Improving piecewise linear snow density models through hierarchical spatial and orthogonal functional smoothing

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.