EQUIVALENT OFFSET MIGRATION

Simple algebraic rationalization of (2) yields  

$$T^2 = T_0^2 + \frac{h_e^2}{v^2},$$ (21)

where  

$$h_e^2 = x^2 + h^2 - \frac{4h^2x^2}{v^2T^2},$$ (22)

is in fact the ''equivalent-offset'' formula of Bancroft and Geiger (1994); Bancroft et al. (1995).

As described by Bancroft et al. 1995, EOM is a PSI type process explained geometrically by Figure . The solid line in this figure represents the hyperbolic equivalent offset curve associated with the point (m,h,T) in the constant time-slice defined by T. The amplitude at time T on a trace with midpoint m and offset h is mapped to every point on this solid curve. Note that EOM is a diffraction-stack process so that each input point at constant time T and fixed offset h corresponds to a hyperbolic trajectory indexed by T with the new offset, h_e, varying as a function of equation (22). Computationally, the algorithm will have characteristics similar to those of standard prestack-Kirchhoff approaches. One can reduce the number of calculations by carefully limiting offset sampling, but the net result is an algorithm whose order is still proportional to n³. That is, the number of calculations required to move input data is directly proportional to the number of points in the output volume, and so is of order n³ for two-dimensional problems.

The presence of v²T² in the denominator of (22) shows that EOM cannot proceed without knowledge or reasonable estimates of the velocity v. Clearly, the accuracy with which v is known has a direct and immediate bearing on the accuracy of the final migration. With the exception of the fact that residual NMO can improve the final image, one can expect the result of EOM to be basically similar to a good quality standard Kirchhoff migration.

 

Fig5
Figure 5 Equivalent-offset migration as a diffraction stack.

To fully understand the similarities between EOM and DMO-PSI, one can attempt to follow the logic of equations (16), (18), (19), and (20). Repeating the exercise with (21) replacing (13) and m₀ = m_e yields  

$$\frac{\partial T}{\partial m} = 4(m - m_e)(1 - \frac{4h^2}{v^2T^2})(v^2T-\frac{4h^2x^2}{v^2T^3})^{-1},$$ (23)

 

$$\frac{\partial T}{\partial h} = 4h(1 - \frac{4(m - m_e)^2}{v^2T^2})(v^2T-\frac{4h^2x^2}{v^2T^3})^{-1},$$ (24)

and  

$$-\frac{\partial h}{\partial m} = \frac{(m - m_e)}{h}(1 - \frac{4h^2}{v^2T^2})(1 - \frac{4(m - m_e)^2}{v^2T^2})^{-1}.$$ (25)

At first glance, the similarity of (25) with respect to (18) gives hope that EOM can be accomplished in the same manner as PSI. This is indeed true in theory. Equation (25) can be solved for (m - m_e) via the standard formula for finding the roots of quadratic polynomials. Ignoring the obvious root-selection analysis that will have to be made, the resulting m_e can be substituted into (22) to specify h_e in terms of m and h. The two equations in m_e, and h_e could in theory be used in the same manner as (19) and (20). Unfortunately, no simple process for performing the indicated operations presents itself. One is still forced to use a diffraction stack methodology.

In the limit, as v²T² approaches infinity, (25) becomes (18). Thus, EOM is asymptotically equivalent to PSI. Although beyond the scope of this paper, one is lead to consider the similarity between EOM and PSI in the absence of DMO and when each is followed by residual NMO. One might expect that for large v²T², differences will be minor. In this case, errors in v will also be inconsequential, so the claim Bancroft (1997) of weak dependence on v is probably valid. On the other hand, the asymptotic equvalence of EOM and PSI suggest that when v²T² is small, DMO-PSI will have an image-quality advantage.

Figures and summarize the basic similarities between EOM and PSI. The solid curves in these figures are the equivalent-offset hyperbola and the PSI circle. The dotted curve in Figure schematically represents the locus of points (m,h,T) which map to (m_e,h_e,T), and is the equivalent of PSI's dotted circle in Figure . That is, the required amplitudes at (m_e,h_e) or (m₀,h₀) can be obtained either by diffracting along the solid curves or by summing over the dotted curves. In the limit, Figure will become exactly equivalent to Figure . Conceptually, these processes are almost identical.

Since it is not obvious how to formulate EOM as (f,k) modeling, it is not clear that it can be implemented as an $O(n \log n)$ algorithm. The net impact is that the current best formulation of EOM on constant time slices is computationally equivalent to standard Kirchhoff migration techniques.

 

Fig6
Figure 6 Equivalent-offset migration.