Theory

The space-domain shot-profile imaging condition including subsurface offset for shot-profile migration Rickett and Sava (2002) is  

$$I(x,h)\vert _{\omega,z}=U(x+h)\,D^*(x-h)\;,$$ (1)

Where I is the migrated image produced by cross-correlating the up-coming, U, and down-going, D, wavefields at every depth and frequency. Both x and h can be areal vectors. ^* represents complex conjugation. To derive the Fourier-domain equivalent, we will perform a piece-wise proof and begin with the Fourier transform D^* to $\hat{D}^*$ (neglecting Fourier scaling)

$$\hat{I}(x,h)=U(x+h)\,\int \hat{D}^*(k_s)\,e^{i\,k_s\,(x-h)}\,dk_s \;.$$ (2)

Continue by Fourier transforming the variable x to find

$$\hat{I}(k_x,h)=\int U(x+h)\,\int \hat{D}^*(k_s) e^{i\,k_s\,(x-h)}\,dk_s e^{-i\,x\,k_x}\,dx \;.$$ (3)

By reordering variables, the equivalent form

$$\begin{eqnarray} \nonumber \hat{I}(k_x,h) & = & \int\hat{D}^*(k_s)\, e^{-i\,k_s\... ...\,h\,(k_x-2\,k_s)} \int U(x')\,e^{-i\,x'\,(k_x-k_s)}\,d x'\,dk_s\end{eqnarray}$$ (4)

is achieved. From here, we can recognize the inner integral is the Fourier transform of the wavefield U which can be replaced directly to yield

$$\hat{I}(k_x,h)= \int\hat{U}(k_x-k_s)\,\hat{D}^*(k_s)\, e^{i\,h\,(k_x-2\,k_s)}\,dk_s \;.$$ (5)

With the use of the definition of offset, $k_h=k_x-2\,k_s$, we can replace several of the above arguments with equivalent expressions to find

$$\hat{I}(k_x,h)= \frac{1}{2}\,\int\hat{U}\left(\frac{k_x+k_h}... ...hat{D}^*\left(\frac{k_x-k_h}{2}\right)\,e^{i\,h\,k_h}\,dk_h \;.$$ (6)

The last integral is recognized as an inverse Fourier transform, this time over the k_h variable. Using this fact, we arrive at the multi-dimensional (over x and h, which can be two-dimensional themselves) Fourier transform of the general shot-profile imaging condition  

$$\widehat{I}(k_x,k_h)=\frac{1}{2}\,\hat{U}\left(\frac{k_x+k_h}{2}\right) \,\hat{D}^*\left(\frac{k_x-k_h}{2}\right)\;.$$ (7)

From this equation, the result that the Fourier-domain equivalent to the conventional space-domain imaging condition for shot-profile migration is again a lagged multiplication of the up-coming and down-going wavefields at each frequency and depth level. Evaluating the arguments inside the wavefields to produce a component of the image shows that the wavefields, in the wavenumber domain, will need to be interpolated by a factor of two to calculate the image space output. The Table 1 showing example calculations of the components of the image space looks like

  

Table 1: Layout of wavenumber components in Fourier-domain imaging conditions