TRANSFER FUNCTION MODELS APPLIED TO WATER TABLE Ilana Arensburg* and Rafael Seoane** Abstract

Transcription

1 TRANSFER FUNCTION MODELS APPLIED TO WATER TABLE Ilana Arensburg* and Rafael Seoane** * Insiuo Nacional del Agua. ** Insiuo Nacional del Agua, Consejo Nacional de Invesigaciones Cieníficas y Técnicas. Argenina. Absrac The objecive of his paper is o presen he resuls obained from he mehodology applied o he analysis of ime series of waer able; and o examine he possibiliy of using linear models of differen complexiy o generae daily forecass of his variable. The waer able series used corresponds o he localiy of Ezeiza, in he Maanza River waershed, Province of Buenos Aires, Argenina (34º48 S, 58º32 O), in a subwe-we region wih waer surpluses in excess of 15 mm per year. The proposed mehodology consiss in applying linear parameric models wih differen exogenous variables o forecas daily waer ables. Transfer funcion models make i possible o assess he dynamic response of he waer able o rainfall and useful waer. The variable accouns for recharge and discharge is useful waer, afer a calculaion of poenial evaporanspiraion values using he Priesley-Taylor equaion esimaed from a daily waer balance. The mehodology proposed by Box and Jenkins includes he following sages: idenificaion of he ime srucure by means of he cross-correlaion funcion; esimaion of parameers wih he maximum likelihood mehod; verificaion by means of residuals analysis and fiing of exra coefficiens (overfing). Finally, daily forecas errors of he seleced models are analyzed. The saisics used o evaluae he esimaed forecas errors are: mean and maximum error, roo of mean square error (RECM), absolue error (EA), and relaive error (RE). Transfer funcion models are used o forecas he daily waer able variable, paricularly he rainfall-waer able model, where he error saisics are comparaively less han hose obained wih he univariae ARMA model (1,1,). Oher resuls show ha he errors saisics of forecas waer able are improved when ransfer funcion model and Naional Oceanic Amospheric Adminisraion rainfall forecass one day in advance are used.

2 1. Inroducion The objecive of his paper is o presen he resuls obained from he mehodology applied o he analysis of ime series of waer able; and o examine he possibiliy of using linear models of differen complexiy o generae daily forecass of his variable. The mehod used is he one proposed by Box and Jenkins (1976). Calibraion, verificaion and forecas of ransfer funcion models are presened. They are idenified from he cross-correlaion funcion of he rainfall-waer able and useful waer-waer able variables. 2. Background informaion The waer able series used corresponds o he localiy of Ezeiza, in he Maanza River waershed, Province of Buenos Aires, Argenina (34º48 S, 58º32 O), in a subwe-we region wih waer surpluses in excess of 15 mm per year. The groundwaer source is a complex aquifer made up of: Puelche Secion wih homonymous sands; Pampeano secion wih limes wih sandy inercalaions; and he waer able wih fine exured sand. 3. Mehodology The proposed mehodology consiss in applying linear parameric models wih differen exogenous variables o forecas daily waer ables. Transfer funcion models make i possible o assess he dynamic response of he waer able o rainfall and useful waer. The mehodology proposed by Box and Jenkins includes he following sages: idenificaion of he ime srucure by means of he cross-correlaion funcion; esimaion of parameers wih he maximum likelihood mehod; verificaion by means of residuals analysis and fiing of exra coefficiens (overfing). Finally, daily forecas errors of he seleced models are analyzed. The ime series analysis mehod is widely used in hydrology for he mahemaical represenaion of differen ypes of processes. Viswanahan (1984) proposes rainfall-waer able ransformaion models hrough dynamic regressions; and Calderón Loaiza and Bermudes (1994) use he ime series analysis o esimae he same variable. 3.1 Inpu and oupu variables Rainfall For he se of daa used for idenificaion of he several models, accumulaed rainfall is mm/year. This value corresponds o a year wih less rainfall han he annual mean (95 mm/year) and an annual index (annual rainfall/mean annual rainfall) less han he uni. Yearly rainfall disribuion is shown in Figure 1. According o his analysis, he rainies season is auumn and he dries is winer. On a daily scale, rainfall has no emporal srucure and behaves like a noise. rainfall (mm) ime (days) Auumn Winer Spring Summer Figure 1. Rainfall disribuion (3/21/92 o 3/2/93). Useful waer The hydrologic cycle is a sysem wih inpu/oupu componens, and he occurrence of meeorological phenomena produces modificaions in he runoff and sorage. Waer able variaions occur when here is a difference beween recharge and discharge volumes. Recharge is produced by rainfall infilraion. The variable accouns for recharge and discharge is useful waer, afer a calculaion of poenial evaporanspiraion values using he Priesley-Taylor equaion (Chow e al., 1988) obained from a daily waer balance:

3 si, P A 1 E A P A 1 E si, P A 1 E 1 (1) 1 si, P A 1 E 1 where: A useful waer on day P rainfall on day E evaporanspiraion on day. According o Thornhwaie-Maher (1957), soil moisure sorage capaciy is he maximum amoun of waer he soil can hold in he rooing zone a field capaciy. In his sudy, i is assumed o be 1 mm since his is a case of fine sandy soil wih pasureland. A waer balance is developed wih daily rainfall values and evaporanspiraion esimaes from which he useful waer variable is obained (equaion 1). Figure 2 shows he behavior of he series esimaed for he useful waer variable. I is a nonsaionary series according o is simple auocorrelaion funcion. Useful waer (mm) ime (days) Auumn Winer Spring Summer Figure 2. Esimaed useful waer series (3/21/92 o 3/2/93) Waer able The behavior of he waer able has flucuaions associaed o he meeorological variables involved in he hydrologic cycle. Evoluion of he waer able hrough ime may be represened wih an auoregressiveinegraed model (Seoane and Arensburg, 1996) of he series in he form of: Nf Nf 1 1Nf 1 1Nf 2 z (2) where: Nf waer able on day 1 auoregressive parameer z residuals of he auoregressive inegraed moving average model, ARIMA (1,1,). Waer able (m) ime (days) Auumn Winer Spring Summer Figure 3. Behavior of he waer able (3/21/92 o 3/2/93) 3.2. Transfer funcion model Transfer funcion models consis of a linear relaionship beween wo saionary ime series. Given saionary X and W, he model is expressed as: W ( B) (3) X where: (B) backward shif operaor ( B) s ( B) B / ( B) b r

4 b s( B) 1 B... B r ( B) 1 1 q X p B... B r s r ( B), ARMA (p*, q*) process ( B) auoregressive process of order p* moving average process of order q* ( B q ( B p ) ) a s, ARIMA (p, d,q ) process independen from X d** number of differencing operaion ha is required o obain he saionary series y a noise (random independen variable idenically disribued wih expeced value zero) Cross-correlaion funcion Under he hypohesis ha wo variables are saionary, wih mean and consan variance, he cross-correlaion funcion for ime-delay k is esimaed as: E ( X r( k) k E( X ))( W E( W )) s s X W (4) where: X inpu variable W oupu variable E expeced value s X and s W sandard deviaions from each variable. The r(k) funcion is significanly differen from zero for a cerain k value when variable X exers a linear influence on W. On he basis of he inpu/oupu variables used, four opions are analyzed: - Cross-correlaion funcion beween rainfall and differenced waer able (d =1). The waer able variable is no saionary because is simple auocorrelaion funcion shows a non-exponenial decrease. This series is differenced and he resuls are shown in Figure 4. - Cross-correlaion funcion beween rainfall and residuals of he ARIMA (1,1,) model esimaed for he waer able. This opion is used when one or boh variables have an auocorrelaion srucure (Granger, 1977). - Cross-correlaion funcion beween differenced useful waer (d =1) and differenced waer able (d =1). - Cross-correlaion funcion beween differenced useful r(k) lag(days) Figure 4. Cross-correlaion funcion beween rainfall differenced waer able (d=1). waer (d =1) and residuals of he ARIMA (1,1,) model esimaed for he waer able series. In all cases, he coefficiens are significanly differen from zero for k =, 1 (Table 1). From he esimaed crosscorrelaion funcions, he orders for he corresponding ransfer funcion models (b,r,s) are idenified. The ransfer funcion models (,,1) are also idenified Transfer funcion esimae Preliminary esimae The preliminary esimae of parameers is made by calculaing he shown in equaion 3 hrough: r ( k) * s / s (5) i w x

5 where: i = i<b i 1 i 1... ai a i b b+a<i<b+s+1, a=1,...,s i 1 i 1... a i a i>b+s In he ransfer funcion models (,,1) equaion 5 is as follows: (B)= + 1 (B) = = r()*s w /s x 1 = 1 = r(1)* s w /s x Table 1 shows he inpu and oupu variables and he cross-correlaion coefficiens ha are used o obain he iniial esimae wih equaion 5. Model Variables Sandard deviaion Significan coefficiens of he cross correlaion funcion r() r(1), (m / mm) 1 (m / mm) I II III IV Inpu Rainfall (X ) 9.31 Oupu Differenced waer able (W ).51 Inpu Rainfall (X ) 9.31 ARIMA (1,1,) Oupu model residuals of waer able (z ).47 Inpu Differenced useful waer (Y ) 7.9 Oupu Differenced waer able (W ).51 Inpu Differenced useful waer (Y ) 7.9 ARIMA (1,1,) Oupu model residuals of waer able (z ) Table 1. Preliminary esimae of parameers. Maximum likelihood esimaes A his sage, he ransfer funcion models are esimaed using he maximum likelihood mehod, and hey are checked for consisency wih daa according o he following hypoheses: H 1 : a is noise; H 2 : X and a are independen. For H 1 i should be ascerained wheher he coefficiens of he simple (fas) and parial (fap) auocorrelaion funcion of he residuals series are no significanly differen from zero. For H 2 i should be ascerained wheher he coefficiens of he cross-correlaion funcion beween X and a are no significan. If boh hypoheses are verified, he model is correc. Oherwise, he ime srucure of he residuals should be idenified and he parameers should be esimaed anew for maximum likelihood. Model I (Table 1), rainfall-differenced waer able is expressed as: W X 1X 1 (6) where: W = Nf -Nf -1, is differenced waer able (d =1) on day. If does no verify he hypohesis on residuals, a model is idenified based on he simple parial auocorrelaion funcion of he series: (7) W X 1X 1 The model parameers are esimaed using maximum likelihood, and he hypoheses are esed again. Table 2 shows he resuls obained wih maximum likelihood using rainfall as he inpu variable and he differenced waer able as he oupu variable. The difference beween he wo models is ha he second verifies he hypoheses on residuals. Model Parameers Model for S res 2

6 W X X 1 1 =.24 s =.2 No ime srucure.12 I 1=.31 s =.2 ( = a ) W X X a a =.24 s =.2 ARIMA (,1,1) wih parameers:.12 1=.31 s 1=.2 1 ** = s 1**=.76 Table 2: Comparison of models esimaed on he basis of he maximum likelihood wih he rainfall and differenced waer able variables. The same procedure is followed wih he oher opions presened in he preliminary esimae (Table 1); he resuls obained are shown in Table 3. Model Parameers Model for S res 2 II z z X X 1 1 X 1X 1 1 1a 1 2a 2 a =.245 1=-.23 =.245 1=.23 s =.2 s 1=.2 s =.2 s 1=.2 No ime srucure ( = a ) ARIMA (,1,2) wih parameers 1 ** = s 1 ** =.54 2 ** = s 2 ** = III W Y Y W 1 1 y 1y 1 1a 1 2a 2 a =.25 1=-.36 =.25 1=-.36 s =.3 s 1=.3 s =.3 s 1=.3 No ime srucure ( = a ) ARIMA (,,2) wih parameers 1 ** = ** = -.59 s 1**=.526 s 2**= IV z Y Y 1 1 =.26 1=-.28 s =.3 s 1=.3 No ime srucure ( = a ).15 ARIMA (1,1,) z Nf Nf Y 1Y 1 1a 1 a 1 1Nf 1 1Nf 2 a =.26 s =.3 ARIMA (,,1) wih parameers 1=-.28 s 1=.3 1 ** =.1841 s 1**= =.3771 s 1= Table 3: Comparison of he ransfer funcion and ARIMA models esimaed based on maximum likelihood. From Tables 2 and 3 i may be inferred ha he simpler rainfall differenced waer able models have he smaller residuals variance esimaors (S res 2 ). A comparison of hese models and ARIMA (1,1,) shows ha when an exogenous variable is inroduced, here is a beer model fi Forecas A forecas consiss in esimaing he waer able on a given day, Nf (1), and in conrasing i wih observaions, Nf +1, for a hiry-day period immediaely following he period used for idenificaion and esimaion. In order o esimae Nf (1) he models wih he bes fi were seleced; i.e., smaller S res 2 and smaller number of parameers in Tables 2 and 3. The saisics used o evaluae he esimaed forecas errors are: mean and maximum error, roo of mean square error (RECM), absolue error (EA), and relaive error (RE).

7 The rainfall forecas esimaor, X (1) is obained from he analysis performed by he Naional Oceanic Amospheric Adminisraion (NOAA) of he corresponding coordinaes one day in advance (X (1) NOAA ) or from he season s daily mean ( X ). The daily rainfall predicive error deermines an EA of.991 and respecively. Therefore, he wo models shown in Table 2 use X (1) NOAA. Table 4 shows he resuls obained using he hisorical rainfall series wih forecas (X (1) = X +1 ) and a weak forecas (X (1) = ) o compare he saisics as a funcion of he rainfall model seleced. ARIMA (1,1,) Transfer funcion model Forecas model of he waer able (one day) Nf( 1) Nf 1Nf 1Nf 1 X (1) =X +1 Nf ( 1) Nf X(1) 1X Nf ( 1) Nf X (1) 1 X 1 a X (1) =X (1) NOAA X (1) = X X (1) = X (1) =X (1) NOAA Maximum Error= ( Nf 1 1 Nf ) Mean RECM = 1/ 2 n 2 ( Nf 1 Nf 1 ) / n 1 EA = n Nf 1 Nf 1 / n / Nf 1 ER = n ( Nf 1 Nf 1 ) / n / Nf Table 4: Saisics on waer able forecas errors. 4. Conclusions The analysis of he resuls obained makes i possible o assess he response of he waer able o some variables of he hydrologic cycle, especially rainfall. Transfer funcion models can be used o forecas he daily waer able variable, paricularly he rainfall-waer able model, where he error saisics are comparaively less han hose obained wih he univariae ARMA model (1,1,); and for simulaion purposes when hisorical rainfall records are longer han he rainfall-waer able simulaneous series. Transfer funcion models of useful waer-waer able models may be beer fied wih he observaions if more complex waer balance mehods are developed. The resuls show ha he errors saisics of forecas waer able are improved when ransfer funcion model and Naional Oceanic Amospheric Adminisraion rainfall forecass one day in advance are used. Bibliography Box, G.E.P. and G.M. Jenkins, Time Series Analysis, Forecasing and Conrol. Holden Day Calderon Loaiza, J. I. and O.N. Bermudes, Aplicación del análisis de series de iempo en Hidrogeología. II Congreso Lainoamericano de Hidrología Suberránea. II, Chile Chow, V. T., D. R. Maidemen and L.W. Mays, Applied Hydrology. McGraw-Hill Granger, C.W.J., and P. Newbold, Forecasing Economic Time Series. Academic Press Thornhwaie, W.C. and J.R. Maher, Insrucions and ables for compuing poenial evaporanspiraion and he waer balance. Dreseel Insiue of Technology, 1(3), New Jersey Seoane, R. and Arensburg, I. Analisis de series emporales de niveles freáicos. Serie Correlación Geológica Faculad de Ciencias Naurales e Insiuo Miguel Lillo, Universidad Nacional de Tucumán, Argenina. 11, Viswanahan, M.N., Recharge characerisics of an unconfined aquifer from he rainfall-waer able relaionship. Journal of Hydrology, 7,