Geostatistics glossary

Clear, one-line definitions of the Geostatistics terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

anti-clustered

A spatial sampling pattern in which sample locations are spaced more evenly than a random Poisson pattern, reducing redundancy between nearby points.

bimodal facies

A rock-type distribution in which two distinct lithologies (e.g. sand and shale) dominate, producing a two-peaked histogram of properties.

block kriging

Kriging that estimates the average value of Z over a block (volume) rather than at a point, by averaging point-to-block covariances.

See: Block kriging and change-of-support

cdf

Cumulative distribution function F(z) = P[Z ≤ z]; estimated via indicator kriging in non-parametric geostatistics.

cell declustering

Declustering that overlays a regular cell grid and weights samples by 1/(number of samples in the cell), then by cell coverage.

See: Cell declustering

change of support

Conversion of statistics from one sample volume to another (e.g. core plug to block) using variance corrections such as the affine or indirect lognormal methods.

co-kriging

Kriging that jointly uses one or more secondary variables, with cross-variograms describing their spatial cross-dependence with the primary.

conditional negative definiteness

The mathematical property a function must satisfy to be a valid variogram; it ensures kriging systems yield non-negative variances.

conditional simulation

Stochastic simulation that honours data values at sample locations exactly while reproducing the chosen histogram and variogram model.

correlogram

The covariance function normalised by C(0), ρ(h) = C(h)/C(0); takes values between -1 and 1.

See: Covariance, correlogram, and variogram, three views of the same thing

covariance function

C(h) = Cov[Z(x), Z(x+h)]; for second-order stationary Z, related to the semivariogram by γ(h) = C(0) − C(h).

cressie-hawkins estimator

A robust experimental-variogram estimator based on fourth-root differences; downweights outliers compared with the classical method-of-moments form.

cross-validation

Leave-one-out re-estimation: each datum is removed in turn, krieged from the rest, and compared to the true value to assess the variogram model.

See: Cross-validation done right, Leave-one-out cross-validation for kriging

cross-variogram

γ_AB(h) = ½·E[(Z_A(x+h) − Z_A(x))(Z_B(x+h) − Z_B(x))]; the bivariate analogue of γ used in co-kriging.

declustering

A weighting procedure that compensates for preferential sampling so that summary statistics represent the domain rather than the clustered samples.

See: Declustering, Cell declustering

drift

The deterministic large-scale trend component m(x) = E[Z(x)] in a non-stationary random function model.

ergodicity

The property that spatial averages over a single realisation converge to ensemble averages as the domain grows; underpins inference from one data set.

See: Stationarity, ergodicity, and the practical compromise

experimental variogram

The empirical estimate γ̂(h) = (1/2N(h)) Σ [Z(xᵢ) − Z(xᵢ+h)]² computed from data before fitting a parametric model.

exponential model

A variogram model γ(h) = c[1 − exp(−3h/a)] that approaches the sill asymptotically; practical range a corresponds to 95% of c.

facies

A rock category defined by lithology, depositional environment, or other geological criterion; commonly modelled with indicator or multipoint methods.

See: Facies Come First, Populate the Facies

gaussian model

A variogram model γ(h) = c[1 − exp(−3h²/a²)] with a parabolic origin behaviour suited to very smooth phenomena.

geometric anisotropy

Anisotropy in which the range depends on direction but the sill is the same in all directions; removable by an affine coordinate rescaling.

h-scatterplot

A scatter plot of Z(x) versus Z(x+h) for a given lag h; its dispersion away from the 45° line illustrates the semivariogram value γ(h).

See: h-scatterplots and lag binning

histogram

An empirical estimate of a probability density from binned counts; used in geostatistics to summarise the marginal distribution of a regionalised variable.

See: Why your sample histogram is biased, Histograms, CDFs, and quantile-quantile plots

indicator kriging

Kriging applied to indicator-transformed data 1{Z(x) ≤ z*} to estimate the local conditional cumulative distribution non-parametrically.

See: Indicator kriging for facies probabilities

indicator simulation

Stochastic simulation of categorical or thresholded continuous variables built from indicator transforms; preserves marginal proportions and indicator variograms.

See: Sequential Indicator Simulation, Sequential Indicator Simulation (SISIM)

indicator transform

The 0/1 mapping I(x; z*) = 1{Z(x) ≤ z*} (continuous) or I(x; k) = 1{facies(x) = k} (categorical) used for non-parametric kriging and simulation.

intrinsic hypothesis

A weaker assumption than second-order stationarity: increments Z(x+h) − Z(x) have zero mean and a variance depending only on h.

jackknife

Re-sampling validation that splits the data into a calibration subset and a test subset; the test subset is krieged from the calibration data only.

See: Jackknife and split-sample validation, Bootstrap, jackknife, and resampling first principles

kriging

A family of best linear unbiased spatial predictors that use a fitted variogram to weight surrounding samples and produce an estimate with a known variance.

See: Kriging Versus Simulation, Kriging: The Best Estimate

kriging variance

The minimised mean-squared estimation error σ²_K produced by the kriging system; a measure of estimate uncertainty that depends on geometry, not on Z values.

See: Calibration of the kriging variance, The kriging variance, what it means and what it doesn't

kriging weight

The coefficient λᵢ assigned to sample i in the kriging linear combination Z*(x₀) = Σ λᵢ Z(xᵢ).

lag

The vector h separating two locations xᵢ and xⱼ; variograms and covariances are indexed by lag magnitude and direction.

log-transform

Mapping Y = log(Z) used to normalise skewed positive variables (e.g. permeability) before variography or kriging; back-transform is non-linear.

lognormal

A distribution whose logarithm is normal; common for permeability, ore grades, and contaminant concentrations.

model-based geostatistics

A parametric approach in which spatial data are modelled as a random field with an explicit covariance model, enabling likelihood-based inference.

multipoint statistics

Simulation framework that infers patterns from a training image rather than from a two-point variogram, enabling curvilinear and complex geological shapes.

See: Capstone 3: Fluvial channel reservoir (multipoint statistics)

nested structure

A variogram model built as a sum of elementary models (e.g. nugget + spherical + exponential) to fit multi-scale spatial behaviour.

See: Nested structures and additive sills

normal-score transform

A quantile-matching transform that maps an empirical distribution to a standard normal one; a prerequisite for sequential Gaussian simulation.

See: Normal-score transform, The Normal-Score Transform

nugget

The semivariogram value at zero lag attributable to measurement error and sub-sample-scale variability; a vertical jump at h = 0.

See: Range, Sill, and Nugget, Nugget effect and short-scale variability

nugget effect

The phenomenon and the parameter c₀ representing variance unresolved by the sampling scale; appears as a discontinuity at the origin of γ(h).

See: Nugget effect and short-scale variability

ooip

Original oil in place: the total reservoir oil volume at discovery, OOIP = (A·h·φ·(1 − Sw))/Boi, often delivered as a P10/P50/P90 distribution.

ordinary kriging

Kriging that assumes an unknown but constant local mean; weights are constrained to sum to one to filter the mean out.

See: Ordinary kriging and the unbiasedness constraint

p10

The 10th-percentile (optimistic) estimate of a resource or reserve; 90% of equally likely outcomes lie below P10 in the usual oil-industry convention.

p50

The median (50th-percentile) estimate of a resource or reserve; equally likely to be exceeded or undershot.

p90

The 90th-percentile (conservative, proven) estimate of a resource or reserve; only 10% of outcomes fall below it.

point kriging

Kriging that estimates Z at a single point location; the limiting case of block kriging as block size → 0.

polygonal declustering

Declustering by assigning each sample a weight proportional to the area or volume of its Voronoi polygon (Thiessen polygon).

See: Polygonal declustering

positive definiteness

The property of a covariance function that guarantees every linear combination Σ λᵢ Z(xᵢ) has non-negative variance; required for valid covariance models.

power model

An unbounded variogram model γ(h) = c·hʷ with 0 < w < 2; models phenomena with no finite sill (non-stationary).

practical range

For models that approach the sill asymptotically (e.g. exponential, Gaussian), the lag at which γ(h) reaches 95% of the sill.

qq plot

A diagnostic comparing the quantiles of two distributions; a straight diagonal indicates a good match between observed and modelled distributions.

random function

A collection of random variables {Z(x): x ∈ D} indexed by spatial location; the formal model behind geostatistical inference.

range

The lag distance at which a semivariogram first reaches (or effectively reaches) the sill; beyond it, samples are essentially uncorrelated.

range anisotropy

A synonym for geometric anisotropy; the correlation length differs by direction while the sill is preserved.

regionalized variable

A spatially distributed attribute (porosity, ore grade, contaminant) modelled as one realisation of a random function.

residual

The local deviation Z(x) − m(x) that remains after subtracting the trend; usually assumed second-order stationary.

See: Residual velocity & higher-order moveout, Deviance, residuals, and GLM diagnostics

screening effect

Tendency for samples close to the target to make more-distant samples in the same direction nearly redundant, driving their kriging weights toward zero.

search ellipse

An anisotropic search neighbourhood whose axes follow the variogram ranges; samples outside the ellipse are excluded from the local kriging system.

See: Anisotropic ellipsoids and the search ellipse

search neighborhood

The local region (radius, ellipse, or octants) within which samples are retained for a kriging or simulation estimate at a given target location.

semivariance

γ(h) = ½·Var[Z(x+h) − Z(x)], the value of the semivariogram at lag h; measures how dissimilar samples become as separation grows.

semivariogram

Half the variogram, γ(h) = ½·Var[Z(x+h) − Z(x)]; the form most often plotted and fitted in geostatistics.

sequential gaussian simulation

An algorithm that generates equiprobable realisations of a multivariate Gaussian field by visiting nodes sequentially and sampling from a kriging-defined conditional distribution.

See: Sequential Gaussian Simulation (SGS)

sequential indicator simulation

A sequential simulation that uses indicator kriging at each node to draw from a non-parametric local conditional distribution; popular for categorical variables.

See: Sequential Indicator Simulation, Sequential Indicator Simulation (SISIM)

sill

The variance value at which a (bounded) semivariogram levels off; equals the total a-priori variance of the random function.

sill anisotropy

A synonym for zonal anisotropy; the total variance differs by direction, often handled by adding a direction-restricted nested structure.

simple kriging

Kriging that assumes a known constant mean m; weights need not sum to one and missing weight is absorbed by m.

See: Simple kriging from first principles

spatial bootstrap

Bootstrap resampling that respects spatial correlation by drawing realisations of the random function rather than re-sampling individual data with replacement.

spherical model

A bounded variogram model γ(h) = c[1.5(h/a) − 0.5(h/a)³] for h ≤ a and γ(h) = c for h > a; reaches the sill exactly at the range a.

stationarity

The assumption that statistical properties of Z(x) are invariant under translation; second-order stationarity requires constant mean and a covariance depending only on h.

See: Trends and Stationarity, Linearity, reciprocity, stationarity

support effect

Statistical properties (especially variance) depend on the volume over which Z is averaged; block values are less variable than point values.

training image

A non-conditional digital model that conveys the spatial patterns the user wishes to reproduce in a multipoint simulation; not directly fit to data values.

See: SNESIM and the role of training images

trend

A smoothly varying large-scale component of a spatial variable, often fitted as a low-order polynomial and removed before residual variography.

universal kriging

Kriging that models the mean as a low-order polynomial trend (drift) and krieges the residual; also called kriging with a trend.

See: Universal kriging and kriging with external drift

variogram

A function 2γ(h) describing how the variance of differences Z(x+h)−Z(x) grows with separation distance h; the workhorse of spatial dependence.

See: Indicator variograms, Robust variogram estimators

voronoi polygon

For a sample location xᵢ, the set of points in the domain closer to xᵢ than to any other sample; used for polygonal declustering and nearest-neighbour estimation.

zonal anisotropy

Anisotropy in which the sill itself differs by direction; not removable by simple coordinate rescaling and modelled with nested structures.