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In this case, we are trying to predict the data sequence *x*_{T} from its lagged
past values. To do that, we are minimizing:
*L* is the lag of the prediction. So we can apply my adaptive
algorithms, taking *y*_{T}=*Z*^{L}*x*_{T}, or *y*(*t*)=*x*(*t*-*L*). To have an adequate
prediction model, I take a marine shot-gather, where I will try to
remove the water-bottom multiples and peglegs. *L* should be the
two-way traveltime in the water trench.
However, the prediction model is not adequate in the time-offset domain,
because the lag of the water-bottom multiples is not constant on a non-0 offset
trace. Actually, this lag is a function of the angle of incidence of
the wavefront at the receiver. As the parameter is constant
along a ray, this lag depends only upon the *p*-parameter of the incident ray.
So, to make the prediction model more appropriate, it is better to transform
the data to the domain with a slant-stack transform, and apply
the prediction process in this domain.

Calling *V*_{W} the water velocity, the variation of the lag *L* with the
*p*-parameter is given by the relationship:

Thus, after having transformed the input gather to the domain,
I will apply the adaptive algorithms on each trace, with *y*_{T}=*Z*^{L}*x*_{T},
and *L* given by the previous relation. However, I recognize that this method
should not be efficient for large *p*-parameters, because the lag *L* will be
very small (close to 1): the process will try at the same time to remove the
multiples and to compress the signal. But for my purpose, which is a comparison
between different adaptive methods, I consider it is sufficient.

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** Up:** EXAMPLE: MULTIPLES REMOVAL
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Stanford Exploration Project

1/13/1998