Migration velocity analysis for anisotropic models

Objective function

As mentioned in the previous section, we estimate the optimum earth model by minimizing a user-defined image perturbation. There are many ways to define the objective function. Here we use the Differential Semblance Optimization (DSO) method (Shen, 2004; Symes and Carazzone, 1991) as the criterion:

$${\bf F}({\bf I}) = ( {\bf 1} - {\bf O}){\bf I},$$ (20)

where ${\bf 1}$ is the identity operator and ${\bf O}$ is the differential operator along the angle axes in the ADCIGs ${\bf I}$ . In the subsurface-offset domain, the objective function (Equation 8) reads:

$$J = \frac{1}{2}\vert\vert{h {\bf I}( {\bf x}, {\bf h})}\vert\vert _{2},$$ (21)

where $h$ is the absolute value of subsurface-offset, and ${\bf I}({\bf x},{\bf h})$ is the image gather in the subsurface-offset domain. This operator is preferred by many researchers since it is a fully automated procedure, with no picking required. However, for isotropic migration velocity analysis, many authors (Vyas and Tang, 2010; Fei and Williamson, 2010) observe undesired artifacts generated by the DSO operator and suggest that a differential operator along $h$ can help compensate for the phase shift caused by the velocity perturbation. Therefore, we use the modified DSO operator as follows:

$$J = \frac{1}{2}\vert\vert{h {\bf D} {\bf I}( {\bf x}, {\bf h})}\vert\vert _{2},$$ (22)

where ${\bf D}$ is a differential operator in ${\bf h}$ . Taking the derivative in the subsurface offset domain is equivalent to an $\alpha$ weighting in the angle domain. Therefore, the objective function (Equation 22) also emphasizes the contribution of the large angle information, which is crucial for velocity analysis.

To guarantee a smooth inversion, we choose a B-spline representation of the model space. The smoothed gradient in the original space is then represented as:

$$\hat{{\bf g}} = {\bf B}{\bf B^*} {\bf g},$$ (23)

where ${\bf g}$ and $\hat{\bf g}$ are the original and the smoothed gradient on the original model grid; ${\bf B}$ is the B-spline projection operator. Then the number and spacing of the B-spline nodes control the smoothness of the model update. Practically, we can choose different B-spline parameters for velocity and $\eta$ .

Migration velocity analysis for anisotropic models