Bounds on the Variance in the Pattern Matching Criteria

The Backus-Gilbert (BG) method is used throughout the satellite community as a noise/pattern adjustment technique. To proceed with the study of the discrete BG (DBG) method, the fundamental properties of the noise/pattern tradeoff need to be studied. This paper investigates the pattern matching criteria the DBG method is based on and how noise minimization is incorporated into the solution. In particular, the bounded nature of the variance and the behavior of the tradeoff curve are shown explicitly. The penalty function is studied, and restrictions on its definition are given; problems with the common choice of J=1 are also presented. The case of a positive semidefinite error covariance matrix is analyzed and the uniqueness of the solution evaluated. This paper serves to provide a thorough theoretical discussion of the noise/pattern tradeoff properties and the practical issues associated with its implementation