CMWRXVI DATA ASSIMILATION IN A FLOOD MODELLING SYSTEM USING THE ENSEMBLE KALMAN FILTER HENRIK MADSEN, JOHAN HARTNACK, JACOB V.T. SØRENSEN DHI Wter & Environment, Agern Allé 5, DK-2970 Hørsholm, Denmr ABSTRACT A dt ssimiltion system bsed on the ensemble Klmn ilter hs been implemented in the MIKE FLOOD modelling system. The modelling system combines 1D nd 2D hydrodynmic modelling, which provides computtionlly eicient pproch or deling with dierent sptil resolutions nd process descriptions in dierent prts o the model domin. It llows modelling the prt o the model domin relted to the min chnnel low in 1D nd the rest o the domin in 2D detil. The implemented dt ssimiltion system cn be used or updting the model bsed on wter level nd lux mesurements in both the 1D nd 2D model domins. To demonstrte the implemented dt ssimiltion system twin test experiment is crried out. The nlysis shows tht the ensemble Klmn ilter is very eective in updting the entire model domin using mesurements rom only ew loctions in the river system. 1. INTRODUCTION The MIKE FLOOD model developed by DHI Wter & Environment (DHI, 2005) is lexible modelling system tht combines one-dimensionl nd two-dimensionl hydrodynmic modelling. The model is bsed on dynmic lining between the well-estblished 1D nd 2D numericl modelling systems MIKE 11 nd MIKE 21 enhnced with etures tht re tilored towrds modelling o loods. Dt ssimiltion cilities hve been implemented in MIKE FLOOD to dptively updte the stte o the system by ssimiltion o lux nd wter level mesurements in both the river system nd in the 2D model domin. The implementtion is bsed on the ensemble Klmn ilter (irst introduced by Evensen, 1994), which hs recently been pplied in opertionl hydrodynmic nd hydrologicl modelling, including the MIKE 11 nd MIKE 21 modelling systems (Mdsen nd Cñizres, 1999; Hrtnc nd Mdsen, 2001; Mdsen et l., 2003). This pper describes the implementtion o the ensemble Klmn ilter in MIKE FLOOD. A twin test exmple is presented to demonstrte the implementtion. 2. MIKE FLOOD MODELLING SYSTEM The MIKE FLOOD modelling system combines one-dimensionl nd two-dimensionl modelling in one modelling system. It llows modelling prt o the model domin in 1D detil (relted to the min river chnnel) nd other prts in 2D, nd hence provides computtionl eicient pproch or deling with dierent sptil resolutions nd process descriptions in dierent prts o the model domin. Furthermore, the combined pproch cilittes implementtion o ine scle structures, such s weirs nd culverts, within 2D domin. MIKE FLOOD cn be used to simulte the detiled looding pttern on loodplins in 2D. 1
Henri Mdsen, John Hrtnc nd Jcob V.T. Sørensen detil while mintining n eicient 1D description or the in-chnnel low. Becuse o its dynmic lining it is lso well suited or modelling coincident river nd storm surge looding in costl res. Three dierent linge types re vilble or dynmiclly lining the 1D nd 2D models: (i) stndrd lin, (ii) lterl lin, nd (iii) structure lin. The stndrd lin type is point-topoint coupling between computtionl point in the 1D model nd computtionl point in the 2D model domin. It cn, or instnce, be used to couple upstrem river brnches to downstrem lood plin. The lterl lin type couples the 1D model with the 2D domin over prt o the river system by lterl inlow/outlow coupling between the two domins. The low is described through weir eqution in ech lined grid point. The lterl lin type cn be used or simultion o overtopping o levees nd low into the loodplin. The structure lin type uses 1D model in 2D domin to model low through structure (deined in the 1D model). This lin type is useul or modelling smll-scle etures which cnnot be dequtely resolved using the typicl grid size in the 2D model. 3. ENSEMBLE KALMAN FILTER The dt ssimiltion system implemented in MIKE FLOOD is bsed on the Klmn ilter. The Klmn ilter lgorithm uses stochstic representtion o the model nd the mesurements to te into ccount the inherent uncertinties in the modelling nd ssimiltion processes. Model uncertinties re relted to errors in the physicl conceptulistions nd mthemticl pproximtions o the governing processes, errors due to the use o non-optiml prmeter vlues, nd errors in the model orcing. Mesurement uncertinties re relted to errors in the mesuring devices nd representtion errors, e.g. cused by using point mesurement to represent grid vlue in the numericl model. It is here ssumed tht model uncertinties re cused only by errors in the model orcing. In this cse the stochstic representtion o the MIKE FLOOD model cn be written x = Φ( x 1, u + ε ) (1) where Φ(.) is the model opertor tht represents the numericl scheme o MIKE FLOOD, x is the stte vector representing the stte o the modelled system t time step, u is the orcing o the system, nd ε is the model error. The stte vector includes dischrges nd wter levels in the computtionl grid o the river system s well s depth-verged x nd y velocities nd wter levels in ech computtionl point o the 2D grid. The orcing includes dierent boundry conditions or the river model (wter level, dischrge, rinll-runo, rting curves) s well s open boundry conditions (prescribed wter levels or luxes) nd wind orcing o the 2D modelling prt. Model errors re ssumed to be correlted in both time nd spce. Time correltion is implemented using irst-order utoregressive model ε = 1 + (2) Aε ζ where A = dig[α 1,α 2,,] is correltion mtrix with utoregressive coeicients α i, nd ζ is white noise. The stte is ugmented with the correlted model errors, which re updted s 2
CMWRXVI prt o the iltering process long with the model stte. The coloured noise ormultion is very eective in correcting erroneous model orcings (Mdsen nd Cñizres, 1999), including correction o model bis (Drécourt et l., 2006). Sptil correltion o errors in the wind orcing nd open boundry conditions is implemented using exponentil correltion models. For updting the model, the dt ssimiltion system utilises wter level nd lux mesurements t speciic loctions in the river system nd 2D model domin. This is ormulted in the mesurement eqution z = C x + η (3) where z is the vector o mesurements, C is mtrix tht describes the reltion between mesurements nd stte vribles (mpping o stte spce to mesurement spce), nd η is the mesurement error, which is ssumed unbised. P re correc- In the Klmn ilter, the model orecst x nd corresponding covrince ted bsed on the mesurements to get n updted stte x nd covrince P x P = x + K ( z C x ) (4) = P KCP (5) where K is the Klmn gin mtrix T [ C P C + R ] 1 T = P C K (6) nd R is the covrince mtrix o the mesurement errors. It is ssumed tht mesurement errors re uncorrelted. In this cse computtionlly eicient nd robust sequentil lgorithm tht process one mesurement t time cn be used or the Klmn ilter updte (Chui nd Chen, 1991). This voids explicit clcultion o the covrince mtrix P nd the mtrix inversion or the clcultion o the Klmn gin in Eq. (6). The Klmn ilter is implemented in MIKE FLOOD using the ensemble Klmn ilter (EnKF) method, irst introduced by Evensen (1994). In the EnKF the stochstic model eqution is represented by n ensemble o stte vectors. In the model orecst, ech o these sttes re propgted ccording to the model dynmics nd orced by model errors, c. Eq. (1). This provides n estimte o the covrince mtrix P. In the mesurement updte, ech stte vector is updted ccording to the Klmn ilter updte scheme. The resulting updted ensemble provides n estimte o the verge stte nd the corresponding covrince mtrix P. It should be noted tht in the implementtion the covrince mtrices re never clculted explicitly. The EnKF hs been widely pplied in ocenogrphy nd meteorology nd more recently in hydrodynmic nd hydrologicl modelling (e.g. Hrtnc nd Mdsen, 2001; Mdsen et l., 2003; Sørensen et l., 2004; Drécourt et l., 2006). The success o the EnKF is minly due to its ese o use nd very lexible implementtion. The EnKF cn be esily implemented in. 3
Henri Mdsen, John Hrtnc nd Jcob V.T. Sørensen numericl engine, bsiclly by ming loop round the one-time-step-hed model prediction, nd it llows deinition o noise in ny component o the numericl model (model prmeters, model orcings, or the model stte itsel). However, one o its drwbcs is relted to the rther slow convergence o the smple pproximtion to the true covrince. For smll ensemble size, spurious correltions my be introduced, resulting in unrelistic updtes nd potentil model instbilities. Temporl nd sptil regulristion techniques hve been introduced to solve these problems (Sørensen et l., 2004), hence ming the EnKF computtionlly esible or rel-time pplictions. 4. TWIN TEST EXPERIMENT To evlute the perormnce o the dt ssimiltion system implemented in MIKE FLOOD twin test experiment hs been crried out. The model being investigted consists o lood plin with min river nd tributry (see Figure 1). The 2D modelling domin covers n re o 13 x 15 m. The rivers hve slope o bout 0.025 % nd cross sections with width o bout 400 meters rom side bn to side bn. The min river length is 23 m nd the tributry length is 11 m. FIGURE 1. Bthymetry o the MIKE FLOOD modelling domin. Mred points re the two mesurement points in the river system (M1 nd M2), two vlidtion points in the river system (R1 nd R2), nd two vlidtion points on the loodplin (F1 nd F2). The min river nd the tributry where the low direction is well deined re modelled using the 1D prt, wheres the remining loodplin is modelled using the 2D component. The 1D nd the 2D domins re lined using lterl inlow/outlow lins. The use o lterl lins ensures physiclly sound pproch to modelling the wter entering the loodplin through the overtopping o the river embnments. The 2D domin is completely closed, i.e. 4
CMWRXVI no open boundries re deined. Thus, the only source o low in the 2D domin is through the spilling rom the 1D component into the 2D re. The 1D component hs two upstrem boundries consisting o dischrge hydrogrph, one t the min river nd one t the tributry. At the downstrem end o the min river rting curve hs been implemented. For the twin test experiment, lood event ws considered with lood hydrogrphs deined t the two upstrem boundries. A reerence MIKE FLOOD simultion ws mde by using the reerence hydrogrph shown in Figure 2 t both boundries. From this reerence run, wter level time series were extrcted t two loctions in the river system, respectively, 8.8 m downstrem o the inlow boundry t the min river, nd 7.3 m downstrem o the inlow boundry o the tributry (see Figure 1). These time series were subsequently used s mesurements or updting model tht ws orced with erroneous lood hydrogrphs t the upstrem boundries (see Figure 2). The ensemble Klmn ilter ws implemented using n ensemble size o 50. Model errors were introduced t the upstrem boundries using coloured noise description with resulting stndrd devition o 10 % o the instntneous dischrge t the boundries. FIGURE 2. Hydrogrphs [m 3 /s] pplied t the two upstrem boundries in the river system. The upper curve is used in the reerence simultions, nd the lower curve is used s the erroneous boundry conditions. Simultion results re shown in Figures 3-5. In the reerence simultion overtopping o the river embnments re primrily occurring t three loctions in the river system (see Figure 3). Overtopping irst occurs in the upstrem prt o the min river, nd the loodplin t the let bn prllel to the min river (right prt o the model domin) is looded. In the tributry, overtopping occurs t the let bn bout 4 m rom the upstrem boundry, inundting n re in the upper prt o the model domin between the tributry nd the min river. Overtopping lso occurs t the right bn in the tributry close to the conluence with the min river, resulting in lrge looding in n re prllel to the downstrem prt o the min river. When the model is orced with the erroneous boundry conditions, much less looding is seen. Besides minor looding close to the river in the upstrem prt o the min river nd t loction t the let bn o the tributry, looding only occurs in the downstrem prt due to overtopping o the river embnments in the tributry close the conluence point (see Figure 3).. 5
Henri Mdsen, John Hrtnc nd Jcob V.T. Sørensen Reerence Flse Updte FIGURE 3. Simulted wter depth t the lood plin t 2004-03-25 18:00 rom, respectively, the reerence simultion, the simultion using erroneous boundry conditions (lse), nd the Klmn ilter updte. As illustrted in Figure 3, dt ssimiltion (using in this cse wter level mesurements rom only two loctions in the river system) is very eective in correcting the model stte in the entire model domin. The updted lood mp is very close to tht obtined rom the reerence run. More detiled simultion results re shown in Figures 4-5 rom two loctions in the river system nd two loctions on the lood plin (loctions re shown in Figure 1). Results re shown or two loctions upstrem o the mesurement points (R1 on the upstrem 6
CMWRXVI prt o the tributry nd F1 on the upstrem prt o the lood plin) nd two loctions downstrem (R2 on the downstrem prt o the min river nd F2 on the loodplin downstrem o the tributry). At the downstrem loctions virtully perect updte is obtined by the Klmn ilter. At the upstrem loctions the updte is less ccurte, but still signiicnt improvement is obtined compred to the simultion orced with erroneous boundry conditions. Note tht t the upstrem vlidtion point on the loodplin, no looding is simulted with the erroneous boundry conditions. FIGURE 4. Simulted wter level [m] t two loctions in the river systems rom, respectively, the reerence simultion (circles), the simultion using erroneous boundry conditions (lower blue curve), nd the Klmn ilter updte (upper red curve). 5. CONCLUDING REMARKS A dt ssimiltion system hs been implemented in the MIKE FLOOD modelling system nd hs been tested using twin test experiment. The dt ssimiltion system is bsed on the ensemble Klmn ilter nd llows ssimiltion o wter level nd lux mesurements in both the 1D nd 2D model domins. The twin test experiment illustrtes tht the dt ssimiltion system is very eective in updting the entire model domin using mesurements rom only two loctions in the river system.. 7
Henri Mdsen, John Hrtnc nd Jcob V.T. Sørensen FIGURE 5. Simulted wter depth [m] t two loctions on the loodplin rom, respectively, the reerence simultion (circles), the simultion using erroneous boundry conditions (lower blue curve), nd the Klmn ilter updte (upper red curve). REFERENCES Chui, C.K. nd G. Chen (1991), Klmn Filtering with Rel-Time Applictions, Springer Series in Inormtion Sciences no. 17 (ed. Hung, T.S.), Second edition, Springer-Verlg, Berlin Heidelberg, Germny. DHI (2005), MIKE FLOOD User Mnul, DHI Sotwre 2005, DHI Wter & Enviroment, Denmr. Drécourt, J.P., H. Mdsen nd D. Rosbjerg (2006), Bis wre Klmn ilters: Comprison nd improvements, Adv. Wter Resour., 707-718. Evensen, G. (1994), Sequentil dt ssimiltion with non-liner qusi-geostrophic model using Monte Crlo methods to orecst error sttistics, J. Geophys. Res., 99(C5), 10,143-10,162. Hrtnc, J. nd H. Mdsen (2001), Dt ssimiltion in river low modelling, 4 th DHI Sotwre Conerence, 6-8 June, 2001, Scnticon Conerence Centre, Helsingør, Denmr. Mdsen, H. nd R. Cñizres (1999), Comprison o extended nd ensemble Klmn ilters or dt ssimiltion in costl re modelling, Int. J. Numer. Meth. Fluids, 31, 961-981. Mdsen, H., D. Rosbjerg, J. Dmgård nd F.S. Hnsen (2003), Dt ssimiltion in the MIKE 11 Flood Forecsting system using Klmn iltering, Wter Resources Systems - Hydrologicl Ris, Mngement nd Development (Proceedings o symposium HS02b held during IUGG2003 t Spporo, July 2003). IAHS Publ. no. 281, 75 81. Sørensen, J.V.T., H. Mdsen nd H. Mdsen (2004), Eicient sequentil techniques or the ssimiltion o tide guge dt in three dimensionl modeling o the North Se nd Bltic Se system, J. Geophys. Res., 109, 10.1029/2003JC002144. 8