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

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

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=(n0r, n0s, n0h, h, s)

)exp()( 0 xxxx DnDn

4/10 )/( xxxx qn

Sensitivity of EnKF analysis to the errors in the microphysical parameters

CNTL

(true parameters)

n0h an order of magnitude lager than true value

n0s an order of magnitude lager than true value

• 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

0 10 010log 1

xn xn

2*2

1

1 M

i ii

J

p p ( is Vr or Z)

Jvr

Jvr

JZ

JZ 1010log 0.5

x x

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

Skji

kjiWkjirEnml),,(

),,(),,(),,(

Parameter Estimation Configurations

• 10log(x) and 10log(n0x) 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

1max | |,| |

2i

t tP i i i iP P P P

• 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 …

Results of single parameter estimation (3 different initial guesses)

n0h

n0s

n0r

h

s

40 ensemble members

Results of single parameter estimation

h

Ensemble Mean RMS Errors (black no error, blue no correction to p. error, red: with p.estimation

s

n0r

Results of single parameter estimation(5 different realizations of parameter perturbations)

n0h s

Estimation of (n0h, h) for 4 initial guesses

n0h

h

40 ensemble members

Estimation of (n0h, n0s, n0r)

n0h

n0s

n0r

40 ensemble members

Error-free obs Obs with errorsaveraged absolute error

8 different initial guess

(no spread)

Estimation of (n0h, n0s, n0r,h)

n0h

n0s

n0r

h

100 ensemble

members

16 initial guesses

very good good poor

very good: 7 cases

good: 5 cases

poor: 4 cases

Estimation of (n0h, n0s, n0r,h)

n0h

n0sn0r

h

Absolute error averaged over 16 cases

Red: error-free data, black: error-containing data

Estimation of (n0h, n0s, n0r,h)

very good

good

poor

Ensemble Mean RMS Errors of State Variables

Estimation of (n0h, n0s, n0r,h,s)

n0h

n0s

n0r

h

s

very good good poor

100 members

32 initial guesses

very good: 4 cases

good: 4 cases

poor: 24 cases

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(n0h, Z) Cor(n0s, Z) Cor(n0r, Z) Cor(h, Z) Cor(s, Z)

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.

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.