Phase calculations

Once the offset ray parameter (p_h), or an approximation of it, is determined, it is then used, along with equation (18), to calculate traveltime (phase shifts). What makes zero-offset migration so trivial is that p_h is known in advance (p_h=0), and therefore, traveltime calculation can be done directly. For prestack migration, we must first estimate p_h then precede with traveltime calculations. Although p_h above can be evaluated using analytical approximations, the phase will always be evaluated by inserting the value of p_h into the exact DSR equation [i.e., equation (18)].

 

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Figure 3 Left: the phase correction, T, as a function p_x using p_h calculated numerically (solid curve), and p_h calculated using the 2-point solution (dashed curve). Right: The absolute difference between each of the two curves (numerical and analytical) on the left. The medium is homogeneous and isotropic with v=2.0 km/s. The black curve corresponds to an offset-to-vertical-time ratio ($\frac{X}{\tau}$) of 1.0 km/s, the dark-gray curve corresponds to $\frac{X}{\tau}$=2.0 km/s, and the light-gray curve corresponds to $\frac{X}{\tau}$=3.0 km/s.

Figures 3 through 7 compares T [where $\omega T$ constitutes the phase of the integral of equation (14)] evaluated using numerical solutions of p_h with those obtained via analytical approximations using either the 2-point, the 3-point, or perturbation-theory solutions of p_h.

For isotropic homogeneous media, T calculated using the 3-point solution of p_h better approximates the T derived from the numerical solutions than that given by the 2-point solution (compare Figures 3 and 4). As expected, errors will increase, for all cases, with increasing offsets. However, because $\frac{X}{\tau}$=3 km/s [offset-to-depth ratio (X/D) of 3] is rarely encountered in practice, and usually falls far within the typical mute zone, these approximations yield fairly accurate phase shifts even for practical large offsets. The irregularity in the T curve in Figure 3, for p_h calculated using the 2-point solution and $\frac{X}{\tau}=3$km/s, is a result of enforcing the condition that $p_x+p_h \le \frac{1}{v}$, which is necessary to avoid imaginary values in the DSR equation.

 

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Figure 4 Same as in Figure 3, but now we are comparing the numerical solution with that obtained for p_h calculated using the analytical 3-point fitting equation.

 

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Figure 5 Same as in Figure 3, but the medium now is VTI with $\eta$=0.2.

In VTI media, only for $\frac{X}{\tau}$=1 km/s, the T derived using the 3-point solution better approximates the numerical solution than T derived using the 2-point solution (compare Figures 5 and 6). The degradation of the 3-point solution at large offsets can be attributed to the failure of the p_x=p_h fitting approximation to improve the results for such offsets. For large offsets we are approaching the validity limit of the 3-point solution. A strict condition is enforced upon the p_h derived from setting p_x=p_h (see Appendix B). Although we have obviously met this condition (or otherwise we would have gotten imaginary solutions), coming close to the limit has clearly negatively effected the solution. Between the 2-point fitting solution and the 3-point one, one might want to implement a weight factor that favors the 3-point solution at near offset and the 2-point solution at far offset.

 

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Figure 6 Same as in Figure 4, but the medium now is VTI with $\eta$=0.2.

Using perturbation theory and Shanks transform (Appendix C; Bender and Orszag, 1978), we can obtain even better 3-point fitting estimations of the stationary point solution for VTI media, much better than the previous 2- or 3-point solutions. In fact, Figure 7 consists of two sets of curves (corresponding to the numerical solutions and analytical ones derived using Shanks transforms) that practically overlap for all p_x. The difference, given by the right curve, shows that the errors are lower than any previously obtained solutions.

 

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Figure 7 Same as in Figure 4, but the medium now is VTI with $\eta$=0.2. Here I am using the perturbation theory and Shank transform.

What's especially remarkable here is the high level of accuracy achieved with imaging horizontal reflectors despite the approximations used in developing p_h for such a reflector dip; the error is almost zero for even the $\frac{X}{\tau}$=3 km/s (X/D=3) example (see Figure 7). The accuracy achieved here far exceeds the ones obtained via Taylor series expansions (or even modifications of it) of traveltime around X=0 (Alkhalifah, 1997b). This analytical result has significant implications for non-hyperbolic moveout inversion in VTI media.