VTI migration velocity analysis using RTM

Preconditioning the DSO gradient

Velocity model building is a highly underdetermined and nonlinear problem. Therefore, prior knowledge of the subsurface is needed to define a plausible subsurface model. In the formulation of Tarantola (1984), prior information is included as the covariance and the mean of the model. In this study, we assume the initial model we use is the mean, and the covariance of the model has two independent components: spatial covariance and collocated cross-parameter covariance (Li et al., 2011). In practice, instead of regularizing the inversion using Tarantola (1984), we use a preconditioning scheme (Claerbout, 2009): smoothing filtering to approximate square-root of the spatial covariance, and a standard-deviation matrix to approximate the square-root of the cross-parameter covariance.

Mathematically, the preconditioned model perturbation $d{\bf n}$ of the subsurface is defined as follows:

$$d {\bf m} = {\bf B} {\bf\Sigma} d {\bf n},$$ (37)

where ${\bf m} = [c ~ \epsilon]^T$ . The smoothing operator ${\bf B}$ is a diagonal matrix:

$${\bf B} = \left \vert \begin{array}{cc} {\bf B}_c & 0 \\ 0 & {\bf B}_{\epsilon} \\ \end{array} \right \vert.$$ (38)

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

$${\bf\Sigma} = \left \vert \begin{array}{cc} \sigma_{cc} & \sigma_... ...\ \sigma_{\epsilon c} & \sigma_{\epsilon \epsilon} \\ \end{array} \right \vert.$$ (39)

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

We call ${\bf n}$ the preconditioning variable, and it relates to the original model ${\bf m}$ as follows:

$${\bf m} = {\bf B} {\bf\Sigma} {\bf n} + ({\bf m}_0 - {\bf B}{\bf\Sigma} {\bf n}_0),$$ (40)

where ${\bf n}_0$ and ${\bf m}_0$ 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 ${\bf n}$ is

$$\nabla_{\bf n} J = (\frac{\partial {\bf m}}{\partial {\bf n}})^* \nabla_{\bf m} J$$

$$= {\bf\Sigma}^* {\bf B}^* \nabla_{\bf m} J,$$ (41)

where $\nabla_{\bf m} J = [\nabla_{c} J ~ \nabla_{\epsilon} J]^T$ .

In a steepest-decent inversion framework, the initial preconditioning model ${\bf n}_0$ is obtained by minimizing the following objective function:

$$J_{\rm init} = \frac{1}{2} \left\langle {\bf m}_0 - {\bf B}{\bf\Sigma} {\bf n}_0, {\bf m}_0 - {\bf B}{\bf\Sigma} {\bf n}_0 \right\rangle.$$ (42)

For the $i_{th}$ iteration

$${\bf n}_{i+1} = {\bf n}_i + \alpha_i \nabla_{\bf n} J,$$ (43)

$${\bf m}_{i+1} = {\bf B} {\bf\Sigma} {\bf n}_{i+1}$$

$$= {\bf B} {\bf\Sigma} {\bf n}_i + \alpha_i {\bf B} {\bf\Sigma} \nabla_{\bf n} J$$

$$= {\bf m}_i + \alpha_i {\bf B} {\bf\Sigma} {\bf\Sigma}^* {\bf B}^* \nabla_{\bf m} J.$$ (44)

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