Newcastle University, Newcastle upon Tyne, United Kingdom

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1 BHS Thid Intenational Sympoium Role o Hydology in Managing Coneuence o a Changing Global Envionment July 2010 Newcatle Univeity, Newcatle upon Tyne, United Kingdom

2 Copyight Bitih Hydological Society ISBN Bitih Hydological Society Thid Intenational Sympoium, Role o Hydology in Managing the Coneuence o a Changing Global Envionment, Newcatle Univeity. Poduced by Celia Kiby on behal o the Bitih Hydological Society Octobe 2010

3 Exploing change in catchment epone chaacteitic: Application o a geneic ilte o etimating the eective ainall and unit hydogaph om an obeved teamlow timeeie B.F.W. Coe National Cente o Goundwate Reeach and Taining, The Autalian National Univeity, Canbea, Autalia Abtact Uncetainty in ainall data can lead to diicultie in model identiication. An appoach to educe the impact o thi ouce o uncetainty ha been developed baed on a baelow ilte compiing a numbe o paallel exponential toe. The appoach give an etimate o the eective ainall (contained to be le than the obeved ainall) and the total unit hydogaph (catchment epone including all low pathway). Thi minimie the impact o uncetainty in the ainall etimate, acilitating moe detailed tudy o the change in the hape o the unit hydogaph though time, om the cale o a ingle event (weathe diven) to longe-cale vaiation (climate diven). The techniue can alo be ued to exploe the elationhip between obeved and eective ainall, a well a pocee not included in many model (e.g. inluence o goundwate abtaction, evapoative lo om ive netwo). The techniue can be ued in ituation whee ainall etimate ae not available, by containing the eective ainall to be zeo i thee ha been no inceae in teamlow om the peviou timetep. Thi aume the ie time o the UH i le than 1 timetep, limiting the ability to detemine the hape o the pea o the UH. Intoduction Rainall-teamlow model typically ue a oil moitue accounting cheme to etimate eective ainall baed on etimated aeal ainall and catchment moitue. Thi i then combined with a unit hydogaph to convet eective ainall into teamlow. The uncetainty in the unit hydogaph i a eult o uncetainty in the input (eective ainall) and the obeved output (teamlow) againt which it i calibated. Fo oil moitue accounting model, the uncetainty in the eective ainall i the eult o uncetaintie in the etimated ainall o the cuent time tep, and the antecedent moitue. The uncetainty in the antecedent moitue i due to uncetainty in the antecedent ainall and evapoation a well a uncetainty in whethe the model tuctue adeuately captue the behaviou o the catchment being tudied. In aeing the uitability o a model tuctue, it i impotant to minimie the noie in the modelled output o that the pat o the obeved output that ae not captued adeuately by the model can be een moe clealy. A one o the main ouce o uncetainty in modelled low i the aeal ainall etimate, an appoach o minimiing the impact that ainall uncetainty ha on the modelled low ha been exploed, baed on a baelow ilte that etimate eective ainall diectly om the obeved teamlow, with the containt that the eective ainall i le than the obeved ainall. Thu the eective ainall i minimally dependent on the etimated ainall. The eult i a modelled low that i not baed on oil moitue accounting, emoving thi component o the model uncetainty om the modelled low. Thi allow the modelle to invetigate moe thooughly the ability o the unit hydogaph (UH) to captue the dynamic o the obeved teamlow data, a well a to exploe the elationhip between the ainall and eective ainall unde the aumption o the adopted UH tuctue. Baelow ilte The popotion o obeved teamlow that deive om baelow can be detemined uing chemical mixing model i uicient wate chemity data ae available, though the uncetainty in the pedicted atio can be conideable (e.g. Adam et al., 2009). In the abence o adeuate chemical data, thee ae eveal method o etimating the baelow component diectly om the obeved low. While ainall teamlow model can be ued to etimate the baelow component; o example, IHACRES (Jaeman et al., 1990, Jaeman and Honbege, 1993), baelow ilte can alo be ued. Such baelow ilte euie ewe paamete than ainall-teamlow model, and ae not dependent on the aumed omulation o the conveion o ainall into eective ainall. A a eult, they potentially have le uncetainty in the etimation o the baelow volume. The implet ae mathematical ilte that do not attempt to epeent the pocee taing place. Thee include the Lyne- Hollic ilte (Lyne and Hollic, 1979), and the Baelow Index (BFI, Gutad et al., 1992). The lac o phyical bai o the Lyne-Hollic ilte euie the ue to detemine when the deived baelow loo ight and doe not inom the ue about the natue o the baelow, but conim hi/he peconceived notion o that natue. The BFI appoach imply attempt to give a lowe envelope o the obeved low, which i then be aumed to coepond to the baelow. An altenative to the BFI i the unning minimum ilte (Coe et al., 2001) which ue a unning minimum ilte o vaiable width (typically 5 day i ued) ollowed by a unning aveage ilte with the ame width. Thi ha the advantage o not being enitive to the tating point, but ha even le phyical bai than the BFI ilte. A moe ophiticated appoach ue an aumed unctional om o the baelow eceion typically an BHS Thid Intenational Sympoium, Managing Coneuence o a Changing Global Envionment, Newcatle 2010 Bitih Hydological Society 1

4 exponential decay. Example o uch om can be ound in Chapman (1999). The implet om i the ilte popoed by Chapman and Maxwell (1996) which ha one paamete (called the C-M ilte hee). The othe ilte dicued by Chapman (1999) wee the two-paamete Boughton ilte (deived om the AWBM model o Boughton, 1993) and the thee-paamete IHACRES ilte (a 2-toe ilte deived om the IHACRES model o Jaeman et al. (1990) and Jaeman and Honbege, (1993). Fuey and Gupta (2001) popoed a new ilte (called F-G hee) baed on thei goundwate dichage model (Fuey and Gupta, 2000). Thee ou ilte can be witten in the ollowing om: (1) (2) whee and ae the uic and low low component, u the eective ainall, α the eceion contant (the ubcipt indicate uic () and low () low component), δ the delay between ainall and it inluence being detected in the team and γ the action o eective ainall that contibute to the baelow and. The aumption made o vaiou ilte ae given in Table 1. Table 1 = (1 + )(1 ) u -, 1 1 u Aumption built into the ilte α γ δ component (ovecome by the evied F-G ilte poviding the additional paamete can be adeuately deined). A ailing in uch ilte i the lac o a containt on the eective ainall. In mot application, thee would be eective ainall (i thi i calculated) o nealy all timetep. Thi limit the phyical bai o the ilte. Coe et al. (2002) developed a geneic omulation o a baelow ilte (baed on the model decibed by Sloan, 2000) whee the eective ainall i contained by eithe the obeved teamlow, o the etimated aeal ainall. The eult i a moe phyically baed model that can be ued to exploe the tuctue o the unit hydogaph (UH) model ued in a ainall-teamlow model, a well a exploing the elationhip between ainall and eective ainall (theeby giving inomation on the wate balance pat o a ainall-teamlow model). Thi pape exploe uing the Coe et al. (2002) ilte o exploing the behaviou o catchment epone, both in tem o the om o the UH and the geneation o eective ainall ued a the input to the UH. Deigning a baelow ilte Auming that teamlow i a combination o baelow () and uiclow () component, then the obeved teamlow () can be witten a: h 1 h h u h u (4) C-M δ Boughton 0 contant δ F-G 0 contant δ +1 IHACRES between -1 and 0 contant δ The F-G ilte wa uthe developed by Fuey and Gupta (2003) to impove it phyical bai by including a oil moitue etimate a an output o the ilte, at the cot o an inceae in the numbe o paamete om thee to ix. Thi enoce a time vaiation in the c1 and c3 paamete, though whethe thi ha the coect unctional om ha not been detemined. In thi pape, only the oiginal F-G ilte will be conideed. The C-M, Boughton and F-G ilte all aume that the uiclow component ha completely dained within the cuent timetep. The IHACRES ilte elaxe thi aumption, uing an exponentially decaying uiclow component. The only dieence between the Boughton ilte and the F-G ilte i the intoduction o a one-timetep delay in the echage, which Fuey and Gupta ague how a lac o claity in the phyical bai in the Boughton ilte. Howeve, while thee may be a delay in the echage with epect to the uiclow component, a timetep could be any ize (depend on the eolution o teamlow data paed though the ilte) and o a one-timetep delay would be a pecial cae. The lac o phyical bai in thi delay doe not mean that the incluion o the delay invalidate the F-G ilte. Rathe, the delay poduce a a moe table ilte. The agument peented o the lac o phyical bai in the IHACRES ilte i alo lawed in that the coect omulation o the uiclow component (uing the ymbol deinition om Fuey and Gupta (2001)) hould be: O j 1 jd1 jd which then ubeuently alte the deived ilte. A poblem common to all the ilte dicued above i the ue o a contant patitioning between uic and low low (3) 2 whee h, h and h epeent the tane unction o uiclow, lowlow and echage epectively, u i the eective ainall (ainall that ultimately become teamlow), i the echage, γ i the action o eective ainall that become ultimately baelow, and * i the convolution opeato: ( a b)() t = a( t )( b t t ) dt. Auming that γ i contant, and h, h and h can be epeented a ingle exponentially decaying toe, then the dicete om o the et o euation in (4) can be witten a: (5) whee δ i the delay between ainall and uiclow epone, δ i the delay between ainall and echage epone, δ i the delay between echage and the baelow epone, and α, α and α ae the eceion contant o uiclow, echage and baelow epectively. Euation 4 can be eaanged to give the baelow timeeie: whee u 1 u u , (7) 1 1 with the paamete o the ilte being α, α, α, γ, δ and ρ. I ρ i le than zeo, then Euation 5 can be ued to calculate the baelow volume o timetep. I ρ = 0, then Euation 5 become: (8) Euation 5 and 7 educe to the Boughton ilte i α = α (6)

5 = δ = ρ = 0; the F-G ilte i α = α = 0 and ρ = -1; and the IHACRES ilte i α = δ = ρ = 0. Fo each ilte, the emaining paamete need to be detemined. Euation 4 can alo be eaanged to give the eective ainall: 1 u 1 1 (9) I ρ i le than zeo, then Euation 9 can be ued to calculate eective ainall. I ρ = 0, then combining Euation 7 and 8 give: u (10) Thi enable the baelow ilte to be contained by eithe the obeved ainall (p) by euiing that u p, o by the obeved low by etting the eective ainall to zeo i < -1 (Thi would mi mall event ollowing lage event, but othewie would give a ueul contain on u ). Thi additional containt give the ilte a tonge phyical bai than the taditional uncontained ilte, at the cot o intoducing the δ paamete (thi can be etimated om the co coelation o ainall with teamlow). While the echage can be included in the ilte omation (Euation 4 and 5), in pactice thi component cannot be identiied om the data, and i uually included a an intantaneou epone. Etimation o paamete value Mot baelow ilte euie the detemination o one o moe paamete value (the exception i the BFI ilte). A dicued above, Fuey and Gupta (2001) developed a method o etimating the paamete om the data, though thee can be conideable uncetainty in ome o the paamete value. One o the advantage o the ilte popoed hee i that, when uing the IHACRES tuctue, by etimating the eective ainall time eie U, the imple eined intumental vaiable (SRIV) calibation techniue (Jaeman, 1979) can be ued to etimate the paamete value o a ilte compiing a modet numbe o exponentially decaying toe. The SRIV etimation method i an unbiaed ecuive etimato that ha the ability to identiy the low low component o the epone cuve at the cot o a modet inceae in computational complexity, but avoiding the complexity o the Reined Intumental Vaiable method (ee Jaeman et al. (1990) o a detailed compaion o the SRIV method with leat uae and othe intumental vaiable method). In thi cae, an iteative pocedue i employed, whee an initial gue at the decay coeicient i made, and the U timeeie deived. The SRIV algoithm i then ued to etimate the decay coeicient uing the timeeie o U. The evied value o the decay coeicient ae then ued to update the U timeeie. The pocedue i iteated until the time contant o the exponential toe convege to within 0.01 timetep. By avoiding the ue o the non-linea module within IHACRES, the uncetainty in the U timeeie i igniicantly deceaed, thu potentially educing the uncetainty in the etimated value o the paamete o the linea module. Application o the ilte The techniue ha been applied to a numbe o gauged ite aco the globe uing data om the Top-Down modelling Woing Goup collection o dataet (tdwg.catchment. og/dataet.html). In Figue 1, the techniue i applied to gauge (Rive Seaton at Tebownbidge, in outh eat England om the National Rive Flow Achive, Figue 1 Gauge (Seaton at Tebownbidge, 39.1 m 2 ) howing ailue to epoduce ome pea due to undeetimation o the ainall with epect to the teamlow. Alo UH ail to epoduce the intemediate low 3

6 Figue 2 Watehed 34 in the Coweeta Expeimental Laboatoy (0.49 m 2 ). Only a ew pea unde-etimated, but evidence o a time vaiable UH. Figue 3 Muindindi Rive, Victoia, Autalia (105 m 2 ). Top panel how the eult when uing modelled low o etimating eective ainall, bottom igue how the impact o uing oecat low (obeved low ued a input athe than the modelled low). 4

7 ac.u/data/na/ive_low_data.html). Thee i a igniicant numbe o event o which the pea low ae not captued, indicating eithe a timing eo (eithe a vaiable delay between ainall and teamlow that i not captued by the model o an eo in the data) o an undeetimation o the ainall with epect to the teamlow o thoe event. In the econd panel, 1 yea o obeved and modelled low ae hown, indicating that the contant coeicient UH model i ove-etimating the ate o decline ollowing mall event, while tending to undeetimating the decline om lage event. Thi ugget a vaiable coeicient model i euied that depend on event magnitude. Figue 2 how the application to watehed 34 in the Coweeta hydologic laboatoy in Noth Caolina, USA (coweeta.uga.edu). Geneally, event pea ae captued moe eliably o thi catchment. The econd panel in Figue 2 how igniicant undeetimation o the decline in low ollowing an event in Octobe 1962, conitent with the impact o highe than uual ainall intenity duing what i peumed to be an intene umme tom. Thi i an example o the impact o uing a coae tempoal eolution (Littlewood and Coe, 2008 and Littlewood, et al., 2010) with daily data not captuing the vaiation in ainall intenity between event. The impact o timing eo can be een in Figue 3. The it panel how the application o the techniue decibed in thi pape to the Muindindi catchment in Victoia, Autalia (gauge ). The econd panel how the eult o including an additional containt on the eective ainall. Thi additional containt i baed on the oecat low at each timetep deived uing the pio obeved low and the etimated eective ainall. The ame appoach can be ued baed on a geneic tane unction appoach. The tane unction o the unit hydogaph i: p a 1 b u a whee p i the oecat low o time tep (e.g. Young, 2001). Compaing p with the obeved low give an etimate o the eective ainall o time tep -δ: u (11) (12) In the econd panel in Figue 3, thi i ued to povide an uppe limit on u -δ obtained by the ilte. In thi cae, the etimate o u -δ wa contained to be le than twice that obtained by Euation 12. The eult i that event o which the ainall i delayed by 1 day appea to be adeuately captued in the top panel (though cloe inpection will how the impact o the timing eo) ae completely mied by the model, poviding a clea indication o event with timing eo. Concluding ema b 0 2 b u p a n m A geneic baelow ilte omulation ha been developed that enable the eective ainall to be etimated. The baelow ilte ha been ucceully applied to a numbe o catchment aco a wide ange o climate, panning Mediteanean to topical climate, and including catchment in cool tempeate climate in Autalia, UK and the USA. By containing the eective ainall by eithe the obeved teamlow o an etimate o the aeal ainall depth, an etimate o the baelow can be made that ha much tonge phyical bai n b m u 5 than peviou baelow ilte. Fo application uing a ingle exponential tem o the baelow, the SRIV algoithm cuently employed in the IHACRES ainall-uno model can be ued to iteatively etimate the paamete value o the baelow ilte. Thi epaate the linea outing module o IHACRES om the non-linea lo module, potentially educing the uncetainty in the etimate o the paamete o the uic and low low component o the unit hydogaph, a well a poviding inomation about eo in the ainall data, and inomation about the patten o eective ainall. By limiting the dependence o the model eult on the ainall, thi appoach can potentially yield a eduction in the uncetainty in the paamete o the UH, a well a poviding a ueul tool o teting the tuctual om o the UH. Reeence Adam, G.A., Conih, P.S., Coe, B.F.W., Hat, M.R., Hughe, C.E. and Jaeman, A.J A new loo at uncetainty in end membe mixing model o teamlow patitioning. In: Andeen, R.S., R.D. Baddoc and L.T.H. Newham (ed), 18th Wold IMACS Conge and MODSIM09 Intenational Conge on Modelling and Simulation. Modelling and Simulation Society o Autalia and New Zealand and Intenational Aociation o Mathematic and Compute in Simulation, July 2009, pp ISBN: og.au/modim09/i3/adam_ga.pd Boughton, W.C A hydogaph-baed model o etimating the wate yield o ungauged catchment, in Hydology and Wate Reouce Sympoium, Intitution o Enginee Autalia, Newcatle, NSW Butaet, W. and Lopez, J.P Bain-cale geohydologic dought low eatue o ipaian auie in the outhen Geat Plain. Wate Reou. Re., 34., Chapman, T.G A compaion o algoithm o team low eceion and baelow epaation. Hydol. Poce., 13, Chapman, T.G. and Maxwell, A I. (1996). Baelow epaation - compaion o numeical method with tace expeiment, in Hydology and Wate Reouce Sympoium, Intitution o Enginee Autalia, Hobat Coe B.F.W., Evan, W.R., Scheide, S. Yu. and Bulle, C Rechage Etimation o Jeabombea Cee Catchment, the Autalian Capital Teitoy. In: F. Ghaemi, P. Whetton, R. Little and M. Littleboy (ed) MODSIM2001, Intenational Conge on Modelling and Simulation, Volume 2, pp Modelling and Simulation Society o Autalia and New Zealand, Decembe, Coe, B.F.W., Smith, A.B. and Jaeman, A.J A One-Paamete Goundwate Dichage Model Lined to the IHACRES Rainall-Runo Model. In: Rizzoli, A.E. and Jaeman, A.J. (ed.), Integated Aement and Deciion Suppot, Poc. 1t Biennial Meeting, Intenational Envionmental Modelling and Sotwae Society, Volume I, Fuey, P.R. and Gupta, V.K Space-time vaiability o low team-low in ive netwo. Wate Reou. Re., 36, Fuey, P. R. and Gupta, V. K. (2001). A phyically baed ilte o epaating bae low om teamlow time eie, Wate Reouce Reeach, 37(11), Fuey, PR. and Gupta, V.K Tet o two phyically baed ilte o bae low epaation. Wate Reou. Re., 39, 1297, doi: /2002wr

8 Gutad, A., Bulloc, A. and Dixon, J.M Low low etimation in the United Kingdom. Intitute o Hydology Repot 108, Intitute o Hydology, UK. Jaeman, A.J Multivaiable intumental vaiable etimato: the choice between altenative. In: R. Iemann, (Ed), Poc. 5 th IFAC Sympoium on Identiication and Sytem Paamete Etimation, Damtadt, Sept Vol 2, Pegamon, Oxod Jaeman, A.J. and Honbege, G.M How much complexity i waanted in a ainall-uno model? Wate Reou. Ree., 29, Jaeman, A.J., Littlewood, I.G.,and Whitehead, P.G Computation o the intantaneou unit hydogaph and identiiable component low with application to two mall upland catchment J. Hydol., 117, Littlewood, I.G. and Coe, B.F.W Data time-tep dependency o conceptual ainall-teamlow model paamete: an empiical tudy with implication o egionaliation. Hydolog. Sci. J., 53, Littlewood, I.G., Young, P.C., and Coe, B.F.W Peliminay compaion o two method o identiying ainall teamlow model paamete inenitive to data time-tep: the Wye at Cen Bwyn, Plynlimon, Wale. Thi volume. Lyne, V.D. and Hollic, M Stochatic time-vaiable ainall-uno modelling. Hydology and Wate Reouce Sympoium, Intitution o Enginee Autalia, Peth Sloan, W.T A phyic-baed unction o modeling tanient goundwate dichage at the watehed cale, Wate Reou. Re., 36, Young, P.C Data-baed mechanitic modelling and validation o ainall-low pocee. In: M.G. Andeon and P.D. Bate (Ed.) Model Validation: Pepective in Hydological Science, Wiley, Chichete