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I am using the basic notion of symmetry in a wave field, or to be more precise
the lack thereof. To expand that idea, let us look at a single
slice through a prestack data volume. Let the data depend on the
following variables:

| |
(1) |

where describes the source behavior,
describes the receiver behavior,
and governs pure propagation effects through the medium.
is an additive noise term.
In an ideal world would be zero, as in Figure .
represents the Green's function of the
medium. It can be shown that
since the anisotropic elastic wave equation is linear and self-adjoint,
the discrete representation
of the continuous function has to be symmetric.
That is,
| |
(2) |

This equation defines a measure for the symmetry of a given data set.
The symmetric part of data is defined as
| |
(3) |

and the antisymmetric part of the data as
| |
(4) |

**matrix
**

Figure 1 Schematic representation of a discrete
symmetric wave field

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Stanford Exploration Project

11/18/1997