PSPM IMPULSE RESPONSE IN MEDIA WITH VARIABLE VELOCITY

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)=v₀+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.

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