  This paper proposes a physical-statistical modeling approach for
spatio-temporal data arising from a class of stochastic convection-diffusion
processes. Such processes are widely found in scientific and engineering
applications where fundamental physics imposes critical constraints on how data
can be modeled and how models should be interpreted. The idea of spectrum
decomposition is employed to approximate a physical spatio-temporal process by
the linear combination of spatial basis functions and a multivariate random
process of spectral coefficients. Unlike existing approaches assuming
spatially- and temporally-invariant convection-diffusion, this paper considers
a more general scenario with spatially-varying convection-diffusion and
nonzero-mean source-sink. As a result, the temporal dynamics of spectral
coefficients is coupled with each other, which can be interpreted as the
non-linear energy redistribution across multiple scales from the perspective of
physics. Because of the spatially-varying convection-diffusion, the space-time
covariance is non-stationary in space. The theoretical results are integrated
into a hierarchical dynamical spatio-temporal model. The connection is
established between the proposed model and the existing models based on
Integro-Difference Equations. Computational efficiency and scalability are also
investigated to make the proposed approach practical. The advantages of the
proposed methodology are demonstrated by numerical examples, a case study, and
comprehensive comparison studies. Computer code is available on GitHub.
