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The above formulation of the DMO operator allows us to
obtain the impulse response of the DMO in a medium with
variable velocity in depth.
The mappings can be evaluated using
ray-tracing for an arbitrary velocity
model. The migration depends on the half offset *h*
and thus must be computed for each common-offset section, while the
modeling is independent from the offset of the data and must be
evaluated only once.
The algorithm used to investigate the DMO operator follows
the definition of the PSPM method and can be divided
in two parts:
- 1.
- Construct the full migration depth model. For a constant
velocity medium this is equivalent to constructing the
migration ellipse for a constant offset section. For a
variable velocity medium, the loci of the points with
equal travel time from source to receiver is a curve resembling
an ellipse or a superposition of several ellipses. See Figures
?? for an example of migration depth models.
- 2.
- Zero-offset modeling. Given the depth model (the reflector
which generates the same impulse in the constant offset section)
we raytrace back at 90 degree from the reflector, simulating
the zero-offset case. The intersection of the ray with the
surface will give the x coordinate of the DMO operator,
while the travel time along the raypath will provide
the time coordinate. See Figures ?? for an example of
DMO impulse responses.

The raytracing algorithm is based on solving the acoustic
wave equation in the high frequency approximation (eikonal
equation). The velocity model has only a depth variation,
though the algorithm can be easily modified to handle lateral velocity
variation as well. The velocity is assumed to be a continuous
function of depth, and the smoothness of the variation can be
controlled by a B spline interpolation subroutine.
For a medium with linear increase of the velocity with depth
*v*(*z*)=*v*_{0}+*az*,
an impulse will map in a elongated ellipse, the elongation being
proportional with the coefficient *a*. The coefficient *a*
varies from 0.3 to 1.3 according to Telford. The deviation from
the constant velocity case is negligible for an coefficient
*a* smaller then 1, when the DMO velocity is chosen correctly.

For media where velocity is not a linear function of depth, the
DMO impulse response may have serious variations from the constant
velocity case. Triplications can occur in the normally
ellipse-shaped DMO impulse response. Figures ?? show an example of
atypical DMO operators.

More here on DMO differences!!!

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** Up:** Popovici and Biondi: PSPM
** Previous:** THE PSPM OPERATOR IN
Stanford Exploration Project

1/13/1998