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 (h_y=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 ($\bf L'$)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 $h_y\neq 0$ to h_y=0. We can think of it in terms of an operator $\bf Z'$which is a sumation over h_y. We allow for some mixing between h_x by expanding our sumation to form h_x=a h_y=0, by summing over all h_y and

 

$$\sum_{a - \Delta h_x}^{a+\Delta h_x},$$ (1)

where $\Delta h_x$ is small.

We can combine these two operators to estimate a 4-D model ($\bf m$) from a 5-D irregular dataset ($\bf d$) through,  

$$\bf m= \bf Z' \bf L' \bf d .$$ (2)

Equation 2 amounts to just running the adjoint of the inversion implied by,  

$$Q(\bf m)= \vert\vert\bf d- \bf L\bf Z\bf m\vert\vert^2 .$$ (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),

$$\bf m= \bf H^{-1} \bf Z' \bf L' \bf d,$$ (4)

where  

$$\bf H^{-1} = \rm diag \left[ \frac{ \bf Z'\bf L'\bf L\bf Zm_{{\rm ref}}}{{\bf m}_{{\rm ref}}} \right].$$ (5)