Is the single scattering model invalid for the coda excitation at long lapse time?

Supposing that the distribution of scatterers in a three-dimensional medium is not uniform but fractally homogeneous with fractal dimensionD, we have made the dimensional analysis for the temporal decay of the multiple scattering energy density at the hypocenter.The number of scatters in a sphere of radiusR is assumed to be proportional toR ^D . Then, the energy density of thekth order scattering decays according to the (D–2)k–3]th power of lapse time. A fractal dimension ofD=3 corresponds to the uniform distribution. If 2<D3, multiple scattering terms of orderk2 dominate over the single scattering term (k=1) at long lapse time. IfD=2, energy density of every order decays according to the — 3rd power of lapse time. The single scattering model survives on conditionD<2; the single scattering term dominates over the higher order multiple scattering terms even at long lapse time, since the negative power of lapse time fork=1 is the smallest of all.