Stability of the P to S Energy Ratio in the Diffusive Regime

In the presence of inhomogeneities the propagation of elastic wave energy can be modeled by radiative transport equations. This is a good approximation when (i) typical wavelengths are short compared to the overall propagation distance, (ii) correlation lengths are comparable to wavelengths so that the inhomogeneities have appreciable effect and (iii) the fluctuations are weak. The relevant transport equations are derived in [1] starting from the elastic wave equations in an unbounded medium. There is one consequence of the transport equations that may be important in understanding from first principles the stability of the P/Lg ratio that has proven so useful in yield estimation [2]. This is the fact that over distances (and times) that are long compared to the transport mean free path (transport mean free time), which is the diffusive regime, the P to S energy conversion by the random inhomogeneities equilibrates in a universal way, independent of the details of the scattering. There is an equipartition of energy [1] that leads to the relation