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  1. Free, publicly-accessible full text available December 1, 2023
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  4. We analyze an inverse problem associated with the time-harmonic Rayleigh system on a flat elastic half-space concerning the recovery of Lamé parameters in a slab beneath a traction-free surface. We employ the Markushevich substitution, while the data are captured in a Jost function, and we point out parallels with a corresponding problem for the Schrödinger equation. The Jost function can be identified with spectral data. We derive a Gel’fand-Levitan type equation and obtain uniqueness with two distinct frequencies.

    Free, publicly-accessible full text available March 7, 2023
  5. SUMMARY Dynamic ray tracing is a robust and efficient method for computation of amplitude and phase attributes of the high-frequency Green’s function. A formulation of dynamic ray tracing in Cartesian coordinates was recently extended to higher orders. Extrapolation of traveltime and geometrical spreading was demonstrated to yield significantly higher accuracy—for isotropic as well as anisotropic heterogeneous 3-D models of an elastic medium. This is of value in mapping, modelling and imaging, where kernel operations are based on extrapolation or interpolation of Green’s function attributes to densely sampled 3-D grids. We introduce higher-order dynamic ray tracing in ray-centred coordinates, which has certain advantages: (1) such coordinates fit naturally with wave propagation; (2) they lead to a reduction of the number of ordinary differential equations; (3) the initial conditions are simple and intuitive and (4) numerical errors due to redundancies are less likely to influence the computation of the Green’s function attributes. In a 3-D numerical example, we demonstrate that paraxial extrapolation based on higher-order dynamic ray tracing in ray-centred coordinates yields results highly consistent with those obtained using Cartesian coordinates. Furthermore, in a 2-D example we show that interpolation of dynamic ray tracing quantities along a wavefront can be done withmore »much better consistency in ray-centred coordinates than in Cartesian coordinates. In both examples we measure consistency by means of constraints on the dynamic ray tracing quantities in the 3-D position space and in the 6-D phase space.« less
  6. SUMMARY Within the field of seismic modelling in anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green’s function. Dynamic ray tracing is based on solving a system of Hamilton–Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (conventional) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between Cartesian and ray-centred phase-space perturbations. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the Cartesian and ray-centred phase-space perturbations. By numerical examples, we (1) visualize the validity limits of the ray-centred coordinate system, (2) demonstrate themore »transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates and (3) address the stability of function value and derivatives of volumetric parameters in a higher-order representation of the subsurface model.« less
  7. Quantitative imaging of subsurface earth properties in elastic media is performed from distributed acoustic sensing data. A new misfit functional based upon the reciprocity gap is designed, taking crosscorrelations of displacement and strain, and these products further associate an observation with a simulation. In comparison with other misfit functionals, this functional has the advantage of only requiring little a priori information on the exciting sources. In particular, the misfit criterion enables the use of data from regional earthquakes (teleseismic events can be included as well), followed by exploration data to perform a multiresolution reconstruction. The data from regional earthquakes contain the low-frequency content that is missing in the exploration data, allowing for the recovery of the long spatial wavelength, even with very few sources. These data are used to build prior models for the subsequent reconstruction from the higher frequency exploration data. This results in the elastic full reciprocity-gap waveform inversion method, and we illustrate its performance with a pilot experiment for elastic isotropic reconstruction.