  This paper investigates the modeling of an important class of degradation
data, which are collected from a spatial domain over time; for example, the
surface quality degradation. Like many existing time-dependent stochastic
degradation models, a special random field is constructed for modeling the
spatio-temporal degradation process. In particular, we express the degradation
at any spatial location and time as an additive superposition of two stochastic
components: a dynamic spatial degradation generation process, and a
spatio-temporal degradation propagation process. Some unique challenges are
addressed, including the spatial heterogeneity of the degradation process, the
spatial propagation of degradation to neighboring areas, the anisotropic and
space-time non-separable covariance structure often associated with a complex
spatio-temporal degradation process, and the computational issue related to
parameter estimation. When the spatial dependence is ignored, we show that the
proposed spatio-temporal degradation model incorporates some existing pure
time-dependent degradation processes as its special cases. We also show the
connection, under special conditions, between the proposed model and general
physical degradation processes which are often defined by stochastic partial
differential equations. A numerical example is presented to illustrate the
modeling approach and model validation.
