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Symmetrizing the wave field

I employ 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 us write the data as

where describes the source behavior, describes the receiver behavior, and governs pure propagation effects through the medium. is an additive noise term. denote convolutional relations. In an ideal world would be zero and 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, if source and receiver axes are discretized in the same manner. Consequently D will then also be symmetric. That is,

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

and the antisymmetric part of the data as

• Symmetric data acquisition