Shot-profile migration

In shot-profile migration each record is migrated independently. The receiver wavefield P^g is downward continued starting from the recorded data. The source wavefield P^sis downward continued starting from an assumed source wavelet. In this case, we assume that the source is a delta function in the space domain and a constant as a function of frequency (impulse at time zero).

Each wavefield is propagated independently by convolution with the Single Square Root operator (SSR). At each depth level that is;  

$$^{\delta}P^s_{z}\left(\omega,x,y;{\bf s}\right)= \delta\left(x-x_s,y-y_s\right) \stackrel{x,y}{\ast} e^{-ik_zz}.$$ (1)

and  

$$P^g_{z}\left(\omega,x,y;{\bf s}\right)= P^g_{z=0}\left(\omega,x,y;{\bf s}\right) \stackrel{x,y}{\ast} e^{ik_zz},$$ (2)

where x and y are defined in the image (model) space, and ${\bf s}=\left(x_s,y_s\right)$ is the location of the shot. The prefix $\delta$ in $^{\delta}P^s_{z}$ indicates that the source function is an impulse. Notice the negative sign in front of the exponential in equation (1). The negative sign is there because the source wavefield propagates downward, as opposed to propagate upward as the receiver wavefield does.

The image cube is formed by cross-correlating along the time axis the two wavefields shifted with respect to each other along the horizontal axes. In the frequency domain the cross-correlation is performed by multiplication with the complex conjugate, and it is evaluated at zero lag by summation over frequencies. The horizontal shift is the subsurface offset $\left({x_h},{y_h}\right)$.The image cube is thus computed as:  

$$I_{\rm shot}\left(z,x,y,{x_h},{y_h}\right) = \sum_{x_s} \su... ...lta}P^s_{z}\left(\omega,x-{x_h},y-{y_h};{\bar {\bf s}}\right) }$$ (3)