Source-Receiver Migration

Traditional source-receiver migration is based on the concept of survey sinking Claerbout (1985). It sorts the recorded data into CMP gathers $P(x,h,z=0,\omega)$ and propagates the CMP gather to the subsurface $P(x,h,z,\omega)$ with the Double Square Root (DSR) equation  

$$\frac{\partial}{\partial z}P(x,h,z,\omega)=\left(\frac{i\ome... ...a^2}\frac{\partial^2 }{\partial x_r^2}} \right)P(x,h,z,\omega),$$ (6)

where x_s=x-h is the shot location, x_r=x+h is the receiver location. The wavefield $P(x,h,z,\omega)$at each depth z is equivalent to the data that would have been recorded when the shots and receivers were located at that depth. Source-receiver migration produces an image by extracting the wavefield at zero subsurface offset $P(x,h=0,z,\omega)$, and stacking over all frequencies. Correspondingly, the stack of $P(x,h,z,\omega)$ along the frequencies is its CIG.

Traditional source-receiver migration assumes that the source is an impulse function at the source location x_s. But in fact, source-receiver migration works for arbitrary sources, such as wavelets, plane waves, and primary reflections as well. For arbitrary sources, the surface wavefield $P(x,h,z=0,\omega)$ is not a simple CMP gather of the recorded data, but the stack of the cross-correlation between the source wavefield and the receiver wavefield at the surface  

$$P(x,h,z=0,\omega)=\sum_s U(x_U=x+h,z=0,\omega,s)\bar{D}(x_D=x-h,z=0,\omega,s).$$ (7)

Then $P(x,h,z=0,\omega)$ is downward continued to the subsurface with the DSR equation (6) and the image is formed by the same method as the traditional source-receiver migration. In fact, traditional source-receiver migration is a special case of source-receiver migration when the source is an impulse function at the source location, and the CMP gather of the recorded data at the surface is the cross-correlation between the impulse function source and the receiver wavefield, which is the recorded data for each shot.