Co-kriging: Multivariate Geostatistical Interpolation
Co-kriging Spatial Interpolation · Also known as: cokriging, co-regionalization kriging, multivariate kriging, CK
Co-kriging is a geostatistical interpolation technique that predicts the spatial distribution of a primary variable by leveraging its spatial cross-correlation with one or more secondary (co-) variables. It extends ordinary kriging to multivariate settings, yielding more accurate predictions when the secondary variable is more densely sampled or spatially correlated with the primary variable of interest.
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When to use it
Use co-kriging when your primary variable of interest is sparsely sampled but one or more secondary variables (cross-correlated with the primary) are more densely measured or easier to obtain — common scenarios include environmental monitoring, soil science, and mineral resource estimation. It is most beneficial when the cross-correlation between variables is strong (|r| > 0.6) and the sampling designs are genuinely complementary. Avoid co-kriging when secondary variables have no meaningful spatial cross-correlation with the primary variable, when all variables are equally dense (ordinary kriging suffices), when you cannot fit a valid linear model of co-regionalization, or when sample sizes are too small to estimate cross-variograms reliably (typically fewer than 50 paired observations per variable).
Strengths & limitations
- Incorporates auxiliary information from secondary variables, reducing prediction error compared to ordinary kriging when cross-correlations are strong.
- Provides full probabilistic prediction maps with spatially varying uncertainty estimates (co-kriging variance).
- Optimal in the mean-squared-error sense among linear unbiased predictors that use all available multivariate spatial data.
- Flexible framework accommodating multiple secondary variables and heterotopic sampling (variables measured at different locations).
- Well-established theoretical foundation in geostatistics with decades of validated applications in mining, soil science, and environmental assessment.
- Requires fitting a valid linear model of co-regionalization, which is technically demanding and computationally intensive, especially with many variables.
- Cross-variogram estimation requires sufficient co-located or near-located data; sparse co-location weakens the cross-variogram estimate.
- Model fitting complexity grows rapidly with the number of co-variables, making large multivariate problems difficult to manage.
- Assumes stationarity of both direct and cross-variograms, which may not hold over large or heterogeneous study areas.
- Computationally heavier than ordinary kriging, particularly when solving large co-kriging systems.
Frequently asked
When does co-kriging outperform ordinary kriging?
Co-kriging outperforms ordinary kriging primarily when the primary variable is sparsely sampled, the secondary variable is densely sampled, and the cross-correlation between them is strong (cross-variogram signal is high). If the secondary variable adds little cross-correlation information, the added model complexity may actually degrade predictions.
What is a linear model of co-regionalization and why is it required?
A linear model of co-regionalization (LMC) expresses all direct and cross-variograms as weighted sums of the same set of basic variogram structures. This constraint guarantees that the full variogram matrix is positive-definite at every lag, which is a mathematical necessity for the co-kriging system to have a valid solution and non-negative prediction variances.
Can I use co-kriging with more than one secondary variable?
Yes, co-kriging generalizes to any number of secondary variables, but each additional variable requires estimating and modeling more cross-variograms, and the co-kriging system grows substantially. In practice, one or two secondary variables with strong cross-correlations typically capture most of the potential gain.
How does co-kriging differ from regression kriging?
Co-kriging uses a geostatistical framework that explicitly models spatial cross-variograms and produces spatially optimal linear predictions. Regression kriging first fits a regression of the primary on secondary variables and then kriging the residuals separately; it is simpler to implement but does not fully exploit the joint spatial structure of the variables.
What sample size is needed for reliable co-kriging?
There is no universal rule, but cross-variogram estimation typically requires at least 50 to 100 pairs of co-located or near-located observations per lag class. Too few paired observations produce noisy cross-variograms that are difficult to fit reliably, undermining the advantage over ordinary kriging.
Sources
- Journel, A. G., & Huijbregts, C. J. (1978). Mining Geostatistics. Academic Press, London. ISBN: 978-0123910561
- Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press, New York. ISBN: 978-0195115383
How to cite this page
ScholarGate. (2026, June 3). Co-kriging Spatial Interpolation. ScholarGate. https://scholargate.app/en/spatial-analysis/co-kriging
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Geographically Weighted RegressionSpatial analysis↔ compare
- Multiscale Geographically Weighted RegressionSpatial analysis↔ compare
- Ordinary KrigingSpatial analysis↔ compare
- Spatial AutocorrelationSpatial analysis↔ compare
- Universal KrigingSpatial analysis↔ compare