RECIPROCITY THEOREMS FOR ONE-WAY WAVE-FIELDS

Acoustic reciprocity theorems have proved their usefulness in the stud y of forward and inverse scattering problems. The reciprocity theorems in the literature apply to the two-way (i.e. total) wavefield, and ar e thus not compatible with one-way wave theory, which is often applied in seismic exploration. By transforming the two-way wave equation int o a coupled system of one-way wave equations for downgoing and upgoing waves it appears to be possible to derive 'one-way reciprocity theore ms' along the same lines as the usual derivation of the 'two-way recip rocity theorems'. However, for the one-way reciprocity theorems it is not directly obvious that the 'contrast term' vanishes when the medium parameters in the two different states are identical. By introducing a modal expansion of the Helmholtz operator, its square root can be de rived, which appears to have a symmetric kernel. This symmetry propert y appears to be sufficient to let the contrast term vanish in the abov e-mentioned situation. The one-way reciprocity theorem of the convolut ion type is exact, whereas the one-way reciprocity theorem of the corr elation type ignores evanescent wave modes. The extension to the elast odynamic situation is not trivial, but it can be shown relatively easi ly that similar reciprocity theorems apply if the (non-unique) decompo sition of the elastodynamic two-way operator is done in such a way tha t the elastodynamic one-way operators satisfy similar symmetry propert ies to the acoustic one-way operators.