Numerical examples

We test the combined conformal mapping and Riemannian wavefield extrapolation approach on a synthetic dataset computed on a rugged topographic surface. The geological model is a merger of common geologic features from the Canadian Foothills in northeastern British Columbia, Canada. The velocity model, shown in Figure , consists of steep thrust fault planes and complex folds typical of a mountainous thrust region. The topographic boundary of interest is demarcated by the velocity model discontinuity nearest to the surface. The total relief of the Earth's surface in this model is approximately 1600 m. Also note that the complex near-surface velocity structure should present a significant imaging challenge Gray and Marfurt (1995).

 

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Figure 5 Foothills velocity model constructed from composite 2-D geologic model from northeastern British Columbia, Canada. Total elevation relief is approximately 1600 m. The topographic boundary of interest is demarcated by the velocity model discontinuity nearest to the surface.

Figure  shows the result of using conformal mapping to construct a coordinate system that incorporates the topography shown in Figure . One important observation is that topography causes focusing of the coordinate system. In particular, the coordinate system compresses under local topographic maxima, and expands beneath local topographic minima. This suggests that Jacobian spreading factor, J, in (3) will be strongly dependent on the local radius of curvature of the topographic surface. However, as the topographic fronts move farther from the surface, the topographic influenced diminishes and the fronts move toward becoming a flat datum. (Hence, this approach could be used for wavefield datuming.)

 

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Figure 6 Topographic coordinate system constructed using conformal mapping. Note the compression of the rays under topographic maxima, and their extension under topographic minima. The influence of topography on the coordinate system diminishes farther from the surface.

A prestack wave-equation imaging test was conducted using a synthetic data set generated by an acoustic, 2-D, finite-difference code through the model shown in Figure . The data set is comprised of 278 shot gathers with a split-spread geophone geometry where absolute offsets range between 15 m and 3600 m. Geophone and source spacing are 15 m and 90 m, respectively. Data were generated on a regular Cartesian mesh. Thus, we interpolated the data to fit on a grid uniform along the topographic surface. Data fidelity may have been lowered by this processing step; however, we emphasize that this step is normally of modest importance since field data likely are nearly uniformly-spaced on the topographic surface.

A sample shot record at horizontal location 14040 m is shown in Figure . Note that the relief causes non-linear moveout of the direct arrival, and a substantial amount of topographic scattering as illustrated by the horizontal banding across the section. No preprocessing of the sections was done to remove these two potential noise sources, and the resulting image is contaminated accordingly.

 

singleshot Figure 7 Shot record from source station 14040 m that shows the influence of topography. Note the non-linear moveout of the direct arrivals, and the significant amount of topographic scattering typified by horizontal streaking across the section.

A preliminary prestack migration image is presented in Figure . The majority of reflectors are well positioned; however, diffractions and discontinuous reflectors exist at locations directly beneath topographic minima and maxima. Although these anomalies may be caused by the data regularization procedure, they more likely arise from limitations imposed by the phase-screen approximation.

Also present are vertical streaks of higher (lower) amplitude directly under local topographic minima (maxima). We attribute these anomalous amplitudes to a combination of: i) the simplicity of the weighing function used in the interpolation of the image between the topographic and Cartesian coordinate systems; and ii) our non-consideration of the dynamic terms in (6). Geological structure poorly imaged or absent include sections of the steeply-dipping fold belt, which is probably due to limitations imposed by both the limited angular bandwidth of the phase-screen approximation, and our use of only one reference medium.

 

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Figure 8 Top: Foothills thrust model velocity model. Bottom: The preliminary prestack migration image using Riemannian wavefield extrapolation on a coordinate system generated through conformal mapping.