Source-receiver migration

Source-receiver migration is based on the concept of survey sinking. After each depth propagation step, the propagated wavefield is equivalent to the data that would have been recorded if all sources and receivers were placed at the new depth level. This task is accomplished by downward continuing all the source and receiver gathers at each depth step. Therefore, the basic downward continuation is performed by convolving with the Double Square Root (DSR) equation, as  

$$\P_{z}\left(\omega,{\bf g},{\bf s}\right)= \P_{z=0}\left(\om... ...el{{\bf g}}{\ast} e^{ik_zz} \stackrel{{\bf s}}{\ast} e^{ik_zz},$$ (4)

where the first convolution downward-continues the receiver wavefield, whereas the second convolution downward-continues the receiver wavefield. Notice the positive sign on both exponentials in equation (4).

At each depth level, the image is extracted from the downward-continued wavefield by evaluating the wavefield at zero time. The image-space coordinates and the source-receiver coordinates are linked by the well-known transformations

$$\begin{eqnarray} x_s=x-{x_h} & \;\;\;\;\;\;\; & x_g=x+{x_h} \nonumber \\ y_s=y-{y_h} & \;\;\;\;\;\;\; & y_g=y+{y_h}. \nonumber \\ \end{eqnarray}$$

The image cube is then computed as  

$$I_{\rm s-g}\left(z,x,y,{x_h},{y_h}\right) = \sum_{\omega} \P_{z}\left(\omega,x+{x_h},y+{y_h},x-{x_h},y-{y_h}\right)$$ (5)