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Conformal Mapping

Conformal mapping is a topic of wide-spread interest in the field of applied complex analysis. Generally, this subject deals with the manner in which point sets are mapped between two different analytic domains in the complex plane. In this paper, we refer only to domains that are simply- (i.e. not multiply) connected. A mapping between complex planes may be thought of as a rule relating how a field of points defined on a domain in the z-plane, z=x+iy, maps to the w-plane, w=u(x,y)+iv(x,y), according to a mapping function, w=f(z) (see the example in Figure [*]). If for each point in the z-plane domain there corresponds a unique number in the w-plane, then the mapping function is analytic. In addition, if for each point in the w-plane there corresponds precisely one point in the z-plane, then the mapping is one-to-one and the transformation is invertible. The Cauchy-Riemann equations Nehari (1975) are the necessary and sufficient conditions for function f(z) to be analytic in a domain of interest.

Figure 1
An example of a conformal mapping between the z-plane, z=x+iy, and the w-plane, w(z)=u(x,y)+iv(x,y), according to mapping function w=z2. In this example, the shaded region in the z-plane maps to the shaded region in the w-plane. Coordinates (u,v) are given by (x2-y2,2xy). Lines in the w-plane: u=1, u=4, v=2, and v=8, map to the following lines in the z-plane: x2-y2=1, x2-y2=4, xy=1. Note also that orthogonality of line intersections in the w-plane are preserved in the z-plane.

A conformal map is distinguishable from other mappings between complex planes by characteristic properties. Most important to this discussion is the following property:

Conservation of Angle:
A conformal mapping of two continuous arcs that locally form an angle $\alpha_0$ in the z-plane will generate two continuous arcs seperated by the same local angle $\alpha_0$ in the w-plane.
Figure [*] illustrates the property that grid lines orthogonal in the w-plane are orthogonal in the z-plane under a conformal map. By extension, non-Cartesian orthogonal coordinate systems can be created in the z-plane (or conversely in the w-plane) by a conformal mapping of a rectangular coordinate system in the w-plane (z-plane).

The first major developments in the theory of conformal mapping originated with the mapping theorem of Riemann (1851), who proved the existence of a unique analytic mapping between any two simply-connected, analytic domains:

Riemann Mapping Theorem:
Let D be a simply-connected region. Then there exists a bijective conformal map $f:D
\rightarrow U$ ,where U is the open unit disk. By extension, if G is a another simply-connected domain, there exists a mapping $g:G
\rightarrow U$. Hence, there exists a composite mapping operation, $f\cdot g^{-1}: D\rightarrow G$, between two arbitrary simply-connected domains.

Figure [*] illustrates the Riemann mapping theorem between three domains pertinent to the current discussion. Figure [*] presents an example of a conformal mapping between a square and unit circule (the mapping g in Figure [*]).

Figure 2
Illustration of the Riemann Mapping Theorem between a physical domain with an undulating upper surface, the unit circle, and a rectangular canonical domain. In this example, a forward mapping function, f, exists between the physical domain and the unit circle and, because the mapping is one-to-one, an inverse mapping f-1 also exists. Forward and inverse mapping functions (g and g-1) also exist between the rectilinear domain and the unit circle. Hence, the composition of functions $f
 \cdot g^{-1}$ denotes a mapping between the physical and canonical domains, while the inverse mapping is given by $g \cdot f^{-1}$. The mapping locations of points labeled 1 through 4 are specified to ensure that the sides in the physical domain correspond to the sides in the canonical domain.

We will use the Riemann mapping theorem to transform the topographic domain to a rectangular computational mesh. Assisting us is an extensive catalog of conformal maps between common geometrical domains. Pertinent to the current discussion are the conformal maps between the unit circle (UC) and the upper half plane (UHP), $f: UC \rightarrow
UHP$ and its inverse $f^{-1}: UC \leftarrow UHP$,
f: & z \rightarrow \frac{z-i}{z+i}, \\ f^{-1}: & i\frac{1+z}{1-z} \leftarrow z, \nonumber\end{eqnarray} (1)
and the mapping between the UHP and a rectangle with sides of arbitrary length, $g: UHP \rightarrow Rect$, and its inverse $g^{-1}:
UHP \leftarrow Rect$,
g: & w(k)=\int_{0}^{z} \frac{ {\rm
 d}\zeta}{\sqrt{1-\zeta^2}\sqrt{1-k^2\zeta^2}} \\ g^{-1}:& sn(w;k), \nonumber\end{eqnarray} (2)
where g is an elliptic integral of the first kind, k is a function of the ratio of the length of the two sides, and sn(w;k) is a Jacobian elliptic function Nehari (1975). Appendix A discusses a method for computing conformal map transforms between arbitrary polygons and the UHP.

Figure 3
Conformal mapping between a square coordinate system and the unit circle.

Table 1 outlines a work flow to generate a topographic coordinate system through conformal mapping. The first step is to define the enclosure of the physical domain where the topographic surface defines the upper boundary. We create the lower boundary by mirroring the topography at twice the maximum extrapolation depth. The side boundaries are defined by straight lines that join the top and bottom segments. We denote the border points ztopobnd, where subscript topo and superscript bnd refer to topography and boundary points, respectively. The four corner points of the physical domain are also specified. The next two steps involve calculating the forward and inverse mapping functions, f and f-1, between the topographic surface and the unit circle. The fourth step is to generate a rectilinear boundary and to define its four corner points. We denote this boundary zrectbnd, where subscript rect refers to rectangle. The next two steps involve calculating forward and inverse mappings functions, g and g-1, between the boundary of the rectangle and the unit circle.

To discern where in the canonical domain to form the coordinate system grid, we need to find the mapping of the topography boundary points on the rectangular domain boundary. This is accomplished by calculating the image of the boundary points under composite mapping operations, $z_{im}^{bnd}=g^{-1} \cdot f(z_{topo}^{bnd})$. A rectangular grid is then set up at the image points to create computational grid, zrectcs, where superscripts cs denote coordinate system.

Table 1. Work flow to calculate topographic coordinates with conformal mapping.

Step Description Notation
1 Define physical domain boundary and 4 corner points ztopobnd
2 Calculate mapping $f: Topo\rightarrow UHP \rightarrow UC$ wtopobnd=f(ztopobnd)
3 Calculate mapping $f^{-1}: UC \rightarrow UHP \rightarrow Topo$ ztopobnd=f-1(wtopobnd)
4 Define canonical domain border and 4 corner points zrectbnd
5 Calculate mapping $g: Rect \rightarrow UHP \rightarrow UC$ wrectbnd=g(zrectbnd)
6 Calculate mapping $g^{-1}: UC \rightarrow UHP \rightarrow Rect$ zrectbnd=g-1(wrectbnd)
7 Find image of topography in the rectangle zimbnd=g-1(f(ztopobnd))
8 Construct rectilinear grid using zimbnd(ztopobnd) zrectcs
9 Map grid zrectcs to physical domain ztopocs=f-1(g(zrectcs))

The final step is to map the rectilinear coordinate system, zrectcs, from the canonical domain back to the topographic coordinates under composite mapping operation, $z_{topo}^{cs}=f^{-1}
\cdot g(z_{rect}^{cs})$. Point set ztopocs defines a coordinate system appropriate for performing wavefield continuation directly from topography at the acquisition locations. The next section details how this point set is used to generate the appropriate extrapolation equations.

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