Wave‐equation tomography

Abstract

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 back‐projection 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 back‐projection patterns that account for the uncertainty inherent in picking traveltimes from bandlimited data.