An innovative use of observations to alleviate weighted-residual asymmetry

Transcription

1 MODFLOW and Moe 2006: Managng Gound-Wate Sstems - Confeence Poceedngs, Poete, Hll, & Zheng - An nnovatve use of obsevatons to allevate weghted-esdual asmmet Glbet Bath, Ph.D. S.S. Papadopulos and Assocates, gbath@sspa.com, Boulde, CO USA ABSTRACT Estmaton of flow o tanspot paametes often nvolves obsevatons such as flow o concentatons that ma be sgnfcant ove multple odes of magntude, tpcall dctatng the use of weghts nvesel popotonal to the poduct of a coeffcent of vaaton multpled b the obsevaton. Ths allows obsevatons of consdeabl dffeent magntude to have smla mpotance n tems of gudng the paamete-estmaton pocess. Howeve, f a smulated value s of smalle absolute magntude than the obseved, the weghted esdual wll be lmted to the nvese of the coeffcent of vaaton, whle weghted esduals of smulated values geate than the obseved ae unbounded. Ths poduces asmmet n the potental contbuton to the objectve functon fom smulated values whose magntudes ae less-than and geate-than the obseved. Ths wok uses a smple tansfomaton to ceate a supplemental obsevaton set. The combned obsevaton set s used to demonstate an objectve functon wth mpoved chaactestcs: ncludng the tansfomed obsevatons esults n a balanced set of objectve-functon contbutons fom smulated values smalle-than and geate-than the obsevaton magntude. The potental fo mpovng the pocess of paamete estmaton, geneatng senstvtes, as well as pecautons fo ntepetng fnal statstcs, ae consdeed. INTRODUCTION Successful paamete estmaton eles on man factos ncludng a set of obsevatons that povde nfomaton suffcent to detemne appopate paamete-value adjustments that mpove the ft between smulated and obseved values. Fo a vald egesson, weghts need to be popotonal to one dvded b the vaance of the obsevaton eo, σ 2, [Dape and Smth, 1998, p. 222] n ode to eflect the uncetant of the measuement. Pope weghtng allows obsevatons of dffeent unts, phenomena, and a wde ange of magntude to be combned, povdng smultaneous feedback fom the ente set of obsevatons on changes n the ft between smulated and obseved values. Weghts can tpcall be lumped nto one of two categoes: fxed-value o vaable-value weghts, FVW and VVW, espectvel. These tems eflect the absolute and elatve natue of the weghts. FVWs tpcall eflect the uncetant of a measuement devce and ae not consdeed n ths wok. VVWs ae elatve to the obsevaton: the ae popotonal to a value, tpcall the obseved value. Obseved values sgnfcant ove a lmted ange, such as goundwate heads, tpcall use FVW. Obsevatons equng VVW nclude concentatons and flows, when the ae sgnfcant ove multple odes of magntude. Ths wok povdes an appoach fo addessng one of the ssues commonl assocated wth VVW appled to obsevatons sgnfcant ove multple odes of magntude: the weghts esult n a lmted magntude of weghted esdual fo postve esduals (obseved - smulated > 0). As a esult, an automated paamete estmaton outne wll tend to adjust paametes to mpove fo an negatve esduals wth lttle, f an egad, fo the postve esduals. Examned ndvduall, esduals fom an obsevaton would povde an ndcaton of the pope paamete adjustment to mpove the ft between the smulated and obseved values. Howeve, t s qute possble fo a few lage weghted esduals, fom negatve esduals, to domnate the objectve functon so that the paamete estmaton algothm accepts the lmted penalt assocated wth man postve esduals that poduce a elatvel mno contbuton to the objectve functon. Pevous wok [Bath and Hll, 2005a and b] exploed ths and othe ssues assocated wth VVW and demonstated pecautons to mpove potental fo successful paamete estmaton and senstvt analss when obsevatons ae sgnfcant ove multple odes of magntude. Ths wok goes beond pevous effots b poposng the use of a smple tansfomaton to ceate addtonal obsevatons fo supplementng the ognal obsevaton set. Usng a smple one-obsevaton example ths wok demonstates the potental benefts of the supplemented obsevaton set: the 305

2 MODFLOW and Moe 2006: Managng Gound-Wate Sstems - Confeence Poceedngs, Poete, Hll, & Zheng - supplemented obsevaton set esults n a moe balanced set of weghted esduals wth espect to postve vesus negatve esduals, can be used to povde a moe appopate set of senstvtes, and elmnates one souce of paamete-estmate bas. A small hpothetcal set of obseved and smulated values ae also used to examne the benefts of supplementng the obsevaton set. Fnall, some consdeatons egadng obsevaton edundanc ae dscussed. Paamete Estmaton and Weghtng METHODS Fo ths wok t s assumed that paametes ae estmated b quantfng the ft between obseved and smulated values, mnmzng wth espect to the paamete values usng a weghted least-squaes objectve functon, S(b ), Equaton (1). The weght matx, ω, and obseved- and smulated-value vectos, and (b ), espectvel, nclude tems fo all the T [ ] [ ( b )] ( b ) ( b ) obsevatons, b s the paamete vecto, and the subscpt ndcates the paamete estmaton teaton numbe. Ths wok assumes that the objectve functon s mnmzed usng a modfed Gauss-Newton method [e.g., Sebe and Wld, 1989; Sun, 1994] as descbed n Hll [1998], howeve the appoach of supplementng obsevatons should beneft a wde ange of objectve-functon-mnmzaton appoaches. When usng VVWs a common appoach s to make the 1 weght nvesel popotonal to the obsevaton ω = whee = 1... ND 2 (2) multpled b a coeffcent of vaaton (Eqn. 2). In (2) ( Cv ) ND s the numbe of obsevatons wth weghts popotonal to the value, Cv s the coeffcent of vaaton, and should be the tue value of the obsevaton. Seveal ssues must be consdeed when applng ths appoach, ncludng ncopoaton of detecton lmts to avod ve small obsevatons, and that s tpcall unknown and appoxmated usng obseved [e.g., Kede and Rosbjeg, 1991] o smulated values [e.g., Wagne and Goelck, 1986]. The smple, one-obsevaton example does not eque a detecton lmt, but a detecton lmt was used n the example dataset examned n ths wok. Ths wok uses obseved values fo n (2). In ths wok the esdual s defned as the obseved mnus the smulated value. The weghted esdual on the th obsevaton (WR ) s defned n (3). Assumng nomalzed values angng fom 0 to 1.0, when ethe ( ) ( ) WR 1 2 = ω = (3) the smulated o obseved value s consdeabl lage Cv than the othe, the absolute value of the esdual appoaches 1.0 and the weghted esdual appoaches WR ω = (4) Cv the squae oot of the weght (4). As a esult, fo obsevatons sgnfcant ove N odes of magntude, thee would tpcall be a dffeence of about N odes of magntude fo postve vesus negatve weghted esduals [Bath and Hll, 2005a]. Fo ths wok the hpothetcal examples nclude flows and concentatons angng ove multple odes of magntude. Fo example, flow obsevaton would be the total volume that flowed n a steam ove the gaton stess peod, e.g., 10 9 ft 3. Values could cove a wde ange dependng on the tpe, unts, and othe factos. Fo smplct the fst example, a sngle-obsevaton example, consdes the stuaton whee the obseved value has been nomalzed so that t equals 1.0. Usng (2) and Cv = 0.1, Fgue 1 shows that the magntude of WR anges fom about 10, fo postve weghted esduals whee s less than about 0.1, to an unbounded amount fo negatve weghted esduals, as gows lage than 10. The use of a log scale s mpotant to the ntepetaton of the esults depcted n Fgue 1. A lnea scale would smpl show an dentcal slope fo both the negatve and postve weghted esduals. Howeve, the obsevaton s sgnfcant ove multple odes of magntude and as a esult, the lnea scale wll tend to mask the fact that postve weghted esduals do not poduce smla nceases n weghted-esdual magntude fo each addtonal ode of magntude devaton fom the obseved value. The log scale, havng ode of magntude ncements, has the capact to coectl dspla and conve ths ssue. S = ω (1) 306

3 MODFLOW and Moe 2006: Managng Gound-Wate Sstems - Confeence Poceedngs, Poete, Hll, & Zheng - Supplemental Obsevatons To elmnate the asmmet depcted n Fgue 1 ths wok poposes a smple tansfomaton of the ognal obsevatons, and appendng the tansfomed values to the ognal obsevaton set. The tansfomaton s smpl takng the multplcatve nvese of each obsevaton,, to ceate addtonal obsevatons, z : z = 1/( ). These new obsevatons, efeed to as nvese obsevatons, ae appended to the exstng set, esultng n a set of 2ND obsevatons. The mplcatons of usng obsevatons twce ae dscussed below. 1.E+03 1.E+01 1.E+00 1.E Log Smulated Value () Fgue 1. Weghted esdual magntudes usng Weghts on z ae smpl calculated n the same smulated, WR, and nvese smulated manne as. The weghted esduals, WRz n WRz values, whee = 1, Cv = 0.1 Equaton (5), ae qute smla to WR. Compason of (3) and (5) eveals that the net esult of addng nvese obsevatons s to poduce addtonal weghted esduals that ae opposte n sgn, and ae nvesel WRz = popotonal to the smulated value, as opposed to the 1 = (5) Cv Cv obseved value. As dscussed below, beng nvesel popotonal to the smulated value s one of the fundamental benefts of usng the nvese obsevatons. Weghted-Resdual Magntude RESULTS AND DISCUSSION WR WRz Fgue 1 also shows the weghted esdual fom the nvese obsevatons (WRz), as a functon of smulated value. Fo nvese obsevatons and Cv = 0.1, negatve weghted esduals have a maxmum value of 10, whle postve weghted esduals ncease wthout bound. The ognal and nvese obsevaton sets complement each othe: the ognal obsevatons poduce sgnfcant weghted esduals fo negatve esduals, whle the nvese-obsevaton weghted esduals ae sgnfcant fo postve esduals. Whle ths does not elmnate the ssue of edundant use of obsevatons, t at least suggests that the contbuton of the ognal and nvese obsevatons have onl mno ovelap fo an gven smulated value. Fgue 2 povdes an example fom a hpothetcal-scenao dataset that ncludes flows and concentatons. Fgue 2a shows the smulated, obseved, and weghts fo the ognal set and the nvese set. Obsevatons 1 5 ae flow n a dan ove the couse of 5 stess peods wth values vang between values of ~10 3 to 10 12, epesentng flow fom the dan dung the successve non-gaton and gaton seasons. The smulated values fluctuate between 10 7, and 10 9, attemptng to epesent low flows n the non-gaton season, and hgh flows dung the gaton season. Obsevatons 6 10 epesent concentatons, n the ange fom 10-5 to The smulated equvalents ae elatvel consstent aound 10-7 o Note that, snce weghts ae nvesel popotonal to the values, the weghts fo the ognal obsevatons ae adjacent to the nvese obsevatons and the weghts fo the nvese obsevatons ae adjacent to the obsevatons. Fgue 2b shows the esdual magntudes fom the ognal and nvese obsevatons, and how the oughl tack each othe. The weghted esduals n Fgue 2c povde a good example of the combned potental of the ognal and nvese obsevatons. Wheneve WR s small due to postve weghted esdual, WRz becomes lage, and vce vesa. When both WR and WRz ae small then the smulated value eall s gettng close to the obseved value. In ths wa, and z complement each othe, povdng moe balanced feedback though the objectve functon egadless of whethe and z ae too lage o too small. 307

4 MODFLOW and Moe 2006: Managng Gound-Wate Sstems - Confeence Poceedngs, Poete, Hll, & Zheng - Geneatng Dmensonless Scaled Senstvtes Whle the combnaton of ognal and nvese obsevatons povdes mpoved potental fo paamete estmaton, ths wok advocates geneatng dmensonless scaled senstvtes (dss) [Hll, 1998] fom the nvese obsevatons alone. Equaton (5) demonstates that nvese obsevatons have weght esduals nvesel popotonal to the smulated value so that dss of the nvese obsevatons have the same benefts as smulated value weghts (SVW) [Bath and Hll, 2005a]. The dss SVW povde a bette epesentaton of the mpact of changes n the paamete on the smulated values [Bath and Hll, 2005a]. Usng onl the nvese obsevatons to geneate senstvtes also elmnates an potental ssues of data edundanc fo the senstvtes. An Appoach fo Usng the Invese Obsevatons Pevous wok [Andeman and Hll, 1999] has demonstated that weghts based on obsevatons ma poduce a based set of paamete estmates whle weghts based on the smulated values poduce unbased paamete estmates. Andeman and Hll [1999] pont out that usng smulated values fom the stat of a paamete estmaton un could be poblematc, and adopted an appoach that stated wth obseved-value weghtng, and swtched to smulated-value weghtng as paamete estmate changes deceased. Ths appoach s documented n the UCODE manual [Poete and Hll, 1998]. Swtchng allows the paamete estmaton to ntall wok wth a consstent (a) 1.E+14 Values 1.E+10 1.E+06 1.E-02 1.E-06 1.E-10 1.E-14 (b) 1.E+14 Resdual Magntude 1.E+10 1.E+06 1.E-02 1.E-06 1.E-10 1.E-14 (c) 1.E+06 Weghted Resdual Magntude 1.E+05 1.E+04 1.E+03 1.E+01 1.E+00 Obsevaton Numbe Fgue 2. (a) Smulated ( and z), obseved ( and z), and weghts (W and Wz), (b) esduals, (c) weghted esduals. set of values and weghts when paametes ae lkel to change sgnfcantl. As the pe-teaton paamete-value changes decease, SVWs ae used to elmnate the potental souce of bas. Togethe, the asmmet of weghted esduals demonstated n ths wok and bas potental dentfed b Andeman and Hll [1999] suggest the need fo modfng the appoach to obsevatons and weghtng. The poposed appoach s as follows o Stat wth the supplemented obsevaton set, whch poduces smmetc weghted esduals about the obseved values. o Dop the ognal obsevatons, leavng onl the nvese obsevatons whch do not have the potental fo a weghtng-nduced bas of the paamete estmates, based on one of the followng decson mechansms: o the pe-teaton changes n paamete values have become small, o o the esdual magntudes have become small. z z W Wz - z-z WR WRz 308

5 MODFLOW and Moe 2006: Managng Gound-Wate Sstems - Confeence Poceedngs, Poete, Hll, & Zheng - Ths appoach ncopoates the balanced weghted-esdual feedback of the supplemented obsevaton set dung ntal teatons and then elmnates the potental paamete-estmate bas b usng onl the nvese obsevatons fo the fnal teatons. In addton, swtchng to nvese obsevatons, as opposed to smulated values, fo the fnal teatons avods the need to geneate new weghts at each paameteestmaton teaton. Ongong eseach focuses on developng ntutve test cases fo demonstatng nvese-obsevaton mpact and testng the two decson mechansms: pe-teaton paamete-estmate changes o esdual magntudes. Addtonal wok ncludes efnement of the esdual-magntude decson mechansm, compang outcomes usng (1) the maxmum esdual, (2) a nom of esduals, (3) esduals of ndvdual obsevatons, o (4) compason of postve and negatve esduals. CONCLUSIONS The demonstated benefts of supplementng the obsevaton set wth nvese obsevatons nclude (1) smmetc weghted esduals, (2) an altenatve method of poducng a moe appopate set of dmensonless scaled senstvtes, and (3) the possblt of unbased paamete estmates. These benefts advocate the development of a complete test case and efnement of the decson mechansm, whch s pat of the ongong eseach. Whle obsevaton-data edundanc s stll a potental ssue, t seems clea that thee ae was to ncopoate the benefts of the nvese obsevatons, and stll geneate vald statstcal summaes when needed. ACKNOWLEDGEMENTS Comments and suggestons fom Steffen Mehl ae geatl appecated. REFERENCES Bath, G.R., Hll, M.C., 2005a. Numecal methods fo mpovng senstvt analss and paamete estmaton of vus tanspot smulated usng soptve eactve pocesses. J. Contam. Hdol. 76, Bath, G.R., Hll, M.C., 2005b. Paamete and obsevaton mpotance n modelng vus tanspot n satuated poous meda nvestgatons n a homogeneous sstem, J. Contam. Hdol. 80, Dape, N.R., Smth, H., Appled Regesson Analss, 3d ed. John Wle and Sons, New Yok. Hll, M.C., Methods and gudelnes fo effectve model calbaton, U.S. Geologcal Suve Wate Resouces Investgaton Repot, , 90 p. Kede, A., Rosbjeg, D., A compason of fou nvese appoaches to goundwate flow and tanspot paamete dentfcaton. Wate Resou. Res. 27 (9), Poete, E. P. and M. C. Hll, Documentaton of UCODE, a compute code fo unvesal nvese modelng, USGS Wate Resouces Investgatons Repot , 123 p. Sebe, G.A.F., Wld, C.J., Nonlnea Regesson. Wle, New Yok, NY. 768 pp. Sun, N-Z, Invese Poblems n Goundwate Modelng. Kluwe Academc Publshes, The Nethelands. Wagne, B.J., Goelck, S.M., A statstcal methodolog fo estmatng tanspot paametes: theo and applcatons to one-dmensonal advectve dspesve sstems. Wate Resou. Res. 22 (8),