On the transient performance of Monte Carlo simulations for initial uncertainty forecasting

The objective of this paper is to evaluate the transient performance of the conventional Monte Carlo method (called FMC in this paper) with a fixed number of samples. The following question is asked: under what conditions could the propagated FMC ensemble have been generated by a direct sampling of the unknown true state-pdf? To answer this question, the propagated ensemble is viewed as the realization of a Markov chain and the FMC process as the evolution of the associated transition kernel. An equation governing the evolution of FMC transition kernel is derived. It is shown that for systems with “zero-divergence” in their force field (∇ · f = 0), the true evolved state pdf is the invariant distribution of the propagated FMC transition kernel at all times. No such equivalence is guaranteed for systems with non-zero divergence. Numerical simulations are provided to support theoretical claims about ensemble quality for both types of dynamic systems.