WAVE-EQUATION TOMOGRAPHY

The relation between ray-trace and diffraction tomography is usually obscured by formulation of the two methods in different domains: the former in space, the latter in wavenumber. Here diffraction tomography is reformulated in the space domain, under the title of wave-equation tomography. With this transformation, wave-equation tomography projects monochromatic, scattered wavefields back over source-receiver wavepaths, just as ray-trace tomography projects traveltime delays back over source-receiver raypaths. Derived under the Born approximation, these wavepaths are wave-theoretic backprojection patterns for reflected energy; derived under the Rytov approximation, they are wave-theoretic back-projection patterns for transmitted energy. Differences between ray-trace and, wave-equation tomography are examined through comparison of wavepaths and raypaths, followed by their application to a transmission-geometry, synthetic data set. Rytov wave-equation tomography proves superior to ray-trace tomography in dealing with geometrical frequency dispersion and finite-aperture data, but inferior in robustness. Where ray-trace tomography assumes linear phase delay and inverts the arrival time of one well-understood event, wave-equation tomography accommodates scattering and inverts all of the signal and noise on an infinite trace simultaneously. Interpreted through the uncertainty relation, these differences lead to a redefinition of Rytov wavepaths as monochromatic raypaths, and of raypaths as infinite-bandwidth wavepaths (Rytov wavepaths averaged over an infinite bandwidth). The infinite-bandwidth and infinite-time assumptions of ray-trace and Rytov, wave-equation tomography are reconciled through the introduction of bandlimited raypaths (Rytov wavepaths averaged over a finite bandwidth). A compromise between rays and waves, bandlimited raypaths are broad backprojection patterns that account for the uncertainty inherent in picking traveltimes from bandlimited data.