Residual moveout-based wave-equation migration velocity analysis in 3-D |

where is the model slowness, and is the prestack image (ADCIGs) migrated with .

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:

in which is the model slowness, is the prestack image migrated with some initial slowness , and is the residual moveout (RMO) function we choose to characterize the kinematic difference (Biondi and Symes, 2004) between and .

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:

where is a local averaging window of length along the depth axis. For the rest of the paper, the summation interval of is always ; thus we can safely omit the summation bounds for concise notation.

We will use gradient-based methods to solve this optimization problem. The gradient given by the objective function 3 is

where can be easily calculated by taking the derivative along the axis of the semblance panel .

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:

where is a simple vertical shift introduced to accommodate bulk shifts in the image introduced by variation in the migration velocity. Notice that the semblance in objective function 3 is independent of because a bulk shift does not affect the power of the stack; therefore we do not include in 3.

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

We differentiate equation 6 with respect to , which gives

(7) |

Now we need to invert a Jacobian to get . We denote to be the first and second order derivatives of image , then define the following:

Let the inverse of matrix be matrix :

Then

Finally, we have the expression for the gradient

The engineering translation of equation 10 is that first we compute the image perturbation , then we backproject this perturbation to model slowness space using the tomography operator .

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 |

2012-05-10