Universal Kriging (Kriging with a Trend)
Also known as: kriging with a trend, kriging with drift, trend kriging, evrensel kriging
Universal kriging generalizes ordinary kriging to data whose mean varies systematically across space — a spatial trend or 'drift'. It models the mean as a function of the coordinates (or covariates) and krigs the residuals, so it can interpolate variables that drift in a preferred direction, such as temperature falling with latitude or a pollutant gradient, while still returning prediction variances.
Key highlights
- Handles non-stationary means (spatial trend/drift) that ordinary kriging cannot.
- Retains kriging's uncertainty quantification via prediction variances.
- Flexible trend specification using coordinates or covariates.
- Reduces to ordinary kriging when the trend is constant.
Intuition
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How it works
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When to use it
Use universal kriging when the variable shows a clear large-scale trend across the study area that violates the constant-mean assumption of ordinary kriging — directional gradients, elevation-driven patterns, or any drift you can express as a function of coordinates or covariates. It gives better predictions than ordinary kriging in such cases and still quantifies uncertainty. Caveats: choosing the trend form is a modelling decision that can be over- or under-specified, the trend/residual-variogram estimation is intertwined and best handled by REML, and extrapolating the trend beyond the data is risky. When the mean is effectively constant, ordinary kriging is simpler; when a fully known covariate drives the mean, kriging with external drift (a special case) is the natural choice.
Strengths & limitations
- Handles non-stationary means (spatial trend/drift) that ordinary kriging cannot.
- Retains kriging's uncertainty quantification via prediction variances.
- Flexible trend specification using coordinates or covariates.
- Reduces to ordinary kriging when the trend is constant.
- Requires choosing a trend model, risking mis-specification.
- Trend and residual-variogram estimation are intertwined and need care (REML).
- Extrapolating the fitted trend beyond the data range is unreliable.
- More complex and assumption-laden than deterministic interpolators.
Common pitfalls
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Applications
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Frequently asked
How does universal kriging differ from ordinary kriging?
Ordinary kriging assumes a constant (but unknown) mean across the area; universal kriging models the mean as a spatial trend (a function of coordinates or covariates) and krigs the residuals. Use universal kriging when the data show a systematic large-scale gradient that the constant-mean assumption would distort.
What is kriging with external drift?
It is a special case of universal kriging where the trend is driven by a known, exhaustively sampled covariate (such as elevation from a DEM) rather than by polynomials of the coordinates. It is a convenient way to fold a strong auxiliary variable into the mean structure.
Why must the variogram be modelled on residuals?
Because the trend inflates and distorts the raw variogram, mixing large-scale drift with local correlation. The spatial correlation of interest is in the residuals after removing the trend, so the residual variogram — estimated jointly with the trend, often by REML — is what the kriging system needs.
Sources
- 1.Matheron, G. (1963). Principles of geostatistics. Economic Geology, 58(8), 1246–1266.
- 2.Cressie, N. A. C. (1993). Statistics for Spatial Data (Revised ed.). John Wiley & Sons.ISBN 978-0-471-00255-0
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ScholarGate. (2026, June 2). Universal Kriging. ScholarGate. https://scholargate.app/spatial-analysis/universal-kriging