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The derivation of the Phase Shift Plus Interpolation (PSPI) algorithm
starts with the scalar wave equation
| |
(1) |

where *p*=*p*(*x*,*z*,*t*) is the pressure field and *v*=*v*(*x*,*z*) is the earth
velocity.
The pressure field *p*(*x*,*z*,*t*) is a finite function and can be therefore
expressed as a double Fourier series

| |
(2) |

Substituting the equation (2) in equation (1) we obtain

which after considering only the equation inside the square parenthesis
becomes
| |
(3) |

valid for all the values of *k*_{x} and .The problem is that in this form, the *x* coordinate in the pressure
field is Fourier transformed and there is no direct
correspondence between a point in the medium of coordinates (*x*,*z*),
the velocity *v*(*x*,*z*), and the corresponding value of *p*(*x*,*z*,*t*)
at that location.
For a laterally invariant velocity and small values of *dz*, we can
consider the velocity constant for a thin depth interval. We write

| |
(4) |

where *k*_{z} is constant for two given values of *k*_{x} and .Equation (3) becomes an ordinary differential equation
| |
(5) |

which has the analytic solution
| |
(6) |

Using equation (5) to downward extrapolate the wave field
in laterally invariant velocity media *v*(*z*) forms the basis of
the classic Phase Shift algorithm (Gazdag, 1978).
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 (*v*_{1}, *v*_{2}, ...) in the interval
[*v*_{min},*v*_{max}] 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 *k*_{x}=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.

** Next:** THE SPLIT-STEP FOURIER METHOD
** Up:** Popovici : PSPI and
** Previous:** INTRODUCTION
Stanford Exploration Project

12/18/1997