Robust Co-Kriging
Robust Co-Kriging Spatial Interpolation · Also known as: robust cokriging, outlier-resistant co-kriging, robust multivariate kriging, RCK
Robust Co-Kriging is a multivariate geostatistical interpolation method that jointly estimates values at unsampled locations using two or more spatially correlated variables, while applying robust estimators for the variogram and cross-variogram to limit the distorting influence of spatial outliers or non-Gaussian measurement errors.
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When to use it
Use Robust Co-Kriging when you need to interpolate a primary spatial variable and a correlated secondary variable is available, but the dataset contains suspected outliers, measurement errors, or heavy-tailed distributions that would compromise classical Co-Kriging. It is particularly valuable in environmental monitoring (soil contamination, air quality), mining grade estimation, and climate modeling where rogue sensor readings or anomalous observations are common. Do not use it when both variables are clean and approximately Gaussian — ordinary Co-Kriging is simpler and equally efficient in that case. Also avoid it when the primary and secondary variables have negligible cross-correlation (below ~0.3), as the additional complexity yields little gain over ordinary robust kriging.
Strengths & limitations
- Outlier-resistant: robust variogram estimators prevent a small fraction of extreme values from distorting the entire fitted covariance structure.
- Multivariate efficiency: leverages cross-correlation between primary and secondary variables to reduce prediction uncertainty compared to univariate robust kriging.
- Provides full uncertainty quantification via the kriging variance at each prediction location.
- Valid positive semi-definite covariance model is guaranteed through the linear model of coregionalization.
- Applicable to irregularly spaced sampling designs common in field surveys.
- Requires collocated or near-collocated measurements of the secondary variable, which may not always be available.
- Fitting the linear model of coregionalization with robust estimates adds computational and modeling complexity compared to univariate approaches.
- Robust estimators reduce but do not eliminate the influence of outliers; very high contamination rates (above ~20%) can still degrade performance.
- Assumes second-order stationarity (or intrinsic stationarity) of the spatial process, which may fail in non-stationary domains.
- Theoretical properties (breakdown point, efficiency) of robust co-kriging predictors are less well characterized than those of the classical estimator.
Frequently asked
How does Robust Co-Kriging differ from standard Co-Kriging?
The prediction equations and multivariate structure are the same. The key difference is how the variogram and cross-variogram are estimated: standard Co-Kriging uses the classical method-of-moments estimator, which is highly sensitive to outliers, while Robust Co-Kriging substitutes a breakdown-resistant estimator such as the Cressie-Hawkins fourth-root formula or a MAD-based version, preventing outliers from warping the fitted covariance model.
When is ordinary robust kriging sufficient, and when do I need the co-kriging extension?
Ordinary robust kriging is sufficient when you have only one spatially varying variable of interest. Add the co-kriging extension when a secondary variable is available, is spatially correlated with the primary variable (cross-correlation above roughly 0.3), and is sampled more densely or at different locations — the extra information can substantially reduce prediction variance.
What is the linear model of coregionalization and why is it required?
It is a constraint that ensures the joint variogram-cross-variogram matrix is positive semi-definite at every lag, which is a mathematical requirement for valid kriging predictions. Without it, the kriging system can produce negative variances or fail to solve. In practice it means fitting each variogram and cross-variogram as a linear combination of the same set of basic structures (e.g., nugget plus spherical model) with a shared range.
Can Robust Co-Kriging handle non-stationary spatial processes?
In its standard form it assumes second-order stationarity. For non-stationary fields (e.g., spatially varying mean), combine robust Co-Kriging with a trend model or use local (moving-window) robust Co-Kriging, fitting variograms within spatial subsets and applying the predictions locally.
Which software packages support Robust Co-Kriging?
The R package gstat supports co-kriging with flexible variogram models; robust variogram estimation is available in the georob package, which implements robust kriging and can be extended to the multivariate case. The geoR package also provides robust variogram estimation tools. Commercial software such as GSLIB and Isatis supports robust variogram options in a co-kriging workflow.
Sources
- Cressie, N. A. C. (1993). Statistics for Spatial Data (Revised ed.). John Wiley & Sons. Chapter 3 covers robust variogram estimation and co-kriging. ISBN: 978-0471002550
- Genton, M. G., & Rousseeuw, P. J. (1995). The Median Absolute Deviation of Spatial Data. Computational Statistics and Data Analysis, 20(4), 385-400. link ↗
How to cite this page
ScholarGate. (2026, June 3). Robust Co-Kriging Spatial Interpolation. ScholarGate. https://scholargate.app/en/spatial-analysis/robust-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.
- Co-krigingSpatial analysis↔ compare
- Ordinary KrigingSpatial analysis↔ compare
- Robust KrigingSpatial analysis↔ compare