The Small BAseline Subset technique provides remote measurements of ground displacement with high spatial resolution, making it a key tool for monitoring geophysical processes in hazard-prone areas. An effective analysis of this type of data requires reliable estimation of their second-order structure, which is difficult to achieve because the measurements are systematically missing over relatively large portions of the investigated areas. We tackle the problem from a functional data analysis perspective and treat the observations as partially observed functional data with two-dimensional domain. To properly characterize the data, we introduce the fragmented regime of partial observation, where parts of the curves are systematically missing across replicates. For this regime, we propose a novel method for covariance estimation, formulating the task as a matrix completion problem with Laplacian regularization. The estimator is nonparametric and free from stationarity or isotropy assumptions. Extensive simulations show that our method achieves consistently low estimation error across a range of covariance structures. Application to ground displacement data relative to the Phlegraean Fields demonstrates its ability to recover meaningful spatial dependence patterns, highlighting its potential for environmental risk assessment and monitoring.
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