Local stability and estimation of uncertainty for inverse problem solvers

Inverse methods have known a very important development in thegeosciences these last years. Most of the techniques currently used areseeking for a mean field as a solution. In practice it is important toknow the response of this solution to a perturbation on the data. This paperis devoted to a quantitative ( i.e.theoretical and algorithmic ) approach of the stability for an inverseproblem. The method is based on methods of statistical physics.The evaluation and the propagation of uncertainties is a veryimportant problem in geophysics. This paper gives a theoretical andalgorithmic solution.The first part of the article is devoted to the theory.The stability of solution ischaracterized by the variance of the deviation, with respect to themaximum likehood estimate, of the true value of the control variable,this quantity is the trace of the Hessan matrix. The estimation ofcovariances are also given. In this part classicalresults in statistical physics are applied.The second part is oriented towardalgorithmic methods for the effective computation of covariances. Thealgorithms are not simple nevertheless they are described step bystep. The last part of the paper illustrates the method with two examples: thefirst example ie a very simple 1-D model and the second one is relatedto the retrieval of the floow field from measurement of temperature andsalinity. The method gives an estimation of the spectrum of the hessianand the observables. Different choices of regularization are used