THE GENERALIZED PHASE-SHIFT METHOD

The generalized phase-shift method is based on the solution of the temporally transformed acoustic wave equation

$${\partial^2 P\over \partial z^2} = -{\omega^2\over c^2}P-{\partial^2 P\over \partial x^2},$$ (1)

where $P(x,z,\omega)$ denotes the temporal transform of the pressure field and c(x,z) is velocity field. As suggested by Kosloff and Baysal(1983), it is convenient to recast equation (1) as a set of two first-order coupled equations given by

$${\partial \over \partial z}\pmatrix{P\cr {\partial P\over \p... ...tial x^2}} & 0\cr}\pmatrix{P\cr {\partial P\over \partial z}}.$$ (2)

The downward continuation in the migration consists of the solution of equation (2) for each frequency at all depths under the initial conditions of the values of P and $\partial P / \partial z$ at the surface z=0 (Kosloff and Baysal, 1983). Equation (2) can be expressed in a more compact form

$${\partial \over \partial z} \pmatrix{P\cr {\partial P\over \partial z}} = [A]\pmatrix{P\cr {\partial P\over \partial z}},$$ (3)

where $[ P ,{\partial P / \partial z}]^T$ denotes a column vector of length 2N_x containing first the N_x pressures $P(idx,z,\omega)$ and then the N_x pressure derivatives ${\partial / \partial z} [P(idx,z,\omega)]$, for i=0,...,N_x-1. As with the ordinary phase-shift method, the solution here is propagated in depth increments. Within each increment z to z+dz, the velocity is assumed to be invariant in the vertical direction although it may vary horizontally. The solution of equation (3) can then be written as

$$\pmatrix{P\cr {\partial P\over \partial z}}_{z+dz} = \exp[Adz]\pmatrix{P\cr {\partial P\over \partial z}}_z.$$ (4)

The solution (4) embodies a phase-shift of the eigenvector coefficients of A. It would therefore seem that a matrix diagonalization would have to be performed before each propagation.