Geostatistics—a branch of statistics tailored for spatial data analysis—has seen significant advancements in recent years. Kriging and Co-Kriging are popular geostatistical techniques for spatial interpolation and prediction. These methods are essential in various fields, including geology, environmental science, agriculture, and more.

Kriging and Co-Kriging help geostatisticians estimate values at unsampled locations based on data collected at neighboring locations. In this blog, we will delve into the differences between Kriging and Co-Kriging, exploring their principles, applications, and when to use each technique.

## Understanding Kriging

Kriging is a geostatistical technique that provides the Best Linear Unbiased Estimate (BLUE) of a variable at an unobserved location based on the spatial correlation of data. Here’s how Kriging works:

1. Spatial Correlation: Kriging is based on the 1st law of geography, which states that nearby data points are more correlated than distant ones. It quantifies this spatial correlation using a semivariogram or covariance function.
2. Weighted Averages: To estimate a value at an unobserved location, Kriging computes a weighted average of nearby sample points, where the weights are determined by the spatial correlation and data geometry (e.g. clustered data receive less weight than isolated observations because of their spatial redundancy).
3. Optimality: Kriging aims to minimize the estimation error variance while ensuring that the estimates are unbiased.

Kriging is versatile and effective for mapping both continuous and categorical variables.

## Exploring Co-Kriging

Co-Kriging is an extension of univariate Kriging that allows for incorporating secondary (auxiliary) variables, in addition to the primary variable of interest. These secondary variables can help improve predictions by providing additional information about the spatial variability.

Co-Kriging predicts the primary variable while considering the cross-covariances between the primary and secondary variables. It shares the same characteristics of minimizing the variance of prediction errors, unbiasedness, and calculating estimates as weighted averages of surrounding data. However, it’s more tedious to implement than kriging as it requires the joint modeling of multiple direct and cross-semivariograms.

## Differences and When to Use Each Method

The main difference between Kriging and Co-Kriging lies in incorporating additional variables. Kriging focuses on predicting a single variable, utilizing spatial autocorrelation, while Co-Kriging integrates multiple variables to enhance predictions.

The choice between Kriging and Co-Kriging depends on your specific data and objectives:

Kriging is suitable when dealing with a single variable or when cross-correlations between variables are negligible or these variables are measured at the same locations (equally sampled case). It’s computationally less intensive and more straightforward to implement than cokriging.

Co-Kriging is the go-to choice when you need to simultaneously predict multiple correlated variables. It can lead to more accurate predictions when:

1. Variables interact significantly
2. Secondary variables are sampled more densely or available at different locations than the primary variable.

Based on the type of unbiasedness constraints (Goovaerts, 1998) two forms of ordinary kriging can be distinguished: traditional and standardized co-kriging, which tends to assign more importance to secondary data. Both versions are available in the Vesta software.

There are alternatives to co-kriging for incorporating secondary information, like kriging with an external drift or residual kriging (Goovaerts, 2000). In particular, the latter allows the straightforward incorporation of a large number of secondary variables that can be categorical or continuous. The trade-off is the requirement that these secondary variables are available at every location being estimated; for example the prediction of rainfall using a digital elevation model and other geomorphometric derivatives (e.g., slope, aspect). These interpolation methods are available in the SpaceStat software and should be implemented in future versions of Vesta.

## Advancing Geostatistics with 2 spatial interpolation methods

Both Kriging and Co-Kriging are powerful tools in the realm of spatial interpolation.

Choosing which method to implement depends on the availability, sampling density, and relevance of additional variables. In particular, the implementation cost of cokriging is not warranted when secondary variables are weakly correlated with the primary variable or available at exactly the same locations. Understanding these methods and their distinctions is essential for effective spatial analysis and informed decision-making.

Our upcoming version of Vesta will be able to perform univariate (kriging) and multivariate (cokriging) spatial interpolation over different spatial supports. Subscribe to our email newsletter to be the first to know when it will be available to the public.

Goovaerts, P. 1998. Ordinary cokriging revisited. Mathematical Geology, 30(1): 21-42.

Goovaerts, P. 2000. Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall. Journal of Hydrology, 228: 113-129.