Use of Power Transform Total Number Concentration as Control Variable for Direct Assimilation of Radar Reflectivity in GSI En3DVar and Tests with Six Convective Storms Cases

When using a double-moment microphysics scheme, both hydrometeor mixing ratios and number concentrations are part of the state variables that are needed to initialize convective-scale forecasting. In the Thompson microphysics scheme, both mixing ratio and total number concentration of rainwater (N_tr) are predicted and they are also involved in the reflectivity observation operator. In such a case, when directly assimilating reflectivity using N_tr as the control variable (denoted as CVnr) within a variational framework, the large dynamic range of N_tr and the nonlinear relationship between reflectivity and N_tr prevent efficient minimization convergence. Using logarithmic N_tr as the control variable (CVlognr) alleviates the problem to some extent but can produce spurious analysis increments in N_tr. In this study, a general power transform of N_tr is proposed as the new control variable for N_tr (CVpnr) where the nonlinearity of transform can be adjusted by tuning the exponent parameter. This formulation is implemented within the Gridpoint Statistical Interpolation ensemble-3DVar system. The performance of CVpnr with an optimal exponent parameter value of 0.4 is compared with those of CVnr and CVlognr for the analysis and prediction of a supercell case of 16 May 2017 in more detail. CVpnr with optimal exponent yields faster convergence of cost function minimization than CVnr. Subjective and objective evaluations of analyzed and predicted reflectivity and hourly precipitation indicate that the optimized CVpnr outperforms the other two methods. When applied to five additional cases from May 2017, overall statistics show that CVpnr produces generally superior forecasts and is therefore the preferred choice.