EVALUATION OF MAXIMUM LIKELIHOOD ENSEMBLE FILTER FOR REAL-TIME ASSIMILATION OF STREAMFLOW DATA IN OPERATIONAL STREAMFLOW FORECASTING

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1 EVALUATION OF MAXIMUM LIKELIHOOD ENSEMBLE FILTER FOR REAL-TIME ASSIMILATION OF STREAMFLOW DATA IN OPERATIONAL STREAMFLOW FORECASTING Arezoo Raieei Nasab 1, Dong-Jun Seo 1, Haksu Lee 2,3, Sunghee Kim 4 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 4 Consultant, Bethesda, MD 1

2 In this presentation Motivation Methodology EnKF, MLEF Problem ormulation Error modeling Model, observation errors Comparative evaluation o EnKF and MLEF Under homoscedastic, heteroscedastic errors Sensitivity analysis Conclusions and uture research recommendations Dec 3, 2012 AGU Fall Meeting 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 Dec 3, 2012 AGU Fall Meeting 3

4 Ensemble Kalman ilter Monte Carlo Approximation X t M t t Z Ht. ( 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) (still assuming Gaussian pd or new model vector) H =

5 Maximum likelihood ensemble ilter Use square-root orecast (prior) 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 N Statespace dimension Minimize cost unction in the 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: Non-dierentiable iterative minimization with superior (Hessian) preconditioning Reduced rank solution in ensemble subspace Estimate o analysis uncertainty available From Zupanski (2005) 5

6 MLEF Extension or model error Zupanski (2005) does not consider model error To account or model uncertainty, augment the square root o the orecast covariance matrix P ( k) M k 1, k P a ( k 1) M T k 1, k Q( k 1) { M k 1, k P a ( k 1)} 1/ 2 { M k 1, k P a ( k 1)} T / 2 Q 1/ 2 ( k 1) Q T / 2 ( k 1) P T / 2 ( k) [{ M k 1, k P a ( k 1)} 1/ 2 Q 1/ 2 ( k 1)] Dec 3, 2012 AGU Fall Meeting 6

7 Fixed-lag smoother ormulation Beginning o the assimilation window End o the assimilation window=prediction time Assimilation window ~ length o unit hydrograph Assimilation cycle ~ nominally once per hour k-l-1 k-l k-l+1 k-1 k k+1 k+2 Time (hrs) All or part o the precipitation, PE, and streamlow data valid within the assimilation window is assimilated. 7

8 Fixed-lag smoother ormulation (cont.) 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) 8

9 Comparatively evaluation o EnKF and MLEF Homoscedastic errors Study basin MTPT2 in WGRFC (435 km 2, time-to-peak o 17 hrs) Nominal parameter settings Streamlow obs error variance: 0.01 (cms) 2 Precipitation obs error variance: 10 (mm/hr) 2 Potential evaporation obs error variance = 1 (mm/hr) 2 Runo obs error variance = 0.1 (mm/hr) 2 Model error (as a raction o soil water bucket size)= 0.05 (i.e. 5%) Ensemble size = 30 Number o streamlow data used in DA = 1 (valid at the end o the assimilation window) The same settings are used or both MLEF and EnKF Dec 3, 2012 AGU Fall Meeting 9

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12 All y-axis units are in mm 12

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14 All y-axis units are in mm 14

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16 All y-axis units are in mm 16

17 Sensitivity analysis Model error (rac) raction o soil water bucket size: 0, 0.05, 5, 25 Streamlow obs error variance ( ) 0.01, 0.1, 1 and 10 (cms) 2 Ensemble size (ns) 9, 30, 50 and 100 members σ 2 q Number o streamlow obs assimilated per cycle (n) 1, 2, 4 and 8 obs within the assimilation window Obs error variances in precipitation, PE and runo Dec 3, 2012 AGU Fall Meeting 17

18 Streamlow Prediction 18

19 σ 2 q = (cms) σ2 Streamlow Prediction q = 0. 1 (cms) σ 2 q = 1(cms) σ 2 q = 10 (cms) 19

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21 Streamlow Prediction 21

22 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 u t τ dτ σ 2 eq = Q obs u t s dsdτ 2 (mm/hr) 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 22

23 Streamlow Prediction 23

24 Conclusions & uture research recommendations MLEF generally improves streamlow prediction over EnKF; the improvement is: very signiicant at short lead times consistent over varying conditions o observational and model errors and ensemble size 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 While the streamlow results appear similar, the soil moisture results are quite dierent between MLEF and EnKF relects large dierences in their solutions Dec 3, 2012 AGU Fall Meeting 24

25 Conclusions & uture research recommendations (cont.) Approximate gradient evaluation in MLEF is not always successul (compared to the adjoint-based) May result in temporal discontinuity streamlow and soil moisture results Improved error modeling may alleviate the problem Need to test on larger-dimensional problems with varying degree o under-determinedness Assess computational requirements as well Assess the quality o analysis (i.e. updated) ensembles via rigorous ensemble veriication or both streamlow and soil moisture Compare with iterative EnKF Dec 3, 2012 AGU Fall Meeting 25

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