Most squares inversion

An important problem in geophysical modeling involves the attempt to find the limits of various model parameters consistent with a set of experimental data. When the agreement between model and data can be described in terms of a quadratic form in the residuals, as is the case whenever linear least squares methods are applicable, then the range of parameter values consistent with the data is easily computed by using a Lagrange multiplier approach. This method results in limits which are different from those implied by the covariance matrix for the least squares solution. The differences are simply calculated but may often be substantial in magnitude. In this paper I derive an expression for the limits, discuss the physical interpretation of these limits and those implied by the parameter covariance matrix, and discuss the extension of linear techniques to quasi-linear techniques.