Next: CONCLUSION Up: Ji and Biondi: Explicit Previous: Lateral velocity variation

WIDE-ANGLE DEPTH MIGRATION

For increasing the accuracy at higher dips, we can use more terms in the Taylor-series expansion. Resulting in a matrix with thicker bands than the tridiagonal matrix. Using the second-order term in the expansion, we see that the approximated square-root operator takes the form
 (21)
where I is the identity matrix, and T represents the tridiagonal matrix that approximates the second partial derivative. Let

with

and

We split the matrix into five pieces, M=Me+Mo+Mt1+Mt2+Mt3, where

and

In the same manner as the tridiagonal matrix, we can approximate the time-stepping operator as
 (22)
Mt1, Mt2, and Mt3 are block diagonal, and the small block matrix F along the diagonal is defined as

Using the series definition of the exponential function, we see that

Exponentiating Mt1, Mt2, and Mt3 amounts to exponentiating F. The terms , and are also block diagonal. Since the eigenvalues of are 1 and with imaginary c, it follows that .As before, to prove the unconditional stability of the algorithm, we need only show that . This follows immediately since

each of which is equal to 1 according to the preceding argument.

In Figure , we compared the impulse responses of first-order approximation with the second-order approximation in the Taylor-series expansion of the square-root operator. We also superposed semicircles which is the theoretical solution of the extrapolation operator on Fig and the higher order approximation shows better fitting to semicircles than the first-first order approximation.

With the same manner as showed in this section, we can get more accurate operator by taking more terms in a Taylor series expansion of the square-root operator. However, it will produce a matrix with increasing width of the band and thus will cause to an increasing in computation cost.

fig3
Figure 3
Impulse response of (a) explicit depth migration with the first-order approximation for the square-root operator and of (b) explicit depth migration with the second-order approximation for the square-root operator.

Next: CONCLUSION Up: Ji and Biondi: Explicit Previous: Lateral velocity variation
Stanford Exploration Project
12/18/1997