ACEE Int. J. on Civil and Environmental Engineering, Vol. 01, No. 01, Feb 2011

© 2011 ACEE 31DOI: 01.IJCEE.01.01.517

Estimation of Weekly Reference Evapotranspirationusing

Linear Regression and ANN ModelsK.Chandra Sekhar Reddy1, S.Aruna Jyothy2, and P.Mallikarjuna3

1 N.B.K.R.I.S.T., Department of Civil Engineering, Vidyanagar, Andhra Pradesh, India.Email: vinod_harsha@yahoo.co.in

2 Sri Venkateswara University, Department of Civil Engineering, Tirupati, IndiaEmail: aruna_jyothy@yahoo.co.in

3 Sri Venkateswara University, Department of Civil Engineering, Tirupati, IndiaEmail : mallikarjuna_perugu@yahoo.co.in

Abstract— The study investigates the applicability of linearregression and ANN models for estimating weekly referenceevapotranspiration (ET0) at Tirupati, Nellore, Rajahmundry,Anakapalli and Rajendranagar regions of Andhra Pradesh.The climatic parameters influencing ET0 were identifiedthrough multiple and partial correlation analysis. Thesunshine, temperature, wind velocity and relative humiditymostly influenced the study area in the weekly ET0 estimation.Linear regression models in terms of the climatic parametersinfluencing the regions and, optimal neural networkarchitectures considering these climatic parameters as inputswere developed. The models’ performance was evaluated withrespect to ET0 estimated by FAO-56 Penman-Monteith method.The linear regression models showed a satisfactoryperformance in the weekly ET0 estimation in the regionsselected for the present study. The ANN (4,4,1) models,however, consistently showed a slightly improved performanceover linear regression models.

Index Terms—Reference evapotranspiration, multiple linearregression, artificial neural network, performance evaluation

I. INTRODUCTION

An accurate estimation of reference cropevapotranspiration (ET0) is of paramount importance fordesigning irrigation systems and managing natural waterresources. Numerous ET0 equations have been developedand used according to the availability of historical and currentweather data. These equations range in sophistication fromempirical to complex equations. The FAO-56 Penman-Monteith (PM) equation [1] is widely used in recent times forET0 estimation. However, the difficulty in using this equation,in general, is the lack of accurate and complete data. Inaddition, the parameters in the equation potentially introducecertain amount of measurement and/or computational errors,resulting in cumulative errors in ET0 estimates. Under theseconditions, a simple empirical equation that requires as fewparameters as possible and, results comparable with Penman-Monteith method is preferable. Owing to the difficultiesassociated with model structure identification and parameterestimation of the nonlinear complex evapotranspirationprocess, most of the models that have been developed maynot yield satisfactory results. ANNs are capable of modellingcomplex nonlinear processes effectively extracting the relationbetween the inputs and outputs of a process without the

physics being explicitly provided to them and also, theyidentify the underlying rule even if the data is noisy andcontaminated with errors [2] and [3]. “Reference [4]”investigated the utility of ANNs for the estimation of dailyET0 and compared the performance of ANNs with PM method.It was concluded that ANNs can predict ET0 better than theconventional methods. “Reference [5]” examined the potentialof artificial neural networks in estimating the actualevapotranspiration from limited climatic data and suggestedthat the crop evapotranspiration could be computed from airtemperature using the ANN approach. “Reference [6]” showedthat ANNs can be used for forecasting ET0 with high reliability.“Reference [7]” derived solar radiation and net radiationbased ET0 equations using multi- linear regression techniqueand concluded that the equations performed better than thesimplified temperature and/or radiation based methods forhumid climates. “Reference [8]” tested the ANNs forestimating ET0 as a function of maximum and minimum airtemperatures and concluded that when taking into accountjust the maximum and minimum air temperatures, it is possibleto estimate ET0. “Reference [9]” tested the ANNs, to estimateET0 as a function of the maximum and minimum airtemperatures in semiarid climate. While comparing with PMmethod, it was concluded that ANN methods are better forET0 estimates than the conventional methods. “Reference[10]” evaluated ANN models for daily ET0 estimation undersituations of presence of only temperature and relativehumidity data. ANNs showed an improved performance overtraditional ET0 equations. “Reference [11]” developedgeneralized artificial neural network (GANN) based referencecrop evapotranspiration models corresponding to FAO-56PM, FAO-24 Radiation, Turc and FAO-24 Blaney-Criddlemethods using the data from California Irrigation Managementand Information System stations. It was concluded that theGANN models can be used directly to predict ET0 under thearid conditions since they performed better than theconventional ET0 estimation methods. “Reference [12]”compared weekly evapotranspiration ANN based forecastswith regard to a model based on weekly averages and foundan improved performance of one week in advance weeklyET0 predictions compared to the model based on means (meanyear model).

ACEE Int. J. on Civil and Environmental Engineering, Vol. 01, No. 01, Feb 2011

© 2011 ACEE 32DOI: 01.IJCEE.01.01.517

II. MATERIALS AND METHODS

The climatic data at Tirupati, Nellore, Rajahmundry,Anakapalli and Rajendranagar meteorological centerscollected from the India Meteorological Department (IMD),Pune, India were used in the data analysis and modeldevelopment. A part of the data was used for the purpose ofdevelopment of models and the rest for validating the modelsdeveloped. The resemblance of the statistical structure interms of mean, variance and skewness of the calibration andvalidation data sets was ensured while making the divisionof the data into training and testing data sets. A briefdescription of the meteorological centers along with the dataperiod is shown in TABLE I. In the present study, an attemptis made to develop simple linear and optimal neural networkmodels considering the climatic parameters influencing theregions selected for the study for weekly ET0 estimation. Thestudy also compares the performance of proposed linearregression and ANN models.

III. MODEL DEVELOPMENT

The weekly reference evapotranspiration model at ameteorological center is developed using the climatic data atthe center. The steps in the modelling include i) identificationof meteorological parameters influencing the region, ii)development of the model and iii) performance evaluation ofthe model developed. The identification of meteorologicalparameters influencing the region is based on multiple andpartial correlation analysis. The linear regression and ANNmodels are developed for the present study. The performanceof the models is verified through selected performanceevaluation criteria.

A. Linear Regression (LR) modelThe objective of the model is the transfer of information

among several variables observed simultaneously and theestimation of the dependent variable from the several otherobserved independent variables.

The weekly reference evapotranspiration (ET0) at ameteorological center is expressed as a simple linear modelas

ET0=C + a1 X1 + a2 X2 +… (1)where a1,a2,…… and C are empirical constants and X1,X2,….are the meteorological parameters influencing the region. Themultiple correlation analysis was carried out using STASTICApackage.

B. Artificial Neural Network model (ANN)A standard multilayer feed-forward ANN with logistic

sigmoid function was adopted for the present study. Aconstant value of 0.1 for learning rate and a constant value of0.9 for momentum factor were considered. The data werenormalized in the range of (0.1, 0.9) to avoid any saturationeffect. Error back propagation which is an iterative nonlinearoptimization approach based on the gradient descent searchmethod [13] was used during calibration. The calibration setwas used to minimize the error and validation set was used toensure proper training of the neural network employed suchthat it does not get overtrained. The performance of the modelwas checked for its improvement on each iteration to avoidoverlearning. The optimal network corresponding to minimummean squared error was obtained through trail and errorprocess. Care was taken to avoid too few and too manyneurons which can respectively cause difficulties in mappingeach input and output in the training set and increase trainingtime unnecessarily, in the process of determination of optimalnumber of hidden layers and nodes in each hidden layer. Theprocess was carried out using MATLAB routines.

IV. PERFORMANCE EVALUATION CRITERIA

The performance evaluation criteria used in the presentstudy are the coefficient of determination (R2), root meansquare error (RMSE) and efficiency coefficient (EC).

TABLE I.BRIEF DESCRIPTION OF METEOROLOGICAL CENTERS

A. Coefficient of Determination (R2)It is the square of the correlation coefficient (R) and the

correlation coefficient is expressed as

where yi and iy are the observed and estimated values

respectively and, y and iy are the means of observed andestimated values and n is the number of observations. Itmeasures the degree of association between the observedand estimated values and indicates the relative assessmentof the model performance in dimensionless measure.

ACEE Int. J. on Civil and Environmental Engineering, Vol. 01, No. 01, Feb 2011

© 2011 ACEE 33DOI: 01.IJCEE.01.01.517

B. Root Mean Square Error (RMSE)It yields the residual error in terms of the mean square error

and is expressed as [14].

RMSE= n

yy ii

n

i

2

1)ˆ(

(3)

C. Efficiency Coefficient (EC)It is used to assess the performance of different models [15].It is a better choice than RMSE statistic when the calibrationand verification periods have different lengths [16]. Itmeasures directly the ability of the model to reproduce theobserved values and is expressed as

EC = 10010

FF

(4)

where F0 = 2

1

)( yy i

n

i

and F = 2

1

)ˆ( ii

n

iyy

A value of EC of 90% generally indicates a very satisfactorymodel performance while a value in the range 80-90%, a fairlygood model. Values of EC in the range 60-80% would indicatean unsatisfactory model fit.

V. RESULTS AND DISCUSSION

The multiple correlation analysis has been carried out toidentify the climatic parameters influencing weekly ET0 in theregions selected for the study. It may be observed frommultiple and partial correlation coefficients presented inTABLE II that the sunshine hours, temperature, wind velocityand relative humidity mostly influence the regions in theweekly ET0 estimation. Linear weekly ET0 regression modelsat the centers have been developed as presented in TABLEIII. The weekly ET0 estimation at the meteorological centershas also been carried out using different artificial neuralnetwork architectures with input nodes ranging from one tofour and, varying the number of nodes in the hidden layer.The ANNs with four input nodes and a hidden layer withfour nodes i.e. ANN (4,4,1) have been identified as optimalarchitectures. The performance indices of linear regression(LR) and ANN (4,4,1) models on comparison of the resultswith those of FAO-56 Penman-Montieth method are presentedin TABLE IV. It may be observed from the results presentedin TABLE IV that the values of R2 and EC of LR models indicatea very satisfactory performance. However, the performancehas improved marginally with optimal artificial neural networkarchitectures.

TABLE IIMULTIPLE AND PARTIAL CORRELATION COEFFICIENTS

TABLE IIILINEAR REGRESSION MODELS

ACEE Int. J. on Civil and Environmental Engineering, Vol. 01, No. 01, Feb 2011

© 2011 ACEE 34DOI: 01.IJCEE.01.01.517

TABLE IV PERFORMANCE INDICES OF LR AND ANN (4,4.1) MODELS

The values of RMSE of ANN(4,4,1) models have also reducedslightly. This may be due to the fact that weekly average ET0values do not exhibit much of nonlinearity. The scatter plots(not shown in the paper) of ET0 values estimated usingPenman-Montieth method against those estimated using LRand ANN (4,4,1) models respectively. The nearly unit slopeand zero intercept of scatter plots (TABLE IV) indicate thecloseness of ET0 values with those of PM method. “Fig. 1”presents the comparison of performance of LR and ANN(4,4,1) models against PM method during testing period. Thestudy reveals that the simple linear regression modelsproposed may be adopted satisfactorily in the weekly ET0estimation at the centers selected for the present study and,the accuracy in the ET0 estimation may further be improvedusing ANN (4,4,1) models.

VI. CONCLUSIONS

The climatic parameters such as sunshine hours,temperature, wind velocity and relative humidity mostlyinfluenced weekly ET0 estimation at Tirupati, Nellore,Rajahmundry, Anakapalli and Rajendranagar regions ofAndhra Pradesh. The linear regression models proposed interms of the climatic parameters influencing the regionsperformed satisfactorily in the weekly ET0 estimation. Theoptimal ANN models proposed showed a marginalimprovement over linear regression models. The linearregression models may therefore be adopted for weekly ET0estimation in the regions with reasonable degree of accuracyand, the accuracy may slightly be improved with ANNarchitectures.

Figure 1. Comparison of average weekly ETO values estimated using LR and ANN models with those estimated byPenman-Monteith method during testing period

ACEE Int. J. on Civil and Environmental Engineering, Vol. 01, No. 01, Feb 2011

© 2011 ACEE 35DOI: 01.IJCEE.01.01.517

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