VTI migration velocity analysis using RTM |

Mathematically, the preconditioned model perturbation of the subsurface is defined as follows:

where . The smoothing operator is a diagonal matrix:

with different smoothing operators for velocity and , according to the geological information in the study area. The standard deviation matrix :

can be obtained by rock-physics modeling and/or lab measurements (Bachrach et al., 2011; Li et al., 2011).

We call the preconditioning variable, and it relates to the original model as follows:

where and are the initial models in preconditioned space and physical space, respectively. Now, the gradient of the objective function 15 with respect to this preconditioning variable is

where .

In a steepest-decent inversion framework, the initial preconditioning model is obtained by minimizing the following objective function:

For the iteration

Equation 44 suggests an interesting consideration in the context of nonlinear inversion: left-multiplying the gradient with a (semi)positive-definite matrix is equivalent to preconditioning with the square-root of the matrix; thus, the resulting direction is still a descent direction (Claerbout, 2009).

VTI migration velocity analysis using RTM |

2012-05-10