Acoustic reciprocity theorems have proved their usefulness in the study of forward and inverse scattering problems. The reciprocity theorems in the literature apply to the two-way (i.e. total) wavefield, and are thus not compatible with one-way wave theory, which is often applied in seismic exploration. By transforming the two-way wave equation into a coupled system of one-way wave equations for downgoing and upgoing waves it appears to be possible to derive 'one-way reciprocity theorems' along the same lines as the usual derivation of the 'two-way reciprocity 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 derived, which appears to have a symmetric kernel. This symmetry property appears to be sufficient to let the contrast term vanish in the above-mentioned situation.The one-way reciprocity theorem of the convolution type is exact, whereas the oneway reciprocity theorem of the correlation type ignores evanescent wave modes. The extension to the elastodynamic situation is not trivial, but it can be shown relatively easily that similar reciprocity theorems apply if the (non-unique) decomposition of the elastodynamic two-way operator is done in such a way that the elastodynamic one-way operators satisfy similar symmetry properties to the acoustic one-way operators.
One of the main benefits of prestack depth migration in seismic processing is its ability to handle complicated medium configurations. When considerable lateral variations in the acoustic parameters are present in the subsurface, prestack depth migration is necessary for optimal lateral resolution. However, most migration algorithms still deal with lateral variations in an approximate manner because these variations are in many cases moderate compared to the profound variations in the depth direction. From other areas of science (e.g., optics, oceanography, and seismology), it is known that lateral variations can be dealt with by a decomposition of the wavefield into wave modes. In this paper, we explore the possibility of applying this concept to the construction of one‐way wavefield operators for depth migration. We expand the Helmholtz operator on an orthogonal basis of wave modes and obtain one‐way wavefield operators that are unconditionally stable and significantly increase the lateral resolution of the result.
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