Attenuation of Multiples

The third term on equation (6) corresponds to the multiples and peg-legs associated with the water bottom and must be removed so that equations (7) and (8) can be used to separate the two upcoming waves. The process used is a simplified version of the method developed by Morley (1982) and described by Wiggins (1988) for the case of a nearly horizontal ocean floor.

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

$$m(x,t) = \alpha(p) {\cal L}(x,t,z) d(x,t) ,$$

where $\alpha(p)$ is the product of the two mentioned reflection coefficients at horizontal slowness p, and z is the water depth. The determination of $\alpha(p)$ and z is achieved by an optimization algorithm that minimizes the objective function

$$\delta(\alpha_{(p)},z) = \sum_x \sum_t \parallel d(x,t) - m(\alpha,z,x,t) \parallel^2 .$$

This optimization is easily performed in the $\omega$-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

$$\delta(\alpha_{(p)},z) = \sum_k \sum_{\omega} \parallel d(k,\omega) - m(\alpha,z,k,\omega) \parallel^2 .$$

The comparison between the computed curve $\alpha(p)$ 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 $\alpha(p)$.

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.