ITERATIVE INVERSION

The statistical approach to seismic inversion considers the fact that the data is imperfect. Random noise, unwanted signals, missing data, and evanescent waves that cannot be backpropagated make a real deterministic inversion impossible. The goal of iterative inversion is to minimize the difference between the registered data vector ${\underline Y}$and the synthetic data computed as the product of the propagator matrix ${\underline A}$ and the earth model ${\underline X}$ step by step in a Gaussian sense. To stabilize the computation, a small term $\epsilon^2 \underline X^2$ has to be added:

$$({\underline Y} - {\underline A} \underline {X})^2 + \epsilon^2 \underline X^2 = min$$ (1)

Linear inversion needs a reasonable a priori velocity model ${\underline X}$as input and to construct the propagator matrix ${\underline A}$. The choice of ${\underline A}$ is, of course, based on wave or ray theory. An iterative inversion algorithm applies migration or tomography in each step of the iteration. In so far, the ``statistical approach'' is also ``deterministic.'' The iteration converges at least after N iterations against the ``true'' earth structure, if N is the number of unknowns to be solved for.

Nonlinear inversion goes one step further because it iterates not only for the model vector ${\underline X}$ but also for the propagator ${\underline A}$. The start propagator matrix is usually the transpose of the forward operator (Claerbout, J.F., 1989). The method of steepest descent or better conjugate gradients are in widespread use to find the minimum. To guarantee convergence a reasonable start model is a necessity.

To make sure that the true earth model can be found, the assumption that the earth behaves Gaussian has to be made. A real statistical approach to seismic inversion should drop this assumption and should use non-Gaussian methods like the Monte Carlo method (Tarantola, A., 1990). Only if we iterate without any a priori information (which can be wrong) can we hope to find the real inverse operator. Unfortunately, this makes the solution space infinite so theoretically we will need a computer with infinite speed. In practice it should be sufficient to combine Gaussian iterative nonlinear inversion with the Monte Carlo method to improve today's results.