  Snow density estimates as a function of depth are used for understanding
climate processes, evaluating water accumulation trends in polar regions, and
estimating glacier mass balances. The common and interpretable
physically-derived differential equation models for snow density are piecewise
linear as a function of depth (on a transformed scale); thus, they can fail to
capture important data features. Moreover, the differential equation parameters
show strong spatial autocorrelation. To address these issues, we allow the
parameters of the physical model, including random change points over depth, to
vary spatially. We also develop a framework for functionally smoothing the
physically-motivated model. To preserve inference on the interpretable physical
model, we project the smoothing function into the physical model's spatially
varying null space. The proposed spatially and functionally smoothed snow
density model better fits the data while preserving inference on physical
parameters. Using this model, we find significant spatial variation in the
parameters that govern snow densification.
