EM-EOF: Gap-Filling in Incomplete SAR Displacement Time Series

An iterative method, namely expectation maximization-empirical orthogonal functions (EM-EOF) is proposed for the first time to retrieve missing values in synthetic aperture radar (SAR) displacement time series. This method decomposes the temporal covariance of a displacement measurement time series into different EOF modes by solving the eigenvalue problem, and then selects the optimal number of EOF modes to reconstruct the time series. After an appropriate initialization of missing values, the EM-EOF method performs: 1) a cross-validation root-mean-square error (cross-RMSE) minimization to find an estimate of the optimal number of EOF modes used in the reconstruction and 2) an iterative update of missing values which gives the best estimate of missing data points according to the cross-RMSE. Synthetic simulations have been first performed to highlight the efficiency of EM-EOF in the case of various displacement signal complexities and different types of noise and gaps, and a thorough error analysis has been conducted to determine the sensitivity of the method to signal-to-noise ratio (SNR), quantity of gaps, and type of noise and gaps. Then, EM-EOF is applied to three displacement measurement time series computed from Sentinel-1 A/B SAR images: two interferograms time series over Gorner and Miage glaciers, and one offset time series over the Argentière Glacier covering a period extending from September 2016 to December 2017. Both synthetic simulations and real data applications demonstrate the ability of EM-EOF to retrieve missing values, even in the cases of frequent data gaps, limited size of the time series, and spatio-temporally correlated noise and gaps.