Velocities of a spatial–temporal stochastic field with embedded dynamics

Abstract

The paper investigates further an approach to modeling dynamically changing Gaussian spatiotemporal fields. In that approach, the dynamics are introduced by embedding deterministic velocities into a stochastic spatiotemporal Gaussian model. In this way, a dynamically inactive stochastic field with given spatial and temporal covariance structure gains dynamics that in general follow a deterministic pattern. Here, we make an important connection between the resulting stochastic field and underlying deterministic dynamics by demonstrating that in the case of isotropic spatial dependencies, the observed random velocities are centered at the velocities of the underlying physical flow. Additionally, we discuss strategies for simulation of such fields and give foundation for fitting and prediction procedures that are based on the obtained results. In an effort to illustrate attractiveness of the approach for modeling environmental phenomena, we consider a parameterized specification of spatiotemporal correlation structure and embed to it the dynamics driven by the shallow water equations. Through simulations, we show how the spatiotemporal behavior of the resulting nonstationary Gaussian field is altered by the embedded dynamics. Copyright © 2012 John Wiley & Sons, Ltd.