Abstract
The first-order (or linear) Rytov or Born approximation is the foundation for formulation of wave-equation tomography and waveform inversion, so the validity of the Rytov/Born approximation can substantially affect the applicability of these theories. However, discussions and research reported in literature on this topic are insufficient or limited. In this paper we introduce five variables in scattering theory to help us discuss conditions under which the Rytov approximation, in the form of the finite frequency sensitivity kernels (RFFSK), the basis of waveform inversion and tomography, is valid. The five variables are propagation length L, heterogeneity scale a, wavenumber k, anisotropy ratio ξ, and perturbation strength ɛ. Combined with theoretical analysis and numerical experiments, we conclude that varying the conditions used to establish the Rytov approximation can lead to uninterpretable or undesired results. This conclusion has two consequences. First, one cannot rigorously apply the linear Rytov approximation to all theoretical or practical cases without discussing its validity. Second, the nonlinear Rytov approximation is essential if the linear Rytov approximation is not valid. Different from previous literature, only phase (or travel time) terms for the whole wavefield are discussed. The time shifts of two specific events between the background and observed wavefields measured by cross-correlation will serve as a reference for evaluation of whether the time shifts predicted by the FFSKs are reasonably acceptable. Significantly, the reference “cross-correlation” should be regarded as reliable only if the condition “two specific similar signals” is satisfied. We cannot expect it to provide a reasonable result if this condition is not met. This paper reports its reliability and experimental limitations. Using cross-correlation (CC) samples as the X axis and sensitivity kernel (SK) or ray tracing (RT) samples as the Y axis, a chart of cross validation among the three is established. According to experimental analysis for a random and deterministic medium, the RFFSK works well for diffraction and geometric optics enclosed by the boundaries ka > 1 and (Λ < 1, ΛΦ < 2π ∪ Λ > 1, Φ < 2π), where Λ is a diffraction variable and Φ is a variable used to measure scattering strength. Both Λ and Φ depend on the five variables mentioned above. However, a large perturbation will limit the applicability of the Rytov approximation and narrow the range in which it is valid, because larger ɛ results in higher-amplitude fluctuations and larger back-scattering. Therefore, large ɛ is important in making SK/RT predictions discrepant from correct CC measurements. We also confirm that the RFFSK theory has wider range of application than ray theory. All the experiments point out the importance, both practical and theoretical, of respecting the conditions used to establish the Rytov approximation.
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We are grateful to the WTOPI Consortium of the University of California, Santa Cruz, for financial support.
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Xu, W., Xie, XB. & Geng, J. Validity of the Rytov Approximation in the Form of Finite-Frequency Sensitivity Kernels. Pure Appl. Geophys. 172, 1409–1427 (2015). https://doi.org/10.1007/s00024-014-0991-8
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DOI: https://doi.org/10.1007/s00024-014-0991-8