Prestack phase-shift migration

The Phase-shift operator in a prestack Fourier-domain is described by the double-square-root equation, which is a function of the angular frequency ${\omega}$,the midpoint rayparameter p_x, the offset rayparameter p_h, and the velocity, v. Constant-velocity prestack migration, with output provided in two-way vertical time ($\tau$), in offset-midpoint coordinates (), is given by:  

$$g(t=0,k_x,h=0,\tau)= {\int d\omega \int d k_h \; e^{i\omega \tilde{p}_{\tau}(p_x,p_h)\tau} F(\omega,k_x,k_h,\tau=0)},$$ (119)

where $F(\omega,k_x,k_h,\tau=0)$ is the 3-D Fourier transform of the field $f(t,y,h,\tau=0)$ recorded at the surface, given by

$$F(\omega,k_x,k_h,\tau=0)= \int dt \; e^{-i\omega t} \int dy e^{i k_xx} \int dh e^{i k_hh} f(t,x,h,\tau=0),$$

and $k_x= 2 \omega p_x$, $k_h= 2 \omega p_h$, and $k_z= 2 \frac{\omega}{v} p_{\tau}$ are the horizontal midpoint wavenumber, the horizontal offset wavenumber, and the vertical wavenumber, respectively. In this paper, I will freely alternate between the half offset, h, and the full offset, X, in representing the offset axis, where X=2h. The phase factor $\tilde{p}_{\tau}(p_x,p_h)$, for isotropic media, is defined as  

$${\tilde{p}_{\tau}(p_x,p_h)} \equiv { \frac{1}{2} (\left[1 - ... ...{1 \over 2} +\left[1 - v^2 (p_x-p_h)^2\right]^{1 \over 2}) },$$ (120)

which is a normalized version of the double-square-root (DSR) equation. The two integrals in ${\omega}$ and k_h in equation () represent the imaging condition for zero offset and zero time (h=0,t=0).

For VTI media, the phase factor is given by a more complicated equation (),  

$${\tilde{p}_{\tau}(p_x,p_h)} \equiv \frac{1}{2} \left(\sqrt{1... ...rt{1-\frac{(p_x-p_h)^2 v^2}{1-2 \eta v^2 (p_x-p_h)^2}} \right).$$ (121)

The dispersion equation, now, includes $\eta$ as well as the velocity.

Kirchhoff migration is typically applied by smearing an input trace, after the proper traveltime shifts, over the output section in a summation process. To obtain the response of inserting a single trace into the prestack phase-shift migration [equation ()], we multiply the input data by a Direc-delta function in midpoint and offset axes as follows  

$$f(t,x,h,\tau=0) = \tilde{f}(t,x,h,\tau=0) \delta(x-x_0,h-h_0),$$ (122)

where x₀ is the midpoint location and h₀ is the offset of the input trace. The Fourier transform of equation () is given by

$$F(\omega,k_x,k_h,\tau=0) = \tilde{F}(\omega,x_0,h_0,\tau=0) e^{i k_x x_0+i k_h h_0}.$$

Inserting this equation into equation () provides us with the migration response to a single input trace given by  

$$g(t=0,x,h=0,\tau)= \int d\omega \tilde{F}(\omega,x_0,h_0,\tau=0) {\int d k_h \int d k_x \; e^{i\omega T}},$$ (123)

where the new phase shift function is given by  

$$T = \frac{1}{2} \left(\sqrt{1-\frac{(p_x+p_h)^2 v^2}{1-2 \et... ...2 \eta v^2 (p_x-p_h)^2}} \right) \tau +2 p_x (x-x_0)+2 p_h h_0.$$ (124)

The number of integrals in Equation () can be reduced by recognizing areas in the integrand that contribute the most to the integrals in k_h and k_x. Since the integrand is an oscillatory function its biggest contributions take place when the oscillations are stationary, when the phase function is either minimum or maximum. This approach is referred to as the stationary phase method (Appendix C). The stationary points (p_x and p_h) correspond to the minimum or maximum of equation (). In fact, the phase has a dome-like shape as a function of p_x and p_h (see Figure ). Thus, to calculate the stationary points, we must set the derivative of equation () with respect to p_h and p_x to zero, and solve the two equations for these two parameters. An easier approach is discussed next and in Appendix A.

 

station
Figure 1 Traveltime, in seconds, as a function offset-midpoint rayparameters (left), and source-receiver rayparameters (right) computed using equation (). All rayparameters have units of s/km. In both plots, the midpoint shift x-x₀=0.1 km, the offset is 0.2 km, and vertical time of 1 s. The medium parameters considered are v=2 km/s, ${\eta=0.2}$.