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 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.

Figures (26)

• Figure 1: Analyzing geophysical random fields on domains with disjoint support. The debiased Whittle maximum likelihood analysis developed in this paper is able to estimate statistical parameters of their covariance structure, suitable for simulation, out-of-sample extension, evaluation, interpretation, and uncertainty appraisal. Bathymetric data GEBCO2024 from the northeast Atlantic (top row). Synthetic realizations that share the first- and second-order statistics of the superjacent data within their window of observation (middle row) and on the joint grid (bottom). In the bottom right, Matérn model parameter estimates (for variance $\sigma^2$, smoothness $\nu$, and range $\rho$) and their one standard deviation uncertainties are displayed for the direct (dir, blue circles), indirect (ind, orange triangles), and merged (mer, yellow stars) data. The parameter estimates for the merged, nonstationary, dataset are not simple averages of those of the underlying component processes, and their uncertainties are irreconcilable due to model misspecification.
• Figure 2: Spatial random fields simulated from special cases of the Matérn class, for a common variance ($\sigma^2_0=1\,\mathrm{m}^2$) and range ($\rho_0=5\,\mathrm{m}$) over a rectangular, regular, grid, $122\,\mathrm{m} \times 141\,\mathrm{m}$, with even $1\,\mathrm{m}$ spacing, for different smoothness parameters $\nu_0$. Top: Smoothness parameter $\nu_0=1/3$, $1/2$, and $1$. Bottom: $\nu_0=3/2$, $5/2$, and $\infty$ (see Table \ref{['tab:MaternSpecial']} for their analytic forms). Blue circles mark the approximate $1/{}3$ decorrelation length of the field by their radii $\pi\rho$, all of which are equal here due to the common range $\rho$ of the six models. Colormap ranges between $\pm2$ standard deviations $\mathsf{s}$ of the fields with means $\mathsf{m}$.
• Figure 3: Realizations of special cases of smoothness $\nu$ of the Matérn covariance class for a common variance (${\sigma^2}=1\, \mathrm{km}^2$) and range ($\rho = 250\, \mathrm{km}$) over a regular rectangular grid, $3840 \,\mathrm{km}$$\times$$3840 \,\mathrm{km}$, with even $30 \,\mathrm{km}$ spacing. We show three cases from Table \ref{['tab:MaternSpecial']}, including $\nu=1/{}2$ (left), $1$ (center), and $3/{}2$ (right). Top: Normalized spectral densities ${\mathcal{S}}_{\boldsymbol{\theta}}(k)/{\mathcal{S}}_{\boldsymbol{\theta}}(0)$. Vertical black lines at $\lambda_\alpha$ correspond to calculated percentages of cumulative spectral variance. Middle: Correlation ${\mathcal{C}}_{\boldsymbol{\theta}}(r)/{\sigma^2}$ as a function of lag distance $r$. Vertical blue lines indicate a distance of $\pi\rho=785\, \mathrm{km}$ where the covariance reaches about one-third of the variance, with an observable dependence on $\nu$. Vertical black lines at $r_\alpha$ correspond to calculated percentages of cumulative spatial variance (eq. \ref{['eq:rootcov']}). Bottom: Spatial simulations generated through circulant embedding of the Matérn spatial covariance model. Blue circle radii of $\pi\rho$ indicate the approximate $1/_{ }3$ correlation length of the fields. Colormaps and subtitles reference the sample mean $\mathsf{m}$ and standard deviation $\mathsf{s}$ of the realizations.
• Figure 4: Maximum-likelihood (eq. \ref{['eq:dwl']}) estimation for a 66.7% observed sampling grid with missingness presented as random deletions. Top left: a single realization of a spatial random field ${\mathcal{H}}(\mathbf{x})$ simulated through the circulant embedding of the Matérn covariance with $\boldsymbol{\theta}= [10\,\mathrm{km}^2\; 1.5 \; 5\,\mathrm{km}]$ on a $319\times326$ grid with $1\,\mathrm{km}\times1\,\mathrm{km}$ spacing. Top center: the empirical periodogram of the field $|H(\mathbf{k})|^2$. Top right: the blurred spectral density $\bar{{\mathcal{S}}}_{\boldsymbol{\theta}}(\mathbf{k})$. Bottom left: the spatial taper $w(\mathbf{x})$ indicating the presence or absence of data in white and black, respectively. Bottom center: the ratio of the average empirical periodogram over 100 realizations to the blurred spectral density. Bottom right: the average empirical periodogram. The empirical periodogram approaches the blurred spectral density in expectation. Boundary effects and sampling irregularities efface the isotropy of both, while keeping their ratio unstructured and uninformative.
• Figure 5: The covariance and pseudocovariance terms of the periodogram covariance (eq. \ref{['eq:percov']}) in the case of random deletions (66.7% observed). The product of the observed grid taper with the spatial autocovariance over all lag interactions (top left) reveals a tartan pattern of negligibly contributing lag pairs that manifests in the spectral-domain terms that include all wavenumber interactions (top middle through right) as elongated hatches of diffuse power in contrast to the fully observed grid, concentric about the intersection of the diagonals of the covariance and pseudocovariance. When $\mathbf{k}=\mathbf{k'}$ (bottom), the random deletions of the observation grid display an oblique symmetry in the periodogram variance.