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When the velocity structure varies arbitrarily in the lateral direction,
the eigenvalues and eigenvectors of matrix A can no longer be obtained by
inspection. It would therefore seem that a matrix diagonalization would
have to be performed before each propagation to equation (11). However,
recent work by Tal-Ezer(1986) indicates how the calculation of
matrix exponential can be done without having to resort to expensive
matrix diagonalizations.
The solution is based on a Chebychev expansion for the function *e*^{x} given by
| |
(18) |

where x is imaginary and *C*_{0}=1 and *C*_{k}=2 for *k*>0.
The polynomials *T*_{k}(*x*) satisfy the recurrence relations
*T*_{0}(*x*) = 1,

*T*_{1}(*x*) = *x*,

and
*T*_{k+1}(*x*)=*T*_{k-1}(*x*)+2*xT*_{k}(*x*)

(Tal-Ezer, 1986). By analogy with equation (18), the exponent in
equation (4) is expanded according to
| |
(19) |

This expansion is valid when the eigenvalues of [*Adz*] are purely imaginary
and *R* must be chosen large enough to span the range of the eigenvalues of
[*Adz*]. It was shown in Tal-Ezer(1986) that for *k* > *R*, the series expansion
converges exponentially. The number of terms required in the sum in
equation (19) will therefore always be finite.
Equation (19) serves as the basis for implementing the generalized phase-shift
migration. First the range *R* of the eigenvalues of [*Adz*] need to be
estimated.
Based on the case of laterally uniform velocity, Kosloff showed the estimate
*r*=*wdz*/*c*_{min} with *c*_{min} denoting the lowest velocity in the strip
(*z*,*z*+*dz*), is sufficient for stable results. The Bessel functions *J*_{k}(*R*) are
computed next.
The solution
is then calculated recursively according to following:
The first two values of *T*_{0}, *T*_{1} needed to initialize the recursion are
given by
and
The next polynomial is generated by the recursion formula
These steps are repeated until a sufficient number of terms have been calculated
in the sum(19). Then the solution is carried out in the next lower level.
When has been calculated completely, the final migrated
section is cumulated by
(Kosloff and Baysal, 1983).

** Next:** GENERATION OF SURFACE VALUES
** Up:** THE GENERALIZED PHASE-SHIFT METHOD
** Previous:** Arbitrary velocity structures
Stanford Exploration Project

1/13/1998