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The technique for splitting waves into up and down components is applied to solutions of the generalized wave equation in three dimensions. In particular, the splitting associated with the principal part of the equation is applied to the fundamental solution (Green function). The equations for the up and down wave components are obtained and the singular parts removed. The use of these reduced wave components together with their equations, boundary data and initial data, in the inverse problem (where the reflected field is produced by a point-impulse source, exterior to the scattering medium) is discussed.
Weston et al. (Wed,) studied this question.