As illustrated in Figure , the ocean-bottom multiples associated with the upcoming wavefield can be predicted using the following process:

- 1.
- Replacement of the air layer by a water layer, above the water surface.
- 2.
- Upward continuation of the wavefield by a distance of two water depths.
- 3.
- Multiplication by a factor that corresponds to the product of the reflection coefficients of the water bottom and water surface.

Similarly, the downgoing wavefield (or ghost), can have its multiples predicted by the same process, except that the wavefield must be continued downward instead of upward. Since the operators to downward-continue a downgoing wave or upward-continue an upcoming wave in a homogeneous media are identical, the predicted wavefields corresponding to the multiples can be expressed by

where is the product of the two mentioned reflection coefficients at horizontal slowness

This optimization is easily performed in the *- k_{x} *
domain because the extrapolation operator has the form of a simple
phase shift, and the decomposition into different values of slowness
is accomplished by simple radial slices within the transform plane.
The objective function on this domain assumes the form

The comparison between the computed curve * *
and the theoretical curve for the ocean-floor reflection
coefficient (Figure ) shows that the computed reflection coefficient
diverges from the theoretical at small and large values
of slowness. The reason for this result is that no primaries are
recorded at small angles (because of the initial offset gap), and no
multiples are recorded with large incident angles. The primaries
that generated the multiples that are present in the near traces
were not recorded, and so, can not be used to predict their
multiples. Another way to see the problem is to recall that the
upward continuation process corresponds to a propagation of
energy from small to large offsets and in the direction of
increasing time. Therefore, the near traces in the model
will not have enough energy to properly represent the multiples.
It is important to notice also that in real data, the source's
directiveness will have a major influence on .

Although the efficiency of the method at small offsets is not critical for the present goals of recovering the converted waves where they are sufficiently strong (large offsets), a possible way to improve the performance at near offsets is the use of an extrapolation process to fill in the gap before the upward continuation.

1/13/1998