COMPARATIVE EVALUATION OF ENKF AND MLEF FOR ASSIMILATION OF STREAMFLOW DATA INTO NWS OPERATIONAL HYDROLOGIC MODELS

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1 COMPARATIVE EVALUATION OF ENKF AND MLEF FOR ASSIMILATION OF STREAMFLOW DATA INTO NWS OPERATIONAL HYDROLOGIC MODELS Arezoo Raieei Nasab 1, Dong-Jun Seo 1, Haksu Lee 2,3, Sunghee Kim 1 1 Dept. o Civil Eng., The Univ. o Texas at Arlington, Arlington, TX 2 NOAA/NWS, Oice o Climate, Water and Weather Services, Silver Spring, MD 3 Len Technologies, Oak Hill, VA 1

2 In this presentation Motivation Methodology EnKF, MLEF Problem ormulation Comparative evaluation o EnKF and MLEF Homoscedastic versus heteroscedastic error modeling Sensitivity analysis Conclusions and uture research recommendations Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 2

3 Motivation Streamlow is the most widely available, high inormationcontent hydrologic data or inerence o soil moisture states o the basin Assimilating streamlow data, however, involves highly nonlinear observation equations Ensemble Kalman ilter (EnKF) Relative simple and easy to implement Optimal only i the observation equation is linear Maximum likelihood ensemble ilter (MLEF) Ensemble extension o variational assimilation (VAR) Can handle nonlinear observation equations No need or adjoint code Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 3

4 Ensemble Kalman ilter Monte Carlo Approximation X t M t t Z H t. ( X ) 1 t X t Recursive updating o each ensemble trace Problem: Nonlinear observation operation Y n = X t Z t Solution?: Augment the state vector x with H(x) The obs eq is linear only in appearance, still assumes linear response o streamlow to soil moisture near the solution H =

5 Maximum likelihood ensemble ilter Use square-root orecast error covariance P p 1 2 [ p1 p2 pn i a M( x p ) M( x) i s ] Ensemble size p i p1, i p2, i ps, i State-space dimension Minimize cost unction (in ensemble subspace) J 1 T -1 x ] P [ x x ] 1 obs [ x 2 1 [ y 2 T - H ( x) ] R [ y - H ( x) ] obs Similar to VAR, but: Uses non-dierentiable iterative minimization with superior (Hessian) preconditioning Provides reduced rank solution in ensemble subspace Estimates analysis uncertainty From Zupanski (2005) 5

6 Fixed-lag smoother ormulation 1) Prescribe the initial background model states and their covariance 3) Solve or the initial model states, biases or precipitation and PE utilizing all available data within the current assimilation window 4) Integrate the model to the end o the assimilation window to obtain the updated IC s valid at the current prediction time, k 2) Propagate the model states and their uncertainty an hour orward k-1 k k+1 k+2 Time (hrs) 6

7 Study area MTPT2 Stream network MTPT2 in WGRFC (435 km 2, time-to-peak ~17 hrs) Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 7

8 Summary o parameter settings Experiment Streamlow error variance (m 3 /s) 2 MAP (mm/hr) 2 Additive error to TCI (mm/hr) 2 Fractional dynamical model error Ensemble size No. o Streamlow data used/cycle 1 2 Streamlow error variance Additive error variance to TCI Homoscedastic error modeling Heteroscedastic error modeling 1, 10, 50, , 0.1, 1, C Q =0.03, 0.3 C P =0.15, 0.25 Function o Z Q Fractional dynamical model error , 0.025, 0.075, Ensemble size , 9, 30, 50 1 No. o streamlow data used per cycle , 2, 4, 8 MAPE error variance = 1 (mm/hr) 2 8

9 Sensitivity to streamlow observation error variance The smaller the error variance, the closer the it through the observed streamlow Smaller RMSE at short lead times but at some expense o larger RMSE at large lead times The DA-aided simulation has slightly larger RMSE than DAless simulation at large lead times uncertainty modeling needs improvement Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 9

10 Heteroscedastic error modeling Error variance in model runo t Q t = {I τ + w(τ)} 0 σ 2 eq = σ2 w 0 t t u(t τ) 0 σ 2 eq = Q obs u t τ dτ u t s dsdτ 2 (cms) 2 Error variance in obs σ 2 q = C q Q obs + additive 2 (cms) 2 σ 2 P = (C p P obs + additive)2 (mm/hr) 2 σ 2 e = 1 (mm/hr)2 10

11 Heteroscedastic error modeling Heteroscedastic modeling o observation errors does not improve DA perormance over homoscedastic modeling 11

12 Summary o parameter settings Experiment Streamlow error variance (m 3 /s) 2 MAP (mm/hr) 2 Additive error to TCI (mm/hr) 2 Fractional dynamical model error Ensemble size No. o Streamlow data used/cycle 1 2 Streamlow error variance Additive error variance to TCI Homoscedastic error modeling Heteroscedastic error modeling 1, 10, 50, , 0.1, 1, C Q =0.03, 0.3 C P =0.15, 0.25 Function o Z Q Fractional dynamical model error , 0.025, 0.075, Ensemble size , 9, 30, 50 1 No. o streamlow data used per cycle , 2, 4, 8 MAPE error variance = 1 (mm/hr) 2 12

13 Model error MLEF raction o soil water bucket size EnKF raction o soil water content Accounting or model errors in soil moisture dynamics improves the perormance o DA signiicantly at short lead times Both MLEF and EnKF achieve their respective best with a raction o Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 13

14 Ensemble size MLEF is not very sensitive to ensemble size The EnKF solution generally improves with increasing ensemble size but does not come close to the MLEF solution even with 50 members The CPU time or MLEF is considerably smaller than that or EnKF Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 14

15 Number o streamlow obs assimilated per cycle Fixed lag smoother MLEF results deteriorate when a larger number o streamlow is assimilated. The perormance o EnKF improves up to 4 streamlow observations assimilated per cycle and then decreases Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 15

16 Example results 16

17 Example results (cont.) 17

18 An example o signiicantly dierent perormance between MLEF and EnKF Comparative Evaluation o EnKF and MLEF 18

19 Conclusions & uture research recommendations MLEF generally improves streamlow prediction over EnKF very signiicant at short lead times At large lead times, EnKF tends to perorm slightly better than MLEF Suggests possible overitting by MLEF Perormance o MLEF is much less sensitive to error modeling and ensemble size than that o EnKF Important consideration or operational applications Computational requirements or MLEF is smaller than those or EnKF While the streamlow results appear similar, the soil moisture results are quite dierent between MLEF and EnKF Relects possible under-determinedness o the problem Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 19

20 Conclusions & uture research recommendations (cont.) Approximate gradient evaluation in MLEF is not always successul (compared to the adjoint-based) May result in temporal discontinuity in streamlow and soil moisture results Need to test on larger-dimensional problems with varying degree o under-determinedness (will be covered by Sunghee Kim on Session 8: Real-world Applications o Data Assimilation in Operational Hydrology) Assess the quality o analysis (i.e. updated) ensembles via rigorous ensemble veriication or both streamlow and soil moisture Sep 11, 2014 CAHMDA-HEPEX/DAFOH Workshop, Austin, TX 20

21 THANK YOU Questions? For more ino, please contact 21

22 Formulation o assimilation problem Lumped SAC - Unit Hydrograph (1-hr timestep) UZTWC K-L UZFWC K-L LZTWC K-L LZFSC K-L LZFPC K-L ADIMP K-L state variables model states 2. update X P and X E and Model Error (k=k-l+1,, K) 1. orecast 3. orecast w/ update K-L K-L+1 K-L+2 K-1 K K+1 time k-1 k time (hr) assimilation window o size L (hrs) Measurements K-L Streamlow Observation P and ET and Model Error (k=k-l+1,, K) K-L+1 K-L+2 K-1 K K+1 assimilation window o size L (hrs) I don t think this slide is necessary time (hr) 22 time streamlow (m 3 /sec)