A fast multistream scattering-based Jacobian for microwave radiance assimilation

The full utilization of satellite-based passive microwave imagery for weather forecasting rests on the ability to assimilate radiances into numerical weather prediction (NWP) models for highly scattering and absorbing hydrometeor states. State vector updates need to be performed rapidly enough to maintain pace with the sensor data stream and require, in particular, rapid calculation of the tangent linear relationship (Jacobian) between the observed antenna temperatures and the NWP prognostic hydrometeor parameters. To facilitate the use of both spaceborne and airborne passive microwave data in numerical forecasting, we present a new rapid multiple-stream discrete-ordinate algorithm for calculating the Jacobian under arbitrary scattering and absorbing conditions. The algorithm is based on the layer-adding method for a plane-parallel atmosphere for which the number of operations required to compute the solution is proportional to the number of layers. A nontrivial aspect of the problem is the stable calculation of the reflectance and transmittance operators in highly scattering layers for which a diagonalization technique and analytical factorization of specific matrices are used to ensure stability. Scaling calculations suggest that the new algorithm will be suitable for use in real-time all-weather microwave radiance assimilation.