- Berryhill, J., 1979, Wave-equation datuming: Geophysics,
**44**, 1329-1344. - Bevc, D., 1997, Floodint the topography: Wave-equation datuming of land data with rugged acquisition topography: Geophysics,
**62**, 1558-1569. - Gazdag, J., and Sguazzero, P., 1984, Migration of seismic data by phase-shift plus interpolation: Geophysics,
**69**, 124-131. - Gray, S., and Marfurt, K., 1995, Migration from topography: Improving the near-surface image: J. Can. Soc. Expl. Geophys.,
**31**, 18-24. - Jiao, J., Trickett, S., and Link, B., 2004, Wave-equation migration of land data: 2004 CSEG Convention, CSEG, CSEG Abstracts.
- Kythe, P., 1998, Computational conformal mapping: Birkhäuser, Boston, MA.
- Nehari, Z., 1975, Conformal mapping: Dover Publications Inc., New York.
- Riemann, B., 1851, Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen grösse (1851) in collected works: Dover Publications Inc, New York.
- Sava, P., and Fomel, S., 2004, Seismic imaging using Riemannian wavefield extrapolation: submitted to Geophysics.
- Sava, P., 2004, High-order kernels for Riemannian wavefield extrapolation: SEP-
**117**.

A

This appendix discusses the conformal mapping between polygons of arbitrary shape and the upper half plane. Assisting us in this transformation is an important conformal map transformation, termed Schwarz-Christoffel mapping, that facilitates solution of a class of BVPs with polygonal boundaries. Figure illustrates the transformation and also illustrates the basic nomenclature.

Figure 9

The formula for calculating the transformation is,

(13) |

The inverse Schwarz-Christoffel transformation is given by,

(14) |

In numerical applications of Schwarz-Christoffel mapping, it is
necessary to determine numerically the (2n+2) parameters (i.e. all
, *x*_{i} and A, B) that appear in equations (13)
and (14). In conformal mapping literature, this problem
is termed the 'parameter problem' Kythe (1998). The numerical
solution to this problem requires selecting 3 points of the x-axis
that map to 3 preassigned points in the u-axis (i.e. *p _{1}*,

(15) | ||

(16) |

(17) |

A solution to the Schwarz-Christoffel integrals begins by expanding the
series of equations (17) in a first order Taylor power
series about initial guesses, *x*_{i}^{(0)}, of the true values,
. This leads to a system of equations that may be solved
using Newton's method:

(18) |

(19) |

10/23/2004