Explaining long-term trends in groundwater hydrographs

Transcription

1 18 h World IMACS / MODSIM Congress, Cairns, Ausralia July 2009 hp://mssanz.org.au/modsim09 Explaining long-erm rends in groundwaer hydrographs Ferdowsian, R. 1 and D.J. Pannell 2 1 Deparmen of Agriculure and Food, Wesern Ausralia 2 School of Agriculural and Resource Economics, Universiy of Wesern Ausralia rferdowsian@agric.wa.gov.au Absrac: An abiliy o undersand and inerpre changes in groundwaer levels is essenial for sound managemen of groundwaer resources. Various saisical mehods have been developed o explain hydrograph rends. Mos of hese operae on he assumpion ha groundwaer rends are linear or bes represened by shor linear segmens. However here is clear evidence ha many hydrograph rends are nonlinear. For example, as he groundwaer level changes, he area of groundwaer discharge may change, dynamically feeding back and alering he rend rae of change. The long-erm underlying rend of groundwaer level over ime may or may no be linear. Three ypes of long-erm rends may be observed: We may observe a rising rend ha reduces over ime and is a common feaure of local groundwaer flow sysems. In hese cases, as groundwaer level rise, he hydraulic gradien and he rae of flow (discharge) boh increase. The resul is ha he rae of groundwaer rise falls sysemaically over ime. There are cases where he long-erm rend in groundwaer level is linear. This is usually he case where aquifers are inermediae o regional, have relaively higher recharge o discharge raios, very lile hydraulic gradien o generae significan flow and groundwaer levels are well-below he soil surface. Occasionally we may observe an increasing rae of groundwaer rise. This may be observed where sagnan aquifers are segmened by obsacles (eg basemen highs). In such cases each segmen will rise unil he area of discharge has increased or groundwaer finds a convenien flow pah o spill ino he lower par of he aquifer. From ha ime, he lower segmen, receiving waer from anoher segmen, will have an increasing rae of groundwaer rise, a leas for a period. We presen hree models for saisically esimaing non-linear rends in groundwaer levels. The firs model is an auoregressive model, in which a pas moving average of he dependan variable is included as an explanaory variable. This approach is useful when regular and frequen waer-level daa is available, alhough i has a few shorcomings. The second model uses ime-relaed spline funcions in he GenSa saisics package. The hird model includes a log ime funcion o capure he non-linear rend. Boh, he spline and log ime models are easy o use and produce realisic rend lines. The spline funcion is a subjecive mehod as he number of splines needs o be seleced. The advanage of he log ime model over he spline mehod is ha i is no subjecive and does no need special sofware; he solver funcion of Microsof Excel may be used o do he analysis. We provide deailed guidance on performing he spline and log ime models. Keywords: Saisics, saliniy, monioring, susainabiliy indicaors, nonlinear rend. 3060

2 1. INTRODUCTION Fresh groundwaer is a valuable resource while saline groundwaer may be a hrea o naural resources. In boh cases, monioring and inerpreing changes in groundwaer levels is essenial for managemen. Hydrographs show changes in groundwaer levels over ime and are ofen he mos imporan source of informaion abou he hydrological and hydrogeological condiions of aquifers. The paern of waer-level change in a hydrograph is governed by physical characerisics of he groundwaer flow sysem, he rainfall paern and he inerrelaion beween recharge o and discharge from an aquifer. Waer level changes in a hydrograph can also be caused by oher managemen opions such as exracion, irrigaion and land use change. Ferdowsian e al. (2001a) presened a new saisical approach for analysing hydrographs, called HARTT (Hydrograph Analysis: Rainfall and Time Trends). The mehod is able o disinguish beween he effec of rainfall flucuaions and he underlying rend of groundwaer levels over ime. Rainfall is represened as an accumulaion of deviaions from average rainfall and he lag beween rainfall and is impac on groundwaer explicily represened. Ferdowsian and Pannell (2001b) showed how o adap he approach o differeniae beween aypical rainfall evens, ime rends, and he impacs of managemens, such as changes in land use, pumping, and drains. The shor-erm underlying rend of groundwaer level over ime is ofen linear. Three ypes of long-erm rends may be observed (Figure 1). Line 1a represens a rising rend ha reduces over ime and is a common feaure of local groundwaer flow sysems. In hese cases, as groundwaer level rise, he hydraulic gradien and he rae of flow (discharge) boh increase. The resul is ha he rae of groundwaer rise falls sysemaically over ime. Line 1b represens a case where he long-erm rend in groundwaer level is linear. This is usually he case where aquifers are inermediae o regional, have relaively higher recharge o discharge raios, very lile hydraulic gradien o generae significan flow and groundwaer levels are well-below he soil surface. Line 1c, represens he case wih an increasing -1 rae of groundwaer rise. This may be observed 1b where sagnan aquifers are segmened by -2 1a obsacles (eg basemen highs). In such cases each -3 segmen will rise unil he area of discharge has increased or groundwaer finds a convenien flow -4 pah o spill ino he lower par of he aquifer. 1c From ha ime, he lower segmen, receiving waer from anoher segmen, will have an increasing rae of groundwaer rise, a leas for a period. Figure 1: Three possible shapes for rend lines 1.1 Meris of deecing he relevan rend Deecing and quanifying he long-erm rend of groundwaer rise or fall is valuable for many reasons: 1. I sheds ligh on he hydrogeological processes operaing a he sie; 2. I helps us o predic he fuure degree of hrea o naural resources; 3. I helps us o predic he exracion capaciy of waer resources; 4. I helps us o undersand he impac of pas land use and waer managemen; and 5. I prevens he risks associaed wih projecing a linear rend oo far ino he fuure. In his paper we presen hree mehods for esimaing he behaviour of groundwaer rend lines in he longerm. We describe he advanages and disadvanages of each mehod and show examples of heir applicaion. 2. METHOD The mehod proceeds in wo sages. In sage 1, he sandard HARTT mehod (Ferdowsian e al, 2001a) is used o esimae he ime-lag beween accumulaive residual rainfall (ARR) and is impac on groundwaer. Two forms of accumulaive residual rainfall may be used (Ferdowsian e al, 2001a): The firs was accumulaive monhly residual rainfall (AMRR; mm): Type of changes in groundwaer levels (m) May-90 May-92 May-94 May-96 May-98 May-00 May-02 May-04 May

3 AMRR = i= 1 ( (1) M i, j M j ) where M i,j is rainfall in monh i (a sequenial index of ime since he sar of he daa se) which corresponds o he jh monh of he year, M is mean monhly rainfall for he jh monh of he year, and is monhs j since he sar of he daa se. The second was accumulaive annual residual rainfall (AARR; mm): AARR = i= 1 ( M A/12) (2) i where A is mean annual rainfall. In sage wo, we esimae a regression model explaining groundwaer level as a funcion of ARR, ime and a hird variable, which varies beween he hree models presened here Auoregressive model wih he dependan variable s pas moving average The firs mehod is a general auoregressive model wih he dependan variable s pas moving average (auoregressive moving average = ARMA) included as he hird variable (Greene, 1993). The advanage of his mehod is ha he non-linear rend is esimaed as a funcion of pas groundwaer levels, which is no he case in he nex wo mehods. This mehod is applicable when regular and frequen waer-level daa is available. The regression model was: Deph = k 0 + k 1 ARR - L1 + k 2 + k 3 MA -L2 (3) where Deph is deph of groundwaer below he ground surface, is monhs since observaions commenced, L1 is lenghs of ime-lag (in monhs) beween rainfall and is impac on groundwaer, MA is pas moving average of groundwaer levels, L2 is lenghs of ime-lag (in monhs) beween pas moving average and he presen groundwaer levels and k 0, k 1, k 2 and k 3 are parameers o be esimaed. Parameer k 0 is approximaely equal o he iniial deph o groundwaer, k 1 represens he impac of above- or below-average rainfall on he groundwaer level, k 2 is he linear rend rae of groundwaer rise or fall over ime and k 3 represens he impac of pas groundwaer levels. The expression k 2 + k 3 MA -L2 represens he long-erm rend of groundwaer rises or falls over ime. The non-linear rend esimaed using his mehod may no vary monoonically. I is likely o rise and fall according o pas rainfall paerns. This may or may no be considered a problem, depending on he purpose of he analysis. A decision is required regarding he ime frame over which he moving average variable (MA) is calculaed. A longer calculaion period reduces he noise associaed wih seasonal flucuaions in groundwaer levels. Here we use one monh. The nex challenge is o selec a suiable delay beween he dependan variable and is pas moving average. The shores delay will resul in he bes fi, bu will mask he impacs of oher facors (rainfall and ime). In his example, we have used a delay of one year Spline funcions The second mehod is o use ime-relaed spline funcion. Splines are funcions buil from segmens of polynomial. The pieces are consrained o be smooh where hey join (known as knos ). We have found ha a quadraic polynomial ha involves he square of ime () o provide a good fi for long-erm hydrographs. Here is an example of a quadraic polynomial equaion. a 2 + b + c (4) The GenSa saisics sofware (GenSa 2008) provides a faciliy o esimae spline funcions. We obained a good resul by esimaing groundwaer level as a linear relaion of ARR and a splined quadraic polynomial funcion for he ime. The funcion was expressed as: Deph = k 0 + k 1 ARR - L + k 2 SSPLINE (Monioring_dae: 2) (5) 3062

4 where Deph is deph of groundwaer below he ground surface, is monhs since observaions sared, L is lenghs of ime-lag (in monhs) beween rainfall and is impac on groundwaer. The expression SSPLINE (Monioring_dae: 2) is he spline submodel. GenSa esimaes he value of his submodel as well as he parameers k 0, k 1 and k 2. The spline funcion is a subjecive mehod as he number of splines needs o be seleced. Anoher drawback of he mehod is ha is rend line is made-up of wo or more segmens. This makes i difficul o use he approach for predicion of he fuure rend The log ime mehod The regression model for his mehod is: Deph = k 0 + k 1 ARR - L + k 2 + k 3 ln (k 4 (+ k 5 )) (6) where Deph is deph of groundwaer below he ground surface, ARR is accumulaive residual rainfall, is monhs since observaions sared, ln is he naural logarihm funcion, L is ime-lag (in monhs) beween rainfall and is impac on groundwaer, and k 0, k 1, k 2, k 3, k 4 and k 5 are parameers o be esimaed. Parameer k 0 is approximaely equal o he iniial deph o groundwaer and k 1 represens he impac of above- or belowaverage rainfall on groundwaer level. The expression k 2 + k 3 ln (k 4 (+ k 5 )) represens he long-erm rend of groundwaer rise or fall over ime. We can use Solver ool in Microsof Excel o esimae he k s based on minimising he sum of squared errors: (Fied-Acual) Advanages and disadvanages of he hree mehods Relaive o he ARMA mehod, he spline funcions and log ime approaches have he following advanages: They do no need subjecive selecion of he period for esimaing he moving average and is lagged impac; They can cope beer wih missing daa because of irregular monioring; and They produce a smooher rend, and may be beer suied for predicing fuure groundwaer levels. In he spline funcion approach, if he number of splines exceeds wo, i oo becomes a curve fiing mehod and ends o lose is predicive value. Secion 4 gives more deails abou he advanages and disadvanages of he hree mehods. 3. RESULTS AND DISCUSSION Here we show hree case sudies. The firs case shows he applicaion of all hree mehods. The second and he hird cases show applicaion of spline funcion and log ime models Case 1 Case 1 is for bore Ah31. This bore is locaed 12km eas of Frankland in Wesern Ausralia and has 18 years of groundwaer daa. The bore has a rend similar o line 1a in Figure 1. Figure 2 shows he raw groundwaer daa for bore Ah31 and he relaed ARR wih one monh delay. We can see ha groundwaer levels have risen despie he ARR having a falling rend (a decline in longerm rainfall). The bore is in a local-scale groundwaer flow sysem. Based on Darcy s Law, he rising rend reduces as groundwaer levels rise. All he hree models explained he variables well (R 2 > 0.9; Table 1). Groundwaer levels (m) Acual GW levels Accummulaive residual rainfall (AAR) /05/90 6/05/92 6/05/94 5/05/96 5/05/98 4/05/00 4/05/02 3/05/04 3/05/06 Figure 2: Unprocessed groundwaer level daa for bore Ah31 and he ARR wih one monh delay. ARR (mm) 3063

5 Table 1: The log ime mehod explained he changes in groundwaer levels of bore Ah31 beer han he oher wo mehods. Sum of squared Mehod residuals R 2 Log ime mehod * Spline funcions wih 2 degrees of smoohing (wo segmens) * Auoregressive model wih 1-year delayed moving average waerlevel * * Noe ha p-value for all parameers (k 0, k 1, k 2 and k 3 ) were < Esimaed long-erm rend lines using each of he -4 hree mehods are shown in Figure 3. Noe ha hese are no fied values, as hey show only he ime-relaed variables. The spline and log ime approaches resul in similar esimaes of he longerm rend, wih groundwaer rising a a -6 Trend by spline funcion Acual waer levels Trend by Ln decreasing rae over ime in each case. The Trend by moving average auoregressive model resuls in a flucuaing rend -7 line (as explained earlier; Figure 3). Is overall rend is more nearly linear han he oher wo, and, in his example, i may over-predic he rae Figure 3: The long-erm rend lines esimaed for case 1 of groundwaer rise in fuure periods. using each of he hree mehods Case 2 The second case is for a bore (Koj 1d88), in a Teriary sand plain (70km norh-eas of Albany in Wesern Ausralia) wihin an inermediae groundwaer flow sysem. This bore is in a sagnan aquifer (low hydraulic gradien and flow) and from experience was expeced o have a linear long-erm rend. Figure 4 shows he acual groundwaer level daa, impac of accumulaive residual rainfall as well as he rend line esimaed using a sandard HARTT model (k 0 + k 1 ARR - L + k 2 ; Ferdowsian e al., 2001a). This model explained 99% of he variance (Table 2). In such sagnan aquifers a sandard HARTT model can esimae he long-erm rend and here are fewer added values in use of he spline funcion or log ime approaches. For example, he k 3 parameer of he log ime model was no significanly differen from zero (p-value was ), so he model collapsed o he sandard HARTT model. Table 2: The percenage of variance accouned for by HARTT and GenSa mehods were more han 99. Mehod Sum of squared residual R 2 Sandard HARTT model * Log ime model ** Spline funcions wih 2 degrees of smoohing (wo segmens) ** * Noe ha p-value for he sandard HARTT model parameers (k 0, k 1 and k 2 ) were < ** Noe ha p-value for k 0, k 1 and k 2 parameers of he log ime and spline models were <0.001 bu p-value for k 3 was >0.5. Deph o groundwaer (m) Groundwaer levels (m) Feb Feb-93 Feb-95 Feb-97 Feb-99 Feb-01 Bore Koj 1d88 Acual GW levels Trend Impac of ARR Feb-03 Aug 87 Aug 89 Aug 91 Aug 93 Aug 95 Aug 97 Aug 99 Aug 01 Aug 03 Aug 05 Aug 07 Feb Figure 4: The raw groundwaer level daa for bore Koj 1d88, he rend (based on sandard HARTT mehod) and he impac of ARR. Feb-07 Impac of ARR (m) 3064

6 3.3. Case 3 Case 3 looks a groundwaer level changes in bore MB2d. This bore which is 66km norh-eas of Esperance in Wesern Ausralia, is in a segmened sagnan aquifer in a Teriary sand plain. I has 24 years of groundwaer level daa. There are basemen highs in his sand plain ha resul in he aquifer being inerruped and pariioned. As groundwaer levels rise, groundwaer finds a convenien flow pah o spill ino he lower segmen. This will cause he lower segmen o show a rend of increasing rae of groundwaer rise. Bore MB2d is locaed in one of hese lower segmens. We noe he groundwaer level in MB2d has risen a an increasing rae in recen years and he increasing rae canno be fully explained by changes in rainfall (Figure 5). Figure 5 shows groundwaer level daa for bore MB2d. Analysis wih he sandard HARTT model wih differen lags indicaes ha here was a hree-monh delay beween he monhly rainfall and is impac on groundwaer levels. Eighy-nine percen of he variance could be Acual GW levels Trend by Ln mehod Trend by spline mehod Impac of ARR explained using he HARTT model (Table 3). The spline and log ime models each explained 95% of variaion in groundwaer levels (Table 3). Figure shows he impac of ARR (wih hree monhs delay) as well as he wo rend lines (for spline and log ime models). Boh of he non-linear models capured he long-erm rend well, showing he recen rise in groundwaer levels following a period of approximaely saic levels. Figure 5: Groundwaer level daa for bore MB2d and he impac of ARR 18/2/82 17/2/85 17/2/88 16/2/91 15/2/94 14/2/97 14/2/00 13/2/03 12/2/06 Table 3: Saisical analysis of groundwaer level for bore MB2d and he power of he log ime and spline models o show he variaion. Mehod Sum of squared residual R 2 Sandard HARTT model * Log ime model* Spline funcions wih 2 degrees of smoohing (wo segmens)* * Noe ha p-value for all seleced parameers were < DISCUSSION AND CONCLUSIONS We have presened hree models for saisically esimaing non-linear rends in groundwaer levels. The firs model is an auoregressive model, in which a pas moving average of he dependan variable is included as an explanaory variable. There are a few challenges o use he approach for predicion of he curren and fuure rends: I is a subjecive mehod because a decision is required regarding he ime frame over which he moving average variable is calculaed. The nex challenge is o selec a suiable delay beween he dependan variable and is pas moving average. The non-linear rend esimaed using his mehod may no vary monoonically. I is likely o rise and fall according o pas rainfall paerns. This may or may no be considered a problem, depending on he purpose of he analysis. The second model uses ime-relaed spline funcions in he GenSa saisics package. This model is easy o use and produces realisic rend lines. The spline funcion mehod is also subjecive as he number of splines needs o be seleced. Anoher drawback of he mehod is ha i requires specialized saisics package. The hird model includes a log ime funcion o capure he non-linear rend. This model is also easy o use and produces a realisic rend line. The advanage of he log ime model over he spline mehod is ha i is no a subjecive model and i does no need special sofware; he solver funcion of Microsof Excel may be used o do he analysis. We provide deailed guidance on performing he spline and log ime models. Groundwaer levels (m) Impac of rainfall (m) 3065

7 ACKNOWLEDGMENTS We hank Andrew Van Burgel from he Deparmen of Agriculure and Food, Wesern Ausralia, and Michael Buron from he Universiy of Wesern Ausralia for saisical assisance. Acknowledgemen is exended o Richard George for his commens on a draf of his paper and o John Simons for providing groundwaer level daa for he sie near Esperance. REFERENCES Ferdowsian, R., Pannell D.J., McCarron, C., Ryder A. and Crossing, L. (2001a), Explaining Groundwaer Hydrographs: Separaing Aypical Rainfall Evens from Time Trends. Ausralian Journal of Soil Research. 39, Ferdowsian, R. Pannell D.J. (2001b), Explaining rends in groundwaer dephs: disinguishing beween aypical rainfall evens, ime rends, and he impacs of reamens. MODSIM 2001 Congress Proceedings, Canberra, December PP (Modelling and Simulaion Sociey of Ausralia and New Zealand INC) Greene, W.H. (1993), Time-Series Models. In Economic analysis. (Second ediion) pp (Prenice- Hall. Inc. New Jersey) GenSa for Windows (2008), VSN Inernaional ld, 11 h ediion. hp:// 3066