Algorithm

Let v_w be the velocity of water. Assume that the seismic velocities of sediments are greater than $\sqrt{2}v_w$.The algorithm of eliminating the water-bottom multiples consists of three steps:

1.

Applying the forward transformation with velocity $\sqrt{2}v_w$ to the shot gather:

$$I(t_0,x)={\bf P}(\sqrt{2}v_w)D(t,x_r).$$ (7)

In the image I(t₀,x), the water bottom multiples are migrated to the left side of the shot while other events stay on the right side of the shot.

2.

Setting I(t₀,x) to zero for all x smaller than x_s:

$$\hat{I}(t_0,x)=\left\{ \begin{array} {ll} 0 & x \leq x_s \\ I(t_0,x) & x \gt x_s \end{array}. \right.$$ (8)

3.

Applying the backward transformation with velocity $\sqrt{2}v_w$ to $\hat{I}(t_0,x)$:

$$\hat{D}(t,x_r)={\bf Q}(\sqrt{2}v_w)\hat{I}(t_0,x)$$ (9)

The final result $\hat{D}(t,x_r)$ will not contain events of velocity v_w.

Until now, I have always assumed that D(t,x_r) is a common shot gather and the subsurfaces are horizontal reflectors. Actually, the same algorithm can be applied to CMP gathers because the travel-time equations for the reflection from a horizontal reflector are identical in a CMP gather and a CSG. For a CSG, the assumption of horizontal reflectors is necessary because the receiver array on the right side of the shot can receive the reflected energy from a dipping bed that is on the left side of the shot. Therefore, after forward transformation, the image of the dipping bed could possibly appear on the left side of the shot. The situation is different when the algorithm is applied to CMP gathers. On a CMP gather, the moveout of a dipping reflector is the same as the moveout of a flat reflector of high stacking velocity. After the forward transformation, the image of a dipping reflector will stay in the positive offsets. Thus the algorithm can adapt the reflection from a dipping layer when it is applied to a CMP gather.