Isotropic scattering coefficient of the solid earth

The isotropic scattering model is a simple mathematical model of the radiative transfer theory (RTT) for the propagation of the wave energy density in random media. There have been many measurements of the isotropic scattering coefficient of the heterogeneous solid earth medium, where the target region varies from the lower and upper mantle, the crust, sediments, volcanoes, mines, rock samples and also the crust and the upper mantle of the moon. Reported isotropic scattering coefficients increase according to some power of frequency with some scatter. We know that the RTT is well approximated by the diffusion equation in the multiple scattering regime, where the equipartition is established. Then, the transport scattering coefficient effectively functions as an isotropic scattering coefficient even if the scattering coefficient derived by the Born approximation for the random velocity fluctuation is anisotropic. Recent review of the power spectral density functions of random velocity fluctuations in the solid earth revealed from various kinds of measurements shows that their spectral envelope is well approximated by the inverse cube of wavenumber for a wide range of wavenumbers (Sato, 2019). The transport scattering coefficient derived from the spectral envelope linearly increases with frequency, which well explains the observed isotropic scattering coefficients for a wide range of frequencies. However, some reported isotropic scattering coefficients show unusual behaviour: the isotropic scattering coefficient increases as depth decreases in the crust and the upper mantle of the earth and the moon, those beneath volcanoes are larger than those in the lithosphere, and that in a sandstone sample with a large porosity is larger than that in a gabbro sample with little porosity. Those differences may suggest possible scattering contribution of pores and cracks widely distributed in addition to the scattering by random velocity fluctuations.