ORIGINAL PAPER

Modeling daily evapotranspiration in hyper-arid environmentusing gene expression programming

A. A. Alazba1,2 & M. A. Yassin2 & M. A. Mattar1

Received: 5 September 2015 /Accepted: 10 December 2015 /Published online: 10 March 2016# Saudi Society for Geosciences 2016

Abstract Accurate estimations of reference evapotranspira-tion (ETref) are extremely important for maximizing the ben-eficial use of water and hydrologic applications, particularly inarid and semiarid regions where water sources are so limited.The aim of this study is to develop mathematical models tocalculate the daily ETref using a gene expression programming(GEP) technique. Eight GEP models (GEP-MOD1–8) weredeveloped from combinations of climatic variables. ThePenman-Monteith equation was considered the referencemethod, with the reference plant height varying from 5 to105 cm in 5-cm increments. Daily climatic variables collectedfrom 13meteorological stations, one station from every regionwithin the Kingdom of Saudi Arabia, covered the 1980 to2010 period. Of the available climatic data, 65 % was usedin the training process for the eight developed GEP models,and 35 % was used in the testing process. The accuracy of theeight developed GEP models to estimate ETref varied in sig-nificance depending on the climatic variables that were in-cluded. As more climatic parameters were input, the accuracyof the GEP model increased. For the testing process, the co-efficient of determination (R2) ranged from a low of 63.4 %for GEP-MOD1 to a high of 95.4 % for GEP-MOD8, and theroot mean square error (RMSE) values ranged from3.19 mm day−1 for GEP-MOD1 to 1.14 mm day−1 for GEP-MOD8. From the spatial evaluation, the values of RMSEranged from 3.27 mm day−1 for GEP-MOD1 to1.21 mm day−1 for GEP-MOD8. In addition, the respective

RMSE values resulting from GEP-MOD8 for plant heights of50 and 12 cm were 0.75 and 0.96 cm. This implies that thedeveloped GEP-MOD8 can be used for any value of the ref-erence plant height ranging from 5 to 105 cm with insignifi-cant errors. Interestingly, solar radiation had an almost insig-nificant effect on ETref in the hyper-arid conditions. In con-trast, wind speed and plant height had a large positive effect onincreasing the accuracy of calculating ETref.

Keywords Evapotranspiration . Penman-Monteithmodel .

Gene expression programming . Arid environment

Introduction

Accurate quantification of evapotranspiration (ET) is neces-sary for different applications because it is generally the larg-est component of water and energy balance (Suleiman andHoogenboom 2009). Direct or indirect measurement methodsincorporating physical and empirical models can be used toestimate ET. The most accurate direct method is measurementwith a lysimeter (Gavilan et al. 2007). However, this proce-dure is difficult, time-consuming, expensive, and unavailablein many locations (Kisi 2007; Wang et al. 2009). EstimatingET through indirect methods involves calculating the refer-ence evapotranspiration (ETref) and then applying a suitablecoefficient. ETref is considered a major component of the hy-drologic process. Therefore, accurate estimation of ETref iscrucial for studies on hydrologic water balance, irrigation sys-tem design and management, plant production, water re-sources planning and management, and environmental assess-ment (Irmak et al. 2003; Temesgen et al. 2005; Chattopadhyayet al. 2009; Kumar et al. 2011). Many factors affect ETref,including daily temperature, relative humidity, wind velocity,hours of sunshine, atmospheric pressure, soil heat flux, and

* A. A. Alazbaalazba@ksu.edu.sa; alazba@hotmail.com

1 Agricultural Engineering Department, King Saud University,Riyadh, Saudi Arabia

2 Alamoudi Water Chair, King Saud University, Riyadh, Saudi Arabia

Arab J Geosci (2016) 9: 202DOI 10.1007/s12517-015-2273-x

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latitude. All of these factors cause estimation of ETref to be oneof the most complex calculations in the meteorology and hy-drology sciences because its estimation depends on the inter-action of several climatic elements (Kumar et al. 2002).Several methods exist to compute ETref using climatologicaldata that may be gathered from empirical relationships basedon temperature, radiation, or mass transfer, or from combina-tion methods based on physical processes. A typical combi-nation method is the physically based, complex Penman(1948) equation. The combination approach links evaporationdynamics with the flux of net radiation and the aerodynamictransport characteristics of a natural surface. Monteith (1965)introduced a surface conductance term to account for the re-sponse of leaf stomata to their hydrological environment. Thismodified form of the Penman equation, widely known as thePenman-Monteith (PM) evapotranspiration model (Monteith1973), is used by the Food and Agriculture Organization ofthe United Nations PM-FAO56 (Allen et al. 1998) and by theAmerican Society of Civil Engineers PM-ASCE70 (Jensen etal. 1990; Walter et al. 2001; ASCE-EWRI 2005) as the onlyaccurate method for calculating ETref and validating otherequations. This equation incorporates thermodynamic andaerodynamic aspects, can be applied in a wide range of cli-matic contexts (Smith et al. 1991; Yin et al. 2008), and re-quires many climatic data inputs including solar radiation,wind speed, air temperature, and humidity. Its disadvantageis that all of the necessary daily climatic variables may not beavailable for a given location. This is especially true in devel-oping countries where reliable weather data sets of radiation,relative humidity, and wind speed are limited (Trajkovic 2010;Tabari and Hosseinzadeh Talaee 2011). Furthermore, the in-stallation and maintenance of weather station equipment canbe expensive and complicated (Sentelhas et al. 2010). Manyresearchers acknowledge that the Penman-Monteith model isthe most promising standardized method to estimate ETref;however, it requires a significant amount of climatic data,which may be unavailable or of low reliability in certain loca-tions, especially within developing countries. In these cases,alternative methods that rely on fewer weather inputs are nec-essary (e.g., George et al. 2002; Xu and Singh 2002;Fooladmand et al. 2008; Sabziparvar and Tabari 2010;Tabari 2010; Trajkovic and Kolakovic 2010).

Over the past decade, intelligent computational models suchas gene expression programming (GEP) have been applied asan alternative method for estimating ETref. This technique, anextension of the genetic programming (GP) of Koza (1992),was developed by Ferreira (2001b). GEP, which is a naturaldevelopment of genetic algorithms, is defined as an algorithmused to implement symbolic regression in an attempt to find amathematical function that fits a data set (Sakthivel et al. 2012).The method is used to solve problems including symbolic re-gression, multi-agent strategies, time series prediction, circuitdesign, and evolutionary neural networks (Samadianfard

2012). GEP has been applied in fields as diverse as artificialintelligence; artificial life; engineering and science; financialmarkets; industrial, chemical, and biological processes; andmechanical models. The capability of GEP to solvehydrological and hydraulic problem modeling has beenexamined by a number of researchers. Guven and Aytek(2009) used the GEP approach to model stage-dischargerelationships and compared the obtained results withconventional methods. They found that the explicit algebraicformulations developed by GEP produced the best results.Similarly, Azamathulla et al. (2011) developed mathematicalmodels for the estimation of stage-discharge relationships forthe Pahang River based on the GP and GEP techniques. Morerecently, Azamathulla and Ahmad (2012) used GEP to predictthe transverse mixing coefficient in open-channel flows.Furthermore, Fernando et al. (2012) introduced an innovativemethod for combining estimated outputs from a number ofrainfall-runoff models using GEP to perform symbolic regres-sion. The results showed that the GEP combination methodproduced a superior river flow forecast. Azamathulla (2013)used GEP to drive a new model to predict the friction factorfor southern Italian rivers.

Only a few of the various published studies on the applica-tion of GEP in hydrological modeling have examined the ap-plicability of GEP for modeling of the evapotranspiration pro-cess. Shiri et al. (2012) introduced a new GEP model to esti-mate daily ETref at four weather stations in northern Spain overa 5-year period (1999–2003). The GEP results were comparedto the Adaptive Neuro-Fuzzy Inference System (ANFIS) andPriestley-Taylor and Hargreaves models using the PM-FAO56equation as the reference. The comparisons showed that theGEP performed better than these three models. Moreover,Traore andGuven (2012) evaluated the ability of GEP tomodelthe ETref of the tropical seasonally dry regions of West Africausing routing meteorological data from Burkina Faso. Theirresults revealed the GEP model to be a fairly promising ap-proach with the advantages of successfully providing simplealgebraic formulas and ease of use without recourse to the fullset of meteorological data required for accurately estimatingETref in sub-Saharan Africa regions. However, other GP tech-niques have been applied in the modeling of the evapotranspi-ration process. Parasuraman et al. (2007) evaluated the perfor-mance of the GP model with artificial neural network (ANN)models and the traditional PM (hourly PM-FAO56) method forestimation of ETref, finding that both of the data-driven models,GP and ANN, performed better than the PM method, and theperformance of the GP and ANN models was comparable.Moreover, Guven et al. (2007) presented GP as a new toolfor the estimation of ETref by using daily climatic variablesobtained from the CIMIS database. Compared to seven con-ventional ETref models, the new equation produced quite satis-factory results and can be used as an alternative to theconventional models. Kisi and Guven (2010) investigated the

202 Page 2 of 16 Arab J Geosci (2016) 9: 202

accuracy of linear genetic programming (LGP), an extension ofthe GP technique, in daily ETref modeling using the PM-FAO56 equation. Comparison of the accuracy of the LGPmod-el with those of support vector regression (SVR), ANN, andfour empirical models showed that the LGP model performsbetter than the other models in estimating total ETref.

As indicated above, the previously developed GEP modelsfor estimating ETref were based on the PM-FAO56 equation.The objective of this study, therefore, was to investigate theaccuracy of developed GEP models to predict daily ETref usingthe ETref standard values calculated by the more generalizedPM equation in which plant height varies. Thirty-one years ofmeteorological data (1980–2010) collected from 19 stations inSaudi Arabia as a representative hyper-arid region were used.In addition, the contribution ratio of each climate parameterused in the calculation of daily ETref was determined.

Materials and methods

Study area and climatic data

The study was carried out in the Kingdom of Saudi Arabia(KSA), a country situated in the far southwest corner of Asia(Fig. 1). KSA is located between latitudes 16° 22′ 46″ N and32° 14′ 00″N and longitudes 34° 29′ 30″ E and 55° 40′ 00″ E.KSA is divided into 13 provinces (Fig. 1). The study coveredthe 13 provinces, arranged in descending order by area inTable 1. The climate of KSA varies from region to regiondepending on the terrain. Generally, the climate is character-ized by hot summers, cold winters, and winter rainfall. Thecentral areas undergo hot and dry summers and cool and drywinters, and the humidity in the coastal areas is high.

For this study, climatic data were collected from 19 mete-orological stations from the 13 provinces of KSA. Nearly halfof the provinces were presented by two stations includingeastern region, Al-Riyadh, Al-Madinah, Makkah, Tabuk,and northern borders (Fig. 1). The climate data provided bythe Presidency of Meteorology and Environment covered31 years of daily meteorological information recorded forthe period from 1980 to 2010. Recorded data for all stationsincluded maximum, minimum, and mean air temperature (Tx,Tn, and Ta); maximum, minimum, and mean relative humidity(Rhx, Rhn, and Rha); wind speed measured at a 2-m height(U2); and solar radiation (Rs). Descriptions of the differentmeteorological stations and the average daily climatic datafrom each station over the study period appear in Table 1.The eight climatic variables, viz., Tx, Tn, Ta, Rhx, Rhn, Rha,U2, and Rs, in addition to the reference plant height (hc), whichvaried from 5 to 105 cm in increments of 5 cm, were themaximum input set variables represented by the GEP models.In the development of the GEP models, 65 % of the climaticdata from the 13 stations (one station from each province) was

used for the training process and 35% was used for the testingprocess. In contrast, the climatic data from the other six sta-tions were used for a spatial assessment of the eight developedGEP models, which was made with hc varying from 5 to105 cm and constant at 50 and 12 cm. The combinations ofclimatic data represented in Table 2 formed the eight GEPmodels for estimation of ETref. The GEP models were giventhe code GEP-MODk, where k represents the model number.As an example, GEP-MOD1 represents the GEP model thatonly includes temperature, whereas GEP-MOD8 representsthe GEP model that includes all climatic variables. Of note,two main variables including air temperature and plant heightwere used as inputs in all GEP models application (Table 2).

Output/targeted data of GEP models

Because ETref is the output variable of the GEP models, a ref-erence value of ETref is needed during the development stage, inaddition to being a value against which the GEPmodels must becompared during assessment. The PMmethod is considered thestandard when no measured lysimeter data is available (Irmaket al. 2003; Gavilan et al. 2006). This method gives optimalresults over all climatic zones (De Souza and Yoder 1994;Chiew et al. 1995; Hupet and Vanclooster 2001; Naoum andTsanis 2003; Irmak et al. 2003; Alazba 2004; Gavilan et al.2006) and has two advantages over many other mathematicalequations. First, it can be used globally with no local calibrationrequired due to its physical basis. Second, it is a well-documented equation validated with a significant amount oflysimeter data (Trajkovic 2010). Many researchers (Kumaret al. 2002; Trajkovic and Kolakovic 2010; Kisi and Ozturk2007; Zanetti et al. 2007) have used the PM equation as a ref-erence and standard equation to evaluate the results of theirmathematical models. Therefore, for this study, the daily ETrefvalues obtainedwith the PM equation were used as output/targetvariables in the GEP models. A generalized form of the PMequation can be written as (Alazba 2004)

ETre f ¼ λ−1 ΔΔþ γ* Rn−Gð Þ þγ

Δþ γ* K es−eað Þ� �

ð1Þ

where

ETref = reference evapotranspiration (mm day−1)

λ= latent heat of vaporization (MJ kg−1)Δ= slope of the saturation vapor pressure-temperaturecurve at mean air temperature (kPa °C−1)γ=psychometric constant (kPa °C−1)Rn =net radiation (MJ m

−2 day−1)G= soil heat flux (MJ m−2 day−1)γ*=modified psychometric constant (kPa °C−1)

K = p a r a m e t e r e q u a l t o 1:854 � 105 λ=raTþ273(MJ m−2 day−1°kPa−1)

Arab J Geosci (2016) 9: 202 Page 3 of 16 202

ra = aerodynamic resistance (s m−1), a function of U2 and

hcT=air temperature (°C)es = saturation vapor pressure at air temperature (kPa)ea = actual vapor pressure (kPa)

Description of GEP

GEP is a new evolutionary artificial intelligence techniquedeveloped by Ferreira (2001a). There are two main factorsin GEP (Ferreira 2006): chromosomes, which are usuallycomposed of more than one gene of equal length, andexpression trees or programs, which are the expressionsof the genetic information encoded in chromosomes. The

chromosomes are composed of multiple genes, each geneencoding a smaller subprogram. In GEP, linear structures,or the chromosomes, represent the genotype, and branchedstructures, or expression trees, represent the phenotype(Ferreira 2001b). GEP computer programs are all encodedin linear chromosomes, which are later translated orexpressed in expression trees. These programs are usuallydeveloped to solve a particular problem and are selectedaccording to their ability to solve that problem (Guvenand Aytek 2009).

Development of the GEP model

Initially, the training set is selected from the whole data,and the testing set uses the remaining data. The modeling

Collected daily climatic data for years 1980 to 2010

Stations for Spatial Validation

Stations for Training (65%) and Testing (35%)

Fig. 1 Map of Saudi Arabia showing provinces and meteorological sites

202 Page 4 of 16 Arab J Geosci (2016) 9: 202

Tab

le1

Meteorologicalstatio

nsitesandaveragedaily

values

oftheclim

aticparameters

Provinces

Areas

a(km

2)

Statio

nsLocationcoordinates

Clim

aticparameters

Longitude

(°)

Latitu

de(°)

Altitude

(m)

Tx(°C)

T n(°C)

T a(°C)

Rh x

(%)

Rh n

(%)

Rh a

(%)

U2(m

s−1)

Rs(M

Jm

−2

day−

1)

Eastern

region

540

Qaisumah

46.13

28.31

355

3219

2577

3050

2.6

21

Dhahran

50.20

26.30

1733

2026

7529

524.2

20

Al-Riyadh

380

Riyadh(north)

46.72

24.93

614

3320

2638

1631

3.9

15

WadiA

l-Daw

asir

45.20

20.50

617

3522

2835

1726

3.4

18

Al-Madinah

150

Al-Madina

39.60

24.47

619

3325

1956

2944

4.2

26

Yanba’

38.10

24.10

129

2217

7823

503.2

29

Makkah

137

Jeddah

39.17

21.40

1234

2822

8137

602.6

23

Al-Ta’if

40.50

21.50

1449

3529

2360

2939

3.2

27

Tabuk

136

Tabuk

36.58

28.38

770

2914

2253

1732

2.9

33

Al-Wajh

36.50

26.20

2028

1018

7022

452.2

29

Najran

130

Najran

44.40

17.60

1214

3529

2560

3344

3.5

28

Ha’il

120

Ha’il

41.70

27.40

1013

3428

2281

3760

2.3

14

Northernborders

104

Turaif

38.65

31.68

854

3529

2360

2939

3.3

29

Rafha

43.50

29.60

447

2914

2253

1732

2.9

22

Al-Jouf

85Al-Jouf

40.10

29.80

689

3014

2248

1831

3.11

25

Asir

80Bisha

42.60

20.00

1157

3317

2547

1529

2.4

28

Al-Qasim

73Al-Qasim

43.80

26.30

650

3218

2544

3018

2.9

27

Jizan

13Jizan

42.60

16.88

336

3025

6134

443.3

36

Al-Bahah

12Al-Baha

41.60

20.30

1656

2916

2256

2238

1.3

28

aSaudiG

eologicalS

urvey(2012),K

ingdom

ofSaudiA

rabia:FactsandNum

bers,editio

n1

Arab J Geosci (2016) 9: 202 Page 5 of 16 202

comprises five major steps in the use of GEP, the detailsof which can be found in Ferreira (2001a, b). The mainpurpose in developing the GEP models is to generate themathematical functions for the prediction of ETref. A setof preparatory model runs were carried out to test theperformance of the models using four possible functionsets. One function set was selected for continued use.All of these procedures were performed on GEP-MOD1,which contained the least number of variables, using theroot mean square error (RMSE) fitness function and theaddition linking function. The results of the investigationof function selection presented in Table 3 show that theoperator function set F4 outperformed all of the otherstructures. The superiority of the F4 function setconfirms the results of Shiri et al. (2012) and Kisi andGuven (2010), who concluded that the GeneXpro defaultoperator function set (F4) performed better than otherapplied function sets for estimating the daily ETref at fourweather stations in the Basque Country, northern Spain,and for estimating the daily suspended sediment load attwo stations on the Cumberland River, USA. Despitethese results, the F3 function set was used in the GEPmodels to develop less complicated mathematical equa-tions and avoid the use of trigonometric functions (sin(x),cos(x), and arctan(x)). There were only slight differencesbetween the F3 and F4 function sets. The parametersused in training the eight models are given in Table 4.The program was run until there was no longer a

significant improvement in the performance of themodels. The maximum number of generations and bestfitness generated during training are listed in Table 5. Inthis study, the general formulation of the GEP modelstakes the following form:

ETre f GEP‐MODkð Þ ¼XNj¼1

gene j ð2Þ

where N is the number of genes.

Performance criteria of the GEP models

Several statistical parameters can be used for compar-ison between referenced and estimated ETref values.The following statistical parameters were used in thisstudy:

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX ni¼1 Ei Cið Þ

2

n

sð3Þ

MAE ¼X n

i¼1 Ei Cij jn

ð4Þ

R2 ¼X n

i¼1 Ei−E� �

Ci−C� �� �2

X ni¼1 Ei−E

� �2:X n

i¼1 Ci−C� �2 ð5Þ

Table 2 The input combinations for forming GEP models

Models Input parameters

Air temperature (°C) Relative humidity (%) Wind speed (m s−1) Solar Radiation(MJ m−2 day−1)

Crop Height (cm)

Tx Tn Ta Rhx Rhn Rha U2 Rs hc

GEP-MOD1 ✓ ✓ ✓ ✓

GEP-MOD2 ✓ ✓ ✓ ✓ ✓ ✓ ✓

GEP-MOD3 ✓ ✓ ✓ ✓ ✓

GEP-MOD4 ✓ ✓ ✓ ✓ ✓

GEP-MOD5 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

GEP-MOD6 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

GEP-MOD7 ✓ ✓ ✓ ✓ ✓ ✓

GEP-MOD8 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Table 3 Preliminary selection ofbasic functions for the expressiontree using the scatter index

Functions RMSE (mm day−1) R2 (%)

F1 +, −, ×, ÷ 3.45 61.3F2 +, −, ×, ÷, sqrt, exp 3.23 63.2F3 +, −, ×, ÷, sqrt, exp, Ln, X2 3.10 64.4F4 +, −, ×, ÷, −, ×, ÷, sqrt, 3Rt, exp, Ln, X2, X3, sin(x), cos(x), arctan(x) 2.94 65.7

Italic values are for the least, moderate and most accurate GEP-MODs

202 Page 6 of 16 Arab J Geosci (2016) 9: 202

where

Ei=value of ETref calculated by the PM modelCi= corresponding ETref value estimated by the GEPmodelsn=number of observationsĒ= average of the calculated valuesC = average of the estimated values

Results and discussion

This section discusses the eight developed GEP models withadditional details oriented to three specific GEPmodels, GEP-

MOD1, GEP-MOD8, and GEP-MOD5. The first modelconsisted of only one climatic parameter, temperature, where-as the second model contained all climatic parameters. Thelatter model produced results in close agreement with GEP-MOD8, even though the Rs parameter was not included. Thespatial assessment of the eight developed GEP models is alsodiscussed. The results of the importance analysis of each ofthe input parameters are presented.

Developed GEP models

Figures 2 and 3, which depict the eight developed GEPmodels for the training and testing stages, respectively, clearlyillustrate the good agreement between the training and testingGEP models. The statistical parameters shown in Table 5

Table 4 Parameters of the GEPmodels Parameter Description of parameter Parameter setting

P1 Number of chromosomes 30

P2 Number of genes 3 (GEP-MOD1 to GEP-MOD7),4 (GEP-MOD8)

P3 Head size 8 (GEP-MOD1 to GEP-MOD7),10 (GEP-MOD8)

P4 Function set +, −, ×, ÷, sqrt, exp, Ln, X2

P5 Linking function Addition

P6 Fitness function Mean squared error

P7 Mutation rate 0.00206

P8 Inversion rate 0.00546

P9 One-point recombination rate 0.00277

P10 Two-point recombination rate 0.00277

P11 Gene recombination rate 0.00277

P12 Gene transposition rate 0.00277

Table 5 Statistical performance of the developed GEP models for the training and testing processes

Model Input combination Training Testing

RMSE(mm day−1)

MAE(mm day−1)

R2 (%) RMSE(mm day−1)

MAE(mm day−1)

R2 (%)

GEP-MOD1 (209,208, 782) Tx, Tn, Ta, hc 3.10 2.29 64.4 3.19 2.40 63.6

GEP-MOD2 (260,672, 794) Tx, Tn, Ta, Rhx,Rhn, Rha, hc

2.85 2.07 72.2 2.94 2.17 71.0

GEP-MOD3 (207,489, 791) Tx, Tn, Ta, U2, hc 2.64 1.98 76.1 2.65 1.98 76.8

GEP-MOD4 (170,031, 805) Tx, Tn, Ta, Rs, hc 2.92 2.11 68.3 2.99 2.21 67.9

GEP-MOD5 (202,825, 831) Tx, Tn, Ta, Rhx,Rhn, Rha, U2, hc

1.98 1.46 89.7 2.09 1.53 89.3

GEP-MOD6 (104,139, 814) Tx, Tn, Ta, Rhx, Rhn,Rha, Rs, hc

2.48 1.73 77.5 2.52 1.81 77.6

GEP-MOD7 (199,602, 845) Tx, Tn, Ta, U2, Rs, hc 2.20 1.64 82.2 2.21 1.63 82.6

GEP-MOD8 (124,547, 915) Tx, Tn, Ta, Rhx, Rhn,Rha, U2, Rs, hc

1.12 0.83 95.5 1.14 0.83 95.4

The numbers between parentheses are the maximum number of generations and the best fitness, respectively

Italic values are for the least, moderate and most accurate GEP-MODs

Arab J Geosci (2016) 9: 202 Page 7 of 16 202

Fig. 2 Estimated daily ETref by GEP versus corresponding calculated daily ETref by the generalized PM model, training process

202 Page 8 of 16 Arab J Geosci (2016) 9: 202

Fig. 3 Estimated daily ETref by GEP versus corresponding calculated daily ETref by the generalized PM model, testing process

Arab J Geosci (2016) 9: 202 Page 9 of 16 202

indicate that the values of R2 for each GEP model during thetraining process are nearly identical to those of the testingprocess. The R2 magnitudes in the training process range from64.4 % for GEP-MOD1 to 95.5 % for GEP-MOD8, whereasthey equal 63.3 and 95.4 %, respectively, for the training pro-cess. The same can be stated for the values of the RMSE andmean absolute error (MAE) parameters. The values of RMSErange from as low as 1.12 and 1.14 mm day−1 to as high as3.10 and 3.19 mm day−1 for the training and testing processes,respectively (Table 5). Also from Table 5, the MAE valuesrange from 2.29 to 0.83 mm day−1 and from 2.40 to0.83 mm day−1 for the respective training and testing process-es. The RMSE and MAE values for the training and testingprocesses clearly show that the GEP models can successfullyestimate ETref with close agreement that depends on the cli-matic parameters included in the GEP model.

As indicated in Fig. 2 and Table 5, the least accurate GEPmodel for the prediction of ETref is GEP-MOD1. This is notsurprising as temperature is the only climatic parameter includ-ed in the model. In contrast, the most accurate GEP model forETref estimation is GEP-MOD8, implying that when all climat-ic parameters are included, GEP can be used with precision.The inclusion of Rs is not essential to predict ETref using GEPmodeling, as shown with GEP-MOD5. The one-to-one curvedepicted in Fig. 2 and the values of R2, RMSE, andMAE listedin Table 5 indicate that GEP-MOD5 is in good agreement withGEP-MOD8. This result may lead to the interesting conclusionthat ETref can be estimated highly accurately in hyper-arid con-ditions even when Rs is unavailable.

Because the ultimate outcome is to predict ETref with theGEP models and for the previously stated justification, thenext section presents the mathematical expressions of threeGEP models, GEP-MOD1, GEP-MOD5, and GEP-MOD8.

Expression trees and mathematical equations of threeGEP models

As shown in Fig. 4, GEP-MOD1 and GEP-MOD5 each con-tain three genes and GEP-MOD8 contains four genes, andeach gene has constants that differ in number and magnitude.These coefficients and their magnitudes are shown in Table 6.From Fig. 4, the algebraic formulations for GEP-MOD1,GEP-MOD5, and GEP-MOD8 can easily be attained.

The algebraic gene expressions for GEP-MOD1 are

gene1 ¼TxC2

−Tn� �

− Tn þ T að Þ

C3−C1ð Þ þ T xhcð6Þ

gene2 ¼ hc−T xð Þ−Tx½ �C1hc

þ C3−hcð Þ þ T aC2 ð7Þ

gene3 ¼ C3−T xð Þ−C1½ �− T a þ hcð ÞC2−hcð Þ þ T a−C1ð Þ ð8Þ

Those for GEP-MOD5 are

gene1 ¼ U2−Rhað Þ � T x½ �= Rhx þ C2ð ÞC1U 2

� �� Rha

ð9Þ

gene2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT x−

U 2 � C1ð Þ � hc � T að ÞRhn

sð10Þ

gene3 ¼�hc � U2

� �−C2

�−C1U 2

� �þ 2Tn

Rhxð11Þ

and those for GEP-MOD8 are

gene1 ¼ C1 þ hc � T að Þ½ �− U 2 þ Rsð ÞC2−2U2

ð12Þ

gene2 ¼ T a � C1ð Þ þ C3C4 þ Rhnð Þ � C2−Rhxð Þ þ U 2 ð13Þ

gene3 ¼ C1− U2C22= Rhx−Txð Þ−Rs½ �

ð14Þ

gene4 ¼ T a � hcð Þ � TnRha−U2ð Þ−C1½ � þ T a−U2ð Þ ð15Þ

Recalling Eq. 2, the mathematical equations for ETrefestimates for these three GEP models can be expressedas follows.

For GEP-MOD1,

ETre f ¼TxC2

−Tn� �

− Tn þ T að Þ

C3−C1ð Þ þ Txhcþ hc−Txð Þ−Tx½ �

C1hc

þ C3−hcð Þ þ T aC2 þC3−Txð Þ−C1½ �− T a þ hcð ÞC2−hcð Þ þ T a−C1ð Þ ð16Þ

and Eq. 16 can be written by replacing the values of the con-stants listed in Table 6 as follows:

ETre f ¼Tx

−5:24−Tn

� �− Tn þ T að Þ

5:13−5:58ð Þ þ T xhc

þ hc−Txð Þ−Tx½ �−6:13hc

þ 0:0687−hcð Þ þ T a3:57

þ 8:98−Txð Þ−2:17½ �− T a þ hcð Þ−0:69−hcð Þ þ T a þ 2:17ð Þ ð17Þ

202 Page 10 of 16 Arab J Geosci (2016) 9: 202

GEP-MOD5

GEP-MOD1

GEP-MOD8

Fig. 4 Expression tree for three developed GEP models for daily ETref estimates

Arab J Geosci (2016) 9: 202 Page 11 of 16 202

For GEP-MOD5,

ETre f ¼ U2−Rhað Þ � Tx½ �= Rhx þ C2ð ÞC1U 2

� �� Rha

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT x−

U 2 � C1ð Þ � hc � T að ÞRhn

s

þ�hc � U2

� �−C2

�−C1U 2

� �þ 2T n

Rhxð18Þ

and by replacing the values of the constants in Table 6, Eq. 18takes the following form:

ETre f ¼ U2−Rhað Þ � Tx½ �= Rhx � 0:78ð Þ−4:70U2

� �� Rha

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT x−

U 2 � −8:46ð Þ � hc � T að ÞRhn

s

þ�hc � U2

� �−1:63

�−0:37

U2

� �þ 2Tn

Rhxð19Þ

For GEP-MOD8,

ETre f ¼ C1 þ hc � T að Þ½ �− U 2 þ Rsð ÞC2−2U 2

þ T a � C1ð Þ þ C3C4 þ Rhnð Þ � C2−Rhxð Þ þ U 2

þ C1− U 2C22= Rhx−Txð Þ−Rs½ �

þ T a � hcð Þ � TnRha−U2ð Þ−C1½ � þ T a−U2ð Þ ð20Þ

and by replacing the values of the constants given in Table 6,Eq. 20 can be rewritten as:

ETre f ¼ 10:40þ hc � T að Þ½ �− U2 þ Rsð Þ−6:35−2U 2

þ T a � 5:30ð Þ−1:607:40þ Rhnð Þ � 1:80−Rhxð Þ þ U2

þ 1:80− U 28:662

.Rhx−T xð Þ−Rs½ �

þ T a � hcð Þ � TnRha−U2ð Þ þ 14½ � þ T a−U 2ð Þ ð21Þ

Equation 21, which includes all climatic parameters in addi-tion to hc, is the best function obtained using GEP to estimateETref.

Spatial assessment of the GEP models

As previously mentioned, the GEP models were spa-tially evaluated using climatic data collected from sixmeteorological stations other than the 13 meteorologi-cal stations used for the development of the GEPmodels. The values of the statistical parametersRMSE, MAE, and R2 are presented in Table 7, whichshows the values for hc5–105, hc50, and hc12. In otherwords, the eight developed GEP models were evaluat-ed for spatial assessment with hc varying from 5 to105 cm in 5-cm increments and with hc equal to thetwo constant values (hc = 12 cm and hc = 50 cm). Theconstant hc values were selected considering the twowell-known PM formulations of PM-ASCE70 and PM-FAO56. The first considers hc equal to 50 cm, and thelatter uses hc equal to 12 cm. With respect to theRMSE, MAE, and R2 statistics shown in Table 7, theeight GEP models can perfectly predict ETref at differ-ent values of hc regardless of the accuracy of eachGEP model. Nevertheless, the best GEP model isGEP-MOD8 because it gives accurate ETref estimates(Table 7) at different values of hc. This can be seenmore clearly by referring to Fig. 4, which depicts therelationship of ETref calculated by Eq. 1, the general-ized PM formula, with ETref predicted by the GEP-MOD8 model.

Equation 21 indicates that the variable hc exists inthe first and fourth gene expressions. Accordingly,Eq. 21 can be expressed for the two well-knownPM-type formulas of PM-ASCE70 and PM-FAO56by simply replacing hc with 50 and 12 cm, respec-tively, in both genes. Along with arithmetic elimina-tion and algebraic manipulation, Eq. 21 can be

Table 6 Constants for GEP-MOD5 and GEP-MOD8 formulations

Constant Gene1 Gene2 Gene3

GEP-MOD1

C1 5.58 −6.13 −2.17C2 −5.24 3.57 −0.69C3 5.13 0.0687 8.98

GEP-MOD5

C1 −4.70 −8.46 0.37C2 −0.78 – 1.63C3 – – –

GEP-MOD8

C1 10.40 5.30 1.80 −14.00C2 −6.35 1.80 8.66 –C3 – −1.6 – –C4 – 7.40 – –

202 Page 12 of 16 Arab J Geosci (2016) 9: 202

written for alfalfa-based and grass-based regions asfollows:

ETre f ¼ 1:8þ U 2 þ 0:013U 2 Tx þ Rs−Rhx½ �

þ 5:3T a−1:67:40þ Rhnð Þ � 1:80−Rhxð Þ −

0:5TnT a2U 2−T a−Rha−14

−10:4þ 0:5T a−U 2−Rs

6:35þ 2U2 ð22Þ

Tab

le7

Statisticalperformance

ofGEPmodelsfortheassessmentp

rocess

insixweather

stations

with

intheperiod

from

1980

to2010

Model

Inputcom

binatio

nh c

5–105

h c50a

h c12b

RMSE

(mm

day−

1)

MAE

(mm

day−

1)

R2(%

)RMSE

(mm

day−

1)

MAE

(mm

day−

1)

R2(%

)RMSE

(mm

day−

1)

MAE

(mm

day−

1)

R2(%

)

GEP-M

OD1

T x,T

n,T a,h

c3.27

2.35

64.0

2.95

2.21

58.9

1.73

1.29

66.1

GEP-MOD2

T x,T

n,T

a,Rh x,

Rh n,R

h a,h

c

3.02

2.04

73.6

2.55

1.81

71.5

1.75

1.27

69.3

GEP-MOD3

T x,T

n,T

a,U2,h c

3.04

2.31

71.3

2.70

2.18

72.4

1.45

1.12

78.3

GEP-MOD4

T x,T

n,T

a,Rs,h c

3.17

2.26

65.4

2.86

2.16

60.8

1.62

1.16

70.8

GEP-M

OD5

Tx,T n,T

a,R

Hx,

RHn,R

Ha,U

2,h

c

2.15

1.62

87.3

1.81

1.42

88.3

1.28

1.02

85.4

GEP-MOD6

T x,T

n,T

a,Rh x,

Rh n,R

h a,R

s,h c

2.54

1.74

81.0

2.30

1.62

78.4

1.46

1.05

82.0

GEP-MOD7

T x,T

n,T

a,U2,Rs,h c

2.63

1.97

78.4

2.25

1.73

78.9

1.77

1.26

76.5

GEP-M

OD8

Tx,T n,T

a,R

Hx,RHn,

RHa,U

2,R

s,h c

1.21

0.89

95.6

0.75

0.55

97.6

0.96

0.76

91.9

aReference

values

forETrefwerecalculated

usingPM-A

SCE70

bReference

values

forETrefwerecalculated

usingPM-FAO56

Italicvalues

arefortheleast,moderateandmostaccurateGEP-M

ODs

Fig. 5 Spatially assessed ETref at different hc considerations for GEP-MOD8

Arab J Geosci (2016) 9: 202 Page 13 of 16 202

Fig. 6 Importance ratio analysis of input variables, climatics, and hc, for GEP models

202 Page 14 of 16 Arab J Geosci (2016) 9: 202

for alfalfa-based and

ETre f ¼ 1:8þ U 2 þ 0:013U 2 Tx þ Rs−Rhx½ �

þ 5:3T a−1:67:40þ Rhnð Þ � 1:80−Rhxð Þ −

0:12TnT a2U 2−T a−Rha−14

−10:4þ 0:12T a−U 2−Rs

6:35þ 2U2 ð23Þ

for grass-based regions.The one-to-one curves in Fig. 5 show good agreement of

the ETref values obtained by GEP-MOD8 and by the general-ized PM method for the three cases of hc, 5 to 105, 50, and12 cm.

Importance ratio analysis of input variables

Importance ratio analysis was also performed usingGeneXproTools software to examine the climatic parametersand the hc variable with the most influence on the eight GEPmodels. Figure 6 shows that hc has an almost identical effecton ETref for all GEP models at a rate of approximately 15–30 %. The temperature effect on ETref is higher when the Rhparameters and/or U2 are included, either as in the case ofGEP-MOD2, GEP-MOD3, GEP-MOD6, and GEP-MOD7or when both are included, as with GEP-MOD1 and GEP-MOD2. In general, Fig. 6 indicates that the parameters of Rhand Rs have the least effect on ETref. In contrast, the U2 pa-rameter has a significant effect on ETref, particularly with theabsence of the Rs parameter in GEP-MOD5. A final conclu-sion on the effects of the input parameters on ETref may not bereached without keeping some parameters constant whilevarying others and/or without the use of special statisticaltools such as cross-product or suitably and appropriately sim-ilar tools. Therefore, an ultimate judgment on the effect ofinput parameters on ETref will require further investigation,which is beyond the scope of this work.

Conclusions

Eight GEP models were developed to estimate the daily ETrefunder hyper-arid environmental conditions. A combination ofclimatic parameters formed the eight GEP models. The inputvariables to the developed GEP models included maximum,minimum, and average air temperature; maximum, minimum,and average relative humidity; and wind speed, solar radia-tion, and plant height. The climatic data were collected from19 meteorological stations in KSA covering the 1980 to 2010period. Data obtained from the 13 meteorological stations

were used to develop the GEPmodels, and data from the othersix stations were used for spatial validation of the eight devel-oped GEP models. In comparison with the PM model, thebest-performing GEP model, GEP-MOD8, included all cli-matic parameters, whereas the worst-performing GEP modelincluded temperature as the only climatic input parameter.Compared to the PM model, GEP-MOD8 with its algebraicequation can perfectly predict ETref for a wide range of hcvarying from 5 to 105 cm. Interestingly, GEP-MOD5, theGEP model without Rs, competed favorably with GEP-MOD8, the most accurate model, to predict the daily ETrefin a hyper-arid environment. This interesting outcome leadsto a possible conclusion that the climatic parameterU2 is moreimportant than Rs within the circumstances of this study. Theimportance analysis of the input parameters was inconclusive,indicating that the ultimate judgment of the effects of variousinput variables on ETref requires further investigation and in-depth analysis.

Acknowledgments The project was financially supported by King Sa-ud University, Vice Deanship of Research Chairs. The climatic data usedin this study were obtained from the Presidency of Meteorology andEnvironment (PME), Kingdom of Saudi Arabia. The authors highly ap-preciate this effort and would like thank the PME for its continued col-laboration in providing the data.

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202 Page 16 of 16 Arab J Geosci (2016) 9: 202

Modeling daily evapotranspiration in hyper-arid environment using gene expression programmingAbstractIntroductionMaterials and methodsStudy area and climatic dataOutput/targeted data of GEP modelsDescription of GEPDevelopment of the GEP modelPerformance criteria of the GEP models

Results and discussionDeveloped GEP modelsExpression trees and mathematical equations of three GEP modelsSpatial assessment of the GEP modelsImportance ratio analysis of input variables

ConclusionsReferences