GENERALIZED RIDGE-REGRESSION WITH APPLICATIONS IN DETERMINATION OF POTENTIAL FIELDS

Tikhonov's regularization techniques are widely applied to geophysical and geodetic inverse problems. A single regularization parameter is frequently adopted and needs to be properly chosen in order to obtain a stable solution. In this paper we generalize the ordinary regularization method by introducing more than one regularization parameter, based on consideration of the minimum mean square error of the estimator. It is shown that the new method results in a smaller mean square error of the estimate than the ordinary regularization method, if the regularization parameters are properly selected. As one example of the most important applications of the proposed method, we discuss the problem of the determination of potential fields using satellite observations. As a result of the theory presented here we expect the following questions can be answered in the positive: (1) Have the methods used conventionally and based upon the use of empirical spectra of the potential coefficients, such as Kaula's rule or modified versions in determination of gravitational models, been adequate in stabilizing the solution in terms of minimum mean square error? (2) Can we solve the unstable problem of determination of potential fields from satellite-tracking data without use of empirical spectra of the potential coefficients? (3) Is it possible to further improve the recently produced potential models, if the proposed method is utilized? Numerical confirmation concerning the size of the improvement will be left to a following contribution, which inevitably requires large scale simulations. The differences of interpreting a geophysical inverse problem between Bayesians and frequentists are also detailed, and the practical implications are especially stressed.