Tarantola (1987) formalizes the geophysical inverse problem from a Bayesian point of view. He gives a theoretical background to compensate for the reflection experiment's deficiencies (acquisition geometry, obstacles, etc.), while being consistent with the acquired data.
Under this scheme, data and model (subsurface image) are assumed to have Gaussian distribution (a priori data and model probability density are Gaussian). The forward operator is assumed to be linear (or weakly non-linear). The resulting subsurface image is the mean of a posterior Gaussian probability density in the model space. Its expression for a general linear (forward modeling) operator is
(1) |
Equation (1) can be solved by using gradient-based methods like steepest descent. This iterative algorithm can be written in the form Tarantola (1987), where the term depends on the gradient of the function to be optimized, a metric in the model space, and an ad-hoc constant.