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Geometry in a generalized 3D Riemannian space is described by a symmetric
metric tensor, g_{ij}=g_{ji}, that relates the geometry in a
nonorthogonal coordinate system, , to an underlying
Cartesian mesh, Guggenheimer (1977). In
matrix form, the metric tensor is written,
 
(1) 
where g_{11}, g_{12}, g_{22}, g_{13}, g_{23} and are functions
linking the two coordinate systems through,
 

 (2) 
(Summation notation  g_{ii} = g_{11}+g_{22}+g_{33}  is used in
equations throughout this paper.) The associated (or inverse) metric
tensor, g^{ij}, is defined by , where
is metric tensor matrix determinant. The associated
metric tensor is given by,
 
(3) 
and has the following metric determinant,
 
(4) 
Weighted metric tensor, , is a useful
definition for the following development.
Next: Acoustic waveequation in 3D
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Stanford Exploration Project
4/5/2006