VTI migration velocity analysis using RTM

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## Objective function test

We test our method on a VTI Marmousi model. First, synthetic Born data is generated using the models in Figures 1 and 2. The maximum offset is 3 km. Next, we perturb the model (Figure 2(a)) by a very smooth field, as shown in Figure 3(a). We change the perturbation from -50% to 50% of the true model, and calculate the corresponding objective function respectively.

Ideally, we'd like to choose an objective function that reaches a local minimum at the correct model and is quadratic around the correct model, so that a gradient-based inversion scheme is guaranteed to converge. Based on the results, we choose an angle domain objective function instead of the DSO objective function (Equation 15):

 (45)

where is the Radon transform operator, and is the derivative operator along the ray-parameter axis.

As shown in Figure 3(b), the angle-domain objective function has a minimum at the correct epsilon model, and has a semi-quadratic shape with respect to the model perturbation. Therefore, this objective function is a good measure of the error in the anisotropic model. Notice that the tilting effect toward negative perturbation is caused by the limited acquisition geometry. This effect is negligible for velocity perturbation, because velocity has a first-order effect on the flatness of the angle gather, while 's effect is second-order. We can increase the acquisition offset to mitigate this tilting effect and help the inversion.

vtrue,ref
Figure 1.
Smooth velocity model (a) in m/s and reflectivity model (b) used to generate the synthetic Born data.

eps,del
Figure 2.
The model (a) and model (b) used to generate the synthetic Born data.

deltaeps,objcurve
Figure 3.
(a) The model to test the objective function. (b) Objective function vs. perturbation. The angle-domain objective function 45 has a minimum at the correct epsilon model, and has a semi-quadratic shape with respect to the model perturbation.

 VTI migration velocity analysis using RTM

Next: Single parameter inversion Up: Numerical test Previous: Numerical test

2012-05-10