Nonstationarity GeostatisticsEdit
Nonstationarity geostatistics is the branch of spatial statistics that tackles the fact that the statistical properties of a spatial process can vary across space and time. In many natural and engineered systems, simple, global assumptions of sameness—where the mean, variance, and correlation structure are the same everywhere—do not hold. Rocks with different mineralogy, soils with distinct textures, urban development, and changing climate or groundwater conditions all produce data whose local behavior departs from a single, universal pattern. Traditional geostatistics rests on stationarity, the idea that a process looks statistically the same when shifted to a different location. When that assumption fails, forecasts, uncertainty quantification, and decision-making can all go off the rails. Nonstationarity geostatistics provides a toolbox to accommodate spatially varying means, variances, and dependence structures, while remaining compatible with the probabilistic machinery that makes geostatistics useful for resource evaluation, risk assessment, and policy-relevant forecasting. geostatistics stationarity nonstationarity
Because landforms, subsurface geology, climates, and human activity are inherently heterogeneous, any practical application—from mineral exploration to groundwater management—benefits from models that reflect spatial variation in how processes behave. A right-of-center approach to this subject emphasizes clarity, cost-efficiency, and the practicalities of decision-making under uncertainty: use models that capture essential nonstationarity without overfitting, employ transparent assumptions, and rely on data that are accessible or economically justified. In this spirit, nonstationarity geostatistics is framed around two core ideas: (1) nonstationarity can enter through the mean function (a spatial trend) or through the covariance structure (changing variability and correlation), and (2) the modeling toolkit ranges from simple, interpretable trend-adjusted methods to sophisticated, compute-intensive constructions that flexibly adapt to spatial context. trend covariance function kriging
Core concepts
Definitions and scope. A spatial process is nonstationary when at least one of its basic statistical descriptors—mean, variance, or the spatial dependence—changes with location or direction. This stands in contrast to stationary processes, where the same probabilistic rules apply everywhere. Nonstationarity can be local (the process behaves differently in different regions), global (a gradual change across the domain), or piecewise (distinct regimes with boundaries). These distinctions matter for how we estimate, predict, and quantify uncertainty. stationarity variogram covariance function
Types of nonstationarity. Broadly, nonstationarity arises from a changing mean (trend nonstationarity) or from a changing covariance structure (covariance nonstationarity). A trend surface might reflect geology, land cover, or dose-response effects in environmental data, while covariance nonstationarity encodes how variability and spatial coherence depend on location (e.g., near a fault, at different depths, or across climate zones). Some approaches explicitly separate the mean from the covariance, while others couple them through shared drivers or deformation ideas. trend intrinsic random function covariance function
Modeling philosophy. The field offers a spectrum of strategies. Some methods extend familiar stationary techniques by incorporating covariates (externally drifted/ universal kriging), others build locally stationary models that piece together region-specific estimates, and still others construct globally defined, nonstationary random fields via mathematical transformations or convolution with spatially varying kernels. The choice often reflects a balance between interpretability, data availability, and computational cost. kriging process convolution deformation method SPDE
Inference and computation. Estimation typically involves a likelihood or Bayesian framework, with attention to identifiability when both mean and covariance vary. Efficient computation is a central concern, especially for large datasets common in environmental monitoring or mining. Modern formulations exploit sparse matrices, finite element methods, or low-rank approximations to keep the problem tractable while preserving essential nonstationary behavior. likelihood Bayesian Gaussian process SPDE
Models and methods
Trend and covariate-driven approaches. Universal kriging and related externally drifted constructions model the mean as a function of location and covariates (such as depth, rock type, or land cover). After removing the estimated trend, a stationary residual field is modeled locally or globally. This class is attractive for its relative interpretability and direct ties to physically meaningful covariates. universal kriging trend covariates
Local stationarity and partitioning. The domain is divided into regions within which the process is assumed to be stationary or near-stationary. Predictions combine region-specific models, often with smooth weighting to avoid discontinuities at boundaries. Local approaches are intuitive and can leverage existing stationary tools, but require careful design of the partition and smoothing to avoid artifacts. local kriging variogram
Deformation methods. The deformation approach maps the original spatial domain into a latent, deformed space where the process becomes stationary. After modeling in the transformed space, predictions are mapped back to the physical space. This strategy is powerful for handling smoothly varying anisotropy and directional effects tied to underlying geology or climate gradients. deformation method sammon mapping
Process-convolution and kernel-based models. A nonstationary field can be built by convolving spatial white noise with location-dependent kernels. The kernel shape and scale may depend on covariates or latent variables, allowing the local correlation structure to adapt to changing conditions. This framework is flexible and conceptually straightforward, though it can be computationally demanding if fully nonparametric. process convolution
SPDE-based nonstationarity. A landmark development links nonstationary Matérn-type random fields to stochastic partial differential equations with spatially varying parameters. When discretized with finite elements, these models produce scalable, interpretable nonstationary covariances that align with physical intuition about diffusion-like processes in heterogeneous media. This approach has become influential in environmental engineering, hydrology, and mining risk assessment. stochastic partial differential equation Matérn covariance function SPDE
Covariate-driven and regime-based covariance modeling. Covariates can drive spatially varying variance and correlation lengths, capturing how environmental or geological context amplifies or dampens spatial dependence. Regime-based modeling allows different covariance regimes in different zones, potentially tied to formation types or management units. covariance function trend
Nonstationary inference in practice. Inference combines model selection, cross-validation, and posterior or likelihood-based evaluation to balance fit, complexity, and predictive performance. Practical guidance emphasizes starting simple (trend plus stationary residuals) and adding nonstationarity only where substantial gains in predictive accuracy or uncertainty calibration are demonstrated on held-out data. cross-validation likelihood
Applications
Mineral exploration and resource estimation. In ore bodies, properties such as grade and texture vary with depth and lithology. Nonstationary models help quantify local ore potential and uncertainty, improving drill targeting, reserve estimation, and decision-making under risk. Such models connect directly with geological interpretation and economic optimization. mineral exploration resource estimation
Groundwater and hydrogeology. Aquifers exhibit spatially variable hydraulic conductivity, porosity, and storativity due to stratigraphy and fracture networks. Nonstationary geostatistics supports better prediction of groundwater levels, contaminant plumes, and risk of saltwater intrusion, informing well field design and water management policies. groundwater hydrogeology
Environmental risk and remediation. Pollutant concentrations may be higher near sources or within heterogeneous soils. Capturing nonstationarity improves plume predictions, exposure assessment, and remediation planning, aligning technical assessments with regulatory expectations and community concerns. environmental risk pollution
Climate, ecology, and agriculture. Climate gradients and land-use change induce spatially varying processes in precipitation, soil moisture, and vegetation indices. Nonstationary models help synthesize remote-sensing data with field observations to guide adaptation strategies, land-management decisions, and agricultural planning. climate ecology precision agriculture
Urban planning and infrastructure. In dense urban environments, material properties, subsurface utilities, and construction histories produce abrupt changes in variability and correlation structures. Nonstationarity-aware models support safer design, risk assessment for subsidence or groundwater impacts, and more reliable forecasts for infrastructure management. urban planning infrastructure
Debates and controversies
Balancing simplicity and realism. A recurring dispute concerns how much nonstationarity to add: too little risks bias and overconfident predictions, too much risks overfitting, identifiability problems, and opaque models. Proponents of parsimonious, covariate-driven nonstationarity argue for tangible gains with transparent interpretation, while advocates of highly flexible, process-convolution or SPDE-based models emphasize predictive performance in complex domains at the expense of interpretability. kriging process convolution SPDE
Computation versus practicality. High-dimensional, nonstationary models can be computationally intensive, particularly for large environmental or industrial datasets. The practical stance prioritizes approaches that scale, exploit sparsity, and deliver timely results for decision-makers, even if that means accepting some modeling approximations. Bayesian likelihood
Identifiability and data requirements. Distinguishing between a slowly varying mean and a changing covariance can be challenging when data are sparse or unevenly distributed. Critics caution against over-interpreting spatial patterns when the data do not justify the complexity, while supporters emphasize the value of incorporating domain knowledge and covariates to guide the modeling. trend variogram
Policy, regulation, and transparency. Regulators and industry clients often prefer models that are traceable and reproducible. Some nonstationary approaches, particularly those with many latent components or strong prior assumptions, raise concerns about reproducibility or the ability to fully defend conclusions in regulatory settings. Advocates counter that clear documentation, sensitivity analyses, and standard validation procedures can mitigate these concerns. regulation transparency
Woke critiques and technical disagreements. From a pragmatic, market-facing viewpoint, some criticisms framed as concerns about environmental or social justice narratives should not obscure the core engineering questions: does a nonstationary model improve predictive accuracy, reduce risk, and justify the cost of data collection and computation? Proponents argue that skepticism about overly ideological critiques helps keep the focus on measurable performance, while critics warn against letting ideology erase legitimate concerns about uncertainty quantification and equity of resource decisions. In technical terms, the debate centers on whether explanatory depth or practical utility should drive model choice, and on how best to communicate uncertainty to stakeholders without triggering misinterpretations. uncertainty risk management
Climate and attribution debates. When nonstationarity is invoked to describe climate- or land-use driven changes, some debate centers on attribution and timescale. Proponents stress that spatially varying processes reflect real, observable heterogeneity that must be modeled for reliable decision-making. Critics may argue that nonstationary models risk conflating natural variability with anthropogenic trends unless supported by rigorous physical interpretation. The prudent path is to integrate nonstationary statistical methods with process-based understanding, and to document assumptions and data limitations clearly. climate process-based