Seismic Transmission Operator Reciprocity – I: A Direct Derivation Showing the Role of Elastic Impedance Operator Symmetry

syeikh muhammad;

The properties of the overburden transmission response in seismic depth imaging are of interest for the analysis of reflectivity illumination or blurring. An elastic transmission-operator reciprocity is derived here by a direct method exploiting a symmetry of the elastic impedance operator at the input and output levels. This impedance-operator symmetry is obtained in a companion paper by assuming the existence and completeness of the lateral modes of the elastic wave equation. The transmission operator, or wave propagator, is a surface-to-surface displacement-to-displacement operator and an appendix here explains heuristically why it does not immediately display its reciprocity property in the way that force-to-displacement Green functions do. Instead, the transmission operator must be augmented by some version of the impedance operator at the input and output levels. We may view this augmentation as re-establishing the connection to force-to-displacement reciprocity or as a separate reciprocity relation for displacements which have been normalized by input and output impedances. Similar statements apply to reflection operators for the region between the two levels. The main transmission-operator reciprocity is an exact property, but a second and approximate symmetry exists when waves that are evanescent in depth are neglected.