A Generalized Karhunen–Loeve Basis for Efficient Estimation of Tropospheric Refractivity Using Radar Clutter

In this paper, we consider the problem of obtaining a reduced-dimension parameterization of a propagation medium for the purpose of estimating the medium from transmission data. The application addressed is microwave remote sensing of tropospheric index-of-refraction profiles over the sea surface, using radar clutter returns. The proposed parameterization balances the desire to represent features prominent in the a priori statistics of the profiles versus the need to capture elements of the profile that significantly affect the observed clutter data. In linear estimation problems, basis vectors for the unknown parameter vector that optimizes this tradeoff have been derived as the reduced-rank Wiener filter or, equivalently, the generalized Karhunen–Loeve transform (GKLT). In this paper, we reinterpret the linear result, producing an extension to the nonlinear refractivity estimation problem. The resulting procedure produces basis vectors for tropospheric refractivity that are less dependent on features that have little effect on the clutter measurements. This results in a more efficient parameterization and reduces mean-square estimation error relative to an approach driven purely by the statistical prior. Application of the generalized KL technique to finding efficient basis vectors for refractivity profiles taken off the southern California coast is presented.