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The analytic transformation between the elliptic cylindrical and
Cartesian coordinate systems (see example in figure ).
A elliptical coordinate system is specified by,
| |
(8) |

where [*x*_{1},*x*_{2},*x*_{3}] are the underlying Cartesian coordinate variables,
the RWE elliptic cylindrical coordinates, and
parameter *a* a stretch parameter controlling the breadth of the
coordinate system.
**ECC
**

Figure 6 Four sample extrapolation steps of an elliptic
cylindrical coordinate system.

The metric tensor describing the geometry of the elliptic coordinate
system is given by,

| |
(9) |

where . The
determinant of the metric tensor is: . The associated
(inverse) metric tensor is given by,
| |
(10) |

and weighted metric tensor () is given by,
| |
(11) |

The corresponding extrapolation wavenumber is generated by using
tensors *g*^{ij} and *m*^{ij} in the general wavenumber
expression for 3D non-orthogonal coordinate system
Shragge (2006). Note that even though the elliptic
coordinate system varies spatially, the local curvature parameters
remain constant: *n*_{1}=*n*_{2}=*n*_{3}=0. Thus, inserting the values of
*g*^{ij}, *m*^{ij} and *n*_{j} leads to the following extrapolation
wavenumber for stepping outward in concentric ellipses ,
| |
(12) |

The wavenumber for 2D extrapolation in elliptic coordinates reduces to
| |
(13) |

Note that equation A-5 does not contain a kinematic
approximation of the extrapolation wavenumber.

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Stanford Exploration Project

5/6/2007