REFERENCES

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.

 

sctran
Figure 9 Schematic of the Schwarz-Christoffel transformation between a polygonal domain in the w-plane and the upper half plane (UHP), $D={z:\Im(z) \gt 0}$, in the z-plane. This transformation maps line segments $\Gamma_i$ that lie on the y=0 line in the z-plane to the line segments $\Gamma_i$ that define the polygonal boundary of the w-plane according to mapping rule f. Exterior angles, $\pi \alpha_i$, are used in the transformation formula, and are defined by $\left\vert \alpha_i \right\vert \gt 1$ and $\sum_{i=1}^{n} \alpha_n = 2$. Points x_i in the z-plane are mapped to points w_i in the w-plane.

The formula for calculating the transformation is,  

$$w=f(z)= A + B \int_{z_0}^{z} \prod_{i=1}^{n} \left( \zeta - x_i \right) ^{-\alpha_i} {\rm d}\zeta,$$ (13)

where A and B are constants that determine the size and position of the polygon $\Gamma$, and $\alpha_i$ denotes the exterior angle (see Figure ). Constants A and B are computed after defining the mapping of 3 points (i.e., known points z₀). The integration is carried out along any path in the domain D that connects known point z₀ and the point in question z.

The inverse Schwarz-Christoffel transformation is given by,  

$$z=f(w)=C_0+C\int_{w_0}^{w}\prod_{i=1}^{n}\left(\zeta-w_i\right)^{-\mu_i} {\rm d}\zeta,$$ (14)

where $\mu_i$ are the interior angles, and integration is carried out along any path that connects known mapping point w₀ with the point in question w.

In numerical applications of Schwarz-Christoffel mapping, it is necessary to determine numerically the (2n+2) parameters (i.e. all $\alpha_i$, 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. p₁, p₂ and p₃). This allows for equation (13) to be segmented into the Schwarz-Christoffel integrals,

$$\begin{eqnarray} I_1= & \int_{p_1}^{p_2} (\zeta-p_1)^{-\alpha_1} (\zeta-p_2)^{-\... ...{-\alpha_{n-1}}(\zeta-p_{n})^{-\alpha_{n}}{\rm d} \zeta \nonumber\end{eqnarray}$$ (15)

Fortunately, the ratio of any two sides of the mapped polygon is independent of scale factors A and B. This allows the parameter problem to be written as the following series of equations:

$$I_j(x_3,x_4,...,x_{n-1})=\lambda_j I_1(x_3,x_4,...,x_{n-1}),\;\;\; j=2,3,...,n-2,$$ (16)

where  

$$\lambda_j = \frac{ \left\vert w_{j+1} - w_j \right\vert}{\left\vert w_2-w_1 \right\vert}\;\;\;j=3,4,...,n-2.$$ (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, $\tilde{x}_i$. This leads to a system of equations that may be solved using Newton's method:

$$I_j^{(0)}+\sum_{\nu=3}^{n-1} h_{\nu}^{(1)} \frac{\partial I... ...rtial I_1^{(0)}}{\partial x_{\nu}} \right], \;\;\;j=2,...,n-2,$$ (18)

where $h_{\nu}$, the correction factors that are being solved for, are applied to yield the next estimate of the vertex corners,

$$x_{\nu}^{(0)} = x_{\nu}^{(0)} + h_{\nu}^{(1)}.$$ (19)

This process is repeated using n^th iterative updates of $h_{\nu}^{(n)}$until the desired tolerance is reached. Finally, the Schwarz-Christoffel integrals are improper because the integrand of each integral becomes unbounded at the two points of integration. Kythe (1998) discusses using the Kantorovich method to regularize these integrals.