A phase transition in the O'Doherty-Anstey model

Kristopher A. Innanen

In the O’Doherty-Anstey (OA) model, a stack of interfaces redistributes the amplitude of a transmitted wave pulse across a range of lags, and does so more and more widely as the reflectivities and number of interfaces grow. This is suggestive of a Maxwell-Boltzmann statistical model involving an ensemble of weighted raypaths, with a particular raypath and its weight playing the role of a system microstate, and lag playing the role of the energy. An artificial temperature, in the same units as the lag, is introduced, which measures in a bulk sense the reflection strengths and interface numbers in the stack. The partition function for such a model requires an estimate of the degeneracy of these states at each lag, which we argue is provided by one of the intermediate calculations in the OA model. Seeded by a real or simulated reflectivity series, a numerical estimate of the partition function allows the average lag to be calculated as a function of temperature. A critical transitional region dividing two distinct temperature regimes (one in which pulses are dominated by small lags and one in which pulses are dominated by lags of roughly the length of the input reflectivity) is observed.