Non-hyperbolic common reflection surface

Analytical Example

crs,ncrs
Figure 1. Relative error in prediction of reflection traveltime in the case of a circular reflector in a homogeneous medium using different approximations: (a) CRS, (b) non-hyperbolic CRS. The model parameters are $m=R=D$ (see Figure B-1.)

The first example we use to compare the accuracy of different approximations is that of a circular reflector under a homogeneous overburden. As shown in Appendix B, the exact traveltime in this case can be derived analytically in a parametric form. Obtaining a non-parametric closed-form expression in this case would require a solution of a high-order algebraic equation (Landa et al., 2010). Figure 1 compares the accuracy of CRS and nonhyperbolic CRS approximations for a range of offsets and midpoints. We display the relative absolute error as a function of offset to depth ratio and midpoint separation to depth ratio for a range of offsets and midpoints. The central midpoint is taken at the same horizontal distance from the center of the circle as the depth. The CRS approximation (8) develops an error both at large offsets and at large midpoint separations. The proposed non-hyperbolic CRS approximation (12) shows a significantly smaller error in the full range of offsets and midpoints. In our experiments, the multifocusing approximation (4) was even more accurate in this example. However, because of its different functional form, we focus our analysis on comparing CRS and non-hyperbolic CRS.

Non-hyperbolic common reflection surface