Arbitrary velocity structures

When the velocity structure varies in the lateral direction, the Fourier transform of velocity is no longer a delta function and the submatrix A₂₁ has the form

$$[A_{21}] =\pmatrix{-{\omega^2 s_0}+k_x^2 & -\omega^2 s_1 & \l... ...x-1} & -\omega^2 s_{N_x-2} & \ldots & -{\omega^2s_0}+k_x^2 \cr}$$ (15)

where

$$s_n(z)={1 \over L}\int_{-L\over 2}^{L\over 2} c^{-2}(x,z)e^{-in\Delta kx}dx$$

Another derivation of submatrix A₂₁ is possible in space frequency domain (Kosloff and Kessler, 1987). Whether the derivative in equation (6) is calculated by the Fourier method as with the ordinary phase-shift method (Gazdag, 1978), or by finite differences with periodic horizontal boundary conditions, it can be written as the cyclic convolution

$${\partial^2 P\over \partial x^2}(idx,z,w)=\sum_{j=0}^{N_x}{P(jdx,z,w)W_{i-j}}$$ (16)

where W_i denotes a convolution operator. For example, for second-order finite differences, $W_0=-{2\over \Delta x^2}$ and $W_{-1}=W_1={1\over \Delta x^2}$, and W_i=0 for |i|>1. For Fourier second-derivative operator,

$$W_0 = -({2 \pi \over{N_x dx}})^2{L(L+1) \over 3}$$

and

$$W_n = {1 \over 2}({2 \pi \over{N_x dx}})^2(-1)^n{{\cos({\pi n\over N_x})}\over {\sin^2({\pi n\over N_x})}}$$

with N_x=2L+1(Kosloff and Kessler, 1987). With the spatial discretization and specification of the second-derivative approximation, the elements of the N_x by N_x submatrix A₂₁ are given by

$$[A_{21}] =\pmatrix{-{\omega^2 \over c_1^2}-W_0 & -W_1 & \ldot... ...{x-1}} & -W_{N_{x-2}} & \ldots &-{\omega^2\over c_{N_x}^2}-W_0}$$ (17)

where c_i=c(idx,z)