Ensemble transform sensitivity for adaptive observations: a general formulation and its practicable implementation

The Ensemble Transformation (ET) method and its variation ET-based Sensitivity (ETS) method have been used in adaptive observation studies. However, the solution of the ensemble transformation matrix in the ET and ETS methods is not unique. A general mathematical formulation for the ensemble transformation matrix is derived and then a generalized equation for the ETS method is derived. It is proved that the previous ETS formulation is a special implementation of the newly derived general formulation. Another practicable implementation of the general ETS formulation that avoids calculating the inverse of some matrices is proposed. This ETS implementation showed physically reasonable statistical sensitivity regions for improving 1-3 day weather forecasts over eastern regions of U.S.A and Beijing region, China.