Some examples of inverse problems in geophysics

Most of the problems arising in geophysics consist in recovering some characteristics of the earth (reflectivity, resistivity,⋯), from some measurements usually made at the top of the earth (seismic data, electromagnetic data,⋯). The observations can be seen as the output data of a noisy (linear or non linear) system, that can be known, or unknown. When the system is unknown, it can be estimated, together with the noise variance, by using a stochastic version of EM algorithm. When the system is identified, some adaptations of MCMC techniques allow one to estimate the a posteriori distribution of the input sequence, according to the prior distribution. This prior can take into account the regularity of the input sequence, as well as the presence of abrupt changes in this sequence.