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## PS geometry regularization

Rosales and Biondi (2002a) introduce the converted-wave azimuth moveout operator. This operator transforms data from a given offset and azimuth to data with a different offset and azimuth. This operator is a sequential application of converted-wave dip moveout and its inverse. PS-AMO reduces to the known expression of AMO for the extreme case when the P velocity is the same as the S velocity. Moreover, PS-AMO preserves the resolution of dipping events and internally applies a correction for the lateral shift between the common midpoint and the common reflection/conversion point. An implementation of PS-AMO in the log-stretch frequency-wavenumber domain is computationally efficient.

For migration efficiency, we want to use a four-dimensional data cube instead of a full five-dimensional data cube. The crossline offset axis is reduced to only one element (hy=0). The traditional process uses Normal Moveout and stacking along the crossline direction to transform the data from an irregular grid to a regular grid with four axes; however, this technique does not consider the dip and the variations along the inline and crossline directions. In this paper, we use the PS-AMO operator to map the data into a regular 4-D mesh. We follow the method described first in Clapp (2006) and extended for PS data by Rosales and Clapp (2006).

flow
Figure 1
Diagram flow for the implementation of the PS-AMO operator.

We use the nearest-neighbor interpolation operator ()to map the data from an irregular mesh into a regular mesh. The PS-AMO operator (diagramed in Figure 1) allows the transformation between various vector offsets. We use PS-AMO to transform data from to hy=0. We can think of it in terms of an operator which is a sumation over hy. We allow for some mixing between hx by expanding our sumation to form hx=a hy=0, by summing over all hy and

 (1)
where is small.

We can combine these two operators to estimate a 4-D model () from a 5-D irregular dataset () through,
 (2)
Equation 2 amounts to just running the adjoint of the inversion implied by,
 (3)
The adjoint solution is not ideal. The irregularity of our data can lead to artifical amplitude artifacts. A solution to this problem is to approximate the Hessian implied by equation 3 with a diagonal matrix based on a reference model Rickett (2001),
 (4)
where
 (5)

Next: PS common-azimuth migration Up: Theory Previous: Theory
Stanford Exploration Project
1/16/2007