Modeling Pan Evaporation for Kuwait using Multiple Linear Regression and Time-Series Techniques

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1 America Joural of Applied Scieces Origial Research Paper Modelig Pa Evaporatio for Kuwait usig Multiple Liear Regressio ad Time-Series Techiques Jaber Almedeij Departmet of Civil Egieerig, Kuwait Uiversity, P.O. Box 5969, Safat 3060, Kuwait Article history Received: Revised: Accepted: Abstract: This study attempts to model evaporatio for Kuwait uder arid coditios by usig a wide rage of mothly evaporatio data, varyig from 0. to 40 mm/day, from Jauary 993 to July 05. Owig to the reaso that the well-kow theoretical evaporatio models preseted i the literature have bee justified for a much shorter data rage, the paper adopts empirical approaches to fit the data. Two evaporatio models are preseted based o classical statistical methods, oe of multiple liear regressio ad aother of time series aalysis. The regressio model, which is a fuctio of temperature, relative humidity ad wid speed, allows differet modificatios i the idepedet variables for more atural evaporatio data sythesis. The time series model, which is a fuctio of time oly, is coveiet for producig forecasts. Both evaporatio models have bee show to produce results that are i reasoable agreemet with observatio values. This study advocates that the specific, rather simple, classical procedures performed to model the evaporatio data ca be effective alteratives to other theoretical ad semi-theoretical methods foud i the literature. Keywords: Meteorological Parameter, Multiple Liear Regressio, Pa Evaporatio, Siusoidal Patter, Arid ad Semi-Arid Zoes Itroductio Estimatio of the water loss by evaporatio is importat for modelig, survey ad maagemet of water resource projects over differet time ad space scales (e.g., Molia Martiez et al., 006; Shirsath ad Sigh, 00). Evaporatio depeds o the supply of heat eergy ad vapor pressure gradiet, which i tur deped o meteorological factors such as temperature, relative humidity, wid speed ad atmospheric pressure (Xu ad Sigh, 998). I practice, potetial evaporatio data may be measured by usig pa evaporimeters ad ca be used for modelig ad aalysis purposes due to their simplicity, low cost ad availability of log historical data series. The data are useful for geeral moitorig ad assessmet of climate coditios, eve if actual evaporatio is overestimated because of differeces i water quality, thermal iertia, advectio ad edge effect (Shaw et al., 00; McMaho et al., 03). Theoretical evaporatio models have bee used by may researchers (e.g., Coulomb et al., 00; Gavi ad Agew, 004; Chag et al., 00; Ershadi et al., 05). The classical Pema (948) equatio may be recommeded as a stadard method. However, may theoretical models require iformatio that might ot be accessible from weather statios, such as the heat storage withi the water body, similar to that foud i Pema equatio, which requires temperature profile measuremets withi the water (Brutsaert, 005). Other models might be applicable to a limited data rage or to coditios that are similar to those withi which the models were derived. For such models, a calibratio process would be ecessary to reder them applicable to the regioal data. Accordigly, simple empirical models have bee developed for regioal evaporatios (Cahoo et al., 99; Crago ad Brutsaert, 99; Xu ad Sigh, 998; Rotstay et al., 006; Shirsath ad Sigh, 00; Tabari et al., 00; Abou El-Magd ad Ali, 0; Kisi, 05; Malik ad Kumar, 05). For example, Cahoo et al. (99) ad Feessey ad Vogel (996) employed regressio methods to develop models for mothly average evaporatio i the USA. Tabari et al. (00) estimated evaporatio i a semi-arid regio of Ira usig both techiques of artificial eural etwork ad of multivariate o-liear regressio. Kim et al. (05) evaluated combied bootstrap resamplig ad eural etwork models for daily evaporatio i the Republic of Korea. Kisi (05) employed a least square support vector machie, multivariate adaptive regressio splies 06 Jaber Almedeij. This ope access article is distributed uder a Creative Commos Attributio (CC-BY) 3.0 licese.

2 Jaber Almedeij / America Joural of Applied Scieces 06, 3 (6): DOI: /ajassp ad M5 Model Tree for evaporatio data from Mersi ad Atalya statios i Turkey. Other similar attempts were made successfully i the arid regio of Saudi Arabia (Yassi et al., 06). Amog the plethora statistical fittig techiques available, regressio ad time series methods are quite commo ad have bee implemeted i may statistical software packages. Regressio is a powerful techique for tred behavior estimatio. The mai advatage of a evaporatio model produced by regressio is to use explicit meteorological parameters as idepedet variables. This approach is useful to apply modificatios for the adopted meteorological variables that yield more atural evaporatio data sythesis. Time series aalysis has the capability of simulatig repeated data variatio patters. A model developed by this techique ca be used to obtai forecasts, because it is a fuctio of time oly. The aim here is to ivestigate the ability of regressio ad time series techiques i order to model a wide rage of mothly evaporatio data obtaied from a weather statio located i Kuwait i a arid eviromet. Iitially, the meteorological data for the case study will be preseted. The the periodic variatio patter of the meteorological data will be examied i the frequecy domai. The evaporatio models will the be developed ad evaluated. Meteorological Data Kuwait, which is about8,000 km, is a desert coutry characterized by log, hot ad dry summers ad short witers. The average depth of aual evaporatio is high, approachig 4000 mm, while that of precipitatio is low, varyig from 50 to 50 mm. Temperatures durig summer, which may vary from average daily temperatures of 43 to 3 C, are cosidered hot, but they ca be worse whe hot wids blow from the desert. Owig to the coastal locatio of the coutry, the heat is ofte redered eve more ucomfortable by high humidity, approachig a daily maximum of 97.5%. Witer temperatures, which vary from average daily temperatures of 5 to 5 C, may be classified as mild, but occasioally become cold whe ortherly or orth-westerly wids brig cold air. The meteorological data used i this study are mothly average measuremets of pa evaporatio (mm/day), temperature at m height ( C), relative humidity (%) ad wid speed at m height (m/s). The effect of precipitatio o evaporatio rates will ot be cosidered i this study due to the rare raifall evets i the coutry. The climatological data adopted are readily available from the Meteorological Departmet of the Directorate of Civil Aviatio, collected at a local weather statio ear Kuwait Airport (Fig. ). These data are of substatial cotiuity coverage, withi a period of ~3 years betwee Jauary 99 ad July 05 ad are cosidered represetative of the climate withi the urba zoe. The reaso for such a meteorological poit estimate to be cosidered represetative is that the urba zoe of the coutry spas a small area, withi latitudes 9 0`N to 9 03`N ad logitudes 47 37`E to 48 0`E ad is characterized by early flat surface elevatios. Table presets statistical descriptio for the daily average measuremets of the climatological parameters. It ca be see that withi the specified time duratio, the measuremets for each parameter are highly variable reflectig the typical arid climate of the coutry. Fig.. Locatio of the weather statio i Kuwait 740

3 Jaber Almedeij / America Joural of Applied Scieces 06, 3 (6): DOI: /ajassp Table. Descriptive statistics of daily average meteorological data for Kuwait from 99 to 05 Evaporatio (mm/day) Temperature ( C) Relative humidity (%) Wid speed (m/s) Mea Stadard deviatio Sample variace Kurtosis Skewess Rage Miimum Maximum Variatio Patter The mothly variatio patters for the meteorological data of evaporatio, temperature, relative humidity ad wid speed are plotted i Fig.. The time series patters ca be cosidered determiistic of a periodic ature with o obvious tred. A apparet correlatio ca be observed such that the variatio of evaporatio is directly proportioal to temperature ad wid speed, but iversely proportioal to relative humidity. The correlatios ca better be described by cosiderig statistics o a seasoal basis. The seasoal mea for the four datasets is obtaied by usig the expressio: j = = N N jν, τ, τ,..., () ν = Fig.. Historical wid speed for the data of Kuwait Where: j = Seasoal time series j = Seasoal mea N = Total umber of years The results are show i Fig. 3. The estimated correlatio coefficiets for the seasoally-averaged evaporatio with temperature, relative humidity ad wid speed are r T = 0.97, r H = ad r u = 0.89, respectively. The high positive correlatio with temperature is a characteristic of desert eviromets. The high egative correlatio with relative humidity reflects the coastal locatio of the coutry alog the Arabia Gulf. Relative humidity refers to the amout of water i the air as a fractio of the total amout saturated air ca hold. 74

4 Jaber Almedeij / America Joural of Applied Scieces 06, 3 (6): DOI: /ajassp Fig. 3. Seasoal mea variatio for mothly meteorological data of Kuwait Fig. 4. Periodograms for mothly meteorological data of Kuwait The more humidity i the air, the less space will be available for evaporatio. Oce the air reaches a upper relative humidity limit of 00%, it is o loger able to hold additioal water molecules. The positive correlatio with wid speed is also sufficietly high. Wid speed affects the rate of evaporatio by sweepig away water particles that are i the air, allowig more particles to evaporate i the space above the water surface. The periodic behavior of the meteorological data ca be ivestigated usig the periodogram techique, which is a Fourier trasform of the autocovariace fuctio represetig a usmoothed spectral plot for examiig the cyclic structure i the frequecy domai (Box et al., 05). This techique is used to reduce the effect of the measuremet oise ad, thus, detect which frequecies withi the rage of time are most resposible for the data 74

5 Jaber Almedeij / America Joural of Applied Scieces 06, 3 (6): DOI: /ajassp patter. Typically, a large peak value show i a periodogram correspods to a period that is strogly represeted i the time series. The periodograms for the meteorological data are plotted i Fig. 4. A domiat aual periodicity of moths is preset i all meteorological data implyig that 6 moths of the year possess cosiderably lower evaporatio, temperature, relative humidity ad wid speed tha the other 6 moths. Aother, but less sigificat, periodicity of 6 moth ca be foud i the data of evaporatio, temperature ad wid speed. This period ca be related to the typical four seasos durig the year. Though other periodicities are preset i all periodograms, they ca be cosidered isigificat. Model Developmet Two models are developed here for the evaporatio data of Kuwait, oe based o regressio techique ad the other o time series aalysis. Regressio Model Regressio techique ca be used to model pa evaporatio data of Kuwait i terms of temperature, relative humidity ad wid speed. Those parameters are importat to cosider because they have direct ifluece o the evaporatio process. Temperature is correlated to solar radiatio, which is the mai source of heat eergy required for vaporizatio. The ability to trasport the vapor away from the evaporative surface is correlated to relative humidity ad wid speed. For multiple liear regressio, the depedet variable y is assumed to be a fuctio of k idepedet variables x, x, x 3,,x k. The model is expressed i the form: y = b + b x b x + e () i 0, i k k, i i where, b 0, b,,b k are fittig costats; y i, x,i,,x k,i represet the ith observatios of the variables y, x,.,x k, respectively; ad e i is a radom error term represetig the remaiig effects o y of variables ot explicitly icluded i the model. For simple regressio models, e i ca be assumed to be a ucorrelated variable with zero mea. The most commo procedure for estimatig b 0, b,,b k is to employ the least squares criterio with the miimum sum of squares of error terms (S); that is, to fid b 0, b,,b k to miimize: j As a result, b 0, b,,b k must satisfy: S ei = ei = 0, j = 0,,..., k (4) b b i = j Ad sice e i = y i observed - y i calculated, the Equatio 4 becomes: S yicalculated = ei = 0, j = 0,,..., k (5) b b j j The meteorological data of Kuwait ca be examied for the suitability of fittig this type of regressio model. The rage of meteorological data from Jauary 99 to December 009 ca be used to perform model fittig. The remaiig data util July 05 ca be used later for verificatio. Figure 5 presets the relatio betwee evaporatio ad the other climatological parameters of temperature, relative humidity ad wid speed. A geeral multiple regressio model expressig the evaporatio i terms of those climatological parameters ca iitially be assumed as: E = bt + b H + b u (6) i i i 3 i Where: E = Pa evaporatio (mm/day) T = Temperature ( C) H = Relative humidity (%) u = Wid speed (m/s) Based o the classical assumptios of multiple regressio modelig, Equatio 6 suggests liear correlatios betwee the evaporatio ad the idepedet variables. However, Fig. 5 shows a defiite curviliear appearace for the relatio betwee evaporatio ad both temperature ad relative humidity. It is see that those relatios are best expressed correspodigly as power ad expoetial fuctios ad the above multiple regressio equatio ca thus be liearized by trasformig the idepedet variables of temperature ad relative humidity as: E = bt * + b H * + b u (7) i i i 3 i Where: * T.69 = T (8) ( iobserved 0, i... k k, i ) i = ( yiobserved yicalculated ) S = y b b x b x = = ei. (3) * H H = e (9) The geeral regressio model with the fitted parameters b, b ad b 3 becomes: H i i i i E = T + e + u (0) 743

6 Jaber Almedeij / America Joural of Applied Scieces 06, 3 (6): DOI: /ajassp Fig. 5. Evaporatio as a fuctio of temperature ad relative humidity for the data of Kuwait from 99 to 009. The solid curves represet suggested relatios fitted by regressio It is worth metioig that Equatio 0 eglects the ifluece of other meteorological parameters o evaporatio rates, as it cosiders a itercept equal to zero, i.e., b 0 = 0. Regardig the effect of wid speed, although Fig. 5 suggests that a model with a o-zero itercept ca accout for a liear correlatio with evaporatio of the form: E = b0 + bu () Regressio might ot result with a sigificat fit for the coefficiets because of the cosiderable data scatter. However, it is possible to assume that the itercept of this correlatio is equal to zero: E = b u () Although this assumptio results with a fittig accuracy less sigificat for the average data preseted o a mothly basis, Fig. 5 shows that the o-zero itercept tred costitutes a possible relatio fitted by eye. Time Series Model A possible applicatio for examiig the periodic behavior of the meteorological data is to employ the detected periodicities i order to provide model forecasts. I geeral, a time series cotaiig a periodic siusoidal compoet with a kow wavelegth ca be modeled usig: k s( t) = R cos f t + i ( π i θi ) (3) Where: s = Periodic siusoidal compoet R i = Amplitude of variatio f i = Frequecy, equal to the iverse of period θ i = Phase agle k = Total umber of periodicities The term (πf i t + θ i ) is measured i radias. That is, the k value for the temperature data is equal to two ad 744

7 Jaber Almedeij / America Joural of Applied Scieces 06, 3 (6): DOI: /ajassp the values of f ad f are set as /6 ad / cycles per moth, respectively. The phase agle θ is ecessary to adjust the model so that the cosie fuctio crosses the mea at the appropriate time, t. The values of θ i ad R i ca be determied by meas of umerical optimizatio. The fitted periodic siusoidal model for the evaporatio data is foud to be: π E( t) =.33.cos t π + 9.7cos t +.7 (4) Model Evaluatio Figure 6 presets the two evaporatio models of regressio (Equatio 0) ad of time series (Equatio 4). For both models, there is a reasoable agreemet betwee observed ad calculated values for the data withi the time from Jauary 99 to December 009, used to perform model calibratio. The remaiig data from Jauary 00 to July 05 were used for model verificatio. The figure shows that both models may be cosidered successful i represetig most of the seasoal variatio patter for the data from Jauary 99 to July 05. Fig. 6. Evaporatio as a fuctio of time i moths for the two models. Calibratio is performed for the duratio from Jauary 99 (correspodig to moth umber ) to December 009 (moth umber 6); verificatio is performed for the duratio from Jauary 00 (moth 7) to July 05 (moth 83); ad forecast is performed for the duratio from August 05 (moth 84) to December 0 (moth 360) 745

8 Jaber Almedeij / America Joural of Applied Scieces 06, 3 (6): DOI: /ajassp Table. Fittig accuracy for the model Accuracy measure Model Techique Rage MAPE RMSE (mm/day) NSE Equatio (0) Regressio Calibratio Verificatio Equatio (4) Time series Calibratio Verificatio The performace of the two models ca be evaluated quatitatively. The followig statistical error tests ca be adopted, which are the Mea Absolute Percetage Error (MAPE), Root Mea Square Error (RMSE) ad Nash-Sutcliffe Equatio (NSE). These idicators are calculated as follows: yicalculated yiobserved MAPE = 00 (5) y iobserved i = icalculated iobserved ( ) (6) RMSE = y y NSE = ( y y ) iobserved icalculated ( yiobserved yi observed ) (7) Respectively, where y is the mea value of the depedet variable. For better data modelig, MAPE ad RMSE statistics should be closer to zero. The NSE criterio compares the model performace to the use of the mea value of the depedet variable as a estimate. Give a perfect fit, the NSE criterio is equal to.0; if the model is worse tha the mea value of the depedet variable, the NSE statistic will be egative. Table presets the accuracy measures, which show that the performaces of the two models are early idetical to each other. Give that the coditios used to derive the model remai the same, Fig. 6 presets the forecasts provided by the time series model for the times pa from August 05 to December 0. Based o the previous verificatio rage, the mea forecastig error is assumed to be early withi accuracy measures of MAPE, RMSE ad NSE equal to 3.9,.76 ad 0.77, respectively. The produced forecasts fall withi the same rage of mothly average historical data ad exhibit a apparet seasoal variatio patter. Coclusio The two derived evaporatio models, oe based o the techique of multiple liear regressio ad the other o time series aalysis, were show to be successful i describig most of the variatio patter for the data obtaied from Kuwait. For the former model, the trasformatio of both temperature ad relative humidity variables, which was cosidered by usig correspodig mathematical fuctios of power ad expoetial forms, improved the correlatio results. Although the employed mathematical fuctios are applicable locally for the meteorological data of Kuwait, it is hoped that this will prompt others to examie whether such correlatios exist uiversally i data collected from other locatios. The time series model, which exploited two typical periods of seasoal ad aual cycles, produced mothly forecasts for the times pa from August 05 to December 0 that ca be of great importace for water resources maagemet applicatios i Kuwait. Ackowledgmet The author is grateful to the Metrological Departmet of the Directorate Geeral of Civil Aviatio of Kuwait, Climatological Divisio, for providig the relevat data of pa evaporatio, temperature, relative humidity ad wid speed. Ethics The author declares that there is o coflict of iterests regardig the publicatio of this paper. Refereces Abou El-Magd, I.H. ad E.M. Ali, 0. Estimatio of the evaporative losses from lake Nasser, Egypt usig optical satellite imagery. It. J. Digital Earth, 5: Box, G.E., G.M. Jekis ad G.C. Reisel, 05. Time Series Aalysis: Forecastig ad Cotrol. 5th Ed., Joh Wiley ad Sos, Hoboke, ISBN-0: , pp: 7. Brutsaert, W., 005. Hydrology: A Itroductio. st Ed., Cambridge Uiversity Press, Cambridge, ISBN-0: , pp: 605. Cahoo, J.E., T.A. Costello ad J.A. Ferguso, 99. Estimatig pa evaporatio usig limited meteorological observatios. Agric. Forest Meteorol., 55: DOI: 0.06/068-93(9)9006-T Chag, F.J., L.C. Chag, H.S. Kao ad G.R. Wu, 00. Assessig the effort of meteorological variables for evaporatio estimatio by self-orgaizig map eural etwork. J. Hydrol., 384: 8-9. DOI: 0.06/j.jhydrol

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