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So long as *R* is isotropic, then it will commute with and there will be a Christoffel-like representation. If *R*
is not isotropic then the following trick will get around the problem.
First, premultiply the equation by *R*^{-1/2}, which I define as the
symmetric (but possibly complex) matrix whose eigenvalues have a positive
real part, and are the inverse of the square-root of the eigenvalues of the
matrix *R*, which, for physical reasons (causality, no energy gain), are
also constrained. So equation (4), dropping the notation, can
be written as,
| |
(5) |

and this can be further simplified by replacing the displacement field
variable, *u*, by an energy-like field variable, *w*, defined equal
to *R*^{1/2}u. Our final form is now,
| |
(6) |

This new field variable is quite similar to the energy flux variable
introduced by Clint Frasier and reported in Aki & Richards (1980),
although our motivations are somewhat different.

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Stanford Exploration Project

11/17/1997