Sensitivity of coda waves to spatial variations of absorption and scattering: radiative transfer theory and 2-D examples

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.