🤖 AI Summary
This study addresses the challenge of modeling uncertainty in spatial random fields arising from irregular sampling and missing data in Earth sciences. The authors propose an asymptotically unbiased spectral maximum likelihood estimation framework that explicitly incorporates the geometry of the sampling design, accommodating complex scenarios such as non-rectangular domains and instrument trajectories. Built upon the assumption of a stationary, isotropic Gaussian field and employing a Matérn covariance model, the method leverages an increasing-domain sampling strategy and provides closed-form uncertainty quantification. Theoretical analysis and numerical experiments demonstrate that increasing-domain sampling substantially reduces both bias and variance in parameter estimates compared to infill (dense) sampling. Furthermore, the work systematically evaluates the influence of covariance priors on field characterization and confirms the superior fitting performance of the Matérn class of models.
📝 Abstract
We study how sampling geometry contributes to uncertainty in modeling spatial geophysical observations as sampled random fields characterized by stationary, isotropic, parametric covariance functions. We incorporate the signature of discrete spatial sampling patterns into an asymptotically unbiased spectral maximum-likelihood estimation method along with analytical uncertainty calculation. We illustrate the broad applicability of our modeling through synthetic and real data examples with sampling patterns that include irregularly bounded contiguous region(s) of interest, structured sweeps of instrumental measurements, and missing observations dispersed across the domain of a field, from which contiguous patches are generally favorable. We find through asymptotic studies that allocating samples following a growing-domain strategy rather than a densifying, infill scheme best reduces estimator bias and (co)variance, whether the field has been sampled regularly or not. As our modeling assumptions, too, shape how (well) an observed random field can be characterized, we study the effect of covariance parameters assumed a priori. We demonstrate the desirable behavior of the general Matern class and show how to interrogate goodness-of-fit criteria to detect departures from the null hypothesis of Gaussianity, stationarity, and isotropy.