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

TL;DR

This paper investigates how different sampling geometries affect the uncertainty in modeling spatial geophysical data using covariance functions, proposing methods to improve estimation and model assessment.

Contribution

It introduces an asymptotically unbiased spectral maximum-likelihood estimation method that incorporates sampling patterns and analyzes their impact on covariance modeling.

Findings

Growing-domain sampling reduces estimator bias and variance more effectively than infill schemes.

The general Matern covariance class exhibits desirable properties under various sampling patterns.

Goodness-of-fit criteria can detect departures from Gaussianity, stationarity, and isotropy.

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.