JJEES Jordan Journal of Earth and Environmental Sciences

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1 JJEES Jordan Journal of Earth and Environmental Science Volume 1, Number 1, Mar ISSN Page Developing Reference Crop Evapotranpiration Time Serie Simulation Model Uing Cla a Pan: A Cae Study for the Jordan Valley /Jordan Mohrik R. Hamdi a, *, Ahmed N. Bdour b, Zeyad S. Tarawneh b a Department of Land Management and Environment, The Hahemite Univerity, Zarqa 13115, Jordan b Department of Civil Engineering, The Hahemite Univerity, Zarqa Jordan Abtract The greatet environmental challenge that Jordan face today i the carcity of water reource. Evapotranpiration (ET) affect water reource and it i conidered an important proce in aridic hydrologic ytem. The etimated long-term average of ET in Jordan i over 90% of the total precipitation; neverthele, there have been no attempt to model reference crop evapotranpiration uing a time erie approach in Jordan. In thi tudy, a eaonal time erie Autoregreive and Moving Average (ARIMA) mathematical model i decribed. It i ued for forecating monthly reference crop evapotranpiration (ET o ) without uing weather data baed on pat hitorical record ( ) of meaured pan evaporation at Central Jordan Valley: an arid to emi-arid region. The developed ARIMA (1, 0, 0) (0, 1, 1) 12 model provide reaonable and acceptable forecat, comparing it performance with a computed reference evapotranpiration from meaured pan evaporation parameter. The forecating performance capability of three tentative ARIMA model wa aeed uing Root Mean Squared Forecating Error, Mean Abolute Forecating Percentage Error, and Maximum Abolute Forecating Percentage Error. The developed model allow local farmer and water reource manager to predict up to 60 month with a percentage error le than 11% of the mean abolute forecating. The potential to make uch prediction i crucial in optimizing the needed reource for effective management of water reource. Furthermore, the developed model offer a imple, accurate, and an eay hort and long-term forecating in the valley. Thi would develop a robut trategy for irrigation water management including ucceful planning, deigning, managing, and operating of water reource ytem. It alo would increae the poitive intended outcome project conducted by many governmental and nongovernmental organization. In addition, it would heighten the efficiency of local and national water reource policie Jordan Journal of Earth and Environmental Science. All right reerved Keyword: Seaonal reference evapotranpiration modeling; ARIMA model; pan evaporation; Jordan; 1. Introduction The Hahemite Kingdom of Jordan (HKJ) i located about 100 km from the outh-eatern coat of the Mediterranean between latitude 29º 11-33º 22 N and longitude 34º 59-39º 12 E. The neighboring countrie to Jordan are Iraq, Paletine, Syria, and Saudi Arabia (Figure 1). The total land area of HKJ i approximately km 2, of which, about 40% i irrigated land. The population of HKJ wa etimated to reach 5, 329, and 000 in 2004, of which 30% reide in rural area (DOS, 2006). The tatitic indicate that the population of HKJ i increaing rapidly, with an etimated annual growth rate of 2.6% (DOS, 2006). ETo : reference crop Evapotranpiration Deir-Alla Figure 1. Map of Jordan how Deir-Alla Weather Station * Correponding author. mohrik@hu.edu.jo

2 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) The irrigation water for agriculture ector repreent about 68% of the total water demand in HKJ. Due to natural and non-voluntary migration, demand on the dometic water upply increaed and accordingly, the per capita hare of frehwater decreaed. Although the population growth rate are declining (DOS, 2006), the increaing population continue to place enormou preure on deciion maker to find new water upplie and develop an updated water conervation policy. The main water upplie in HKJ are groundwater and urface water. However, HKJ hare mot of it urface water reource with neighboring countrie, which make it one of the world' hot pot in regard to water dipute. The groundwater reource repreent about 65% of total available water in HKJ. Becaue of the hortage of the urface water reource, groundwater ha been extenively ued in the lat twenty year; it i overdrafted and become uceptible to alt water intruion. Thi immene water conumption lead to a decline in water acceibility over the lat few decade and i conidered one of the mot important environmental problem facing deciion-maker in HKJ. The extraction of groundwater and contaminant problem comprie a major deterrent to utain a progreive economy in the region. It may lead to evere limitation in the agricultural and indutrial progre in Jordan Jordan Valley ha long been ued for agriculture and i one of the main irrigated agricultural area in the HKJ. The irrigated area i more than 34,000 ha, and the agricultural ector conume about 68% of Jordan' water reource. Thi percentage embodie the larget ue of water in Jordan. In agriculture, water requirement are linked to irrigation ue (Allen et al., 1998). Due to Jordan carcity of water, the efficient utilization of water reource including the ue of irrigation water i a major national concern. There are three type of irrigation method in HKJ. Thee method are urface irrigation, prinkler irrigation, and drip irrigation. Irrigation efficiency i a ignificant dimenion of the water preervation iue that need to be invetigated to reach a better water reource management in HKJ. For example, irrigation efficiencie of the drip irrigation ytem in the Central Jordan Valley varied from 34% to more than 90% (DAI report, 1995). Furthermore, irrigation water i wated through runoff, and evaporation; the latter loe about 67% of applied irrigation in the Jordan Valley. In pite of the ignificance of evaporation a a component in the irrigation efficiency, it wa not given a well-deerved attention. Mot often, applied irrigation in agricultural i baed on farmer experience. Evaporation can either meaured or etimated. Pan A Cla can be ued for meauring daily evaporation. Evaporation can be predicted by uing imulation model. Thee model range from imple to more ophiticated uch a ARIAM. The evaporation component hould be included in the development of any irrigation water management plan in Jordan Valley. Potential evapotranpiration (ET p ) i defined a the rate at which evapotrapration would occur from a large area completely and uniformly covered with growing vegetation which ha acce to an unlimited upply of oil water and without advection or heat torage effect (Dingman, 2002). Modeling and predicting the evapotranpiration rate i required for reliable and effective planning, deign, managing, and operation of water-reource project; managing water quality, determining afe yield from aquifer, planning for flood control and foret fire prevention, proper irrigation cheduling, economic of building water-upply reervoir, determining the water budget at field cale, aeing oil moiture, predicting climate change and ecoytem repone to climate change, etimation of water available to human ue and it management a well a playing an important role in other environmental iue and concern (Jenen, et al., 1990; Singh and Jaiwal, 2006; Fardou et al., 2001; Mazahrih et al., 2001). Irrigation cheduling for crop in Jordan Valley i quite empirical and could lead to a great lo of irrigation water and low irrigation efficiencie. Evapotranpiration i the mot important variable ubequently to rainfall in the context of irrigation to crop and it i a multivariate phenomenon a it i affected by many hydrological variable (Mohan and Arumugam, 1996). Evapotranpiration include evaporation from open-water, bare wet oil near the plant, tranpiration from within the leave of plant, the ue of water by the vegetation to build new plant tiue, evaporation from the moit membrane urface of the vegetation a well a ublimation from ice and now urface (Blaney and Hanon, 1965). The objective of thi tudy i to develop a time erie model to imulate the Reference Crop Evapotranpiriation for the Jordan Valley. Specifically, the objective are to ue hitorical information of ET o to calibrate the developed model, and generate a imulation projection for ET o for the next five year. If the calibrated model will, approximately duplicate meaured ET o, then, the local farmer communitie and other water authoritie will ue it to predict evapotranpiration, which will improve water reource management in the Jordan Valley. Thi can be done by mean of the Box-Jenkin parametric modeling to identify eaonality effect and conduct trend analyi and forecating a well a to develop a eaonal forecat for future event in the time erie. Furthermore, uing autoregreive integrated moving average (ARIMA) model (Box and Jenkin, 1976), thi tudy meant to develop the ARIMA model for forecating the monthly value of ET o and to analyze the predictability and the performance of thee model. In Jordan, there have been no attempt to model reference crop ET data uing a time erie technique. Thi i the firt tudy of it kind conducted in Jordan, thu the carcity of available data wa a challenge. The focal point of thi paper i retricted to improving the long-term prediction uing pat E pan data that wa converted to ET o. With thi objective, the Jordan Valley wa elected for the tudy a it contribute the larget proportion of irrigated area; irrigated area being 32.4% of the total cultivated area in Jordan (DOS, 2006) Climate and Topographic Feature of the Study Area Jordan i located at the eatern margin of the Mediterranean climatic zone with highly variable and irregular rainfall. Thi climate i characterized by hot, dry ummer and cool, wet winter. The etimated long-term

3 2008 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) 35 average of ETmight reache 93% of the total precipitation (Taha, 2006; MWI, 2004). During 2004, it wa etimated that 7334 MCM were returned to the atmophere by evaporation and tranpiration from the urface of Jordan (MWI, 2004). However, the etimated long term average potential ET in the Jordan Valley range between 1289 mm/yearn in the north at Baqoura and increae in the outh at Ghor Al-Safi to reach 1545 mm/y (DOS, 2006)(Figure 1). Jordan valley lie between m below ea level, extending from Lake Tiberia in the North to the Dead Sea, with a length of 104 km and a width of between 4-to16 km; it i urrounded in the eat and wet by high mountain. The Jordan Valley conit of the Northern Ghor ( ha), Middle Ghor (7875 ha) and the Southern Jordan Valley ( ha). Jordan Valley produce 80% of the national agricultural production and i conidered the mot important agricultural area, a there i a permanent ource of water from the Yarmuk River and ide dam for the Jordan River. In the Jordan Valley, rainfall decreae from approximately 300 mm in the north to 102 mm in the outh. The climate of the valley i characterized by very dry, hot ummer with average temperature of 31.5 o C and cool, wet winter with average temperature of 14 o C. The relative humidity range between 64% in the winter to 27% during the ummer. Due to it poition below ea level and high temperature (microclimate), Jordan Valley i conidered the mot important winter vegetable producing area (85%), with citru and banana production (10%), becaue of it tropical climate and the limitation of irrigation water. Forage crop (5%) are grown on a very limited cale (Abu-Zanat, 1995). Thee crop and vegetable are exported abroad to Europe and other countrie. Therefore, the valley i conidered a food baket for all riparian countrie due to it unique climate and agricultural environment. Mot cultivable land in Jordan Valley are irrigated where 73% of the total irrigation ector exit. The majority of holding are between (3-4 ha). Farmer ue modern agricultural technique in irrigation, production, and marketing. 2. Methodology and Aumption It i eential for a ucceful water reource management in HKJ to evaluate crop water requirement on monthly bae becaue it i included in any long-term water management operation of water upply and torage ytem. The crop water requirement i related to crop evapotranpiration, ET c, of the crop being grown. Therefore, it i realitic to provide one forecat of a reference crop evapotranpiration rate, ET o, for a region. Subequently the ET rate for each crop growing in the region can be forecated. A time erie i a et of meaurement of a variable taken over time at equally paced time interval. Additional valuable information could be offered during time erie analyi. Analyi of time erie involve analyi of the tatitical manner of a erie of data over time, giving that the record i complete, and continuou along with aumption of negligible variability of phyical condition over the period of analyi. If change of thee condition occur over a long time, one hould conider it prior to the analyi. Reference crop evapotranpiration i the rate of evapotranpiration from an extended urface of 8 to 15 cm tall, with green cover of uniform height, actively growing, completely hading the ground under, and no deficiency of water. Although the etimation of ET o can be done eaily, but the objective of the tudy i to forecat the ET o, i.e., to predict future value to be able to improve planning and managing of the water reource and to tet the predictability of the developed model. To forecat the ET o rate, one can ue either relationhip that rely on forecat of phyical weather parameter or one can conider a mathematical method that eek to predict future ET o rate baed on the pat hitory of the ET o rate in a certain region. In thi manucript, we will focu on the latter cae. Pruitt developed different method and Doorenbo (1977) for etimating ET o in a region uing relationhip relating phyical parameter incorporated in the ET proce. Of thee relation are the Blany-Criddle equation; the Penman equation, the Radiation equation; or the Pan Evaporation method. Pan provide a meaurement of the integrated effect of radiation, wind, temperature, and humidity on the evaporation from an open water urface. They concluded that mitaken forecat of the mean wind peed are the main ource of difference between the predicted and meaured reference crop ET. Moreover, the phyically baed relationhip for forecating evapotranpiration have ome limitation. However, in 1990, the International Commiion adopted the Penman- Monteith combination method a a tandard for Reference crop Evapotranpiration for Irrigation, Drainage, and World Meteorological Organization. A direct meaurement of Evapotranpiration i cotly and i not eay; therefore, uing hitorical information of ET o, time erie can be an alternative method for forecating ET o. Pan Evaporation (E p ) data wa recorded at Deir-Alla Weather Station and obtained from Applied Meteorological (Diviion Jordan Department of Meteorology, 2006). The Station i located in the Central Jordan Valley, at latitude of 32 o 13 N, 35 o 37 Eatlongitude with an elevation of 224 meter below the ea level. ET o wa predicted uing 30 year of pat record of weather value of pan evaporation from the Station. The value of ET o for thi period were produced by pan evaporation method (E p ). In pite of the difference between pan evaporation and the evapotranpiration of cropped urface, the ue of pan evaporation may be warranted to predict ETo for period of 10 day or longer (Allen et al., 1998; Abdo, F., Peronal communication. Senior Agronomit, Jordan Department of Meteorology, Amman, Jordan). Uing pan evaporation to predict ET o for period of 10 day i an international convention that ha been reliably put into practice due to the fact that a 10 day i a reaonable time to put any agricultural activity or problem that might occur into operation to be benefited or corrected by the farmer eaily and atifactorily. Pat ET o data i ued to get the long-term ET o etimate by modeling the time erie uing uitable ARIMA technique known a Box Jenkin model which i very popular type of time erie model ued in hydrological forecating. The longet available data et wa ued to

4 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) avoid mileading reult when uing ARIMA procedure (Box and Jenkin, 1976). The data et wa divided into two ection: the firt ection, compoed of twenty-five year ( ) (300 month) of data that wa ued for calibration to identify and develop a univariant eaonal ARIMA model, the econd, compoed of five year ( ) (60 month) wa ued to validate and tet the model performance and it predictability. Variou time erie model were developed and teted uing monthly averaged ET o for the tudy area for the period1973 to The pan ha proved it practical value and ha been ued uccefully to etimate reference evapotranpiration by oberving the evaporation lo from a water urface and applying the pan coefficient to relate pan evaporation to ET o (Allen et al., 1998). Mazahrih et. al (2001) how a good relationhip (R 2 =0.72) between evaporation from cla-a pan and the meaured crop evapotranpiration of pepper inide a platic houe. In thi manucript, the variable K p method to derive the 30-year reference evapotranpiration from E p data wa ued. The pan coefficient (K p ) wa calculated from cla-a pan located at Deir-Alla uing tabulated value (table 5 in FAO Irrigation and Drainage Paper 56) or table 7 (regreion equation) of the method decribed by Allen et al. (1998) in chapter 4. Then the ET o wa calculated uing the following relation (Allen et al., 1998) ET o = K p E p (1) Where ET o i reference crop evapotranpiration (gra) [mm/day], K p i pan coefficient [-], and E p i pan evaporation [mm/day] Model and ARIMA Development The mot common approache to forecating evapotranpiration i extrapolation or the prediction method which i baed on an inferred tudy of pat data behavior over time. In time erie analyi, the obervation taken at a contant interval of time are conidered random variable. Any particular oberved erie i upported to be the only realization of all poible erie that could be generated under the ame et of condition. ARIMA model in time erie analyi can atifactorily explain uch procee according to Box and Jenkin (1976). The Box Jenkin model authorize u not only to expoe the hidden pattern in the data but alo to generate forecat of the future baed excluively on hitorical value of the dependent variable. Moreover, the accuracy of forecat of time erie model are good, convenient to ue when eaonal or monthly pattern mut be taken into account, upple enough to be modified when trategy change occur, the leat data-intenive compared to many other model, and eaily developed by mean of variou tandard oftware package. In addition, eaonal ARIMA model allow for randomne in the eaonal pattern, unlike the claical method approach baed on linear regreion. However, they are inaccurate when coniderable change in determining variable occur in the future and can be uceptible to their tarting value, when carrying the greatet weight in the forecat. A general ARIMA model contain autoregreive (AR) and moving average (MA) part. The AR part decribe the relationhip between preent and pat obervation, wherea the MA part characterize the autocorrelation tructure of the error or diturbance erie. In thi paper, time erie analyze; reference crop evapotranpiration (ET o ) or {Y t } for forecating and modeling a a function of time. A time erie i uually repreented by {Y t } where the dependent erie {Y t } = Y 1, Y 2, Y 3.Y t. When t=1 the obervation i Y 1, and o on. Then the ARIMA model of Y t can be repreented by ARIMA (p,d,q), where p i the order of non-eaonal autoregreive operator; d i the order of the non-eaonal difference paing operator and typically have a value of 0 or 1, and eldom greater than that; q i the order of non-eaonal moving average operator applied in non-eaonal modeling proce. If we can expre the variable Y at time t a the um of reidual at previou time, then the moving average model can be written a: Yt = C + θq (B) εt (2) Where C i a contant term; θ q (B) = 1-θ 1 (B) - θ 2 (B 2 )-..- θ q (B q ) i the moving average operator of order q; B i backward hift operator (By t = y t-1 and B y t = y t-, y t i the current value of the time erie examined). q i the order of moving average operator which conidered the number of lagged period correlated to preent value of the time erie. The value of q can be determined from the characteritic of the erie of Autocorrelation Function (ACF). The ACF value hould be reduced gradually and diappeared after a time lag of q. If the variable Y at time t can be written a the um of the weighted variable of previou time, then the auto regreion model can be expreed a: φp (B)Yt = C + εt (3) Where φ p (B) = 1-φ 1 (B) - φ 2 (B 2 )-..- φ p (B p ) i the autoregreive operator of order p (the number of lag period where the error term i correlated to the time erie). Gradually reducing ACF value and diappearing of the PACF after a time lag of p. The combined model can be written a: φp(b)yt = C + θq(b)εt (4) For a eaonal time erie to be modeled, the relationhip at the eaonal lag mut be incorporated. The lag in thi tudy i 12 for the monthly data collected monthly. Thee relation can be repreented by including of AR and MA part at that lag. The ame procedure i applied to determine the tentative value of P, D, and Q, where P i the order of eaonal autoregreive operator; D i the order eaonal difference paing operator and typically have a value of 0 or 1, and eldom greater than that; Q i the eaonal moving average operator applied in eaonal modeling proce. In thi cae, the time erie regarded a non-tationary. Differencing technique (the integration component of ARIMA) are often ued to tranform a non-tationary time erie into a tationary time erie. Viual inpection of the plot of ET o time erie point out a periodic trend (Figure 2). Now conidering that y 1, y 2... y t are the differenced value of ET o of time erie data, uing the backhift operator B" hifting the ubcript of a time erie obervation backward in time by one period give: By t = y t-1 (5)

5 2008 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) 37 Then the eaonal operator will be: = l-b (6) Where i the periodicity or eaonality of the erie (12 in thi tudy), Uing the general tationary tranformation, the tranformed time erie explained a: Z t = D y t (7) Evapotranpiration (mm) Where (D) y t i the lag-12 difference operator Time (month) Figure 2. Time erie of reference crop evapotranpiration of the tudied area between 1973 and 2002 Therefore, the general full general multiplicative eaonal and non-eaonal ARIMA Box-Jenkin model uing backhift operator i (Pankratz, 1983): d D ϕ ( B ) ϕ ( B )(1 B) (1 B ) Z = C + θ ( B) θ ( B ) ε P P t q Q t t = 1, 2, n (8) Thi can be ummarized a ARIMA (p, d, q) (P, D, Q) 12, where p, d, q, P, D, and Q a defined earlier. It i worth noting that the polynomial Φ (B ) and Θ (B ) capture the eaonal behavior of the erie (P- and Q-order eaonal AR and MA operator, repectively) and C i contant with no pecific meaning. Beide, the tationary and invertibility (tationary condition for the MA part of the ARIMA model) condition demand that all the root of the characteritic equation Φ (B ) =0, φ (B) = 0, Θ (B ) =0, θ (B) =0, lie outide the unit circle in the complex plane (Box and Jenkin, 1976). The following operator can be ued to decribe Box-Jenkin model a non-eaonal and eaonal autoregreive operator of order P and moving average operator of order Q repectively. procedure i conducted. Thee tep are identification of a tentative model, etimation of model parameter by Maximum Likelihood (ML) Technique, diagnotic checking, and teting the adequacy of thi model and providing neceary modification of the model if the tentative model i poor. The final model i then ued for forecating purpoe at the fourth tage. Thi iterative procedure neceitate viual aement of intermediate tatitical reult, and the model i ultimately developed baed on the expert judgment of the author Preparing Data for Analyi and Model Identification The Model identification tep in the Box-Jenkin iterative modeling prove to be the mot complicated and hard tak, particularly if the time erie i eaonal or periodic. Seaonal time erie might be caued by the nature of the annual weather cycle a in the cae of the ET o time erie data of thi tudy (Figure 2). Since Box-Jenkin aume a tationary time erie, therefore tationary nature of the proce that generated the time-erie i one of the mot coniderable condition that have to be impoed on the development of an ARIMA model in order to improve the forecat. The firt part of thi tep conit of checking whether the variation in the time erie i untable with time. Tranformation mut be done on untable erie. If the data fluctuate with increaing variation, a pre-differencing treatment (e.g., logarithmic or quare root tranformation) i required to tabilize the variance of the time erie. Moreover, a trend exiting in time erie data require differencing. A time erie may be conidered tationary if the mean, variance, and it covariance of the time erie are contant through time (Box and Jenkin, 1976). Box- Jenkin time erie analyi wa applied to the monthly ET data et. The plot of autocorrelation function (ACF) for the overall data et i preented in Figure 3. The continuou line in the graph repreent the confidence limit. Value of the ACF within thee limit are not ignificantly different from zero. p φp ( B) = (1 φ1b φ2b... φ pb ) (9) 2 P φp( B ) = (1 φ1, B φ2, B... φp, B ) (10) 2 q θq( B) = (1 θ1b θ2b... θqb ) (11) 2 Q θq( B ) = (1 θ1, B θ2, B... θq, B ) (12) Where Ф and θ are unknown parameter, which repreent the autoregreive and moving average coefficient at different lag of t time. In thi tudy, ET o i conidered a eaonal time erie. The eaonal time erie have relationhip at a definite lag (12 for the data collected monthly). To develop a uitable model for forecating ET o, an iterative procedure involving a four-tep modeling Figure 3. Autocorrelation function of the monthly averaged reference crop evapotranpiration between By viual inpection of the ample ACF of the original ET o time erie plot, an evidence of periodicity or eaonality exit indicating trong eaonal erial dependencie and ome correlation at lag up to 12. In order to remove the erial dependency and to

6 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) model/forecat the ET o (Y), time erie analyi begin by tranforming Y to enure it being tationary. Examining the characteritic and tatitic of both ACF and PACF (PACF not hown to reduce the ize of the manucript) of the tranformed erie i the econd part of tep one. The purpoe of thi part i to determine if the erie need additional differencing to remove the trend or eaonality to make the erie tationary by eliminating eaonal and non-tationary behavior of the ET time erie. The ACF of a tationary time erie how a quick decay for moderate and large lag. A ditinctive feature of the data that ugget the convenience of differencing the original time erie i a lowly decaying poitive ACF. The plot above illutrate thi behavior, which clearly indicate eaonal periodicity in the tudied data. To eliminate thi periodicity trend, the time erie mut be differenced until a rapidly decaying ACF (Brockwell and Davi, 1996). The ACF plot of the overall time erie how that the data are eaonal due to the peak of the ACF at lag that are multiple of 12. The peak preented in the plot how a correlation in the data every 12 lag. Thi mean that the order of the eaonal differencing i one (D=1). Removing thi eaonal component of period 12 from the erie Y t, the tranformed erie Z t = Y t -Y t-12 = (1-B 12 ) Y t wa generated where B i the backward hift operator a indicated earlier (Box and Jenkin, 1976). In thi tudy, variou type of differenced erie for both noneaonal and eaonal pattern are plotted and examined until mot of erial dependencie have diappeared. Differencing create a new data erie, {Z t }, which become input for the Box-Jenkin analyi; the ARIMA (Figure 4 and 5). Then, the tranformed time erie i fitted with an ARIMA model where the current value of the time erie, {Y t }, i expreed a a linear combination of p earlier value and a weighted um of q earlier deviation (original value minu fitted value of previou data) plu a tochatic random proce or error, ε t that are independently and identically ditributed with a normal ditribution N(0, σ a 2 ). Figure 4. Sample autocorrelation function of tranformed reference crop evapotranpiration time erie (D=1). A verification of eaonal removal, i conducting by replotting the ACF to ee the elimination of the peak. After the elimination of eaonal and/or cyclical component, the reulting time erie {Y t } could be non-tationary. Thu, the time erie can be tranformed into a tationary one, {Z t }, differencing recurively D time until the ACF reduced ignificantly (Brockwell and Davi, 1996). The ample time erie ACF, Figure 3 already how decay for moderate lag, being no neceary more differencing i required. We now have a tationary and eaonal time erie, {Z t }, and the identification of the ARIMA (p, d, q)*(p, D, Q) 12 model order hould be undertaken. Figure 5. Sample partial autocorrelation function of tranformed reference crop evapotranpiration time erie (D=1) 3. Reult 3.1. Parameter Etimation of the Model Before we can pecify the parameter to be etimated, the formulation of ARIMA model ha to be identified baed on the tranformed ET o time erie data. Here we determine the value of the parameter (θ and φ) of the propoed model of both eaonal and non-eaonal time erie after determining the type of a tentative ARIMA model, i.e. p, d, q; P, D, and Q uing tatitic interpretation and plot of ACF and PACF. The parameter are etimated uing maximum likelihood approache. Thee method are frequently ued and would yield efficient parameter etimate compared to other approache. Furthermore, ML can tolerate miing data in the erie in pite of no miing value in our ET o data (Shumway and Stoffer, 2000). In thi manucript, the computation were done uing SPSS computer oftware (SPSS, 2005). In general, ample autocorrelation function and the ample partial autocorrelation function are ued for the detection of variou type of autocorrelation. Thi wa done by uing the behavior of both function at the noneaonal level to tentatively identify non-eaonal and eaonal model decribing the time erie value (Brockwell and Davi, 1996). Then, the parameter of the propoed tentatively identified model are etimated followed by etimating t value for diagnotic and adequacy checking of thoe model and, if required, to ugget an improved model. The judgment of model election entail not only knowledge but alo a good deal of experimentation with alternative model a well a the technical parameter of ARIMA (Bowerman and

7 2008 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) 39 O Connell, 1993). In the parameter etimation tep uing the iterative ML method, the optimization criterion i baed on the minimizing of reidual um of the quared between the oberved data and the etimated one. Beide, the etimation wa conducted uing a a minimal iteration value and a a minimal change between iteration for the um of the quared reidual. Many model were identified for detailed aement. Three candidate model were elected for diagnoi; ARIMA (2,0,0) (0,1,1), ARIMA (1,0,0) (2,1,1) and ARIMA (1,0,0) (01,1). limit indicated by the upper and lower horizontal line hown in Figure 6, 7 and 8 which reveal that the reidual are not correlated and all the ACF value can be neglected Diagnotic Checking of the Model Two objective hould be fulfilled in thi tep. The firt i to tet hypothei proce by checking the ignificant of the parameter for each propoed model. Checking the normality of the reidual ditribution i the econd tak. In addition to the viual inpection of the reidual, variou diagnotic (i.e., t-ratio, Q tat, Akaike Information Criteria AIC, etc) are ued to check the adequacy of the tentatively identified model and, if neceary, to propoe an enhanced model. Between competing model, the model that produce reidual with ACF that are not ignificantly different from zero at all lag and have maller tandard error will be elected. The model (1,0,0)(0,1,1) ha the mallet error magnitude, and the autocorrelation coefficient of the prediction error were not tatitically ignificant, i.e., all value of reidual correlation were cloe to 0 and inide the confidence limit to 95%; there wa no erial dependency between reidual. Moreover, correlation analyi of the reidual in the plot of the ACF and the goodne of fit uing chi quare tet i conducted. If the model doe not how a correlation in the reidual, then the reidual are white noie indicating the adequacy to repreent the time erie. ACF of Reidual of the three elected tentative model i hown in Figure 6, 7 and 8. The time erie of error aociated with the initial model forecat wa analyzed uing the ACF analyi tool. Thi procedure wa repeated until the error of the forecating model were reduced to white noie with no ignificant correlation. Figure 6, 7, and 8 indicate that no ignificant ample autocorrelation of the reidual erie are found baed on the value of ACF reidual. Baed on Box-Pierce chi-quare tatitic, all the value can be conidered negligible and reidual are not correlated. Therefore, the elected two candidate ARIMA model are adequate and the ET o time erie i the white noie. The reult for detailed evaluation and verification of thee three model are preented in Table 1. Baed on Box and Jenkin (1976), a good model hould require the mallet poible number of etimated parameter of an adequate repreentation of the pattern in the available data. The overall time a eaonal model that include one non-eaonal autoregreive and one eaonal moving average term of order1 decribe erie. Table 1 alo preent the reult of parameter value of the tentative model, Box-Pierce and, Box-Pierce chiquare and tandard error. The Box-Pierce i baed on the computation of ACF reidual. All ACF lie within the Figure 6. Autocorrelation function of reidual of ARIMA (1, 0, 0) (2, 1, 1)12 Figure 7. Autocorrelation function of reidual of ARIMA (2, 0, 0) (0, 1, 1)12 Figure 8. Autocorrelation function of reidual of ARIMA (1, 0, 0) (0, 1, 1)12 A high tandard error correpond to a higher uncertainty in parameter etimation, which querie the tability of the model. If the reult of the ratio between the parameter value to the tandard error i larger than two, then the model i adequate. The Akaike Criteria (AIC) and the reidual variance are additional helpful parameter in electing the bet model.

8 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) Table 1. Reult of Model Etimation and Verification Tentative ARIMA Model I ARIMA Model II ARIMA Model III Model (200)(011)12 (100)(211)12 (100)(011)12 Parameter Parameter Parameter Tet AR(1) AR(2) SMA(1) AR(1) SAR(1) SAR(2) SMA(1) AR(1) SMA(1) Parameter value Standard error t-ratio Q-value Although the model 1, 2, and 3 have no major difference in term of the parameter hown in table 1, model No 3 i preferable. Baed on the lower value of Q, lower number of parameter, t value and other verification reult, model 3 i recommended and i therefore more uitable than model 1 and 2 in forecating. Furthermore, model 2 doe not meet the t-tat condition, unlike model 1 and 3. Table 2 how imple tatitic for both the original time erie of the evapotranpiration and the predicted time erie uing the bet-diagnoed model. The value of Model III are cloet to thoe tatitic that we elected: mean variance and kew ne. the mean and maximum acro all model, we can get an indication of the uncertainty in the prediction. Table 2. Simple tatitic of the bet-diagnoed model Model Mean Variance Skewne Original ETo Time Serie Predicted ETo Time Serie, Model I (2,0,0)(0,1,1) Predicted ETo Time Serie, Model II (1,0,0)(2,1,1) Predicted ETo Time Serie, Model II (1,0,0)(0,1,1) 3.3. Model Forecating We fitted on the firt 300 data point, uing the Q-tat, t ratio, AIC criterion, and tandard error. The bet obtained model i for p = ١, d=0, q = 0, P = ٠, D=1, Q = 1. The final model can be written in the following form: ETot, = 0.35ETot, 1 + ETot, 12 (13) 0.35ET 0.914ε + ε ot, 13 t 12 t Thi model i ued to forecat future value of the tranformed time erie. Latly, the previou tranformation are undone, in order to obtain the future value of the original time erie, {Y t }. All the tep were done in an iterative fahion. The proce of forecating uually require a great deal of experience and teting alternative model. The reulted forecat are hown in Figure 9. Once a final atifactory ARIMA model wa elected, the tentative model were ued to forecat monthly value of ET o. The forecating period wa 1 month ahead for 60 month that cover the period 1998 to 2002 baed on the previou 300 month. The forecating performance capability for the pot ample period of the tentative ARIMA model wa aeed uing Root Mean Squared Forecating Error (RMSFE), Mean Abolute Forecating Percentage Error (MAFPE), and Maximum Abolute Forecating Percentage Error (MXAFPE). By examining Figure 9. Comparion of predicted to actual reference crop evapotranpiration of the two tentative ARIMA model Typically, the RMSFE i defined a the error accumulated in the forecated obervation. RMSFE = 1 N N t= 1 ( Y t Y t ) 2 (14) The mean percentage abolute forecating error i the abolute error in the deired prediction length, which conidered a meaure of how much a dependent erie varie from it model-predicted level. The mathematical repreentation of thi meaure i: N 1 et MPAFE = *100 (15) N Y t= 1 t Where e t i the abolute error between oberved and etimated ETo, Yt baed on an etimated model at time i and N i the number of forecat ued for thi purpoe. The Maximum Abolute Forecating Percentage Error i the larget forecated error, which i conidered a meaure ueful in imagining a wort-cae cenario for the forecat. Auming that the etimated model i repreentative of the forecating period, the pot-ample RMSFE hould be conitent with the reidual tandard error of the etimated model. A a reult, comparion of forecat performance baed on the RMSFE, MAFPE, and MAXAFP are made. It i worth mentioning that for evaluation intention, the pot-ample period i employed to provide reaonable cro-validation and to hun the potential mileading idea that the fit i better than it really i due to over-fitting the training erie. The out-of-ample error indicator for both candidate ARIMA model are preented in table 3. In general, the bet forecating for the overall time erie came from the ARIMA model 3. Overall, the ARIMA

9 2008 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) 41 approach in forecating reference evapotranpiration gave very good reult for monthly data (Mariňo et al. 1993: Hameed et al., 1995; and Trajkovic, 1999) which can be eaily implemented following the aforementioned tep. Table 3. Sample error indicator for the three candidate ARIMA Model # ARIMA RMSFE MXAFPE MAFPE I Model (200)(011) II III Model (100)(211) Model (100)(011) 4. Dicuion After a complete evaluation of model identification, etimation, diagnotic checking, and forecating, time erie analyi i eventually etablihed for reference crop evapotranpiration. ACF and PACF, a element in time erie analyi play an important role in thi matter. Both function were teted to calculate the ignificant autocorrelation exiting in the reference crop evapotranpiration data and to identify the component of the ARIMA model. The application of ARIMA model on a time erie hould obey tationary criterion. A periodic ET o time erie a hown in Figure 2, mandate tranforming the data to be tationary by differencing once, reducing ACF of the time erie ignificantly. Throughout the dicuion of thi data et, 5/6 of the total data i ued to etablihed and identify the model, and the remaining 1/6 of the total i ued to validate the model. Twelve different type of ARIMA model are equentially teted for the ET o time erie data. Baed on the exploration of the nature of the time erie data (i.e., the identification phae of ARIMA), a eaonal ARIMA and non-eaonal ARIMA with lag 12 are run on the data and both autoregreive and moving average ARIMA parameter are etimated. Table 3 lit three final model prepared for the tudied area. To elect the bet-developed model for forecating of the crop evapotranpiration, an aement of the performance of thee model wa conducted a meaurement of how cloely two independent data et match. Thi evaluation wa done uing Root Mean Square Forecating Error a a determinitic approach in addition to the Mean Abolute Forecating Percentage Error and the Maximum Abolute Forecating Percentage Error. The reult of thi evaluation are hown in Table 3. The three candidate model demontrated good performance. The eaonal ARIMA model (1,0,0)(0,1,1) 12 fit lightly better and give a maller confidence interval than the model (1,0,0)(2,1,1) 12 and (2,0,0)(0,1,1). The model diagnotic how that the reidual do pa the tet for normality (not hown). The finet ARIMA model wa identified calculating the t tatitic and other tatitical tool. The RMSFE, MAXAFP, and MAFPE were minimum for the cae of the model (1, 0, 0) (0, 1, 1) 12. The elected ARIMA model with two parameter eem to forecat the time erie data very well (Figure 9). Thi reult i upported by the work done by Marin o et al., 1993; Hameed et al., 1995; and Trajkovic, Thee invetigator have obtained good reult when ARIMA model wa compared with different time erie and conventional method of evapotranpiration etimation. Figure 9, however, how that predicted ET o gave reaonable agreement for the ARIMA (1, 0, 0) (0, 1, 1) 12 up to rate of 200 mm month -1 but inignificant underetimation at higher rate. Yet, the ARIMA (1, 0, 0) (0, 1, 1) model forecat ET o better than model I and II. To reflect the uncertainty in the forecat, thi analyi therefore follow that an approximate 95% prediction interval for the hitorical and forecating value may be made in the ame manner a in general leat-quare regreion iue. Furthermore, the model produce imilar ACF when predict the ET o during the period 1998 to 2002 a indicated in Figure 10. For the ame period, by viual inpection, Figure 11 howed an excellent correlation between the oberved and the forecated (calculated) value of ET o which upport our reult. Figure 10. Autocorrelation function of the predicted reference crop evapotranpiration uing ARIMA (1, 0, 0) (0, 1, 1) Figure 11. Oberved veru forecated reference evapotranpiration between 1998 and 2002 It mut be pointed out that the predicted time erie value of reference crop evapotranpiration are relatively lower than the oberved value obtained from the Deir- Alla weather tation on the year Thi might be due to a drought period. Drought period might lower the performance of the forecating technique. To examine the drought period, ET o etimation for the 1998 drought year

10 Jordan Journal of Earth and Environmental Science. All right reerved - Volume 1, Number 1 (ISSN ) wa at relatively higher variance with the reported ET. Thi variation contributed ignificantly to increaing the RMSFE and, in turn, lowering the forecating performance. Moreover, the reference crop ET relating to the evere drought year act a an outlier and influence the characteritic (e.g. being tationary) of the erie. Therefore, we expect a longer erie would lead to improved forecat. Thi i due to enduring the negative effect of drought period on the performance of a forecating technique. The lack of water in arid and emi-arid region contitute a major deterrent to utainable development of thee area. To meet demand for water for a multitude of ue, there i a continuing truggle. Inufficient water at the right place, at the right time, and with the right quality require more than ever before improved management, efficient utilization, and increaed conervation of limited frehwater reource. Thi manucript aim at providing a time erie forecating approach preentation, which can identify the need for future development aociated with water reource development, utilization, management, and conervation in arid and emi-arid region. Thi would reflect into the real world application where time erie modeling may erve a a convenient tool for the prediction of ET o when water reource are of a paramount importance a in the cae of Jordan. Succeful forecating of evapotranpiration would play a vital role in the irrigated agriculture ector. Thi will help in water reource management, which will contribute in trengthening Jordan economy. Therefore, adequate etimation and forecating of evapotranpiration are required epecially in irrigated agriculture. Evapotranpiration i one of the mot ignificant hydrologic procee affected by human activitie that alter the type and extent of vegetative cover. Moreover, knowledge of ET on not only a local but alo a regional cale enable hydrologit to perform water balance calculation and undertand hydrological cycle. ET forecating hall be ueful in irrigated agriculture, i.e., for water and agricultural planner: it will allow agronomit and farmer to ae crop water requirement in Deir-Alla Irrigated agriculture i a trade of Jordanian lineage practiced in the Jordan Valley, which contribute to the production of food and job opportunitie in direct and indirect agricultural employment and upporting ervice. It alo augment the environment and help apprehend deertification indirectly. Thi enhance and help in increaing on-farm irrigation efficiency and maximizing the agricultural output of a unit of land area per unit flow of irrigation water. In addition, in order to determine the volume of replenihment water that i needed for irrigation, evapotranpiration rate need to be forecated to wet root zone and to utain a low drainage rate. ET information i needed for drainage deign and drain networking to be help in intalling drainage ytem in the valley where natural drainage i not ufficient to erve thi purpoe. Thi encourage community farmer who uually need additional water for oil leaching from alt in the tudied area to et up a drainage ytem in their farm. Forecating ET would help in that propective and through minimizing the reue of treated water a upplement water in irrigation. Therefore, the water urplu that can be aved by knowing ET priori can be ued to either irrigate extra arable land to maximize productivity and increaing cropping intenitie or ue it wherever hortage occur in any of the other waterconumed ector: municipality and indutry. It i worth mentioning that irrigation water conume about three-fourth of the available freh water reource in Jordan. Managing the irrigation water ue under geographic, ocio-economic, and demographic contraint i of a vital importance to Jordan. Thi cae tudy how that forecating ET can be incorporated in irrigation water management by proper choice of crop and farming pattern. Furthermore, manager who manipulate oil-plant ytem in the Jordan Valley hould have a good fundamental undertanding of the proce of ET and the factor that influence it magnitude. Controlling the fate of water and achieving a proper management of the carce water reource in Jordan would diminih the exploitation of water reource, which i conidered a real threat to peace and future development proce in the region. Poible conflict over water in the region might retard any integrated and utainable development plan. Moreover, improved and efficient water reource management could help in utaining the tourim ince the Jordan Valley contain many tourim attraction and place like the Dead Sea, Jordan River and many other. It ha been demontrated by many tudie in the world that the tourim water ue produce higher net revenue compared to irrigation ue (The PRIDE Report, 1992). Potential evaporation i extremely high in the tudied area. Evapotranpiration i conidered on of the mot important factor in managing water reource in the Jordan Valley. Poor water management can reult in dieaecauing pollution, lo of topoil from eroion, damage to animal habitat and to foret and other. Wort of all i the damage to irrigated agriculture that now provide Jordan' food upply and mut be relied upon to provide more ince rainfed agriculture ha reached a ceiling and can produce little more than it already i producing. 5. Concluion On a per capita bai, Jordan i one of the lowet ranked countrie in water reource with the per capita hare of water being le than 175 for all ue. Thi place Jordan at only 20 percent of the water poverty level, which reflect the doughtine level. Evapotranpiration i one of the mot important indice in the drought equation, which exceed 90% in Jordan. Therefore, time erie forecating of evapotranpiration wa conducted to help the deciion maker and water ytem manger etablih appropriate trategie to utain and manage water reource. Time erie aume that hitory repeat itelf, o that by tudying the pat, better deciion, or forecat, can be made for the future. In thi paper, an endeavor i made to obtain a long-term forecat of monthly averaged reference crop ET without uing weather data. Thirty year of ET data wa ued in thi tudy to enure a atifactory etimate of monthly value. Neverthele, the change of inherent characteritic in the ET o may occur very lowly and time-erie model