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The analytic transformation between the oblate spheroidal and
Cartesian coordinate systems (see example in figure ). A
elliptical coordinate system is specified by,
| |
(14) |

where, again, *a* is a stretch parameter controlling the breadth of the coordinate system.
**OSC
**

Figure 7 Three sample extrapolation steps of an oblate
spheroidal coordinate system.

The metric tensor *g*_{ij} describing the geometry of oblate
spheroidal coordinates is given by,

| |
(15) |

where and
. The determinant of the
metric tensor is: . The associated (inverse) metric
tensor is given by,
| |
(16) |

and weighted metric tensor is given by,
| |
(17) |

The corresponding extrapolation wavenumber is generated by inputting
tensors *g*^{ij} and *m*^{ij} into the generalized wavenumber
expression for 3D non-orthogonal coordinate systems
Shragge (2006). Unlike in elliptic cylindrical
coordinates, though, the oblate spheroidal system has
non-stationary *n*^{i} coefficients: , *n*^{2}=0 and . The resulting
extrapolation wavenumber is
| |
(18) |

The wavenumber for 2D extrapolation in elliptic coordinates reduces to
| |
(19) |

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Stanford Exploration Project

5/6/2007