Generating orthogonal coordinate system meshes from PFs requires ascribing a physical interpretation to equipotentials: they are equivalent to extrapolation steps. Similarly, the characteristics of the PF gradient field are intrinsically related to geometric coordinate system rays. Figure illustrates these concepts for the example of 2D wave-equation migration from topography.

topocoord
Scenario of migration from
topography. The upper and lower topographic surfaces are denoted
and , respectively. Connecting
vertical lines are denoted (left) and
(right). The upper and lower surfaces have
equipotential values of and ,
respectively, whereas both side boundaries have zero normal
derivatives.
Figure 1 |

This scenario requires extrapolating a wavefield comprised of *M*
traces into the subsurface a total of *N* steps, which ideally
occurs directly from the topographic surface.
The upper boundary of the computational domain, denoted
in this figure, is the acquisition surface. The lower
boundary, denoted , is the desired flat subsurface datum.
These two bounding surfaces are connected by two curves,
and , extending between the first and
last extrapolation levels.

Solving for a PF satisfying Laplace's equation first requires specifying appropriate boundary conditions. Because distinct upper and lower equipotentials are desired, these two surfaces must have different constant values. Thus, I choose the following boundary conditions,

(4) |

The Laplace's equation defined by the boundary conditions in Equation (4) is representable by a system of equations similar those commonly solved with conjugate gradient methods Claerbout (1999),

(5) |

(6) |

I use the following algorithm to obtain PF solution, :

- 1.
- Map the irregular topographic domain to a Cartesian mesh to generate vector ;
- 2.
- Fix the PF values on the boundary of, and external to, the mapped domain using the mask operator ;
- 3.
- Initialize the model vector with a starting guess (i.e., ) exploiting the smooth variation of between the upper and lower surfaces (i.e., through linear interpolation of on [1,0] between and );
- 4.
- Solve system of equations in Equation (5) using a conjugate gradient algorithm Claerbout (1999), by allowing the solver to iterate until convergence is reached.

5/3/2005