BAYESIAN STATISTICS OF NONLINEAR INVERSE PROBLEMS - EXAMPLE OF THE MAGNETOTELLURIC 1-D INVERSE PROBLEM

We present a practical algorithm for determining the Bayesian solution of non-linear inverse problems with a limited number of parameters. This approach allows the use of very general conditional probability density functions (pdfs of the data given the model parameters) and a priori pdfs (prior beliefs upon the parameters). The results consist in the a posteriori marginal pdfs of the model parameters. The marginal pdfs describe the additional information (if any) brought by the data to the prior knowledge about the model parameters. We have paid a special attention to the numerical calculation of the a posteriori pdfs and we propose a solution which enables the simultaneous determination of the numerical estimates of the a posteriori pdfs and their uncertainties. With the help of a practical example (the 1-D magnetotelluric inverse problem) with synthetic and real data, we show that in most cases, the a posteriori marginal pdfs are complicated functions and cannot be predicted from the data pdf or the a priori pdfs. Consequently, the standard analyses applied to the same data sets such as the maximum likelihood techniques or the asymptotic estimation technique would lead to severely biased estimates of the parameters. This remark is likely to be true for any non-linear inverse problem.