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Geostatistics

Geostatistics is a branch of statistics dedicated to the analysis, modeling, and prediction of spatial or spatiotemporal datasets, where the location of observations plays a crucial role in understanding patterns and variability in phenomena such as mineral deposits, pollutant distributions, or environmental attributes.^[1] It addresses the challenge of incomplete spatial information by employing probabilistic models to characterize uncertainty and interpolate values at unsampled locations, distinguishing it from classical statistics that often assume data independence.^[2] The field originated in the mining industry during the mid-20th century, with pioneering work by South African engineer D.G. Krige in the 1950s, who developed early interpolation methods for ore grade estimation.^[3] These ideas were formalized and expanded in the 1960s by French mathematician Georges Matheron at the Fontainebleau School of Mines, who coined the term "geostatistics" and established its theoretical foundations, including the integration of random function theory for spatial prediction.^[2] Matheron's contributions built on earlier stochastic processes from fields like meteorology and hydrology, transforming practical mining problems into a rigorous statistical framework.^[4] At its core, geostatistics relies on key concepts such as regionalized variables, which represent spatially continuous phenomena that vary gradually but exhibit local heterogeneity, and the variogram, a tool that quantifies spatial dependence by measuring how dissimilarity between data points increases with distance.^[5] The hallmark method, kriging, uses variogram models to produce best linear unbiased predictions (BLUPs) at unknown locations, incorporating measures of estimation variance to assess reliability—extending beyond simple interpolation to account for spatial autocorrelation where nearby points tend to have similar values.^[4] Variations like ordinary kriging or universal kriging adapt to trends or secondary data, enhancing accuracy in complex scenarios.^[1] Geostatistics finds wide application across disciplines, including resource evaluation in mining and petroleum engineering for reserve appraisal, environmental science for mapping soil contaminants or groundwater quality, precision agriculture for nutrient distribution, meteorology for precipitation forecasting, and public health for disease incidence modeling.^[2] These methods enable decision-making under uncertainty, such as optimizing extraction in ore bodies or assessing pollution risks, and are often integrated with geographic information systems (GIS) for visualization and further analysis.^[1]

History and Background

Origins and Early Applications

Geostatistics is a branch of applied statistics that specifically addresses the analysis and prediction of spatial or spatiotemporal data, with a core emphasis on modeling uncertainty associated with spatially correlated phenomena.^[6] This discipline arose from the need to handle data exhibiting spatial continuity, such as mineral grades or environmental attributes, where traditional statistical methods often failed due to the inherent spatial dependencies.^[7] The origins of geostatistics trace back to the South African gold mining industry in the 1950s, where early applications focused on estimating ore reserves amid sparse and irregularly distributed sampling data from boreholes.^[8] Miners employed manual estimation techniques, such as distance-weighted averages, to predict gold grades in unsampled areas of the Witwatersrand and Orange Free State goldfields, aiming to reduce financial risks from biased valuations.^[7] These practices addressed initial challenges like handling irregularly spaced data points and the regionalized variables—spatially varying attributes like ore concentration that defied simple averaging due to local trends and variability.^[8] A pivotal contribution came from Danie G. Krige, whose 1951 master's thesis at the University of the Witwatersrand, titled "A statistical approach to some mine valuation and allied problems on the Witwatersrand," introduced empirical statistical methods for gold ore reserve estimation.^[8] Published in the Journal of the Chemical, Metallurgical and Mining Society of South Africa (vol. 52, pp. 119–139), Krige's work applied regression analysis to extrapolate from known assays to unsampled blocks, incorporating polynomial drift models to account for systematic spatial trends in grade distribution. This approach marked a shift toward quantifying conditional bias in block valuations, laying groundwork for handling spatial prediction in mining contexts.^[7] By the early 1960s, these empirical mining practices in South Africa transitioned toward formalized statistical methods, influenced by Krige's innovations and prompting further theoretical development to generalize spatial estimation beyond ad hoc techniques.^[8]

Key Developments and Pioneers

Georges Matheron, a French mathematician and geologist, was instrumental in the theoretical development of geostatistics during the 1960s while affiliated with the Centre de Morphologie Mathématique at the École des Mines de Paris in Fontainebleau. In 1960, he coined the term "kriging" to honor Danie G. Krige's pioneering empirical work on spatial estimation in South African gold mines, thereby acknowledging the foundational contributions to unbiased prediction methods in resource evaluation.^[9] This naming decision, first appearing in Matheron's publications that year, integrated Krige's ideas into a rigorous probabilistic framework, marking a shift from ad hoc techniques to formalized spatial statistics.^[10] Matheron's seminal 1963 publication, Traité de géostatistique appliquée (Volume II: Le krigéage), established the theory of regionalized variables, which treats spatially distributed data as realizations of random functions varying continuously across a region. This work also introduced the intrinsic hypothesis, positing that spatial increments follow a stationary distribution, enabling the analysis of non-ergodic phenomena without assuming full second-order stationarity.^[11] These concepts provided the mathematical backbone for handling heterogeneity in geological datasets, influencing subsequent advancements in spatial modeling. The founding of the International Association for Mathematical Geosciences (IAMG) on August 22, 1968, during the International Geological Congress in Prague, Czechoslovakia, facilitated the global dissemination of geostatistical ideas. With initial involvement from scientists across multiple countries, the IAMG organized early conferences, such as sessions at subsequent geological congresses and the Príbram Mining Congress in the 1970s, which highlighted geostatistical applications and spurred interdisciplinary dialogue.^[12] These events, alongside the launch of journals like Mathematical Geology in 1969, promoted the adoption of Matheron's methods beyond academic circles. By the 1970s, geostatistics extended its reach from mining to petroleum engineering and environmental sciences, where it proved valuable for reservoir characterization and pollution mapping. This expansion was supported by influential texts, including A. G. Journel and Ch. J. Huijbregts' 1978 Mining Geostatistics, which synthesized practical implementations and case studies from the Centre de Morphologie Mathématique, making the discipline accessible to broader applications.^[13] Concurrently, the integration of Gaussian process theory bolstered geostatistics by framing spatial data as realizations of Gaussian random fields, facilitating the early adoption of linear prediction models that ensure unbiased estimation under normality assumptions.^[13]

Fundamental Concepts

Spatial Random Fields and Stationarity

In geostatistics, spatial data are modeled using the probabilistic framework of spatial random fields. A spatial random field Z(\mathbf{x}) is defined as a collection of random variables indexed by spatial locations \mathbf{x} in a domain D \subseteq \mathbb{R}^d, where each Z(\mathbf{x}) represents the value of a spatially continuous phenomenon at position \mathbf{x}. This formulation allows geostatisticians to treat observed spatial measurements as outcomes from an underlying stochastic process, enabling inference about unobserved locations. Regionalized variables, a concept introduced by Georges Matheron, serve as the bridge between deterministic spatial observations and this probabilistic model. A regionalized variable is a spatial attribute, such as mineral grade or pollutant concentration, that varies continuously across a region but is only sampled at discrete points; it is conceptualized as a single realization of the random field Z(\mathbf{x}).^[14] This perspective shifts the analysis from fixed values to statistical properties inferred from the ensemble of possible realizations, facilitating the handling of uncertainty in spatial prediction.^[14] Central to the random field model are assumptions about stationarity, which ensure that statistical properties are translation-invariant. Second-order stationarity, also known as weak stationarity, requires that the mean E[Z(\mathbf{x})] = \mu is constant across D and that the covariance \text{Cov}(Z(\mathbf{x}), Z(\mathbf{x} + \mathbf{h})) = C(\mathbf{h}) depends solely on the spatial lag \mathbf{h}, not on the absolute position \mathbf{x}. This assumption implies finite second moments and allows the use of covariance functions to quantify spatial dependence. A weaker condition, intrinsic stationarity, relaxes the requirements by focusing on increments rather than absolute values. It assumes that the expected increment E[Z(\mathbf{x} + \mathbf{h}) - Z(\mathbf{x})] is constant (often zero) and that the variance of the increment \text{Var}(Z(\mathbf{x} + \mathbf{h}) - Z(\mathbf{x})) is finite and depends only on \mathbf{h}.^[15] Intrinsic stationarity is sufficient for many geostatistical analyses, as it permits the definition of dependence measures like the variogram without needing full second-order properties.^[15] The practical application of these models often relies on ergodicity, which posits that ensemble averages (over multiple realizations) can be estimated from spatial averages within a single realization of the random field.^[16] Under ergodicity, statistics such as the mean or covariance derived from sampled data in the domain D approximate the true process parameters, justifying the use of one observed spatial dataset for inference.^[16] This assumption is particularly crucial in geostatistics, where multiple realizations are rarely available, but it requires careful validation to avoid bias in non-ergodic settings.

Variogram and Covariance Functions

In geostatistics, the semivariogram, often denoted as \gamma(\mathbf{h}), serves as a fundamental measure of spatial dependence, defined as half the variance of the difference between values of a spatial random field Z(\mathbf{x}) at locations separated by lag vector \mathbf{h}:

\gamma(\mathbf{h}) = \frac{1}{2} \mathrm{Var}[Z(\mathbf{x}) - Z(\mathbf{x} + \mathbf{h})].

This definition, introduced by Matheron in the early 1960s, quantifies how dissimilarity between observations increases with separation distance, assuming the intrinsic hypothesis of stationarity where only the increments need finite variance.^[17]^[18] The experimental semivariogram, \hat{\gamma}(\mathbf{h}), is estimated from data by averaging squared differences for pairs of observations separated by approximately \mathbf{h}:

\hat{\gamma}(\mathbf{h}) = \frac{1}{2 |N(\mathbf{h})|} \sum_{N(\mathbf{h})} [Z(\mathbf{x}_i) - Z(\mathbf{x}_j)]^2,

where N(\mathbf{h}) is the set of pairs with lag \mathbf{h}, and |N(\mathbf{h})| its cardinality. This nonparametric estimator provides an empirical depiction of spatial structure but requires fitting to a theoretical model for inference, as detailed in foundational geostatistical texts.^[18] Theoretical variogram models are parametric functions fitted to the experimental semivariogram to ensure positive definiteness and interpretability. Common models include the spherical, which rises smoothly to the sill and is given by

\gamma(h) = \begin{cases} c \left[ \frac{3h}{2a} - \frac{1}{2} \left( \frac{h}{a} \right)^3 \right] & 0 \leq h \leq a, \\ c & h > a, \end{cases}

the exponential, \gamma(h) = c [1 - \exp(-h/a)], and the Gaussian, \gamma(h) = c [1 - \exp(-(h/a)^2)], where h = \|\mathbf{h}\| for isotropic cases. These models are characterized by three key parameters: the nugget effect c_0, representing microscale variability or measurement error at h=0; the sill c + c_0, the plateau value approached as h increases, equaling the process variance under stationarity; and the range a, the distance beyond which observations are uncorrelated. Selection among models depends on data characteristics, with the spherical often favored for its finite range in natural resource applications.^[18] Under second-order stationarity, where the mean is constant and the covariance depends only on lag, the semivariogram relates directly to the covariance function C(\mathbf{h}) = \mathrm{Cov}[Z(\mathbf{x}), Z(\mathbf{x} + \mathbf{h})] via \gamma(\mathbf{h}) = C(0) - C(\mathbf{h}), with C(0) as the sill. This equivalence allows covariance-based formulations in some analyses, though variograms are preferred for their robustness to non-ergodic processes. Variograms often exhibit anisotropy, where dependence varies by direction, modeled by scaling range and sill parameters along principal axes or using directional variograms. For instance, in geological settings, elongation along stratigraphic layers is common, requiring experimental variograms computed in multiple directions before fitting.^[18] Practical fitting of theoretical models to experimental variograms employs methods like weighted least squares, which minimizes a weighted sum of squared residuals between observed and modeled values, with weights inversely proportional to variance estimates for robustness. Alternatively, maximum likelihood estimation maximizes the likelihood of the data under a Gaussian process assumption, incorporating spatial correlations for parameter inference, particularly useful in large datasets.^[19] Model adequacy is assessed through diagnostics such as leave-one-out cross-validation, where each observation is predicted from the rest using the fitted variogram, evaluating residuals for bias (mean error near zero), precision (low mean squared error), and sharpness (low mean squared deviation ratio). These metrics ensure the model captures spatial structure without overfitting.^[18]

Estimation Methods

Kriging Principles

Kriging serves as the best linear unbiased prediction (BLUP) method in geostatistics, providing optimal spatial interpolation by minimizing the prediction variance while ensuring unbiasedness under second-order stationarity assumptions.^[20] This approach treats the spatial variable as a random field and derives weights for observed data points to estimate values at unsampled locations, outperforming simpler interpolators like inverse distance weighting in accounting for spatial correlation.^[21] In simple kriging, the predictor at an unsampled location x_0 assumes a known constant mean \mu and is formulated as

Z^*(x_0) = \sum_{i=1}^n \lambda_i Z(x_i),

where Z(x_i) are the observed values at n sampled locations, and the weights \lambda = (\lambda_1, \dots, \lambda_n)^T are obtained by solving the linear system C \lambda = c_0, with C as the n \times n covariance matrix whose entries are c(x_i - x_j) and c_0 as the vector of covariance values c(x_0 - x_i). The covariance function is related to the variogram by c(h) = c(0) - \gamma(h).^[18] These weights ensure the predictor is unbiased and has minimum variance, leveraging the variogram to quantify spatial dependence.^[22] Ordinary kriging extends this framework when the mean \mu is unknown but constant, incorporating a Lagrange multiplier \nu to enforce the unbiasedness constraint \sum_{i=1}^n \lambda_i = 1. The system becomes an augmented (n+1) \times (n+1) matrix equation:

\begin{pmatrix} \Gamma & \mathbf{1} \\ \mathbf{1}^T & 0 \end{pmatrix} \begin{pmatrix} \lambda \\ \nu \end{pmatrix} = \begin{pmatrix} \gamma \\ 1 \end{pmatrix},

where \mathbf{1} is a vector of ones, \Gamma is the variogram matrix with entries \gamma(x_i - x_j), and \gamma is the vector \gamma(x_0 - x_i), solving simultaneously for the weights \lambda and \nu.^[18] This adjustment maintains the BLUP property without requiring prior knowledge of the mean, making it widely applicable in practice.^[21] The kriging variance, which measures prediction uncertainty at x_0, is given by \sigma_K^2(x_0) = \lambda^T \gamma(x_0). The nugget effect in the variogram model accounts for variability at small distances or measurement error.^[18] This variance depends solely on the spatial configuration and variogram model, not on the observed data values, providing a reliable uncertainty estimate independent of specific realizations.^[23] Kriging's linearity requires only second-order moments (mean and covariance or variogram), without assuming normality of the underlying distribution; however, for Gaussian random fields, it yields the full conditional distribution, enabling probabilistic inferences beyond point estimates.^[24]

Kriging Variants and Extensions

Universal kriging extends ordinary kriging to scenarios where the spatial mean is not constant but varies with location according to a known trend model, typically expressed as \mu(\mathbf{x}) = \mathbf{X}(\mathbf{x}) \boldsymbol{\beta}, where \mathbf{X}(\mathbf{x}) is a vector of known functions (e.g., polynomials for trend) and \boldsymbol{\beta} is a vector of unknown coefficients. The estimation involves first using generalized least squares to estimate \boldsymbol{\beta} from the data, accounting for spatial correlations via the covariance matrix, and then applying kriging residuals around this trend to predict values at unsampled locations.^[25] This approach, introduced by Georges Matheron in 1969, allows for handling non-stationary processes in applications like mining grade estimation where underlying drifts are present.^[25] Co-kriging represents a multivariate extension of kriging that incorporates auxiliary variables correlated with the primary variable of interest, enhancing prediction accuracy when direct data are sparse.^[26] It relies on cross-variograms, such as \gamma_{12}(\mathbf{h}), which measure the spatial covariance between the primary variable Z_1(\mathbf{x}) and secondary variable Z_2(\mathbf{x}), integrated into the kriging weight system to solve for optimal linear combinations.^[27] Developed as part of Matheron's theory of regionalized variables in 1971, co-kriging is particularly useful in environmental monitoring, where secondary data like elevation or remote sensing imagery inform predictions of pollutants or soil properties.^[27] Indicator kriging addresses categorical or non-Gaussian data by transforming the variable into a set of binary indicator variables I(\mathbf{x}; z_k) = 1 if Z(\mathbf{x}) \leq z_k and 0 otherwise, for multiple thresholds z_k, then applying ordinary kriging to each indicator variogram separately.^[28] Predictions are obtained by back-transforming the kriged indicator probabilities to recover the conditional cumulative distribution function (ccdf) at unsampled points, enabling estimation of local means, medians, or quantiles without assuming normality.^[28] This nonparametric method, formalized by André G. Journel in 1983, is widely applied in hydrogeology for delineating contaminant plumes or ore boundaries from threshold-based classifications.^[28] Block kriging adapts point kriging for estimating averages over finite volumes or blocks, such as mining panels, by modifying the right-hand side of the kriging system to account for the block support through an averaging kernel integrated against the variogram. This adjustment captures smoothing effects due to larger support sizes, reducing variance compared to point estimates and providing more stable predictions for volume-based decisions. Originating in early geostatistical practices for resource evaluation, block kriging ensures unbiased block averages while respecting spatial continuity models. Disjunctive kriging provides a nonlinear framework for estimating transformed variables, particularly useful when the goal is to predict functions like exceedance probabilities or smooth indicators, by expanding the process in orthogonal Hermite polynomials to achieve point support under multigaussian assumptions.^[29] The method decomposes the nonlinear estimator into a sum of point krigings of these polynomial components, preserving the minimum variance property while allowing for complex, non-additive spatial relationships.^[29] Introduced by Matheron in 1976, disjunctive kriging facilitates advanced applications in grade-tonnage curves for mineral resources, where nonlinear recovery functions are critical.^[29]

Simulation Methods

Monte Carlo Simulation in Geostatistics

Monte Carlo simulation in geostatistics involves generating multiple realizations of spatial random fields to quantify uncertainty in spatial predictions and propagate it through models for risk assessment. This approach aims to reproduce key spatial statistics, such as the variogram, while honoring observed data constraints at sampled locations, enabling the evaluation of probabilistic outcomes in applications like resource estimation. Unlike deterministic methods, it provides a stochastic framework for modeling spatial variability, where realizations are drawn to reflect the underlying spatial continuity and heterogeneity of the phenomenon. The Monte Carlo framework in geostatistics relies on repeated sampling from a prior probability distribution that is conditioned on available observations to generate equiprobable realizations of the spatial field. This process begins with modeling the spatial dependence structure, typically via a fitted variogram or covariance function, and then simulates values across a grid or domain to capture the joint distribution of the field. By averaging over numerous simulations, estimates of uncertainty metrics, such as confidence intervals for spatial averages or extremes, can be derived, facilitating informed decision-making under uncertainty. Unconditional simulation generates realizations directly from the fitted variogram model without incorporating specific data values, thereby reproducing the overall spatial statistics and variability of the field on average across multiple runs. This method is foundational for exploring the inherent spatial patterns and testing model assumptions independently of local observations. It serves as a building block for more complex simulations by providing a baseline representation of the random field's unconditional distribution. In contrast, conditional simulation adjusts unconditional realizations to exactly match observed data at sampled locations, often by adding kriging-estimated residuals to ensure the simulations honor the data constraints while preserving spatial continuity. Kriging is used here as a conditioning tool to compute these residuals, aligning the simulated field with the posterior distribution given the observations. This step ensures that each realization is both spatially consistent and data-compatible, allowing for realistic propagation of uncertainty. A primary advantage of Monte Carlo simulation over traditional estimation techniques like kriging is its ability to yield full probability distributions for derived quantities, such as ore volumes or extreme values, rather than point estimates or means alone. This enables comprehensive risk assessments by quantifying the likelihood of various scenarios, including tails of distributions that indicate rare but critical events. Such probabilistic outputs support better management of spatial variability in fields with limited data.

Sequential and Object-Based Simulations

Sequential Gaussian simulation (SGS) is a pixel-based algorithm that generates multiple realizations of a Gaussian random field conditioned to data, by sequentially simulating values at grid nodes while honoring the mean, variance, and spatial continuity defined by a variogram model. Introduced by Gómez-Hernández and Journel in 1993, SGS addresses the smoothing effect of kriging by producing equiprobable realizations that capture local uncertainty and variability in geological properties such as porosity or permeability. The method assumes an underlying Gaussian distribution after transformation, making it suitable for continuous variables in stationary fields. The SGS process begins with a normal score transform, where original data are ranked and mapped to a standard Gaussian distribution to ensure normality, followed by back-transformation of simulated values at the end. A random path is then defined through the simulation grid, visiting each unsampled node in sequence, often using a space-filling random sequence to avoid bias. At each node, simple kriging or ordinary kriging is performed using previously simulated values and conditioning data as inputs, estimating the conditional mean and variance; a random draw from this Gaussian conditional distribution is added to the mean to simulate the value, after which the node is marked as "visited" and included in the kriging system for subsequent nodes. This local conditioning ensures ergodicity and reproduction of the input variogram, with computational efficiency achieved through updating the kriging matrix incrementally. Multiple-point statistics (MPS) extends sequential simulation paradigms like SGS to capture complex, non-stationary spatial patterns beyond two-point variograms, using training images that exemplify geological continuity.^[30] Pioneered by Guardiano and Srivastava in 1993, MPS scans the training image for multi-point patterns matching local data configurations, then replicates analogous patterns in the simulation grid to condition realizations.^[30] Algorithms such as SNESIM sequentially visit grid nodes, search the training image for replicates of the data event (a local template of informed neighbors), and select patterns based on their frequency to draw categorical or continuous values, enabling simulation of curvilinear structures like channels or faults that variogram-based methods struggle to reproduce. Object-based simulation models geological heterogeneity by explicitly placing discrete objects, such as sediment bodies or facies units, rather than pixel-by-pixel assignment, using Boolean or marked point processes to generate realistic geometries. Originating from Boolean models in geostatistics as described by Chautru in 1989, the approach involves germ points distributed via a Poisson process, around which objects (e.g., ellipsoids representing channels) are placed with random orientations, sizes, and positions, followed by property filling using secondary simulations like SGS within objects. Marked point processes extend this by associating attributes (e.g., permeability) to points, allowing hierarchical modeling of nested structures; rules for object interaction prevent overlaps, ensuring geological plausibility.^[31] This method excels in reproducing connectivity and volumes of discrete features in fluvial or turbidite reservoirs. Validation of these simulations involves verifying that ensembles reproduce key statistics: the histogram for marginal distributions, the variogram for two-point spatial structure, and connectivity measures for higher-order patterns like channel continuity. Realizations are assessed by averaging multiple simulations to match data histograms and variograms within acceptable error, while connectivity is checked via percolation analysis or object counts to confirm realistic flow paths.

Applications and Advances

Traditional Uses in Earth Sciences

Geostatistics has been traditionally applied in earth sciences to address spatial variability in resource distribution, enabling more accurate estimation and simulation for decision-making in extraction industries. In mining and petroleum engineering, kriging methods provide unbiased predictions of ore grades or reservoir properties, while variograms quantify spatial dependence to support block modeling and uncertainty assessment. These techniques, rooted in the work of pioneers like D.G. Krige and G. Matheron, have become standard for optimizing resource recovery and minimizing operational risks in pre-2000 earth science practices.^[32] In ore reserve estimation, kriging is widely used to predict metal grades and tonnages within block models, which represent discretized volumes of the deposit for mine planning. Ordinary kriging, in particular, weights nearby samples based on spatial covariance to produce local estimates with minimal variance, facilitating the calculation of recoverable resources. Mining software such as Surpac integrates these geostatistical tools to generate three-dimensional block models, allowing engineers to delineate economic ore zones and assess dilution risks. For instance, in nickel deposits, ordinary kriging has been applied to interpolate grades from drillhole data, yielding reserve estimates that align closely with validation samples and support cut-off grade optimization.^[33]^[34] Petroleum reservoir modeling employs co-kriging to integrate multiple variables, such as porosity and permeability, during history matching processes that calibrate models to observed production data. Co-kriging extends ordinary kriging by accounting for cross-correlations between secondary variables like seismic attributes and primary petrophysical logs, improving the spatial distribution of flow properties across the reservoir. This approach helps quantify uncertainty in production forecasts by generating multiple realizations of permeability fields, which are essential for probabilistic assessments of recovery factors. In one application, co-kriging of acoustic impedance with neutron porosity logs enhanced areal predictions of reservoir heterogeneity, leading to more reliable simulations of fluid flow and ultimate oil recovery.^[35]^[36] In hydrogeology, universal kriging is utilized for interpolating groundwater levels, incorporating trends such as topography as deterministic components to model non-stationary spatial patterns. This variant detrends the data using external covariates like elevation, then applies kriging to the residuals, resulting in smoother and more accurate contour maps of aquifer heads. By accounting for topographic gradients, universal kriging reduces interpolation biases in hilly terrains, aiding in the delineation of recharge zones and sustainable pumping strategies. Studies on regional groundwater datasets have shown that universal kriging outperforms simpler methods in capturing elevation-driven variability, with cross-validation errors minimized for piezometric surface estimation.^[37]^[38] Soil science leverages variograms to map nutrient variability, such as phosphorus and potassium levels, enabling site-specific management in precision agriculture. Experimental variograms model the semivariance of soil samples as a function of separation distance, revealing nugget, sill, and range parameters that guide sampling density and interpolation. These maps inform variable-rate fertilizer application, optimizing yields while reducing environmental impacts from over-application. In farm-scale studies, variogram-based geostatistics has delineated nutrient hotspots and depleted areas, supporting management zones that improve phosphorus distribution accuracy by integrating terrain and land-use factors.^[39]^[40] A seminal case study from the Witwatersrand gold fields in South Africa illustrates the impact of early kriging applications, where D.G. Krige's statistical methods reduced estimation errors compared to traditional polygonal approaches. By regressing sample values onto block supports and applying ordinary kriging, error variances dropped from approximately 0.31 (orthodox methods) to 0.095, representing a substantial improvement in grade prediction reliability for tabular ore bodies. This work, conducted in the 1950s, laid the foundation for geostatistics in mining and demonstrated how accounting for spatial correlation could enhance reserve valuation accuracy. Kriging for block support was briefly referenced in these analyses to adjust point estimates to larger volumes, preserving mass balance in tonnage calculations.^[32]^[41]

Modern Extensions and Interdisciplinary Applications

Modern extensions of geostatistics have integrated advanced computational techniques to address complex spatial processes, expanding its utility beyond traditional earth sciences into fields like environmental monitoring and public health. These advancements, particularly post-2000, leverage Bayesian frameworks, machine learning, and scalable algorithms to handle large-scale, non-stationary data while incorporating uncertainty quantification. Such developments enable real-time analysis and interdisciplinary applications, such as modeling climate impacts and epidemiological risks.^[42] Bayesian geostatistics incorporates prior knowledge through hierarchical modeling of spatial processes, often fitted using Markov chain Monte Carlo (MCMC) methods to estimate posterior distributions of parameters. This approach allows for flexible incorporation of covariates and non-stationarity, improving predictions in scenarios with sparse data. For instance, meshed Gaussian processes within Bayesian hierarchies enable scalable inference for massive datasets by partitioning domains and using low-rank approximations.^[43]^[44]^[45] Machine learning hybrids have reframed classical geostatistical methods, with Gaussian process regression serving as a kernelized form of kriging that facilitates non-linear predictions via flexible covariance kernels. This equivalence allows geostatisticians to borrow from machine learning toolkits for optimization and hyperparameter tuning. Additionally, deep learning enhances multipoint statistics by training neural networks on training images to simulate complex spatial patterns, outperforming traditional sequential simulations in capturing geological heterogeneity. Recent applications include hybrid geostatistical CNN-RNN models for geochemical prediction in mine tailings, achieving over 97% accuracy in concentration mapping and reducing errors by 88-91% compared to ordinary kriging.^[46]^[47]^[48] Handling big data in geostatistics employs scalable kriging variants, such as fixed-rank kriging, which uses a basis function expansion to approximate covariance matrices with sparse representations, reducing computational complexity from O(n³) to O(n). This is particularly useful for interpolating satellite imagery in remote sensing, where high-resolution raster data from sources like Landsat require efficient spatial prediction over vast areas.^[49]^[42] In environmental applications, geostatistics supports climate variable downscaling by integrating coarse global model outputs with local observations through methods like direct sampling kriging, ensuring physical consistency in variables such as temperature and soil moisture. Pollution mapping utilizes indicator kriging to estimate exceedance probabilities of contaminant thresholds, providing probabilistic risk assessments for groundwater and air quality without assuming normality.^[50]^[51]^[52] To address gaps in handling non-Gaussian data, trans-Gaussian kriging applies transformations like Box-Cox to induce normality before kriging, followed by back-transformation to recover original margins, thus accommodating skewed distributions common in environmental datasets. Real-time geostatistics in IoT sensor networks processes streaming spatiotemporal data via distributed computing frameworks, enabling on-the-fly variogram estimation and kriging for dynamic monitoring of phenomena like urban air quality. Simulations in these contexts quantify uncertainty in environmental risk assessments, such as flood probabilities.^[53]^[54]^[55]

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[PDF] Another Look at the Kriging Equations | CCG
Kriging is an essential element of modern geostatistics. This paper presents a thorough review of the random function, stationarity, and derivation of the ...
[21]
Principles of geostatistics | Semantic Scholar
Principles of geostatistics · G. Matheron · Published 1 December 1963 · Geology · Economic Geology.Missing: original | Show results with:original
[22]
[PDF] Applied geostatistics Lecture 5 – Spatial prediction from point ...
Feb 6, 2016 · This formula has a simple interpretation: the kriging variance is the weighted sum of the semivariances between each sample point and the ...
[23]
ORDINARY KRIGING
The Lagrange method of multipliers allows the problem to be solved by reducing it to an unconstrained problem by means of a new objective function called the ...
[24]
Matheron, G. (1969) Le krigeage universel (Universal kriging) Vol. 1 ...
This study used modelling techniques to determine how changes in climate could affect vegetation productivity in the northern part of Nigeria.Missing: original | Show results with:original
[25]
https://www.scirp.org/reference/referencespapers?referenceid=2057609
[26]
[PDF] MUCK - A 8ovel Approach to Co-Kriging - Dr Isobel Clark
The term “co-kriging” appears to have been coined by Matheron himself in his original work,. The Theory of Regionalized Variables and Its Applications (1971) ...
[27]
Nonparametric estimation of spatial distributions
May 25, 1982 · Nonparametric estimation of spatial distributions. Published: June 1983. Volume 15, pages 445–468, (1983); Cite this ...
[28]
Disjunctive Kriging - jstor
Disjunctive kriging ... probabilities. The method. Matheron (1976) described the underlying mathematics of disjunctive kriging in his original paper.
[29]
https://www.jstor.org/stable/20002916
[30]
Combining geologic‐process models and geostatistics for ...
May 29, 2010 · We present an aquifer modeling methodology that combines geologic-process models with object-based, multiple-point, and variogram-based ...
[31]
[PDF] PROF. D. G. KRIGE, S - SAIMM
African gold mines that the reduction in the error variance of the ordinary kriged estimates was significantly better than that which would have been ...<|control11|><|separator|>
[32]
[PDF] Geostatistical Modeling using Ordinary Kriging for Estimating Nickel ...
Jan 12, 2025 · Ordinary kriging is now a well-accepted method in mining grade control and mine reserve estimation [21]. The ordinary kriging method is a ...
[33]
(PDF) Conventional and Computer Aided Ore Reserve Estimation
May 27, 2022 · Surpac software as one of the best and handy of these packages has remarkably helped the geologists and mining engineers in ore body modelling ...
[34]
Dynamic Reservoir Model Supports Reservoir Management
Seismic-interpreted acoustic impedance data were co-kriged with neutron porosity and magnetic resonance imaging well logs to provide an improved areal ...
[35]
History matching of petroleum reservoirs employing adaptive ...
Dec 15, 2015 · History matching, a process in which certain input parameters to the reservoir simulator such as porosity, permeability, thickness, saturations, ...Introduction · Genetic Algorithm · Workflow Of Genetic...
[36]
On the optimal selection of interpolation methods for groundwater ...
In addition, some studies used multivariate kriging methods like universal Kriging (UK) and co-Kriging (CoK), to incorporate the influence of the topography on ...
[37]
Spatial analyses of groundwater levels using universal kriging
Apr 18, 2007 · This study aims to determine which of these empirical semivariogram models will be best matched with the experimental models obtained from groundwater-table ...
[38]
[PDF] Soil Phosphorus and Potassium Mapping - Using a Spatial ...
Precision agriculture often incorporates precise, spatial information about soil properties and/or nutrients (e.g., phosphorus and/or potassium) across farm ...
[39]
Geospatial variability mapping of soil nutrients for site specific input ...
Jan 28, 2022 · The ability to define site-specific variability affects the acceptability of precision farming. A study was conducted to determine the soil ...
[40]
A statistical approach to some basic mine valuation problems on the ...
Ap,PROACH TO SOME BASIC MINE VALUATION . PROBLEMS ON THE WITWATERSRAND .•. By D. ·G. KRIGE. (Published in the.Journal, December 1951).
[41]
A review of geostatistical simulation models applied to satellite ...
Jun 15, 2021 · We review recent geostatistical simulation models relevant to satellite remote sensing data and discuss the characteristics and advantages of each approach.Missing: origins | Show results with:origins
[42]
Bayesian Modeling and Analysis of Geostatistical Data - PMC - NIH
We attempt a review of a fully model-based perspective for such data analysis, the approach of hierarchical modeling fitted within a Bayesian framework.Missing: seminal papers
[43]
Hierarchical modeling for spatial data problems - ScienceDirect.com
This short paper is centered on hierarchical modeling for problems in spatial and spatio-temporal statistics. It draws its motivation from the ...
[44]
Highly Scalable Bayesian Geostatistical Modeling via Meshed ...
We introduce a class of scalable Bayesian hierarchical models for the analysis of massive geostatistical datasets. The underlying idea combines ideas on ...
[45]
Traditional kriging versus modern Gaussian processes for large ...
Jul 21, 2023 · The canonical technique for nonlinear modeling of spatial/point-referenced data is known as kriging in geostatistics, and as Gaussian ...
[46]
Integrating Multimodal Deep Learning with Multipoint Statistics for ...
In this study, we integrate deep learning, geostatistical methods, and multi-source heterogeneous data fusion techniques to advance the theory and methodology ...
[47]
Fixed Rank Kriging for Very Large Spatial Data Sets
In the spatial case, Johannesson and Cressie (2004a) achieved speed-ups of the order of 10 8 over directly solving the kriging equations. They could compute ...
[48]
Demonstration of a geostatistical approach to physically consistent ...
Jan 24, 2013 · The approach is used here to downscale climate variables including skin surface temperature (TSK), soil moisture (SMOIS), and latent heat flux ( ...
[49]
Application of geostatistics with Indicator Kriging for analyzing ...
A geostatistical analysis with Indicator Kriging (IK) was employed to investigate the regionalized variation of As concentration. The IK prediction map shows a ...
[50]
Combining geostatistics and simulations of flow and transport to ...
Jan 10, 2020 · Kriging is used to map contamination in soils and groundwater as it provides linear and unbiased estimates of pollutant concentration at ...
[51]
Improving kriging of groundwater level data using nonlinear ...
We investigate three established normalizing methods, Gaussian anamorphosis, trans-Gaussian kriging and the Box-Cox method to improve the estimation accuracy.
[52]
Copula-based multiple indicator kriging for non-Gaussian random ...
In this paper, we introduce a copula-based multiple indicator kriging model for the analysis of non-Gaussian spatial data by thresholding the spatial ...
[53]
Geostatistics on Real-Time Geodata Streams—An Extended ... - MDPI
Feb 22, 2023 · In this paper a methodology is developed to apply analyses of spatiotemporal autocorrelation directly to geodata streams through a distributed streaming ...Missing: 2010 | Show results with:2010