Irregularly and incompletely sampled random fields in the Earth sciences: Analysis and synthesis of parameterized covariance models

Summary Geophysical data observed in the Earth, planetary, and lunar sciences are spatially limited to sampling plans designed to span a feature of interest whose resolution and regularity in discretization are bound by site, observational, and instrumental constraints. Common wisdom tells us that the more observations we make, the less variable our estimates of statistical properties of measured physical quantities will be. However, given the limited spatial extent and possibly irregular sampling of any dataset, how confident can we be in our estimates of the parameters of statistical processes underlying such fields? 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 regions of interest, structured sweeps of instrumental measurements, and missing observations dispersed across the domain of a field, which spur behaviors in the estimator. 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 Matérn class and show how to interrogate goodness-of-fit criteria to detect departures from the null hypothesis of Gaussianity, stationarity, and isotropy.