Migration velocity analysis for anisotropic models

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# 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:

 (20)

where is the identity operator and is the differential operator along the angle axes in the ADCIGs . In the subsurface-offset domain, the objective function (Equation 8) reads:

 (21)

where is the absolute value of subsurface-offset, and 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 can help compensate for the phase shift caused by the velocity perturbation. Therefore, we use the modified DSO operator as follows:

 (22)

where is a differential operator in . Taking the derivative in the subsurface offset domain is equivalent to an 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:

 (23)

where and are the original and the smoothed gradient on the original model grid; 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 .

 Migration velocity analysis for anisotropic models

Next: Numerical test Up: Li and Biondi: Anisotropic Previous: Migration Velocity Analysis for

2011-05-24