The Field-wise Wind Retrieval Objective Function

The Field-wise Objective Function is an important concept in eldwise wind retrieval. A discussion on the purpose of the eld-wise objective function is presented. Several variations of the eld-wise objective function are compared. The gradient vectors and Hessian matrices are derived for each variation. 1 An Overview of Field-wise Objective Functions An objective function is an error metric, providing a scalar value quantifying the distance between an estimate and the observed measurements. Thus, the estimate that minimizes the objective function is considered \closest" to the observed measurements. The well-known \least-squares" problem is an implementation of a minimized objective function. In the case of eld-wise wind retrieval, the error metric is a function of the model parameters. A model parameter vector x parameterizes an M N wind eld w through the linear model w = Fx. An objective function J(x) measures the di erence between the wind eld w and the observed measurements. A direct error metric between the wind eld w and the observed radar backscatter ( o) is meaningless, because w is not in the o measurement space. To create a metric between w and o, they must be transformed into the same space. As there is no model to transorm o into the wind vector space, all of the objective functions employ a metric in the o measurement space. The Geophysical Model Function (GMF) enables w to be transformed into the o measurement space. The GMF returns the o value that would result from examining a wind vector under a given set of measurement conditions (i.e., instrument azimuth and incidence angles, and ). The trans-