The pressure field p(x,z,t) is a finite function and can be therefore expressed as a double Fourier series
Substituting equation (2) in equation (1) we obtain
For a constant velocity we write
However there are several ambiguities in equation (7) that we need to discuss. One is the question of time propagation. If we know the pressure field (or wavefield) at a certain depth we can propagate it forward in time or backward in time. We can also propagate it up (along the z-axis) or down. To understand how we determine the propagation direction we have to analyze the values and sign of kz. There are several restrictions on the values of kz. Equation (6) has the solution (7) only for real values of kz which imposes the condition
The solution represented in equation (7) is the Fourier transform of the wavefield. The general solution in time-space coordinates is obtained by summing all the Fourier coefficients obtained from equation (7)
represents a plane wave. Equation (8) sums many plane waves to obtain the general solution. We examine the sign of kz necessary to upward propagate a wave by examining each plane wave solution. If we ignore kx x, which determines the lateral variation, we can write
The phase is constant along a plane wave, and we write
for the phase of a particular plane wave. The plane wave is moving downward when kz has the same sign with because z increases with t in order to keep the phase constant. So for the upward moving waves we need to have opposite signs of kz and (z is decreasing when t is increasing). We have now figured out that in order to have only upgoing waves we have to look at the sign of and assign to kz the opposite sign.
Next we correlate this information with the depth level where we want to find the wavefield. If the known wavefield is at depth z0 and we want to find the wavefield at depth z0+z, then we have to propagate the wavefield back in time (toward t=0) because we know that the wavefield travels upward. If the known wavefield is at depth z0 and we want to find the wavefield at depth z0-z, then we are propagating the wavefield forward in time. This is the direction we are interested in for modeling.
However for depth varying velocity v(z) we have kz approximately constant only for small depth intervals () where we can consider the velocity constant. Therefore equation (7) becomes
For laterally variant media, Gazdag and Sguazzero (1984) propose to downward extrapolate the wavefield one depth interval at a time with several velocities. We can apply the same idea to upward propagate the wavefield with several velocities. We consider several velocities (v1, v2, ...) in the interval [vmin,vmax] and upward propagate the wavefield to with each velocity. We can afterward inverse Fourier transform in x the resulting wavefields
To obtain a single upward propagated wavefield, at each point associated with a velocity , the value of the resulting wavefield in this point is interpolated between the two wavefields with closest velocities ().
In addition we need to implement a technique (Gazdag and Sguazzero, 1984) in the PSPI modeling algorithm to ensure that all the zero dips (corresponding to the case kx=0) are upward propagated without distortion. The technique consists of multiplying the wavefield with
As it will be seen later when the PSPI results are compared to the Split-step Fourier results, this supplemental phase subtraction and addition improves the accuracy in the migration case but actually produces worse results in the modeling case. Figure shows the flow of the PSPI migration and modeling algorithms.