The pressure field p(x,z,t) is a finite function and can be therefore expressed as a double Fourier series
Substituting the equation (2) in equation (1) we obtain
which after considering only the equation inside the square parenthesis becomes
For a laterally invariant velocity and small values of dz, we can consider the velocity constant for a thin depth interval. We write
For laterally variant media, Gazdag (1984) proposes to downward extrapolate the wave field one depth interval at a time with several velocities, then to inverse Fourier transform in x the extrapolated field and finally to interpolate each point along the X axis. In other words starting with equation (3) we consider several constant velocities (v1, v2, ...) in the interval [vmin,vmax] and downward extrapolate the wavefield to with each velocity. The resulting wavefields
are then inverse Fourier transformed in the direction of the X axis to obtain
To obtain a single downward extrapolated wavefield, for each point of X coordinates associated with a velocity v(z,x), the value of the resulting wavefield in this point is interpolated between the two wavefields with closest velocities.
In addition to this idea, Gazdag (1984) implements a technique in the PSPI algorithm to ensure that all the zero dips (corresponding to the case kx=0) are downward continued without distortion. The technique consists of multiplying the wavefield with
prior to the Fourier transformation along the X axis and multiplying the downward extrapolated wavefield by
where the subscript j denotes the index of the constant velocity used in the downward extrapolation step. As it will be seen later in the Split-step Fourier method, this supplemental phase subtractieon and addition improves the accuracy of the results.