1. Simultaneous Estimation of Microphysical Parameters and State Variables with Radar data and EnSRF – OSS Experiments Mingjing Tong and Ming Xue School of Meteorology and Center for Analysis and Prediction of Storm University of Oklahoma EnKF Workshop April 2006

2. Introduction Model error can impact the estimation of flow-dependent multivariate error covariances An important source of model error for convective-scale data assimilation and prediction is microphysical parameterization Question – Can we correct model error using data? A possible solution – parameter estimation

3. Uncertain microphysical parameters chosen for this study Marshall-Palmer exponential drop size distribution (DSD) of 3-ice single- moment Lin et al (1983) scheme x is r (rain), s (snow), or h (hail) P=(n 0r, n 0s, n 0h,  h,  s )

4. Range of variations of the parameters Parameter p i n 0h (m -4 ) 4  10 3 4  10 6 4  10 4 n 0s (m -4 ) 5  10 5 1  10 8 3  10 6 n 0r (m -4 ) 3×10 6 8  10 7 8  10 6  h (kg m -3 ) 400913  s (kg m -3 ) 20400100

5. Sensitivity of EnKF analysis to the errors in the microphysical parameters CNTL (true parameters) n 0h an order of magnitude lager than true value n 0s an order of magnitude lager than true value

6. Z is more sensitive to P than Vr Limit of estimation accuracy Unique global minimum Sensitivity of EnKF analysis to the errors in the individual microphysical parameters (  is Vr or Z) J vr JZJZ JZJZ

7. Initialization of Ensemble An environmental sounding + smoothed random perturbations with specified covariances. Perturbation at (l,m,n) is All model variables, except for p, are perturbed. Microphysical variables are perturbed based on the observed echo and only at levels where non-zero values are expected 40 to 100 ensemble members

8. Parameter Estimation Configurations 10log(  x ) and 10log(n 0x ) as additional control parameters Initial parameter ensemble is sampled from a normal prior distribution with Reflectivity > 10 dBZ only are used for parameter estimation. Both Vr and Z data are used for state estimation. The estimation of a parameter vector starts from different initial guesses of the parameter vector with different random realization of the initial ensemble and observation error

9. A data selection procedure is applied. Only 30 reflectivity data are used, where the absolute values of background error correlation are among the top 30. To compensate the quick decrease of the parameter ensemble spread, a minimum standard deviation is pre- specified, which is upper bound of the error of each parameter with negligible impact on model state estimation Parameter Estimation Configurations- continued …

10. Results of single parameter estimation (3 different initial guesses) n 0h n 0s n 0r hh ss 40 ensemble members

11. Results of single parameter estimation hh Ensemble Mean RMS Errors (black no error, blue no correction to p. error, red: with p.estimation ss n 0r

12. Results of single parameter estimation (5 different realizations of parameter perturbations) n 0h ss

13. Estimation of (n 0h,  h ) for 4 initial guesses n 0h hh 40 ensemble members

14. Estimation of (n 0s,  s ) for 4 initial guesses n 0s ss Gray: 40 ensemble members, Black: 100 ensemble members

15. Estimation of (n 0h, n 0s, n 0r ) n 0h n 0s n 0r 40 ensemble members Error-free obsObs with errors averaged absolute error 8 different initial guess (no spread)

16. Estimation of (n 0h, n 0s, n 0r,  h ) n 0h n 0s n 0r hh 100 ensemble members 16 initial guesses very good goodpoor very good: 7 cases good: 5 cases poor: 4 cases

17. Estimation of (n 0h, n 0s, n 0r,  h ) n 0h n 0s n 0r hh Absolute error averaged over 16 cases Red: error-free data, black: error-containing data

18. Estimation of (n 0h, n 0s, n 0r,  h ) very good good poor Ensemble Mean RMS Errors of State Variables

19. Estimation of (n 0h, n 0s, n 0r,  h,  s ) n 0h n 0s n 0r hh ss very goodgoodpoor 100 members 32 initial guesses very good: 4 cases good: 4 cases poor: 24 cases

20. Correlations between Z and P at 70 min Model response to the errors of different parameters can cancel each other. Certain combination of the multiple parameters can result in good fit of the model solution to the observations. Cor(n 0h, Z)Cor(n 0s, Z)Cor(n 0r, Z) Cor(  h, Z)Cor(  s, Z)

21. Conclusions EnKF can be used to correct model errors resulting from uncertain microphysical parameters through simultaneous state and parameter estimation Data selection based on correlation information is found to be effective in avoiding the collapse of parameter ensemble hence filter divergence. When error exists in only one of microphysical parameters, the parameter can successfully estimated without exception When errors exist in multiple parameters, the estimation becomes more difficult, although for most combinations the estimation can still be successful. The identifiability of the microphysical parameters is ultimately determined by the uniqueness of the inverse solution. Unique minima of the response functions are shown to exist in the cases of individual parameter estimation which seem to guarantee convergence of the estimated parameters to their true values.

22. Conclusions … continued The difficulty in identifying multiple parameter set arises from the fact that different combinations of the parameter errors may result in very similar model response, so that the solution of the parameter estimation problem may be non-unique. The identifiability of the microphysical parameters also depends on the quality of data. Parameter estimation is found to be most sensitive to the realization of initial parameter ensemble, especially in the multiple-parameter estimation cases. The identifiability of the microphysical parameters may be case dependent. Estimation using additional polarimetric radar data that contain microphysical information has shown promise. The ability of such parameter estimation procedure for real cases where many sources of model errors may co-exist remains to be investigated.