A phase transition in the O’Doherty-Anstey model

In the ODoherty-Anstey model, the spectrum of a seismic wave transiting and reverberating within a stack of interfaces is estimated in terms of the reflectivity. At the centre of the model is a discrete combinatorical calculation, in which contributing ray-paths at each lag are weighted and counted. The model maps naturally into an analysis based on statistical mechanics, with ray-paths playing the role of system configurations, and lag playing the role of the system energy. This leads to an expression for the probability of finding a contribution to the wave at a given lag, determined up to a partition function and a scattering parameter analogous to temperature. The partition function is estimated within the ODoherty-Anstey model itself, and the scattering parameter can be adjusted to describe geological media with increased or decreased scattering potential. The average contributing lag deriving from this analysis exhibits a continuous phase transition, separating two distinct scattering regimes, one in which the direct wave dominates, and the other in which scattering dominates. The weighting produced by the reflectivity appears to govern the smoothness of the transition.