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

Abstract: We study how sampling geometry contributes to uncertainty in modeling spatial 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.  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. We study the effect of covariance parameters assumed a priori. We demonstrate the desirable behavior of the general Matern class of covariance functions and show how to interrogate goodness-of-fit criteria to detect departures from the null hypothesis of Gaussianity, stationarity, and isotropy.