However, in practice wavefield continuation is seldom applied directly to data sets acquired on topography without significant preprocessing. The predominant challenge is that the metric of source and geophone arrays seldom conform to a regular computational mesh. Rather, due to instrument cabling, geophone arrays are more likely to uniformly sample the topographic surface. Two common solutions to this problem are either to employ a migration procedure involving wavefield injection Jiao et al. (2004), or to perform an upward-datuming prior to migration Bevc (1997). Migration by wavefield injection commences at the global topographic maximum where the data recorded at this station are injected into the wavefield. The wavefield is then continued downward and data are injected into the wavefield whenever the extrapolation step reaches the height of the topography. Two drawbacks of this approach are that data need to be regularized beforehand to a uniform grid usually through interpolation, and that the additional number of fine-scale extrapolation steps significantly increase cost. Upward wavefield datuming or ``flooding the topography'' procedures are employed to generate a regular wavefield above the highest point. This processing step can be done successfully with Kirchhoff or other migration operators. One downside of this approach is, again, the increased preprocessing cost. In general, although these methods produce good results, a significant amount of data preprocessing is required to render Cartesian-based wave-equation migration approaches applicable and, as a result, data fidelity may be compromised.
In this paper, we argue that many of the difficulties with state-of-the-art migration from topography technology could be precluded by abandoning the Cartesian coordinate system for one conformal with the topographic surface. To find such a method, we observe that wave-equation imaging is a specific example of a boundary value problem (BVP) that has a solution domain defined by a polygonal boundary. (Images are the superposition of the monochromatic solutions to a number of BVPs of different frequency.) This observation motivates us to examine the results of other applied fields that routinely solve BVPs, such as aerospace and mechanical engineering.
One method routinely employed to help solve BVPs is conformal mapping. This procedure defines how to transform the physical solution domain to a more symmetric canonical domain through mapping in the complex plane Kythe (1998). Relating this concept to wave-equation imaging from topography, we suggest using conformal mapping to transform the topographically-influenced physical domain to a canonical domain characterized by a rectangular computational grid. We term this new orthogonal calculation mesh a ``topographic'' coordinate system. Moreover, the forward and inverse conformal map transforms are also used in defining the wavefield extrapolation equations appropriate for the canonical domain. Consequently, we are both able to perform wavefield extrapolation and to apply the imaging condition in the topographic coordinate domain. The final image is generated by mapping the topographic coordinate image to the physical domain using the inverse conformal mapping transform.
We begin the paper with an overview of conformal mapping illustrated by some simple examples. We then review Riemannian wavefield extrapolation Sava and Fomel (2004) and the steps required to generate appropriate wavefield extrapolation equations. Prestack migration results are presented for a data set acquired over a 2-D geological model characterized by severe elevation relief, strong near-surface velocity contrast, and complicated folding and faulting. The paper concludes with a discussion on the relative merits and drawbacks of the proposed approach.