Krzysztof Podgórski

The paper investigates further an approach to modeling dynamically changing Gaussian spatio-temporal fields. In that
approach, the dynamics are introduced by embedding deterministic velocities into a stochastic spatio-temporal 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
parametrized specification of spatio-temporal correlation structure and embed to it the dynamics driven by the shallow
water equations. Through simulations, we show how the spatio-temporal behavior of the resulting non-stationary Gaussian
field is altered by the embedded dynamics.