Hydrol. Erth Syst. Sci., 16, 451 455, 2012 www.hydrol-erth-syst-sci.net/16/451/2012/ doi:10.5194/hess-16-451-2012 Author(s) 2012. CC Attriution 3.0 License. Hydrology nd Erth System Sciences Technicl Note: Anlyticl sensitivity nlysis of two prmeter recursive digitl seflow seprtion filter K. Eckhrdt University of Applied Sciences Weihenstephn-Triesdorf, Weidench, Germny Correspondence to: K. Eckhrdt (klus.eckhrdt@hswt.de) Received: 19 Septemer 2011 Pulished in Hydrol. Erth Syst. Sci. Discuss.: 25 Octoer 2011 Revised: 25 Jnury 2012 Accepted: 30 Jnury 2012 Pulished: 13 Ferury 2012 Astrct. A sensitivity nlysis for well-estlished seflow seprtion technique, two prmeter recursive digitl filter, is presented. The sensitivity of the clculted seflow index to errors or uncertinties of the two filter prmeters nd of the initil seflow vlue is nlyticlly scertined. It is found tht the influence of the initil seflow vlue is negligile for long time series. The propgtion of errors or uncertinties of the two filter prmeters into the seflow index is expressed y dimensionless sensitivity index, the rtio etween the reltive error of the seflow index nd the reltive error of the respective prmeter. Representtive index vlues re derived y ppliction of the resulting equtions to 65 North Americn ctchments. In the men the prmeter, the recession constnt, hs stronger influence on the clculted seflow index thn the second filter prmeter mx. This is fvourle in tht cn e determined y recession nlysis nd therefore should e less uncertin. Whether this finding lso pplies for specific ctchment cn esily e checked y mens of the derived equtions. 1 Introduction 1.1 The two prmeter recursive digitl filter The im of seflow seprtion is to distinguish two stremflow components: seflow (groundwter dischrging into the strem) nd quick flow (surfce runoff nd interflow). In the pst, mny seprtion methods hve een proposed, mongst them the two prmeter recursive digitl filter of Eckhrdt (2005), which hs since een pplied in numerous studies, sometimes under the nme of Eckhrdt filter. The eqution of this low-pss filter is k () k 1 + (1 ) mx y k (1) suject to k y k, where is the seflow, y is the stremflow, nd k is the time step numer. The filter hs two prmeters: the recession constnt nd the mximum vlue mx of the seflow index (the long-term rtio of seflow to totl stremflow) tht cn e modelled y the lgorithm. A key question is how errors nd uncertinties in these two prmeters ffect the results of the seprtion. A first ttempt to nswer this question ws the empiricl sensitivity nlysis y Eckhrdt (2005). An empiricl sensitivity nlysis consists of three steps, which re repeted severl times: 1. Input of model (consisting in generl of one or more equtions) is vried. 2. The model is run. 3. The model output is nlysed. One cn lso spek of n experimentl sensitivity nlysis. However, n empiricl sensitivity nlysis is only mkeshift if n nlyticl sensitivity nlysis, tht is n nlyticl clcultion of the error propgtion through the model, is not fesile. In the cse of Eq. (1), such clcultion of the error propgtion is possile nd will e presented in Sect. 2. 1.2 Comprison with other filters Eqution (1) represents whole clss of filter lgorithms which re sed on the widely ccepted liner storge model (Eckhrdt, 2005). Exmples re the lgorithms of Chpmn nd Mxwell (1996) nd Boughton (1993). The filter of Chpmn nd Mxwell (1996) k 2 k 1 + 1 2 y k (2) Pulished y Copernicus Pulictions on ehlf of the Europen Geosciences Union.
452 K. Eckhrdt: Anlyticl sensitivity nlysis of two prmeter recursive digitl seflow seprtion filter is derived from Eq. (1) for mx 0.5. The filter of Boughton (1993) k 1 + C k 1 + C 1 C y k (3) is equivlent to Eq. (1) with C (1 ) mx. (4) Filter lgorithms which rely more on physics hve een presented y Furey nd Gupt (2001) nd Huyck et l. (2005). In the lgorithm of Furey nd Gupt (2001) k is function of k d 1 nd y k d 1. Four prmeters hve to e specified: the time dely d etween precipittion nd groundwter rechrge, the rtio c 1 of overlnd flow to precipittion, the rtio c 3 of groundwter rechrge to precipittion, nd the recession constnt. Required re time series of stremflow nd precipittion. In the lgorithm of Huyck et l. (2005) k is function of k 1, k d, k d 1, y k d, nd y k d 1. Twelve prmeters hve to e specified: d, c 1, c 3, nd nine other prmeters descriing hydrulic chrcteristics nd the shpe of the quifer. Required re not only time series of stremflow nd precipittion, ut lso digitl elevtion model nd informtion on the drinle porosity of the soil. Eqution (1) hs only two prmeters nd requires only stremflow dt. Eqution (1) nd the lgorithms of Furey nd Gupt (2001) nd Huyck et l. (2005) exemplify fundmentl prolem in hydrology: The sounder the physicl sis is, the more complex is the model nd the greter is the numer of its prmeters. Anlyticl expressions my exist for some or ll of the prmeters, yet the prolem of prmeter uncertinty persists. Both Furey nd Gupt (2001) nd Huyck et l. (2005) empiriclly nlyse the impct of prmeter uncertinty on the results of their lgorithm. The incresing physicl reliility does not diminish the need for sensitivity nlysis, ut enhnces it. 2 Anlyticl sensitivity nlysis 2.1 Prmeters nd mx The nlyticl sensitivity nlysis of the filter Eq. (1) requires the clcultion of the prtil derivtives of k with respect to nd mx : () ( k y k ) ( ) 2 (5) mx ( 1) ( k 1 y k ) ( ) 2 (6) (see Appendix A). In the following, the considered model output is the seflow index k y k y where denotes the sum of seflow nd y the sum of stremflow over the whole period of the ville stremflow mesurements. The error propgtion into the model output is descried y the prtil derivtives of with respect to nd mx : mx (7) ( ) 2 ( + 0 n mx y) (8) 1 ( ) 2 [ ( + 0 n ) y] (9) (see Appendix A). In order to get representtive vlues, the filtered hydrogrphs should e long. In this cse the term 0 n in the Eqs (8) nd (9) cn e neglected: mx ( ) 2 ( mx y) (10) 1 ( y). (11) 2 ( ) Now, the question of how n error in the filter prmeter propgtes into the clculted seflow index cn e nswered. Smll errors in cuse n error in of ( ) 2 ( mx y). (12) Correspondingly, smll errors mx in the filter prmeter mx cuse n error in of mx mx mx 1 ( ) 2 ( y ) mx. (13) As mesure for the sensitivity of the seflow index with respect to the prmeters nd mx, dimensionless sensitivity index S is clculted s the rtio etween the reltive error of nd the reltive error of the respective prmeter. The sensitivity index for the prmeter is Hydrol. Erth Syst. Sci., 16, 451 455, 2012 www.hydrol-erth-syst-sci.net/16/451/2012/
K. Eckhrdt: Anlyticl sensitivity nlysis of two prmeter recursive digitl seflow seprtion filter 453 S( ) / () ( mx ) ( ) 2 (14) (see Appendix B). In this nottion, S stnds for sensitivity index, the first symol in the prentheses (here ) indictes the output tht is ssessed, nd the second symol (here ) the uncertin input. Sometimes this dimensionless index is lso clled elsticity index. The sensitivity index for the prmeter mx is S( mx ) mx / mx mx ( 1) ( 1) mx ( ) 2 (15) (see Appendix B). For specific vlues of, mx, nd, the sensitivity indices S( ) nd S( mx ) cn now e clculted nd compred. 2.2 Initil vlue 0 The user of the filter lgorithm hs to choose n initil seflow vlue 0. In the present section, the sensitivity of the seflow index to this vlue will e scertined. Eqution (1) cn e written s k A k 1 + B y k (16) with A () (17) nd B (1 ) mx. (18) The seflow k t the time step k is now trced ck to the initil vlue 0 : k A k 1 + B y k A (A k 2 + B y k 1 ) + B y k A 2 k 2 + A B y k 1 + B y k... A k 0 + B k A i 1 y k i+1. (19) i1 Thus, the prtil derivtive of k with respect to 0 is A k. (20) The prtil derivtive of with respect to 0 is y 1 y A k. (21) For long time series (numer n of oservtions ): 1 y lim n A k. Becuse of A < 1, the limit of the geometric series for n is 1 y A 1 A : A 1 A A k ( ). (22) 1 The longer the time series, the greter is the sum y of the mesured stremflow vlues nd the smller is the right side of Eq. (22). In other words: for long time series the influence of the initil vlue 0 on the seflow index ecomes negligile. 3 Appliction 3.1 Dt nd results An empiricl sensitivity nlysis requires severl runs of the filter over the hydrogrph of specific strem, ech one with different vlues of the two filter prmeters. Susequently, the resulting time series of seflow hve to e nlysed to scertin how the seflow index vries. This finlly llows the clcultion of the sensitivity indices. Alterntively, only one filter run nd clcultion of the seflow index is sufficient, if Eqs. (14) nd (15) re used for ssessing the sensitivity indices. This method hs een pplied to the 65 ctchments whose seflow indices were clculted y Eckhrdt (2008). The results re summrised in Tle 1. Two ctchment types re distinguished: ctchments with perennil strem nd porous quifer, nd ctchments with n ephemerl strem nd porous quifer. Eckhrdt (2005) suggested ttriuting mx vlue of 0.80 to the former nd of 0.50 to the ltter. The recession constnt of ech ctchment ws determined y recession nlysis s descried y Eckhrdt (2008), the respective sensitivity indices were clculted with Eqs. (14) nd (15). 3.2 Discussion The nlyticl sensitivity nlysis shows tht the recession constnt influences the clculted seflow index more thn the filter prmeter mx. In the cse of the ctchments with perennil strem nd porous quifer, for exmple, the vlue S( ) 0.77 signifies tht reltive error of www.hydrol-erth-syst-sci.net/16/451/2012/ Hydrol. Erth Syst. Sci., 16, 451 455, 2012
454 K. Eckhrdt: Anlyticl sensitivity nlysis of two prmeter recursive digitl seflow seprtion filter Tle 1. Results of the nlysis of 65 North Americn ctchments (men vlues nd their stndrd devition). ctchment type numer mx S ( ) S ( mx ) perennil strem, 60 0.970 0.80 0.67 0.77 0.26 porous quifer ±0.001 ±0.01 ±0.07 ±0.01 ephemerl strem, 5 0.961 0.50 0.42 0.42 0.11 porous quifer ±0.004 ±0.04 ±0.22 ±0.03 X percent in cuses reltive error of 0.77 times X percent in. A reltive error of X percent in mx cuses only reltive error of 0.26 times X percent in. This is good news ecuse the recession constnt cn e determined y recession nlysis wheres n optiml mx vlue cnnot e derived from the stremflow mesurements lone. Therefore, the vlue of mx will e more uncertin thn the vlue of. At first glnce, the finding tht the prmeter hs stronger influence on the clculted seflow index thn the prmeter mx seems to contrdict the result of the empiricl sensitivity nlysis of Eckhrdt (2005) which used hydrogrphs of two ctchments not elonging to the pool of the 65 ctchments nlysed for the present pper. For ctchment with perennil strem nd porous quifer, nd ssuming vlues of 0.925 nd mx 0.75, the seflow index ws found to e 0.72 nd sensitivity indices S( ) 0.55 nd S( mx ) 0.96 were clculted (: men vlue of the seflow). For ctchment with perennil strem nd hrd rock quifer, nd ssuming vlues of 0.925 nd mx 0.25, the seflow index ws found to e 0.25 nd the sensitivity indices were S( ) 0.00 nd S( mx ) 0.98. Therefore, the conclusion ws tht mx is the more criticl prmeter. Indeed, this is confirmed if we insert only the two fore-mentioned sets of vlues into Eqs. (14) nd (15). With 0.925, mx 0.75, nd 0.72 one gets S( ) 0.10 nd S( mx ) 0.28. With 0.925, mx 0.25, nd 0.25 one gets S( ) 0.00 nd S( mx ) 0.10. This, however, is oviously nonrepresenttive result. In the cse of the first ctchment, the min reson is tht the vlue 0.925 of the recession constnt, which ws ritrrily choosen s strting point for the empiricl sensitivity nlysis y Eckhrdt (2005), is too smll. If the recession nlysis of Eckhrdt (2008) is pplied, it is found tht the ctul recession constnt for this ctchment is out 0.96. According to Eq. (14), however, the sensitivity index S( ) is the greter the greter is. With 0.925 the sensitivity index S( ) ws underrted. In the cse of the second ctchment, the finding S( ) 0.00 is consequence of the specil sitution tht mx equls. We hve here further rgument for the nlyticl sensitivity nlysis: ecuse it requires less effort thn n empiricl sensitivity nlysis once the Eqs. (14) nd (15) re derived, more ctchments cn e included nd hence more relile conclusion cn e drwn. 4 Conclusions The finding tht mx is the less criticl prmeter in the filter Eq. (1) is fvourle in tht mx cnnot e derived from the stremflow mesurements nd therefore is more uncertin thn the other filter prmeter, the recession constnt. Optiml mx vlues hve to e found y clirtion. Gonzles et l. (2009), for exmple, hve clirted the filter (Eq. 1) y mens of trcer-sed seprtion using dissolved silic nd found n optiml mx vlue of 0.92 for Dutch ctchment. Eckhrdt (2005) suggested mx 0.80 for such ctchment with perennil strem nd porous quifer. Thus there my e n uncertinty of out 0.15 or 19 % in the filter prmeter mx. The sensitivity index S( mx ) 0.26 indictes tht such n error leds to men error in the clculted seflow index of only 0.26 19 % 5 %. For ctchments with ephemerl strem nd porous quifer, the uncertinty is smller yet. Of course, these vlues only chrcterise men conditions derived for the 65 North Americn ctchments presented y Eckhrdt (2008). The seflow index of specific ctchment cn show nother sensitivity to uncertinties in the filter prmeters. However, this cn esily e checked y mens of Eqs. (14) nd (15), which herewith provide importnt dditionl informtion to this seflow seprtion technique. Hydrol. Erth Syst. Sci., 16, 451 455, 2012 www.hydrol-erth-syst-sci.net/16/451/2012/
K. Eckhrdt: Anlyticl sensitivity nlysis of two prmeter recursive digitl seflow seprtion filter 455 Appendix A Clcultion of the prtil derivtives ( ) k 1 + (1 ) mx y k ( ) k 1 + mx y k 1 ( ) k 1 1 ( ) 2 + mx y k mx 1 ( ) 2 () ( k y k ) ( ) 2 (A1) mx k 1 mx mx + (1 ) y k mx k 1 1 ( ) 2 mx + (1 ) y k 1 ( ) 2 ( 1) ( k 1 y k ) ( ) 2 (A2) y ( ) ( k y k ) ( ) 2 (see Eq. A1) ( ) 2 ( k y k ) ( ) 2 ( + 0 n mx y) (A3) mx ( 1) ( k 1 y k ) ( ) 2 (see Eq. A2) 1 ( ) 2 ( k 1 y k ) 1 ( ) 2 [ ( + 0 n ) y]. (A4) Appendix B Clcultion of the sensitivity indices S( ) / () ( mx y) y ( ) 2 (see Eq. 12). With y (Eq. 7) one cn lso write S( ) () ( y mx y) y ( ) 2 () ( mx ) ( ) 2 S( mx ) / mx mx mx ( 1) ( y) y ( ) 2 mx mx mx (see Eq. 13) Edited y: N. Verhoest References (B1) ( 1) ( 1) mx ( ) 2. (B2) Boughton, W. C.: A hydrogrph-sed model for estimting the wter yield of unguged ctchments. Hydrology nd Wter Resources Symposium, Institution of Engineers Austrli, Newcstle, 317 324, 1993. Chpmn, T. G. nd Mxwell, A. I.: Bseflow seprtion Comprison of numericl methods with trcer experiments, Hydrologicl nd Wter Resources Symposium, Institution of Engineers Austrli, Hort, 539 545, 1996. Eckhrdt, K.: How to construct recursive digitl filters for seflow seprtion, Hydrol. Process., 19, 507 515, 2005. Eckhrdt, K.: A comprison of seflow indices, which were clculted with seven different seflow seprtion methods, J. Hydrol., 352, 168 173, 2008. Furey, P. nd Gupt, V. K.: A physiclly sed filter for seprting se flow from stremflow time series, Wter Resour. Res., 37, 2709 2722, 2001. Gonzles, A. L., Nonner, J., Heijkers, J., nd Uhlenrook, S.: Comprison of different se flow seprtion methods in lowlnd ctchment, Hydrol. Erth Syst. Sci., 13, 2055 2068, doi:10.5194/hess-13-2055-2009, 2009. Huyck, A. A. O., Puwels, V. R. N., nd Verhoest, N. E. C.: A se flow seprtion lgorithm sed on the linerized Boussinesq eqution for complex hillslopes, Wter Resour. Res., 41, W08415, doi:10.1029/2004wr003789, 2005. www.hydrol-erth-syst-sci.net/16/451/2012/ Hydrol. Erth Syst. Sci., 16, 451 455, 2012