University of Nis

Transcription

University of Nis

University of Nis
Faculty of Civil Engineering and Architecture
Slavisa Trajkovic
Estimating Reference Evapotranspiration
by Artificial Neural Networks
0.8
0.7
Salton Sea West
CIMIS #127
April 06, 1965
ETo
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(mm h )
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ETo_ly
ETo_annthr
ETo_anntr
ETo_pm70
ETo_pm42
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Hours
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MMIX
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Author: Slavisa Trajkovic, Ph. D.
Reviewers:
Ozgur Kisi, Ph.D., Civ. Eng., Hydraulic Division, Civil Engineering Department,
Engineering Faculty, University of Erciyes, Turkey
Mladen Todorovic, Ph. D., Civ. Eng., International Center for Advanced
Mediterranean Agronomic Studies (CIHEAM) - Mediterranean Institute of
Agriculture of Bari, Land and Water Division, Valenzano, Italy
Dragan Arandjelovic, Ph. D., Civ. Eng., Faculty of Civil Engineering and
Architecture, University of Nis, Serbia
Published by: Faculty of Civil Engineering and Architecture, University of Nis
For publisher: Dragan Arandjelovic, Ph. D.
Approved to be printed on November 7, 2008 by the Decree of the TeachingScientific Council of the Faculty of Civil Engineering and Architecture, University
of Nis, Serbia
Cover design: Zoran Stefanovic
Printed by: M KOPS CENTAR, Carnojevica 10a, Nis
Press: 100 copies
ISBN 978-86-80295-84-8
Estimating reference evapotranspiration by artificial neural networks
i
CONTENTS
List of Tables
ii
List of Figures
iv
Preface
v
Acknowledgements
vi
I. Introduction
1
II. Estimating FAO-24 coefficients by RBF networks
7
Estimating FAO-24 Blaney-Criddle b factor by RBF networks
7
Estimating FAO-24 Pan Kp factor by RBF networks
14
Estimating FAO-24 Radiation c factor by RBF networks
18
Estimating FAO-24 Penman c factor by RBF networks
21
III. Forecasting of ETo by RBF networks
25
Forecasting of reference evapotranspiration by adaptive RBF network
25
Forecasting of ETo by sequentially adaptive RBF network
32
IV. Estimating reference evapotranspiration by sequentially adaptive
RBF networks
Estimating hourly reference evapotranspiration from limited weather data
by sequentially adaptive RBF networks
Comparison of RBF networks and empirical equations for converting from pan
evaporation to reference evapotranspiration
Temperature-based approaches for estimating reference evapotranspiration
37
37
45
55
V. Conclusions
67
Notation
69
References
71
About the author
80
About the reviewers
81
ii
Contents
List of Tables
1.
FAO-24 Blaney-Criddle b factors
11
2.
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13
4.
Comparison of models used to estimate FAO-24 Blaney-Criddle b
factors
Estimation of monthly ETo (mm) in Nis, Serbia, with b factors were
obtained by RBF network, regression equation or interpolation
FAO-24 pan Kp factors
5.
Comparison of models used to estimate FAO-24 pan Kp factor.
16
6
FAO-24 pan Kp factors obtained by RBF network, regression equation
and table interpolation
FAO-24 radiation adjustment c factors
16
Structure of various RBF networks used to estimate FAO-24 radiation
adjustment c factors
Comparison of various RBF networks used to estimate FAO-24
radiation adjustment c factors
FAO-24 Penman c factors
19
24
12.
Comparison of various methods used to calculate FAO-24 Penman
c factors
Statistic properties of ARIMA and ANN forecasting models at Griffith
13.
Statistic properties of ANN forecasting model at Nis, Serbia
36
14.
Daily micrometeorological and lysimeter data at Davis, CA
38
15.
Statistical summary of hourly ETo estimates at Davis, CA
41
16.
Average weather parameters at Kimberly, Idaho, during July
47
17.
Data requirements of the ETo equations
50
18.
Summary statistics of ETo equations at Policoro, Italy
51
19.
Summary statistics of ETo equations at Novi Sad, Serbia
52
20.
Summary statistics of ETo equations at Kimberly, Idaho
54
3.
7.
8.
9.
10.
11.
14
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20
23
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iii
21.
Summary of selected weather stations in Serbia
56
22.
Statistical summary of ETo estimates for seven locations in Serbia
59
23.
Statistical summary of calibrated ETo equations and RBF network for
five locations in Serbi
Statistical summary of adjusted ETo equations and RBF network for
five locations in Serbia
62
24.
64
iv
Contents
List of Figures
1.
Structure of RBF network used to estimate FAO-24 Blaney-Criddle b
factor
Reference evapotranspiration at Griffith, Australia, from 1962 to 1972
10
31
4.
Comparison of observed ETo (ETo_obs) and forecasted ETo using RBF
network (ETo_ann) and ARIMA model (ETo_arima) at Griffith
Structure of RBF network
5.
Growth pattern
34
6
Comparison of observed ETo (ETo_obs) and forecasted ETo using RBF
network (ETo_ann) at Nis, Serbia
Comparisons between estimated and measured ETo at Davis on July 14,
1966
Daily ETo estimated by RBF network versus lysimeter ETo at Policoro,
Italy
Comparison of mean daily ETo calculated for four growing seasons at
Novi Sad, Serbia using FAO-56 PM equation, RBF network (ETo_ann),
FAO-24 pan equation (ETo_pan) and Christiansen equation (ETo_chr)
Comparison of daily ETo computed for 4 years at Bari, Italy using RBF
network (ETo_ann) and FAO-56 Penman-Monteith equation (ETo_pm)
35
2.
3.
7.
8.
9.
10.
25
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43
52
53
65
v
PREFACE
Reference evapotranspiration is a complex nonlinear process which depends on
several climatological factors. Accurate estimation of reference evapotranspiration
is needed for many studies such as hydrological water balance, irrigation system
design, irrigation scheduling, and water resources planning and management.
Artificial neural networks (ANNs) are nonlinear mathematical structures, which are
very appropriate for the modeling of nonlinear processes. During the last decades
there has been a significant increment in their application in wide variety of
scientific fields due to the development of computer technologies. Artificial neural
networks have been used for pattern identification, modeling processes, and time
series analysis in different areas such as ecological research, financial research,
hydrology, meteorology, agronomy, and engineering research studies.
However, the application of ANNs for estimating reference evapotranspiration has
been less frequent than in other fields of knowledge. The main objective of this
publication is to provide project managers, consultants, irrigation engineers,
hydrologists, meteorologists, agronomists, and students with an ANN approach for
estimating reference evapotranspiration.
The author has been into estimating reference evapotranspiration by artificial neural
networks for ten years and all the most significant accomplished results have
published in this publication. The author would feel that the publication has served
a purpose if it leads to initiation of studies to improve the estimation of reference
evapotranspiration.
vi
Preface&Acknowledgements
ACKNOWLEDGEMENTS
The author is particularly grateful toward Professor William O. Pruitt (University of
Davis, CA, United States), Dr. Mladen Todorovic (Mediterranean Agronomic
Institute of Bari, Italy), Professor Angelo Caliandro (University of Bari, Italy), and
Dr. Vladimir Stojnic (LinkWater O&M Alliance, Spring Hill, Australia) for
providing the agrometeorological and lysimetar data of Davis, CA, United States
and Policoro, Italy and agrometeorological data of Griffith, NSW, Australia,
respectively.
The author would like also to acknowledge and thank Professor Branimir Todorovic
(University of Nis, Serbia) for explaining the procedures used by RBF networks and
Professors Miguel A. Marino and Robert H. Shumway (University of Davis, CA,
United States) for explaining the procedures used by ASTSA computer package.
The reviewers, selected by the publisher, were: Professor Ozgur Kisi (University of
Erciyes, Turkey), Dr. Mladen Todorovic (Mediterranean Agronomic Institute of
Bari, Italy); and Professor Dragan Arandjelovic (University of Nis, Serbia). The
author would like to thank all of reviewers whose comments and suggestions
resulted in significant improvements to this publication.
The author wishes to express his gratitude to Professors Vidosava Momcilovic
(University of Nis, Serbia), Srdjan Kolakovic (University of Novi Sad, Serbia),
Stevan Prohaska (University of Belgrade, Serbia) and Miomir Stankovic
(University of Nis, Serbia) for their encouragement and support. Special thanks are
due to Mr Goran Stevanovic (University of Nis, Serbia) for his patience and
valuable assistance in the preparation of this publication.
To researchers cited in this publication, the author offers his thanks and
appreciation for their contributions. Without their work, this publication could not
have been written. Finally, the author would like to thank his family for providing
continuing support and long patience during the completion of this publication.
Slavisa Trajkovic
I.
1
INTRODUCTION
Evapotranspiration (ET) is a physical process in which water passes from liquid to
gaseous state while moving from the soil to the atmosphere. It refers both to
evaporation from soil and vegetative surface and transpiration from plants. These
two separate processes (evaporation and transpiration) occur simultaneously and
there is no easy way of distinguishing one from the other.
Evapotranspiration is one of the major components in the hydrological cycle, and its
reliable estimation is essential to water resources planning and management.
A common procedure for estimating evapotranspiration is to first estimate reference
evapotranspiration (ETo) and to then apply an appropriate crop coefficient.
Reference evapotranspiration is defined in Allen et al. (1998) as "the rate of
evapotranspiration from hypothetical crop with an assumed crop height (0.12 m)
and a fixed canopy resistance (70 s m-1) and albedo (0.23) which would closely
resemble evapotranspiration from an extensive surface of green grass cover of
uniform height, actively growing, completely shading the ground and not short of
water". Crop coefficients, which depend on the crop characteristics and local
conditions, are then used to convert ETo to ET. This publication addresses only the
estimation of ETo.
Reference evapotranspiration can be measured by lysimeters. However, the use of
lysimeters is generally limited to specific research purposes due to difficult and
expensive construction, and this requires special care for operation and
maintenance. These limitations make more attractive the application of indirect
methods of measurements which are based on easy-to-obtain weather data.
Numerous equations, classified as temperature-based, radiation-based, pan
evaporation-based and combination-type, have been developed for estimating
reference evapotranspiration (ETo), most of which are complex and require
numerous weather parameters. Relationships were often subject to rigorous local
calibrations and proved to have limited global validity. In many areas, the necessary
data are lacking, and new techniques are required.
Reference evapotranspiration is the complex and nonlinear process because it
depends on several interacting weather parameters. Artificial neural networks
(ANNs) are efficient tools to model nonlinear processes. An artificial neural
network is a mathematical construct whose architecture is essentially analogous to
the human brain. Basically, the highly interconnected processing units arranged in
layers are similar to the arrangements of neurons in the brain. ANNs offer a
relatively quick and flexible means of modeling, and as a result, application of
ANN modeling is reported in the evapotranspiration literature.
Trajkovic et al. (2000a) presented the application of Radial Basis Function (RBF)
network to estimate the FAO Blaney-Criddle b factor. The b values obtained by
RBF networks were compared to the appropriate b values produced using
regression equations. The RBF network predicted b values better than the
2
Introduction
regression equations. An example is given to illustrate the simplicity and accuracy
of the RBF network for ETo estimation. There was a good agreement of the factors
obtained by RBF network and table interpolation, which automatically lead to
agreement in estimated evapotranspirations. The relative difference was less than
0.4% (3 mm/year) in evapotranspiration estimates.
Odhiambo et al. (2001) used ANN as a part of fuzzy-neural model from which the
results can be compared to the results of the standard Penman-Monteith equation.
Their model had four inputs: solar radiation, relative humidity, wind speed, and
temperature difference.
Kumar et al. (2002) used six climatic parameters for the calculation of ETo: Tmax,
Tmin, RHmax, RHmin, U2, and Rs. Several issues associated with the use of ANNs
were examined including different learning methods, number of neurons, number of
hidden layers, and number of learning cycles. Three learning methods (the standard
back-propagation with learning rates of 0.2 and 0.8, and back-propagation with
momentum) were used for the training of 36 ANNs with a single hidden layer and
18 ANNs with two hidden layers. The best architecture was selected from the total
set of 162 ANNs ((36+18) x3) on the basis of minimum weighted standard error of
estimate (WSEE). The best ANN had six input neurons, seven hidden neurons, and
one output neuron.
Sudheer et al. (2003) demonstrated the potential of RBF neural network for
estimating the rice crop ET using limited climatic data. The number of hidden
neurons increased with the reduction in the number of variables in the input set.
There were twelve hidden neurons in the RBF network with temperature input set.
Trajkovic et al. (2003) applied a sequentially adaptive Radial Basis Function (RBF)
network to the forecasting of reference evapotranspiration (ETo). Sequential
adaptation of parameters and structure was achieved using extended Kalman filter.
Criterion for network growing was obtained from the Kalman filter's consistency
test. Criteria for neuron/connections pruning were based on the statistical parameter
significance test. The monthly evapotranspiration data were available at Nis, Serbia
from 1977 to 1996. Network was learned to forecast ETo,t+1 based on ETo,t-11 and
ETo,t-23. The results showed that the ANNs can be used for the forecasting of
reference evapotranspiration with high reliability.
Kisi and Yildirim (2005) wanted to know which criteria the authors of previous
paper had used for the selection of the ANN input vector. Trajkovic et al. (2005)
explained that the ANN inputs were chosen on the basis of information obtained
from the auto-correlation function (ACF) and partial auto-correlation function
(PACF). The ACF plot showed a significant auto-correlation at a lag of twelve
months (ETo,t-11). The PACF plot presented a significant auto-correlation at a lag of
twelve (ETo,t-11) and twenty four months (ETo,t-23).
The basic goal of the Trajkovic (2005) was to examine whether it is possible to
attain the reliable estimation of ETo only on the basis of the temperature data. This
goal was reached by the evaluation of the reliability of four temperature-based
3
approaches (RBF network, Thornthwaite, Hargreaves, and reduced set PenmanMonteith equations) as compared to the FAO-56 Penman-Monteith equation.
The seven weather stations selected for this study are located in Serbia. In this
study, equations were calibrated using the standard FAO-56 PM method. However,
the RBF network better predicted FAO-56 PM ETo than calibrated temperaturebased equations at most locations. It gives reliable results in all locations and it has
proven to be the most adjustable to the local climatic conditions. These results are
of significant practical use because the adaptive temperature-based RBF network
can be used when relative humidity, radiation and wind speed data are not available.
In Parasuranam et al. (2006), a novel neural network model called the spiking
modular neural networks (SMNNs) was proposed. A novel network consists
of an input layer, a spiking layer, and an associator neural network layer.
The modular nature of the SMNN helps in finding domain-dependent relationships.
The performance of the SMNN model was evaluated using case study and involved
modeling of eddy covariance-measured evapotranspiration. Results from the study
demonstrate that SMNNs performed better than regular feed forward neural
networks (FFNNs). In modeling evapotranspiration, it is found that net radiation
and ground temperature alone can be used to model the evaporation flux effectively.
Kisi (2006a) used two different feed-forward neural network algorithms,
Levenberg–Marquardt (LM) and conjugate gradient (CG), for estimation of daily
reference evapotranspiration (ET) from climatic data. The performances of the
LM and CG algorithms in estimating ET were analyzed and discussed and various
combinations of weather data as inputs to the artificial neural network (ANN)
models are examined in the study so as to evaluate the degree of the effect of each
of these variables on ET. The LM and CG training algorithms were compared with
each other according to their convergence velocities in training and estimation
performances of ET. The results of the ANN models were compared with those of
multi-linear regression (MLR) and the empirical models of Penman and
Hargreaves. Based on the comparisons, it was found that the neural computing
technique could be employed successfully in modelling evapotranspiration process
from the avaliable climatic data.
Kisi (2006b) investigated the potential of the generalized regression neural
networks (GRNN) technique in modeling of reference evapotranspiration (ETo).
Various combinations of daily climatic data were used as inputs to the ANN so as to
evaluate the degree of effect of each of these variables on ETo. A comparison was
made between the estimates provided by the GRNN and those obtained by the
common empirical equations. The empirical equations were calibrated using the
standard FAO Penman-Monteith ETo estimates. The GRNN estimates were also
compared with those of the calibrated models. Based on the comparisons, it was
found that the GRNN technique whose inputs were solar radiation, air temperature,
relative humidity and wind speed could be employed successfully in modeling the
ETo process.
4
Introduction
One of the most interesting points of the paper by Kisi (2006b) was the comparison
of ANN results with empirical equations. Aksoy et al. (2007) and Koutosoyiannis
(2007) wondered if it was correct to compare an ANN fitted on a specific site to an
empirical equations applied on this site. Kisi (2007a; 2007b) did not agree with
previous discussants’ remarks that the ability of the ANN to perform better than the
empirical equations is very natural and hence not surprising. The GRNN was
calibrated on the FAO-56 PM itself. However, the empirical models were also
calibrated using the FAO-56 PM ETo data. Kisi (2007b) agreed with Trajkovic
(2005) in that people should adapt all calculations to their local conditions and they
should use their own judgment for the results based on their local experiences and
not take the results blindly.
Zanetti et al. (2007) tested an artificial neural network (ANN) for estimating the
reference evapotranspiration (ETo) as a function of the maximum and minimum air
temperatures in the Campos dos Goytacazes County, State of Rio de Janeiro.
The ANNs (multilayer perceptron type) were trained to estimate ETo as a function
of the maximum and minimum air temperatures, extraterrestrial radiation, and the
maximum daylight hours; and the last two were previously calculated as a function
of either the local latitude or the Julian date. According to the results obtained in
this ANN testing phase, it is concluded that when taking into account just the
maximum and minimum air temperatures, it is possible to estimate ETo in Campos
dos Goytacazes.
Parasuraman et al. (2007) investigated the utilization of genetic programming (GP)
to model the evapotranspiration process. The performance of the GP model was
compared with artificial neural network (ANN) models and the traditional PenmanMonteith (PM) equation. Results from the study indicate that both the data driven
models, GP and ANNs, performed better than the PM equation.
The accuracy of an adaptive neurofuzzy computing technique in estimation of
reference evapotranspiration (ETo) was investigated in Kisi and Ozturk (2007).
The daily weather data used as inputs to the neurofuzzy model to estimate ETo
obtained using the FAO-56 Penman–Monteith equation. A comparison was made
between the estimates provided by the neurofuzzy model and some empirical
models. The empirical models were calibrated using the standard FAO-56 PM ETo
estimates. The estimates of the neurofuzzy technique were also compared with
those of the calibrated empirical models and artificial neural network (ANN)
technique. The comparison results reveal that the neurofuzzy models could be
employed successfully in modeling the ETo process.
Kisi (2007c) investigated the modelling of evapotranspiration using the feedforward artificial neural network (ANN) technique with the Levenberg-Marquardt
(LM) training algorithm. The LM was used for the optimization of network
weights, since this algorithm is more powerful and faster than the conventional
gradient descent. Various combinations of daily weather data were used as inputs to
the ANN so as to evaluate the degree of effect of each of these variables on
5
evapotranspiration. A comparison is made between the estimates provided by the
ANN and some common empirical models. Based on the comparisons, it was found
that the neural computing technique could be employed successfully in modeling
evapotranspiration process from the available weather data.
In Gonzalez-Camacho et al. (2007), a feedforward backpropagation artificial neural
network (ANN) was trained to estimate the ETo from weather data. An ANN with
four neurons in the hidden layer, one neuron in the output layer, and hyperbolic
tangent transfer functions was applied to weather database from an automated
meteorological station located at Sinaloa, Mexico. The supervised training
algorithm of Levenberg-Marquardt allowed a good performance of the ANN to
estimate the ETo in all the scenarios considered in terms of mean square error and
determination coefficient.
The objective of Khoob (2008a) was to compare Hargreaves and ANN approaches
for estimating ETo only on the basis of the temperature data. The twelve weather
stations selected for this study are located in Iran. The Hargreaves equation mostly
underestimated or overestimated ETo obtained by the standard Penman-Monteith
equation. The ANN predicted ETo better than Hargreaves equation at all sites.
Kim and Kim (2008) developed the generalized regression neural networks model
(GRNNM) embedding the genetic algorithm (GA) in order to estimate and calculate
the pan evaporation (PE) and the alfalfa reference evapotranspiration (ETr) in the
Republic of Korea. An uncertainty analysis was used to eliminate the climatic
variables of the input layer nodes and to construct the optimal COMBINEGRNNM-GA. It was found that optimal COMBINE-GRNNM-GA can estimate the
PE and the alfalfa ETr.
In addition to the use of classic ETo equations, the adoption of artificial neural
network (ANN) models for the estimation of daily ETo has been evaluated in
Landeras et al. (2008). Seven ANNs (with different input combinations) have been
implemented and compared with ten locally calibrated empirical ETo equations. The
comparisons have been based on statistical error techniques, using standard FAO-56
PM daily ETo values as a reference. ANNs have obtained better results than the
locally calibrated ETo equations.
The potential of three different artificial neural network (ANN) techniques, the
multi-layer perceptrons (MLPs), radial basis neural networks (RBNNs) and
generalized regression neural networks (GRNNs), in modelling of reference
evapotranspiration (ETo) is investigated in Kisi (2008). It is found that the MLP and
RBNN techniques could be employed successfully in modelling the ETo process.
Kumar et al. (2008) was carried out to develop artificial neural network (ANN)
based reference crop evapotranspiration models corresponding to the ASCE’s best
ranking conventional ETo estimation methods (Jensen et al. 1990). The ANN
architectures corresponding to these less data-intensive equations were developed
for four CIMIS (California Irrigation Management Information System) stations.
Daily meteorological data for a period of more than 10 years were collected and
6
Introduction
were used to train, test, and validate the ANN models. Two learning schemes,
namely, standard back-propagation with learning rate of 0.2 and standard backpropagation with momentum having learning rate of 0.2 and momentum term of
0.95 were considered. ETo estimation performance of the ANN models was
compared with the FAO-56 PM equation. It was found that the ANN models gave
better closeness to FAO-56 PM ETo than the best ranking method in each category.
Thus these models can be used for ETo estimation in agreement with climatic data
availability, when not all required climatic variables are observed.
The objective of Khoob (2008b) was to test an artificial neural network (ANN) for
converting pan evaporation data (Ep) to estimate reference evapotranspiration (ETo)
as a function of the maximum and minimum air temperature. The FAO-24 pan
equation was also considered for the comparison. The ANN has been evaluated
under semi-arid conditions in Iran, comparing daily estimates against those from the
FAO-56 Penman–Monteith equation (PM), which was used as standard. The
comparison shows that, the FAO-24 pan equation underestimated ETo obtained by
the PM equation. The ANN gave better estimates than the FAO-24 pan equation
that requires wind speed and humidity data.
Chauhan and Shrivastava (2008) is an attempt to find best alternative method to
estimate ETo when input climatic parameters are insufficient to apply standard
FAO-56 Penman–Monteith equation. The ANNs, using varied input combinations
of climatic variables have been trained using the backpropagation with variable
learning rate training algorithm. ANNs were performed better than the climatic
based equations in all performance indices. The analyses of results of ANN suggest
that the ETo can be estimated from maximum and minimum temperature using
ANN approach.
In Trajkovic (2009a), the Radial Basis Function (RBF) network is applied for pan
evaporation to evapotranspiration conversions. The obtained RBF network,
Christiansen, FAO-24 pan, and FAO-56 Penman-Monteith equations were verified
in comparison to lysimeter measurements of grass evapotranspiration. Based on
summary statistics, the RBF network ranked first with the lowest RMSE value. The
RBF network was additional tested using mean monthly data collected in Novi Sad,
Serbia, and Kimberly, Idaho, U.S.A. The overall results recommended RBF
network for pan evaporation to evapotranspiration conversions.
This publication is organized as following. After introduction, in the second
chapter, the brief overview of the estimating FAO-24 coefficients by non-adaptive
RBF networks is presented. In the third chapter, the application of adaptive and
sequentially adaptive RBF structures and ARIMA models in forecasting reference
evapotranspiration is described. In the fourth chapter, the sequentially adaptive RBF
approach has been implemented and compared with empirical ETo equations.
Finally, the conclusions are presented in the fifth chapter.
7
II. ESTIMATING FAO-24 COEFFICIENTS BY RBF NETWORKS
II.1. Estimation of FAO-24 Blaney-Criddle b Factor by RBF Network
II.1.1. Introduction
The Food and Agricultural Organization of the United Nations (FAO-24) BlaneyCriddle formula has the form (Doorenbos and Pruitt 1977):
ETo = a + b[ p (0.46T + 8.13)]
(1)
where ETo = reference crop evapotranspiration (mm day-1); a and b = adjustment
factors; p = mean daily percentage of total annual daytime hours; and T = mean
daily air temperature, (oC). The a factor is estimated as:
a = 0.0043RH min − (n / N ) − 1.41
(2)
where RHmin = minimum daily relative humidity (%); and n/N = mean ratio of
actual to possible sunshine hours. Parameters p and N can be obtained from tables
for specific latitudes and months (Doorenbos and Pruitt 1977; Allen and Pruitt
1986); or they may be calculated using equations (Allen et al. 1989; Jensen et al.
1990).
Tabular b values are given in Appendix II of Doorenbos and Pruitt (1977).
The b factor is shown as a function of minimum daily relative humidity, RHmin;
mean daytime wind speed, Ud; and mean ratio of actual to possible sunshine hours,
n/N. The b value can be obtained using table interpolation. However, it is necessary
to make seven interpolations in order to obtain that value. Using this approach can
lead to a considerable error. Besides that, more time is needed to obtain a b value
(a few minutes). The second approach requires the use of regression equations first
introduced by Frevert et al. (1983) and later improved by Allen and Pruitt (1991).
Frevert et al. (1983) equation can be expresses as:
b = 0.81917 − 0.0040922 ⋅ RH min + 1.0705
− 0.0059684 ⋅ RH min
n
+ 0.065649 ⋅ U d
N
(3)
n
− 0.0005967 ⋅ RH min ⋅ U d
N
Allen and Pruitt (1991) presented a more accurate equation:
b = 0.908 − 0.00483 ⋅ RH min + 0.7949
− 0.0038 ⋅ RH min
− 0.0038 ⋅ RH min
n
2
+ 0.0768[ln(U d + 1)]
N
n
n
− 0.000433 ⋅ RH min ⋅ U d + 0.281 ⋅ ln(U d + 1) ⋅ ln( + 1)
N
N
n
n
− 0.000433 ⋅ RH min ⋅ U d + 0.281 ⋅ ln(U d + 1) ⋅ ln( + 1)
N
N
(4)
8
Estimating FAO-24 coefficients by RBF networks
In spite of the improvements, the error in estimating the b factor as compared to
tabular values may be as high as 10%. The aim of this section is to present a new
approach based on the RBF networks.
II.1.2. RBF networks
The RBF network is a feed-forward type of Artificial Neural Network (ANN).
In last decade, these networks have been used in hydrology (Mason et al. 1996; Han
and Felker 1997, Fernando and Jayawardena 1998, Trajkovic et al. 2000a,
Trajkovic 2005, Kisi 2008). The property of locality is the main reason why the
RBF network can be learned much faster than the multilayer perceptron (Park and
Sandberg 1991). An arbitrary function can be approximated by the linear
combination of locally tuned factorable basis functions. The RBF network has the
input layer with r neurons, the hidden layer with s neurons and the output layer with
t neurons. The internal units form a single layer of s receptive fields which can give
the localized response function in the input space (Figure 1). The output of the
RBF network is obtained by the following equations:
λ
( )= A ⋅ g (x )
λ
y =ϕ x
[
= [y
λ
λ
λ
(5)
]
,..., y ]
x = x1λ , x 2λ ,..., x rλ
y
λ
λ
1
, y 2λ
(6)
λ
(7)
t
( ) = [g (x ), g (x ),..., g (x )]
gx
λ
λ
1
where x
λ
λ
2
λ
(8)
s
λ
= real-valued vector in the input space; y = vector of output neurons
( ) = vector of response functions of the i-th receptive field;
activites; g x
λ
λ = number of samples, λ = 1, 2, ..., Λ; and A = [a ki ] , k = 1, 2, ..., t; i = 1, 2, ..., s;
are the output coefficients. In this section the Gaussian radial basis function is used
as response function:
( )
⎛ r ⎛ xλ − m ⎞2 ⎞
j
ij
⎜
⎟ ⎟
g i x = exp⎜ − ∑ ⎜
(9)
⎜
⎟ ⎟⎟
σ
⎜ j =1 ⎝
ij
⎠ ⎠
⎝
where mij and σij , = center and the width of the radial basis function gi for
λ
j-th input .
Tasks that the learning algorithm should perform can be formulated as follows.
Given Λ input/output data and the specified model error ε > 0, obtain the optimal
solution for network parameters, aki, mij and σij, k = 1, 2, ..., t; i = 1, ..., s; j = 1, ..., r,
which satisfies the inequality:
E=
y
*,λ
*, λ
1 Λ λ
y −y
∑
2 λ =1
2
<ε
(10)
]
(11)
[
= y1*,λ , y 2*,λ ,..., y t*,λ
where y
*, λ
9
is a vector of desired output neurons activities. Parameter tuning
process is based on the gradients ∂E / ∂a ki , ∂E / ∂m ij , ∂E / ∂σ ij , k = 1, ..., t; i =
1, ..., s; j = 1, ..., r, derivation:
Λ
E = ∑ Eλ , Eλ =
λ =1
(
1 t
y kλ − y k*,λ
∑
2 k =1
)
(12)
( )
(13)
2
λ
∂E λ ∂E λ ∂y kλ
= λ
= ( y kλ − y k*,λ )⋅ g i x
∂a ki ∂y k ∂a ki
λ
λ
∂E λ
∂E λ ∂g i ( x ) ⎛⎜ t ∂E λ ∂y kλ ⎞⎟ ∂g i ( x )
=
= ∑ λ
λ
⎜ k =1 ∂y k
∂mij ∂g ( x λ ) ∂mij
∂g i ( x ) ⎟⎠ ∂m ij
i
⎝
λ
(14)
λ
∂E λ
∂E λ ∂g i ( x ) ⎛⎜ t ∂E λ ∂y kλ ⎞⎟ ∂g i ( x )
=
= ∑ λ
λ
⎜ k =1 ∂y k
∂σ ij ∂g ( x λ ) ∂σ ij
∂g i ( x ) ⎟⎠ ∂σ ij
i
⎝
(15)
t
∂E λ ∂y kλ
(y kλ − y k*,λ )⋅ a ki
=
∑
∑
λ
λ
k =1 ∂y k ∂g ( x )
k =1
i
(16)
t
λ
λ
λ x j − m ij
∂g i ( x )
= 2g i (x )
∂mij
σ ij2
(17)
λ
λ (x j − m ij )
∂g i ( x )
= 2g i (x )
∂σ ij
σ ij3
λ
2
(18)
Tuning of the parameter q using the gradient method with fixed step size is defined
by the following iterative process:
q(n +1) = q(n ) + Δq(n )
Δq(n ) = −η
∂E (n )
∂q (n )
(19)
(20)
An enhanced version of back-propagation uses a momentum term and flat regions
elimination. The momentum term introduces the old parameter change as a
10
parameter for the computation of the new weight change. The momentum term is
used in this section, so as to avoid the oscillation problems common with the
regular back-propagation algorithm when the error surface has a very narrow
minimum area. The new parameter update is computed by:
Δq (n + 1) = −η
∂E (n )
+ αΔq(n )
∂q (n )
(21)
where α is the momentum and η is the learning step. This adaptation of the step
size increases learning speed significantly.
II.1.3. Estimation of factor b
RBF networks used for estimation of the b factor have the following structure
(Figure 1). There are three neurons in the input layer, defined by the fact that the
values of the b factor depend on three variables (Ud, n/N and RHmin). The number of
neurons in the hidden layer varies from 10 to 70. There is one neuron in the output
layer of the networks.
y=b
a11
g1
a 12
g2 ...
x1 = Ud
a1i
g i ...
x 2 = n/N
a 1s-1
a1s
gs-1
gs
x3 = RHmin
Figure 1. Structure of RBF network used to estimate calibration factor b
Samples (216 tabular b factors) were divided into two groups. For the RBF
networks training, 186 randomly chosen training samples (nonunderlined values in
Table 1) were used. For verification of networks, obtained in a stage of training,
all samples (216 tabular b factors) were used. Thus, the b values produced by RBF
networks can be compared to the regression estimates and table values. Thirty table
b values, found in the verifying set only, are used for controlling the ability of the
networks to generalize the knowledge obtained during the training stage.
n/N
11
Table 1. FAO-24 Blaney-Criddle b factors (Doorenbos and Pruitt 1977)
RHmin (%)
0
20
40
60
80
100
Ud
(m s-1)
0.0
0.2
0.4
0.6
0.8
1.0
0.84
1.03
1.22
1.38
1.54
1.68
0.80
0.95
1.10
1.24
1.37
1.50
0.74
0.87
1.01
1.13
1.25
1.36
0.64
0.76
0.88
0.99
1.09
1.18
0.52
0.63
0.74
0.85
0.94
1.04
0.38
0.48
0.57
0.66
0.75
0.84
0
0.0
0.2
0.4
0.6
0.8
1.0
0.97
1.19
1.41
1.60
1.79
1.98
0.90
1.08
1.26
1.42
1.59
1.74
0.81
0.96
1.11
1.25
1.39
1.52
0.68
0.8
0.97
1.09
1.21
1.31
0.54
0.66
0.77
0.89
1.01
1.11
0.40
0.50
0.60
0.70
0.79
0.89
2
0.0
0.2
0.4
0.6
0.8
1.0
1.08
1.33
1.56
1.78
2.00
2.19
0.98
1.18
1.38
1.56
1.74
1.90
0.87
1.03
1.19
1.34
1.50
1.64
0.72
0.87
1.02
1.15
1.28
1.39
0.56
0.69
0.82
0.94
1.05
1.16
0.42
0.52
0.62
0.73
0.83
0.92
4
0.0
0.2
0.4
0.6
0.8
1.0
1.18
1.44
1.70
1.94
2.18
2.39
1.06
1.27
1.48
1.67
1.86
2.03
0.92
1.10
1.27
1.44
1.59
1.74
0.74
0.91
1.06
1.21
1.34
1.46
0.58
0.72
0.85
0.97
1.09
1.20
0.43
0.54
0.64
0.75
0.85
0.95
6
0.0
0.2
0.4
0.6
0.8
1.0
1.26
1.52
1.79
2.05
2.30
2.54
1.11
1.34
1.56
1.76
1.96
2.14
0.96
1.14
1.32
1.49
1.66
1.82
0.76
0.93
1.10
1.25
1.39
1.52
0.60
0.74
0.87
1.00
1.12
1.24
0.44
0.55
0.66
0.77
0.87
0.98
8
0.0
0.2
0.4
0.6
0.8
1.0
1.29
1.58
1.86
2.13
2.39
2.63
1.15
1.38
1.61
1.83
2.03
2.22
0.98
1.17
1.36
1.54
1.71
1.86
0.78
0.96
1.13
1.28
1.43
1.56
0.61
0.75
0.89
1.03
1.15
1.27
0.45
0.56
0.68
0.79
0.89
1.00
10
12
A network with twenty neurons in the hidden layer, which gave the minimum error
at the verifying stage, was chosen for use. An analysis and comparison of the results
of 216 b values produced by the RBF network, rather than from the regression
equations, is given in Table 2. In this table, DEV is the standard deviation of
absolute relative error and R2 is the coefficient of determination. The mean absolute
relative error (MARE) of the RBF network is 0.34%, while the corresponding error
of the regression equations is 1.69% (Allen and Pruitt 1991) and 3.07%
(Frevert et al. 1983). The RBF network has maximum absolute relative error
(MXARE) less than 2%, while in the regression equations it is over 11% and 14%,
respectively. The number of samples with an error greater than 2% (NE>2%) in the
regression models is 64 and 128, respectively, while in the RBF network there is
not a factor with such a large error.
Table 2. Comparison of models used to estimate FAO-24 Blaney-Criddle b factors
Model
MARE
MXARE
NE>2%
DEV
R2
(%)
(%)
(%)
Frevert et al. (1983)
3.07
14.4
126
2.72
0.989
Allen and Pruitt (1991)
1.69
11.8
64
1.68
0.998
RBF Network
0.34
1.8
0
0.31
1.000
II.1.4. Application
The example data set is for monthly weather data at Nis, Serbia, during 1991.
The b factor was produced by RBF network (brbf), Allen's and Pruitt's regression
equation (breg) and table interpolation (bti). The parameters p and N were obtained
by table interpolation. The latitude of Nis is 43.3 oN. Reference crop
evapotranspiration was estimated using the FAO-24 Blaney-Criddle equation.
Table 3 gives values of weather data, b factors and reference evapotranspiration.
There is a good agreement of the factors obtained by RBF network and table
interpolation, which automatically lead to agreement in estimated
evapotranspirations. The relative difference is less than 0.4% (3 mm/year) in
evapotranspiration estimates. The greatest difference of 1.2 mm occurs in July.
On the other hand, b factors produced by the regression equation are underestimated
2-5% when compared to values obtained using table interpolation, which can lead
to significant differences in reference evapotranspiration estimates. The yearly
difference is 41.5 mm. The greatest monthly difference of 6 mm occurs in July.
13
Table 3. Estimation of monthly reference evapotranspiration (mm) in Nis, Serbia, with
b factors were obtained by RBF network, regression equation or, table interpolation
months
parameters
II
III
IV
V
VI
VII VIII IX
X
XI year
1.8
8.3
10.5
12.7
20.6
21.4
19.6
17.9
11.6
7.8
o
T ( C)
RHmin (%)
65
50
50
61
45
55
56
43
55
63
U2 (m/s)
1.4 1.89 1.65 1.6 0.77 1.17
1
1.25 1.44 1.34
n (h)
81.4 135.2 156.9 141.6 292.6 250.5 221 236.6 110.6 69.5
n/N
0.276 0.366 0.390 0.311 0.636 0.535 0.51 0.626 0.323 0.238 p
0.24 0.27 0.3 0.33 0.347 0.337 0.31 0.28 0.25 0.22
brbf
0.823 1.012 1.017 0.884 1.165 1.053 1.022 1.202 0.93 0.811 breg
0.788 0.977 0.981 0.846 1.136 1.015 0.982 1.179 0.889 0.775 bti
0.821 1.011 1.016 0.886 1.162 1.047 1.017 1.199 0.928 0.813 (brbf-bti)/bti 0.2% 0.1% 0.1% -0.2% 0.3% 0.6% 0.5% 0.3% 0.2% -0.2% (breg-bti)/bti -4.0% -3.4% -3.4% -4.5% -2.2% -3.1% -3.4% -1.7% -4.2% -4.7% ETo_rbf (mm) 10.1 52.8 71.1 81.1 157.9 144.5 116.3 109.7 50.7 21.4 815.6
ETo_reg (mm) 8.0 49.3 66.9 75.7 152.6 137.3 109.8 106.5 46.4 18.6 771.1
ETo_ti (mm) 10.0 52.7 70.9 81.4 157.4 143.3 115.5 109.3 50.5 21.6 812.6
PErbf
0.1
0.1
0.2 -0.3 0.5
1.2
0.8
0.4
0.2 -0.2 3.0
PEreg
-2.0 -3.4 -4.0 -5.7 -4.8 -6.0 -5.7 -2.8 -4.1 -3.0 -41.5
PErbf = ETo_rbf – ETo_ti ; PEreg = ETo_reg – ETo_ti
II.1.5. Conclusions
This section presents a new approach for estimating the b factor of the FAO-24
Blaney-Criddle equation. The RBF network estimated b factors better than the
regression equations. Improved estimates of the b factors do reduce the difference
between evapotranspiration estimates. The use of RBF network is very simple and
does not require any knowledge of ANNs.
14
II.2.
Estimation of FAO-24 pan Kp Factor by RBF Network
Irrigation projects require an accurate estimation of evapotranspiration for their
effective planning, design, and operation. Evapotranspiration can be estimated using
pan evaporation data. Evaporation pans are used to estimate reference crop
evapotranspiration by multiplying pan evaporation amounts by pan factor (Kp).
The model relating the pan evaporation (Epan) to the reference crop
evapotranspiration (ETo) is:
ETo = K p E pan
(22)
Table 4. FAO-24 pan Kp factors (Doorenbos and Pruitt, 1977)
U2 (km day-1)
F(m)
RHmean (%)
<40
40-70
>70
a
a
(40 )
(55 )
(70a)
0.65
0.75
1
0.55
<175
(175a)
10
0.65
0.75
0.85
100
0.70
0.80
0.85
1000
0.75
0.85
0.85
1
0.50
0.60
0.65
175-425 (300a)
10
0.60
0.70
0.75
100
0.65
0.75
0.80
1000
0.70
0.80
0.80
a
0.45
0.50
0.60
1
425-700 (562 )
10
0.55
0.60
0.65
100
0.60
0.65
0.70
1000
0.65
0.70
0.75
1
0.40
0.45
0.50
>700
(700a)
10
0.45
0.55
0.60
100
0.50
0.60
0.65
1000
0.55
0.60
0.65
a Representative (mean) value
Tabular pan factors are given in Table 4. The pan factor is shown as a function of
mean daily relative humidity, RHmean; wind speed, U2; and upwind fetch of lowgrowing vegetation, F. The pan factors can be obtained using regression equations
first introduced by Frevert et al. (1983) and later improved by Snyder (1992).
Frevert et al. (1983) equation can be expressed as:
k p = 0.475 − 2.35 ⋅ 10 −4 U 2 + 5.16 ⋅ 10 −3 RH mean + 1.18 ⋅ 10 −3 F − 1.63 ⋅ 10 −5 ( RH mean ) 2 +
−3
−5
−9
1.18 ⋅ 10 F − 1.63 ⋅ 10 ( RH mean ) − 9.07 ⋅ 10 ( RH mean ) F
2
2
(23)
15
Snyder (1992) presented a simpler, more accurate equation. This equation has the
form:
k p = 0.482 + 0.024 ln( F ) − 0.000376U 2 + 0.0045 RH mean
(24)
In spite of the improvements, the error in estimating the pan factor as compared to
tabular values may be as high as 10%. A great problem at defining models for the
estimation of pan factor is represented by the absence of precisely determined input
values given in the table in broad limits.
Frevert and Snyder used the representative (mean) values of each range of relative
humidity and wind speed. However, the use of RHmean and U2 values different from
representatives results in underprediction or overprediction of pan factor (Trajkovic
et al. 2000b). The aim of this section is to present a new approach for estimating Kp
factor based on the RBF networks. The new approach uses ranges of relative
humidity and wind speed as qualitative variables.
II.2.2. Estimation of Kp factor
In this secton, the Radial Basis Function (RBF) network from Trajkovic et al.
(2000b) was applied for estimation of the pan factor. RBF networks used for
estimation of the pan factor have the following structure. There are eight neurons in
the input layer (F, U2<175, 175<U2<425, 425<U2<700, U2>700, RHmean<40,
40<RHmean<70, and RHmean>70). There are three active input neurons (F,
appropriate U and RHmean neurons). Value 1 is brought at the input of active U2 and
RHmean neurons, and value 0 at nonactive neurons. For instance, for wind speed
equal to U2=181 km day-1 value 1 is brought at U2175-425 neuron, while value 0 is
brought at other U2 neurons. The number of neurons in the hidden layer varies from
8 to 20. There is one neuron in the output layer of the networks.
Samples (48 tabular factors) were divided into two groups. For the RBF networks
training, 40 randomly chosen training samples (nonunderlined values in Table 1)
were used. For verification of networks, obtained in a stage of training, all samples
(48 tabular factors) were used. Thus, the pan factors produced by RBF networks can
be compared to the regression estimates and table values. Eight table values, found
in the verifying set only, are used for controlling the ability of the networks to
generalize the knowledge obtained during the training stage. A network with ten
neurons in the hidden layer, which gave the minimum error at the verifying stage,
was chosen for use.
An analysis and comparison of the results of 48 pan factors produced by the RBF
network, rather than from the regression equations, is given in Table 5. In this table,
DEV is the standard deviation of absolute relative error and R2 is the coefficient of
determination.
16
Table 5. Comparison of models used to estimate FAO-24 pan Kp factor
Model
MARE MXARE NE>4% DEV
R2
(%)
(%)
Frevert et al. (1983) 6.43
20.0
29
0.051 0.874
Snyder (1992)
3.27
10.0
18
0.025 0.950
RBF
0.42
3.6
0
0.009 0.997
The mean absolute relative error (MARE) of the RBF network is 0.42%, while the
corresponding error of the regression equations is 6.43% (Frevert et al. 1983) and
3.27% (Snyder 1992). The RBF network has maximum absolute relative error
(MXARE) less than 3.6%, while in the regression equations it is 20% and 10%,
respectively. The number of samples with an error greater than 4% (NE>4%) in the
regression models is 29 and 18, respectively, while in the RBF network there is not
a factor with such a large error.
II.2.3. Application
The following section includes examples for applying a new approach for the
estimation of FAO-24 pan Kp factor. The evaporation date came from pans at
Kimberly, Idaho, USA (July); Griffith, NSW, Australia (January); and Novi Sad,
Serbia (July). The pan factors were produced by RBF network, Snyder's regression
equation and table interpolation. Table 6 gives values of pan factors.
Table 6. Pan factors obtained by RBF network, regression equation and interpolation
Model
U2
F
RHmean
Kp
Difference
(km day-1)
(m)
(%)
Kimberly, Idaho, USA, July
Snyder
190
20
58
0.743
5.3%
RBF
190
20
58
0.709
0.4%
Table
190
20
58
0.706
Griffith, NSW, Australia, January
Snyder
155
100
41
0.719
-10.2%
RBF
155
100
41
0.802
0.3%
Table
155
100
41
0.8
Novi Sad, Vojvodina, Serbia, July
Snyder
181
10
65
0.762
8.8%
RBF
181
10
65
0.704
0.6%
Table
181
10
65
0.7
-
17
Snyder's regression equation is practically inapplicable for conditions existing in
Griffith (Trajkovic and Stojnic 2008) and Novi Sad. The estimation of pan factors
by regression equation for Kimberly is a more accurate because the values of wind
speed and relative humidity are close to mean values used in the development of the
regression equation. The RBF network estimates the pan factors for all three
locations with greater accuracy than the regression equation (Trajkovic and
Kolakovic 2009d).
II.2.4. Conclusion
This section presents the application of radial basis function (RBF) network to
estimate the FAO-24 pan Kp factor. The application of the regression equations, in
spite of the improvements done by Snyder, does not always give satisfactory
results. The RBF network estimated pan factors better than the regression equations.
All advantages of the RBF network over the regression equation were demonstrated
by means of a particular example.
18
II.3.
Estimation of FAO-24 radiation Cr Factor by RBF Network
The estimation of evapotranspiration is an important component in agricultural
water research, management and development. The FAO-24 equations are
recognized as the international standard for predicting crop water requirement and
have been used worldwide by hydrologists (Jensen et al. 1990; Chiew et al. 1995).
FAO-24 Radiation equation (Doorenbos and Pruitt 1977) can be used as a surrogate
for FAO Penman-Monteith approach (Allen et al., 1998) for areas where wind
speed and humidity data are not available. In this eqaution, relationship is given
between radiation term (WRs) and reference evapotranspiration (ETo).
The relationship recommended is expressed as:
ETo = C r (W ⋅ Rs )
(25)
where: Cr = FAO-24 radiation adjustment factor, W = weighting factor, and
Rs = solar radiation (mm day-1). The weighting factor (W) can be obtained from
table for specific altitude and temperature (Doorenbos and Pruitt, 1977) or it may be
calculated using equations (Allen and Pruitt, 1991). Tabular adjustment factors are
given in Appendix II of Doorenbos and Pruitt (1977). The c factor is shown as a
function of mean daily relative humidity (RHmean) and daytime wind speed at 2 m
(U2). The adjustment factor can be obtained using table interpolation. However, it is
necessary to make three interpolations in order to obtain that value. Using this
approach can lead to a considerable error. Besides that, more time is needed to
obtain an adjustment factor. The aim of this section is to present a new approach
based on the neural networks.
II.3.2. Estimation of FAO-24 radiation adjustment c factor
In this section, the Radial Basis Function (RBF) network from Trajkovic et al.
(2003a) was applied for estimation of the FAO-24 Radiation c factor. The neural
networks used for estimation of the c factor have the following structure. There are
two neurons in input layer. The number of hidden neurons values is from six to
thirty. There is one neuron in the output layer. Samples (110 tabular c values) were
divided into two groups. For the networks training, 90 randomly chosen training
samples were used. For verification of eleven neural networks, obtained in a stage
of training, a group of twenty test samples is used (underlined values in Table 7).
On the basis of the root-mean-square error and number of test samples with error
less than 0.5%, four networks were chosen for further testing by using all samples
(110 tabular c values). Thus, the c factors obtained by neural networks can be
compared to the table values. Twenty table c values, found in the verifying set only,
are used for controlling the ability of the networks to generalize the knowledge
obtained during the training stage. Structure of the chosen networks is given in
Table 8 in which abbreviation NIN denotes the number of input neurons, NHN is
19
the number of hidden neurons, NON is the number of output neurons, and NI the
number of iterations.
U2
(m s-1)
0
1
2
3
4
5
6
7
8
9
10
Table 7. FAO-24 radiation adjustment Cr factors
RHmean (%)
10
20
30
40
50
60
1.04
1.02
0.99
0.95
0.91
0.87
1.09
1.07
1.04
1.00
0.96
0.91
1.13
1.11
1.08
1.04
0.99
0.94
1.17
1.15
1.11
1.07
1.02
0.97
1.21
1.18
1.14
1.10
1.05
0.99
1.24
1.21
1.17
1.13
1.07
1.01
1.27
1.24
1.20
1.15
1.09
1.03
1.29
1.26
1.22
1.17
1.11
1.05
1.31
1.28
1.24
1.19
1.13
1.07
1.34
1.30
1.26
1.21
1.15
1.09
1.36
1.32
1.28
1.23
1.17
1.10
70
0.82
0.85
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.01
1.02
80
0.76
0.79
0.81
0.83
0.85
0.87
0.89
0.91
0.92
0.93
0.94
90
0.70
0.73
0.74
0.76
0.78
0.80
0.81
0.83
0.84
0.85
0.86
100
0.64
0.66
0.67
0.69
0.70
0.72
0.73
0.74
0.75
0.76
0.77
Table 8. Structure of RBF networks used to estimate FAO-24 radiation adjustment factors
RBF network
NIN
NHN
L/U BIWN
NON
NI
c2.net
2
20
10/10
1
200
c6.net
2
20
5/10
1
300
c9.net
2
10
1/2
1
400
c10.net
2
10
2/5
1
300
Chosen networks have twenty hidden neurons (c2.net and c6.net) and ten hidden
neurons (c9.net and c10.net). The lower/upper bound of initial width of neurons
(L/U BIWN) is 10/10, 5/10, 1/2, and 2/5, respectively. Training of networks was
completed after 200, 300, 400, and 300 iterations, respectively. Differences between
results of chosen RBF networks and tabular c values are shown in Table 9.
Comparison of the neural networks is carried out according to the following: mean
absolute relative error (MARE), maximum absolute relative error (MXARE), rootmean-square error (RMSE) and number of samples with error greater than 0.5%
(NE>0.5%).
The best results are provided by c9.net. Mean relative error is 0.16%. None of the
samples has an error greater than 1%, and only five samples have an error which is
greater than 0.5%. Maximum relative error is 0.72%.
20
Table 9. Comparison of RBF networks used to estimate FAO-24 radiation Cr factors
RBF network
MARE
MXARE
RMSE
NE>0.5%
(%)
(%)
c2.net
0.23
0.71
0.000386
7
c6.net
0.21
0.68
0.000356
5
c9.net
0.16
0.72
0.000211
5
c10.net
0.15
0.75
0.000271
3
II.3.3. Conclusions
This section presents the application of an artificial neural network (ANN) to
estimate the FAO-24 radiation adjustment factor. The RBF networks provide
improvements over tabular intepolation and adequately estimate values of
adjustment factors introduced by Doorenbos and Pruitt (1977). Improved estimates
of the adjustment factors do reduce the difference between evapotranspiration
estimates.
II.4.
21
Estimation of FAO-24 Penman c Factor by RBF Network
The Penman equation is used worldwide in evapotranspiration estimation.
A frequently used version is (Doorenbos and Pruitt 1977):
ETo = c[WRn + (1 − W )0.27(1 + 0.01U 2 )(ea − ed )]
(26)
where ETo = reference evapotranspiration (mm day-1); c = FAO-24 Penman
adjustment factor; W = weighting factor; Rn = net radiation (mm day-1); U2 = mean
wind speed at 2 m (km day-1); ea = saturation vapor pressure (mbar); and ed = actual
vapor pressure (mbar).
Tabular values of the c factor are given in Appendix II of Doorenbos and Pruitt
(1977). The c factor is shown as a function of daily global solar radiation, Rs;
maximum daily relative humidity, RHmax; mean daytime wind speed, Ud; and the
ratio of daytime to nighttime wind speeds Ud/Un.
Weighting factor is the weighting factor for the effect of radiation on reference
evapotranspiration. A table defining W is provided in Table 4 of Doorenbos and
Pruitt's paper. Net radiation Rn is the difference between all incoming and outgoing
radiation and estimated as a function of the extraterrestrial radiation, Ra, and the
maximum sunshine hours, N. Parameters Ra and N can be obtained from tables for
specific latitude and months (Doorenbos and Pruitt, 1977); or it may be calculated
using equations (Allen et al., 1989), (Jensen et al., 1990). Values of saturation vapor
pressure ea can be determined from Table 5 of Doorenbos and Pruitt's paper.
The values of parameters W, Ra, N and ea can be easily obtained by table
interpolation. Most often, only one interpolation is needed to obtain the accurate
values of the appropriate parameter. The value of c factor can also be obtained
using table interpolation. However, it is necessary to make 15 interpolations in
order to obtain that value, which requires the introduction of 45 different numbers
into the calculation. This way of calculation, with 15 interpolations, can lead to the
possibility of making an error. Besides that, more time is needed to obtain c value
(a few minutes).
The second approach that is used for estimating c values requires the use of
regression expressions first introduced by Frevert et al. (1983) and later improved
by Allen and Pruitt (1991). Frevert et al. (1983) equation can be expressed as:
U
c = 0.6817006+ 0.0027864⋅ RH max + 0.0181768⋅ Rs − 0.0682501⋅U d + 0.0126514 d
Un
U
U
+ 0.0097297⋅U d d + 4.3025⋅10−5 ⋅ RH max ⋅ Rs ⋅U d − 9.2118⋅10−8 ⋅ RH max ⋅ Rs d
Un
Un
(27)
Allen and Pruitt (1991) presented a more accurate equation. This equation has the
form:
22
U
c = 0.892− 0.0781⋅U d + 0.00219⋅U d ⋅ Rs + 0.000402⋅ RHmax ⋅ Rs + 0.000196 d U d ⋅ RHmax
Un
+ 0.0000198
⎛U
Ud
2
U d ⋅ RH max ⋅ Rs + 0.00000236 ⋅ U d ⋅ RH max ⋅ Rs − 0.0000086⎜⎜ d
Un
⎝ Un
2
⎞
⎟⎟ U d ⋅ RH max
⎠
(28)
Ud
2
U d 2 ⋅ RH max
− 0 . 0000000292
⋅ Rs − 00000161 ⋅ RH max ⋅ Rs 2
Un
Despite these improvements, the error in calculating FAO-24 Penman c Factor may
be as high as 30%. In Kotsopoulos and Babajimopoulos (1997) analytical
expression for computing FAO-24 Penman c Factor has been proposed.
The proposed equation has the following form:
U
c = 1.5033− 1.5904⋅ ( RH max ) −0.125 + 0.3216⋅ ( Rs ) 0.2 − 0.2454⋅ (U d ) 2 / 3 + 0.03985 d (U d ) 0.4
Un
+ 0.02215(U d ) 0.55 (RHmax ) 0.45 + 0.002548(Rs )1.45 (U d ) 2 / 3 − 2.3464⋅10−6 (RHmax )1.5 (Rs )1.5 (U d ) 0.4
U
Ud
− 8.15849 ⋅10 − 6 ( RH max ) 1.5 (U d ) 0.4 d
Un
Un
Ud
U
+ 1.19257 ⋅10 − 6 ( RH max ) 1.5 ( R s ) 1.5 (U d ) 0.4 d
Un
Un
− 1.01086 ⋅10 − 7 ( RH max ) 1.5 ( R s ) 1.5
− 0.000496( R s ) 1.5 (U d ) 0.4
(29)
In spite of the improvements, the number of c factors with an error bigger than 4%
is still a large one. It shows that it is necessary to develop a new approach of
estimating the values of c factors. The aim of this section is to present new approach
based on the RBF networks that would be simple to use, because it wouldn't
demand from a user any background knowledge of Artificial Neural Networks
(ANNs).
II.4.2. Estimation of factor c
In this section, the Radial Basis Function (RBF) network from Trajkovic et al.
(2001) was applied for estimation of the FAO-24 Penman c factor. The RBF
networks used for estimation of the factor c have the following structure. There are
four neurons in the input layer. Their number is defined by the fact that the values
of the c factor depend on four variables (RHmax, Rs, Ud, and Ud/Un). The number of
neurons in the hidden layer varies from 6 to 60. There is one neuron in the output
layer of the RBF networks.
Samples (192 tabular values of factor c) are divided into two groups. For the RBF
network training, 168 randomly chosen training samples (no underlined values in
Table 10) were used. All samples (192 tabular values of the c factor) are used for
verification of RBF networks, obtained in a stage of training. Thus, the c values
produced by RBF networks can be compared to the regression estimates and table
values. Twenty-four table c values found in the verifying set only were used for
controlling the ability of the networks to generalize the knowledge obtained during
the training stage.
23
A network with ten neurons in the hidden layer, which gave the minimum error at
the verifying stage, was chosen for use. Table 2 shows the comparison of the
c values produced by RBF networks or regression expressions with 192 tabular
c values. In this table MARE denotes mean absolute relative error; MAXPRE the
maximum positive relative error; MAXNRE the maximum negative relative error;
NE the number of test samples with error greater than 4% (NE>4%); DEV is the
standard deviation of absolute relative error and R2 is the coefficient of
determination.
Table 10. FAO-24 Penman c factor (Doorenbos and Pruitt 1977)
SOLAR RADIATION (mm day-1)
Ud
RHmax = 30%
RHmax = 60%
(m s-1) 3
6
9
12
3
6
9
12
3
RHmax = 90%
6
9
12
0
3
6
9
0.86
0.79
0.68
0.55
0.90
0.84
0.77
0.65
1.00
0.92
0.87
0.78
1.00
0.97
0.93
0.90
(a) Ud/Un = 4
0.96 0.98 1.05
0.92 1.00 1.11
0.85 0.96 1.11
0.76 0.88 1.02
(b) Ud/Un = 3
0
3
6
9
0.86
0.76
0.61
0.46
0.90
0.81
0.68
0.56
1.00
0.88
0.81
0.72
1.00
0.94
0.88
0.82
0.96
0.87
0.77
0.67
1.05
1.06
1.02
0.88
1.05
1.12
1.10
1.05
1.02
0.94
0.86
0.78
1.06
1.04
1.01
0.92
1.10
1.18
1.15
1.06
1.10
1.28
1.22
1.18
0
3
6
9
0.86
0.69
0.53
0.37
0.90
0.76
0.61
0.48
1.00
0.85
0.74
0.65
1.00
0.92
0.84
0.76
(c) Ud/Un = 2
0.96 0.98 1.05
0.83 0.91 0.99
0.70 0.80 0.94
0.59 0.70 0.84
(d) Ud/Un = 1
1.05
1.05
1.02
0.95
1.02
0.89
0.79
0.71
1.06
0.98
0.92
0.81
1.10
1.10
1.05
0.96
1.10
1.14
1.12
1.06
0
3
6
9
0.86
0.64
0.43
0.27
0.90
0.71
0.53
0.41
1.00
0.82
0.68
0.59
1.00
0.89
0.79
0.70
0.96
0.78
0.62
0.50
1.05
0.99
0.93
0.87
1.02
0.85
0.72
0.62
1.06
0.92
0.82
0.72
1.10
1.01
0.95
0.87
1.10
1.05
1.00
0.96
0.98
0.96
0.88
0.79
0.98
0.86
0.70
0.60
1.05
0.94
0.84
0.75
1.05
1.19
1.19
1.14
1.02
0.99
0.94
0.88
1.06
1.10
1.10
1.01
1.10
1.27
1.26
1.16
1.10
1.32
1.33
1.27
Mean absolute relative error of the RBF network is 0.54%, while the corresponding
error of the regression expressions is 2.93% (Allen and Pruitt 1991), 3.69% (Frevert
et al. 1983) and 1.69 (Kotsopoulos and Babajimopoulos 1997). The RBF network
has maximum error less than 3%, while in the regression expressions it is over 32%,
26%, and 5%, respectively. Number of samples with error greater than 4% in the
24
regression models is 36, 58, and 11, respectively, while in the RBF networks there
is not a factor with such a large error.
Table 11. Comparison of various methods used to calculate FAO-24 Penman c factor
MARE MAXPRE MAXNRE NE>4% DEV R2
Model
(%)
(%)
(%)
(%)
Frevert et al. (1983)
3.69
26.2
13.0
58
3.744 0.955
Allen and Pruitt (1991) 2.93
32.4
13.3
36
3.981 0.979
Eq (29)
1.69
5.6
5.4
11
1.266 0.989
RBF network
0.54
2.8
2.9
0
0.523 0.999
II.4.3. Application
The following section includes examples for applying a new approach for factors
estimation. The example data set is for average weather data at Beograd, Serbia
during April, July and September from 1971 to 1975. The use of a trained RBF
network is very simple and does not require any knowledge of ANN.
The agreement between the c values produced by trained RBF network and the
table interpolation is great. Using the RBF network for the April, where variables
RH=72.4%, Rs=6.34 mm day-1, Ud= 3.14 m s-1 and Ud/Un = 1.13, c is equal to 0.886.
The interpolated c value from Table 1 for the same data is 0.896 (difference of
1.1%). Applying the RBF network for the July, where variables RH=76.4 %,
Rs=8.74 mm day-1, Ud=2.22 m s-1 and Ud/Un =1.22, c is equal to 0.991. The
interpolated c value from Table 1 for the same data is 1.009 (difference of 1.7%).
Using the RBF network for the September, where variables RH=84.2%, Rs=5.68
mm day-1, Ud=2.34 m s-1 and Ud/Un = 1.05, c is equal to 0.934. The interpolated
c value from Table 1 for the same data is 0.931 (difference of 0.3%). RBF networks
in comparison with table interpolation obtain the c values twenty times faster.
II.4.4. Conclusions
The validity of evapotranspiration calculation by FAO-24 Penman equation is
increased with the accurate estimation of c factors. The determining of c values by
table interpolation should be avoided because of its long procedure that can lead to
a high error, which is directly transferred to the estimated evapotranspiration (see
the equation (26)). The application of the regression equations, in spite of the
improvements done by Allen and Pruitt, does not always give satisfactory results.
The comparative analysis showed that RBF networks guarantee a more accurate
estimation of c factors when compared to regression equations.
III.
25
FORECASTING OF ETo BY RBF NETWORKS
III.1. Forecasting of reference evapotranspiration by adaptive RBF networks
III.1.1. Introduction
The ability to forecast reference evapotranspiration is of utmost importance for
operating irrigation systems effectively in agricultural areas where crop production
is the principal user of water. There are several methods for forecasting
evapotranspiration. Tracy et al. (1992) showed the unreliability of the forecast
obtained by the simple Yearly Differencing (YD) or Monthly Average (MAV)
models. Several investigators have found the seasonal autoregressive integrated
moving average (SARIMA) model provides better agreement with the observed
time series in comparison to the YD and MAV models (Marino et al. 1993;
Trajkovic 1998). Hameed et al. (1995) tested the possibility of using the transfer
function noise (TFN) model. They obtained results similar to the SARIMA model.
In last decade artificial neural networks (ANNs) have been successfully applied to
the forecasting of hydrology time series (Fernando and Jayawardena 1998;
Elshorbagy et al. 2000, Coulibaly et al. 2000, Trajkovic et al. 2003c, GonzalesCamacho et al. 2008, Kisi 2008). The objectives of this study were: first, to present
an adaptive RBF network for forecasting reference evapotranspiration; second, to
evaluate the reliability of two approaches (RBF network and SARIMA model) for
forecasting ETo.
III.2.1. Materials and methods
III.2.1.1. ETo data
Reference evapotranspiration data for Griffith, Australia are used in this study.
A plot of the reference evapotranspiration in Griffith is shown in Figure 2.
350
Griffith, NSW, Australia
ETo
-1
(mm month )
300
250
200
150
100
50
Months
0
0
12
24
36
48
60
72
84
96
108 120 132
Figure 2. Reference evapotranspiration at Griffith, Australia from 1962 to 1972
26
Forecasting of ETo by RBF networks
A total of 132 mean monthly ETo values from January 1962 to December 1972 are
available for use in this study. The data were divided into two groups. A calibration
set is used to calibrate the forecasting models and encompasses the first 120
months. A validation set is used for model validation and encompasses the last 12
months.
III.2.1.2. ARIMA model
The Box-Jenkins method is one of the most popular time series forecasting methods
in hydrology (Marino et al. 1993, Maier and Dandy 1996, Trajkovic 1998 and Jain
et al., 1999). The method uses a systematic procedure to select an appropriate model
from a rich family of models (ARIMA models).
AutoRegressive (AR) models estimate values for the dependent variable Xt as a
regression function of previous values Xt-1, ..., Xt-p plus some random error et.
Moving Average (MA) models give a series value Xt as a linear combination of
some finite past random errors, e t-1,..., et-p. p and q are referred as orders of the
models. AR(p) and MA(q) models can be combined to form an ARMA(p,q) model.
This model can provide additional flexibility in describing of the time series.
However, a large number of time series is nonstationary and for the modeling of
such time series, simple AR, MA or ARMA models are not appropriate. Box and
Jenkins (1976) suggested that a nonstationary series can be transformed into a
stationary one by differencing. The ARMA models applied to the differenced series
are called integrated models, denotes by ARIMA (AutoRegressive Integrated
Moving Average) models.
A time series involving seasonal data will have relations at a specific lag s which
depends on the nature of the data, e.g. for monthly data s =12. Such series can be
successfuly modeled only if the model includes the connections with the seasonal
lag as well.
The general multiplicative seasonal ARIMA (p,d,q)(P,D,Q)s model has the
following form :
φ p ( B)Φ P ( B s )(1 − B) d (1 − B s ) D xt = c + θ q ( B)Θ Q ( B s )et
(30)
where C=constant; B=a backshift operator defined as BsZt=Zt-s; d=order of
nonseasonal difference operator; D=order of the seasonal difference operator;
p=order of nonseasonal AR operator; P=order of seasonal AR operator; q=order of
nonseasonal MA operator; and Q=order of seasonal MA operator.
φ p ( B ) = 1 − φ 1 B − φ 2 B 2 − ... − φ p B p
(31)
Φ P ( B s ) = 1 − Φ 1 B s − Φ 2 B 2 s − ... − φ P B Ps
(32)
27
θ q ( B) = 1 − θ 1 B − θ 2 B 2 − ... − θ q B q
(33)
Θ Q ( B s ) = 1 − Θ1 B s − Θ 2 B 2 s − ... − Θ Q B Qs
(34)
The conditions of stationarity and invariability are met only if all the roots of the
characteristics equation φ p ( B ) = 0, Φ P ( B s ) = 0 , θq ( B ) = 0 , ΘQ ( B ) = 0 lie outside the
unit circle. The Box-Jenkins method performs prediction through the following
process:
1.
Model Identification: The orders of the model are determined.
2.
Model Estimation: The linear model coefficients are estimated.
3.
Model Validation: Certain diagnostic methods are used to test the suitability
of the estimated model.
4.
Forecasting: The best model chosen is used for forecasting.
One of the basic conditions for applying the ARIMA model on a particular time
series is its stationarity. A time series with seasonal variation may be considered
stationary if the theoretical autocorrelation function (ρk) and theoretical partial
autocorrelation function (ρkk) are zero after a lag k=2s+2. It is considered that rk and
rkk equal zero if:
ρ k = 0 ako rk ≤ 2 /(T ) 0.5
(35)
ρ kk = 0 ako rkk ≤ 2 /(T ) 0.5
(36)
where rk = sample autocorrelation at lag k; rkk =sample partial autocorrelation at lag
k; and T=number of observations.
The sample autocorrelation function (ACF) of analysed series does not meet the
above condition already mentioned decreasing extremely slowly in a sinusoidal
fashion. That is why the time series is being transformed into a stationary one using
differencing (d=0, D=1, s=12) according to the following equation:
yt = (1 − B ) d (1 − B s ) D xt = (1 − B12 ) ETo ,t
(37)
On the basis of the information obtained from the ACF and the PACF of the
differenced data set, the several forms of the ARIMA model were identified
tentatively. ARIMA models are developed with the aid of the ASTSA computer
package (Hameed et al. 1995). The parameters of model were calculated by
maximum likelihood estimation. The data from 1962 to 1971 were used for
estimating the unknown parameters.
Once a model has been selected and the parameters have been calculated, the
adequacy of the model has to be checked. This process is also called the diagnostic
checking. The Box-Pierce method is used for this purpose, as well as the
Portmanteau lack-of-fit test and t-statistics.
28
The Box-Pierce method is based on the calculation of ACF residuals. If the model is
adequate for describing the behavior of the time series, the residuals are not
correlated, i.e. all ACF values lay within the limits included in the equations (35)
and (36). The Portmanteau lack-of-fit test investigates the first m ACF values of the
residuals using Box-Pierce chi-square statistics which is given in the following
expression:
Q = (T − d )
m
∑ rj2
(38)
j =1
where m = the number of residual autocorrelation used in the estimation of Q
(m=3s); rj=autocorrelation at lag j. ARIMA model is adequate if Q < χ20.5 (m-np)
where np=the number of model parameters.
The third test of model adequacy is the examination of standard errors of the model
parameters. A high standard error in comparison with the parameter values, points
out a higher uncertainty in parameter estimation which questions the stability of the
model. The model is adequate if it meets the following condition:
t = cv / se > 2
(39)
where cv=parameter value and se=standard error.
If several tentative models pass the diagnostic checking, AIC (Akaike Information
Criteria) or BIC (Bayes Information Criteria) is applied to select the best model. On
the basis of minimum AIC value, a seasonal ARIMA (1,0,0).(0,1,1) model was
selected and it is given in this form:
ETo ,t = 0.4714 ETo ,t −1 + ETo ,t −12 − 0.4714 ETo ,t −13 − 0.6839 et −12 + et
(40)
VAR = 568.5; AIC c = 6.889
where VAR= residual variance.
The ACF for the residuals can be considered to be negligible. The Q statistics at lag
36 also show that the residuals are not correlated (Q=36.2<48.3= χ20.5 ), and it is
concluded that the time series et is the white noise. Values of t-statistic (tAR(1) =5.29
and t SMA(1)=8.25) show that the model is stable.
III.2.1.3. Adaptive RBF networks
In this section, the adaptive Radial Basis Function (RBF) network from Trajkovic et
al. (2002) was applied. The output of the RBF network is obtained by the equations
(5) – (8). The Gaussian radial basis function is used as response function (Eq. 9).
Several algorithms such as K-means clustering method or Gram-Schmidt
ortogonalisation procedure have been proposed to identify a nonlinear system by
RBF. However in these methods relatively large number of the basis function is
required, since the tuning parameters are limited to only the coefficients of RBF.
29
Self generating algorithm by Maximum Absolute Error (MXAE) selection method
can be used as a design method for RBF. This method as it will be shown below
satisfies the specified model error with a relatively small number of basis functions.
Tasks which the self generating algorithm should perform can be formulated as
follows. Given Λ input/output data and the specified model error ε > 0 obtain the
minimal number s of RBF and optimal solution for network parameters, aki, mij
and σij , k = 1, 2, ..., t; i = 1, ..., s; j = 1, ..., r, which satisfies the inequality (10).
The MXAE method consists of the following two processes:
a) A parameter tuning process with a fixed number of radial basis function,
b) Architecture adaptation process - new basis function generation.
The parameter tuning process is based on the gradients ∂E / ∂aki , ∂E / ∂mij , ∂E / ∂σij ,
,k = 1, ..., t; i = 1, ..., s; j = 1, ..., r, derivation (Eqs. (12) - (18))
Appropriate gradient methods can be used for parameter tuning such as the steepest
descent method, conjugate gradient method, and quasi-Newton method, or heuristic
methods qualitatively based on the "Manhattan" update rule which is used in this
paper. Only the sign of the gradient is needed to obtain the change of any
parameter. Let p be any of adjustable network parameters aki , mij , σ ij . Tuning of
parameter p is defined by the following iterative process:
p(n +1) = p(n ) + Δp(n )
⎛ ∂E (n ) ⎞
⎟⎟
Δp(n ) = − sgn ⎜⎜
⎝ ∂p(n ) ⎠
(41)
(42)
where
if x > 0
⎧1
⎪
sgn( x) = ⎨− 1 if x < 0
⎪0
else
⎩
(43)
and Δ(n) = local update value at step n.
The adaptation of local updates value is defined by:
⎧
⎪Δ (n − 1) ⋅η +
⎪
⎪
Δ (n) = ⎨Δ (n − 1) ⋅η −
⎪
⎪0
⎪
⎩
if
if
∂E (n)∂E (n − 1)
>0
∂p(n)∂p (n − 1)
∂E (n)∂E (n − 1)
<0
∂p (n)∂p (n − 1)
(44)
else
When the value of the error function (10) takes the minimum during the tuning
process for a fixed number of radial basis functions, algorithm generates a new
30
basis function. A new basis function is generated in the way that the center is
located at the point were the maximum of absolute error occurs in the input space.
An ANN model which used in this paper can be described as:
(45)
ETo ,t = g non ( ETo ,t −1 , ETo ,t − 2 ,...., ETo ,t − r ) + et
where gnon( ) is the unknown nonlinear mapping function, et is the unknown
mapping error (to be minimized), and r is (unknown) number of input. This model
structure is represented by the notation ANN(r, s, t), where r is the input neurons, s
is the number of neurons in the hidden layer, and t is the number of output neurons
(t = 1 in our case).
To identify an ANN model, a number of neurons in the input layer must be selected,
and the number of hidden neurons and the values for the network weights, the
centers and the widths of the radial basis functions must be estimated so that the
forecasting error is minimized. In this study, the number of input neurons was
varied over the range 12 to 24. The self generating algorithm was used to estimate
the number of hidden neurons and the values for the network weights, the centers
and the widths of the radial basis functions using the calibration data. Ten years
(from 1962 to 1971) is used for model identification. One year of data (1972) is
used for model validation. RBF network with twelve neurons in input layer and the
five neurons in the hidden layer (the best fit network) was selected as a
representative of this model.
III.2.1.4. Results and discussion
The evaluation of the model performances is based on the ability of the model to
forecast the validation data, and the error statistics associated with this portion of
the data. The prediction of evapotranspiration for the validation data using the ANN
and ARIMA models are presented in Figure 3 along with the actual observations.
Table 12 summarizes the validation statistics.
Table 12. Statistic properties of ARIMA and ANN forecasting models at Griffith
Model
RMSE MXAE MAE
R2
-1
(mm day ) (mm) (mm)
ARIMA(1,0,0)(0,1,1)
0.85
2.07
0.66
0.90
ANN (12,5,1)
0.65
1.59
0.47
0.92
The forecasting errors are within an acceptable range of accuracy for most practical
purposes. The RBF network used to predict ETo has the root mean squared error
(RMSE) of 0.65 mm day-1, the maximum absolute error (MXAE) is 1.59 mm day-1,
and the mean absolute error (MAE) is 0.47 mm day-1.
A seasonal ARIMA model is also developed for the ETo data. Based on the
preceding results it can be said that the RBF networks are superior as compared to
the ARIMA models.
12
Griffith, NSW, AUS
1972
ETo
-1
10
31
(mm day )
8
6
4
2
ETo_obs
ETo_ann
ETo_arima
Months
0
1
2
3
4
5
6
7
8
9
10
11
12
Figure 3. Comparison of observed ETo (ETo_obs) and forecasted ETo using RBF network
(ETo_ann) and ARIMA model (ETo_arima) at Griffith, NSW, Australia
III.2.1.5. Conclusions
The potential of ANN models for forecasting of reference evapotranspiration has
been presented in this section. For forecasting, time series analysis and adaptive
RBF network were used. It was found that the ETo values are better forecasted
through the ANN model. The RBF network with self generating algorithm appears
to be a viable alternative approach.
These results are of significant practical use because the RBF network could be
used to forecast ETo. Although the RBF networks exhibit a tendency to obtain a
generalized architecture, application of this RBF network to other areas needs to be
studied.
32
III.2. Forecasting of ETo by sequentially adaptive RBF network
III.2.1. Introduction
Forecasting of reference evapotranspiration (ETo) is important for adequate
management of irrigation systems. There are several methods for forecasting
evapotranspiration. The objective of this study is to present a sequentially adaptive
RBF network for forecasting reference evapotranspiration.
III.2.2. Materials and methods
III.2.2.1. ETo data
The weather parameters data (air temperature, relative humidity, wind speed and
sunshine) were available at Nis, Serbia from January 1977 to December 1996. Nis
(latitude 43.3 oN, altitude 202 m) is located in the center of a humid agricultural
production area. However, there is no lysimeter in Nis, so that the monthly
reference evapotranspiration data were produced by FAO-56 Penman-Monteith
equation which is proposed as the sole standard equation for the computation of the
reference evapotranspiration (Allen et al. 1998). The reference evapotranspiration
ranged from 15.8 to 161.5 mm, and the average was 71.3 mm.
III.2.2.2. Sequentially adaptive RBF network
The applications of artificial neural networks (ANNs) are based on their ability to
construct a good approximation of functional relationships between past and future
values of time series. It has been proven that radial basis function (RBF) networks
possess the best approximation property (Girrosi and Poggio 1990). The output of
RBF network is given by:
h( u) = θ +
NH
∑ ai φi ( u, mi , σi )
(46)
i =1
where h(u) = output of RBF network, θ = bias, ai = the weight of the i-th Gaussian
basis function φi ( u , mi , σi ) , i=1,...,NH, NH = number of hidden neurons.
The output of i-th hidden neuron with Gaussian basis function is given by:
⎛ NI ⎛ u − m
ij
φ i (u; mi ;σ i ) = exp⎜ − ∑ ⎜⎜ j
⎜ j =1
⎝ σ ij
⎝
⎞
⎟
⎟
⎠
2
⎞
⎟
⎟
⎠
(47)
where u j = j-th input, j=1,...,NI, NI = number of input neurons, mij = center of i-th
basis function for j-th input, σij = width of i-th basis function for j-th input.
In this section, a sequentially adaptive RBF network is applied to the forecasting of
reference evapotranspiration (ETo). Sequential adaptation of parameters and
structure is achieved using the extended Kalman filter (EKF). A criterion for
network growing is obtained from the Kalman filter's consistency test (Todorovic et
al. 2000).
33
At the moment when a new hidden neuron is added, the parameters of its input and
output links are not adapted, and the level of knowledge represented by those
parameters is low. During the adaptation, the level of knowledge increases along
with the arrival of the new data. The neuron, whose parameters have accumulated a
certain level of knowledge (this knowledge cannot be significantly improved by the
new data), is called a specialized neuron. The new hidden neuron is then added if
the sample, which brings about the neuron’s onset, does not activate any of the
nonspecialized neurons. In such a way the neurons are given time for the adaptation
of their parameters before the new neuron is asked for assistance. In this section, the
moment of the neuron specialization is determined by the number of samples that
bring about the activation of the neurons. If this number is low in comparison to the
total number of samples taken from the moment when the neuron is added, the
neuron is considered insufficiently specialized.
The optimal brain surgeon (OBS) and optimal brain damage (OBD) pruning
methods are derived for networks whose parameters are estimated by the EKF.
Criteria for neurons/connections pruning are based on the statistical parameter
significance test.
The specialized or insufficiently specialized hidden neuron should be pruned if all
of its output connections have statistically insignificant parameters or at least one of
its connections has small, insignificant width. Statistically insignificant width of
any connection between the input and hidden neuron, as the result of parameter
adaptation reflects the tendency to diminish as much as possible the area of the
input space for which the hidden neuron is active. Such neurons often degrade the
continuity of RBF network output and because of that they should be pruned.
III.2.3. Results and discussion
The sequence of 240 samples of ETo was scaled between [-1,1]. The network was
learned to forecast ETo,t+1 based on ETo,t-11 and ETo,t-23. The ANN inputs were
chosen on the basis of information obtained from the autocorrelation function
(ACF) and partial autocorrelation function (PACF). The ACF plot showed a
significant auto-correlation at a lag of 12 months (ETo,t-11). The PACF plot
presented a significant auto-correlation at a lag of 12 (ETo,t-11) and 24 months (ETo,t23). Further details may be found in the papers by Marino et al. (1993), Trajkovic
(1998) and Trajkovic et al. (2000c).
The adaptive RBF network learns by the data, which arrive continually and are
shown in the network only once. The RBF network simultaneously forecasts and
learns. On the basis of the forecast error on the last sample, the parameters and
structure of the RBF network change, and the changed network gives the forecast
for the next sample, where another error is obtained, which again changes the
parameters and the structure of the RBF network. In such a manner the RBF
network gives the realistic forecast of the analyzed time series.
34
After the completed training, the RBF network has the following structure: the input
layer contains two neurons that receive information on the ETo,t-11 and ETo,t-23
values; the hidden layer contains two neurons; and in the output layer contains one
neuron giving the ETo,t+1 value. Figure 4 is a schematic presentation of the RBF
network, and Figure 5 shows the change in the RBF network structure that is
expressed by the change of number of the hidden neurons during the training.
ETo,t+1 = h(u1, u2)
θ
a1
m1j
σ1j
(i=1)
(j=1)
ETo,t-11 = u1
a2
(i=2)
m2j
σ2j
(j=2)
ETo,t-23 = u2
Figure 4. Structure of RBF network
Figure 5. Growth pattern
Evapotranspiration is, on the basis of Eqs. (46) and (47), obtained from the
following equation:
2
⎡ ⎛ ⎛ ET
− mi 2
⎛ ET
o ,t −11 − mi1 ⎞
⎜
⎟⎟ + ⎜⎜ o ,t − 23
= θ + ∑ ai ⋅ exp ⎢− ⎜⎜
σ i1
σ i2
⎢ ⎜⎝ ⎝
i =1
⎠ ⎝
⎣
2
ETo ,t +1
⎞
⎟⎟
⎠
35
2
⎞⎤
⎟⎥
⎟⎥
⎠⎦
(48)
where ai = weight of the i-th Gaussian basis function, mi1 = center of i-th basis
function for first input, σi1 = width of i-th basis function for first input, mi2 = center
of i-th basis function for second input, σ i2 = width of i-th basis function for second
input. The forecasting of reference evapotranspiration is shown in Figure 6.
160
ETo (mm)
120
80
40
ETo_obs
ETo_for
Samples
0
0
50
100
150
200
250
Figure 6. Comparison of observed ETo (ETo_obs) and forecasted ETo using RBF network
(ETo_ann) at Nis, Serbia
The forecast ETo obtained by the RBF network are compared to the observed ETo
from 1980 to 1996. Table 13 summarizes the error statistics, where the RMSE is the
root mean square error of the test samples, MAE is the mean absolute error,
ETo_for/ETo_obs is the ratio of forecast and observed evapotranspiration,
pETo_for/ETo_obs is the ratio of forecast and observed evapotranspiration in the month
of maximum water use (July) and R2 is the coefficient of determination.
The values of the RMSE and MAE are within the acceptable range for the most of
the practical applications. The ETo_for/ETo_obs statistics show that the average
forecast reference evapotranspiration differs from the average observed reference
evapotranspiration by only 0.6%, while the difference for the peak month (July) is
4.9%. These results indicate that the ANNs can be used for forecasting reference
evapotranspiration with high reliability. Trajkovic et al. (2005) compared the RBF
model to the seasonal autoregressive integrated moving average (SARIMA) model.
36
The SARIMA model was chosen for comparison with the ANN model because
Marino et al. (1993) and Trajkovic (1998) demonstrated that the SARIMA model is
advantageous in comparison to other, simpler methods. The SARIMA model used
for ETo forecasting gave the mean square error (MSE) of 213 mm2, and the mean
absolute error (MAE) was 11.2 mm. The ANN model gave the MSE of 131 mm2
and the MAE of 8.9 mm. Based on the preceding results, it can be said that RBF
network is superior when compared to the SARIMA models. Similar results for
Griffith (New South Wales, Australia) were obtained in Trajkovic et al. (2000c).
Table 13. Statistic properties of ANN forecasting model at Nis, Serbia
R2
MAE
ETo_for/ETo_obs pETo_for/ETo_obs
RMSE
-1
-1
(mm month )
(mm month )
11.45
8.90
0.994
0.951
0.95
III.2.4. Conclusions
This section presents the potential of a sequential adaptive RBF network for the
forecasting of reference evapotranspiration. Along with time-varying parameter
estimation using the extended Kalman filter, growing and pruning have been
combined to obtain a sequential adaptive RBF network. Statistical criteria for
growing and pruning are derived using the Kalman filter's estimate of parameters
and innovations statistics. Using a statistically based criterion, pruning methods
similar to OBS and OBD were derived for the neural network, whose parameters
are estimated by the EKF. The results suggest that the sequential adaptive RBF
networks are a promising approach to forecasting reference evapotranspiration.
37
IV.
ESTIMATING REFERENCE EVAPOTRANSPIRATION
BY SEQUENTIALLY ADAPTIVE RBF NETWORKS
IV.1.
Estimating hourly reference evapotranspiration from limited weather
data by sequentially adaptive RBF network
IV.1.1. Introduction
Accurate estimates of hourly reference evapotranspiration (ETo) are important for
adequate management of irrigation systems. In the past several years many papers
have evaluated various equations for calculating the hourly ETo (Ventura et al.
1999, Lecina et al. 2003, Berengena and Gavilan 2005, Allen et al. 2006, LopezUrrea et al. 2006b, Gavilan et al. 2007). These studies have indicated the superiority
of the Penman-Monteith equation for estimating hourly ETo. The Penman-Monteith
equation has two advantages over many other equations. First, it can be used
globally without any local calibrations due to its physical basis. Secondly, it is a
well documented equation that has been tested using a variety of lysimeters. The
FAO-56 Penman-Monteith combination equation (FAO-56 PM) has been
recommended by the Food and Agriculture Organisation of the United Nations
(FAO) as the standard equation for estimating reference evapotranspiration (ETo).
The FAO-56 PM equation requires numerous weather data: air temperature, relative
humidity, wind speed, net radiation and soil heat flux. The main shortcoming of this
equation is that it requires numerous weather data that are not always available for
many locations.
The purpose of this paper is to develop an adaptive Radial Basis Function (RBF)
networks for hourly estimation of ETo from limited weather data and to be able to
accurately estimate hourly values of ETo compared against lysimeter data.
In this paper, two sequentially adaptive RBF networks with different number of
inputs (ANNTR and ANNTHR) and two FAO-56 Penman-Monteith equations with
different canopy resistance values (PM42 and PM70) were evaluated against hourly
lysimeter data from Davis, California.
IV.1.2. Materials and methods
IV.1.2.1. Study area and data collection
The Campbell Tract research site in Davis (38o32' N; 121o46' W; 18 m above sea
level) is characterized with the semiarid Mediterranean climate. Lysimeters in use at
Davis consist of the two units. The weighting lysimeter was installed in 1958-59.
This lysimeter is circular, 6.1 m in diameter, and a depth of 0.91 m. The floating
drag-plate lysimeter, identical in size to the earlier one, was installed in 1962. In the
period 1959-67 both lysimeters were in grass (perennial ryegrass, 1959-63; alta
fescue, 1964-67) and were located about 52 m apart near the middle of 5.2 ha grass
field. The soil in and around the lysimeters was disturbed Yolo loam. The grass was
maintained at height between 8 and 15 cm until optimal water conditions.
38
Estimating ETo by sequentially adaptive RBF networks
Irrigations were applied following a 0.075 m depletion of soil moisture. The ETo
data were measured in kg of weight loss from the weighting lysimeter and
converted to standard units (1 kg h-1= 0.008554 mm h-1). Comparison was made for
the 1966-67 data with ET from the floating drag-plate lysimeter, and agreement
within 2% was usual.
The micrometeorological data were taken from smoothen profiles (at heights of 50,
100, 140, and 200 cm) of temperature, humidity and wind. Wet- and dry-bulb
thermopile sensors gathered the profile data for temperature and humidity.
A separate system measured profiles of absolute humidity using an infrared
hydrometer as the sensor. Thornthwaite cup anemometers gathered wind profile
data. Net radiation was measured at 2 m above the grass surface with a forcedventilated radiometer. The soil heat flux was measured as the mean of three heat
flux plates buried at 0.01 m depth in the soil.
The available data were collected at half-hour intervals during 1962-63 and 1966-67
(Pruitt and Lourence 1965; Morgan et al. 1971). Nineteen days of
micrometeorological and lysimeter data were used for training and testing RBF
networks (Table 14). There were few nighttime data provided, so only data during
daylight hours were analyzed. This data set had a total of 436 patterns.
Table 14. Daily micrometeorological and lysimeter data at Davis, CA
Date
30/07/62
31/07/62
31/08/62
30/10/62
14/08/63
15/08/63
01/06/66
02/06/66
03/06/66
12/07/66
13/07/66
14/07/66
02/05/67
03/05/67
04/05/67
05/05/67
09/05/67
28/09/67
29/09/67
Time
14.00-20.00
06.00-18.30
07.00-19.00
10.00-17.00
06.00-20.00
06.00-19.30
14.30-20.00
06.00-20.00
06.00-20.00
10.00-20.00
06.00-20.00
06.00-20.00
09.00-19.00
12.30-19.00
07.00-19.00
06.30-17.00
06.00-18.00
10.00-20.00
06.30-19.30
Number
of
patterns
12
23
23
15
29
28
12
29
29
21
29
29
21
14
25
22
25
21
27
Training
/Testing
T
o
( C)
RH
(%)
Training
Training
Training
Training
Training
Training
Testing
Training
Training
Testing
Testing
Testing
Training
Testing
Training
Testing
Testing
Testing
Testing
26.5
24.2
26.2
20.4
27.1
29.7
18.8
17.7
19.3
21.1
20.9
21.0
18.7
19.0
16.2
13.9
14.5
25.3
22.5
38.5
42.4
41.6
65.6
36.6
31.1
40.9
43.3
37.8
56.4
56.5
51.9
47.4
46.4
65.0
71.6
73.6
51.7
60.7
Rn
U2
ETo_lys
(kJm-2s- (m s-1) (mm day1
1
)
)
0.234
4.0
5.11
0.382
3.0
11.14
0.333
1.2
8.32
0.236
1.4
3.70
0.290
2.0
11.76
0.304
2.4
12.80
0.211
5.7
4.39
0.343
2.9
11.60
0.326
2.8
11.20
0.354
3.0
9.01
0.324
3.4
12.18
0.324
2.5
11.82
0.385
2.6
8.31
0.296
2.5
5.33
0.359
3.1
8.70
0.240
3.5
4.87
0.184
5.5
4.94
0.228
4.3
8.22
0.213
3.7
8.79
39
IV.1.2.2. FAO-56 Penman-Monteith equation
The FAO-56 PM equation for hourly calculations can be expressed as (Allen et al.
1998):
ETo =
37
U 2 (e a − e d )
T + 273
r
Δ + γ (1 + c )
ra
0.408Δ ( Rn − G ) + γ
(49)
where ETo = reference evapotranspiration (mm h-1); Δ = slope of the saturated vapor
pressure curve (kPa oC-1); Rn =net radiation (MJ m-2 h-1); G =soil heat flux
(MJ m-2 h-1); γ = psychrometric constant; T = mean air temperature (oC); U2 = wind
speed at a 2 meters height (m s-1); (ea-ed) = vapor pressure deficit (kPa),
ra = aerodynamic resistance (s m-1) and rc = canopy resistance (s m-1).
The Allen et al. (1998) recommended the use of rc = 70 s m-1 for hourly time period.
However, using canopy resistance equal 42 s m-1, FAO-56 PM equation (PM42)
best matched measured evapotranspiration in Davis (Ventura et al. 1999; Pruitt,
personal communication, 2000). Todorovic (1999) found out that when the canopy
resistance is calculated for Davis data by his model, the rc values resulted in an
average value of 40 s m-1for most days.
IV.1.2.3. Artificial Neural Networks
ANNs offer a relatively quick and flexible means of modeling, and as a result,
application of ANN modeling is widely reported in the evapotranspiration literature
(Trajkovic et al. 2000; Kumar et al. 2002; Kisi 2006; 2007). Recent papers have
reported that ANNs may offer a promising alternative for estimation of daily
evapotranspiration from limited weather data (Sudheer et al. 2003; Trajkovic 2005,
2009b, 2009c; Zanetti et al. 2007). In this study, a sequentially adaptive Radial
Basis Function (RBF) network from Trajkovic et al. (2003c) was applied to
estimating hourly ETo.
Data set (436 patterns) was divided into two groups. For the RBF network training,
ten randomly chosen days (234 patterns) were used (Table 1). For verification of
RBF network, obtained in a stage of training, the remaining nine days (200 patterns)
were used.
The RBF networks were trained with weather data as inputs, and ETo as output.
Two RBF networks with different number of inputs (ANNTHR and ANNTR) were
considered. Air temperature, humidity, and (Rn-G) term were used as inputs in
ANNTHR. As opposed to the Penman-Monteith equation, the ANNTHR did not
use the wind speed for the ETo calculation. After the completed training, ANNTHR
has the following structure: in the input layer, there are three neurons which receive
information on air temperature (Ta), humidity (H), and (Rn-G) term, in the hidden
layer, there are four neurons, and in the output layer, there is one neuron giving the
ETo value.
40
⎡ ⎛⎛ T − m ⎞2 ⎛ H − m
4
i1
i2
⎟⎟ + ⎜⎜
ETo ,annthr = ∑ a i exp ⎢− ⎜ ⎜⎜ a
⎜
σ
σ
⎢ ⎝⎝
i =1
i1
i2
⎠ ⎝
⎣
2
⎞ ⎛ ( R n − G ) − mi 3 ⎞
⎟⎟ + ⎜⎜
⎟⎟
σ i3
⎠ ⎝
⎠
2
⎞⎤
⎟⎥ + Θ
⎟⎥
⎠⎦
(50)
where ai = weight of the i-th Gaussian basis function, mi1 = center of the i-th basis
function for first input, σ i1 = width of the i-th basis function for first input, m i2 =
center of the i-th basis function for second input, σ i2 = width of the i-th basis
function for second input, m i3 = center of the i-th basis function for third input, σ i3
= width of the i-th basis function for third input, and θ = bias (θ = 0.06035 for the
ANNTHR).
The ANNTR requires only two parameters (air temperature and net radiation) as
inputs. ANNTR did not use wind speed, relative humidity and soil flux density for
estimating ETo. After the completed training, ANNTR has the following structure:
in the input layer, there are two neurons which receive information on air
temperature and net radiation, in the hidden layer, there are five neurons, and in the
output layer, there is one neuron giving the ETo value.
⎡ ⎛⎛ T − m ⎞2 ⎛ R − m
i1
i2
⎟⎟ + ⎜⎜ n
= ∑ ai exp ⎢− ⎜ ⎜⎜ a
⎜
σ
σ
⎢
i =1
i1
i2
⎠ ⎝
⎣ ⎝⎝
5
ETo ,anntr
⎞
⎟⎟
⎠
2
⎞⎤
⎟⎥ + 0.4146
⎟⎥
⎠⎦
(51)
IV.1.2.4. Evaluation Parameters
Several parameters can be considered for the evaluation of ETo estimates. In this
study the following statistic criteria were used: root mean squared error (RMSE)
and daily deviation (D). The RMSE values were calculated as:
RMSE =
1 n
∑ ( ETo _ est ,i − ETo _ ly ,i ) 2
n i =1
(52)
where ETo_est,i = estimated half-hourly ETo, ETo_ly,i = half-hourly lysimeter ETo, and
n is number of observations. The RMSE value less than 0.074 mm h-1 is acceptable
for most practical purposes (Ventura et al. 1999).
Daily deviation is estimated using equation:
⎛ ETo _ est
⎞
− 1⎟100
D=⎜
⎜ ET
⎟
o _ ly
⎝
⎠
(53)
where ETo_est = daily sum of half-hourly ETo estimates, ETo_ly = daily sum of halfhourly lysimeter measurements.
41
IV.1.3. Results and discussion
Two sequentially adaptive RBF networks with different number of inputs (ANNTR
and ANNTHR) and two Penman-Monteith (PM) equations with different surface
resistance values (PM42 and PM70) were compared against hourly lysimeter data
from verification data set (nine days).
Table 15. Statistical summary of hourly ETo estimates at Davis, CA
Date
01/06/66
ETo_ly =
4.387 mm day-1
12/07/66
ETo_ly =
9.010 mm day-1
13/07/66
ETo_ly =
12.182 mm day-1
14/07/66
ETo_ly =
11.817 mm day-1
03/05/67
ETo_ly =
5.328 mm day-1
05/05/67
ETo_ly =
4.866 mm day-1
09/05/67
ETo_ly =
4.941 mm day-1
28/09/67
ETo_ly =
8.215 mm day-1
29/09/67
Parameters
ETo_est mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
ANNTHR
3.774
-14.0
0.062
8.744
-2.9
0.039
11.510
-5.5
0.064
11.783
-0.3
0.040
5.006
-6.4
0.052
4.816
-1.0
0.024
4.913
-0.6
0.028
7.306
-11.1
0.095
8.921
ANNTR
3.629
-17.3
0.071
9.311
+3.3
0.037
12.320
+1.1
0.051
12.245
+3.6
0.033
5.129
-3.7
0.041
5.014
+3.0
0.026
4.975
+0.7
0.032
8.483
+3.3
0.080
9.651
PM70
3.618
-17.5
0.075
7.616
-15.5
0.086
10.591
-13.1
0.081
10.510
-11.1
0.068
4.298
-19.4
0.096
4.224
-12.8
0.044
4.084
-17.4
0.051
7.314
-11.0
0.073
7.438
PM42
4.168
-5.0
0.026
8.370
-7.1
0.049
11.734
-3.7
0.040
11.388
-3.6
0.037
4.680
-12.2
0.075
4.877
+0.2
0.024
4.911
-0.6
0.030
8.094
-1.5
0.044
8.335
ETo_ly =
8.806 mm day-1
Average
ETo_ly =
7.728 mm day-1
D (%)
RMSE (mm h-1)
ETo_est mm day-1
D (%)
RMSE (mm h-1)
+1.3
0.079
7.419
-4.0
0.058
+9.6
0.061
7.862
+1.7
0.050
-15.5
0.073
6.635
-14.1
0.071
-5.3
0.039
7.395
-4.3
0.041
42
The results of this comparison are presented in Table 15. The ANNTHR performed
reasonable well for most days. This approach underestimated hourly ETo for the
second half of June 1, 1966, and midday of September 28, 1967, and overestimated
first half of September 29, 1967. The D statistic was -14%, -11.1% and 1.3%,
respectively. RMSE values were within acceptable range for all days excluding the
September 28, 1967 (RMSE=0.095 mm h-1), and September 29, 1967
(RMSE=0.079 mm h-1). On average, ANNTHR underestimated hourly ETo_ly by
about 4% with RMSE value equal 0.058 mm h-1.
Estimates by ANNTR were in closest agreement with the grass ET for most days.
ANNTR underestimated hourly ETo_ly for the second half of June 1, 1966, and
overestimated first half of September 28, 1967, and September 29, 1967 with
D value of -17.3%, 3.3%, and 9.6%, respectively. RMSE values were within
acceptable range for all days excluding the September 28, 1967 (RMSE=0.080 mm
h-1). On average, this approach showed slight deviation of 1.7% relative to the
ETo_ly with RMSE value equal to 0.050 mm h-1.
The deviation of ANNTHR and ANNTR on June 1, 1966, September 28, 1967, and
September 29, 1967 may be partly due to high wind speed (average wind speed was
5.7, 4.3 and 3.7 m s-1, respectively) and low net radiation (average net radiation was
0.211, 0.228, and 0.213 kJ m-2 s-1, respectively). The average wind speed only in
one of ten training days exceeded 3.1 m s-1, and the average net radiation was not
less than 0.234 kJ m-2 s-1 in any training day.
The ANNTHR and the ANNTR were especially successful on May 5, 1967, and
May 09, 1967. These days had extreme values of micrometeorological data (the
lowest air temperature, the highest relative humidity, very low net radiation and
high wind speed). The ANNTHR and the ANNTR had the negligible departures
from the ETo_ly, even though the existence of the cloudiness produced high
variations of the grass evapotranspiration during the day. The success is even
greater, if it is emphasized that during the training days there were no days with
such extreme values of the meteorological data.
The FAO-56 Penman-Monteith equation using the surface resistance rc = 70 s m-1
(PM70) was the poorest in estimating ETo of all equations evaluated. The PM70
consistently underestimated hourly ETo_ly for all days by about 14%. The RMSE
values varied from 0.044 (May 5, 1967) to 0.096 mm h-1 (May 3, 1967). These
results strongly support the introduction of new value for surface resistance in the
hourly FAO-56 PM equation recommended by Allen et al. (2006).
The PM42 yielded the excellent estimate of the grass ET for most days. This
method underestimated ETo_lys during July 12, 1966, and May 3, 1967 with daily
deviation of -7.1% and 11.7%, respectively. RMSE value for May 3, 1967 slightly
exceeded acceptable level of 0.074 mm h-1(RMSE = 0.075). The PM42 consistently
underestimated peak hourly ETo_lys for all days by about 10%. On average, this
method underestimated ETo_ly by 4.3% with RMSE value equal to 0.041 mm h-1.
43
The overall results indicate that ANNTR, ANNTHR, and PM42 give acceptable
estimates of hourly ETo. The ANNTR and PM42 were slightly better than
ANNTHR at matching ETo_ly. Figure 7 shows a comparison between estimated and
measured ETo on July 14, 1966.
0.8
Davis, CA
July 14, 1966
ETo
-1
0.7
(mm h )
0.6
0.5
0.4
ETo_ly
ETo_annthr
ETo_anntr
ETo_pm70
ETo_pm42
0.3
0.2
0.1
Hours
0
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Figure 7. Comparisons between estimated and measured ETo at Davis on July 14, 1966
All of the equations paralleled ETo_ly through the day. The PM70 consistently
underestimated ETo_lys. ANNTR, ANNTHR, and PM42 followed hourly ETo_lys
quite closely through the day. The ANNTHR slightly overestimated ETo_lys in
morning hours, and underestimated ETo_ly in midday. The ANNTR slightly
overestimated ETo_lys in morning hours. The PM42 underestimated ETo_ly in
midday.
IV.1.4. Conclusions
Two sequentially adaptive RBF networks with different number of inputs (ANNTR
and ANNTHR) and two Penman-Monteith equations with different surface
resistance values (PM42 and PM70) were tested against hourly daytime lysimeter
data from Davis, CA.
The ANNTR requires only two parameters (air temperature and net radiation) as
inputs. Air temperature, humidity, and (Rn-G) term were used as inputs in
ANNTHR. PM equations use air temperature, humidity, wind speed, net radiation
and soil heat flux as inputs. The results reveal that ANNTR and PM42 were
generally the best in estimating hourly ETo. The ANNTHR performed less well, but
the results were acceptable for estimating ETo.
44
This study indicates that the RBF network using limited weather data was able to
reliably estimate hourly ETo for a well-irrigated grass under different atmospheric
conditions. The calculation of the hourly ETo is possible only on the basis of the air
temperature and the net radiation, without using the wind speed, humidity and soil
flux density. These results are of significant practical use because the RBF network
with air temperature and net radiation as inputs could be used to estimate hourly
ETo when relative humidity and wind speed data are not available. Although the
RBF networks exhibit a tendency to obtain a generalized architecture, application of
ANNTR to other areas needs to be studied.
IV.2.
45
Comparison of RBF networks and empirical equations for converting
from pan evaporation to reference evapotranspiration
Evapotranspiration is one of the major processes in the hydrological cycle, and its
reliable estimation is essential to water resources planning and management.
A common practice for estimating evapotranspiration is to first estimate reference
evapotranspiration (ETo) and then to apply a corresponding crop coefficient.
Numerous equations have been developed for estimating ETo, most of which are
complex and require numerous weather parameters. In many areas, the necessary
data are lacking, and simpler techniques are required. Evaporation pans (U.S.
Weather Bureau Class A pan) are used throughout the world because of the
simplicity of technique, low cost, and ease of application in determining crop water
requirements for irrigation scheduling (Stanhill 2002). Evaporation pan data are
easy to obtain and can be very reliable if the evaporation site is properly
maintained.
The objectives of this study were: first, to develop an adaptive RBF network for pan
evaporation to reference evapotranspiration conversions using pan evaporation and
lysimeter data from Policoro, Italy; second, to evaluate the reliability of three panbased approaches (RBF network, Christiansen, FAO-24 pan), and FAO-56
Penman-Monteith equation for estimating ETo as compared against lysimeter data;
third, to test the applicability of obtained RBF network and pan-based equations at
other locations.
IV.2.2.1. Study areas and weather data collection
The three weather stations selected for this study are Policoro, Italy; Novi Sad,
Serbia and Kimberly, Idaho, U.S.A. The adaptive RBF network was developed
using daily data collected at Policoro, Italy. The RBF network obtained on the basis
of the daily data from Policoro, Italy was additional tested using monthly data
collected in Novi Sad, Serbia, and Kimberly, Idaho, U.S.A.
IV.2.2.1.1. Policoro
Daily lysimeter and weather data (minimum and maximum air temperature,
minimum and maximum relative humidity, wind speed, sunshine, and pan
evaporation) were collected at the experimental field "E. Pantanelli" of Bari
University, located in the area of Policoro (Province of Matera), along the Western
Ionian Coast, about 3 km from the sea. The experimental site is characterized with
the Mediterranean semiarid climate with 40o17' N, 16o40' E, and altitude 15 m
above sea level. The long-term average values of the major weather parameters are
presented below: minimum and maximum air temperature are 11.0 and 21.4 oC,
respectively; minimum and maximum relative humidity are 52 and 87 %,
46
respectively; sunshine is 6 h 36 min; wind speed is 2.3 m s-1; and Class A pan
evaporation is 5.2 mm day-1 (Caliandro et al. 1990).
The agrometeorological station was equipped with a Class A evaporation pan and a
4 m2 (2x2 m) wide and 1.3 m deep weighting lysimeter covered by fescue grass.
The lysimeter was situated near the center a 60x60 m grass field. The site was
maintained under optimal water conditions. The fescue grass was periodically
mowed to keep the height between 8 and 15 cm. Irrigations were applied with a
frequency from 3 to 5 days.
The various instruments were located about 30 m from the lysimeter. The data for
temperature and humidity were gathered by bimetallic thermograph and hair
hydrograph, respectively. The wind speed was measured by propeller anemograph
3.5 m above the grass. Campbell-Stokes sunshine recorder gathered bright sunshine
duration (Todorovic 1999). The integrity of data was assessed by comparison with a
nearby station through "double mass analysis". According to Allen (1996)
procedure, the solar radiation data of Policoro were tested using solar radiation
envelope curve. Todorovic (1999) showed that solar radiation values estimated by
Angstrom formula from sunshine hours were below the solar radiation envelope
curve and he used the adjustment factor of 1.11 for correction of solar radiation.
The adjusted Rs data were used in this study.
The raw Policoro data set included lysimeter and weather data from May 15, 1981
to December 18, 1984 (A. Caliandro, University of Bari, personal communication,
2000). However, problems existed on some days, so not all days were selected for
analysis. Days with irrigations or rainfall, grass cuttings, and problems associated
with irrigations or equipment were omitted from analysis. The final data set used in
this study had a total of 497 patterns distributed over all seasons. In this study, the
fetch distance used for Policoro was 30 m.
IV.2.2.1.2. Novi Sad
Novi Sad, Serbia (latitude 45o20'N; longitude 19 o51'E; and altitude is 86 m above
sea level) is located in the center of the spacious humid agricultural area. The area
of Novi Sad has following long-term average values of the weather parameters
(Smith 1993): maximum and minimum air temperature are 16.7 and 5.9 oC,
respectively; relative humidity and wind speed are 79 % and 1.9 m s-1, respectively;
bright sunshine is 5.8 hours. The annual rainfall is 647 mm. Nearly 54 % of the
annual rainfall occurs during the growing season (from April to September).
Monthly pan evaporation data along with weather data on air temperature,
humidity, wind speed, and sunshine were collected at Novi Sad during the period
April-September from 1981 to 1984. The fetch distance used for Novi Sad was 10
m. The data set had 23 patterns. The pan evaporation from June 1981 and May 1983
were not available and the pan evaporation from October 1984 was annexed to data
set.
47
IV.2.2.1.3. Kimberly
Data from Kimberly, Idaho, U.S.A. were also used to check the applicability of
RBF network for estimating ETo. Kimberly (latitude 42o33' N; longitude 114o21' W;
and altitude is 1,207 m above sea level) is located in a semiarid irrigation
environment. The average weather data for the peak month (July) were presented in
Table 16 (Jensen et al. 1990 and Allen and Pruitt 1991). The lysimeter at Kimberly,
Idaho, U.S.A. was situated near to center of a 2.6 ha alfalfa plot. The steel lysimeter
soil tank was 1.83 m square and 1.22 m deep (Wright 1988). ETo data were selected
from days when the alfalfa crop was well watered, actively growing, and at least 0.3
m tall. Irrigations were applied when the tensiometers at the 0.45 m depth exceeded
60 kPa (Katul et al. 1992).
Table 16. Average weather parameters at Kimberly, Idaho, during July (Jensen et al. 1990)
Parameter
Average value
o
29.5
Maximum temperature ( C)
o
11.8
Minimum temperature ( C)
9.7
Dewpoint temperature (oC)
2.2
Wind speed adjusted to a height of 2 m (m s-1)
Solar radiation (MJm-2 day-1)
26.96
Extraterrestrial radiation (MJm-2 day-1)
40.24
Pan evaporation (mm day-1)
8.4
Lysimeter evapotranspiration (mm day-1)
7.87
The weather data for Kimberly were obtained from the National Weather Service
weather station located about 800 m from the lysimeter field site. The pan was
centered in a 45x36 m irrigated clipped grass plot. Irrigated field plots planted to
various crops surrounded the station. Jensen et al (1990) and Allen and Pruitt
(1991) used fetch distance of 1,000 m for Kimberly. However, the fetch distance
estimated for Kimberly was 20 m in Katul et al. (1992).
IV.2.2.2. Sequentially Adaptive RBF Network
In this paper, a sequentially adaptive Radial Basis Function (RBF) network from
Trajkovic (2008) was applied to estimating reference evapotranspiration (ETo).
Policoro pan evaporation and lysimeter data from 1981 to 1983 were used for
training of RBF network. The training data set had a total of 385 patterns. The RBF
network was trained with pan evaporation and extraterrestrial radiation data as input
and the lysimeter data as output. Extraterrestrial radiation data are computed as a
function of the local latitude and Julian data (Allen et al 1998). The sequence of 385
samples was scaled between –1 and 1. The RBF network was trained to estimate
ETo based on Epan and Ra. The adaptive RBF network is trained by the data, which
arrive continually and are shown in the network only once. The RBF network
simultaneously estimates and learns. On the basis of the estimate error on the last
48
sample, the parameters and structure of the RBF network change. The network
changed in that way gives the estimate for the next sample, where another error is
obtained, which again changes the parameters and the structure of the RBF network.
Readers are referred to Trajkovic et al. (2003c) for a detailed description of this
neural network.
After the completed training, the RBF network has the following structure: in the
input layer, there are two neurons that receive information on the Epan and Ra, in the
hidden layer there are two neurons, and in the output layer, there is one neuron
giving the ETo value. The reference evapotranspiration is obtained from following
equation:
⎡⎛ ⎛ E − m ⎞ 2 ⎛ R − m
pan
i1
i2
⎟⎟ + ⎜⎜ a
ETo = Θ + ∑ ai exp ⎢⎜ ⎜⎜
⎜
σ
σ
⎢
i =1
i1
i2
⎠ ⎝
⎣⎝ ⎝
2
⎞
⎟⎟
⎠
2
⎞⎤
⎟⎥
⎟⎥
⎠⎦
(54)
where ai = weight of the i-th Gaussian basis function, mi1 = center of the i-th basis
function for first input, σ i1 = width of the i-th basis function for first input, m i2 =
center of the i-th basis function for second input, σ i2 = width of the i-th basis
function for second input, and θ = bias (θ = 0.39777 for this RBF network).
IV.2.2.3. Christiansen Equation
Christiansen equation for estimating ETo from Class A pan evaporation and several
weather parameters was presented in Jensen et al. (1990) as follows:
ETo = 0.755 E pan ⋅ C t ⋅ C u ⋅ C h ⋅ C s
(55)
where ETo = reference evapotranspiration (mm day-1), Epan = measured Class A pan
evaporation (mm day-1). The coefficients are dimensionless.
⎛T
C t = 0.862 + 0.179⎜⎜
⎝ To
⎞
⎛T
⎟⎟ − 0.041⎜⎜
⎠
⎝ To
⎞
⎟⎟
⎠
2
(56)
where T = mean air temperature (oC) and To = 20 oC.
⎛U
C u = 1.189 − 0.240⎜⎜
⎝Uo
⎞
⎛U
⎟⎟ + 0.051⎜⎜
⎠
⎝Uo
⎞
⎟⎟
⎠
2
(57)
where U = mean wind speed at 2 m (km h-1) and Uo = 6.7 km h-1.
⎛ H
C h = 0.499 + 0.620⎜⎜
⎝ Ho
⎞
⎛ H
⎟⎟ − 0.119⎜⎜
⎠
⎝ Ho
⎞
⎟⎟
⎠
2
where H = mean relative humidity, expressed decimally and Ho = 0.6.
(58)
⎛ S
C s = 0.904 + 0.008⎜⎜
⎝ So
⎞
⎛ S
⎟⎟ + 0.088⎜⎜
⎠
⎝ So
⎞
⎟⎟
⎠
49
2
(59)
where S = percentage of possible sunshine, expressed decimally and So = 0.8.
IV.2.2.4. FAO-24 Pan Equation
Doorenbos and Pruitt (1977) provided guidelines for using Class A pan data to
estimate ETo.
The relation between pan evaporation and reference
evapotranspiration can be described by following equation:
ETo = K p E pan
(60)
where ETo = reference evapotranspiration (mm day-1), Kp = pan coefficient, Epan =
pan evaporation (mm day-1). The pan coefficient depends on the upwind fetch
distance (F), mean daily wind speed (U2), and mean daily relative humidity
(RHmean) associated with the sitting of the evaporation pan (Doorenbos and Pruitt
1977). Several Kp equations have been developed in the past 20 years, including
Frevert et al. (1983), Allen and Pruitt (1991), Snyder (1992), and Raghuwanshi and
Wallander (1998). Irmak et al. (2002) reported that Frevert et al.'s 1983 Kp equation
provided more accurate ETo estimates compared to Snyder's 1992 Kp equation
under the humid climatic conditions. Grismer et al. (2002) compared several
Kp equations using data from California. Frevert et al.'s 1983 and Snyder's 1992 Kp
equations most closely approximated the average measured ETo. Gundekar et al.
(2008) found that Snyder's 1992 Kp equation provided more accurate ETo estimates
compared to other Kp equations for semi-arid climate. Trajkovic et al. (2000b)
presented the RBF network for estimating Kp. The RBF network predicted Kp
values better than Frevert et al.'s 1983 and Snyder's 1992 Kp equations. Kp values
obtained by RBF network from Trajkovic et al. (2000b) were used in this study.
IV.2.2.5. FAO-56 Penman-Monteith Equation
The Food and Agriculture Organisation of the United Nations (FAO) has proposed
using the Penman-Monteith equation as the standard method for estimating
reference evapotranspiration, and for evaluating other ETo equations. Many studies
have confirmed the superiority of this equation (Ventura et al. 1999; Lecina et al.
2003, Berengena and Gavilan 2005, Lopez-Urrea et al. 2006a; Gavilan et al. 2007).
The Penman-Monteith equation has two advantages over many other equations.
First, it can be used globally without any local calibrations due to its physical basis.
Secondly, it is a well documented equation that has been tested using a variety of
lysimeters. FAO-56 Penman-Monteith (FAO-56 PM) equation was presented in
Allen et al. (1998) as follows:
50
ETo =
900
U 2 (e a − e d )
T + 273
Δ + γ (1 + 0.34U 2 )
0.408Δ ( Rn − G ) + γ
(61)
where ETo = grass reference evapotranspiration (mm day-1); Δ = slope of the
saturation vapor pressure function (kPa oC-1); Rn = net radiation (MJ m-2 day-1);
G =soil heat flux density (MJ m-2 day-1); γ = psychrometric constant (kPa oC-1);
T = mean air temperature (oC); U2 = average 24-hour wind speed at 2 meters height
(m s-1); (ea-ed) = vapor pressure deficit (kPa).
IV.2.2.6. Data requirements
The data requirements of the selected ETo equations are presented in the Table 17.
The RBF network required measurements of only one weather parameter, pan
evaporation. Christiansen, FAO-24 pan and FAO-56 Penman-Monteith equations
used seven, four, and six weather parameters for estimating ETo, respectively. The
RBF network and Christiansen equation required using of one astronomic
parameter (extraterrestrial radiation (Ra) and maximum sunshine duration (N),
respectively). The FAO-56 Penman-Monteith equation used two astronomic
parameters (Ra and N). The FAO-24 pan equation did not use the astronomic
parameters. However, this equation used the fetch distance (F) in estimating ETo.
Table 17. Data requirements of the ETo equations
Equation
Epan Tmax Tmin RHmax RHmin U
RBF network
*
Christiansen
*
*
*
*
*
*
FAO-24 pan
*
*
*
*
FAO-56 PM
*
*
*
*
*
n
Ra
*
*
N
F
*
*
*
*
*
IV.2.3.1. Policoro
The adaptive RBF network, FAO-24 pan, Christiansen, and FAO-56 PenmanMonteith equations were tested by comparing with lysimeter measurements of grass
evapotranspiration using data collected at Policoro, Italy from February 25 to
December 18, 1984. The test data set had a total of 112 patterns.
Summary statistics of daily ETo are given in Table 18. In this Table, RMSE is the
root mean square error, R2 is the coefficient of determination, and ETo_eq/ETo_ly is
the ratio of mean annual estimated ETo and lysimetar ET data, pETo_eq/ETo_ly is the
ratio of mean estimated ETo and lysimetar ET data for days in the peak month
(July). Based on summary statistics, the RBF network ranked first with the lowest
RMSE value (0.433 mm day-1), the highest coefficient of determination (0.950), and
the closest ETo estimates to lysimeter ETo data for all days and other equations
ranked in decreasing order are: FAO-24 pan, FAO-56 Penman-Monteith, and
51
Christiansen. It should be noted that the RBF network provided better results
compared to Christiansen, FAO-24 pan, and FAO-56 Penman-Monteith equations,
even though it uses the least number of parameters (only one weather parameter,
Epan).
Table 18. Summary statistics of ETo equations at Policoro, Italy
Equation
Number of
RMSE
R2
ETo_eq/ETo_ly pETo_eq/ETo_ly
-1
(%/100)
(%/100)
parameters (mm day )
RBF network
2 (1+1)a
0.433
0.950
1.018
1.049
Christiansen
8 (7+1)
0.722
0.923
1.084
1.158
FAO-24 pan
5 (4+1)
0.550
0.935
1.048
1.088
FAO-56 PM
8 (6+2)
0.558
0.921
0.966
0.953
a Numbers in parentheses denote the required number of climatic and other parameters, respectively.
The FAO-24 pan equation was nearly as good, making it the reliable equation with
the second lowest RMSE value (0.550 mm day-1). However, this equation
overestimated lysimeter data by 4.8 and 8.8 % for all days and days in peak month,
respectively.
The FAO-56 Penman-Monteith equation provided reliable ETo estimates with
RMSE value equal to 0.558 mm day-1. It was quite sensitive to effects of changes in
weather parameters on lysimeter measurements. On average, ETo estimates by the
FAO-56 Penman-Monteith equation were 3 to 5 % lower than lysimeter
measurements.
The Christiansen equation was the poorest equation evaluated for estimating daily
ETo with RMSE value of 0.722 mm day-1. On average, this equation overestimated
lysimeter measurements by about 8 and 16 %, for all days and days in the peak
month, respectively. The poor performance of the Christiansen equation was due
primarily to the fact that this equation was developed for monthly ETo estimates.
The ETo estimates by RBF network (ETo_ann) were plotted against lysimeter
measurements (ETo_ly) in Figure 8. Locations of points close to the 1:1 line indicate
remarkable agreement between estimates and lysimeter measurements.
52
9
ETo_ann
8
(mm day )
-1
7
6
5
4
3
2
ETo_ly
1
-1
(mm day )
0
0
1
2
3
4
5
6
7
8
9
Figure 8. Daily ETo estimated by RBF network versus lysimeter ETo at Policoro, Italy
IV.2.3.2. Novi Sad
The RBF network obtained on the basis of the daily data from Policoro, FAO-24
pan, and Christiansen equations were compared to the FAO-56 Penman-Monteith
equation. The FAO-56 PM has been used as substitute for measured ETo data and
that is the standard procedure when there is no measured lysimeter data (Irmak et al.
2003; Martinez-Cob and Tejero-Juste 2004; Landeras et al. 2008, Khoob 2008).
Summary statistics of monthly ETo data are presented in Table 19.
Table 19. Summary statistics of ETo equations at Novi Sad, Serbia
RMSE
R2
ETo_eq/ETo_pm pETo_eq/ ETo_pm
-1
(mm day )
(%/100)
(%/100)
RBF network
0.311
0.901
1.043
1.008
Christiansen
0.395
0.821
1.026
1.003
FAO-24 pan
0.439
0.764
1.041
1.010
Equation
The RBF network (ETo_ann) best matched ETo estimates by Penman-Monteith
equation (ETo_pm) with lowest RMSE values of 0.311 mm day-1 and the highest
coefficients of determination (0.901). The RBF network overestimated ETo_pm by
4.3 and 0.8 % for all months and the peak month (July), respectively.
The Christiansen equation (ETo_chr) was slightly better than FAO-24 pan equation
(ETo_pan) at matching ETo_pm. These equations had RMSE values of 0.395 and 0.439
mm day-1, respectively.
53
The mean daily ETo values for each month, as estimated by the selected equations
were plotted in Figure 9. The ETo estimates paralleled ETo_pm fairly well with the
exception of August 1981, and July 1983. The poor fit in these months may be
partly due to errors in measuring the pan evaporation.
5.5
ETo_pm
ETo_ann
ETo_pan
ETo_chr
ETo
-1
(mm day )
4.5
3.5
2.5
1.5
1981
1982
1983
Novi Sad, Serbia
1984
Figure 9. Comparison of mean daily ETo calculated for four growing seasons at Novi Sad,
Serbia using FAO-56 PM equation (ETo_pm), RBF network (ETo_ann), FAO-24 pan
equation (ETo_pan) and Christiansen equation (ETo_chr)
IV.2.3.3. Kimberly
ETo values computed by RBF network obtained on the basis of the daily data from
Polocoro, ASCE Penman-Monteith, FAO-56 Penman-Monteith, FAO-24 pan using
the fetch distance of 20 m and fetch distance of 1,000 m, and Christiansen equations
were compared to the mean measured evapotranspiration (ETo_ly) for the peak
month (July).
In these calculations, estimates of reference evapotranspiration for grass and
hypothetical crop were adjusted upward by a factor of 1.32 (adjETo) for comparison
with alfalfa ET (Allen et al. 1989). Results of comparison were given in Table 20.
All equations gave acceptable estimate of mean peak ETo with the exception of the
FAO-24 pan equation with fetch distance equal to 1,000 m that overestimated ETo
by 12.7%. FAO-24 pan equation with F = 20 m were in closest agreement with
measured ETo. An adjusted ETo estimate by these equations was 0.5 % lower than
measured ETo. These results indicate that a fetch distance of 20 m would have been
more appropriate for estimating ETo from pan data at Kimberly. The ASCE
Penman-Monteith and FAO-56 Penman-Monteith equations underestimated
54
measured ETo by 1.5 and 3.5 %, respectively. Adjusted ETo estimates by the RBF
network and Christiansen equation were 3.9 and 3.5 % higher than measured ETo,
respectively.
Table 20. Summary statistics of ETo equations at Kimberly, Idaho
ETo
Reference crop
adjETo adjETo_eq/ETo_ly
(mm day-1)
(mm day-1)
(%/100)
Lysimeter
7.87
Alfalfa
7.87
RBF network
6.19
Grass
8.18
1.039
Christiansen
6.17
Grass
8.14
1.035
FAO-24 pan (F=1000 m)
6.72
Grass
8.87
1.127
FAO-24 pan (F=20 m)
5.93
Grass
7.83
0.995
FAO-56 PM
5.75
Hypothetical crop
7.60
0.965
ASCE PM
7.75
Alfalfa
7.75
0.985
Method
IV.2.4. Conclusions
In this section, three pan-based approaches (RBF network, FAO-24 pan,
Christiansen) are used for estimating reference evapotranspiration. The obtained
results demonstrate that RBF network and empirical equations using data from
well-maintained Class A pans can be successful alternative to FAO-56 PenmanMonteith equation for estimating reference evapotranspiration.
The FAO-24 equation gives the acceptable daily and monthly estimates of ETo.
The obtained results demonstrate that this equation is very sensitive to errors in
determining the fetch distance. The Christiansen equation provides reliable monthly
ETo estimates. However, this equation cannot be recommended for daily estimating
reference evapotranspiration.
The FAO-24 pan and Christiansen equations used four and seven weather
parameters for estimating ETo, respectively. The basic obstacle to using these
equations widely is the numerous required weather parameters. In many areas, the
necessary data are lacking, and simpler techniques are required.
The results indicate that the RBF network is able to estimate daily and monthly ETo
for a well-irrigated reference crop under different climatic conditions. It gives the
reliable estimation in all locations. The RBF network consistently provided better
results compared to FAO-24 pan and Christiansen equations, although required
measurements of only one weather parameter, pan evaporation.
The fact that the RBF network developed for the daily estimating ETo in the
semiarid climate yields the reliable calculation of the monthly ETo at other semiarid
and humid locations has a special significance. The results recommended adaptive
RBF network for pan evaporation to reference evapotranspiration conversions.
The use of the RBF network is very simple and does not require any knowledge of
ANNs.
55
IV.3. Temperature-based approaches for estimating ETo
Evapotranspiration (ET) is one of the major components of the hydrologic cycle.
Accurate estimates of ET are important for planning, design, and operation of
irrigation systems. A common procedure for estimating ET is to first estimate
reference ET (ETo). Crop coefficients, which depend on the crop characteristics and
local conditions, are then used to convert ETo to the ET. This study addresses only
the estimation of ETo.
The Committee on Irrigation Water Requirements of the American Society of Civil
Engineers (ASCE) has analyzed the properties of 20 different equations against
carefully selected lysimeter data from 11 stations located worldwide in different
climates (Jensen et al. 1990). The Penman-Monteith equation ranked as the best
equation for estimating daily and monthly ETo in all the climates.
The International Commission for Irrigation and Drainage (ICID) and Food and
Agriculture Organisation of the United Nations (FAO) have proposed using the
Penman-Monteith equation as the standard equation for estimating reference
evapotranspiration, and for evaluating other equaitons (Allen et al. 1994 a, b).
The FAO-56 Penman-Monteith (FAO-56 PM) equation requires numerous weather
data: maximum and minimum air temperature, maximum and minimum relative air
humidity (or the actual vapor pressure), wind speed at 2-m height, solar radiation
(or sunshine hours). The basic obstacle to widely using the FAO-56 PenmanMonteith equation is the numerous required data that are not available at many
weather stations (Trajkovic et al. 2004). A serious problem is the quality of the data.
Solar radiation data are not always reliable (Llasat and Snyder 1998). Wind speed at
2 m height may be site specific, or of questionable reliability (Jensen et al. 1997).
Measurement of relative humidity by electronic sensors is commonly plagued by
errors (Allen 1996). Where radiation data are lacking, or are of questionable quality,
the difference between the maximum and minimum temperature can be used for the
estimation of solar radiation (Hargreaves et al. 1985; Allen 1997). Estimates of
actual vapor pressure (ed) can be obtained from minimum air temperature (Jensen et
al. 1997; Kimball et al. 1997; Allen et al. 1998; Thornton et al. 2000). Where no
wind data are available an average wind speed value of 2 m s-1 can be used as
acceptable for most locations (Jensen et al. 1997; Allen et al. 1998). On the basis of
the previous analysis it can be stated that maximum and minimum air temperatures
constitute the minimum set of weather data necessary to estimate ETo.
The basic goal of this section is to examine whether it is possible to attain the
reliable estimation of ETo only on the basis of the temperature data. This goal was
reached in two steps: first, by the development of an adaptive temperature-based
radial basis neural (RBF) network for estimating reference evapotranspiration;
second, by evaluation of the reliability of four temperature-based approaches (RBF
network, Thornthwaite, Hargreaves, and reduced set Penman-Monteith equations)
56
as compared to the FAO-56 PM equation. In this section, the FAO-56 PM has been
used as substitute for measured ETo data and that is the standard procedure when
there is no measured lysimeter data (Irmak et al. 2003; Martinez-Cob and TejeroJuste 2004; Utset et al. 2004; Vanderlinden et al. 2004, Khoob 2008).
IV.3.2.1. Description of data
The seven weather stations selected for this study are located in Serbia.
These locations are Palic, Belgrade, Novi Sad, Negotin, Kragujevac, Nis, and
Vranje. Temperature, wind speed, relative humidity, actual vapor pressure, and
sunshine hours were collected at these stations for different time periods.
The description of the different weather stations along with the observation periods,
number of patterns and mean weather data is given in Table 21.
Station
Palic
Novi Sad
Belgrade
Negotin
Table 21. Summary of selected weather stations in Serbia
Latitude Altitude Period Patterns Tmax Tmin RH
U2
ETo_pm
(oN)
(m)
(oC) (oC) (%) (m s-1) (mm d-1)
Nis
46.1
45.3
44.8
44.2
44.0
43.3
102
86
132
42
190
202
Vranje
42.6
433
Kragujevac
1977-83
1981-84
1977-84
1971-74
1981-84
1977-84
1993-96
1971-74
84
48
96
48
48
96
48
48
15.5
16.2
16.5
16.3
16.4
17.0
18.4
15.9
6.1
6.3
7.9
5.9
6.0
6.2
6.8
5.7
74.3
73.8
69.4
73.9
74.8
71.1
68.1
71.5
1.7
1.9
2.0
1.7
1.1
1.0
1.1
1.5
2.22
2.32
2.50
2.34
2.09
2.20
2.38
2.32
These locations were chosen because: first, they represent all the climatic types
existing in Serbia; second, they cover all the latitudes in Serbia (from 42o30’ N to
46 o10’ N); and third, they are situated at different elevations above the sea level.
Differences in the mean weather data for these locations are not very significant.
The mean annual maximum and minimum temperatures (Tmax and Tmin) for most
locations varied between 15.5 and 17.0 oC and 5.7 and 6.8 oC, respectively, and they
were highest at Nis (1993-96; 18.4 oC) and Belgrade (1977-84; 7.9 oC),
respectively. The mean maximum and minimum temperatures for the peak month
(pTmax and pTmin) for these locations ranged from 26.1 to 28.4 oC and from 13.9 to
16.0 oC, respectively. The mean relative humidity for the peak month (pRH) varied
between 64.8 and 71.0% for all locations except for Nis (1993-96) where it was
55.3%. The mean annual wind speed (U2) was the lowest at Kragujevac (1981-84;
1.1 m s-1) and Nis (1977-84 and 1993-96; 1.0 and 1.1 m s-1, respectively); it varied
for all other locations between 1.5 and 2.0 m s-1. The mean annual and peak
monthly estimates by the FAO-56 PM equation (ETo_pm and pETo_pm) ranged from
2.09 to 2.50 mm day-1 and 4.18 to 4.83 mm day-1, respectively.
57
IV.3.2.2. Temperature-based equations for estimating ETo
The temperature-based equations use only temperature and latitude data for
estimating ETo. The FAO-56 PM equation that uses only maximum and minimum
air temperatures for estimating ETo was called reduced set FAO-56 PM (PMt).
Solar radiation was obtained from following equation:
R s = K (Tmax − Tmin ) 0.5 Ra
(62)
where Rs = solar radiation (MJ m-2 day-1); Ra = extraterrestrial radiation
(MJ m-2 day-1); Tmax and Tmin = maximum and minimum temperature (oC),
respectively; and K = adjustment coefficient. Hargreaves et al. (1985)
recommended using K = 0.16 for "interior" locations and K = 0.19 for coastal
locations. Estimates of actual vapor pressure (ed) were obtained from minimum air
temperature:
⎡ 17.27Tmin ⎤
ed = 0.611exp⎢
⎥
⎣ Tmin + 237.3 ⎦
(63)
and the value of 2 m s-1 was adopted for wind speed.
The Hargreaves equation is one of the simplest equations used to estimate ETo. It is
expressed as (Hargreaves et al. 1985, Hargreaves and Allen 2003):
⎛ T + Tmin
⎞
ETo = 0.0023Ra ⎜ max
+ 17.8 ⎟ Tmax − Tmin
2
⎝
⎠
(64)
Thornthwaite (1948) correlated mean monthly temperature with ET as determined
by east-central United States water balance studies. The Thornthwaite equation used
in this paper is:
12
ETo ,k
⎛
⎜
16 ⋅ N k ⎜
10 ⋅ Tk
=
12
360 ⎜
1.514
⎜ ∑ (0.2 ⋅ Tk )
⎝ k =1
⎞
⎟
⎟
⎟
⎟
⎠
0.016⋅ ∑ (0.2⋅Tk )1.514 + 0.5
k =1
(65)
where ETo,k = ETo in the k-th month (mm day-1); Nk = maximum possible duration
of sunshine in the k-th month (hours); Tk = mean air temeperature in the k-th month
(oC); k = 1, 2, …12.
IV.3.2.3. Artificial Neural Networks (ANNs)
This study investigates the utility of an adaptive RBF network with Tmax and Tmin as
inputs for estimating reference evapotranspiration and compares the performance of
the RBF network with the FAO-56 PM equation.
In this study, a sequentially adaptive radial basis function (RBF) network was
applied to estimating reference evapotranspiration (ETo). Sequential adaptation of
parameters and structure was achieved using the Extended Kalman filter (EKF).
58
A criterion for network growth is obtained from the Kalman filter's consistency test
(Todorovic et al. 2000). Readers are referred to Trajkovic et al. (2003c) for a
detailed description of this neural network.
Data from Palic (84 patterns; 1977-83), Belgrade (96 patterns; 1977-84), and Nis
(96 patterns; 1977-84) were used for training of the RBF network. The training data
set had a total of 276 patterns. The FAO-56 PM estimated ETo values were
employed as substitute for measured ETo data and used for training of RBF
network. The RBF network was trained with weather data (Tmax and Tmin) and
astronomic datum (maximum possible duration of sunshine - N) as inputs and the
FAO-56 PM estimated ETo (ETo_pm) as output (Trajkovic et al. 2003b). Parameter N
depends on the position of the sun and is hence a function of latitude and date
(month).
The sequence of 276 samples of ETo was scaled [-1, 1]. The network learned to
estimate ETo based on Tmax, Tmin, and N. The adaptive RBF network learns from the
data, which arrive continually and are shown in the network only once. The RBF
network simultaneously estimates and learns. On the basis of the estimate error on
the last sample, the parameters and structure of the RBF network change, and the
changed network gives the estimate for the next sample, where another error is
obtained, which again changes the parameters and the structure of the RBF network.
Training is over when all the samples pass through network. After the training is
over, the weights, number of hidden neurons, and radial basis functions of the
network are frozen. The obtained adaptive temperature-based RBF network had the
following structure: in the input layer, there were three neurons that receive
information on the Tmax, Tmin and N values, in the hidden layer, there were two
neurons, and in the output layer, there was one neuron giving the ETo value.
IV.3.3.1. Estimating monthly ETo by temperature-based equations
Three temperature-based (PMt, Hargreaves and Thornthwaite) equations were used
to estimate monthly ETo at seven humid locations. The ETo estimates (mm day-1)
was computed for each month using weather data for that month in the ETo
equations. The ETo values estimated by the three temperature-based equations
(ETo_eq) were compared with estimates by the standard FAO-56 PM equation
(ETo_pm). The statistical summary of ETo estimates for seven locations in Serbia is
presented in Table 22. In this table the following abbreviations are used: the
ETo_eq/ETo_pm is the ratio of mean annual temperature equations estimated ETo and
FAO-56 PM estimated ETo, pETo_eq/ETo_pm is the ratio of temperature equations
estimated ETo and FAO-56 PM estimated ETo in the peak month (July), MXAE is
the maximum absolute error, MAE is the mean absolute error, the RMSE is the root
mean square error.
59
Table 22. Statistical summary of ETo estimates for seven locations in Serbia
MAE
RMSE
Equation ETo_eq/ETo_pm pETo_eq/ETo_pm MXAE
(%/100)
(%/100)
(mm day-1) (mm day-1) (mm day-1)
Palic (1977-83)
PMt
1.045
0.999
0.450
0.134
0.161
Harg
1.137
1.117
0.746
0.306
0.371
Thw
0.873
0.993
1.189
0.387
0.485
Novi Sad (1981-84)
PMt
1.053
1.046
0.663
0.155
0.173
Harg
1.146
1.166
0.907
0.344
0.448
Thw
0.849
0.993
1.320
0.454
0.571
Belgrade (1977-84)
PMt
0.922
0.937
0.791
0.208
0.263
Harg
1.013
1.049
0.626
0.193
0.232
Thw
0.816
0.961
1.622
0.506
0.636
Negotin (1971-74)
PMt
1.090
1.016
0.768
0.237
0.300
Harg
1.182
1.125
1.163
0.442
0.524
Thw
0.841
0.962
1.114
0.425
0.511
Kragujevac (1981-84)
PMt
1.187
1.107
0.873
0.394
0.445
Harg
1.286
1.224
1.296
0.601
0.704
Thw
0.915
1.004
1.036
0.335
0.415
Nis (1977-84)
PMt
1.194
1.126
1.193
0.432
0.513
Harg
1.289
1.243
1.420
0.640
0.773
Thw
0.894
0.994
1.348
0.416
0.514
Nis (1993-96)
PMt
1.187
1.165
1.013
0.449
0.525
Harg
1.274
1.270
1.486
0.659
0.792
Thw
0.876
1.009
1.102
0.459
0.545
Vranje (1971-74)
PMt
1.080
1.033
0.533
0.217
0.251
Harg
1.168
1.141
0.866
0.393
0.463
Thw
0.807
0.927
1.318
0.474
0.573
The temperature-based equations were generally poor in estimating ETo. The PMt
equation overestimated mean annual ETo_pm by 4.5% at Palic (1977-83) to as much
as 19% at Nis in both periods (1977-84 and 1993-96) and underestimated ETo_pm by
7.8% at Belgrade (1977-84). The PMt equation gave acceptable estimates of mean
peak ETo at Palic (1977-83), Novi Sad (1981-84), Negotin (1971-74) and Vranje
(1971-74) with deviations of -0.1 to +4.6% relative to the ETo obtained by the FAO56 PM equation. The RMSE values ranged from 0.161 to 0.525 mm day-1 for all
seven locations.
60
The Hargreaves (Harg) equation overestimated mean ETo_pm for the peak month by
about 5% at Belgrade (1977-84) to as much as 27% at Nis (1993-96). According to
the MXAE, MAE, RMSE statistics, this equation was in the last place in ranking at
Negotin (1971-74), Kragujevac (1981-84) and Nis for both periods (1977-84 and
1993-96). The RMSE values for all seven locations varied from 0.232 to 0.792
mm day-1.
The Thornthwaite (Thw) equation underestimated ETo_pm at all locations. On an
annual basis, the Thornthwaite equation underpredicted ETo_pm by 8.5% at
Kragujevac (1981-84) to as much as 19.3% at Vranje (1971-74). This equation was
the best in estimating the mean ETo for the peak month for all locations except
Vranje (1971-74, deviation of -7.3%) with deviation of -3.9 to +0.4% relative to the
ETo obtained by the FAO-56 PM equation. However, according to the statistics, the
Thornthwaite equation was the poorest at Palic (1977-83), Novi Sad (1981-84),
Belgrade (1977-84), and Vranje (1971-74).
The Thornthwaite and Hargreaves equations yielded similar RMSE values.
The poor results for both the Hargreaves and Thornthwaite equations were in good
agreement with data reported by Jensen et al. (1990) and Amatya et al. (1995).
Statistics of temperature-based equations obtained at Nis for both periods (1977-84
and 1993-96) were very similar.
The temperature-based equations mostly underestimated or overestimated ETo
obtained by the FAO-56 PM equation. In those cases, Allen et al. (1994a)
recommended that empirical equations be calibrated using the standard PM
equation. ETo is calculated as:
ETo = a + bETo _ eq
(66)
where ETo = grass reference ET defined by the FAO-56 PM equation; ETo_eq =
ETo estimated by the temperature-based equation; a and b = calibration factors,
respectively.
The data used for the training of RBF network (weather data from Palic
(84 patterns; 1977-83), Belgrade (96 patterns; 1977-84), and Nis (96 patterns; 197784)) were used for calibration of temperature-based equations. Thus, the calibrated
ETo estimates can be compared to the ETo values produced by RBF network and
FAO-56 PM estimates.
The calibration temperature-based equations are obtained from the following
equations:
ETo _ cpmt = 0.949 ETo _ pmt + 0.013
(67)
where ETo_cpmt = ETo estimated by the calibrated PMt equation; and ETo_pmt =
ETo estimated by the PMt equation.
ETo _ ch arg = 0.817 ETo _ h arg + 0.320
(68)
61
where ETo_charg = ETo estimated by the calibrated Hargreaves equation; and ETo_harg
= ETo estimated by the Hargreaves equation.
ETo _ cThw = 0.880 ETo _ Thw + 0.565
(69)
where ETo_cthw = ETo estimated by the calibrated Thornthwaite equation; and ETo_thw
= ETo estimated by the Thornthwaite equation.
IV.3.3.2. Estimating monthly ETo by RBF network and calibrated
temperature-based equations
The calibrated temperature-based equations and the RBF network were tested by
comparing them with the estimated FAO-56 PM data for Novi Sad (1981-84),
Negotin (1971-74), Kragujevac (1981-84), Nis (1993-96), and Vranje (1971-74).
The test data set had a total of 240 patterns that were not used for training or
calibration. The statistically processed data are presented in Table 23.
The calibrated PMt equation predicted FAO-56 PM ETo best at Novi Sad.
This equation gave the lowest RMSE (0.145 mm day-1). The RMSE was similar for
the RBF network (0.161 mm day-1). The calibrated Hargreaves and Thornthwaite
equation gave poor agreement with FAO-56 ETo estimates.
The RBF network was ranked at the top at Negotin with the lowest RMSE of
0.221 mm day-1. The RMSE was similar for the calibrated PMt equation
(0.235 mm day-1). The calibrated Hargreaves and Thornthwaite equation yielded
the poorest correlation with FAO-56 ETo estimates.
The RBF network predicted FAO-56 PM ETo best at Kragujevac. This approach
gave the lowest value of RMSE (0.230 mm day-1). The calibrated temperaturebased ETo equations consistently overestimated ETo.
The RBF network was ranked at the top at Nis with the lowest RMSE of
0.232 mm day-1. Estimates by the calibrated PMt and Hargreaves equations
consistently overestimated ETo. The calibrated Hargreaves equation yielded the
highest RMSE (0.456 mm day-1).
The calibrated PMt estimates were in closest agreement with FAO-56 PM estimates
at Vranje. This equation gave the lowest RMSE of 0.205 mm day-1. The RBF
network was ranked at the second place with RMSE of 0.266 mm day-1.
This method consistently slightly underestimated ETo. This underestimation may
occur due to the high site elevation (433 m). There were no locations with such
elevation in training locations. The calibrated Thornthwaite equation gave the
highest RMSE (0.399 mm day-1).
62
Table 23. Statistical summary of calibrated ETo equations and RBF network for five
locations in Serbia
MAE
RMSE
Equation ETo_eq/ETo_pm p ETo_eq/ETo_pm MXAE
(%/100)
(mm day-1) (mm day-1) (mm day-1)
(%/100)
Average Serbia
cPMt
1.067
1.022
0.871
0.215
0.272
cHarg
1.128
1.039
0.897
0.317
0.363
cThw
1.001
0.986
0.958
0.315
0.378
RBF
0.992
1.001
0.850
0.172
0.224
Novi Sad (1981-84)
cPMt
1.005
0.996
0.497
0.110
0.145
cHarg
1.073
1.026
0.538
0.198
0.230
cThw
0.991
1.004
0.959
0.319
0.400
RBF
0.972
1.015
0.482
0.130
0.161
Negotin (1971-74)
cPMt
1.040
0.967
0.549
0.192
0.235
cHarg
1.102
0.985
0.603
0.271
0.321
cThw
0.982
0.963
0.644
0.294
0.336
RBF
0.974
0.951
0.556
0.175
0.221
Kragujevac (1981-84)
cPMt
1.133
1.054
0.725
0.282
0.324
cHarg
1.203
1.077
0.813
0.425
0.451
cThw
1.075
1.019
0.690
0.297
0.355
RBF
1.063
1.050
0.499
0.183
0.230
Nis (1993-96)
cPMt
1.132
1.108
0.871
0.324
0.383
cHarg
1.176
1.104
0.897
0.419
0.456
cThw
1.009
1.005
0.705
0.339
0.395
RBF
1.007
1.030
0.753
0.179
0.232
Vranje (1971-74)
cPMt
1.030
0.983
0.401
0.167
0.205
cHarg
1.092
1.004
0.559
0.275
0.307
cThw
0.953
0.942
0.901
0.325
0.399
RBF
0.949
0.965
0.850
0.193
0.266
Calibration enhances the performance of the temperature-based equations.
The calibrated PMt estimates correlated well with the mean annual and peak
monthly FAO-56 PM estimates for all sites with wind speed between 1.5 and 2 m s-1.
This method yielded the best estimate of the mean annual ETo both at Novi Sad and
Vranje. However, the calibrated PMt method gave unsatisfactory results in
Kragujevac and Nis. This method overestimated mean annual ETo_pm by about
13% at Kragujevac and Nis. This overestimation may be due to low wind speeds at
these locations.
63
The calibrated Hargreaves equation was very poor in estimating the mean annual
ETo. The RMSE values varied from 0.230 to 0.456 mm day-1. This equation
overestimated ETo_pm by about 7 % at Novi Sad to as much as 20 % at Kragujevac
with an average of 12.8 % for all sites. The calibrated Hargreaves equation
performed very well in estimating peak monthly ETo both at Vranje and Negotin
with deviations of +0.4 and -1.5% relative to the ETo obtained by the FAO-56 PM
equation, respectively.
On average, estimates with the calibrated Thornthwaite equation were in closest
agreement with mean annual FAO-56 PM estimated ETo values. This equation was
the best in estimating the mean ETo for the peak month in Novi Sad, Kragujevac
and Nis. On average, however, according to the MXAE, MAE, and RMSE values,
this equation was on the last place in ranking. The calibrated Thornthwaite equation
underpredicted mean daily ETo_pm in the first half of the year and overpredicted
ETo_pm in the second half of the year. That is why its MAE and RMSE values are
poor. The RBF network provided very good estimate of both peak and mean annual
ETo at all locations. This method yielded the best estimate of the mean annual ETo
both at Kragujevac and Nis. On average, the RBF network values were in the
closest agreement with annual and peak monthly FAO-56 PM estimated ETo values
with deviation of -0.8 and +0.1 %, respectively. The RMSE averaged 0.224
mm day-1 over all sites.
The adaptive temperature-based RBF network is simpler in structure, in comparison
with the ANNs from Kumar et al. (2002) and Sudheer et al. (2003). It uses fewer
training samples and is trained faster. The adaptive RBF approach uses only one
network, which in the course of training changes its structure while the Kumar et al.
(2002) approach selects the best ANN from the set of 162 ANNs.
Kumar et al. (2002), in the second part of this paper, have showed that the local
ANN model can predict lysimeter ETo slightly better than the standard FAO-56 PM
equation. However, if the local calibration of the global PM equation had been
conducted, probably the results more favorable for the PM equation would have
been obtained. On the other hand, the PM equation proved itself as a global model
and thus can be applied even for the sites for which it is not trained which is not
true for the local ANN models.
The recent research has indicated that adjusted Hargreaves equation (Trajkovic
2007) and reduced-set FAO-56 PM approaches with local or regional default wind
speed value (Trajkovic and Kolakovic 2009c) gave slightly better results in
comparison to temperature-based RBF network with partial exception of Novi Sad.
The results of comparison are given in Table 24. However, if the new wind speed
input had been introduced, probably the results more favorable for the RBF network
would have been obtained. The wind-adjusted Turc equation (Trajkovic and
Kolakovic 2009a) consistently gave the poorest results. Readers are referred to
Trajkovic (2007) and Trajkovic and Kolakovic (2009a, 2009c) for detailed
description of these equations.
64
Table 24. Statistical summary of adjusted ETo equations and RBF network for five locations
in Serbia
PMt,r
cTURC AHARG
RBF
Approach
PMt,l
Vranje 1971-74
RMSE (mm day-1)
0.186
0.194
0.290
0.234
0.266
ETo_eq/ETo_pm
1.027
0.984
0.924
0.971
0.949
Nis
RMSE (mm day-1)
0.205
0.293
0.352
0.213
0.232
ETo_eq/ETo_pm
1.042
1.089
0.916
1.050
1.007
Kragujevac
RMSE (mm day-1)
0.217
0.266
0.250
0.209
0.230
ETo_eq/ETo_pm
1.069
1.098
0.989
1.069
1.063
Negotin
RMSE (mm day-1)
0.240
0.196
0.255
0.214
0.221
ETo_eq/ETo_pm
1.056
1.006
0.944
0.980
0.974
Novi Sad
RMSE (mm day-1)
0.179
0.142
0.323
0.184
0.161
ETo_eq/ETo_pm
1.043
0.974
0.936
0.953
0.972
IV.3.3.3. Estimating daily ETo by RBF network
The obtained temperature-based RBF network was additionally tested using daily
FAO-56 PM ETo data from the experimental field "E. Pantanelli" of Bari University
with 40o17' N, 16o40' E, and altitude 15 m above sea level. The long-term average
values of the major weather parameters are presented below: minimum and
maximum air temperature are 11.0 and 21.4 oC, respectively; minimum and
maximum relative humidity are 52 and 87 %, respectively; sunshine is 6 h 36 min;
wind speed is 2.3 m s-1 (Caliandro et al. 1990).
The data set used in this study had 497 patterns distributed from May 15, 1981 to
December 18, 1984 (A. Caliandro, University of Bari, personal communication,
2000). The RBF network was compared to the FAO-56 PM equation. The daily ETo
values as estimated by the RBF network (ETo_rbf) and FAO-56 PM method (ETo_pm)
are plotted in Figure 10.
ETo (mm day-1)
8
65
Policoro, Italy
6
4
2
Days
0
1
ETo_ann
ETo_pm
1314
Figure 10. Comparison of daily ETo computed for four years at Policoro, Italy using RBF
network (ETo_ann) and FAO-56 Penman-Monteith equation (ETo_pm)
Figure 10 illustrates the close relationship between ETo from the FAO-56 equation
and from the RBF network. The ETo for the RBF network and FAO-56 PM
averaged 3.67 mm day-1 for both methods over a 4 year period. The ratio of the RBF
network to the FAO-56 PM equation during days of the peak month (July) equals
0.94. The results suggest that the daily ETo could be computed from air temperature
using the RBF network.
IV.3.3.4. Conclusions
Some temperature-based equations significantly underestimated or overestimated
mean annual and peak monthly FAO-56 Penman-Monteith estimates. Calibration
enhanced the performances of temperature-based equations. However, some
equations greatly underestimated or overestimated ETo_pm even after calibration.
The reduced set FAO-56 Penman-Monteith equation with 2 m s-1 wind speed (PMt)
is not recommended, even in the calibrated form, in the locations that have
significantly different wind speed than 2 m s-1.
The calibrated Hargreaves equation (cHarg) overestimated ETo_pm even after the
calibration. So, this equation cannot be recommended for utilization. The calibrated
Thornthwaite estimates correlated well with the mean annual and peak monthly PM
estimates for all sites. However, this equation significantly underpredicted mean
daily ETo in the first half of the year and overpredicted ETo in the second half of the
year. So, the calibrated Thornthwaite method may be recommended only for
estimating ETo for the peak month.
66
The RBF network provides the quite good agreement with the evapotranspiration
obtained by the FAO Penman-Monteith equation. It gave reliable estimation at all
the locations and it has proven to be the most adjustable to the local climatic
conditions. The RBF network mostly provided better results compared to calibrated
temperature-based methods. These results recommend the temperature-based RBF
network for estimating reference evapotranspiration. The overall results are of
significant practical use because the temperature-based RBF network can be used
when relative humidity, radiation and wind speed data are not available.
67
V. CONCLUSIONS
In this publication, RBF networks have been used for estimating FAO-24
evapotranpiration coefficients, forecasting of reference evapotranspiration and
estimation of reference evapotranspiration.
The validity of evapotranspiration calculation by FAO-24 equations is increased
with the accurate estimation of evapotranspiration coefficients. The determining of
these coefficients by table interpolation should be avoided because of its long
procedure that can lead to a high error, which is directly transferred to the estimated
evapotranspiration. The second approach requires the use of regression equations
first introduced by Frevert et al. (1983) and later improved by many researchers.
However, the application of the regression equations did not always give
satisfactory results. The comparative analysis showed that RBF networks guarantee
a more accurate estimation of FAO-24 evapotranspiration coefficients when
compared to regression equations. Improved estimates of the FAO-24 adjustment
factors do reduce the error in estimating reference evapotranspiration.
All advantages of the RBF network over the regression equation were demonstrated
by numerous examples.
The potential of two types (adaptive and sequentially adaptive) of RBF networks
for forecasting of reference evapotranspiration has been presented in this
publication. Self generating algorithm by maximum absolute error (MXAE)
selection method is applied as a design method for adaptive RBF network.
This publication presents the application of adaptive radial basis function (RBF)
structures in forecasting of reference evapotranspiration at Griffith, Australia.
A seasonal autoregressive integrated moving average (SARIMA) model is also
developed for the evapotranspiration data. According to the comparison results it
can be said that RBF networks are superior as compared to SARIMA model.
A sequentially adaptive Radial Basis Function network is applied to the forecasting
of reference evapotranspiration at Nis, Serbia. Sequential adaptation of parameters
and structure is achieved using extended Kalman filter. Criterion for network
growing is obtained from the Kalman filter's consistency test. Criteria for
neuron/connections pruning are based on the statistical parameter significance test.
The results show that the ANNs can be used for the forecasting of reference
evapotranspiration with high reliability.
In this publication, sequentially adaptive RBF networks were used for estimating
hourly, daily and monthly reference evapotranspiration. The RBF network using
limited weather data was able to reliably estimate hourly ETo for a well-irrigated
grass under different atmospheric conditions at Davis, CA, United States.
The calculation of the hourly ETo is possible only on the basis of the air temperature
and the net radiation, without using the wind speed, humidity and soil heat flux.
This publication presents that the pan-based RBF network can be successful
alternative to FAO-56 Penman-Monteith equation for estimating daily reference
68
Conclusions
evapotranspiration. The RBF network consistently provided better results compared
to FAO-24 pan and Christiansen equations, although required measurements of only
one weather parameter, pan evaporation.
The temperature-based RBF network provides the quite good agreement with the
monthly reference evapotranspiration obtained by the FAO-56 Penman-Monteith
equation for five humid Serbian locations. It gave reliable estimation at all the
locations and it has proven to be the most adjustable to the local climatic conditions.
The RBF network mostly provided better results compared to calibrated
temperature-based equations. These results recommend the temperature-based RBF
network for estimating reference evapotranspiration. The overall results are of
significant practical use because the temperature-based RBF network can be used
when relative humidity, radiation and wind speed data are not available.
This publication demonstrates that the correct application of ANN vs. FAO-56
Penman-Monteith is the development of the ANN models for estimating reference
evapotranspiration from limited climatic data, especially in the areas where there
are no measurements of all the weather data required from the FAO-56 PM.
The FAO-56 PM is still a guide and people should adapt all calculations to their
local conditions. The people should use their own judgment for the results based on
their local experiences and not take the results blindly.
Finally, it can be concluded that artificial neural networks are the promising
approach to estimating reference evapotranspiration.
NOTATION
a = calibration factor;
ai = the weight of the i-th Gaussian basis function φi ( u , mi , σi ) , i=1,...,NH;
b = calibration factor;
Cr = FAO-24 radiation adjustment factor,
c = FAO-24 Penman adjustment factor;
E = sum of squares of errors;
Epan = pan evaporation;
ea = saturation vapor pressure;
ed = actual vapor pressure;
ea-ed = vapor pressure deficit;
ETo = reference crop evapotranspiration;
F = upwind fetch of low-growing vegetation;
G =soil heat flux;
gi = activation function of i-th hidden neuron;
h(u) = output of RBF network;
K = adjustment coefficient;
Kp = pan factor;
mij = center of i-th basis function for j-th input;
N = maximum sunshine hours;
NH = number of hidden neurons;
NI = number of input neurons;
n/N = mean ratio of actual to possible sunshine hours;
n = actual sunshine hours;
p = mean daily percentage of total annual daytime hours;
q = parameter;
Ra = extraterrestrial radiation;
Rs = solar radiation;
Rn = net radiation;
ra = aerodynamic resistance;
rc = canopy resistance;
RHmin = minimum daily relative humidity;
T = mean air temperature;
69
70
Tmax = maximum air temperature;
Tmin = minimum air temperature;
U2 = mean wind speed at 2 m;
Ud/Un = ratio of daytime to nighttime wind speeds;
u j = j-th input, j=1,...,NI;
W = weighting factor;
xj = value of j-th input;
yk = k-th output from network;
y*k = desired k-th output;
α= momentum;
Δ = slope of the saturated vapor pressure curve;
ε = specified model error;
γ = psychrometric constant;
η = learning step;
λ =number of samples, (λ = 1, 2, ..., Λ);
θ = bias; and
σij = width of i-th basis function for j-th input.
Notation
71
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80
About…
ABOUT THE AUTHOR
Slavisa Trajkovic
B.Sc. in Civil Engineering at the University of Nis, Serbia (1990). M.Sc. in Civil
Engineering at the University of Nis, Serbia (1995). Ph.D. in Civil Engineering at
the University of Nis, Serbia (2002).
Associate professor at the Faculty of Civil Engineering, University of Nis, Serbia.
The domains of his scientific activities and expertise are: evapotranspiration,
application of ANNs in water resources research, hydrometeorology, water balance
modeling, irrigation scheduling optimization, implementation of EU Water
Framework Directive.
Participant in numerous international development projects (International
Management Group (IMG)-Towns and Schools for Democracy (TSfD) Programme,
UN Development Programme (UNDP)-Rapid Employment Programme (REP) in
South Serbia, Cooperative Housing Foundation (CHF)-Community Revitalization
through Democratic Action (CRDA), Cooperazione Internazionale (COOPI)Improvement of management and control of hydro and environmental resources in
the City of Nis).
Co-editor of one international scientific monograph „Sicevo and Jelasnica gorges
environmental status monitoring” financed by the Italian Ministry of Foreign
Affairs and co-author of one textbook. His scientific papers have been cited several
scores of times in numerous international scientific journals. Reviewer of several
scientific journals cited by Science Citation Index in the field of hydrology,
irrigation and water resources.
81
ABOUT THE REVIEWERS
Ozgur Kisi
B.Sc. in Engineering Faculty (Department of Civil Engineering) at the Cukurova
University, Turkey (1997), M.Sc. in Institute of Science and Technology
(Hydraulics Division) at the Erciyes University, Turkey (1999), Ph.D. in Institute of
Science and Technology (Hydraulics Division) at the Istanbul Technical University,
Turkey (2003). His research fields are: relatively new mathematical tools applied to
hydrological sciences, sometimes called hydroinformatics, suspended sediment
modeling, evapotranspiration estimation, river flow forecasting techniques.
Participant in numerous national research projects. Supervisor of several MSc and
PhD Works. Author of several peer-reviewed scientific publications. In editorial
board for World Applied Sciences Journal (included in ISI database), Journal of
Engineering and Applied Sciences and The Open Hydrology journal. Reviewer of
several scientific journals cited by Science Citation Index in the field of hydrology,
irrigation and water resources. 2006 International Tison Award (given by the
International Association of Hydrological Sciences (IAHS), URL:
http://www.cig.ensmp.fr/~iahs/Tison/kisi.htm).
Mladen Todorovic
B.Sc. in Civil Engineering at the University of Belgrade, Serbia (1987). M.Sc. in
Irrigation at the CIHAM-IAMB, Italy (1993). Ph.D. in Agro-meteorology at the
University of Bologna, Italy (1998). Senior Research Scientist and lecturer at the
International Center for Advanced Mediterranean Agronomic Studies (CIHEAM) Mediterranean Institute of Agriculture of Bari. Since 2000, scientific tutor of the
international post-graduate programme on “Land and water resource management:
irrigated agriculture”. The domains of his scientific activities and expertise are:
agro-meteorology, water saving, water balance modeling, evapotranspiration,
application of GIS and spatial modeling in water resources management, irrigation
scheduling optimization. Consultant on EU/DG I Regional (Mediterranean) Action
Programme “Water Resources Management”. Participant in numerous international
and national research projects. Supervisor of several MSc and PhD works. Member
of ASCE-EWRI Committee on Evapotranspiration in Irrigation and Hydrology and
of Technical Committee on revision of Kc. Author of several peer-reviewed
scientific publications, co-editor of one book on “Halophytes Uses in Different
Climates (Backhuys Publishers, Leiden, The Netherlands) and co-editor of a special
issues of Physics and Chemistry of the Earth on “Water Resources Assessment for
Catchment Management” (Elsevier Science Ltd., The Netherlands) and two Options
Mediterraneennes, Serie B: Studies and Research, No 48 and No 52 (CIHEAM).
Reviewer of several international scientific journals in the field of irrigation water
management. Best research paper Award of ASCE Journal of Irrigation and
Drainage Engineering in 1999.
82
About…
Dragan Arandjelovic
B.Sc. in Civil Engineering at the University of Nis, Serbia (1972). M.Sc. in Civil
Engineering at the University of Belgrade, Serbia (1976). Ph.D. in Civil
Engineering at the University of Belgrade, Serbia (1981). Professor at the Faculty
of Civil Engineering, University of Nis, Serbia. The domains of his scientific
activities and expertise are: fluid mechanics, hydrometric research, water balance
modeling, ground water research. Participant in numerous international and national
research projects. Supervisor of several MSc and PhD works. Since 2003, Dean at
the Faculty of Civil Engineering and Architecture, University of Nis.
CIP – Каталогизација у публикацији
Народна библиотека Србије, Београд
551.573
556.131
TRAJKOVIĆ, Slaviša, 1965 Estimating Reference Evapotranspiration by
Artificial Neural Networks / Slavisa
Trajkovic. – Nis: Faculty of Civil
Engineering and Architecture, 2009 (Nis : M
kops centar). – VI, 82 str. : graf. prikazi,
tabele ; 24 cm
Na vrhu nasl. str.: University of Nis. - Tiraž
100. – About of Author: str. 80. –
Bibliografija: str. 71 – 79.
ISBN:978-86-80295-84-8
а) Евапотранспирација – Прорачун
COBISS.SR-ID 168055820

University of Nis

Transcription

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