Kriging Interpolation

Introduction

Kriging is an advanced geostatistical procedure that generates an estimated surface from a scattered set of points with z-values. Using geostatistical techniques, you can create surfaces incorporating the statistical properties of the measured data. Kriging is based on statistics. These techniques produce not only prediction surfaces but also error or uncertainty surfaces, giving you an indication of how good the predictions are. More so than other interpolation methods, a thorough investigation of the spatial behavior of the phenomenon represented by the z-values should be done before you select the best estimation method for generating the output surface.

Many kriging methods are associated with geostatistics, but they are all in the kriging family. Ordinary, simple, universal, probability, indicator, and disjunctive kriging, along with their counterparts in cokriging, are all available in the Geostatistical Analyst. Not only do these kriging methods create predictions and error surfaces, but they can also produce probability and quantile output maps depending on user needs.

Let’s focus more on ordinary kriging (OK) interpolation in this section. This form of kriging usually involves four steps:

• Removing any spatial trend in the data (if present).
• Computing the experimental variogram, γγ, which is a measure of spatial autocorrelation.
• Defining an experimental variogram model that best characterizes the spatial autocorrelation in the data.
• Interpolating the surface using the experimental variogram.
• Adding the kriged interpolated surface to the trend interpolated surface to produce the final output.

Kriging is the estimation procedure using known values and a semi-variogram to determine unknown values. The procedures involved in kriging incorporate measures of error and uncertainty when determining estimations. Based on the semi-variogram used, optimal weights are assigned to unknown values in order to calculate the unknown ones. Since the variogram changes with distance, the weights depend on the known sample distribution.

The basic equation used in ordinary kriging is as follows:

where, d is the distance between known points; n is the number of pairs of samples separated by d; Z is the attribute value (elevation of known points).

The equation indicates that the semi-variance is expected to increase as d increases. One of the most popular approaches is ordinary kriging, which will be applied in this study. Ordinary kriging assumes the model:

Z(s) = μ + ε(s),

where μ is an unknown constant.

One of the main issues concerning ordinary kriging is whether the assumption of a constant mean is reasonable. Sometimes there are good scientific reasons to reject this assumption. However, as a simple prediction method, it has remarkable flexibility.

Advantages of Kriging

The main advantages of kriging include:

1. Rigorous statistical framework: Kriging is a rigorous statistical method that incorporates a measure of uncertainty and provides confidence intervals for the estimated values.
2. Optimal interpolation: Kriging provides the optimal linear unbiased estimate of the variable of interest, given the assumptions about the data distribution and spatial autocorrelation.
3. Ability to incorporate auxiliary data: Kriging can incorporate auxiliary data such as covariates or secondary data sources to improve the accuracy of the estimates.
4. Flexibility: Kriging can be applied to various types of data, including continuous, categorical, and count data.

Applications of Kriging

The main applications of kriging include:

1. Geology and mining: Kriging can be used to estimate the distribution of minerals or other resources in the subsurface based on sparse sampling.
2. Environmental monitoring: Kriging can be used to estimate the concentration of pollutants or other environmental variables such as soil properties or water quality.
3. Agriculture and forestry: Kriging can be used to estimate soil properties and vegetation characteristics such as biomass and tree volume.
4. Hydrology and water resources: Kriging can be used to estimate precipitation, streamflow, and groundwater levels.

Limitations of Kriging

However, kriging also has limitations that should be considered:

1. Assumptions about data distribution and spatial autocorrelation: Kriging assumes that the data are stationary, normally distributed, and have a specific spatial autocorrelation structure. Violations of these assumptions can lead to inaccurate estimates.
2. Sensitivity to sample size and distribution: Kriging estimates are sensitive to the number and distribution of sample points. Sparse sampling or clustering of sample points can result in inaccurate predictions.
3. Computationally intensive: Kriging can be computationally intensive, especially for large datasets, and may require specialized software and hardware.
4. Difficulty in incorporating non-linear relationships: Kriging assumes a linear relationship between the variable of interest and the covariates. Non-linear relationships may require more complex modeling approaches.

Overall, kriging is a powerful and widely used method in GIS for spatial interpolation, but its limitations should be considered in the context of the specific dataset and research question. Other interpolation methods, such as inverse distance weighting or spline interpolation, may be more appropriate in some situations.