A general random medium could have separate perturbations in P velocity, S velocity, and density, but in practice a common simplification is to assume a linear scaling relationship among the perturbations (e.g., Sato, 1990) and/or to assume zero density perturbations.

Single-scattering theory provides equations that give the average scattered power as a function of the incident and scattered wave types (i.e., P or S), the power of the incident wave, the local volume of the scattering region, the bulk and statistical properties of the random medium, the scattering angle (the angle between the incident wave and the scattered wave), and the seismic wave number ( k = 2π/Λ, where Λ is the wavelength). A review of the properties (elasticity, conductivity, permeability) and statistics of random heterogeneous materials is given in the text by Torquato (2002). Detailed descriptions of Born scattering theory for elastic waves are contained in Wu and Aki (1985a, 1985b), Wu (1989), and Sato and Fehler (1998). Single-scattering theory is sometimes called Chernov theory (after Chernov, 1960). Thus this approximation is only valid when the scattered waves are much weaker than the primary waves, which is the case in the Earth when the velocity and density perturbations are relatively small (quantifying exactly how small depends upon the frequency of the waves and the source-to-receiver distance). The total energy in the seismic wavefield therefore increases by the amount contained in the scattered waves and energy conservation is not obeyed. The mathematics in this case is greatly simplified if we assume that the primary waves are unchanged by their passage through the scattering region (the Born approximation).

Shearer, in Treatise on Geophysics, 2007 1.20.2.1 Single-Scattering Theory and Random Mediaįor sufficiently weak velocity and density perturbations, most scattered energy will have experienced only one scattering event and can be adequately modeled using single-scattering theory. These methods have been widely used around the world (Biswas and Aki, 1984 Dewberry and Crosson, 1995 Hartse et al., 1995 Phillips and Aki, 1986). The ratio of coda amplitudes of different earthquakes at a single station gives the relative source radiation, since the site factor is common. He also found less variability in site effect measurements made for a single site using coda waves from various sources than from direct waves for the same sources. Tsujiura (1978) found that estimates of site effects made using coda are similar to those obtained from direct wave measurements. The ratio of coda amplitudes at different sites gives the relative site amplification factor since the source factor is common. Aki (1980) first used this method for the estimation of (8) against r, the gradient gives the attenuation per travel distance. We may smooth out the radiation pattern differences when the measurements are made over a large enough number of earthquakes.

Coda meaning