AUGMENTING SHORT HYDROLOGICAL RECORDS TO IMPROVE WATER RESOURCES STUDIES
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1 AUGMENTING SHORT HYDROLOGIAL REORDS TO IMPROVE WATER RESOURES STUDIES Zead S. Tarawneh ivil Engineering Deparmen The Hashemie Universi Zarqa Jordan ABSTRAT haracerizing exreme hdrological evens depending on shor hdrological records ma resul in unreliable esimaes of he exreme even characerisics. The linear regression model has been widel used o exend shor hdrological records, however exensions produced using he linear regression model usuall fail o preserve he serial correlaion of he shor hisorical record. This paper presens a new record exension model ha can be used o augmen shor hdrological records while preserving he serial correlaion of he shor record. Using generaed daa of a normall disribued bivariae ( and x ), he performance of he new model has been esed. ompared o he simple linear regression model, he new model performed quie well in erms of mainaining he characerisics of he shor record. The new model was used o exend a shor precipiaion record a Rabba gauging sie in Jordan back o he ear 194 for he purpose of analzing drough evens. The analsis shows he occurrence of 19 drough evens of duraion ranging beween 1 6 ears, he 1 ear drough even is he mos common. In Rabba region, if a drough condiion has jus ended, he analsis shows ha he average waiing ime for he nex drough even o occur is nearl ears. Kewords: Record augmenaion, Precipiaion, Droughs in Jordan. 1. INTRODUTION Waer resources sudies relaed o he planning, design, and managemen of surface waer ssems depend on he properies of exreme hdrological evens like he magniude of he flood and is reurn period, he magniude of he drough and is duraion. Such and oher properies can be esimaed benefiing of exised hisorical hdrological records (precipiaion or sreamflow), however he reliabili of hese properies increases as he number of he observed exreme evens increases, which could no be he case when shor hdrological records are used (Salas e al., 005). In general, using shor records o esimae he populaion parameers of cerain hdrological saisics ma resul in poor esimaes of hese parameers (Fiering, 1963).
2 Tree ring indices have been widel used in lieraure as predicor variables o augmen shor hdrological records of precipiaion or sreamflow. Exending he lengh of shor hdrological records using ree ring indices was made for several purposes like deecing he climae change (e.g. Gou e al., 007; Lara e al., 008; Davi e al., 009), and predicing he occurrence of exreme droughs (e.g. Touchan e al., 1999; Gonzales and Valdes, 003; Akkemik e al., 008; Woodhouse e al., 009; Tarawneh and Hadadin, 009; Fang e al., 010). Excep in Gonzales and Valdes (003) where he used an exension model ha includes a sochasic erm and a erm o accoun for he serial correlaion, he majori of he previous sudies have emploed ree ring indices as predicor variable and used he radiional linear regression model o augmen he shor precipiaion or sreamflow records x n 1 n 0 ime Figure 1 Schemaic plo of he shor record ( ), he long record (x ), he shor period n, he exension period n 1, and he exensions (dashed series) In lieraure, here are several heoreical models ha can be used o exend he lengh of shor hisorical records. Such models were developed iniiall o exend he shor records argeing wo main hemes. The firs heme was o improve he esimaes of he populaion parameers like he mean and he variance of he shor record (e.g. Maalas and Jacobs, 1964; Vogel and Sedinger, 1985), while he second heme was o mainain he esimaes of he parameers of he shor record (e.g. Hirsch, 198; Grgier and Sedinger, 1989; Gonzales and Valdes, 003). However few exension models (e.g. Grgier and Sedinger, 1989; Gonzales and Valdes, 003) include a erm ha accouns for preserving he serial correlaion of he record o be exended. The record exension model in Gonzales and Valdes (003) conains also a sochasic erm (whie noise), herefore i can no produce a unique exension. The model proposed b Grgier and Sedinger (1989) produces exensions ha preserve he serial correlaion, bu such exensions usuall overesimae he variance of he original shor record speciall
3 when he cross correlaion beween he shor record and he long record (predicor) is low. The radiional linear regression model ha has been widel used o exend he shor records relies on ransferring informaion from he long record (x ) o he shor record ( ) over he exension period. The usefulness of he informaion being ransferred (he amoun of he explained variance) depends on he cross correlaion ( ) beween he variables x and over heir common period (n). The common period n is called he calibraion period and usuall has he lengh of he shor record (Figure1). If n 1 is he lengh of he exension period (Figure1), hen simple linear regression model expresses he exensions ( ) over he exension period n 1 as follows: b x x = 1,,, n 1 (1) where and x are he mean values of he shor record and he long record x over he calibraion period n respecivel, and b is he model parameer. Johnson and Wichern (005) expressed he leas square esimaor of he model parameer b as: b () where is he lag-0 covariance beween he variables x and, and is he lag-0 covariance of x, i.e. he variance of x. The linear regression model (Equaion 1) produces exensions of he shor record such ha he mean of he shor variable and he cross correlaion beween and x are mainained. However when he shor record is seriall correlaed hen here is no guaranee ha he linear regression model will produce exensions ha preserve he serial correlaion of he shor record, consequenl he ime dependence srucure of he exended record and he shor record ma be differen. Beond o wha he radiional linear regression model can produce, he main objecive of his paper is o propose a new model ha can be used o exend a shor hdrological record while preserving mos of he characerisics of he shor record. Such characerisics o be mainained are he mean, he variance, he serial correlaion of he shor record, and he cross correlaion beween he shor and he long record. The performance of he new model will be esed and compared wih he performance of he linear regression model using snheic daa. Laer he new model will be used o exend a shor precipiaion record a Rabba gauging sie in Jordan. The combined record, e.g. he shor hisorical and he exended par, will be used o characerize droughs in Rabba region in Jordan.
4 . MODEL FORMULATION AND PERFORMANE The developmen of he new record exension model sars from he general shape of he radiional simple linear regression model (Equaion 1). The formulaion of he new model will conain a erm ha will care abou he serial correlaion of he shor hisorical record..1 Model Formulaion Given he shor record and he long record x, he new model ha can be used o produce exensions of he shor record is: a bx x 1 = 1,,, n 1 (3) where is he exension a he ime sep, and x are he mean values of and x over he calibraion period n respecivel, a and b are he model parameers. Using he mehod of momens (Appendix), he momen esimaors of a and b are: a ( ) (4) and a b (5) where and as defined previousl, - is he lag-1 covariance beween x and wih lagged one ime sep behind x, and - are he lag-0 and lag-1 covariance of he variable, respecivel. omparing he new model (Equaion 3) wih he simple linear regression (Equaion 1), i is expeced ha he addiion of he erm a 1 will conribue in producing exensions ha will preserve he serial correlaion of he shor record.. Tesing he Model Performance To es he performance of he new model versus he widel used simple linear regression model a huge amoun of snheic daa were used samples of a normall disribued bivariae ( and x ) were generaed a each of he cross correlaion ( ρ ) values of 0.5, 0.65, 0.8, and 0.9. Such correlaion range is common beween naural daa. Each sample of he bivariae ( and x ) was generaed o a lengh of 150 daa poins. A mean of 100 and sandard deviaion of 30 were seleced arbiraril for
5 each of he generaed and x. The serial correlaion coefficien of he shor variable was kep a a value of 0.3 (significan ime dependence), while he serial correlaion coefficien of he predicor variable x was kep a a value of 0.0 (ime independen) like mos of he ree ring indices ha are used as predicors o exend shor precipiaion or sreamflow records. The reason behind selecing ime independen variable x while is ime dependen is o show he capabili of he new record exension model o produce exensions ha will preserve he serial correlaion of he shor hroughou he addiion of he erm a 1 alhough he predicor variable x is ime independen. Since he linear regression model lacks of a erm ha accouns for he ime dependence hen i produces exensions ha will highl likel fail o preserve he serial correlaion of he original variable speciall when he predicor variable x is ime independen. Ou of he 150 daa poins generaed for, he las 50 daa poins ( = 101, 10,, 150) were kep a side for validaion purposes, herefore he variable is considered a shor record of a lengh of 100 daa poins (n = 100). The new model and he simple linear regression model were used o esimae for = 101, 10,, 150 o replace he 50 daa poins ha were kep a side. If he exended par for = 101, 10,, 150 is combined wih he shor par ( = 1,,,100), hen he combined record ~ is obained: ~ 1,,...,100 (6) 101,...,150 For each of he 1000 generaed samples, he mean, he variance, he lag-1 serial correlaion coefficien of he shor record for = 1,,, 100, and he cross correlaion ρ beween and x for = 1,, 100 were calculaed. Also he mean ~, he variance ~, he lag-1 serial correlaion coefficien ~ of he combined record ~, and he cross correlaion ρ x~ beween ~ and x for = 1,,, 150 were calculaed. Moreover he sum of he squared errors ( e ) beween he exensions and he e. replaced 50 daa poins of were also calculaed noing ha Defining he performance saisics ~ /, ~ /, ~ / and ρ x~ / ρ he performance of boh exension models, i.e. he new model and he simple linear regression, can be esed. A each ρ value of 0.5, 0.65, 0.8, and 0.9, here were 1000 generaed samples of he bivariae and x, hence here were 1000 value for each of performance saisics b which he expecaions E[ ~ / ], E[ ~ / ], E[ ~ / ], E[ ρ x~ / ρ ], and E[ e ] were calculaed. Tables 1,, 3, and 4 show he expecaions of he performance saisics a ρ values of 0.5, 0.65, 0.8, and 0.9 respecivel.
6 Table 1 The performance saisics a value of 0.5 Performance saisics New model Simple linear regression E[ ~ / ] E[ / ] ~ E[ ~ / ] E[ ρ x~ / ρ ] E[ e ] Table The performance saisics a value of 0.65 Performance saisics New model Simple linear regression E[ ~ / ] E[ / ] ~ E[ ~ / ] E[ ρ x~ / ρ ] E[ e ] Table 3 The performance saisics a value of 0.8 Performance saisics New model Simple linear regression E[ ~ / ] E[ / ] ~ E[ ~ / ] E[ ρ x~ / ρ ] E[ e ] Table 4 The performance saisics a value of 0.9 Performance saisics New model Simple linear regression E[ ~ / ] E[ / ] ~ E[ ~ / ] E[ ρ x~ / ρ ] E[ e ]
7 3. RESULTS AND DISUSSION Tables 1 4 show he expecaion of he saisics ~ /, ~ /, ~ /, and ρ x~ / ρ suggesed here o judge he performance of boh models as he cross correlaion varies (as he relaion beween and x varies). If he expecaion of a seleced saisic as calculaed from he exensions produced b a record exension model is 1, hen ha model produces exensions ha are able o preserve he seleced saisics. In relaion o preserving he mean value of he shor record, Tables 1 4 show ha boh models performed quie well, i.e. E[ ~ / ] = 1, which means ha boh models produced exensions ha well preserved he mean value of he shor record regardless of he cross correlaion value. In erms of he amoun of he explained variance, boh models produced exensions ha alwas underesimaed he variabili of he shor record, i.e. he value of E[ ~ / ] is alwas is less han 1 regardless of he model ha was used o produce he exensions, herefore less amoun of he variance was explained b he exensions produced b boh models. However as he cross correlaion ρ increases, he exensions produced b boh models showed an increased amoun of he explained variance, i.e. E[ ~ / ] has increased, which is expeced due o he increased relaion beween and x. Tables 1 4 show he resuls in relaion o he power of boh models o produce exensions ha preserve he lag-1 serial correlaion coefficien of he shor record. Regardless of he cross correlaion beween and x, he expecaion E[ ~ / ] of he saisic ~ / as esimaed from he exensions produced b he new model is nearl 1, which means ha such exensions have a serial correlaion ha is so close o he shor record serial correlaion, i.e. he exended par well reproduced he serial correlaion of he shor record. On he oher hand, he expecaion E[ ~ / ] as esimaed from he exensions produced b he linear regression model is alwas much less han 1, i.e. such exensions alwas underesimaing he serial correlaion of he shor record. Such resul is expeced since he formulaion of he simple linear regression model (Equaion 1) ensures he producion of exensions ha onl will preserve he mean value of he shor record and he cross correlaion beween he shor and he predicor variable, bu no he serial correlaion. Exensions produced b boh models behaved similarl in erms of preserving he cross correlaion beween and x. A low cross correlaion values (Table 1), he exensions produced b boh models slighl overesimaed he original cross correlaion ( ρ ) beween and x, however as he cross correlaion increases he expecaion E[ ρ ~ / ρ ] ges closer o 1, which means ha boh models evenl gave exensions ha closel reproduced he cross correlaion beween he shor and he predicor variable x.
8 The sum of he squared errors is a crierion ha is used o validae exensions produced b a record exension model. I measures how much he produced exensions deviae from here exac values. Over he 1000 generaed samples a differen cross correlaion values, he expecaion of he sum of he squared errors E[ e ] is shown in Tables 1 4 as calculaed from he exensions produced b boh models. Generall he exensions produced b boh models depared similarl from here exac values, however as he cross correlaion increases he sum of he squared errors decreases indicaing ha he exensions produced b boh models become more close o here exac values. The new model shown b Equaion (3) was used o exend he precipiaion record a Rabba gauging sie in Jordan (Figure ) for he purpose of drough analsis. In Rabba region, crops like whea are usuall culivaed depending on he rain season ha sars b Ocober of he curren ear and ends b April of he nex calendar ear. The monhl precipiaion record used in his sud for Rabba gauging sie sars he ear 195 and ends he ear The annual rain season precipiaion is he sum of precipiaion for he monhs Ocober o April. The mean, he variance, and he serial correlaion of he rain season precipiaion ( ) a Rabba saion are: 339.5mm, 181.9, respecivel. Truncaing he annual rain season series ( ) a he level of he long erm mean for drough analsis shows he occurrence of 11 drough evens of duraion ranging beween 1 6 ears. The nearb precipiaion gauging sie of a relaivel long record is Amman airpor gauging saion. The annual rain season precipiaion as he sum of Ocober o April monhl precipiaion spans over he period The mean, he variance, and he serial correlaion of he rain season precipiaion a Amman airpor gauging saion are: 71.7mm, 8443., respecivel. The cross correlaion beween he precipiaion a he wo gauging sies over he calibraion period ( ) is This longer precipiaion series was used as a predicor in he new model o exend he precipiaion record a Rabba gauging sie over he period The model parameers a and b were esimaed using Equaions (4) and (5) having values of and respecivel. The combined precipiaion record a Rabba gauging sie (he exended over he hisorical over ) was used o analze droughs. The combined precipiaion record a Rabba gauging sie has a mean, variance, and serial correlaion of 344.4mm, , and respecivel. omparing such saisics wih hese obained from he shor hisorical record i can be seen ha he exensions obained from he new record exension model well reproduced he mean and he serial correlaion while he variance was underesimaed. Such resul ver well mach wih he resuls obained from he analsis of he 1000 snheic samples shown b Tables 1 4. Figure 3 shows he hisorical precipiaion a boh gauging sies and he exended precipiaion a Rabba sie.
9 Figure Rabba and Amman airpor precipiaion gauging sies. The conours show he spaial disribuion of he annual precipiaion. Figure 4 shows he combined precipiaion record a Rabba gauging sie runcaed a he level of he long erm mean of mm for he purpose of drough analsis. ompared o jus 11 drough evens obained b analzing he shor hisorical precipiaion record ( ), he analsis of he longer combined precipiaion record ( ) shows he occurrence of 19 drough evens of duraion ranging beween 1 6 ears. Such drough evens are disribued as: 63.% for he 1-ear even, 15.8% for he -ears even, 5.% for he 3-ears even, 0% for he 4-ears even, 10.5% for he 5-ears even, and 5.% for he 6-ears even. In Rabba region, he average waiing ime beween he end and he sar of subsequen drough evens is 1.75 ears, i.e. given ha a drough siuaion has jus ended hen on average he ime needed for he nex drough even o show up is nearl ears.
10 700 Precipiaion amoun (mm) Years Figure 3 The hisorical precipiaion (solid line) and he exended precipiaion (small dashed line) a Rabba gauging sie versus he hisorical precipiaion a Amman airpor gauging sie (large dashed line) Precipiaion (mm) Exended Hisrorical Years Figure 4 The combined (hisorical + exended) precipiaion a Rabba gauging sie runcaed a he level of mm (dashed line)
11 4. ONLUSIONS urren record exension models like he widel used linear regression model generae exensions ha ma no reproduce all he saisics of he shor hisorical records speciall he serial correlaion. This paper presens heoreical aspecs for a new model ha can be used o exend shor hdrological records such ha mos of he shor record saisics are well preserved focusing on he reproducion of he shor record serial correlaion. The model was esed versus he simple linear regression model using simulaed normall disribued bivariae samples a differen cross correlaions. In conclusion he new model preformed quie well compared o he simple linear regression in erms of preserving he serial correlaion. The new model was hen used o exend he hisorical precipiaion record a Rabba gauging sie in Jordan back o he ear 194. The combined series of he hisorical and he exended precipiaion was used o analze drough evens in Rabba region in Jordan. The analsis showed he occurrence of 19 drough evens of duraion ranging beween 1 6 ears. Drough of 1 ear duraion was found o be he mos common even. Assuming ha drough condiion has jus ended in Rabba region, hen he average waiing ime for he nex drough even o occur is nearl ears. APPENDIX The new record exension model is: bx x a 1 (1a) Re-wriing Equaion (1a) as: bx x a 1 (a) Mulipling boh sides of Equaion (a) b 1 and aking he expecaion on boh sides: E a E b Ex x (3a) The expecaion E variable, i.e. -, likewise he expecaion E 1 and he expecaion Ex x becomes: 1 is esimaed b he lag-1 covariance of he 1 is esimaed b, 1 is esimaed b, herefore Equaion (3a) a b (4a) Mulipling boh sides of Equaion (a) b x x and aking he expecaion on boh sides and replacing he expecaions b here esimaes (covariances) hen:
12 a b (5a) Solving Equaions (4a) and (5a) for a and b, hen: a ( ) and b a REFERENES Akkemik, U., D'Arrigo, R., herubini, P., Kose, N. and Jacob, G., 008. Tree-ring reconsrucions of precipiaion and sreamflow for norh-wesern Turke, Inernaional Journal of limaolog, Vol. 8, pp Davi, N., Jacob, G., D'Arrigo, R., Baaarbileg, N., Jinbao, L. and uris A., 009. A ree-ring-based drough index reconsrucion for far-wesern Mongolia: , Inernaional Journal of limaolog, Vol. 9, pp Fang, K., Davi, N., Gou, X., hen, F., ook, E., Li, J. and D'Arrigo, R., 010. Spaial drough reconsrucions for cenral high Asia based on ree rings, limae Dnamics, Vol. 35, pp Fiering, M., Use of correlaion o improve esimaes of he mean and variance, Geological Surve Professional Paper Gonzales, J. and Valdes, J., 003. Bivariae drough analsis using ree ring reconsrucion, Journal of Hdrological Engineering, Vol. 8, No.4, pp Gou, X., hen, F., ook, E., Jacob G. and Yang, M., 007. Sreamflow variaions of he Yellow River over he pas 593 ears in wesern hina reconsruced from ree rings, Waer Resources Research, Vol. 43, pp. Grgier, J. and Sedinger, J., A generalized mainenance of variance exension procedure for exending correlaed series, Waer Resources Research, Vol. 5, No. 3, pp Hirsch, R., 198. A comparison of four sreamflow record exension echniques. Waer Resources Research, Vol. 18, No., pp Johnson, R. and Wichern, D., 005. Applied mulivariae saisical analsis, Pearson Educaion, India, pp. 358.
13 Lara, A., Villalba, R. and Urruia, R., 008. A 400-ear ree ring record of he Puelo river summer-fall sreamflow in he Valdivian rainfores eco-region, hile, limae change, Vol. 86, pp Maalas, N. and Jacobs, B., A correlaion procedure for augmening hdrological daa, Geological Surve Professional Paper 434-E. Salas, J.D., Fu,., ancelliere, A., Dusin, D., Bode, D., Pineda, A. and Vincen, E., 005. haracerizing he severi and risk of drough in he Poudre River, olorado, Journal of Waer Resources Planning and Managemen, Vol. 131, No. 5, pp Tarawneh, Z. and Hadadin, N., 009. Reconsrucion of he rain season precipiaion in cenral Jordan, Hdrological Sciences Journal, Vol. 54, No. 1, pp Touchan, R., Meko, D. and Hughes, M., A 396-ear reconsrucion of precipiaion in souhern Jordan, Journal of he American Waer Resources Associaion, Vol. 35, No. 1, pp Vogel, R. and Sedinger J., Minimum variance sreamflow record augmenaion procedures. Waer Resources Research, Vol. 1, No. 5, pp Woodhouse,., Russel, J. and ook, E., 009. Two modes of norh American drough from insrumenal and paleoclimaic daa, Journal of limae, Vol., pp
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