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

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## Extension to 3-D

As we shift from 2-D seismic surveys to 3-D ones, two extra dimensions (cross-line axis and subsurface azimuth ) are added to our seismic image. Therefore we denote the prestack image as . Assuming there are values for , then . Now the maximum-stack-power objective function 1 can be generalized to

 (11)

in which we stack the gather along both and axes.

The next step is to choose a proper residual moveout parameterization for the 3-D ADCIGs, in which the moveout is a surface (defined by ) rather than a curve (of ). There are certainly more than one way to design such parameterization. As an initial attempt, we choose a straightforward approach, in which we separate the moveout surface into individual curves by azimuth . For each azimuth angle , we assign the curvature parameter and the static shift parameter , respectively. Notice that all the curves share the same parameter, because the center of the move-out surface at ( ) is shared by all curves.

Under this parameterization, the 3-D counterpart of objective function 3 would be

 (12)

where becomes a vector, . The gradient formula 4 now turns into

 (13)

Because each is treated separately, we can compute in exactly the same way as we do in the 2-D case.

Analogously, we can define an auxiliary objective function for each image point that uncovers the relationship:

 (14)

Using the same trick of finding partial derivatives for implicit functions, equation 6 is generalized as

 (15)

We differentiate equation 15 with respect to :

 (16)

We can calculate the by Jacobian matrix elements and the right-hand side based on equation 14:

 (17)

Denoting matrix to be the inverse of the Jacobian, then

 (18)

Finally, plugging equation 18 and 17 back into the model gradient expression 13, we get

 (19)

in which

 (20)

In practice, there are some caveats when taking the inverse of the Jacobian matrix. The Jacobian can be ill-conditioned when all elements in one row or column are close to zero. For example, if the image point is not illuminated from a certain azimuth direction , i.e. , then the row and column of the Jacobian would be zero. In order to avoid numerical overflow under this circumstance, we pre-exclude those azimuth angles with poor illumination energy from the Jacobian, and we invert a subset of the original Jacobian that contains only image gathers at well-illuminated azimuth angles.

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

Next: Gaussian anomaly example Up: 3-D RMO WEMVA method Previous: 2-D Theory Review

2012-05-10