Far field elastodynamic Born scattering revisited

• Published in: Journal of applied geophysics Vol. 89; pp. 141 - 163
• Main Authors: Geerits, Tim W., Veile, Ines, Hellwig, Olaf
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2013
• Subjects: Body waves; Inverse theory; Wave propagation; Wave scattering and diffraction; Wave scattering and diffraction; Wave propagation; Body waves; Inverse theory
• ISSN: 0926-9851; 1879-1859
• DOI: 10.1016/j.jappgeo.2012.11.012

This paper summarizes part of an ongoing feasibility study that investigates the possible use of the full elastic Born approximation in multipole borehole acoustics. As a first step we exclude the fluid-filled borehole with the motivation that one or two wavelengths away from the fluid-filled borehole, radiating borehole mode amplitudes (e.g., Stoneley wave, formation dipole wave, etc.) are small compared to body wave amplitudes (P-, SV- and SH-waves). Consequently, for scatterers one or two wavelengths away from the fluid-filled borehole, it suffices to only consider their interaction with body waves. In this paper we apply the contrast-source stress-velocity forward scattering (integral equation) formulation for solid configurations in its first order (Born-) approximation (De Hoop, 1995) assuming a multipole force source excitation in a zero-offset configuration. To scrutinize the validity of the Born approximation, we consider the simplest type of scatterer, i.e., one characterized by a (Heaviside) step function change in one or more of the contrast (perturbation) parameters and we derive analytic zero-offset formulas for the scattered wave particle velocity and displacement in both the space–frequency and space–time domain, respectively. We assume the scatterer to be located in the far-field. More complicated layered configurations can easily be derived by superposition of the given solution types. Explicit results are given for the dipole and quadrupole excitation, where the former is allowed to have an arbitrary orientation relative to the scatterer and where the latter one is located in a plane perpendicular to the scatterer. In the time domain it is shown, how the scattered wave field decomposes in a specular and diffuse wave field (two terms borrowed from ‘Optics’), where the former contribution vanishes in the absence of an imaging condition and where the latter is always present. For the dipole case, we subject our results to a sensitivity analysis with respect to the three independent perturbation parameters (i.e., density and two compliance parameters) and we compare these results to a full waveform benchmark code that has implemented the reflectivity method (Kennett, 1983), for a ‘horizontally’ stratified elastic medium. This allowed us to pinpoint the root cause of the observed (small) differences. As it turns out these deviations could be traced back to the inaccurateness of the first order Born scattering coefficients. An additional confirmation of this fact is provided through a comparison between the zero-offset scattering coefficients and the corresponding Zoeppritz reflection coefficients. Most notably, it was found that the PP first order scattering coefficient needs a higher than quadratic correction in two of the three independent perturbation parameters, i.e., the two compliance parameters, δΛ and δM. With respect to the density perturbation parameter (δρ) the PP scattering coefficient correction is quadratic with respect to this perturbation parameter, as is to be expected for a first order approximation. Moreover, also the SS first order scattering coefficient only needs a quadratic correction with respect to its associated perturbation parameters, i.e., δρ and δM. Finally, we give a brief outline on how to numerically implement the Born approximation (employing arbitrary offsets) in a configuration where a source–receiver pair is moving continuously relative to the ‘contrast’ (Geology), as is the case in borehole acoustic applications. ► Far field Born approximation assuming multipole force source excitation. ► Analytic zero-offset formulas assuming a Heaviside step in contrast parameters. ► Full waveform benchmarking and sensitivity analysis for zero-offset dipole excitation. ► Comparing zero-offset Born scattering coefficients with Zoeppritz reflection coefficients. ► Dissecting scattered wavefield in specular and diffused contribution.