.081
0.177
-0.051
-0.009
0.026
0.205
0.118
0.325
0.045
-0.029
0.201
0.365
0.146
0.126
-0.020
0.132
0.115
0.181
0.149
0.231
0.293
0.142
0.412
0.157
0.190
0.113
0.121
0.187
0.047
0.175
0.248
0.216
0.085
0.318
0.203
0.379
0.178
0.181
0.211
0.253
0.158
0.033
0.140
0.190
0.081
0.143
0.197
0.299
0.100
0.064
0.222
0.391
0.143
0.219
1.367
0.235
1.358
0.100
0.484
2.954
0.968
0.282
2.833
0.244
0.347
0.248
0.715
2.475
0.695
0.218
0.557
0.223
0.942
1.811
0.281
2.383
1.030
0.902
0.481
0.854
0.220
1.725
1.013
0.923
0.405
0.271
0.285
0.987
0.264
0.579
0.300
2.885
Table 11.2 Four stations out of 94 showing large discordancy values
River
Name
Sluice
Griffeen
Dodder
Station
8005
9002
9010
D Value
6.16
6.26
8.94
Observation on discordancy
Large Cv of 0.6
Large Cv of 0.8
Outlier due to Hurricane Charlie
9010 RIVER DODDER @ W ALDRON'S BRIDGE
8005 RIVER SLUICE @ KINSALEY HALL
300
10
w inter peak
6
summer peak
4
A M F (m 3/s )
AMF(m3/s )
EV1
250
8
2
2
0
5
10
25
50
100
200
w inter peak
150
summer peak
100
50
500
2
0
-2
-1
0
1
2
EV 1 y
3
4
5
6
7
-2
-1
0
5
1
10
2
25
3
50
4
100
500
5
6
EV1 y
30
AM F (m 3/s )
25
20
w inter peak
15
summer peak
10
5
2
0
-2
-1
0
5
1
10
2
25
3
50
4
100
500
5
6
7
EV 1 y
Figure 11.1EV1 probability plot for the discordant stations
11.2 Effect of catchment type on pooled growth curve estimates.
Catchment have been divided into subgroups on the basis of
•
catchment storage as measured by FARL values
•
peat content of catchment as measured by % Peat value
•
size of catchment as measure by area
•
location of catchments in one of 4 regions (see in Figure 11.2)
In all cases the effect on X100 has been assessed for simplicity rather than try to compare
entire growth curves simultaneously. The exception is that X50 is used in the case of
estimates from decadal data because of the obvious smaller amount of data available.
104
7
The growth curve ordinate X100 has been estimated for each of the 85 gauging stations
from its own pooling group based on the distance measure defined in equation (10. 2)
and using the 5T rule for number of data included in each. Estimates were obtained by
both GEV and GLO estimating equations.
´
Legend
!
(
Gauging station
!
(
SHANNON
(
!!
(
!
(
!
(
!
(
!
(!
(
!
(
!
(
!
(
!
(
!
(
WEST
!
(
!
(
!
(
!
(
!
!
((
!
(
!
(
!
(
!
(!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
EAST
!
(
!
(
!
(
!
(
!
(!
(
!
!
( (
!
( !
(
!
(!
(
(
!!
((
!!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
!
(
SOUTH-WEST
!
(
!
(
Figure 11.2 Location of the 85 stations and the four regions
105
The effects of catchment storage and of percentage peat are shown in Figure 11.3 and
Figure 11.4 in box plot form. Each box plot in Figure 11.3 summarises values of X100
while Figure 11.4 summarise the corresponding Hosking- Wallis homogeneity statistics
H1 and H2. The leftmost box plot represents all 85 stations in the study.
Pooling Growth Curve(GEV)
2.4
2.2
X100
2
1.8
1.6
1.4
1.2
1
All st(85)
≥5%
peat(44)
≥10%
peat(30)
≥15%
peat(22)
≥20%
peat(14)
farl
≤.98(30)
farl
≤.95(23)
farl
≤.90(19)
farl
≤.85(15)
farl
≤.95(23)
farl
≤.90(19)
farl
≤.85(15)
Figure 11.3(a)
Pooling Growth Curve(GLO)
2.4
2.2
X100
2
1.8
1.6
1.4
1.2
1
All st(85)
≥5%
peat(44)
≥10%
peat(30)
≥15%
peat(22)
≥20%
peat(14)
farl
≤.98(30)
Figure 11.3 Box plots of X100 values from region of influence pooling estimates of X100
obtained by (a) GEV, (b) GLO estimating equations showing effect of %peat and FARL.
The numbers in brackets (85, 44, 30…) indicate the number of X100 values contributing to
each box plot.
106
Homogeneity Statistics(H1)
14
12
10
H1
8
6
4
2
0
-2
All st(85)
≥5%
≥10%
≥15%
≥20%
farl
farl
farl
farl
peat(44) peat(30) peat(22) peat(14) ≤.98(30) ≤.95(23) ≤.90(19) ≤.85(15)
Figure 11.4(a)
14
12
10
H2
8
6
4
2
0
-2
All st(85)
≥5%
≥10%
peat(44) peat(30)
≥15%
≥20%
farl
peat(22) peat(14) ≤.98(30)
farl
farl
≤.95(23) ≤.90(19)
farl
≤.85(15)
Figure 11.4 Values of (a) H1 (b) H2 for the analyses summarised in Figure 11.3
The median X100 values differ only slightly with variation in peat percentage and
without a definite pattern. The median X100 values show a slight monotonically
decreasing trend with increasing amount of storage i.e. smaller FARL values. The
results are substantially the same for both GEV and GLO estimation. In nearly all cases
107
the median H1 values are greater than 2 while nearly all the median H2 values are less
than 2 and those few that do exceed 2 do so by a very small amount.
The effects of geographical region and of catchment area are shown in Figure 11.5 and
Figure 11.6. In terms of geographical region South-West has the lowest median X100 of
approximately 1.52 while the East has the highest value of approximately 1.9. However
catchments of all sizes, degree of FARL and of percentage peat are mixed together in
these subjectively selected geographical groupings. The median X100 values shows a
monotonic decrease with increase in catchment area. Figure 11.6 display H1 and H2
values that are slightly more dispersed than those in Figure 11.4, but with similar overall
results, showing median values of H2 less than 2.
108
2.4
2.2
X100
2
1.8
1.6
1.4
1.2
1
All st(85)
West(17)
SouthWest(18)
East(24)
Shannon A≤100(19) A~101200(20)
(26)
A~201- A≥500(22)
500(24)
Figure 11.5(a)
2.4
2.2
X100
2
1.8
1.6
1.4
1.2
1
All st(85)
West(17)
SouthWest(18)
East(24)
Shannon A≤100(19) A~101(26)
200(20)
A~201- A≥500(22)
500(24)
Figure 11.5 Box plots of (a)X100 (GEV) (b) X100 (GLO) showing effect of geographical area
and catchment size
109
12
10
8
H1
6
4
2
0
-2
All st(85)
W est(17)
SouthW est(18)
East(24)
Shannon (26) A≤100(19)
A~101200(20)
A~201500(24)
A≥500(22)
A~101200(20)
A~201500(24)
A≥500(22)
Figure 11.6(a)
12
10
8
H2
6
4
2
0
-2
All st(85)
W est(17)
SouthW est(18)
East(24)
Shannon
(26)
A≤100(19)
Figure 11.6 Values of (a) H1, (b) H2 for the analyses summarised in Figure 11.5
110
11.3 Temporal effect on pooled growth curve estimates
Pooled growth curves have been obtained from different periods of record as follows:
•
distinguishing between early and late halves of each record and
•
distinguishing also by decades, 1972 - 81, 1982 – 91 and 1992 – 2001 (50
stations) and
•
distinguishing by decades, 1952 - 61, 1962 - 71, 1972 - 81, 1982 – 91 and 1992
– 2001 (for which data are available for 26 stations).
•
distinguishing by decades 1957-66, 1967-76, 1977-86, 1987-96 and 1997-2006
where 2006 is the most recent year of record in the data set. There are 34
records available for all 5 decades and 68 records available for the last 3 such
decades.
This part of the study was completed using a total of 90 gauging stations before the data
provider had indicated some of these are unsuitable for detailed statistical analysis.
Hence the number 90 is greater than 85 mentioned above in Section11.1. It is believed
that the overall nature of the findings would not be greatly altered if the study had been
repeated with just 85 stations. This would be specially true if the five omitted stations
had reasonable water level records and were rejected on the basis of doubts about the
reliability of rating curves.
The effects of period of record (1st half of each record compared with 2nd half) and of
successive decades are shown in
Figure 11.7 and Figure 11.8. H1 an H2 values are shown in Figure 11.8. Median X100
shows a decrease between the earlier and later halves of the records while the decade
based values of X50 show no particular pattern. Indeed the 3rd decade value of median
X50 is not much different to the 1st decade median X50 value.
111
X100 (X50 in case of decade analysis)
2.4
2.2
2
1.8
1.6
1.4
1.2
1
All record(90)
1st half
record(90)
2nd half
record(90)
Decade70(52)
Decade80(83)
Decade90(82)
Figure 11.7(a)
X100 (X50 in case of decade analysis)
2.4
2.2
2
1.8
1.6
1.4
1.2
1
All record(90)
1s t half
record(90)
2nd half
record(90)
Decade70(52) Decade80(83) Decade90(82)
Figure 11.7 Variation of X100 with period of record (first half versus second half of each
record) and variation of X50 with decades, as estimated by (a) GEV, (b) GLO from each
individual stations pooling group using the 5T rule
112
12
10
8
H1
6
4
2
0
-2
A ll record(90)
1st half
record(90)
2nd half
record(90)
Decade70(52)
Decade80(83)
Decade90(82)
Decade80(83)
Decade90(82)
Figure 11.8(a)
Hom ogeneity Statis tics (H2)
12
10
8
H2
6
4
2
0
-2
A ll record(90)
1st half
record(90)
2nd half
record(90)
Decade70(52)
Figure 11.8 Values of (a) H1, (b) H2 for the analyses summarised in Figure 11.7
113
There are 50 stations for which decade estimates are available for all 3 decades. The
values of X50 for each decade for each station are displayed in Figure 11.9(a). From this
it can be seen that the largest X50 values occur in the middle decade, 1982-91, in every
case. The corresponding Qmed values, standardized at each station by the 1972-81
value of Qmed, are shown in Figure 11.9(b). In most cases the Qmed values for the 2nd
and 3rd decade exceed the 1st decade value and while the 3rd decade displays a number
of remarkable Qmed values, only about one third of the stations have their largest Qmed
in the 3rd decade. Figure 11.9(c) shows the estimated Q50 values, standardized at each
station, by the 1972-81 value. The majority of Q50 values in the 2nd and 3rd decades
exceed those of the 1st decade, with more that half of these exceedances occurring
during the 2nd decade.
There are 26 stations for which decade estimates are available for 5 decades. At some
stations there are not a full 10 years of data available for the 1st decade. Only stations
which have a minimum of 7 years of data available for the 1st decade are included. The
results of the analyses are examined in the same way as the 3 decades result above and
presented in Figure 11.10. The 2nd decade (1960s) value of X50 is larger than those of
the other decades at every station with an average value of approximately 1.7. This
contrasts strongly with the results of the last decade (1990s) where the average value is
approximately 1.4. Figure 11.10(b) shows the decadal standardized Qmed values. This
shows that the largest values tend to be from the 2nd (1960s) and the last (1990s)
decades. The Q50 values in Figure 11.10(c) also conveys the impression that the largest
Q50 values occur during the 2nd (19960s) and last (1990s) decades.
The results for the decades ending in 1966, 1976, 1986, 1996 and 2006 show results that
are similar to those described in the last paragraph. Results are shown in Figure 11.11
which is analogous to Figure 11.10.
114
115
30061
34001
34009
35001
35002
36011
36015
36018
36019
30061
34001
34009
35001
35002
36011
36015
36018
36019
26007
26006
26005
26002
25030
25029
25027
25023
25016
25014
25006
25005
25003
25001
29004
0
29004
0.2
27002
0.4
27002
0.6
27001
0.8
27001
1
26022
1.2
26022
1.4
26019
1.6
26019
Decadal Q50 value
26018
2
26018
Figure 11.9(b)
26008
Station No.
26008
26007
26006
26005
26002
25030
25029
25027
25023
25016
25014
25006
25005
Decade90
25003
Decade90
23001
1.8
25001
19001
18005
18004
16009
16008
16004
16003
16002
16001
15003
Station No.
Figure 11.9 (a)Column chart of X50 (GEV) values (b) Column chart of Qmed values (c)
Column chart of Q50 values for the decadal data (‘70, ‘80 and ‘90) of 50 stations
360 19
360 18
360 15
360 11
350 02
350 01
340 09
340 01
300 61
290 04
270 02
270 01
260 22
260 19
260 18
260 08
260 07
260 06
260 05
260 02
250 30
250 29
250 27
250 23
250 16
250 14
250 06
250 05
250 03
250 01
230 01
190 01
180 05
180 04
160 09
160 08
160 04
160 03
160 02
160 01
150 03
140 19
140 18
140 06
Decade90
23001
19001
18005
18004
16009
16008
16004
16003
16002
16001
14019
Decade80
15003
14018
14006
140 05
120 01
110 01
90 01
60 26
60 11
X50
Decade80
14019
Decade80
14018
14006
Decade70
14005
12001
11001
9001
6026
6011
Q m e d (s ta n d a r d i s e d b y Q m e d o f
decade70)
Decade70
14005
Decade70
12001
11001
9001
1.8
6026
6011
Q 5 0 (s ta n d a rd is e d b y Q 5 0 o f
decade70)
1.9
1.8
Pooling Growth Curv e (GEV)
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
Station No.
Figure 11.9(a)
1.6
Decadal Qmed value
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Pooling Growth Factor
2.00
1.80
X 50
1.60
1.40
Decade50
1.20
Decade60
Decade70
1.00
Decade80
0.80
Decade90
0.60
0.40
0.20
36018
36011
27002
26019
26008
26007
26006
26005
26002
25023
25014
25006
25005
25003
25001
18005
16009
16008
16004
16003
16002
15003
14019
14018
14006
14005
0.00
Station No.
Figure 11.10(a)
1.8
Qmed Value
Q m ed ( stan d ard ised b y Q m ed o f
d ecad e50)
1.6
1.4
1.2
Decade50
1
Decade60
Decade70
0.8
Decade80
0.6
Decade90
0.4
0.2
36018
36011
27002
26019
26008
26007
26006
26005
26002
25023
25014
25006
25005
25003
25001
18005
16009
16008
16004
16003
16002
15003
14019
14018
14006
14005
0
Station No.
Figure 11.10(b)
1.80
Q50 Value
Q 5 0 ( s ta n d a r d i s e d b y Q 5 0 o f
decad e50)
1.60
1.40
1.20
Decade50
Decade60
1.00
Decade70
0.80
Decade80
Decade90
0.60
0.40
0.20
Station No.
Figure 11.10(a)Column chart of X50 (GEV) values (b)Column chart of Qmed values (c)
Column chart of Q50 values for the decadal data (’50, ’60 ‘70, ‘80 and ‘90) of 26 stations
116
36018
36011
27002
26019
26008
26007
26006
26005
26002
25023
25014
25006
25005
25003
25001
18005
16009
16008
16004
16003
16002
15003
14019
14018
14006
14005
0.00
117
26008
26018
26019
27002
35005
36011
36018
36019
26018
26019
27002
35005
36011
36018
36019
25006
25003
25001
19001
26008
0
26007
0.2
26007
0.4
26006
0.6
26006
1
26005
0.8
26005
Decade 97-06
26002
1.2
26002
Decade 87-96
25030
Decade 77-86
25030
Decade 67-76
25023
Decadal Q50 Value
25023
Figure 11.11(b)
25014
Station No.
25014
25006
25003
25001
Decade 57-66
19001
2
18005
Decade 57-66
18005
2
18004
16009
16008
16003
16002
15003
14019
14018
14006
14005
Station No.
Figure 11.11 (a) Column chart of X50 (GEV) values (b) Column chart of Qmed values (c)
Column chart of Q50 values for the decadal data (’57-66, ’67-76 ’77-86, ’87-96 and ’97-06)
of 34 stations
36019
36018
36011
35005
27002
26019
26018
26008
26007
26006
26005
26002
25030
25023
25014
25006
25003
25001
19001
18005
18004
16009
16008
16003
16002
15003
14019
14018
14006
14005
12001
9001
6026
57-66
67-76
77-86
87-96
97-06
18004
16009
16008
16003
16002
15003
14019
14018
1.4
14006
1.6
14005
1.8
12001
1.4
12001
1.6
9001
6011
X50
Decade
Decade
Decade
Decade
Decade
9001
1.8
6026
6011
Qmed( standardised by Qmed of decade5766)
2.5
6026
6011
Q50( standardised by Q50 of decade 57-66)
3
Growth Factor X50
2
1.5
1
0.5
0
Station No.
Figure 11.11(a)
Decade 67-76
Decadal Qmed Value
Decade 77-86
Decade 87-96
1.2
Decade 97-06
0.8
1
0.6
0.4
0.2
0
11.4 Arterial drainage effect on pooled growth curve estimates
Pooled growth curves have been obtained from the pre and post drainage records of 16
gauging stations, 8 of which are located in the Boyne catchment. These 16 stations are
those that have their pre- and post-drainage annual maximum series listed in Table 11.3
and displayed on probability plots in Appendix 4D. The descriptor BFI is needed in the
pooling procedure but 3 of these stations have not had a BFI value provided. Two of
these are on the Boyne catchment (7007 and 7011) so the values at neighboring gauging
stations on the same river have been used for these instead – 7003 for 7007 and 7004 for
7011. The third station lacking a BFI value, namely 30004, was assigned the BFI value
for gauging station 30005 even though this is not on the same stream. A visual check
was made on the appearance of yearly hydrographs of these two gauging stations to
check if it is reasonable to make this assignment. The agreement is reasonable but not
perfect. The probability plots in Appendix 4D show a clear increase in magnitude at all
return periods for only 9 gauging stations. These are 3051, 7003, 7004, 7010, 7011,
7012, 24001, 24013 and 30004. One gauging station's plot, 30001, shows quite the
opposite effect while the remaining 6 show a variety of cross-over effects namely that
the pre-drainage large floods exceed the post drainage large floods or else lines fitted to
each period of record might cross over.
The results of the pooled estimation of X50, Qmed and Q50 for these stations, from
both pre-drainage and post-drainage data are shown in Figure 11.12. These show that
the pre-drainage growth factor exceeds the post-drainage growth factor in every case,
whereas the Qmed values are greater in the post-drainage period than in the predrainage period in every case except station for the above mentioned 30001. The
estimated Q50 values for the post-drainage period exceed the pre-drainage values in 10
cases, 9 of them being those noted above in relation to their probability plot behaviour,
the 10th one being 26012 which shows just a small increase.
It should be noted that the extensive studies have already been carried out by OPW
(1981) on the effect of catchment drainage on flood magnitude return period
relationships as well as on unit hydrograph characteristics
118
Table 11.3 List of 16 no. Pre and post- drainage stations
Station
Number
Station Name
River Name
Catchment
Record
length
3051
FAULKLAND
BLACKWATER
Blackwater
26
7002
Killyon
Boyne
46
7003
Castlerickard
Boyne
46
7004
Stramatt
Deel
Blackwater
(Enfield)
Blackwater
(Kells)
Boyne
48
7005
Trim
Boyne
Aqueduct
Boyne
Boyne
47
Boyne
45
Boyne
46
7011
Liscartan
O'Dalys
Bridge
Boyne
Blackwater
(Kells)
Blackwater
(Kells)
Boyne
44
7012
Slane Castle
Boyne
Boyne
65
23002
Listowel
Feale
Feale
59
24001
Croom
Maigue
Maigue
51
24013
Rathkeale
Deel
Maigue
46
26012
Tinacurra
Boyle
Boyle
48
30001
Cartronbower
Aille
Cara
48
30004
Corrofin
Clare
Cara
35
30005
Foxhill
Robe
Cara
45
7007
7010
119
Timespan
19752004
19532004
19532004
19562004
19532004
19532004
19532004
19582004
19402004
19462004
19532003
19532004
19572004
19522001
19522003
19532004
Drainage
works
Missing
year
86-'89
3/744/79
19701975
19731981
19711974
1986-89
1956,70-74
73-78
19821986
1973-78
1970,71,8285
80- 83
1980-82
69-86
None
51-59
None
75- 76
19631968
None
1963-68
82-91
None
70-71
58-64
1970-71
195873,79,86
73-78
1961,73-78
1973-78
1973-81
1971-74
2.5
Pooling Growth Factor
Pre-Drainage
Post-Drainage
2
X50
1.5
1
30005
30004
30001
26012
24013
24001
23002
7012
7011
7010
7007
7005
7004
7003
3051
0
7002
0.5
Sta tion No.
Qmed value
Pre-Drainage
Post-Drainage
3
2.5
2
1.5
1
30005
30004
30001
26012
24013
24001
23002
7012
7011
7010
7007
7005
7004
7003
0
7002
0.5
3051
Qmed (standardised by Qmed of predrainage series)
3.5
Sta tion No.
3
Q50 value
Post-Drainage
2.5
2
1.5
1
Station No.
Figure 11.12 (a) Column chart of X50 (GEV) values (b) Column chart of Qmed values
(c)Column chart of Q50 values for the pre- and post-drainage data of 16 stations
120
30005
30004
30001
26012
24013
24001
23002
7012
7011
7010
7007
7005
7004
7003
0
7002
0.5
3051
Q50 (standardised by Q50 of
pre-drainage series)
Pre-Drainage
11.5 Implications for Flood Frequency Estimation
The FARL value, catchment area and geographical location appear to have an effect on
the X100 growth factor with percentage peat having virtually no noticeable effect. In
temporal terms the middle of the 3 decades between 1972 and 2001 produces the largest
X50 values as a whole, while the same decade displays many, but not all, of the largest
Q50 values. The 1972-81 decade displays the smaller values of X50, Q50 and Q50 values
with just a small number of exceptions. When the smaller number of cases for which 5
decades of data are available the impression is that the 2nd (1960s) and last (1990s)
decades produce the largest floods.
In view of the effect of FARL, catchment area and geographical location on XT the
question arises as to whether the catchments which are allowed into a particular pooling
group should be screened or filtered in advance of pooling for any particular subject
site. The disadvantage of this is to reduce the pooling group constituency from which
the pooling group members are to be drawn. Given that the region of influence based
pooling method takes catchment area into account in the distance measure dij this
variable's effect on XT is partially taken into account. The geographical location may be
partially taken into account by SAAR but not uniquely so while FARL is not taken into
account in any manner.
It may be best to draw users' attention to these issues and to leave it open to users to
decide for themselves whether to screen the data or not. It is likely that users will want
to take geographical location explicitly into account in some cases, especially for east
coast locations. Users will have to advised that there is a trade-off involved between
tailoring the pooling group constituency on the one hand and loss of station years and
lack of precision on the other.
It should also be noted that the effects refereed to above have not been statistically
tested to determine whether these relatively small observed effects would in fact reject
the null hypothesis of no effect.
With regard to temporal effects and whether only recent records should be used in flood
estimation the evidence is not quite as clear cut that only recent data should be included.
121
This may conflict with other published views that the past is not a good guide to the
future, cf "The past is longer the key to the future, and the future is uncertain" in Ireland
at Risk, published by Irish Academy of Engineering, 2007, p8. However this latter view
conflicts with the well established practice of using design flood estimates based on
historical records.
122
12 Selection of distance measure variables
As stated earlier "The region of influence approach selects stations, which are nearest to
the subject site in catchment descriptor space, to form the pooling group for that subject
site." Equation (10. 2) is one definition of a distance measure used for this purpose
which has been used in the investigations described in the last subsection, 1. However
other catchment descriptors could also be used in the definition, fewer or more of them
could be used and each component could have different weights.
12.1 Selecting variables for pooling
In this study four catchment descriptors, AREA, SAAR, BFI and FARL have been
considered in eight different combinations as described later below:
The objective is to find which of these combinations leads to pooling groups which are
most homogeneous and therefore most effective at exploiting the information about the
flood distribution contained in the pooling groups. This is examined with the help of
two statistics
•
H1, the Hosking- Wallis homogeneity statistic based on similarity of L-CV
values in the pooling group, and
•
PUM100, the FEH defined Pooled Uncertainty Measure of the 100 year quantile:
The general form of the similarity measure used for selecting members of a pooling
group, is
n
d ij =
∑ W (X
k
− X k, j )
2
k ,i
--- (12. 1)
k =1
where, n is the number of catchment descriptors, X k ,i is the value of kth catchment
descriptor at the ith site and Wk is the weight applied to descriptor k reflecting the
relative importance of that catchment descriptor. The subscript j applies to the ‘subject
site’ and the subscript i applies to the ith ‘donor site’.
123
In choosing a distance measure d ij a decision has to be made about which catchment
descriptors are to be included in the distance measure and what weightings are to be
applied to them and whether logarithms or other transformations are to be used.
The Flood Estimation Handbook (1999) gives a number of useful maxims for choosing
a distance measure. It uses pooled uncertainty measure (PUM) for selecting the final set
of pooling variables. For UK data the distance measure is taken as equation (10. 1),
recalled here again:
dij =
1  ln Ai − ln A j

σ ln A
2 
2
  ln SAARi − ln SAAR j
 + 
σ ln SAAR
 
2
  BFIHOSTi − BFIHOST j
 + 
σ BFIHOST
 



2
The weight of 0.5 is given to AREA only to allow SAAR and BFIHOST to play a
slightly more significant role in forming the pooling groups in the distance measure.
Recently a revised distance measure has been developed for UK data (Kjeldsen et al.,
2008) where two new variables FARL and FPEXT (the extent of flood plains in a
catchment) has been included and the variable BFIHOST has been omitted:
2
2
2
 ln A − ln Aj 
 ln SAARi − ln SAARj 
 FARLi − FARLj 
 FPEXT
i − FPEXT
j  --- (12. 2)
 + 0.5
 + 0.1
 + 0.2

dij = 3.2 i
1
.
28
0
.
37
0
.
05
0
.
04








12.2 Pooled uncertainty measure, PUM
The PUM is a weighted average of the differences between each site growth factor and
the pooled growth factor measured on a logarithmic scale. The pooled uncertainty
measure for return period T, PUMT is defined by (FEH, 1999) as
2
M long
∑ n (ln x
i
PUM T =
Ti
− ln x
p
Ti
)
i =1
--- (12. 3)
M long
∑n
i
i =1
124
where Mlong is the number of long-record sites, ni is the record length of the ith site, x Ti is
the T-year site growth factor for the ith site, and ln x p Ti is the T-year pooled growth factor
for the ith site. PUM is a measure of how effective a pooling method is at identifying a
homogeneous region. A good pooling method will yield low values of PUM.
12.3 Selecting pooling variables for Irish data
For Irish conditions, four catchment descriptors have been selected for the potential
pooling variables. They are AREA, SAAR, BFI and FARL. These variables have been
found to be effective in explaining the observed variation in Qmed values in regression
studies.
In FEH pooling variables were chosen based on the pooled uncertainty measure (PUM).
In this study PUM as well as heterogeneity measure (H) is used to select the final set of
pooling variables.
In this study PUM has been evaluated at 100-year return period for catchments whose
record length is 20 or more years of data. 90 sites of category A have been considered
for the pooling analysis. Among them 85 sites have 20 or more years of data. For each
of these catchments, the pooling group was selected from 90 sites. For determining the
size of the pooling group the 5T rule has been used here.
Generalised Extreme Value Distribution (GEV) has been used to calculate the pooled
and single site growth factors. In FEH the subject site was not included in its own
pooling group when PUM was evaluated. In this study PUM is evaluated in both the
presence and absence of the subject site.
The following combinations of the four variables have been tested based on PUM and H
lnAREA
lnAREA,lnSAAR
lnAREA,lnSAAR,BFI
lnAREA,lnSAAR,BFI,FARL
lnSAAR
125
BFI
lnAREA+BFI
lnAREA,lnSAAR,lnBFI
The data sets that have been used in the study are summarised Table 12.1.
Table 12.1Summary of AMF data sets used in dij study
Number of stations
Shortest record length
Longest record length
Mean record length
Number of AMF events
90
18
55
37.47
3372
Initially all weights Wk in equation (12. 1) were set to unity. Figure 12.1 show the
variation in 100-year PUM value for different sets of catchment descriptors used in
equation (12. 1) in box-plot form. In Figure 12.1(a) the PUM value is evaluated without
inclusion of subject site in its own pooling group while in Figure 12.1(b) the subject site
is included. Figure 12.2(a, b) summarise the corresponding heterogeneity statistics H1
and H2.
Further Table 12.2 summarises the corresponding mean variation of PUM100, H1 and
H2 values for different sets of pooling variables which shows that the numerical
measures of effectiveness vary by very little between rows. As the standard errors of
these measures are unknown it is not possible to state whether the values in any one
column are significantly different from one another in the statistical sense.
From Table 12.2 it is shown that the set of two variables lnAREA and lnSAAR and the
set of one variable lnSAAR alone performed best in terms of providing the lowest PUM
value. In terms of H1 heterogeneity the set consisting of lnAREA and lnSAAR comes
second best to the set consisting of lnAREA, lnSAAR, BFI and FARL and in terms of
H2 heterogeneity it comes third behind lnAREA on its own and behind lnSAAR o n its
own.
Overall the set of variables comprised of lnAREA and lnSAAR can be considered as
being the most suitable set of pooling variables for Irish condition. However if there is
also a desire to incorporate another physical catchment effect then BFI could be
126
included with these two, as was done in equation (10. 2) above. While inclusion of just
one or two catchment descriptors may be indeed be best there is an attraction in
representing some descriptor of catchment response also even at a small apparent loss in
effectiveness.
Table 12.2 Variation in the mean PUM100, H1 and H2 values for different sets of pooling
variables
PUM(without taking
subject site into
account)
PUM( taking
subject site into
account)
lnAREA
lnAREA,lnSAAR
0.1956
0.1951
6.1670
2.7697
0.1929
0.1924
5.2276
2.7684
lnAREA,lnSAAR,BFI
lnAREA,lnSAAR,BFI,FARL
lnSAAR
BFI
lnAREA+BFI
lnAREA,lnSAAR,lnBFI
0.1958
0.1951
5.4381
2.8232
0.1966
0.1959
5.0160
2.9584
0.1913
0.1929
5.5279
2.7258
0.2053
0.2048
7.1408
3.2138
0.2044
0.2023
6.1348
2.9501
0.1958
0.1953
5.5582
2.8771
Variables used in model
H1
H2
An extension to this investigation was done with varying values of weights Wk in
equation (10. 2). The results of all variations examined are not reported in detail here.
It was found that the weights 1.7, 1.0 and 0.2 for ln AREA, ln SAAR and BFI
respectively gave PUM100 = 0.1889 and 0.189, H1 = 5.20 and H2 = 2.62 which offer
small improvements on the Wk = 1.0 values used in the calculations on which Table
12.2 are based.
Thus either equation (10. 2) or the following one is considered suitable for calculating
the distance measure.
dij =
1.7 
σ ln AREA

2
  ln SAARi − ln SAARj
 + 
σ SAAR
 
127
2

 BFIi − BFI j
 + 0.2 
σ BFI


2

 --- (12. 4)

Pooled Uncertainty Measure(PUM)
0.35
0.30
P U M 100
0.25
0.20
0.15
0.10
0.05
0.00
lnA
lnA+lnS
lnA+lnS+B
lnA+lnS+B+F
lnS
B
lnA+B
lnA+lnS+lnB
B
lnA+B
lnA+lnS+lnB
Figure 12.1(a)
Pooled Uncertainty Measure (PUM)
0.35
0.30
P UM 100
0.25
0.20
0.15
0.10
0.05
0.00
lnA
lnA+lnS
lnA+lnS+B
lnA+lnS+B+F
lnS
Figure 12.1 Box-plot of 100-year PUM values for different sets of catchment descriptors
used in defining the distance measure dij. (a) subject site is not included in its own poolinggroup when PUM is evaluated (b) subject site is included in its own pooling-group when
PUM is evaluated.
128
14
12
10
H1
8
6
4
2
0
lnA
lnA+lnS
lnA+lnS+B
lnA+lnS+B+F
lnS
B
lnA+B
lnA+lnS+lnB
B
lnA+B
lnA+lnS+lnB
-2
Figure 12.2(a)
Homogeneity Statistics (H2)
14
12
10
H2
8
6
4
2
0
-2
lnA
lnA+lnS
lnA+lnS+B
lnA+lnS+B+F
lnS
Figure 12.2 Box-plot of (a) H1 values, (b) H2 values for different sets of catchment
descriptors used in defining the distance measure dij.
129
13 Standard errors of Qmed, QT and XT
The standard error se of an estimate of QT is an indication of how reliable that estimate
is. It is based on the assumption that the data upon which the estimate is based are
randomly drawn from a single population – an assumption that cannot easily be proven.
If an infinite number of similarly sized data sets were to be drawn from the same
population and the value of QT obtained from each set by the same procedure then the se
is defined as
se(QT ) = Standard deviation of all the possible set of QT values.
This measure only represents the degree of scatter of the several estimates and does not
refer to whether the mean of these equals the true value in the population. If this
equality holds then the procedure whereby QT is calculated is said to be unbiased –
otherwise it is said to be biased.
In flood hydrology randomness of annual maximum floods and lack of trend with time
is generally assumed. Likewise it is assumed that a single form of statistical distribution
describes all the AM flood series in a region or country. Such assumptions cannot be
proven and there is some evidence to suggest that Irish flood data are not entirely trend
free. The presence of low or high outliers in some data sets also makes interpretation
difficult.
If the data are drawn are not truly from a unique homogeneous parent
population then the se cannot strictly be defined. However the se concept is widely
used in flood hydrology.
Its value is determined on the assumption of a unique
homogeneous parent and the value of se so obtained may be considered as a lower
bound on what the true value, if it could be defined, actually is. Hence it provides a
useful guide to the precision of the QT value obtained in any situation.
The aim in this section is to provide expressions or graphs which give an indication of
the order of magnitude of se(QT) in at-site estimation and in pooling group based
estimation of QT. This will be restricted to the EV1 and GEV cases on the assumption
130
that the GLO/LO and LN3/LN2 standard errors would be of the same order of
magnitude.
13.1 Standard Error of Qmed
This has been investigated in section 9.1 above and the result, adapted to Irish AM flood
condiditons, is reproduced here as follows:
se(Qmed ) =
0.36
N
Qmed
se(Qmed) ≈ 0.161 Qmed
( not tabulated in FEH,3, Table 2.2)
If N = 5,
13.2 Standard Error of At-site estimate of QT
Theoretical expressions for L-Moment based estimates of QT in both EV1 and GEV
cases have been given by Lu & Stedinger (1992) as follows.
EV1:
[ ]
SE Qˆ T =
α
n
[1.1128 + 0.4574 y + 0.8046 y ]
2
--- (13. 1)
where Q̂T is the estimate for the T-year flow event ; α is the EV1 scale parameter;
y = yT = − ln (− ln (1 − 1 / T )) ; and n is the number of observations in the sample.
[ ]
GEV: se Qˆ T =
α
[exp{a0 (T ) + a1 (T )exp(− k ) + a 2 (T )k 2 + a3 (T )k 3 }]1 / 2 --- (13. 2)
n
where Q̂T is the estimate for the T-year flow event; a0 (T ) , a1 (T ) , a2 (T ) , and a3 (T )
are coefficients that depend on the return period, T ; α and k are scale and shape
parameters respectively; and n is the number of observations in the sample. The values
131
for constant coefficients for different return periods are tabulated in Lu and Stedinger
(1992). For example,
for T=10: a0= -2.667, a1= 4.491, a2= -2.207, a3= 1.802
for T=100: a0= -4.147, a1= 8.216, a2= -2.033, a3= 4.780
The percentage standard error, se(QT)/QT % is, in particular, focused on here. In the
EV1 case this percentage can be expressed as a function of the ratio α/u and yT or
equivalently as a function of Cv and yT or of L-Cv and yT. Note that L-Cv ≈ 0.5 Cv in
the EV1 case.
Equation (13. 1) has been applied to the data of 85 A1 and A2 stations in which the
values of u, α and QT were estimated from each site's data separately and the ratio
se(QT)/QT % formed. By using estimated values of parameters in these expressions the
true value of se(QT) is not obtained but it is considered useful to examine the range of
values so obtained because part of their scatter is caused by inter-site heterogeneity,
assuming such exists, as well as by random sampling effects and unequal record lengths
(with an average of 37 years). A box plot of these 85 values for 6 different return
periods is shown in Figure 13.1. It is suggested that the upper and lower values
indicated by the box plot i.e. the middle 50% of values, ought to give a good indication
of the range of values in which the true se values fall. The extreme high and low values
can be considered to be unrepresentative of the whole and hence the suggestion of
focusing on the boxes.
In Figure 13.1(b) the same results are re-presented along with the theoretical EV1
percentage standard errors of equation (13. 1) for the average sample size of 37 years
and for a range of L-Cv values that covers the range of L-Cv values experienced among
Irish AM flood data. [The difference in appearance of the boxes in Figure 13.1(b) from
those shown in Figure 13.1(a) arises from a technical difficulty in graphing two
different types of series.]
What these show, under the EV1 assumption, is that the se can be assumed to be
between 5% and 7% for the 10 year quantile and between 6.5% and 10% for the 100
year quantile.
132
Corresponding results for the GEV case are presented in Figure 13.2. The theoretical
figures in Figure 13.2(b) are now for a single value of shape parameter k = -0.1 which it
is felt ought to cover the most extreme underlying population case that would arise in
Irish conditions. From the box plots we can conclude, using the same arguments as
above, that under the GEV assumption, that the se can be assumed to be between 6%
and 8% for the 10 year quantile and between 8% and 15% for the 100 year quantile.
However it can be seen that the most extreme values are extremely large but these are
not representative of the true se value.
Relative Standard Error (At site/EV1)
18
16
14
se(Q ^T)/Q T %
12
10
8
6
4
2
0
T=5
T=10
T=25
T=50
T=100
T=500
Figure 13.1(a)
18
Relative Standard Error (Atsite/EV1)
16
14
LCV=0.3
se(Q T)/Q T%
12
LCV=0.25
10
LCV=0.2
LCV=0.15
8
LCV=0.1
6
4
2
0
T=5
T=10
T=25
T=50
T=100
T=500
Figure 13.1 (a) Box-plot of percentage standard error of flood quantile estimates for EV1
distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100, 500-year
return period. (b) The theoretical EV1 percentage standard errors for the average sample
size of 37 years and for a range of L-Cv values
133
Relative Standard Error (At site/GEV)
100
90
80
se(Q ^T)/Q T %
70
60
50
40
30
20
10
0
T=5
T=10
T=25
T=50
T=100
T=500
T=100
T=500
Figure 13.2(a)
Relative Standard Error(At-site/GEV)
50
K=-0.1
se(Q T)/Q T
40
30
20
10
LCV=0.3
LCV =0.25
LCV =0.2
LCV =0.15
LCV =0.1
0
T=5
T=10
T=25
T=50
Figure 13.2 (a) Box-plot of percentage standard error of flood quantile estimates for GEV
distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100, 500-year
return period. (b)The theoretical GEV (k=-0.1) percentage standard errors for the average
sample size of 37 years and for a range of L-Cv values
134
13.3 Standard Error of pooled estimate of XT and of QT
A simulation based estimate of the order of magnitude of the pooled estimate of XT,
se(XT), for selected values of T, was obtained by simulation for each of the 85 stations.
Each gauging station was selected in turn as the subject site and the following procedure
applied.
1. Identify the gauging stations in the subject site's pooling group using dij values
of equation (10. 2) and with a minimum of 500 station years of data in the
pooling group, which satisfies the 5T rule for the 100 year quantile.
2. Random samples are drawn from EV1 populations for the subject site and for
each site in the pooling group. For each site the sample size is taken as equal to
the length of the observed historical record at the site and the parameter values u
and α are those estimated from the observed record by L-Moments.
3. The sample Qmed is obtained for the subject site.
4. The L-Cv value is obtained for each sample in the pooling group and the average
of these is calculated to represent the pooled average L-Cv.
5. The pooled average L-Cv is used to determine the pooling group's EV1 growth
curve parameter β
6. The subject site's xT = 1 + β (ln(ln 2) − ln(− ln(1 − 1 / T )) ) is calculated for T = 5,
10, 25,………500 years
7. The subject site's QT = Qmed XT is calculated for T = 5, 10, 25,………500
years
8. Steps 2 to 7 are repeated 1000 times to provide 1000 values of XT and QT at the
subject site and the se(XT) and se(QT) are calculated for the subject site by the
following equations:
N
1
SE ( X T )[%] =
N
∑
1
SE (QT )[%] =
N
N
i =1
∑
i =1
2
1
M
 Xˆ T − Xˆ T
i
 i, m
∑
T

X

m =1
i





1
M
 Qˆ T − Qˆ T
i
 i, m
∑
T

Qi
m =1 



 × 100

M
M
× 100
--- (13. 3)
2
135
--- (13. 4)
where Xˆ iT, m and Qˆ iT, m are the estimated T-year growth factor and T-year flood quantile
respectively at a site i at mth repetition; X̂ iT and Q̂iT are the mean of those estimated
measures; X iT and QiT are the assumed true T-year growth factor and T-year quantile at
site i; N is the number of sites in the pooling group and M is the number of repetitions.
These expressions average the standard error over all the sites in the pooling group to
take heterogeneity in the pooling group into account. It should be noted that the actual
standard error for any individual site could be either smaller or larger than the calculated
value depending on the at site value of L-Cv and L-skewness. The extreme values in the
Box plots illustrate the range that could occur.
Steps 1 to 8 were repeated for each of the 85 stations and the percentage standard errors
for XT and QT are presented in box plot form in Figure 13.3 and Figure 13.4. The same
remarks apply to the interpretation of these plots as for those of the at-site standard
errors above.
The values of percentage se for XT vary from 1.1 to 1.2 at T = 10 to 1.8 to 2.0 for T =
100 while the corresponding percentage se values for QT remain relatively constant over
the whole range of T value, varying between from 3.7 to 8.5 at T = 10 and from 4.0 to
9.0 at T = 100.
The simulation procedure described above was also applied with the GEV distribution
used instead of EV1 in steps 2 to 6. Because the shape parameter k can be very variable
and hence unreasonably large or small when estimated from individual flow records a
single value of k = -0.1 was used in all simulations. This is a conservative (worst case
from the hydrological point of view) choice in the context of Irish flood data. In step 4
L-Skewness and average L-Skewness is calculated as well as L-Cv. In step 5 the GEV
growth curve parameters β and k are calculated by equations (10. 6) and (10. 7); and in
step 6 the expression for XT given in equation (10. 5) is used.
The results are presented, in box plot form, in Figure 13.5 and Figure 13.6. The values
of percentage se for XT vary from 1.4 to 1.8 at T = 10 to 3.7 to 5.0 for T = 100 while the
136
corresponding percentage se values for QT show a increasing trend, varying between 3.8
to 9.0 to at T = 10 and from 4.6 to 10.6 at T = 100.
These ranges for all return periods and both distributions, EV1 and GEV, are
summarized in tabular form in Table13.1. Since the 100 year flood is used for design
purposes far many major projects in Ireland, such as major bridges, it is worth noting
that its estimate's standard error can be considered to be approximately 10% of Q100.
Table13.1 Ranges of percentage se for growth factors, XT, and quantile estimates, QT.
T
5
10
25
50
100
500
Se(XT)%,EV1
0.7 to 0.8
1.1 to 1.2
1.4 to 1.6
1.6 to 1.8
1.8 to 2.0
2.1 to 2.3
Se(QT)%,EV1
3.7 to 8.4
3.7 to 8.5
3.8 to 8.7
3.9 to 8.9
4.0 to 9.0
4.1 to 9.3
Se(XT)%,GEV
0.8 to 1.0
1.4 to 1.8
2.3 to 3.0
3.0 to 3.9
3.7 to 5.0
5.4 to 7.3
137
Se(QT)%,GEV
3.7 to 9.0
3.8 to 9.0
4.1 to 8.9
4.3 to 9.6
4.6 to 10.6
5.7 to 12.2
Relative Standard Error of Growth Curve (Pooled/EV1-Simulation)
3.0
2.5
se(X ^T)/X T %
2.0
1.5
1.0
0.5
0.0
T=5
T=10
T=25
T=50
T=100
T=500
Figure 13.3 Box-plot of percentage standard error of growth factors estimates for EV1
return period.
Relative Standard Error of Quantile Estima te (Pooled/EV1-Simulation)
35
30
se(Q ^T)/Q T %
25
20
15
10
5
0
T=5
T=10
T=25
T=50
T=100
T=500
Figure 13.4 Box-plot of percentage standard error of quantile estimates for EV1
return period.
138
Rela tive Standard Error of Grow th Curve (Pooled/GEV-Simula tion)
12
10
se(X ^T)/X T %
8
6
4
2
0
T=5
T=10
T=25
T=50
T=100
T=500
Figure 13.5 Box-plot of percentage standard error of growth factors estimates for GEV
return period.
Relative Standard Error of Quantile estimates(Pooled/GEV-Simulation)
45
40
35
se(Q ^T)/Q T %
30
25
20
15
10
5
0
T=5
T=10
T=25
T=50
T=100
T=500
Figure 13.6 Box-plot of percentage standard error of quantile estimates for GEV
return period.
139
13.4 Standard Error of QT based on ungauged catchment estimate of
Qmed and pooled estimate of XT
Using the index flood method
Qˆ T = Qˆ med × xˆT
--- (13. 5)
The variance of the T-year event is approximated using Taylor series expansion as:
Var (Qˆ T ) = Var (Qˆ med × xˆT )
≈ E ( xˆT ) 2 Var (Qˆ med ) + E (Qˆ med ) 2 Var ( xˆT ) + 2 E (Qˆ med ) ⋅ E ( xˆT ) ⋅ cov(Qˆ med ⋅ xˆT ) --- (13. 6)
Ignoring effects arising from
-
bias in the sample median
-
bias in the sample L-moment ratio
-
covariance between the sample median and the estimated regional growth curve
leads to
Var (Qˆ T ) ≈ ( xT ) 2 | m Var (Qˆ med ) + (Qmed ) 2 | m Var ( xˆT )
--- (13. 7)
The expression for var(QT) is dominated by the var(Qmed) term as var(XT) affects only
the 3rd or 4th decimal point in the value of the expression. Hence the expression
effectively reduces to
se(QT ) ≈ X T ⋅ se(Qmed )
se(QT ) QT ≈ X T ⋅ se(Qmed ) ( X T ⋅ Qmed )
se(QT ) QT ≈ se(Qmed ) Qmed
se(QT ) QT ≈
0.36 Qmed
⋅
N Qmed
se(QT ) QT ≈
0.36
N
--- (13. 8)
140
Therefore, se(QT ) QT is independent of T but dependent on value of N associated with
the worth of a catchment descriptor based estimate of Qmed which is considered to be
equivalent to either 1 year of AM data or perhaps a little more (e.g. Nash and Shaw,
1975; FSR, 1975, I, eq 4.15; Hebson and Cunnane, 1987).
Hence for ungauged catchments, with N=1, se(QT ) QT =
0.36
= 0.36= 36%.
1
This result applies regardless of whether the growth curve is obtained using EV1 or
GEV. This may seem counter intuitive but it is caused by the fact that the uncertainty in
QT is dominated by the uncertainty in Qmed. The 36% value of percentage standard
error in this derivation is consistent with FSE =1.36 in equation (14. 9).
141
14 Guidelines for determining QT
14.1 Introduction
QT may be determined from at-site data if sufficient data exist and the required T value
is relatively small. In some instances although adequate at-site data may exist
interpretation may be problematical e.g. an irregular shaped probability plot, in which
case it may be necessary to use a combined at-site/pooling group approach.
For large return periods a combined at-site/pooling group approach is recommended.
When no data exist at the subject site then the site's Qmed must be estimated from the
catchment descriptor equation and enhanced where possible by use of suitable donor
site data.
While most hydrologists are greatly influenced by the appearance of at-site data on
probability plots when making choices about which procedures or estimates to adopt it
must be borne in mind that random samples from any particular statistical population
show tremendous inter-sample variation when displayed on probability plots. Some
such samples do not always display convincing straight line behaviour even when the
parent distribution is represented by a straight line on such a plot. Figure 4.1 show nine
random samples from an EV1 population with Cv = 0.33, a value which is slightly
higher than the average Cv for Irish A1 station data. Sample sizes are 25 and 50. It can
be seen that in the smaller samples the departures from straight line behaviour is more
striking than in the larger samples. Some of the sample of 25 would have been given a
low score in the assessment of probability plots described earlier in (Chapter 4).
However most of the size 50 samples do not depart too markedly from straight line
behaviour.
A series of factors must be borne in mind when selecting a design flood magnitude
including
142
(i)
The at-site or pivotal station's annual maxima probability plot on ordinary
and on log scales
(ii)
The range of variation that can occur between random samples drawn from a
single population – which emphasizes that a single at-site estimate may
differ from the true value.
(iii)
One or more pooled growth curves
(iv)
Preference for 2 parameter over 3 parameter distributions in at-site mode
(v)
The overall statistical superiority of the at-site Qmed plus the pooled growth
curve estimation method in homogenous pooling groups (c.f.Figure 10.1 (d))
(vi)
The robustness of the at-site Qmed plus pooled growth curve estimation
method to small departures from pooling group homogeneity
(vii)
Consideration of straight line versus concave downwards relations including
the issue of upper bounds on flood magnitudes
(viii)
Comparison of estimated QT and corresponding water levels with ground
truth i.e. does the proposed estimate make sense in the context of what has
been previously observed or not observed in the vicinity?
(ix)
Careful consideration of possible underestimation when the at-site Qmed
plus the pooled growth curve estimate of QT is smaller than some observed
at-site floods
(x)
The credibility of large outliers and how they might be viewed, assessed or
accommodated.
14.2 Determining QT from at-site data
The required flood quantile may be estimated from a single record of at-site data
providing
•
the record length N is at least 10 years long
•
the required quantile return period T is less than N, or not appreciably larger
than N and certainly not more than 2N
Ordinarily a 2 parameter distribution should be used, either the EV1 or LN2. In the case
of EV1 the parameters should be estimated by the method of probability weighted
moments (or the L-moment equivalent). In the case of the LN2 the parameters should
143
be estimated from the mean and standard deviation of the logarithms of the data. Use of
L-moments, as in the EV1 case, is statistically efficient as well as being simple to use.
Use of ordinary of the logarithms of the data, as in the LN2 case, is also statistically
efficient and simple to use and indeed simpler than use of L-moments in this case.
It is assumed that the basic statistics of the at-site data will have been calculated and that
dimensionless statistics have been examined to see where they fall in the range of
observed values among all gauged catchments. In other words it is assumed that the user
has carried out a thorough check on the at-site data in the context of the overall data set
for the country and especially in the context of data for similar type catchments. (We
may define similarity using the distance measure dij in pooling group nomenclature, e.g.
equation (10. 2) above).
The at-site data should be displayed on an EV1 based probability plot, using Gringorten
Plotting Positions, and also on a lognormal probability plot, using Blom Plotting
Positions as follows:
Gringorten Plotting Positions:
y i = − ln(− ln( Fi )) where Fi =
i − 0.44
, i = 1,2,........N
N + 0.12
--- (14. 1)
Blom Plotting Positions:
y i = Φ −1 ( Fi )) where Fi =
i − 3/8
, i = 1,2,........N
N + 1/ 4
--- (14. 2)
and Φ(y) is the Standardised Normal Distribution Function.
EV1 parameters u and α are estimated as
α =
2 M 110 − M 100
L2
=
ln(2)
ln(2)
--- (14. 3)
144
u = M 100 − 0.5772α = L1 − 0.5772α
--- (14. 4)
Then
QT = u + αyT
--- (14. 5)
where yT = − ln(− ln(1 −
1
))
T
LN2 parameters are estimated as
µz =
1 n
∑ z i where zi = log10(Qi)
n i =1
--- (14. 6)
1/ 2
 1 n

σ z =
( zi − z ) 2 
∑
 n − 1 i =1

--- (14. 7)
Then
QT = 10 ZT
--- (14. 8)
1
where Z T = Φ −1 (1 − ) and Φ −1 ( F ) is the standardised inverse Normal variate.
T
14.3 Determining QT from Pooled data
14.3.1 Data available at the subject site
14.3.1.1
Determining Qmed
If a gauging station exists at the subject site then Qmed is obtained from the AM series
of floods at that site.
If the record is short Qmed so obtained is subject to larger standard error than if the
record is long. However since gauged data are so much better that information obtained
from a catchment descriptor based formula it is recommended to base the Qmed value
on the observed data even if the record is short. The data of upstream and downstream
gauging stations on the same river as well as on other neighbouring stations should be
checked to see if the Qmed values obtained from the same period of record as those at
145
the subject site are smaller or larger than the long term Qmed at these sites. The ratio of
long term Qmed to short term Qmed could be applied to the subject site's Qmed as a
correction factor.
14.3.1.2
Determining XT
The growth factor should be determined from that the data of a pooling group selected
by means of the distance measure dij of equation (10. 2) or (12. 4) as desired. If the 5T
rule is applied to the 100 year flood estimation then a minimum of 500 station years of
data should be included in the pooling group. It is suggested that such a pooling group
be used for all return periods rather than constructing a separate group for every return
period of interest. If the hydrologist wishes to depart from this advice and to use
geographical regions or exclude certain types of catchment then the reasons for doing so
will be expected to be documented. Consideration must also be given to choice of a 2
parameter or a 3 parameter distribution in line with the discussion below in Section 14.6
below. In the event that the at-site estimate of Q-T relation is steeper that the pooled one
then consideration will have to be given to using a combination of the at-site estimate
and the pooled estimate for design flood estimation. While there is a possibility that this
could lead to over design it avoids the less desirable outcome of underdesign.
14.3.2 Subject site is ungauged
14.3.2.1
Determining Qmed
The value of Qmed will have to initially be obtained from the catchment descriptor data
using either the WP2.3 Rural Equation
Qmed rural = 1.083 × 10 −5 AREA 0.938 BFIsoils −0.88 SAAR 1.326 FARL2.233 DRAIND 0.334
× S1085 0.188 (1 + ARTDRAIN 2) 0.051
(14. 9)
or a modified version which takes urbanization into account through an urban
adjustment factor, UAF. The Factorial Standard Error (FSE) of the above equation is
1.36.
Use of a donor catchment, as described above in Section 9.3, is recommended or in the
absence of a donor catchment on the same river a suitably chosen analog catchment
146
might be used. However an unsuitable analog catchment may add very little useful
information. The equation for Qmed is applied with the catchment descriptors of the
donor catchment and the adjusted Qmed is obtained at the subject site by use of
equation (9. 2).
As shown in Section 9.3 use of equation (9. 2) introduces a noticeable improvement in
the estimate of Qmed.
14.3.2.2
Determing XT
The same remarks apply as in the Section 14.3.1.2 above.
14.4 Some examples
Example 1 – Data series shows good straight line behaviour on a probability plot
60
25016 RIVER CLODIAGH @ RAHAN
50
A MF data
A t-site EV 1 fit
AMF(m3/s)
40
Pooled EV 1 fit
EV 1
pooled GEV fit
30
20
10
2
0
-2
-1
0
5
1
10
2
25
50
EV31 y
4
100
500
5
6
7
8
Figure 14.1 At-site and pooled quantiles for station 25016
A good example of such behaviour is shown by the River Clodiagh at Rahan (Stn No
25016) with 42 years of record. In Table 4.3 this station was assigned scores of 4 and
L2 in the linear and non-linear pattern assessments where 4 means “good agreement
with straight line” and L2 means “little deviation from straight line”. The Cv = 0.22 and
147
Hazen Skewness = 0.63 which is less than the theoretical value of 1.14 in the EV1
distribution. It seems reasonable to suggest that the 50 year or even 100 year return
period flood could be estimated by the EV1 distribution fitted to the at-site data in this
case. The standard error of Q100 in such a case would be approximately 6% to 10%,
indicated in Figure 13.1 providing the underlying parent distribution is EV1. The at-site
Cv is lower than the national average. The pooled growth curves shown in Figure 14.2
are in general agreement with the at-site data and in this case the pooled estimate should
be used for design flood determination.
Example 2 – Data series shows good straight line behaviour on a probability plot but
lack of agreement between at-site and pooled estimates
A very good example of such behaviour is shown by the River Ryewater at Leixlip (Stn
No 9001) with 48 years of record. The Cv = 0.44 and Hazen Skewness = 1.17 which is
practically equal to the theoretical value of 1.14 in the EV1 distribution. It seems
reasonable to suggest that the 50 year or even 100 year return period flood could be
estimated by the EV1 distribution fitted to the at-site data in this case. The standard
error of Q100 in such a case would be approximately 4% to 8%, indicated in Table13.1
providing the underlying parent distribution is EV1. The at-site Cv is higher than the
national average and higher than that of other stations in the hinterland. Hence the
pooled growth curves are lower than the at-site curve which is shown in Figure 14.2.
However much faith is place in the superiority of the pooled estimate in general, it is a
fact in this case that the pooled estimate of Q50 has been exceeded 5 times in 48 years
already. This evidence cannot be overlooked when a final design flood is being selected
even for long return periods. It could be that the observed sample displays a steepness
on the probability plot that is at the upper end of the scale of steepness and that the
sample actually has come from a population with a less steep growth curve i.e. with
lower Cv. However no practical design project would ignore the at-site data in such a
case.
148
140
9001 RIVER RYEW ATER @ LEIXLIP
120
A MF data
A t-site EV 1 fit
AMF(m3/s)
100
Pooled EV 1 fit
Pooled GEV f it
80
60
40
20
2
5
10
25
50
100
500
0
-2
-1
0
1
2 EV 1 y 3
4
5
6
7
8
Other very good examples of straight line behaviour are shown by 14006 Barrow at
Pass Bridge (N = 51, Cv = 0.20 and Hazen Skewness = 1.46) and 26007 Suck at
Bellagill (N = 53, Cv = 0.19, Hazen Skewness = 1.05). Because these have low Cv
values the pooled growth curves are lower than the at-site one.
Caveat: Random samples drawn from an EV1 distribution do not always display
straight line behaviour on probability plots. The likelihood of this happening decreases
as the sample size gets larger as seen in Figure 4.1(a, b). Conversely some random
samples drawn from non-EV1 distributions could display straight line behaviour on
probability plots. The proportion which does so decreases of course as the departure of
the parent distribution from EV1 increases.
Example 3 – Data series shows concave upward behaviour on a probability plot or
displays one or more high outliers.
A very good example of such behaviour is shown by the River Dodder at Waldron's
Bridge (Stn No 9010) even though it has a relatively short 19 years of record. Even if
the underlying parent growth curve is a straight line the short record makes it much
more likely to observe departure from linearity on the probability plot. The Cv = 0.86
and Hazen Skewness = 3.24, both of which are among the highest values recorded
among the entire FSU data set. While a three parameter GEV distribution provides a
149
good graphical fit (see in Figure 14.3) in this case it has to be borne in mind, from a
theoretical point of view, that GEV at-site estimates have high standard error especially
with short data sets. That means that other similar sized later records might not show
the same pattern or be so steep. Hence, notwithstanding the good at-site fit, it would
normally be recommended that a regional pooling estimate be used for flood estimation
at this site except for low, T < 25 years. Because most other stations included in the
pooling group will more than likely have smaller Cv and skewness than that of the atsite data the pooling group based growth curve is lower than the at-site growth curve.
However the pooled growth curve looks extremely low in comparison with the observed
data.
The hydrologist then has to balance the weight of evidence provided by several
publications relating to the advantages of the regional pooling method over the at-site
method and his/her own beliefs about the representative nature of the observed data and
whether the behaviour shown could be repeated in another similar length of record. For
instance the at-site GEV estimate of Q100 ≈ 350 cumec. One has to ask if this is
credible. The largest flood on record (269 cumec) was caused by Hurricane Charlie in
1986 and its counterpart on the neighbouring Dargle catchment had earlier been
equalled twice in the previous 80 years. Undoubtedly therefore the 269 cumec on the
Dodder could occur again but this does not necessarily support the figure of Q100 = 350
cumec. On the other hand the pooling group estimate of Q100 is seen on the plot to have
been exceeded 4 times in 19 years which indicates that in this case the pooling based
curve is definitely too low.
This sort of dilemma always presents itself when the subject site record has a Cv and
skewness which is considerably larger than the regional average, as has occurred above
in Example 2. In such cases a prudent approach has to be adopted where "Ground
Truth" cannot be overlooked. In such cases it would be wise not to place too much faith
in estimates of QT for large T.
150
400
9010 RIVER DODDER @ W ALDRON'S BRIDGE
350
A MF data
300
A tsite GEV f it
AMF(m3/s)
Pooled GEV fit
250
Pooled EV 1 fit
200
150
100
50
2
0
-2
-1
0
5
1
10
25
2 EV 1 y 3
50
100
4
500
5
6
7
8
Other very good examples of this behaviour are 29011 River Dunkellin at Kilcolgan
Bridge (N = 22, Cv = 0.30, Hazen Skewness = 3.37) and 36015 Finn at Anlore (N = 33,
Cv = 0.32, Hazen Skewness = 2.62).
It is noticeable that these three examples are based on records which considerable
shorter that the longest records available in the study and also that the Dodder
Catchment is more urbanized than many other Irish catchments.
Example 4 – Data series shows concave downward behaviour on a probability plot or
displays a number of high values which are almost equal and possibly some very low
values among the smallest values in the series.
A very good example of such behaviour is shown by the River Newport at Barrington's
Bridge (Stn No 25002) with 51 years of record. The Cv = 0.16 and Hazen Skewness = 0.65 both of which are among the lowest values recorded among the entire FSU data set.
While a three parameter GEV distribution provides a very good graphical fit (see in
Figure 14.4) in this case it has to be borne in mind, from a theoretical point of view, that
GEV at-site estimates have high standard error especially with short data sets although
51 years is not considered short. In addition the k value is positive which implies an
upper bound on the fitted distribution. This is not reasonable from a hydrological point
of view and while it might be satisfactory to use the fitted distribution up to return
periods of 25 years it is recommended that a regional pooling estimate be used for flood
151
estimation at this site. Because most other stations included in the pooling group will
more than likely have larger skewnesses than that of the at-site data the pooling group
based growth curve is steeper than the at-site growth curve. This may require that the
pooling based estimate for high return periods be interrogated further in the context of
"Ground Truth" considerations referred to above. It is also known that this river as well
as the neighbouring Malkear river has been embanked since the 1920’s and these
embankments have an effect on flood flows as they are overtopped in one out of every 5
years. See also further discussion below on the issue upper bounds.
160
AMF(m3/s)
25002 RIVER NEW PORT @ BARRINGTONS BRIDGE
140
A MF data
120
A tsite GEV f it
Pooled GEV fit
100
Pooled EV 1 fit
80
60
40
20
2
0
-2
-1
0
5
1
10
25
2 EV 1 y 3
50
100
4
500
5
6
7
8
Other very good examples of this behaviour are provided by 25021 River Little Brosna
at Croghan (N = 44, Cv = 0.14, Hazen Skewness = -0.13) and 34024 Pollagh at
Kiltimagh (N = 29, Cv = 0.0.16, Hazen Skewness = -2.095). It is noticeable that two of
these three examples are based on records which are considered to be among the longest
in the study.
Example 5 – Data series contains one large outlier or two large outliers which are
noticeably out of line from the other data points in a probability plot.
A very extreme example of such behaviour is provided by 8009, River Ward at
Balheary where 10 of 11 values available are less than 12 cumec and the largest is
152
recorded as 53.6 cumec which was a summer occurrence in the year 1992-93. The atsite and pooled quantile curves are given in Figure 14.5.
The small sample size
exacerbates the problem of trying to estimate a design flood for this location. This
station has by far the highest recorded skewness, 5.58, of the whole data set. It has to be
acknowledged that skewness measured from such a short record is unreliable. The
median value, which is not affected by the largest flood, is 5 cumec and even assuming
a conservative growth factor of 2.5, the Q100 estimate would be 12.5 cumec. The
probability of obtaining a largest value in 15 years which is more than 4 times the
estimated Q100 is virtually zero.
As a result of this statistical incompatibility there can be no unique recommendation
made in this case. The following points should be considered
•
The meteorological conditions leading to the 1993 summer storm should be
examined and the likelihood of these being repeated anywhere in the region
(including the subject site) should be assessed
•
Even if the rating curve or other features of the measurement are less than
satisfactory the water level achieved in the locality will be known. This could be
used as the basis of design of important works and floor levels of dwellings.
•
The hydrologist has to balance between the belief that the 1992 flood was so
large that it could never happen again or the belief that if it happened once it
could happen again. For instance the Hurricane Charlie 1986 flooding in Bray
was so large that many believed it was unrepeatable until it was discovered that
almost exactly similar events had occurred in 1904 and 1932.
Hence for small return period events a standard at-site Qmed and a pooled estimate of
XT could be used, but for long return periods the above points have to be taken into
account.
153
90
A tsite EV 1 fit
70
2
5
10
25
50
100
500
Pooled EV 1 f it
60
AMF(m3/s)
8009 RIVER W ARD @ BALHEARY
A MF data
80
Pooled GEV f it
50
40
30
20
10
0
-2
-1
0
1
2
EV 1 y 3
4
5
6
7
8
Further examples of outliers but not of such extreme nature are provided by 36021
River Yellow at Kiltybarden , by 36031 River Cavan at Lisdarn and by 26006, River
Suck at Willsbrook. The same points as made above apply to these cases also.
Example 6 – Data series shows irregular behaviour of one sort or another e.g an
elongated S shape or a noticeable change of slope between lower and upper segments of
the graph.
Examples are 9002 River Griffeen at Lucan (N = 25 years), 15001 Kings River at
Annamult ( N = 42 years) and 26008 River Rinn at Johnston's Bridge (N = 50 years).
The at-site and pooled quantile curves for these stations are displayed in Figure 14.6,
Figure 14.7 and Figure 14.8 respectively.
The River Griffeen is particularly problematical as there is a distinctive increase in slope
caused by the 4 largest floods in the record. It should be noted also that it has
experienced an increase in its urban fraction over the past 2 decades especially. Qmed
is 5.25 and again taking a conservatively large value of X100 = 2.5 gives Q100 = 13
cumec, a value which has been well exceeded 3 times in 25 years. Additional local
knowledge must be gained about the meteorological and physical conditions leading to
these floods. The observed largest water levels will also have to act as a guide as to
what design conditions to adopt.
154
The 15001 case is one of a probability plot with reverse curvature or elongated S shape.
However the overall appearance is too far from a straight line, in the context of Figure
4.1(a, b) and hence application of a pooled growth factor should provide a satisfactory
QT estimate.
The 26008 case is a more extreme version of 15001. While no single distribution could
describe the at-site probability plot the upper end is not too dissimilar to some of the
samples in Figure 4.1(a, b). Hence application of a pooled growth factor should provide
a satisfactory estimate of QT.
40
A MF data
35
AMF(m3/s)
A tsite EV 1 fit
30
Pooled EV 1 f it
25
Pooled GEV f it
20
15
10
5
2
0
-2
-1
0
5
1
10
25
2 EV 1 y 3
50
100
4
500
5
6
7
8
7
8
250
15001 RIVER Kings @ Annamult
AMF data
200
At-site EV1 fit
AMF(m3/s )
Pooled EV1 fit
150
Pooled GEV fit
100
50
2
5
10
25
50
3 y
EV1
4
100
500
0
-2
-1
0
1
2
155
5
6
60
26008 RIVER RINN @ JOHNSTON'S BRIDGE
A MF data
50
A t-site EV 1 fit
Pooled EV 1 fit
AMF(m3/s)
40
Pooled GEV f it
30
20
10
2
0
-2
-1
0
5
10
1
25
2 EV 1 y 3
50
100
4
500
5
6
7
8
14.5 Flood growth curves with an upper bound or without an upper
bound?
Three parameter distributions, such as GEV and GLO, have an upper bound if the shape
parameter k >0. This does not seem hydrologically realistic regardless of whether fitted
to at-site or pooled data because some unprecedented rainfall could occur in the future
which could have a runoff or routing mechanism different to all previous floods and
which could therefore produce a flood much in excess of the largest flood on record.
When k>0 a two parameter distribution growth curve (i.e. a straight line growth curve)
should be estimated instead.
In such a case it is nevertheless necessary to use
judgement, following assessment of all relevant physical factors and water levels
reached during previous floods, before a design flood can be specified. There is a
possibility that a straight line two parameter distribution, when extrapolated to estimate
very rare floods, may produce flood estimates that may seem implausibly large in the
context of all relevant physical factors and the hydrologist's experience i.e. positive bias
as in Figure 10.1(b). Hence the hydrologist will have to apply judgement to the
estimation process in such a case.
156
14.6 Choice between at-site and pooled estimates and between 2 and 3
parameter pooled estimates
In ordinary circumstances a three parameter distribution should not be used with at-site
data. An exception could be made if the data series is very long, say > 50 years, and the
required return period is small, say ≤ 25 years. A three parameter distribution is more
flexible and may give a better fit visually on a probability plot but the estimation of a 3rd
parameter has the effect of increasing the standard error of estimate of the estimated
quantile, see Figure 10.1(c). The contrary can also occur where a three parameter
distribution, fitted to individual at-site data, can give a Q-T relation that is not
intuitively acceptable because of extreme upwards or downwards curvature.
In the event that the probability plot of available at-site data shows a marked departure
from straight line behaviour, either by way of upward or downward curvature or
because of some form of an elongated S shape, then consideration has to be given to
using a pooling group estimate of the growth factor XT for use with the at-site value of
Qmed. However even this may not remove all doubts about the suitability of the
method chosen as the pooled growth curve may disagree significantly from whatever
general pattern is shown by the at-site data as in Examples 1 and 2 above. In such a
case a decision has to be made whether to trust the pooled growth curve as a matter of
good practice or principle, based on the well documented reduced standard error of
estimate and robustness of the pooling method, e.g. Figure 10.1(d). However when the
at-site data display Cv and skewness greatly in excess of the pooling group average it is
not always easy to trust the pooled growth curve.
If the available at-site record is long it may be worthwhile displaying the maximum of
non-overlapping pairs or triplets of years to see what kind of pattern is displayed on a
probability plot
If a very large flood is observed during the period of record the question arises as to
whether it should it over-ride any more modest estimate of QT obtained by a pooling
group approach or whether a weighted combination of the pooling group estimate and
the at-site estimate should be adopted. If a combination is used the weights to be given
157
to the two components of the combination cannot be specified by any rule based on
scientific evidence but must be chosen in an arbitrary, however one would hope a
reasonable, way.
In the case of a very large observed flood it is also possible, under certain assumptions,
to calculate the probability that such a large flood could occur. For instance if a sample
of 50 floods are drawn randomly from an EV1 distribution with Cv = 0.3, then the
probability that the largest flood would exceed 1.25Q100, where Q100 is the population
value of the 100 year return period flood, is less that 7% and the probability that it
would exceed 1.5Q100 is less than 1%.
In practice the true value of Q100 is unknown
and has to be replaced by an estimate which makes these percentage probabilities less
reliable. Nevertheless such calculations could be adapted to provide an informal test of
the hypothesis that the observed outlier is consistent with the assumed parent
population. [The quoted probabilities are based on the fact that if Q is EV1(u,α) then
Qmax in N years is EV1(u+αlnN,α) ].
If a data series shows negative skewness and 3 or 4 largest floods that differ from one
another by only a very small amount then the data series will usually exhibit downwards
curvature on a probability plot. Any 3 parameter distribution fitted to the at-site data
will usually have an upper bound which is usually not very much larger than the largest
recorded flood. However it is unreasonable to expect that there is an upper bound of
such modest size and in such a case the advice is to using the pooled growth curve
rather than the at-site one. If the pooled growth curve is also concave downwards then a
two parameter distribution should be fitted to the pooled data so as to avoid the upper
bound. However in that case care has to be taken so as not to specify design floods, for
values of T ≈ N, which are greatly in excess of the observed floods and judgement has
to be used in such cases. On the other hand some unknown large rainfalls could occur
in the future which would cause the catchment to respond differently to those storms
which caused the recorded floods and it might be unwise not to accept the two
parameter based estimates for higher return periods, T>>N.
158
15 Summary and Conclusions
15.1 Data
Data from approximately 200 gauging stations in the Republic of Ireland, from the
arcives of OPW, EPA and ESB were available. Of these 115 were of quality Grade A
while 67 were Grade B.
Data of 17 stations which had both pre and post drainage
records were also available. Summary statistics and probability plots were prepared for
all stations. However no further inferences were drawn from the Grade B stations data.
15.2 Descriptive Statistics
Examination of the data for trend and randomness shows that up to 10% of records
display some trend or lack of randomness. The usual assumption that annual maximum
floods are statistically independent and identically distributed,
was nevertheless
assumed as in most other national flood studies supported by the fact that 90% of
records did not show significant trend or lack of randomness.
The descriptive statistics show that Irish AM data have low Cv (average of 0.3) and low
skewness (average H-Skewness of 1.0) whether measured by the traditional statistics or
by L-moment statistics. Examination of probability plots and moment ratio diagrams
suggest that among 2-parameter distributions the Extreme Value Type I (EV1) and
lognormal (LN2) are the most appropriate, the exception being for a number of stations
that display very low skewness values which in some cases is caused by the 3 or 4
largest values in the record not being appreciably different from another. Explanations
for this latter phenomenon were sought by considering flood volumes and by referring
to the catchment descriptor, Flood Attenuation Index, but to no avail.
15.3 Seasonal Analysis
Seasonal analysis shows that two thirds of AM flows occur during the winter months of
October to March and at 11 stations no AM floods occurred during the summer months.
July is the least likely month in which AM floods occur while August and September
159
are the most likely summer months to have AM floods. At 20 out of the 202 stations
examined the largest flow on record occurred during summer, 8 of them in August,
while 45% of stations had their largest flood on record during December.
15.4 m-year Maxima
As well as annual maxima, maxima of no-overlapping 2 and 3 year periods were also
examined on probability plots. The pattern seems to be that the behaviour of m-year
maxima i.e. straight line, concave upwards or concave downwards, is very much
predictable from the annual maximum behaviour and no new more stable pattern could
be gleaned from the m-year maxima than from the annual maxima.
15.5 Estimation of Design Flood
The estimation of the T year flood quantile is considered for each of the full range of
possibilities that can arise in practice, from gauged to ungauged catchments and from
short to long flow records. The index flood method is recommended whereby QT is
expressed as
QT = Qmed XT
where QT is the flood of return period T years, Qmed is the median of the annual
maximum at the subject site and XT is the growth factor appropriate to the subject site,
estimated either from at-site data or from data of an appropriate homogeneous pooling
group.
Where at-site data are available Qmed is estimated from those data. If the data record is
long the Qmed obtained from it is used on its own. If the available flow record is short
or if Qmed is obtained, in the ungauged case, from a catchment descriptor based
equation then the Qmed may be adjusted with the assistance of data from a suitable
donor or analog site. Tests conducted on the data of 38 gauging stations located in 10
different catchments were used to assess the efficacy of different data transfer methods
for improving the Qmed estimate. Data transfer from downstream donor stations was
found to be a little superior to data transfer from upstream donor stations and both were
160
found to be superior to use of analog stations. Use of the regression based adjustment
technique was found to be superior to use of area based adjustment. This reflects the
fact that flood magnitude depends on more than catchment area alone.
Estimation of XT may be based on at-site data if a sufficiently long data record exists at
the subject site.
Otherwise XT is estimated from the dimensionless L-Cv and L-
Skewness values obtained by averaging these quantities obtained over the stations in a
homogenous pooling group of stations. The members of the pooling group are chosen
with the help of a Euclidean distance or similarity measure, dij. Tests have been carried
out and reported on into the effect on the estimated value of XT of catchment area, peat
coverage, lakiness (as measured by FARL), geographical location and period of record
varying from the 1950s to the 1990s. None of these were judged to be sufficiently
influential to make special provision for them. Also, tests were carried out and reported
on about the effectiveness of different combinations of catchment descriptors in the
definition of dij and the most effective ones were AREA, SAAR and BFIsoils.
The standard errors associated with estimates of XT and QT have also been investigated.
The standard error of QT estimated by the index flood method is dominated by se(Qmed)
and se(QT), expressed as a percentage, varies only slightly with T. When Qmed is
estimated from at-site data and XT is estimated from a pooling group containing
approximately 500 station years of data then se(QT) is of the order of 5% to 10 % QT
regardless of return period. If Qmed is estimated from a catchment descriptor based
formula alone and XT is estimated from a pooling group containing approximately 500
station years of data then se(QT) is of the order of 36% QT.
Chapter 14 deals with guidelines for the estimation of QT, from both at-site and pooled
data. Several points that need to be taken into account in practical cases are outlined
and the point is emphasised that the blind use of a prescribed method can sometimes
lead to a QT estimate which is not always in accordance with “ground truth”. This is
especially the case when the at-site data have a higher than average Cv or skewness. A
number of examples are discussed, most of which throw up practical problems of the
type met in practice where difficult choices have to be considered. Discussion is also
provided on the issues of flood distributions with upper bounds and on the choice
between two parameter and three parameter distributions. A number of examples are
161
discussed, most of which throw up practical problems of the type met in practice where
difficult choices have to be considered. In some of these cases the user may be
recommended to consider using an at-site based estimate in preference to the generally
recommended regional pooling based method.
What this indicates is that flood
estimation cannot be reduced to a single strict formula based procedure but that
individual analysts must make choices which depend on the problem circumstances and
which take into account their own knowledge and experience.
162
References
Burn, D.H.,(1990),Evaluation of regional flood frequency analysis with a region of
influence approach. Wat. Res. Res. 26(10), 2257-2265.
Dalrymple,T., (1960), Flood frequency methods. U.S. Geol. Survey, Water Supply
Paper 1543 A, pp. 11–51, Washington.
Foster, H. A. (1924), Theoretical frequency curves and their application to engineering
problems. Trans. ASCE, 87, 142-173.
Flood Studies Report (1975), Natural Environment Research Council, London, vol. I.
Flood Estimation Handbook (FEH), (1999), Institute of Hydrology, U.K.
Fiering, (1963), Use of correlation to improve estimates of the mean and variance.
Statistical studies in Hydrology, Geological Survey professional paper 434-C, US Gov.
Printing Office.
Greis, N.P. and Wood, E.F., (1981), Regional flood frequency estimation and network
design. Water Resour. Res. , 17(4) , 1167 – 1177.
Hazen, A.,(1930), Flood flows, John Wiley, New York.
Hebson, C.S. and Cunnane, C., (1988), Assessment of Use of At-site and Regional
Flood Data for Flood Frequency Estimation. In V.P. Singh (Editor), Hydrologic
Frequency Modelling, D. Reidel, Publ. Co., Dordrecht, pp. 433-448.
Hosking, J.R.M., Wallis, J.R. and Wood, E.F., (1985a), An appraisal of the regional
flood frequency procedure in the U.K. Flood Studies Report. Hydrol. Sci. J., 30(1), 85109.
Hosking, J.R.M., Wallis, J.R. and Wood, E.F., (1985b), Estimation of the generalised
extreme value distribution by the method of probability weighted moments.
Technometrics, 27(3), 251-261.
Hosking, J.R.M. and Wallis, J.R., (1997), Regional flood frequency analysis: An
approach based on L-Moments, Cambridge Univ. Press.
Irish Academy of Engineering (2007), Ireland at Risk – Impact of Climate Change on
the Water Environment. Proc Workshop held at RDS, Ballsbridge, Dublin, May
2007(www.iae.ie/publications).
Kjeldson, T., Jones,D., Bayliss, A., Spencer, P., Surendran, S., Laeger,S., Webster, P.
and McDonald, D.,(2008), Improving the FEH Statistical Method . In: Flood &
Coastal Management Conference 2008, University of Manchester, 1-3 July 2008.
Environment Agency/Defra. Available at UK Natural Environment Research Council
Open Research Archive, http://nora.nerc.ac.uk/3545/ , 10 pages.
163
Lettenmaier, D.P., Wallis, J.R. and Wood, E.F.,(1987), Effect of heterogeneity on Flood
frequency estimation. Water Resour. Res., 23(2), 313-323.
Lu & Stedinger (1992), Variance of two-and three-parameter GEV/PWM quantile
estimators: formulae, confidence intervals, and a comparision. J. of Hydrol., 138(1-2),
247-267.
Nash, J.E. and Shaw, B.L.,(1965), Flood frequency as a function of catchment
characteristics. Institution of Civil Engineers, Symposium on River Flood Hydrology,
(published 1966), 115-136.
OPW, (1981), Estimating Flood Magnitude/ Return Period Relationships and the Effect
of Catchment Drainage.
WMO, (1989), Operational Hydrology Report no. 33: Statistical Distributions for Flood
Frequency Analysis.
164
Appendix 1
TREND ANALYSIS
ANNUAL MAXIMUM FLOWS IN IRISH RIVERS
See separate volume
165
Appendix 2
Assessment of effectiveness of Donor and Analog
Catchments in adjusting the Catchment Descriptor
based estimate of Qmed
166
St.No.
Obs
Qmed
Area km2
Adj.Qmed
%obs
Qmed
Area Adjustment
16008
16009
Subject_site
16002
Donor_up
16004
52.7
512
52.7
21.4
236
46.4
92.3
1120
42.2
100
88
80
Average difference via donors
Donor_down3
16011
Analog
25021
162.2
1602
51.8
223.0
2173
52.5
28.6
493
29.7
48.3
98
100
56
92
0
20
8
8
16002 Suir @ Be ak s tow n, Are a=512 k m 2
(Are a Adjus tm e nt)
120
100
1602
2173
236
%obs Qmed
1120
80
493
60
40
20
0
Donor_up
Donor_dow n1
Donor_dow n2
Donor_dow n3
Data Transf er Category
(a)
167
A nalog
Regression
Regression
16008
16009
92.3
162.2
1120
1602
43.3
53.0
Subject_site
16002
52.7
512
52.7
Donor_up
16004
21.4
236
34.4
100
65
82
1
35
17
8
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs
Qmed
Donor_down3
16011
223.0
2173
59.7
Analog
6013
27.3
309
40.5
Regression
113
77
92
101
16002 Suir @ Be ak s tow n, Are a=512 k m 2
(Re gre s s ion Adjus tm e nt)
120
2173
1602
100
%obs Qmed
1120
309
80
236
60
40
20
0
Donor_up
Donor_dow n1
Donor_dow n2
Donor_dow n3
A nalog
Regression
Figure A2. 1 Performance of data transfer method of estimating Qmed at Station 16002,
Suir at Beakstown, (a)using area based adjustment (b) using regression based adjustment.
168
48.3
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs Qmed
Subject_site
16008
92.32
1120
92.32
100
Donor_up2
16004
21.4
236
101.43
110
Area Adjustment
Donor_up1 Donor_down1
16002
16009
52.7
162.2
512
1602
115.20
113.41
125
123
Donor_down2
16011
223
2173
114.94
124
Analog
14006
80.52
1096
82.3
89
10
25
20
12
16008 Suir @ New Bridge, Area=1120 km2
(Area Adjustment)
140
512
120
1602
2173
236
%obs Qmed
100
1096
80
60
40
20
0
Donor_up2
Donor_up1
169
Analog
Regression
Regression
103.1
112
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs Qmed
Subject_site
16008
92.32
1120
92.32
100
Donor_up2
16004
21.4
236
73.35
79
Donor_up1 Donor_down1
16002
16009
52.7
162.2
512
1602
112.34
112.97
122
122
Donor_down2
16011
223
2173
127.44
138
Analog
26007
88.1
1207
83.4
90
21
38
26
12
16008 Suir @ New Bridge, Area=1120 km2
(Regression Adjustm ent)
160
2173
140
512
1602
112
%obs Qmed
120
100
1207
236
80
60
40
20
0
Donor_up2
Donor_up1
Analog
Regres s ion
Suir at Newbridge, (a)using area based adjustment (b) using regression based adjustment.
170
Regression
103.1
112
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs qmed
Subject_site
18050
124.5
244.6
124.5
100
Donor_up
18016
79.7
113
172.4
138
Area Adjustment
18048
18006
220
286
881
1058
61.1
66.1
49
53
Donor_down3
18002
344
2333
36.0
29
Analog
16003
29.9
246
29.8
24
38
71
52
33
18050 Blackw ate r @ Duarrigle, Are a=244.6 km2
(Are a Adjustment)
160
140
113 km2
%obs qmed
120
100
80
60
881 km2
1058 km2
40
2333 Km2
246 km2
20
0
Donor_up
(a)
171
Analog
Regression
Regression
82.9
67
St.No.
Obs Qmed
Area
Adj.Qmed
%obs qmed
Subject_site
18050
124.5
244.6
124.5
100
Donor_up
18016
79.7
113
121.2
97
18048
18006
220
286
881
1058
100.9
130.1
81
104
Donor_down3
18002
344
2333
124.2
100
Analog
35011
115.3
293
76.1
61
0
19
7
33
18050 Blackw a ter @ Duarrigle , Are a=244.6 km2
(Re gression Adjustment)
160
140
%obs qmed
120
100
1058 km 2
113 km 2
2333 Km 2
881 km 2
80
67
293
60
40
20
0
Donor_up
Analog
Regres s ion
Blackwater at Duarrigle, (a)using area based adjustment (b) using regression based
adjustment.
172
Regression
82.9
67
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs qmed
Subject_site
24008
119.1
805
119.1
100
Area Adjustment
Donor_up Donor_down1
24004
24082
50.6
140
246
764
165.7
147.5
139
124
Analog
36010
66.8
774
69.4
58
Regression
96.2
81
24
39
31
19
24008 M aigue @ Cas tle robe rts , Are a=805 k m 2
(Are a Adjus tm e nt)
160
140
246
764
%obs qmed
120
100
80
774
60
40
20
0
Donor_up
Donor_dow n1
A nalog
173
Regression
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs qmed
Subject_site
24008
119.1
805
119.1
100
24004
24082
50.6
140
246
764
113.1
139.1
95
117
Analog
14006
80.5
1063
73.6
62
Regression
96.2
81
5
17
11
19
24008 M aigue @ Cas tle robe rts , Are a=805 k m 2
(Re gre s s ion Adjus tm e nt)
160
140
764
%obs qmed
120
100
246
80
1063
60
40
20
0
Donor_up
Donor_dow n1
A nalog
Regression
Maigue at Castleroberts, (a) using area based adjustment (b) using regression based
adjustment.
174
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs Qmed
Subject_site
26002
56.6
626
56.6
100
Area Adjustment
26006
26005
24.2
93.2
182
1050
83.3
55.6
147
98
Donor_down2
26007
88.2
1184
46.6
82
Analog
35005
75.4
642
73.5
130
2
47
22
30
26002 Suck @ Rockw ood, Area=626 km2
(Area Adjustment)
160
182
140
642
642
Analog
Regres s ion
%obs Qmed
120
1050
100
1184
80
60
40
20
0
Donor_up
175
Regression
73.8
130
St.No.
Obs Qmed
Area km2
Adj.Qmed
%obs Qmed
Subject_site
26002
56.6
626
56.6
100
Donor_up Donor_down1 Donor_down2
26006
26005
26007
24.2
93.2
88.2
182
1050
1184
54.7
53.9
59.7
97
95
105
Analog
30007
62.9
152
75.1
133
Regression
73.8
130
3
5
4
30
26002 Suck @ Rockw ood, Are a=626 km2
(Re gression Adjustme nt)
160
140
152
%obs Qmed
120
100
1184
182
1050
80
60
40
20
0
Donor_up
Analog
Regres s ion
suck at Rockwood, (a) using area based adjustment (b) using regression based adjustment.
176
Appendix 3
Flood volumes associated with largest peaks on convex
probability
177
7006 RIVER MOYNALTY @ FYANSTOWN
Annual Maximum Floods 1986 to 2004.(no missing years)
Mean
26.726
Median
27.933
Std.Dev.
5.493
A(km2)=
176.00
N=
CV
0.206
HazenS.
46
-1.088
40
EV1
35
AMF(m3/s)
30
'00
'04
'93
'95
w inter peak
summer peak
25
20
15
10
5
2
5
10
25
50
100
500
0
-2
-1
0
40
1
2
EV1 y
3
4
5
6
Hydrograph during Maximum Flood in the year 1993
35
Discharge (m3/s)
30
25
20
15
10
5
0
02/10/1993 00:00
04/10/1993 00:00
35
06/10/1993 00:00
08/10/1993 00:00
Tim e
10/10/1993 00:00
12/10/1993 00:00
30
Discharge (m 3/s)
25
20
15
10
5
0
22/11/1995 00:00
24/11/1995 00:00
26/11/1995 00:00
28/11/1995 00:00
30/11/1995 00:00
Tim e
178
02/12/1995 00:00
04/12/1995 00:00
06/12/1995 00:00
7
35
30
Discharge (m3/s)
25
20
15
10
5
0
31/10/2000 00:00
02/11/2000 00:00
25
04/11/2000 00:00
06/11/2000 00:00
Tim e
08/11/2000 00:00
10/11/2000 00:00
Volume of hydrographs of different year during max peak
2000
20
1995
Mill. cu. meter
1993
15
10
5
0
1
2
Days
7
14
30
Comments:
1. One day flood volumes are noticeably similar for these three events. For 2, 7 ,14
and 30 days the 1995 event shows slightly higher volume than the 2000 event
but the 1993 event, that with the largest peak, gives the least volume among the
three; in fact it is 50% of the volume of 1995 event for the 7, 14 and 30-day
volume.
2. The 2000 and 1995 flood events display multiple peaks probably explain why
they have greater volumes than the 1993 event.
179
A(km2)=
Mean=
141.820
Median=
150.378
Std.Dev.=
30.597
298.00
N=
CV=
0.216
HazenS.=
'97
'85
-0.996
EV1
200
'98
'68
150
A M F ( m 3 /s )
46
w inter peak
summer peak
100
50
2
5
10
25
50
100
500
0
-2
-1
0
1
2
EV1 y
3
4
5
6
7
200
Y ellow zone= V ol. of 2day
Blue z one= V ol. of 1w eek
Discharge (m3/s)
150
100
50
0
22/08/1986
00:00
23/08/1986
00:00
24/08/1986
00:00
25/08/1986
00:00
26/08/1986
00:00
Time
27/08/1986
00:00
28/08/1986
00:00
29/08/1986
00:00
30/08/1986
00:00
200
Blue zone= V ol. of 1w eek
Discharge (m3/s)
150
100
50
0
14/11/1997
00:00
15/11/1997
00:00
16/11/1997
00:00
17/11/1997
00:00
18/11/1997
00:00
Time
180
19/11/1997
00:00
20/11/1997
00:00
21/11/1997
00:00
22/11/1997
00:00
100
Hydrograph during Max imum Flood in the ye ar 1998
Red zone= V ol. of 1day
Blue zone= Vol. of 1w eek
90
80
Discharge (m3/s)
70
60
50
40
30
20
10
0
15/09/1999 00:00
17/09/1999 00:00
50
Mill. cu. meter
21/09/1999 00:00
Time
23/09/1999 00:00
25/09/1999 00:00
27/09/1999 00:00
45
1998
40
1997
35
19/09/1999 00:00
1985
30
25
20
15
10
5
0
1
2
Da ys
7
14
30
Comments:
1. There is a noticeable growth of volume with peak flood magnitude at 1 and 2day durations and slight growth of volume at 7 days. However this is reversed
for the larger durations where the largest flood peak (1985) is associated with the
smallest volume.
181
16008 RIVER SUIR @ NEW BRIDGE
Mean=
90.663
Median=
92.323
Std.Dev.=
1120.00
CV=
0.125
'68
'60
11.350
N=
46
HazenS.=
-0.324
EV1 Plot
120
100
A M F ( m 3 /s )
A(km2)=
'87
'83
w inter peak
summer peak
80
60
40
20
2
5
10
25
50
100
500
0
-2
-1
0
120
1
2
EV1 y
3
4
5
6
7
Red z one= V ol. of 1day
100
Discharge (m3/s)
80
60
40
20
0
26/11/1960
00:00
28/11/1960
00:00
30/11/1960
00:00
140
02/12/1960
00:00
04/12/1960
06/12/1960
00:00
00:00
Tim e
08/12/1960
00:00
Hydrograph during Max im um Flood in the ye ar 1968
120
10/12/1960
00:00
12/12/1960
00:00
14/12/1960
00:00
Discharge (m3/s)
100
80
60
40
20
0
16/12/1968
00:00
18/12/1968
00:00
20/12/1968
00:00
22/12/1968
00:00
182
24/12/1968
00:00
Time
26/12/1968
00:00
28/12/1968
00:00
30/12/1968
00:00
01/01/1969
00:00
120
100
Y ellow z one= V ol. of 2day
Discharge (m3/s)
80
60
40
20
0
29/01/1984
00:00
31/01/1984
00:00
120
02/02/1984
00:00
04/02/1984
00:00
06/02/1984
08/02/1984
00:00
00:00
Tim e
10/02/1984
00:00
12/02/1984
00:00
100
14/02/1984
00:00
16/02/1984
00:00
Discharge (m3/s)
80
60
40
20
0
26/01/1988
00:00
28/01/1988
00:00
250
30/01/1988
00:00
01/02/1988
00:00
03/02/1988
05/02/1988
00:00
00:00
Tim e
07/02/1988
00:00
09/02/1988
00:00
11/02/1988
00:00
1987
200
1983
1968
Mill. cu. meter
1960
150
100
50
0
1
2
Days
7
14
30
Comments:
1. The flood hydrographs of these four events display very similar shape and that reflect
to the calculation of volume as well. There is hardly any difference found among the
flood volume of these four events particularly at the 1, 2 and 7-day durations.
183
16009 RIVER SUIR @ CAHIR PARK
Annual Maximum Floods 1953 to 2004.(no missing year)
Mean=
159.294
Median=
162.214
Std.Dev.=
A(km2)=
1602.00
CV=
0.172
27.397
N=
46
HazenS.=
-0.409
250
EV1 Plot
200
AMF(m3/s)
'04
'68 '00
'89
'60
150
100
50
2
5
10
25
50
100
500
0
-2
-1
250
0
1
2
EV 1 y
3
4
Hydrograph during Max im um Flood in the ye ar 1989
5
6
7
Discharge (m3/s)
200
150
100
50
0
30/01/1990
00:00
250
01/02/1990
00:00
03/02/1990
00:00
05/02/1990
00:00
07/02/1990
09/02/1990
00:00
00:00
Time
11/02/1990
00:00
Hydrograph during Max imum Flood in the ye ar 1960
13/02/1990
00:00
15/02/1990
00:00
Discharge (m3/s)
200
150
100
50
0
25/11/1960 27/11/1960 29/11/1960 01/12/1960 03/12/1960 05/12/1960 07/12/1960 09/12/1960 11/12/1960 13/12/1960 15/12/1960
00:00
00:00
00:00
00:00
00:00
00:00
00:00
00:00
00:00
00:00
00:00
Time
184
250
Discharge (m3/s)
200
150
100
50
0
29/10/2000
00:00
31/10/2000
00:00
02/11/2000
00:00
04/11/2000
00:00
06/11/2000
00:00
08/11/2000
00:00
10/11/2000
00:00
12/11/2000
00:00
Tim e
250
Y ellow z one= V ol. of 2day
Discharge (m3/s)
200
150
100
50
0
20/10/2004
00:00
22/10/2004
00:00
350
300
24/10/2004
00:00
26/10/2004
00:00
28/10/2004
30/10/2004
00:00
00:00
Tim e
01/11/2004
00:00
03/11/2004
00:00
05/11/2004
00:00
V olum e of hydrographs of diffe re nt ye ar during m ax pe ak
2004
Mill cu. meter
2000
250
1960
200
1989
150
100
50
0
1
2
Days
7
14
30
Comments:
1. This too displays similar kind of hydrographs and the volumes are also found
similar in magnitude particularly for the one and two-day volume.
2. At the 30-day duration, the event with the largest peak (1989) gives the largest
flood volume which is found more than double than the volume of 2004 event.
185
24013 RIVER DEEL @ RATHKEALE
A(km2)=
Annual Maximum Floods 1953 to 2004.(missing years: 63-68)
Mean=
95.213
Median=
101.757
Std.Dev.=
29.944
CV=
426.00
N=
0.314
HazenS.=
46
-0.908
160
EV 1 Plot
140
'94
120
'88 '80
'73
'98
AMF(m3/s)
100
80
60
40
20
2
5
10
25
50
100
500
0
-2
-1
0
1
2
EV 1 y
3
4
5
6
7
160
Hydrogr aph during M axim um Flood in the ye ar 1973
140
Discharge (m3/s)
120
100
80
60
40
20
0
26/11/1973
00:00
27/11/1973
00:00
28/11/1973
00:00
29/11/1973
00:00
30/11/1973
00:00
01/12/1973
00:00
02/12/1973
00:00
03/12/1973
00:00
04/12/1973
00:00
05/12/1973
00:00
06/12/1973
00:00
Tim e
160
140
Discharge (m3/s)
120
100
80
60
40
20
0
25/12/1998 00:00
27/12/1998 00:00
29/12/1998 00:00
31/12/1998 00:00
Tim e
186
02/01/1999 00:00
04/01/1999 00:00
160
Hydrograph during Maxim um Flood in the year 1980
140
Discharge (m3/s)
120
100
80
60
40
20
0
28/10/1980 00:00
30/10/1980 00:00
01/11/1980 00:00
140
03/11/1980 00:00
Tim e
05/11/1980 00:00
07/11/1980 00:00
09/11/1980 00:00
Hydrograph during Maxim um Flood in the year 1988
120
Discharge (m3/s)
100
80
60
40
20
0
15/10/1988 00:00
17/10/1988 00:00
19/10/1988 00:00
21/10/1988 00:00
23/10/1988 00:00
25/10/1988 00:00
27/10/1988 00:00
29/10/1988 00:00
Tim e
80
70
V olume of hydrograph of dif f erent years during max peak
1988
1980
Million cu.m
60
50
1998
1973
40
30
20
10
0
1
2
Days
7
14
30
Comments:
1. Not much difference is shown among the volumes of hydrographs of four events
for the one and two-day volume.
2. The two largest peak events, 1973 and 1998 are found to be delivered 30% more
volume than the volume of 1980 and 1988 events in the case of 7-day volume.
187
A(km2)=
Annual Maximum Floods 1977 to 2004.( no missing year )
Mean=
135.469
Median=
140.010
Std.Dev.=
35.757
CV=
764.00
N=
0.264
HazenS.=
46
-0.216
250
EV 1 Plot
200
'89
'88
AMF(m3/s)
'98
'00
150
100
50
2
5
10
25
50
100
500
0
-2
-1
0
250
1
2
EV 1 y
3
4
Hydrogra ph during Max imum Flood in the yea r 1989
5
6
7
Discharge (m3/s)
200
150
100
50
0
01/02/1990 00:00
03/02/1990 00:00
200
05/02/1990 00:00
07/02/1990 00:00
Time
09/02/1990 00:00
180
11/02/1990 00:00
160
Discharge (m3/s)
140
120
100
80
60
40
20
0
31/10/2000 00:00
02/11/2000 00:00
04/11/2000 00:00
06/11/2000 00:00
Time
188
08/11/2000 00:00
10/11/2000 00:00
12/11/2000 00:00
200
180
160
Discharge (m3/s)
140
120
100
80
60
40
20
0
14/10/1988
00:00
16/10/1988
00:00
200
18/10/1988
00:00
20/10/1988
00:00
22/10/1988
00:00
Time
24/10/1988
00:00
180
26/10/1988
00:00
28/10/1988
00:00
Red z one= Vol. of 1day
Yellow z one= Vol. of 2day
Blue z one= Vol. of 1week
160
Discharge (m3/s)
140
120
100
80
60
40
20
0
23/12/1998 00:00
25/12/1998 00:00 27/12/1998 00:00
160
140
29/12/1998 00:00 31/12/1998 00:00
Time
02/01/1999 00:00 04/01/1999 00:00
V olum e of hydrogr aphs of diffe r e nt ye ar during m ax pe ak
1998
1988
Mill cu. meter
120
100
2000
1989
80
60
40
20
0
1
2
Days
7
14
30
Comments:
1. Not much difference is shown among the volumes of hydrographs of these four
events for the one and two-day volume.
2. The highest peak event (1989) is found to be given the largest volume for the 7,
14 and 30-day volume and in the case of 30-day volume it gives 30% more flood
volume than the volume of the1998 event.
189
25017 RIVER SHANNON @ BANAGHER
Annual Maximum Floods 1950 to 2004.(no missing year)
Mean=
95.213
Median=
101.757
Std.Dev.=
A(km2)=
N/A
CV=
0.314
29.944
N=
46
HazenS.=
-0.908
700
EV 1 Plot
600
'94
AMF(m3/s)
500
'01 '89
'54
'99
400
300
200
100
2
5
10
25
50
100
500
0
-2
-1
0
600
1
2
EV 1 y
3
4
5
6
7
Hydrograph during Maximum Flood in the ye ar 1999
500
Discharge (m3/s)
400
300
200
100
0
16/12/1999 00:00
21/12/1999 00:00
26/12/1999 00:00
31/12/1999 00:00
05/01/2000 00:00
10/01/2000 00:00
Time
600
Hydrogra ph during Ma x imum Flood in the ye a r 1989
500
Discharge (m3/s)
400
300
200
100
0
22/01/1990 00:00
27/01/1990 00:00
01/02/1990 00:00
06/02/1990 00:00
11/02/1990 00:00
Time
190
16/02/1990 00:00
21/02/1990 00:00
26/02/1990 00:00
600
Discharge (m 3/s)
500
400
300
200
100
0
03/02/2002
00:00
05/02/2002
00:00
07/02/2002
00:00
600
09/02/2002
00:00
11/02/2002
00:00
13/02/2002
00:00
Time
15/02/2002
00:00
17/02/2002
00:00
19/02/2002
00:00
21/02/2002
00:00
23/02/2002
00:00
Hydrogra ph during Ma ximum Flood in the yea r 1994
500
Discharge (m 3/s)
400
300
200
100
0
21/01/1995 00:00
26/01/1995 00:00
31/01/1995 00:00
05/02/1995 00:00
10/02/1995 00:00
15/02/1995 00:00
Time
1400
1200
1000
Volum e of hydr ograph of diffe r e nt ye ar s during m ax pe ak
1994
2001
1989
Mill. cu. meter
1999
800
600
400
200
0
1
2
Days
7
14
30
Comments:
1. All the four events display similar kind of hydrographs and eventually they give
similar amount of flood volumes.
191
25021 RIVER LITTLE BROSNA @ CROGHAN
Mean=
28.028
Median=
28.578
Std.Dev.=
A(km2)=
493.00
N=
CV=
0.141
HazenS.=
3.939
46
-0.134
45
EV1 Plot
40
35
'00
AM F (m 3/s )
30
'94
'62
'99
w inter peak
summer peak
25
20
15
10
5
2
5
10
25
50
100
500
0
-2
-1
0
40
1
2
EV1 y
3
4
5
Hydrogra ph during Ma x imum Flood in the ye a r 1962
6
7
35
Discharge (m3/s)
30
25
20
15
10
5
0
31/10/1962 00:00
02/11/1962 00:00
40
04/11/1962 00:00
06/11/1962 00:00
Time
08/11/1962 00:00
35
10/11/1962 00:00
12/11/1962 00:00
Discharge (m3/s)
30
25
20
15
10
5
0
29/10/2000
00:00
31/10/2000
00:00
02/11/2000
00:00
04/11/2000
00:00
06/11/2000 08/11/2000
00:00
00:00
Time
192
10/11/2000
00:00
12/11/2000
00:00
14/11/2000
00:00
40
35
30
Discharge (m3/s)
25
20
15
10
5
0
16/12/1999
00:00
18/12/1999
00:00
20/12/1999
00:00
40
22/12/1999
00:00
24/12/1999
00:00
26/12/1999
00:00
Time
28/12/1999
00:00
30/12/1999
00:00
35
01/01/2000
00:00
03/01/2000
00:00
30
Discharge (m3/s)
25
20
15
10
5
0
21/01/1995
00:00
23/01/1995
00:00
70
60
25/01/1995
00:00
27/01/1995
00:00
29/01/1995
00:00
Time
31/01/1995
00:00
02/02/1995
00:00
04/02/1995
00:00
V olum e of hydr ogr aphs of diffe r e nt ye ar dur ing m ax pe ak
1994
1999
Mill cu. meter
50
2000
1962
40
30
20
10
0
1
2
Days
7
14
30
Comments:
1. Though the 1962 flood event possesses largest peak discharge, it gives considerably
lower flood volume than the volume of other events for the 7, 14 and 30-day durations.
193
Appendix 4
Probability Plots
See separate volume
194

FREQUENCY ANALYSIS

Transcription

Similar documents

Sony CDX

Rapping up Safety!

2016 TheWeatherChannel

Swallow Rocks to Big Mountain

What you have told us

Floodwatch®

Grooves Info Sheet - Tin Can And String Radio Network

January 12th - Groton Public Schools

p - Garden Commerical Properties