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Scattering attenuation in 3D

Different scattering regimes exist when waves propagate in heterogeneous media, according to the ratio of the wavelength, $\lambda $, to the size, $b$, of heterogeneities. The formalism including the different scattering regimes, when heterogeneities are modeled by spherical inclusions, is the Mie scattering theory. Recent experimental results (Le Gonidec and Gibert, 2007) on sonic-wave reflectivity in a granular medium, made up of beads of size $b$ in a water tank, illustrate this classification :

  1. for low frequencies, when $\lambda>\pi b$, backward scattering is dominant, the Born approximation can be used, and the regime is Rayleigh scattering;
  2. for wavelengths similar in size to heterogeneity, when $\pi b>\lambda>\pi b/2$, lateral scattering is important, multiples should not be neglected, and the regime is called resonant scattering; and
  3. for high frequencies, when $\pi b/2>\lambda$, waves are scattered mainly forward, and localization theory and the Rytov formalism are appropriate.
Experiments have demonstrated that wave reflectivity strongly decreases when the wavelength of the incident wave is twice the diameter of the beads, for which lateral scattering starts to be dominant. Mean wavefield formalism is valid only for a low frequency, when the wavelength is larger than the size of heterogeneities because of the assumptions in the derivation (Karal and Keller, 1964). The Born approximation can describe the Rayleigh regime and the approach of resonant phase scattering (Frankel and Clayton, 1986; Sato and Fehler, 1998). For wavelengths shorter than the size of heterogeneities, artificial decoherence by phase randomization occurs (Sato and Fehler, 1998; Wu, 1982). We intend to describe the 3D attenuation in a stochastic fractal medium, when $kb\leq1$, which is relevant for seismic survey frequencies. The limit of validity corresponds to wavelengths approaching the size of heterogeneities. We assume heterogeneities to be isotropic. The schemes in Figure 5 compare a realistic geological structure with two different end-member models. Scattering calculations in 1D underestimate the scattering loss by small-scale heterogeneities.

schemescatter3dgg schemescatterd schemescatter3d
Figure 5.
Schematic comparison of single scattering effects, during a vertical wave propagation in sediments, between a realistic geological structure (a) and two end-member models: horizontal layers with propagation including 1D scattering (b) and isotropic heterogeneities with 3D scattering (c).
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Next: Low-frequency waves in 3D Up: Browaeys & Fomel: Fractals Previous: Seismic-scale heterogeneities