Heuristic Continuous Base Flow Separation

Transcription

1 Heurstc Contnuous Base Flow Separaton Jozsef Szlagy 1 Abstract: A dgtal flterng algorthm for contnuous base flow separaton s compared to physcally based smulatons of base flow. It s shown that the dgtal flter gves comparable results to model smulatons n terms of the multyear base flow ndex when a flter coeffcent s used that replcates the watershed-specfc tme delay of model smulatons. Ths way, the applcaton of the heurstc dgtal flter for practcal contnuous base flow separaton can be justfed when auxlary hydrometeorologcal data such as precptaton and ar temperature typcally requred for physcally based base flow separaton technques are not avalable or not representatve of the watershed. The flter coeffcent can then be optmzed upon an emprcal estmate of the watershed-specfc tme delay, requrng only the dranage area of the watershed. DOI: / ASCE :4 311 CE Database subject headngs: Base flow; Dgtal flters; Runoff; Streamflow; Algorthms. Introducton Detaled knowledge of groundwater contrbuton to streams,.e., base flow, s mportant n many water management areas: water supply, wastewater dluton, navgaton, hydropower generaton Dngman 1994 and aqufer characterzaton Brutsaert and Neber 1977; Troch et al. 1993; Szlagy et al. 1998; Brutsaert and Lopez Also, base flow can drectly be related to aqufer recharge Brtles 1978; Wttenberg and Svapalan 1999; Szlagy et al. 2003, whch s crucal n ascertanng safe yelds of water development schemes, such as rrgaton plannng n the Great Plans Sophocleous The mportance of havng knowledge of base flow s reflected n the number of publshed works, as revewed by Tallaksen Wth the wdespread use of PCs, tradtonal, event-based methods that contan varyng degrees of subjectvty, such as graphcal base flow separaton Barnes 1939; Hewlett and Hbbert 1963; Szlagy and Parlange 1998, have been replaced by automated technques that can result n contnuous base flow modelng. Present-day automated technques consst manly of two types: dgtal flterng methods Nathan and McMahon 1990; Arnold et al. 1995; Arnold and Allen 1999 and conceptual hydrologc models e.g., Jakeman et al. 1990; Szlagy and Parlange The former have no true physcal bass Arnold and Allen 1999 but have the dstnct advantage of requrng only streamflow measurements. The latter are physcally based but requre precptaton data as a mnmum n addton to measured streamflow. Often, avalable precptaton data are nsuffcent because the precptaton staton s ether not located wthn the watershed, or t s wthn the watershed but not at a representatve locaton. In larger catchments, more than one staton s typcally 1 Conservaton and Survey Dvson, Insttute of Agrculture and Natural Resources, Unv. of Nebraska-Lncoln, 114 Nebraska Hall, Lncoln, NE E-mal: jszlagy1@unl.edu Note. Dscusson open untl December 1, Separate dscussons must be submtted for ndvdual papers. To extend the closng date by one month, a wrtten request must be fled wth the ASCE Managng Edtor. The manuscrpt for ths paper was submtted for revew and possble publcaton on December 16, 2002; approved on October 28, Ths paper s part of the Journal of Hydrologc Engneerng, Vol. 9, No. 4, July 1, ASCE, ISSN /2004/ /$ needed to obtan a good estmate on the amount of water avalable to runoff. Many tmes the precptaton record has dscontnutes that can easly thwart efforts to perform contnuous base flow separaton usng physcally based technques. Clearly, there s a practcal need for a technque that uses the most basc nformaton avalable: streamflow and the correspondng dranage area. The dgtal flterng technque of Nathan and McMahon 1990 s such a mnmalst approach. Because ther method s not based on any physcal law, a queston arses whether the ensurng base flow hydrograph s realstc at all, or, n other words, can the results be backed by a more complex, physcally based approach? Unfortunately, there s no trval way of valdatng the results of the flter algorthm by measurements. Isotope or chemcal tracer technques may one day prove useful n valdaton efforts n spte of the currently exstng dscrepancy n base flow nterpretaton between physcal and tracer technques Rce and Hornberger Baseflow recesson can generally be descrbed by the followng equaton Brutsaert and Neber 1977 : dq dt aq (1) where a L 3(1 ) T 2 and ( ) constants; Q b L 3 T 1 the groundwater dscharge to the stream. Under smplfyng assumptons Brutsaert and Lopez 1998, the theoretcal value of durng recesson may change from three to unty. When reaches unty, the aqufer behaves as a lnear reservor, and a then equals k 1, the nverse of the storage coeffcent T n the lnear storage equaton S kq b, where S L 3 s water volume n storage. Naturally, not all aqufers behave as lnear reservors, even after a suffcent perod of streamflow recesson Brutsaert and Neber 1977; Szlagy and Parlange 1998; Troch et al. 1993; Wttenberg and Svapalan 1999, but many do, as reported by Vogel and Kroll 1992 ; Jakeman and Hornberger 1993, and Brutsaert and Lopez The analyss that follows s strctly vald for watersheds that exhbt ths latter type of base flow recesson property, although the results and conclusons can straghtforwardly be generalzed to a fully nonlnear aqufer case as well, where s always larger than unty. Jakeman and Hornberger 1993 ponted out that the nformaton content of a ranfall-runoff model allows for only a handful of model parameters to be optmzed. Perrn et al. 2001, na JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2004 / 311

2 study nvolvng 429 catchments worldwde, demonstrated that very smple models can acheve a level of performance almost as hgh as models wth more parameters. In fact, nadequate complexty typcally results n model over-parameterzaton and parameter uncertanty Perrn et al In the lght of these fndngs, the smplest possble physcally based model for base flow smulaton was sought. The model of Jakeman et al and Jakeman and Hornberger 1993, from now on referred to as the Jakeman model, meets ths crteron. Methodology Followng Jakeman et al and Jakeman and Hornberger 1993, any nonlnearty n the ranfall-runoff relatonshp can be dealt wth by the transformaton of the observed precptaton seres nto excess or effectve ranfall u LT 1 va an antecedent precptaton ndex s( ) s c r 1 1 r r 2... (2) where r LT 1 observed ranfall; ( ) the rate at whch the catchment wetness declnes n the absence of precptaton; tme ndex ncremented on a daly bass ; and c TL 1 a normalzng parameter that ensures that the excess ranfall volume equals the volume of total runoff over the calbraton perod. Excess ranfall s obtaned by u r s (3) Seasonal changes n evapotranspraton are descrbed by 0 e f 30 t (4) where f t 1 a temperature modulaton factor; t temperature C ; and 0 the rate at whch the catchment wetness declnes at 30 C. Fg. 1. Schematc representaton of the Jakeman model Effectve ranfall s routed through two parallel lnear reservors representng quck and slow.e. base flow storm responses. The unt mpulse response h( ) of a lnear reservor n dscrete tme s O Connor 1976 h 1 1 k k 1 k 0,1,2,... (5) from whch the mpulse response of the two parallel dscrete lnear reservors follows as h h q h b v q kq 1 k q 1 k q v b k 1 k b b 1 k b 0,1,2,... (6) where the subscrpts q and b represent quck and base flow storm responses, respectvely. Note that n dscrete tme the storage coeffcents k b and k q become untless. The volumetrc throughput coeffcents v q and v b ( ), add up to unty. The model response (Q m LT 1 ) to effectve ranfall s obtaned va the convoluton summaton m Q m 0 h u m, m 0,1,2,... (7) Fg. 2. Model response to fctve precptaton wth arbtrary parameters: k q 1 (day), k b 30 (day), f 1 ( C 1 ), 0 1, v q 0.5. Sold lne s modeled base flow; ntermttent lne s modeled total runoff. 312 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2004

3 where the dfference operator s for tme shftng,.e., g g g 1, where g s an arbtrary dscrete functon. Upon nvertng the resultng transfer functon H(z) Fg. 3. Schematc representaton of the modfed Jakeman model Altogether, the model has seven parameters f, 0, k q, k b, v q, v b, and c. A schematc of the system s shown n Fg. 1. As a demonstraton, the model response to fctve precptaton s shown wth arbtrarly assgned parameter values n Fg. 2. The base flow peaks occur almost smultaneously wth the total runoff peaks. Ths s because effectve ranfall s splt nto two parts and routed drectly through the two lnear reservors representng quck and base flow responses, wthout any tme delay n the latter case. In realty, there generally s a tme lag between the two peaks Plgrm and Cordery 1993; Szlagy and Parlange 1998, dependng on how long t takes the nfltrated water to reach the saturaton zone. The present modfcaton of the orgnal Jakeman model can account for ths possble tme lag by ncorporatng a thrd lnear reservor wth a storage coeffcent k s ) representng sol storage Besbes and de Marsly 1984; Wu et al. 1997; Wttenberg and Svapalan A schematc of the model arrangement can be seen n Fg. 3. The unt mpulse response of two seral dscrete lnear reservors s obtaned va the Z-transform of the dfference equaton Sngh k s 1 k b Q u (8) H z 1 k s k b k s k b z 2 (9) k s k b 2k s k b z k s k b the dscrete unt mpulse response results as 1 k b k s k s k s 1 k s k s k s 1 k s k b k b k b 1 k b k h b k b k s k 2 s k b k 2 s k b k 2 b k 2 b k s (10) By addng a sol-storage component to the Jakeman model, the number of parameters has ncreased through k s, by one, from seven to eght. The sol-storage component delays the base flow peak as well as flattens t, thus makng t look more realstc, as s seen n Fg. 4, where a k s 2 day was added to the prevously prescrbed model parameter set. In the last modfcaton of the model, the changng effect of the exponent n Eq. 1 s beng nvestgated. Rght after the start of the base flow recesson, the exponent may reach a value of three, provded the aqufer became close to full saturaton. Fg. 5 from Szlagy 1999 demonstrates ths case, wth the lower envelopes that are thought to represent pure groundwater dscharge of the data ponts expressng a slope of three and unty. Numercal and analytcal solutons of the Boussnesq equaton that descrbe groundwater dranage also confrm Brutsaert and Neber 1977; Szlagy 1999 ths change of the exponent n Eq. 1. A tmevaryng exponent n Eq. 1 can only be modeled va a general nonlnear reservor, S k Q n b, f n changes wth tme as well. Alternatvely, rather than changng n through tme, k may be changed wth tme n the lnear reservor representaton, as was done by Aksoy et al The crtcal base flow dscharge z 2 Fg. 4. Response of the modfed Jakeman model to fctve precptaton wth arbtrary parameters: k q 1 (day), k s 2 (day), k b 30 (day), f 1 ( C 1 ), 0 1, v q 0.5. Sold lne s modeled base flow; ntermttent lne s modeled total runoff. JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2004 / 313

4 Fg. 5. Measured daly dscharge versus change n dscharge between consecutve days, 6 days after ran. (Q b0 ), when ths change starts Fg. 5, s obtaned by solvng Eq. 1 smultaneously for the two lower envelope lnes as Q b0 (a 1 /a 3 ) 0.5, where a 1 and a 3 are wth 1 and 3, respectvely. For convenence, t s assumed here that k changes lnearly from a maxmum value of k b ( a 1 1 ), when Q b Q b0, to a mnmum value of 0.5 k b when the aqufer becomes close to saturaton. Under smplfyng assumptons Brutsaert and Lopez 1998, dranable water storage at full saturaton, S max, can be estmated as S max 1.97 A(a 1 a 3 ) 1/2 1, where A dranage area of the watershed. Snce k s changng wth tme now, a smple convoluton cannot be mantaned; nstead, base flow s smulated Fg. 6 by numercally solvng the lnear storage equaton wth a tme-varyng storage coeffcent. Ths means that through the calculaton of S at each tme step, the correspondng k(s) value s Fg. 6. Response of the modfed Jakeman model wth tme-varyng storage coeffcent to fctve precptaton wth arbtrary parameters: k q 1 (day), k s 2 (day), k b 30 (day), f 1 ( C 1 ), 0 1, v q 0.5. Sold lne s modeled base flow; ntermttent lne s modeled total runoff. Q 0 and S max are assumed to be 0.05 (mm d 1 ) and 10 mm, respectvely. 314 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2004

5 Fg. 7. a Frst year of the smulated streamflow ntermttent lne and base flow values; b base flow hydrographs of the same perod. Intermttent lne s the flter result. Here k q 1 (day), k s 1 (day), k b 60 (day), v q 0.2. obtaned wth the help of the maxmum and mnmum values of k, as k c 1 S c 2, where c 1 k b /2(S 0 S max ), and c 2 k b S 0 c 1. S 0 s the dranable water storage at Q b Q b0. Ths last modfcaton of the Jakeman model MJ wll be used for the valdaton of the dgtal flter algorthm Nathan and Mc- Mahon 1990, whch estmates base flow (Q b )as Q b pq b 1 1 p Q 2 Q 1 (11) from measured or modeled streamflow Q, where p s the flter parameter. The resultng base flow values are constraned by the concurrent streamflow values, so that whenever Q b Q, the Q b value s replaced by Q. The valdaton s done by runnng the MJ model wth Monte Carlo-smulated daly precptaton values n combnaton wth determnstc daly temperature values, followng Mlly 1994 and Szlagy The daly values of precptaton (P d L ) are assumed to follow an exponental dstrbuton P d e P d (12) where 1 P a /(365.25*SF), wth P a L denotng the mean annual precptaton, and SF T 1 the mean storm frequency. SF s calculated as 2 P d 2 /var(p d ), where the angular brackets denote temporal averagng, and var denotes the varance. The number of nterstorm days ( d ) s assumed to follow a Posson dstrbuton Fg. 8. a Frst year of the smulated streamflow ntermttent lne and base flow values; b base flow hydrographs of the same perod. Intermttent lne s the flter result. Here k q 1 (day), k s 2 (day), k b 30 (day), v q 0.8. JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2004 / 315

6 Fg. 9. a Frst year of the smulated streamflow ntermttent lne and base flow values; b base flow hydrographs of the same perod. Intermttent lne s the flter result. Here k q 2 (day), k s 2 (day), k b 30 (day), v q 0.8. where SF 1. Results and Dscusson P d N N! e (13) Rather than fttng the MJ model to measured streamflow and comparng the flter results to the MJ-model-obtaned base flow, a Monte Carlo-type smulaton wth the MJ model was preferred due to the much greater flexblty the latter approach offers. By makng sure that the model-prescrbed parameters are physcally meanngful and drvng the model wth realstc precptaton and temperature nputs, realstc model smulatons of base flow can be expected and compared to flter results. The MJ model, even n ts orgnal, smplest form, performed qute effectvely n smulatng daly streamflow of small catchments n the U.S., Europe, Asa, and n Australa Jakeman et al. 1990; Jakeman and Hornberger The modfed MJ model was run n a Monte Carlo smulaton mode wth daly precptaton and daly mean temperature nputs, characterstc of a mld contnental clmate of central Europe, wth a mean annual precptaton of 600 mm evenly dstrbuted.e., no seasonal cycle throughout the year, a mean annual temperature of 11 C, and a mean storm frequency of /day. A choce of 0 1 and f 1 C 1, n combnaton wth a 5th order polynomal n Eq. 2, resulted n a 7% runoff rato, whch s typcal of the lowland regons n central Europe. The daly mean temperatures C followed the mean monthly temperatures n the model startng wth January: 1.1, 1, 5.8, 11.8, 16.8, 20.2, 22.2, 21.4, 17.4, 11.3, 5.8, and 1.5. Each model smulaton represented 10 years. The quck storm response parameter k q and the sol storage coeffcent k s were each assgned two values: 1 and 2 d. The base flow storage coeffcent k b was allowed to have values of 30 and 60 d. The volumetrc throughput parameter v q was assgned the followng values: 0.2, 0.4, 0.6, and 0.8. Note that v b 1 v q, correspondngly. The values of the above parameters are representatve of the catchments reported by Jakeman and Hornberger Wth decreasng v q values, groundwater contrbuton to the streamflow ncreases, requrng ncreased subsurface storage capablty n the watershed. Ths s accommodated for n the model by ncreasng the value of S max and Q b0 n the model accordngly, such as 5, 0.03, 10, 0.06, 15, 0.09, and 20, 0.12, where the frst value n each parenthess s S max mm, the second one s Q b0 mm/day, and the frst parenthess corresponds to v q 0.8. The S max and Q b0 values are representatve of the small catchments of the Washta Expermental Watershed complex n Oklahoma Brutsaert and Lopez The three storage coeffcents and the v q values amount to 32 dfferent and unque combnatons. Wth each combnaton of the model parameters, the MJ model was run for 10 years n daly tme ncrements. From the resultng base flow hydrograph, the Table 1. Model Smulaton and Optmzed Flterng Results (N d, p, BFI flt /BFI). k q (d) 1 2 k s (d) k b (d) V q / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2004

7 Fg. 10. Hstograms of the a BFI ratos; b optmzed p; and c the tme delay parameter values mean watershed-specfc tme delay Lnsley et al N d (d) could be calculated. N d s the mean elapsed tme between the peak of streamflow and the frst nstant when streamflow becomes domnated by base flow. The crtcal pont when ths latter happens was calculated as Q b /Q v q, whch results n a Q b /Q rato of 80% when v q s 0.8 and 95% when v q s 0.2. A more strngent crtcal value s necessary when base flow domnates streamflow. Note that when v q 0.2, 80% of the streamflow s made up by base flow on a long-term bass, whch means that the base flow ndex, BFI ( Q b / Q, where the angle brackets denote temporal averagng s 0.8 or 80%, as well. Note also that the use of such a crtcal value s not necessary wth the flter algorthm because of the constrant appled there, whch makes streamflow become base flow fully overnght. In the MJ model, ths can never happen due to the exponental decay n the quck flow component. Wth the known N d value from the MJ model, the flter parameter p was systematcally changed untl the flter model gave the closest possble matchng value of N d wth the MJ model, whch was generally wthn 1%. Fgs. 7, 8, and 9 dsplay hydrographs for small ( 3.15d), medum ( 4.16d), and large N d ( 6.13d) cases, respectvely. The resultng p, N d, and BFI flt /BFI values are lsted n Table 1. As t can be seen, the MJ-smulated watershed-specfc tme delays ranged between 3.15 and 6.21 d, the flter parameter value p ranged from to 0.999, and the BFI flt /BFI ratos changed between 0.48 and Fg. 10 dsplays the dstrbuton of the values. Fg. 10 shows that the long-term base flow ndex, gven by the flter algorthm s wthn 20% of the modeled BFI value n 80% of the cases consdered, wth a mean value of only 6% less than the modeled mean BFI value. Ths suggests that the flter algorthm of Nathan and McMahon 1990 s of practcal value, provded one can estmate the watershed-specfc tme delay N d for real watersheds. Fortunately, ths s possble by the applcaton of Lnsley s emprcal equaton Lnsley et al N d A 0.2, where N d s n days and A, the dranage area of the watershed, s n square mles. When applyng the flter algorthm, the flter parameter must be adjusted untl the resultng N d value becomes suffcently close to Lnsley s value. Ths has been done by Szlagy et al for 100-plus gaugng statons n Nebraska where the spatal dstrbuton of the long-term BFI ndex was of nterest. In concluson, t can be stated that the flter algorthm, n spte of ts lack of any physcal bass, can have ts place n practcal applcatons when more complex and/or physcally based base flow separaton methods are hndered by data avalablty. The flter algorthm, wth ts suggested optmzaton, based on the watershed-specfc tme delay, requres only the most basc data: streamflow and the correspondng dranage area. Of course, at best, the practcal value of the flter algorthm s only as good as the emprcal equaton of Lnsley et al. 1958, whch has been frequently used n a wde varety of applcatons n the past 4 decades. As llustrated prevously wth the help of model smulatons, t gave comparable results to a more complex, physcally based base flow separaton technque under a varety of sol and aqufer propertes characterstc of small watersheds n Oklahoma and North Carolna. Acknowledgment The wrter s grateful to Charles Flowerday for hs edtoral help. The vews, conclusons, and opnons expressed n ths paper are solely those of the wrter and not the Unversty of Nebraska, state of Nebraska or any poltcal subdvson thereof. References Aksoy, H., Bayazt, M., and Wttenberg, H Probablstc approach to modellng of recesson curves. Hydrol. Sc. J., 46 2, Arnold, J. G., and Allen, P. M Automated methods for estmatng baseflow and ground water recharge from streamflow records. J. Am. Water Resour. Assoc., 35 2, Arnold, J. G., Allen, P. M., Muttah, R., and Bernhardt, G Automated base flow separaton and recesson analyss technques. Ground Water, 33 6, Barnes, B. S The structure of dscharge recesson curves. Transact. AGU, 20, JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2004 / 317

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