Kinematic operators

Let D(t,x_r) be a common shot gather (CSG). Define ${\bf P}(v)$to be a velocity-dependent operator that transforms the data D(t,x_r) into its image I(t₀,x).

$$\begin{eqnarray} I(t_0,x) & = & {\bf P}(v) D(t,x_r) \\ & = & \sum_{\tau} \sum_{x_r} W_{D}(\tau,x_r)D(\tau,x_r)\delta(\tau-t),\end{eqnarray}$$ (3) (4)

where W_D is a weighting function and t is defined in equation (1). Notice that the operator ${\bf P}(v)$ is actually a Kirchhoff migration operator of constant velocity v. Similarly, the backward transformation from I(t₀,x) to D(t,x_r) is

$$\begin{eqnarray} D(t,x_r) & = & {\bf Q}(v) I(t_0,x) \\ & = & \sum_{\tau} \sum_{x} W_{I}(\tau,x)I(\tau,x)\delta(\tau-t_0),\end{eqnarray}$$ (5) (6)

where t₀ is defined in equation (2). ${\bf Q}(v)$ may be the inverse or transpose of ${\bf P}(v)$ depending on the choice of weighting function W_I. The parameter v of the operators plays an important role in both forward and backward transformations. For the purpose of imaging, we set v to be the velocity of the medium. As a result, I(t₀,x) will be the precise image of the earth. For other values of v, however, I(t₀,x) will be a distorted image. Our goal is to select a special value of v such that after the forward transformation with operator ${\bf P}(v)$ the images of the water bottom multiples are as separable as possible from the images of other events.