Tarantola (1987) formalizes the geophysical inverse problem by giving a theoretical approach to compensate for experimental deficiency (e.g., acquisition geometry, complex overburden), while being consistent with the acquired data. His approach can be summarized as follows: given a linear modeling operator , compute synthetic data d using where m is a reflectivity model. Given the recorded data , a quadratic cost function,
(1) |
(2) |
The main difficulty with this approach is the explicit calculation of the Hessian inverse. In practice, it is more feasible to compute the least-squares inverse image as the solution of the linear system of equations,
(3) |