Residual moveout-based wave-equation migration velocity analysis in 3-D |
Objective functions defined this way are prone to cycle-skipping (Symes, 2008). To tackle this issue, we approximate objective function 1 with the following one:
The meaning of equation 2 can be easily explained. As the model changes from to , it leads to the change of the image kinematics between and , where the differences are characterized by the moveout parameter . Since will be kinematically the same as being applied moveout , if we substitute the former image ( ) with the latter one, we transit from equation 1 to equation 2. Notice that the new objective function is expressed as a function of only the moveout parameter , while the parameter is then related to the model slowness .
Furthermore, notice that equation 2 weights the strong-amplitude events more heavily. To make the gradient independent from the strength of reflectors, we further replace 2 with the following semblance objective function:
We will use gradient-based methods to solve this optimization problem. The gradient given by the objective function 3 is
To evaluate the derivative of the moveout parameter with respect to the slowness model , we define an auxiliary objective function in a fashion similar to the one employed by Luo and Schuster (1991) for cross-well travel-time tomography. The auxiliary objective function is defined for each image point ( ) as:
The explanation for equation 5 is as follows: The moveout parameters and are chosen to describe the kinematic difference between the initial image and the new image . In other words, if we apply the moveout to the initial image, the resulting image will be the same as the new image in terms of kinematics; this is indicated by a maximum of the cross-correlation between the two.
Given the auxiliary objective function 5, can be found using the rule of partial derivatives for implicit functions. We compute the gradient of 5 around the maximum at and ; consequently
(7) |
Then
Since we can compute the gradient in equation 10, any gradient-based optimization method can be used to maximize the objective function defined in equation 3. Nonetheless, in terms of finding the step size, it is more expensive to evaluate equation 3 (which is an approximation of equation 1 purely based on kinematics) than to evaluate the original objective function 1. In our implementation we choose 1 as the maximization goal while using the search direction computed from equation 3.
Residual moveout-based wave-equation migration velocity analysis in 3-D |