Equivalence of source-receiver migration and shot-profile migration

For the sake of simplicity, I demonstrate the equivalence by showing that the images obtained by migrating a single shot record are the same. The linearity of both migrations with respect to the input wavefield makes the extension to the full data set obvious.

A crucial observation for proving the equivalence of the two migration methods is that the downward continuation of the sources commutes with the downward continuation of the receivers. This property is obvious for vertically layered media where downward continuation can be performed in the wavenumber domain. However, it is also valid in presence of lateral velocity variations, because the wavefield is downward-continued along each direction by a convolution that is independent from the other direction. For example, the sources are downward continued by convolving each receiver gather with a convolutional operator that is non-stationary along the source axis, but is independent of the location of the receiver gather.

The wavefield at the surface for one single shot gather is given by the products of two functions: the first is independent of the source-coordinate ${\bf s}$(the recorded data $P^g_{z=0}\left(\omega,{\bf g};{\bar {\bf s}}\right)$), the second is independent of the receiver-coordinate ${\bf g}$(a delta function at ${\bar {\bf s}}$).

The wavefield at depth obtained by survey sinking can thus be expressed as

$$\begin{eqnarray} \lefteqn{ \P_{z}\left(z,{\bf g},{\bf s}\right) } \nonumber \\ &... ...left(\omega,x_s,y_s;{\bar {\bf s}}\right) } \right]. \\ \nonumber \end{eqnarray}$$ (6)

Imaging is performed by evaluating the downward-continued wavefield at the appropriate locations, as described by equation (6). If we apply this imaging condition to the wavefield in (7), we obtain

$$I_{\rm s-g}\left(z,x,y,{x_h},{y_h}\right) = \sum_{\omega} \... ...z}\left(\omega,x-{x_h},y-{y_h};{\bar {\bf s}}\right) } \right],$$ (7)

that is exactly the same image cube as the image cube obtained by shot-profile migration [equation (3)].