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# 15-DEGREE DEPTH MIGRATION

The first-order approximations of the one-way wave equation by both continued-fraction expansion and Taylor-series expansion result in the same equation: the conventional 15-degree equation. The 15-degree migration equation in the domain (Claerbout, 1985) is given by
 (6)
After a second difference approximation to the second partial derivative and subsequent simplification, the equation (6) becomes
 (7)
where the matrix M takes the form

Rearranging,

with

and

The solution to equation (7) is
 (8)
The calculation of the exponentiated matrix requires matrix diagonalization, which is numerically expensive. However, we can approximate the exponential of the matrix by writing (Richardson, et al., 1991)
 (9)
where the matrix M is split into two matrices M=Me+Mo, with

and

The approximate depth-stepping operator,
 (10)
forms the basis for our unconditionally stable, explicit algorithm for migration. To compute the matrix exponentials, notice that both Me and Mo are block diagonal and we need only consider the exponential of the 2 by 2 matrix

By using the eigenvalue decomposition of the matrix, we see that

where Q represents the matrix whose columns are eigenvectors of the matrix E, and is the diagonal eigenvalue matrix.

Since both Me and Mo are block diagonal, exponentiating them amounts to exponentiating E. Both and are also block diagonal. Since the eigenvalues of are and with a and b imaginary, it follows that and .To prove the unconditional stability of the algorithm we need only show that . This follows immediately since , each of which 1 according to the preceding discussion.

In Figure , the impulse responses of both implicit and explicit operators of 15-degree migration are compared for the same extrapolation step . In the case of the implicit scheme, we used a Crank-Nicholson implementation which has an accuracy on the order of .For the explicit scheme, we performed a second-order approximation using a split M = Me + Mo whose accuracy is on the order of .We see that the impulse response of the explicit method has the shape of an ellipse, which is characteristic of the 15-degree extrapolation operator. The impulse response of the explicit method shows some dispersion than that of the implicit scheme due to a poor approximation of the matrix exponent. The way to get a more accurate approximation is discussed in the following section.

fig1
Figure 1
The impulse responses (a) of explicit 15-degree migration with split M = Me + Mo and (b) of implicit 15-degree migration with Crank-Nicholson approximation. Both extrapolations are performed with the same extrapolation step .

Next: Accuracy Up: Ji and Biondi: Explicit Previous: ONE-WAY WAVE EQUATION AND
Stanford Exploration Project
12/18/1997