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Stochastic models are often used to describe the spatial structure of atmospheric phase delays in differential interferometric synthetic aperture radar (DInSAR) data. Synthetic aperture radar interferograms often exhibit anisotropic atmospheric signals. In view of this, the use of anisotropic models for atmospheric phase estimation is increasingly advocated. However, anisotropic models lead to increased computational complexity in estimating the correlation function parameters with respect to the isotropic case. Moreover, the performance is degraded when dealing with DInSAR techniques involving only a few sparse points usable for computations, as in the case of persistent scatterer interferometry applications, particularly when this estimation has to be done in an automated way on many interferograms. In the present work, we propose some observations about the actual advantage given by anisotropic modeling of atmospheric phase in the case of sparse-grid point-target DInSAR applications. Through analysis of simulated data, we observe that an improvement in the performances of kriging reconstruction approaches can be obtained only when sufficient sampling densities are available. In critical sampling conditions, automated methods with reasonable computational cost may improve their performance if external information on the atmospheric phase screen field is available.