A derivation of the seismic representation theorem using seismic reciprocity

The seismic representation theorem is very useful in determining seismic displacements: (1) from extended and moving sources; (2) for boundary-value problems; and (3) for diffraction by vanishingly thin screens. Derivations of the representation theorem for anisotropic and homogeneous elastic media have been given by Knopoff [1956] and deHoop [1958] starting from first principles. These derivations, while elegant, are difficult. The seismic representation theorem is derived in a simple straightforward manner using the principle of seismic reciprocity as the starting point. The derivation is simplified further by the use of the polyadic notation. The representation theorem obtained is valid for anisotropic, inhomogeneous, linear elastic media. The utility of the representation theorem is demonstrated by considering illustrative examples concerning: (1) the radiation fields from various force couples; (2) the radiation from moving forces; and (3) the diffraction of elastic waves by thin screens.