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

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

where

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:

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

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:

Denoting matrix to be the inverse of the Jacobian, then

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 |

2012-05-10