Spatial data in the Earth and environmental sciences acquired by instrument collection or simulation are constrained to finite, discrete, (ir)regular grids whose geometry is delineated by a boundary within which missingness, either random or structured, may exist. We model (ir)regularly sampled Cartesian spatial data as realizations of discrete two- and three-dimensional random fields whose covariance structure we estimate parametrically with a spectral-domain maximum-likelihood estimation strategy using the debiased Whittle likelihood, which efficiently counters the effects of aliasing and spectral leakage that arise from finite sampling and boundary effects. We work with the general, flexible Matérn class of covariance functions, which characterizes the shape of a field through three parameters that quantify its amplitude, smoothness, and correlation length. We quantify parameter covariance analytically and asymptotically based on the parametric model and sampling grid alone, agnostic of observed data. Our uncertainty quantification allows us to study how sampling geometry imparts uncertainty on a covariance model and provides a path for optimizing the design of a sampling grid to reduce error for an anticipated model. We formulate several approaches for interrogating our model residuals to interpret where real Earth data depart from the null hypotheses of Gaussianity, stationarity, and isotropy. We explore select case studies that demonstrate the broad applicability of our models across Earth science disciplines and develop software in MATLAB and Python for implementation by domain scientists, in hydrology, and elsewhere.