Validation of nonlinear inverse algorithms with Markov chain Monte Carlo method

Abstract

The data processing of remote sensing measurements usually involves solving a nonlinear inverse problem. The error characterization is typically done by assuming linearity around the estimated likelihood point, and a covariance matrix approximating a Gaussian uncertainty is given. However, often the solutions are not Gaussian. We describe here a methodology for validating the traditional error characterization by applying the Markov chain Monte Carlo (MCMC) technique. The validation methodology is applied to the inverse problems of the Global Ozone Monitoring by Occultation of Stars (GOMOS) stellar occultation instrument. The adaptive Metropolis algorithm that is used makes the implementation of the MCMC technique effortless. We demonstrate the approach by using real GOMOS data from two occultations, using a bright and a dim star. It is shown that the covariance matrices that are computed in the operational GOMOS data processing characterize well the error structure of the GOMOS measurements if we assume that noise is Gaussian and no prior information is used. However, we also observe that a significant improvement in retrieved parameters is achieved if a positivity prior is introduced. Such prior information can be correctly modeled with the MCMC technique. The sensitivity of the retrieval on the noise characterization is studied by comparing the results obtained with the operational algorithm with a more robust inversion based on ℓ1 norm. At low altitudes a more robust inversion improves the results. In addition, we demonstrate how the MCMC technique can be used to study the modeling errors in the case of nonlinear problems.