Figure 1

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 in the
*z*-plane will generate two continuous arcs seperated by the same local angle in the*w*-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 ,where U is the open unit disk. By extension, if G is a another simply-connected domain, there exists a mapping . Hence, there exists a composite mapping operation, , 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

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*), and its inverse ,

(1) | ||

(2) | ||

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

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 *z*_{topo}^{bnd}, 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

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,
. A rectangular grid is
then set up at the image points to create computational grid,
*z*_{rect}^{cs}, 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 | z_{topo}^{bnd} |

2 | Calculate mapping | w_{topo}^{bnd}=f(z_{topo}^{bnd}) |

3 | Calculate mapping | z_{topo}^{bnd}=f(^{-1}w_{topo}^{bnd}) |

4 | Define canonical domain border and 4 corner points | z_{rect}^{bnd} |

5 | Calculate mapping | w_{rect}^{bnd}=g(z_{rect}^{bnd}) |

6 | Calculate mapping | z_{rect}^{bnd}=g(^{-1}w_{rect}^{bnd}) |

7 | Find image of topography in the rectangle | z_{im}^{bnd}=g(^{-1}f(z_{topo}^{bnd})) |

8 | Construct rectilinear grid using z_{im}^{bnd}(z_{topo}^{bnd}) |
z_{rect}^{cs} |

9 | Map grid z_{rect}^{cs} to physical domain |
z_{topo}^{cs}=f(^{-1}g(z_{rect}^{cs})) |

The final step is to map the rectilinear coordinate system,
*z*_{rect}^{cs}, from the canonical domain back to the topographic
coordinates under composite mapping operation, . Point set *z*_{topo}^{cs} 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.

10/23/2004