Propagation of elastic waves in a homogeneous medium

I compute the propagation of waves in $(\omega,p_x,p_y)$ space in a homogeneous medium by phase shifting in depth. Given a wavefield at some depth z₀, $P(\omega,p_x,p_y,z_0)$ the wavefield at another depth, z, can be expressed as a phase shifted version of this wavefield.

$$P(\omega,p_x,p_y,z) = P(\omega,p_x,p_y,z_0) e^{i{\bf A}(z-z_0)}$$

In general the operator ${\bf A}$ is a matrix operator but if the wavefield, P is expressed as a vector of amplitudes of the different wavetypes the operator A is diagonal. The different wavetypes are the eigensolutions of the Christoffel equation (see appendix) so the operator is diagonalized. For each of the six wavetypes the extrapolation in depth can be expressed as,

$$P_i(\omega,p_x,p_y,z) = P_i(\omega,p_x,p_y,z_0) e^{i\omega p_{z_i}(z-z_0)} \ \ \ \ i=1,\ldots,6$$

Thus, once I have calculated the wave amplitudes at the top of any layer I can calculated the amplitudes at the bottom of the layer by phase shifting the data. The value of p_z is the same for all frequencies but the phase shift through the layer for each frequency is given by.

$${\bf d_{bottom}} = \pmatrix{ e^{i\omega p_z(1)( z_{bottom}- ... ...i\omega p_z(3)(z_{bottom}- z_{top})} \cr } \cdot {\bf d_{top}}$$

This calculation is different for each frequency but on a vector computer it is fully vectorized over frequency.

If I am ``turning off'' propagation of a particular wavetype it is at this stage that I do it. I can simply zero the amplitude of the wave instead of applying the shifting operator.