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The derivation of the Phase Shift Plus Interpolation (PSPI) modeling
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 equation (2) in equation (1) we obtain

| |
(3) |

Equation (3) should hold for any values of *k*_{x} and .This is possible only if each term inside the square parenthesis
is zero. This reasoning is similar to the condition that if
a polynomial is zero for any values of *x*, the coefficients of the
polynomial are zero.
Equation (3) becomes
| |
(4) |

valid for all 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 (*x*,*z*) in the medium,
the velocity *v*(*x*,*z*), and the corresponding value of *p*(*x*,*z*,*t*)
at that location.
For a constant velocity we write

| |
(5) |

where *k*_{z} is constant for two given values of *k*_{x} and .Equation (5) is the well known **dispersion **
relation.
Equation (4) becomes an ordinary differential equation
| |
(6) |

For a constant *k*_{z} equation (6) has the analytic solution
| |
(7) |

Equation (7) shows how in constant velocity media we
can find the pressure field at a certain depth level ,
if we know it at any other depth level .
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 *k*_{z}.
There are several restrictions on the values of *k*_{z}. Equation
(6) has the solution (7) only for real
values of *k*_{z} 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)
| |
(8) |

The function
represents a plane wave. Equation (8) sums
many plane waves to obtain the general solution.
We examine the sign of *k*_{z} necessary to upward
propagate a wave by examining each plane wave solution.
If we ignore *k*_{x} *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 *k*_{z} 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 *k*_{z} 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
*k*_{z} 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 *z*_{0} and we want to
find the wavefield at depth *z*_{0}+*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 *z*_{0} and we want to
find the wavefield at depth *z*_{0}-*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 *k*_{z}
approximately constant only
for small depth intervals () where we can consider the velocity constant.
Therefore equation (7) becomes

| |
(9) |

and can be used to downward or upward extrapolate
the wave field for a small depth interval.
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 (*v*_{1}, *v*_{2}, ...) in the interval
[*v*_{min},*v*_{max}] and upward propagate the wavefield
to with
each velocity.
We can afterward inverse Fourier transform in *x* the resulting wavefields

to obtain
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 *k*_{x}=0) are upward propagated without distortion. The technique
consists of multiplying the wavefield with

| |
(10) |

after the Fourier transformation in *x*
and multiplying the upward propagated wavefield
by
| |
(11) |

where
the subscript *j* denotes the index of the constant velocity used in the
upward propagation step.
For a zero dip plane wave (i.e. *k*_{x} = 0) we have
| |
(12) |

and the plane wave can be upward propagated with the velocity
*v*(*x*,*z*) corresponding to the point coordinates (*x*,*z*) instead
of the constant velocity *v*_{j}.
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.

**pspiboth
**

Figure 1 Phase Shift Plus Interpolation (PSPI) migration and the
conjugate transpose PSPI modeling algorithm.

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

11/18/1997