Generalized Low-Flow Frequency Relations for

b
.
..'.ii11
,..
VOL. 26, NO.2
WATER RESOURCES
BULLETIN
AMERICAN WATER RESOURCESASSOCIATION
APRIL 1990
-
GENERALIZED
LOW -FLOW
FREQUENCY
RELATIONSmpS
FOR UNGAGED
SITES IN MASSACHUSETTSl
Richard
M. Vogel and Charles N. Kroll2
ABSTRACT: Regional hydrologic procedures such as generalized
least squaresregressionand streamflow record augmentation have
been advo~a~ed
for obtaining e~timates ~f both flood-flow and lowflow statistics at ungaged sites. WhIle such procedures are
extremely useful in regional flood-flow studies, no evaluation of,
their merit in regionallow-flow estimation has been made using
actual streamflow data. This study developsgeneralized regional
regressionequationsfor e~tim~ting the d-day, T-year low-flow dis-
coincident with streamflow gaging stations; therefore,
accurate estimation
of Iow-flow statistics at ungaged
sites is a fundamental
and widespread problem.
Th
'd 1
d .d
f 1
fl
' th
e most Wl, e y use
m ex o ow- ow m
e
Umted States IS the seven-day, ten-year low-flow
(Q7 10), defined as the annual minimum
seven-day
ave~age daily
discharge
that is expected to be
charge,
exceeded
and
~,T,
30
days.
atungaged
A
Sites
two-parameter
mMassachusett?
lognormal
w~ere.d
dlstnbutlon
=.3,7,14,
IS
fit
to
in
nine
out
of
every
,
ten
years
(Riggs
..
app I lcatlons
et
al.,
~ft ~n reqwre
.
sequencesof annual minimum d-day low-flows and the estimated
1980). Smce water reso~rce
parameters of the lognormal distribution are then related to two
drainage basin characteristics: drainage area and relief. The
resulting modelsare genersl, simple to use, and about as preciseas
mos~ ~revious models that o~ly provide e~timates of. a single
statistic such as Q7 10. Compansonsare proVIdedof the Impact of
using ordinary least squaresregression,generalized least squares
regression,and streamflow record augmentation proceduresto fit
regionallow-f1owfrequencymodelsin Massachusetts.
(KEY TERMS: low-flows; regional regression; low-flow frequency
analysis; water quality management; Massachusetts; low-flow
management; regional hydrology; generalized least squares
regression;streamflowrecord augmentation; droughts.)
knowledge of the magnItude of annual mmlmum lowflows for recurrence intervals (T) other than 10 years,
and for durations
(d) other than 7 days, this study
takes a more general approach by specifying
the
general d-day, T-year event Qd,T.
STUDY
Th e regIona
'
1 regressIon
.
REGION
.
deve 1ope d In
. th IS
equa t Ions
'
study are based on data obtained from twenty-three
U .S. Geological Survey streamflow-gaging
stations in
or near Massachusetts. Table 1 provides the U,S. Geo-
INTRODUCTION
logical Survey streamflow-gaging
station numbers,
record lengths, and selected basin characteristics,
In
addition, site numbers used in this and other studies
(Vogel and Kroll,
1989, 1990; Vogel et al., 1989,
Fennessey and Vogel, 1990) are provided in Table 1,
The location of those stations is shown in Figure 1.
All of the rivers are perennial; that is, all streamflows are greater than zero. No significant
withdrawals, diversions,
or artificial
recharge areas are
within any of the basins; hence, the streamflows are
considered to be unregulated
(natural). The twentythree gaging stations employed here are about the
only long-term, unregulated,
continuous flow records
available in or near Massachusetts.
The Massachusetts
Department
of Environmental
Protection is currently making hundreds of estimates
oflow-flow statistics each year at ungaged sites for the
purposes of determining
waste-Ioad allocations, issuing and/or renewing
National
Pollution
Discharge
Elimination
System (NPDES) permits,
siting treatment plants and sanitary landfills,
and for decisions
regarding interbasin transfers of water and allowable
basin withdrawals.
In addition, low-flow statistics are
used widely in Massachusetts,
and elsewhere,
to
determine
minimum
downstream
release requirements from hydropower, irrigation,
water-supply
and
cooling-plant facilities. Unfortunately,
most locations
where such low-flow statistics
are required are not
IPaper No.89079 of the WaterResourcesBulletin. DiscussioDB are open until December I, 1990.
2Respectively, Assistant Professor, Department of Civil Engineering, Tufts University, Medford, Massachusetts
Engineer, Goldberg Zoino & Associates, Inc., 320 Needham St., Newton Upper Falls, Massachusetts 02164.
241
02155; and Staff
WATER RESOURCES
BULLETIN
.f,
-Generalized
ww-FlowFrequency
Relationships
for UngagedSitesin Massachusetts
;
...;
plot correlation coefficient test statistics to compare
annual minimum flow series fit to 2- and 3-parameter
lognormal, 2- and 3-parameter Weibull, and the log
Pearson Type III distributions. On the basis of the
previous studies, it was assumed in this study that all
flow series considered here can be described by a twoparameter lognormal distribution.
year event. In a comparative study of alternative
estimators of quantiles of a two-parameter lognonnal
distribution, Stedinger (1980) advocated the use of the
maximum likelihood estimators shown above when
one's primary interest is in estimating the lower
quantile of the distribution. An excellent approximation to ~ is obtained using a formula proposed by
Tukey (1960)
TABLE1. U.S.
Survey
GagingStationNumbers
andGeological
Selected
Basin
Characteristics.
Record
Length
(yrs)
73
34
37
16
34
23
19
11
22
38
31
44
20
17
48
52
20
34
Drainage
Area
A
(mi2)
52.70
63.69
8.01
41.39
3.39
8.68
51.00
5.46
2.85
21.30
7.35
89.00
16.02
24.09
94.00
40.90
12.31
42.60
USGS
GageNo.
01180500
01096000
01106000
01170100
01174000
01175670
01198000
01171800
01174900
0ll0l000
01187400
01169000
01111300
01169900
01181000
01332000
01097300
01333000
Site
No.*
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
01165500
01171500
19
20
65
45
12.10
54.00
01176000
01162500
01180000
21
22
23
71
63
28
150.00
19.30
1.73
Basin
Relief
H
(ft)
1765
1161
240
1873
531
417
1317
530
585
277
877
1667
393
1298
1739
2068
248
2658
~ = 4.91
0.14
(
-I-T
1
) 0.14J
(2)
as long as 1.01 ~ T ~ 100 years.
Figure 2 displays the fit of a two-parameter lognormal distribution to the twenty-three series of annual
minimum 7-day low-flows shown in Table 1 using
computer generated lognormal probability paper. The
.
d db
b b .l . t
1t
1 t.
SIteSare or ere y pro a 1 1 y p 0 corre a IOn coeffi1cient test statistics, i.e., site 1 displays the worst fit
and site 23 displays the best fit to a two-parameter
lognormal distribution. The empirical probability plot
is shown using open circles and the fitted lognormal
d . t . b t ..
th
l .d I .
IS n u Ion IS e so I me. Th e fitt
I ed t wo-parame t er
lognormal distribution provides an excellent representation of each empirical cumulative density function.
Each empirical probability plot is obtained by plot.ting the ordered observations q. versus Blom's
...(I)
plottmg posItIon (Blom, 1958)
.
797
1476
P{Q ~ q.) = 1- 3/8
I
n + 1/4
801
718
643
(3)
on computer generated lognormal probability plotting
paper. Blom's plotting position is unbiased when
*SeeFigure1 for the locationof eachsite.
observations arise from a normal distribution, which
is the case for the log transformed streamflows yj. The
subscript i corresponds to the rank order for each
streamflow,
i=1 values,
and i=nrespectively.
corresponding
to and
the
smallest
and with
largest
Vogel
~a~Imum lIkelIhood estI~ates of th~ low-flow
statIstIcs at each of the gaged SIteSare obtamed from
~.T = exp(~(d) + ZTCry(d»
[ (T1 )
Kroll (1989) described procedures for generating prob-
(1)
ability plots on a computer.
where
n
~(d) = 1 L yj(d)
n j=I
REGIONAL LOW-FLOW FREQUENCY
MODELS IN MASSACHUSETTS
n
Cr2(d) = -1- L(Y{d) -~(d»2
y
n-l j=1 I
Many investigators have developed regional models
for estimating low-flow statistics at ungaged sites
from readily available geomorphic, geologic, climatic,
yj(d) = In[Qj(d)]
and topographic
characteristics
for have
example,
Thomas
and Benson,
1970). Such (see,
models
been
developed
use at ungaged
many regions
in
the
United forStates.
Usually, sites
theseformodels
take the
ti
~j(d).IS the mmImum d-day average dIschar~e m year
I, n IS the number of years of record, and ZT IS a stan-
dard normal random variable corresponding to the T-
o~ --h
~.T -~1
243
"
v bIVbn..P~
.."l ~A3 ...
WATER RESOURCES
(4 )
BULLETIN
VogelandKroll
where ~.T is obtained from gaged streamflow records,
the X; are easily measured drainage basin and climatic
cparacteristics, and the bi are parameter estimates
obtained from the use of multivariate
regression
procedures.
.Recently,
Vogel et al. (1989) presented a regional
.low-flow model that describes the time response of
runoff from a watershed
during dry periods.
Describing the watershed as a system of linearized
Dupuit-Boussinesq aquifers, they found that water-
~(d)
crq(d) = exp(2~(d) +cry2(d))[exp(cry2(d))-1)]
(5)
where k is the hydraulic conductivity of the aquifer, A
is drainage area, H is watershed relief, d is drainage
density, ~ is the dimensionless baseflow recession
constant and t is time. The product Hd is an approximation to the average basin slope. Here, as in
Zecharias and Brutsaert (1985) the basin relief, H, is
defined as the difference in elevation between the
basin summit and the basin outlet where the basin
summit elevation is the average of the highest peak
and the two adjacent peaks on either side of it. The
values of drainage area and relief are summarized in
Table 1.
Unfortunately, accurate estimates of k and ~ are
difficult to obtain at an ungaged site; hence, those
characteristics are ignored here. The inclusion of
drainage density, d, did not result in improvements in
our ability to estimate low-flow statistics at ungaged
sites hence that basin characteristic is ignored here
(see Kroll, 1989, Table 5.4). Instead, this study
assumes that low-flows can be characterized by watershed area and basin relief alone.
~'(d)
= In[bo] + b1In[A] + ~ln[H]
In [crq(d)] = Inllio] + b1In[A] + Ea
(8b)
(9)
where regression estimates of the moments in log
space are obtained using the relations
.'(d)
~'(d)
(6a)
= In
[[1 + (crq'(d)/~'(d))2]1/2
.~~~)
J
cry'2(d) = In [ 1 + ( ~)2]
(6b)
Estimates of the moments of the annual minimum dday streamflows in real space are obtained from the
moments in log space using the relations
WATER RESOURCES BULLETIN
(8a)
Q'd,T= exp(~'(d) + ¥y'(d))
+ ElL
(7b)
Table 2 summarizes the regional regression equations in Eq. (8) for the annual minimum d = 3- , 7- ,
14- , and 3O-day low-flows. In general, the regression
equations for liq'(d) have significantly higher coefficients of determination
(R2), and lower standard
errors of estimate (SE%), than the regression
equations for crq'(d). In all cases, the model residuals
(E), are approximately
normally
distributed
as
evidenced by the high computed probability plot
correlation coefficient test statistics re (see Vogel and
Kroll (1989) for discussion of these statistics). The tratios of the estimated model parameters, defined as
each parameter estimate divided by its standard
deviation, are always in excess of 7.2, 15.2, and 3.5 for
lnllio], b1 and ~, respectively. Using ordinary least
squares regression procedures, estimates of In[bo] in
Eq. (6) are unbiased, and corresponding estimates of
bo may contain slight bias. Only t-ratios for the
unbiased parameter estimates, In[bo], are quoted
here. The high t-ratios are evidence ofboth the significance and relatively high precision associated with all
of the model parameter estimates in Table 2.
The generalized regional low-flow frequency relations are obtained using
and standard deviation, crq(d), of the annual minimum d-day low-flow series are obtained by using ordinary least squares regression procedures to fit the
models
In[~(d)]
= bo A b1H~
crq'(d) = bo A b1
Ordinary Least Squares Regional
Regression Procedures
Regional regression equations for the mean, ~(d),
(7a)
where liy(d) and cry(d) are obtained using the
estimators shown in Eq. (1). The resulting regional regression estimators for the moments in real space, denoted ~'(d) and crq'(d),may be rewritten as
shed runoff at time t after the hydrograph peak may
be described by
Qt ==4 k A (Hd)2 ~ t
= exp(~(d) + 1/2&q2(d))
248
(lOa)
(lOb)
Genon"'od "'w.FloWF,..,uon'Y..Iatio~hip' fu, Ung...d SI", 10Mu~,hu~t"
TABLE2 SU=a",orOnlina", r.~t Squ~
UlonEqnatw~ fu, Moment.orthOAnnaIMini=m
d.day"'w.Fww.in...1 Sp~fu,tho 23 StudySi"'
",,'(d) = boAblHb,
"01d) = boAb,
MODEL
hO
h,
.,
...13)
00031
(-77)0
L152
(152)
0454
(35)
351
957
09£7
1'«.(7)
000444
(-75)
L132
(156)
0430
(35)
336
958
0968
...114)
00065
(-74)
Ll00
(160)
.All
(35)
3L8
980
0978
...130)
00136
(-72)
L065
(178)
0369
(36)
n5
967
0988
..13)
00429
(-141)
L168
(189)
408
928
0983
..17)
00558
(-130)
Ll22
(163)
406
923
0976
..114)
00759
(-lL2)
L082
(152)
422
9L2
0978
..130)
t
I
I
0
R2'
01226
L058
355
932
(-107)
(174)
Critical voluoorr,i. 0963noing5%.ignifi~nooI~t ',i. tho probabilitypwtro~l.twn~mciont "'t .tatI.tk('..9"
Thev"~. orR' .rea..n.tod furthOnUmb~or-orrreodom
whi,hremaw al1.,p.,omotero..lm.twn
CompntoduoingSE%
= 100[oxP(0,'}-1]"',whore..' I. tbovarianooortho ~Idn".o, d.rmedin(6"
V"u~ inparen..o~. ~t.ntw.oro.timatod modelpanmo"n In[boJ,b" andb,
where Ii..'(d) and ".'(d) are given hy Eq (8) and
ZT is obtained from Eq. (2). Figure 2 summarize5
the regre88ion e8timates of the distribution of annu81
minimum 7-day low-flow5 at the twenty-three 5ites
u8ingdashed line8. Overall, the re8ult5 are quite good
except for five sites (site numbers I, 5, 8, 14, and 15)
where the regression estimates are only fair approximations to both the fitted lognormai distribution
(LN2) and the empirical distribution
shown using
open circles. In general, the lack-of-fit is probably due
to model error that results from not including additional basin characteristics such as the hydraulic conductivity and baseflow recession constant, both of
which describe the geohydrologic response of each
watershed. Riggs (1961), Vogel et al. (1989), and Kroll
(1989) describe the improvement in regional low-flow
frequency models that result from the inciusion of the
baseflow recession constant.
D
SE%I
/I
.1
.
etectlOno{ I n uentlO
Ob
r.'
0979
Vogcl.ndKroII.
regional regression model If one or more sites 8re par.
ticularly Influential, the re5ulting model would be
que9tionable and more site5 would be required to bolster our confidence in the regres5ion model5. Draper
and Smith (1981, Section 312 and Section 6,8) summarize a number of stand8rd St8tistiCS that can be
employed to detect unusually influential observation5.
As an alternative to these standard influence
statistics, we use a graphical approach that is easier
to explain. Each of the regression equations in Eq. (6)
was fit to all sites except the one being validated, and
the resulting estimates of Q7,T are shown as dotted
lines in Figure 2. (Each dotted line repre5ents the
regional estimate of Q7,T when that site is dropped
from the original sample of 23 sitesJ In all cases, the
subsample (validation) estimates are almost indistinguishable from the regression estimates based on all
23 sites. We conclude that none of the twenty-three
sites employed in this study has an unusually large
impact on the estimation of any of the parameters of
the models in Eq. (6).
servatlOns
An experiment was performed to detect whether
the deletion of any one of the sites affects the fit of the
249
WATER RESOURCES BULLETIN
VogelandKroll
The Im~act of Generalized Least Squares
RegressLonProcedures
ciated t-ratios obtained using OLS and GLS procedures are almost indistinguishable. However, because
the GLS procedures account for the heteroscedastic
structure of the errors and the cross-correlation of the
flow-series used to estimate the dependent variables,
they result in smaller model errors than the OLS
models. Tasker and Driver (1988) reached similar
conclusions for regional urban runoff water quality
models.
The total prediction error is the sum of the model
error, sampling error, and measurement error. The
latter is ignored here. GLS procedures document that
the total prediction errors are larger than OLS procedures because GLS procedures account for the varying
sampling errors associated with the dependent variabIes, but even GLS estimates of total error are low
because they ignore measurement errors. The OLS
and GLS model results in Table 3 show that the total
prediction errors are largely due to model error; sampIing error remains insignificant by comparison. If
measurement errors are ignored, these results indicate that future research in regional low-flow
frequency analysis should concentrate primarily upon
reducing model error .
In the previous section ordinary least squares
(OLS) regression procedures were employed to esti': mate the parameters of the regional model (eq. 6).
Stedinger and Tasker (1985) observed that most
regionalization problems, where the dependent variabIes are both cross-correlated and based on different
record lengths, violate the commonly made assumptions that the model residuals are homoscedastic and
independently distributed. They developed generalized
least squares (GLS) regression procedures that
account for the fact that the dependent variables are
both cross-correlated and are based on samples with
widely varying record lengths. A computer program
which implements the GLS procedures is available
(Tasker et al., 1987) and has already been employed in
regional hydrologic studies for flood frequency analysis (Tasker et al., 1986; Potter and Faulkner, 1987)
and urban runoff quality (Tasker and Driver, 1988).
The authors are unaware of any previous applications
of these procedures to regional low-flow frequency
analysis.
GLS regression procedures were used to fit the
following regional models for annual minimum 7-day
average streamflows
In(~7,T) = In[bo] + b1ln[A] + ~ In[H] + v
The Impact of Streamflow Record
Augmentation Procedures
(12)
where
Streamflow record augmentation procedures developed by Fiering (1963) and Vogel and Stedinger
(1985), exploit the cross-correlation between concurrent flow records to obtain improved estimates of the
~7T = exp(~(7) + ZTcr (7»
.y
The residual error V is made up of model error £
and sampling error 11.The sampling error is due to the
finite and varying length of records used to estimate
Q7,T. In addition, sampling error in the dependent
variable occurs because the sequences of annual minimum 7-day low-flows are cross-correlated in space.
The average cross-correlation of the sequences was
0.35 and this value was used to characterize the covariance matrix of the residuals. An alternative would
have been to model the cross-correlation offlow-series
as a function of distance between sites as suggested by
Tasker et al. (1987); however, we could not find any
significant relations between cross-correlation and
distance.
Table 3 compares the parameter estimates and
residual errors in Eq. (12) using both OLS and GLS
regression procedures. The total prediction error v is
equal to the sum of the model error £ and the sampIing error 11.Stedinger and Tasker (1985) and Tasker
et al. (1986) describe procedures for estimating these
error components using both GLS and OLS procedures. The model parameter estimates and their assoWATER RESOURCES BULLETIN
moments of the flows at a short-record site. U sing the
procedures outlined by Vogel and Stedinger (1985)
and Vogel and Kroll (1990), record augmentation
estimates of the mean and variance of the logarithms
of the annual minimum 7-day low-flow series were
obtained.
Streamflow record augmentation requires the selection of a nearby hydrologically similar basin with a
longer record than the site in question. For each of the
23 basins described in Table 1, the basin that led to
the largest increase in effective record length (see
Vogel and Kroll, 1990) was considered the long-record
neighbor to be exploited in the record-augmentation
procedure. This amounted to choosing a neighbor with
simultaneously
a high cross-correlation and long
record length. The record-augmentation estimates of
the .moments in
log space, denoted
liy*(7)
and O"y*(7),were transformed into real space using
the standard moment transformations
for the
lo~orrnal distribution gi'Yen in eq. 7, .the resulting
estimates are denoted J.1q*(7) and O"q*(7). The
record-augmentation
estimators liq*(7) and crq*(7)
250
VogelandKroll
CONCLUSIONS
years of data. In addition, the large model error component of the total prediction errors implies that the
.This
study describes a procedure for developing
sampling errors had only a marginal impact on the
generalized regional regression equations for estimatanalysis. In general, GLS procedures will have signifiing the d-day, T-year low-flow statistic (~ T) at uncant advantages over OLS procedures in studies
gaged sites in Massachusetts for d = 3, 7, 14, and 30
which seek to include very short-records such as at
days. The resulting models (equations 8 9 and 10)
partial-record sites. In such instances, GLS proceonly require estimates of the drainage ar~a ~d basin
dures can lead to significant improvements because
relief of the ungaged site. The procedure for estimatthe number of sites included in the analysis can be
ing QdT at an ungaged site consists of substitution of
increased considerably.
the dr;Unage area A, and basin reliefH into Eq. (8) to
In this study, record lengths ranged from 11 to 73
obtain Aq'(d) and <rq'(d), which are then substiyears and the average cross-correlation among concurtuted into Eq. (10) to obtain A;(d) and <r;(d),
rent traces of annual minimum 7-day lo",:-flows was
which in turn are substituted into Eq. (9) to obtain
only 0.35. If the average cross-correlatlon among
the final regression estimate Q'd T. The low-flow
concurrent low-flow series were higher and if sites
predictions from this model are ~omparable with
with record lengths smaller than, say, 10 years were
previous regionallow-flow
investigations in terms of
used, then GLS procedures would likely result in
average prediction errors. ~e models developed~e
signific~ntly different and more precise model param~re useful in practice due t~ !~ei!.-e~~~..SJL~.pp-tic~!-!9-n eter estImates than OLS procedures.
and their ab~tyto ge~~-g~-~~!~li~~~~es
Both OLS .and GLS procedures document that only
and magnitudes oflow-flowsatungaged
sites.
a small fractIon of the total model error was due to
--""Re~on-alnydro1ogic.models similar to those develsampling error; thus, efforts to reduce sampling error,
oped here, known as "state equations," are in
such as the application of streamflow record augmenwidespread use for estimating peak floodflows at
tation procedures, led to only modest improvements in
ungaged sites. to~ e~a~ple. NewtQn and Herrin
the models described he~e: In the future, attention
(1983) recommend such statistically based regional
should focus on the remammg sources of model error:
reeTe;s~O~e~uations ~-of-~~.s.e:d
me~s~rement error, mode~ form, and ~h~ inclusion of
determInIstIc mo els for estimating floodflows at
addItIonal explanatory basm charactenstIcs. After all,
t,lEgagedsites.-Their recommend&t1O-iisarebased-tipon
as Wallis (1965) so clearly showed, one cannot hope to
a- ~r~e nationWide ~dy using~mp-eti~ds
uncover basic physical relationships using multivarid~~p~~!!1~
federal ~g~!!~ies. Unforate stati~tical pr~cedu~es without prior knowledge of
tunately, most studies, such as this one, which have
the physIcal relatIonshIps.
attempted the same approach to estimate low-flow
statistics at ungaged sites have met with only limited
success. For example, Thomas and Benson (1970)
show that average prediction errors for low-flow
ACKNOWLEDGMENTS
regional regression models in the Potomac river basin
are at least twice as large as for analagous floodflow
regional regression models in the same basin. Riggs
(1961) Vo el et al. (1989)
d Kr II (1989) h
th t
,
g
..'
an ..O
S OW. a
further reductIons In the predIctIon errors assocIated
with such regionallow-flow models will likely result
from the inclusion of additional watershed characteristics such as basin hydraulic conductivity, average
basin slope, and the baseflow recession constant.
This study has also compared the use of general-
This researchwas supportedby a cooperativeagreement
betwe?nTufts University and the U.S..~.ological Surveyw!th
matchi~ fundsfromthe Massachus.etts
DIVlSIOn
of Wat.erPollutIon
ControlIn the Departmentof EnvIronmentalProtection.We are
gratefulto KatherineDriscollMartel for providingthe estimates
of
reliefin Table1.
LITERATURECrrED
izedS)
least squares
(GLS)
squares
.
d and ordinary least.Blom,
G., 1958. Statistical Estimates and Transformed Beta
Variables. John Wiley and Sons,New York, N.Y., pp. 68-75and
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( OL
regressIon proce ures. GLS regressIon procedures led to almost identical regional regression model
parameter estimates when compared with the OLS
procedures. Although
GLS procedures
led to only.
marginal
gains in the prediction
errors assoc. t d
t .la
he
. th }
fl
.
1
.
WI
OW- OW reglona
regressIon
equa Ions, t at
result only reflects the fact that all sites had at least
eleven years of data, and most had more than twenty
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