Simple KrigingEdit
Simple Kriging is a geostatistical method used to estimate the value of a spatially varying quantity at an unsampled location by leveraging nearby observations and a model of how the quantity varies in space. It sits within the broader framework of geostatistics and the kriging family of estimators, and it is particularly straightforward when the mean of the underlying process is known. See Kriging and Geostatistics for the broader context, and Variogram for the core tool used to describe spatial dependence.
The essential idea is to form a linear predictor of the form Z*(s0) = m + Σi=1..n λi [Z(si) − m], where Z(si) are the observed values at the n nearby sample locations si, m is the known mean of the process, λi are weights to be determined, and s0 is the location where the prediction is sought. The weights are chosen to minimize the estimation variance Var[Z(s0) − Z*(s0)] subject to the constraint implied by the known mean. Because the mean is assumed known, simple kriging avoids the more restrictive constraint used in some other kriging variants and relies on the spatial covariance structure to determine the weights. See Covariance and Variance (statistics) for related concepts.
In practice, the weights are obtained by solving a linear system built from the spatial covariance (or the semivariogram) model. If C(si, sj) denotes the covariance between observations at si and sj, and c(si) denotes the covariance between si and the prediction location s0, the weights λ satisfy the system C λ = c. The kriging predictor is then Z*(s0) = m + Σi λi [Z(si) − m]. The corresponding kriging variance, which reflects the uncertainty of the prediction, is Var[Z(s0) − Z*(s0)], computed from the same covariance model.
Theoretical foundations
Random fields and second-order structure: Simple Kriging rests on a model where the mean m is known and the second-order structure (covariances or, equivalently, the semivariogram) describes how values co-vary with distance or direction. See Stationarity (statistics) for the assumptions underlying this approach.
Kriging system and weights: The prediction at s0 is a linear combination of nearby observations with weights chosen to minimize estimation variance under the known-mean assumption. The key equations link the weights to the covariance between observations and to the covariance between each observation and the prediction location: ∑j C(si, sj) λj = C(si, s0) for i = 1,…,n.
Assumptions and limitations: Simple Kriging requires the mean m to be known exactly. If m is misspecified, the resulting estimator can be biased. In many applied settings, m is estimated from data or treated as a drift that varies over space, which leads practitioners to consider other kriging variants such as ordinary kriging (where the mean is unknown but constant) or universal kriging (where a trend or drift is modeled explicitly). See Mean (statistics) and Kriging with external drift for related ideas.
Practical considerations
Model choice: The variogram or covariogram model used to describe spatial dependence is central. Choices range from simple isotropic models to anisotropic forms that reflect directional structure (e.g., along or across a river or fault line). See Variogram and Anisotropy (geostatistics).
Neighborhood and data quality: The selection of which nearby observations to include affects both accuracy and computation. Too-small a neighborhood may ignore informative data; too-large a neighborhood may dilute local structure or introduce bias if the mean is not truly constant. Cross-validation and diagnostic checks are standard parts of the workflow. See Cross-validation (statistics) and Spatial interpolation for broader methods.
Comparison with other kriging approaches: If the mean is unknown or varies over space, ordinary or universal kriging may be preferable. External drift kriging uses covariate information to model the drift. Each variant has trade-offs in bias, variance, interpretability, and computational effort. See Ordinary Kriging, Universal Kriging, and Kriging with external drift for contrasts.
Extensions and applications
Extensions to non-stationarity: While classical simple kriging assumes a constant, known mean and stationary covariance, practitioners extend the approach with drift components or by switching to methods designed for non-stationary fields. See Non-stationarity and Kriging with external drift for generalized strategies.
Applications: Simple Kriging is used in a range of fields where a known or reliably estimated mean is justified and computational simplicity is valued. Common domains include groundwater resource assessment, mineral resource estimation, environmental monitoring, and soil property mapping. See Hydrogeology and Mining for typical contexts, and Spatial statistics for related techniques.
Relation to predictive uncertainty: The kriging variance provides a formal measure of prediction uncertainty that can be propagated into decision-making, resource planning, or risk assessment. This part of geostatistics emphasizes transparent uncertainty quantification, a hallmark of kriging methods. See Uncertainty for broader concepts.