  Statistical methods are required to evaluate and quantify the uncertainty in
environmental processes, such as land and sea surface temperature, in a
changing climate. Typically, annual harmonics are used to characterize the
variation in the seasonal temperature cycle. However, an often overlooked
feature of the climate seasonal cycle is the semi-annual harmonic, which can
account for a significant portion of the variance of the seasonal cycle and
varies in amplitude and phase across space. Together, the spatial variation in
the annual and semi-annual harmonics can play an important role in driving
processes that are tied to seasonality (e.g., ecological and agricultural
processes). We propose a multivariate spatio-temporal model to quantify the
spatial and temporal change in minimum and maximum temperature seasonal cycles
as a function of the annual and semi-annual harmonics. Our approach captures
spatial dependence, temporal dynamics, and multivariate dependence of these
harmonics through spatially and temporally-varying coefficients. We apply the
model to minimum and maximum temperature over North American for the years 1979
to 2018. Formal model inference within the Bayesian paradigm enables the
identification of regions experiencing significant changes in minimum and
maximum temperature seasonal cycles due to the relative effects of changes in
the two harmonics.
