The solution to equation (7) is
The approximate depth-stepping operator,
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.