Shot-profile Migration

In shot-profile migration, each shot is treated as an independent physics experiment. Each shot is migrated separately and the images of all the shots are then stacked to generate the final image. The source for shot-profile migration is not necessarily an impulse function. Actually, it is a wavelet for a point shot in practice. The source can also be a plane wave, a primary reflection Guitton (2002), or some other configurations. We assume that the source wavefield is a down-going wavefield and the receiver wavefield is an up-going wavefield. These two wavefields are downward continued independently. Let $U(x_U,z=0,\omega,s)$ be the receiver wavefield and $D(x_D,z=0, \omega,s)$ be the source wavefield at the surface for shot s. The source wavefield at the subsurface $U(x_U,z,\omega,s)$ can be obtained by extrapolating $U(x_U,z=0,\omega,s)$ with the up-going wave equation  

$$\frac{\partial}{\partial z}U(x_U,z,\omega,s)= \frac{i\omega... ...\omega^2}\frac{\partial^2}{\partial x_U^2}}U(x_U,z,\omega,s) ,$$ (1)

and the source wavefield at subsurface $D(x_D,z,\omega,s)$ can be obtained by extrapolating $D(x_D,z=0, \omega,s)$ with the down-going wave equation  

$$\frac{\partial}{\partial z}D(x_D,z,\omega,s)= \frac{-i\omeg... ...{\omega^2}\frac{\partial^2}{\partial x_D^2}}D(x_D,z,\omega,s) .$$ (2)

where $\sqrt{1+\frac{v^2(x_D,z)}{\omega^2}\frac{\partial^2}{\partial x_D^2}}$ is a pseudo-partial differential operator Zhang (1993). The image for shot s is formed by cross-correlating the source wavefield $D(x_D,z,\omega,s)$and receiver wavefield $U(x_U,z,\omega,s)$ along the time axis at all depths and evaluating this at zero time lag Claerbout (1971). Stacking the images of all the shots, we can get the image of frequency $\omega$  

$$I(x,z,\omega)=\sum_s U(x_U=x,z,\omega,s)\bar{D}(x_D=x,z,\omega,s),$$ (3)

and the image of all frequencies

$$I(x,z)=\sum_{\omega} I(x,z,\omega).$$ (4)

To perform a velocity analysis, Rickett and Sava (2002) developed the Common Image Gather (CIG) for shot-profile migration, calculated by  

$$I(x,h,z)=\sum_{\omega}I(x,h,z,\omega)=\sum_{\omega}\sum_s U(x_U=x+h,z,\omega,s)\bar{D}(x_D=x-h,z,\omega,s).$$ (5)