State Upgrades and Natural Rate of Cities

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1 Rsk n Water Resources Management (Proceedngs of Symposum H03 held durng IUGG20 n Melbourne, Australa, July 20) (IAHS Publ. 347, 20). 07 Assmlaton of streamflow dscharge nto a contnuous flood forecastng model YUAN LI,2, DONGRYEOL RYU, Q. J. WANG 2, THOMAS PAGANO 2, ANDREW WESTERN, PRASANTHA HAPUARACHCHI 2 & PETER TOSCAS 3 Department of Cvl and Envronmental Engneerng, The Unversty of Melbourne, Parkvlle, Vctora 300, Australa y.l40@pgrad.unmelb.edu.au 2 CSIRO Land and Water, Hghett, Vctora 390, Australa 3 CSIRO Mathematcs Informatcs and Statstcs, Clayton, Vctora 368, Australa Abstract Four state updatng schemes are explored to ntegrate the observed dscharge data nto a flood forecastng model. Hourly streamflow dscharge measured n the Ovens Rver catchment, Australa, s assmlated nto the Probablty Dstrbuted Model (PDM) usng the ensemble Kalman flter. The results show that the overall forecast accuracy mproves when the dscharge observatons are ntegrated, manly due to better ntalsaton of the model. Settng error covarance proportonal to each state varable gves better results than settng error covarance as a constant value. Updatng routng states of PDM affects dscharge predcton nstantly, whle the effect of sol mosture updatng results n a lagged response n dscharge leadng to a poorer update performance. However, durng the forecast lead tme, updatng sol mosture results n slower degradaton of the forecast accuracy, whch s manly because the sol mosture store s the only state nfluencng dscharge volume, whle the routng storages only descrbe the flow delay. Key words dscharge assmlaton; flood forecastng; ensemble Kalman flter; state updatng INTRODUCTION Flood has destructve mpacts on people and ther lvng envronment. Flood forecastng, wth suffcent lead tme and accuracy, has great sgnfcance for effectve flood warnng and emergency response. However, forecastng models, the core of the quanttatve flood forecastng systems, are stll far from perfect, although they have mproved consderably from conceptual to process-based ones to date. Moreover, useful nformaton exstng n the montorng data collected durng the antecedent perods and any response observed up to the start of the forecast perod need to be utlzed. Therefore, data assmlaton, amng to merge model forecasts and observaton data to reduce forecastng uncertanty, receves growng attenton as a way to further mprove the accuracy of the forecastng system. In general, hydrologcal data assmlaton can be separated nto three categores: error predcton, parameter estmaton, and state updatng (Anctl et al., 2003; Sene, 2008). Error predcton, regarded as a post-processng method, s to correct the errors between model outputs and observed values (Anctl et al., 2003), whle parameter estmaton can adjust parameters to acheve an mproved forecast (Vrugt et al., 2006). Accordng to recent work, state updatng s known to be more effectve for the real-tme forecastng process (Clark et al., 2008; Komma et al., 2008; Sene, 2008; Seo et al., 2009; Threl et al., 200), and there s some research showng the benefts of updatng states and parameters smultaneously (Moradkhan et al., 2005; Vrugt et al., 2006; Salamon & Feyen, 2009). State updatng nvolves assmlatng avalable observatons nto forecastng models to mprove the overall predcton. Gauged dscharge data are effectve and commonly preferred for assmlaton, snce the purpose of real-tme updatng n flood forecastng s to obtan better dscharge forecasts. Whle sol mosture and evapotranspraton assmlatons are also becomng ncreasngly mportant, partcularly n other areas of hydrology, they are stll not easy to effectvely use n operatonal flood forecastng. There are varous knds of state updatng methods. The most wdely used methods are sequental data assmlaton and varatonal data assmlaton. Varatonal data assmlaton s essentally a smoothng method that requres a set of observatons durng a perod of tme, whle sequental data assmlaton, such as the Kalman flter (KF), s more sutable for real-tme updatng as t can adjust state varables whenever a new observaton s Copyrght 20 IAHS Press

2 08 Yuan L et al. avalable. The extended Kalman flter (EKF) and the ensemble Kalman flter (EnKF) are two wdely-used sequental methods, adapted from the standard Kalman flter for nonlnear dynamc systems. Both requre estmates of model and observaton error to determne the optmal combnaton of model and observatons. The EKF estmates model error by propagatng error covarance matrx usng a lnearzed operator (Clark et al., 2008). Ths approach s neffcent for complex systems and unstable when appled to hghly nonlnear models. On the other hand, the EnKF does not requre lnearzaton of the models to propagate ensemble states and to estmate model error covarance at updatng tme steps (Evensen, 994). It has proven to be relatvely effcent and sutable for complex nonlnear models. There have been a number of streamflow forecastng studes based on the EnKF n the past decade (Moradkhan et al., 2005; Vrugt et al., 2005, 2006; Weerts & El Serafy, 2006; Andreads et al., 2007; Clark et al., 2008; Komma et al., 2008; Pauwels & De Lannoy, 2009). These studes argue that, despte the more detaled specfcaton of physcal processes n process-based models, smple models can run more effcently and the state updatng can be mplemented more easly wth them. In fact, current flood forecastng systems used n most countres are manly conceptual models (Sene, 2008). Although the conceptual models are rather smple compared to the physcally-based hydrologcal models, ntegratng state updatng nto the models s stll challengng because of some techncal ssues. One of the most notable problems s to quantfy error covarance and select approprate state varables to update. In ths paper we test four updatng schemes to demonstrate the mpacts of dfferent error quantfcaton and choce of states to update on real-tme forecastng. DATA AND METHODS Data The study area s the Ovens Rver catchment upstream regon of the Myrtleford observaton staton n northeast Vctora, Australa. The catchment area s 20 km 2 and the stream channel network s about 8000 km long. The catchment drans the northern slopes of the Great Dvdng Range. Data used n ths paper nclude hourly averaged precptaton from the rangauges, monthly averaged potental evapotranspraton, and hourly dscharge at the catchment outlet. Wth these data, model parameter calbraton s carred out from January 999 to 2 July 2004 usng the Shuffled Complex Evaluaton (SCE) algorthm (Duan et al., 992). State updatng s carred out for two contnuous flood events from 22 July to 2 August 2004 usng EnKF. Based on the updated state varables, the forecastng s mplemented for the next two flood events from 3 to 25 August All the data used are contnuous observatons. P Drect runoff Surface storage E S 2 S 22 q s Surface runoff Probablty-dstrbuted sol mosture storage S Recharge q q b Baseflow S 3 Groundwater storage Fg. The PDM model wth two lnear storages for surface runoff routng.

3 Assmlaton of streamflow dscharge nto a contnuous flood forecastng model 09 PDM model The Probablty Dstrbuted Model (PDM) s a conceptual ranfall runoff model whch calculates the sol mosture storage by a spatal probablty dstrbuton and routes the surface flow and the subsurface flow by lnear and nonlnear routng methods, respectvely (Moore, 2007). In ths paper, we use a cascade of two lnear storages as the surface routng model and a cubc nonlnear storage as the groundwater routng model. The model structure s shown n Fg.. More detaled descrpton of the PDM model can be found n Moore (2007) and Srkanthan et al. (2008). EnKF When usng the ensemble Kalman flter for sequental state updatng, there are two man steps: model predcton and state updatng. In the model predcton step, state varable ensembles are propagated wthn the state space and nto the observaton space through the followng two equatons: x θ + ω, =,... n () + t+ = f ( xt, ut, ) ˆ t y t+ = h( xt+, θ) (2) where xt+ s the predcted state ensemble, x + t s the updated ensemble, u t s the nput data vector, θ s the model parameter vector, ω t s the error of model wth a Gaussan probablty dstrbuton, yˆ t + s the predcton vector, t s the tme step and s the ensemble member number. In the state updatng step, predcted state ensembles are updated as the followng equaton: + = x + K y yˆ ) (3) x + t+ t+ ( t+ t+ y where t + s the observaton vector generated by addng Gaussan nose representng the observaton error, K t+ s the Kalman gan matrx whch can be calculated by the followng equaton: xy yy y t+ = Σt+ [ Σt+ + Σt+ ] K (4) where Σ xy yy t+ s the cross covarance matrx of the state varables and predcton, Σ t+ s the error covarance matrx of the predcton, and Σ y t+ s the observaton error covarance matrx. Equatons (3) and (4) show the method of mplementng the dfference between dscharge observaton and predcton to update state varables n ths paper. For more detals about the applcaton of EnKF n hydrologcal forecastng, refer to Moradkhan et al. (2005) and Srkanthan et al. (2008). RESULTS AND DISCUSSION As mentoned above, among the key nformaton requred for optmal state updatng are the errors n model predctons. Model uncertanty comes from nput data, ntal states, model parameters and model structure. Errors from these parts wll transfer nto the state varables and then to the output data of subsequent tme steps as the model runs forward. Therefore, n usng the ensemble Kalman flters we can update state varables of the current tme step by perturbng state varables and sometmes also nput ranfall data. Updatng all the state varables s possble but one thng we should notce s that the error of ranfall runoff states, such as sol mosture, wll transfer nto, and may be the man source of catchment routng states. Thus, as mentoned n the PDM model (Moore, 2007), many applcatons only focus on updatng sol mosture or catchment routng states. The forecastng model used n ths paper has four state varables: the sol mosture storage S; the surface routng storage S2 and S22; and the subsurface routng storage S3. In order to

4 0 Yuan L et al. demonstrate the mpact of the dfferent state varables and state error quantfcaton on updatng performance, we desgned four updatng schemes: () update S22 and S3 wth a fxed magntude of error; (2) update S22 and S3 wth varyng errors as a specfed fracton of the state varables; (3) update S wth a fxed magntude of error; and (4) update S wth varyng errors as a specfed fracton of the varable. The standard devatons are shown n Table. Table Standard devatons of state varables for perturbaton. Standard devaton Scheme Scheme 2 Scheme 3 Scheme 4 S (mm) 0 S 0% S22 (mm) 0.8 S22 0% S3 (mm) 0. S3 0% S, sol mosture; S22, the second surface routng storage, S3, subsurface routng storage. The values of the states used n Scheme 2 and Scheme 4 are the mean of ensemble states. Model performance durng updatng perods Fgure 2 demonstrates the results of four contnuous updatng schemes. Wthn less than 20 hours, the predctons wth updatng of all the four schemes reach a hgh precson compared wth the observatons. Schemes and 2 outperform schemes 3 and 4. However, ths s not surprsng as S22 and S3 are the state varables that drectly determne the surface dscharge and baseflow dscharge, whle S only affects dscharge through routng storages. Ths s also why the predctons of updatng scheme and scheme 2 mprove faster than scheme 3 and scheme 4 and why updatng sol mosture results n lags durng peak perods. From the left panel of Fg. 2, we can see that scheme 2 performs well durng the whole perod, whlst scheme s worse durng peak perods. Ths s manly because of the assumed constant error of S22 and S3 n scheme. Durng the peak perods, the perturbatons of states are not suffcent for the predcton to be adjusted fully to the observatons n scheme. The predcatons of scheme 3 and scheme 4 have lttle dfference whch s mostly because the sol mosture does not vary much durng the whole updatng perods. Fg. 2 The contnuous updatng performance. Obs, observed dscharge; Sm, smulated dscharge wthout updatng; Scheme to Scheme 4: the smulated dscharge usng the four updatng schemes n turn. Forecast based on state updatng As noted before, quantfyng state errors as certan ratos of state varables s more ratonal and gves better results n the updatng perod. Consequently, usng the updated state varables

5 Assmlaton of streamflow dscharge nto a contnuous flood forecastng model obtaned from updatng scheme 2 (forecast ) and scheme 4 (forecast 2), we ran forecasts for 24 days that nclude the next two runoff events (Fg. 3). Observed ranfall was used to represent the forecast ranfall, whch assumes the ranfall forecastng s as accurate as the observatons of ranfall. The results show that durng the subsequent days, the mprovement from updated ntal states s sgnfcant but the benefts decrease as lead tme ncreases. The benefts from routng storages updatng decrease more rapdly compared wth the sol mosture updatng, even though updatng routng storages affects the predcted dscharge more drectly and gves better analyss results than updatng sol mosture (Fg. 2). Ths s probably partly because there s shorter memory n the routng storages, partcularly the surface routng, whch s mportant durng events, compared wth the sol mosture storage. It s also partly due to the sol mosture havng a drect mpact on the volume of water generated durng forecast events, whlst routng storages are only used to descrbe the delay of water flow and the effects could be drect but short. Thus updatng routng states wthout updatng sol mosture wll not provde much beneft at longer forecast lead tmes. The Fg. 3 Short tme forecasts based on state updatng. Obs, observed dscharge; Sm, smulated dscharge wthout updatng; Forecast, forecast dscharge based on updatng scheme 2; Forecast 2, forecast dscharge based on updatng scheme 4. Fg. 4 Short tme forecasts based on state updatng usng perturbed ranfall observatons. Forecast 3 (left panel): forecast dscharge based on updatng scheme 2 usng perturbed ranfall; Forecast 4 (rght panel): forecast dscharge based on updatng scheme 4 usng perturbed ranfall.

6 2 Yuan L et al. model forecastng performance ndcates that for flood forecastng purposes, updatng sol water states s more mportant than updatng routng states. In order to smulate the real-tme forecastng as realstcally as possble, we used the perturbed ranfall to forecast dscharge based on state updatng schemes 2 and 4, resultng n forecast 3 and forecast 4, respectvely. The standard devatons were set to be 30% of ranfall observatons and a zero mean temporally ndependent normally dstrbuted error was added to the observed hourly ranfall. As shown n Fg. 4, the degradaton of the nput data nfluences the forecasts to some extent. The degradaton of the ranfall nput had surprsng lttle mpact on the qualty of the forecast n both cases. The results ndcate that the choce of update scheme has more mpact than the ranfall forecase error; however, ths may not hold wth more realstc forecast errors that could have a bas over a whole event. The slght mprovement n forecast 4 compared wth forecast 2 s probably a chance outcome assocated wth the partcular realsaton of ranfall errors. CONCLUSION Wth the contnuous catchment observaton data we calbrated the PDM model usng SCE. Usng EnKF, we tested four contnuous state updatng schemes. Based on the updated state varables, we ran the model for streamflow forecastng and analysed the mpacts of dfferent updatng schemes and ranfall perturbaton schemes on forecastng results. The results show that contnuous state updatng can sgnfcantly mprove the model performance. Updatng routng states affect dscharge predcton drectly, whle the effect of sol mosture updatng has a tme lag, whch mples that updatng the sol mosture may only not result n the prompt mprovements of dscharge as much as t s seen for the routng states updatng. Settng error covarance as a rato of each state varable for state perturbaton gves better results than settng error covarance as a constant value. Despte the tme lag, sol mosture updatng gves better forecasts after the updatng perod than routng states updatng. That s manly because the sol mosture store s the only state nfluencng runoff volume, whle the routng storages only descrbe the flow delay. When usng the perturbed precptaton for forecastng, the error of ranfall wll add more uncertanty to forecasted dscharge but state updatng stll makes sense for the forecastng wthn the lead tme of a few days. REFERENCES Anctl, F., Perrn, C. & Andreassan, V. (2003) Ann output updatng of lumped conceptual ranfall/runoff forecastng models. J. Am. Water Resour. Assoc. 39(5), Andreads, K. M., Clark, E. A., Lettenmaer, D. P. & Alsdorf, D. E. (2007) Prospects for rver dscharge and depth estmaton through assmlaton of swath-altmetry nto a raster-based hydrodynamcs model. Geophys. Res. Lett. 34(0), L0403, do:0.029/2007gl Clark, M. P., Rupp, D. E., Woods, R. A., Zheng, X., Ibbtt, R. P., Slater, A. G., Schmdt, J. & Uddstrom, M. J. (2008) Hydrologcal data assmlaton wth the ensemble Kalman flter: Use of streamflow observatons to update states n a dstrbuted hydrologcal model. Adv. Water Resour. 3(0), Duan, Q., Sorooshan, S. & Gupta, V. (992) Effectve and effcent global optmzaton for conceptual ranfall runoff models, Water Resour. Res. 28(4), Evensen, G. (994) Sequental data assmlaton wth a nonlnear quas-geostrophc model usng Monte Carlo methods to forecast error statstcs. J. Geophys. Res. 99(C5), Komma, J., Bloschl, G., & Reszler, C. (2008) Sol mosture updatng by ensemble Kalman flterng n real-tme flood forecastng. J. Hydrol. 357(3 4), Moore, R. J. (2007) The PDM ranfall runoff model. Hydrol. Earth System Sc. (), Moradkhan, H., Sorooshan, S., Gupta, H. V. & Houser, P. R. (2005) Dual state-parameter estmaton of hydrologcal models usng ensemble Kalman flter. Adv. Water Resour. 28(2), Pauwels, V. R. N. & De Lannoy, G. J. M. (2009) Ensemble-based assmlaton of dscharge nto ranfall runoff models: A comparson of approaches to mappng observatonal nformaton to state space. Water Resour. Res. 45, W08428, do:0.029/2008wr Salamon, P. & Feyen, L. (2009) Assessng parameter, precptaton, and predctve uncertanty n a dstrbuted hydrologcal model usng sequental data assmlaton wth the partcle flter. J. Hydrol. 376(3 4), Sene, K. (2008) Flood Warnng, Forecastng and Emergency Response. Sprnger, Berln, Germany. Seo, D. J., Cajna, L., Corby, R. & Howeson, T. (2009) Automatc state updatng for operatonal streamflow forecastng va varatonal data assmlaton. J. Hydrol. 367(3 4),

7 Assmlaton of streamflow dscharge nto a contnuous flood forecastng model 3 Srkanthan, R., Amrthanathan, G. & Kuczera, G. (2008) Applcaton of ensemble Kalman flter for flood forecastng n Australan rvers. Australan J. Water Resour. 2(3), Threl, G., Martn, E., Mahfouf, J. F., Massart, S., Rcc, S. & Habets, F. (200) A past dscharges assmlaton system for ensemble streamflow forecasts over France Part : Descrpton and valdaton of the assmlaton system. Hydrol. Earth System Sc. 4(8), Vrugt, J. A., Gupta, H. V. & Nuallan, B. O. (2006) Real-tme data assmlaton for operatonal ensemble streamflow forecastng. J. Hydromet. 7(3), Vrugt, J. A., Dks, C. G. H., Gupta, H. V., Bouten, W. & Verstraten, J. M. (2005) Improved treatment of uncertanty n hydrologc modelng: Combnng the strengths of global optmzaton and data assmlaton. Water Resour. Res. 4(), W007, do:0.029/2004wr Weerts, A. H. & El Serafy, G. Y. H. (2006) Partcle flterng and ensemble Kalman flterng for state updatng wth hydrologcal conceptual ranfall runoff models. Water Resour. Res. 42(9), W09403, do:0.029/2005wr