ALBERT

Description: 〈span〉〈div〉SUMMARY〈/div〉To describe the energy transport in the seismic coda, we introduce a system of radiative transfer equations for coupled surface and body waves in a scalar approximation. Our model is based on the Helmholtz equation in a half-space geometry with mixed boundary conditions. In this model, Green’s function can be represented as a sum of body waves and surface waves, which mimics the situation on Earth. In a first step, we study the single-scattering problem for point-like objects in the Born approximation. Using the assumption that the phase of body waves is randomized by surface reflection or by interaction with the scatterers, we show that it becomes possible to define, in the usual manner, the cross-sections for surface-to-body and body-to-surface scattering. Adopting the independent scattering approximation, we then define the scattering mean free paths of body and surface waves including the coupling between the two types of waves. Using a phenomenological approach, we then derive a set of coupled transport equations satisfied by the specific energy density of surface and body waves in a medium containing a homogeneous distribution of point scatterers. In our model, the scattering mean free path of body waves is depth dependent as a consequence of the body-to-surface coupling. We demonstrate that an equipartition between surface and body waves is established at long lapse-time, with a ratio which is predicted by usual mode counting arguments. We derive a diffusion approximation from the set of transport equations and show that the diffusivity is both anisotropic and depth dependent. The physical origin of the two properties is discussed. Finally, we present Monte Carlo solutions of the transport equations which illustrate the convergence towards equipartition at long lapse-time as well as the importance of the coupling between surface and body waves in the generation of coda waves.〈/span〉

Description: 〈span〉〈div〉Summary〈/div〉To describe the energy transport in the seismic coda, we introduce a system of radiative transfer equations for coupled surface and body waves in a scalar approximation. Our model is based on the Helmholtz equation in a half-space geometry with mixed boundary conditions. In this model, Green’s function can be represented as a sum of body waves and surface waves, which mimics the situation on Earth. In a first step, we study the single-scattering problem for point-like objects in the Born approximation. Using the assumption that the phase of body waves is randomized by surface reflection or by interaction with the scatterers, we show that it becomes possible to define, in the usual manner, the cross-sections for surface-to-body and body-to-surface scattering. Adopting the independent scattering approximation, we then define the scattering mean free paths of body and surface waves including the coupling between the two types of waves. Using a phenomenological approach, we then derive a set of coupled transport equations satisfied by the specific energy density of surface and body waves in a medium containing a homogeneous distribution of point scatterers. In our model, the scattering mean free path of body waves is depth dependent as a consequence of the body-to-surface coupling. We demonstrate that an equipartition between surface and body waves is established at long lapse-time, with a ratio which is predicted by usual mode counting arguments. We derive a diffusion approximation from the set of transport equations and show that the diffusivity is both anisotropic and depth dependent. The physical origin of the two properties is discussed. Finally, we present Monte-Carlo solutions of the transport equations which illustrate the convergence towards equipartition at long lapse-time as well as the importance of the coupling between surface and body waves in the generation of coda waves.〈/span〉

Description: 〈span〉〈div〉Abstract〈/div〉In wave physics, the geometrical limit is defined as a propagation regime where the scattering cross‐section σ of an object becomes independent of its internal structure and tends to twice its geometrical cross section σg as the frequency goes to infinity. This is a result that is particularly well documented in the field of optics. Following the classification of 〈a href="https://pubs.geoscienceworld.org/bssa#rf20"〉Wu and Aki (1985b)〈/a〉, we study the high‐frequency scattering limit for velocity‐type and impedance‐type elastic perturbations. Although velocity‐type scatterers do follow the geometrical limit of σ→2σg, the scattering cross section of any impedance‐type scatterer depends on both its density and elastic properties at all frequencies. These results are illustrated using the example of a spherical inclusion that exhibits a small contrast of properties with its environment. We derive simple asymptotic formulas that show good agreement with exact solutions of the boundary value problem (BVP). Our results confirm the distinct behavior of velocity‐type versus impedance‐type perturbations at all frequencies.〈/span〉

Description: Coda-wave interferometry is a technique which exploits tiny waveform changes in the coda to detect temporal variations of seismic properties in evolving media. Observed waveform changes are of two kinds: traveltime perturbations and distortion of seismograms. In the last 10 yr, various theories have been published to relate either background velocity changes to traveltime perturbations, or changes in the scattering properties of the medium to waveform decorrelation. These theories have been limited by assumptions pertaining to the scattering process itself—in particular isotropic scattering, or to the propagation regime—single-scattering and/or diffusion. In this manuscript, we unify and extend previous results from the literature using a radiative transfer approach. This theory allows us to incorporate the effect of anisotropic scattering and to cover a broad range of propagation regimes, including the contribution of coherent, singly scattered and multiply scattered waves. Using basic physical reasoning, we show that two different sensitivity kernels are required to describe traveltime perturbations and waveform decorrelation, respectively, a distinction which has not been well appreciated so far. Previous results from the literature are recovered as limiting cases of our general approach. To evaluate numerically the sensitivity functions, we introduce an improved version of a spectral technique known as the method of ‘rotated coordinate frames’, which allows global evaluation of the Green's function of the radiative transfer equation in a finite domain. The method is validated through direct pointwise comparison with Green's functions obtained by the Monte Carlo method. To illustrate the theory, we consider a series of scattering media displaying increasing levels of scattering anisotropy and discuss the impact on the traveltime and decorrelation kernels. We also consider the related problem of imaging variations of scattering properties based on intensity perturbations observed in the coda. The impact of anisotropy is particularly pronounced for the scattering and decorrelation sensitivity kernels, which probe spatial/temporal changes in the scattering properties of the medium. Compared to the isotropic case, scattering anisotropy strongly increases the sensitivity of coda waves in the vicinity of the single-scattering ellipse, which may have important implications for imaging applications. In addition to demonstrating the impact of non-isotropic scattering on the sensitivity kernels of coda waves, our work offers a practical solution to model this process accurately.

Description: Seismogram envelopes recorded at Campi Flegrei caldera show diffusive characteristics as well as steep amplitude increases in the intermediate and late coda, which can be related to the presence of a non-uniformly scattering medium. In this paper, we first show the results of a simulation with a statistical model considering anisotropic scattering interactions, in order to match coda-envelope duration and shape. We consider as realistic parameters for a volcanic caldera the presence of large square root velocity fluctuations (10 per cent) and two typical correlation lengths for such an heterogeneous crust, a  = 0.1 and 1 km. Then, we propose the inclusion of a diffusive boundary condition in the stochastic description of multiple scattering, in order to model intermediate and late coda intensities, and particularly the sharp intensity peaks at some stations in the caldera. Finally, we show that a reliable 2-D synthetic model of the envelopes produced by earthquakes vertically sampling a small region can be obtained including a single drastic change of the scattering properties of the volcano, that is, a caldera rim of radius 3 km, and sections varying between 2 and 3 km. These boundary conditions are diffusive, which signifies that the rim must have more scattering potential than the rest of the medium, with its diffusivity 2–3 orders of magnitude lower than the one of the background medium, so that the secondary sources on its interface(s) could enhance coda intensities. We achieve a good first-order model of high-frequency (18 Hz) envelope broadening adding to the Monte Carlo solution for the incident flux the secondary source effects produced by a closed annular boundary, designed on the caldera rim signature at 1.5 km depth. At lower frequencies (3 Hz) the annular boundary controls the intermediate and late coda envelope behaviour, in a way similar to an extended diffusive source. In our interpretation, the anomalous intensities observed at several stations and predicted by the final Monte Carlo solutions are mainly due to the diffusive transmission reflection from a scattering object of increased scattering power, and are controlled by its varying thickness.

Description: We analyse the statistics of phase fluctuations of seismic signals obtained from a temporary small aperture array deployed on a volcano in the French Auvergne. We demonstrate that the phase field satisfies Circular Gaussian statistics. We then determine the scattering mean free path of Rayleigh waves from the spatial phase decoherence. This phenomenon, observed for diffuse wavefields, is found to yield a good approximation of the scattering mean free path. Contrary to the amplitude, spatial phase decoherence is free from absorption effects and provides direct access to the scattering mean free path.

Description: We investigate the impact of spatial variations of absorption and scattering properties on the energy envelopes of coda waves. To model the spatiotemporal distribution of seismic energy, we employ a scalar version of the radiative transfer equation with spatially dependent absorption and scattering quality factor. The scattering pattern which describes the angular distribution of energy upon scattering is assumed to be statistically isotropic, independent of position, but otherwise arbitrary. Further assuming that the spatial variations of the governing parameters are sufficiently weak, we employ perturbation theory to derive linearized relations between the absorption/scattering properties of the medium and the intensity detected in the coda. These relations take the form of weighted integrals where so-called scattering/absorption sensitivity kernels play the role of weighting function. The kernels depend on the type of perturbation (scattering or absorption), the lapse-time in the coda, and require the knowledge of the complete angular dependence of the specific intensity describing the flow of energy in a given direction at a given location. In the long lapse-time limit, we establish simplified formulae which depend on the first two angular moments of the specific intensity only. As an illustration of the theory, we calculate the absorption and scattering sensitivity kernels in a 2-D isotropically scattering medium at different lapse-times in the coda, and discuss their singularities in detail. The sensitivity kernels are then employed to calculate the relative intensity variations of the coda caused by a localized Gaussian absorption/scattering anomaly. We find that the dominant effect of absorption anomalies is to modify the decay rate of the coda, while scattering anomalies have a more complex signature, causing either positive or negative deflection of the energy envelope, depending on their location and the lapse-time. Our results suggest the possibility to locate and discriminate between scattering and absorption anomalies from the energy envelope of coda waves.