Choosing the Right Interpolation Method: Simple Kriging, Ordinary Kriging, and Universal Kriging

When conducting spatial interpolation analysis, selecting the appropriate method can significantly impact the accuracy and reliability of your results. This article explores the principles and applications of simple kriging, ordinary kriging, and universal kriging, detailing their differences and limitations.

The Context of Kriging

Kriging, a sophisticated geostatistical technique, was developed in the 1960s when computational resources were limited. Initially, interpolation tasks were often carried out manually, with a wide range of techniques available. Today, with the advent of powerful computing tools, the landscape of spatial data analysis has drastically changed. Kriging has become a widely used method for interpolating spatial data, but it requires a deep understanding of its underlying principles and limitations.

Spatial Interpolation and Kriging

At its core, Kriging is a method that minimizes the variance of the error in the estimated value by finding the optimized weights of the sample points' influence on the value of the estimated point. The process is designed to provide the best linear unbiased estimate, which means that the method aims to minimize the variance while maintaining an unbiased result.

Simple Kriging vs. Ordinary Kriging

The primary difference between simple kriging and ordinary kriging lies in the constraint of the weights. Simple kriging requires that the sum of the weights equals 1, which ensures that the estimated value is unbiased, given that the mean of the data is known. In contrast, ordinary kriging does not have this constraint and can be more flexible in fitting the data, although it still minimizes the variance.

It is often the case that simple kriging and ordinary kriging yield similar results, but the choice depends on the specific requirements of the analysis and the data at hand.

Handling Trends in Data

One significant limitation of kriging is its ability to handle trends in the data. When data exhibit a general trend, the assumption of stationarity (i.e., consistent variance over the spatial domain) is violated. This is where universal kriging comes into play.

Universal kriging generally involves two steps: (1) removing the trend from the data, (2) applying ordinary kriging to the residuals, and (3) replacing the trend after the interpolation. However, in practice, universal kriging can sometimes yield inconsistent results. Therefore, a common approach is to separate the process into three distinct steps: (1) detrending the data, (2) calculating the experimental variogram, (3) performing ordinary kriging, and (4) adding the trend back in.

The Importance of Variogram and Interpolation Range

The accuracy of Kriging highly depends on the accuracy of the interpolated variogram used to model the data's variances. The range of the variogram, in particular, is crucial because exceeding this range leads to the spread of errors across the entire estimated area, reducing the accuracy of the interpolation.

In my experience, fine-tuning the variogram and interpolation range visually in software can yield the best results. Accurate modeling of spatial variability is key to successful Kriging.

Modern Interpolation Techniques

With the advent of powerful computing tools, other interpolation methods have emerged as alternatives to Kriging. Cubic spline interpolation, for instance, can often yield results that are essentially the same as carefully applied kriging processes. This makes cubic splines a highly efficient and accurate alternative for many applications.

When computational power was limited, kriging with interpolated variograms in different directions was often the best solution, as it was better than simple linear interpolation. However, with modern computing, cubic spline interpolation has become a preferred choice for many users due to its efficiency and accuracy.

The Bottom Line

When using Kriging, it is crucial to remember that it can be risky if you do not thoroughly understand the underlying principles and limitations. Modern software tools allow for visual manipulation of variogram and interpolation range, making it easier to achieve accurate results. In the end, if you are not certain about your methods, it is best to rely on simpler and more robust techniques like cubic spline interpolation.

So, the takeaway is: always be cautious when using Kriging, and consider alternatives if you are not confident in your skills or data. Ensuring accurate and reliable spatial interpolation analysis is key to making informed decisions based on your data.