By Richard Lane

A key feature problem in geological modelling is how to take scattered measurements and use these to make predictions at locations where there are no measurements. The data may be measurements on the surface, or samples taken from drilling, or channel samples taken while excavating. Figure 1 shows the basic problem. Solving this problem is fundamental to how Leapfrog software works, and it underpins the geological and mineralization models that are produced.

Figure 1: A simple scattered data problem. Estimate the value at the red cross from the blue samples.
Figure 1: A simple scattered data problem. Estimate the value at the red cross from the blue samples.

The basic process of estimating the value where there is no data is called interpolation. There are many well-known interpolation techniques, such as nearest neighbour, inverse distance weighting and radial basis functions. What is less well known is that the geostatistical procedure of Kriging can also be described as an interpolation procedure. When Kriging is implemented as an interpolation it is known as Dual Kriging, and its application in the mining industry has been known since the 1980’s.

This article is a non-rigorous description of the relationship between radial basis functions and Kriging and has been simplified in the interests of accessibility. For a more formal discussion of the relationship see the references provided at the end of the article.

Dual Kriging

Figure 2
Figure 2.

Figure 2: The fundamental problem, estimate the value q from the 3 samples Math-1.

Kriging in its simplest form estimates a value at a point as the weighted sum of the known samples, and the weights are determined mathematically by the distribution of the samples relative to the point an estimate is to be made. If we look at the simple example in Figure 2, the estimate Math-q is made by:


Clearly this can be used as a means of interpolation, because it’s possible to use the procedure to estimate at any arbitrary position where there is no data. The only problem is that when you make an estimate at another location you need to recalculate the weights, and this is a time consuming procedure.

It was noticed that it was possible to rearrange the Kriging equations in the following form:


This is a process known as dual Kriging. While this way of describing Kriging appears at first sight to be considerably more complicated than the simple sum of weighted samples, in reality it is also quite easy to understand.

The quantity Math-4 describes how the sample located at position Math-x1 influences the estimated value throughout the nearby space. In this form Math-5 is equivalent to the variogram. In this approach the weighted influences of all the samples are added to produce something which can be used to make a prediction at any location Math-x. A prediction made in this way produces the same result as one computed by using the traditional weighted sum of the samples. Figure 3 shows a 1 dimensional example of this approach.

Fig 3: A simplified demonstration of a radial basis function in 1 dimension.
Fig 3: A simplified demonstration of a radial basis function in 1 dimension.

So this leads to the obvious question, “If the answers are the same why bother with something that is harder to understand?” The answer is simply speed. Every time you make an estimate at a new position using the weighted sum of samples you have to recalculate the weights. With the dual approach where you sum the influences of the samples you only need to calculate the weights once.

Radial Basis Functions

Leapfrog uses Fast Radial Basis Functions as its primary computation engine. Effectively RBFs are a way of implementing dual Kriging, a fact noted by Matheron as far back as 1980. However, rather than solve the equations directly which is slow, Leapfrog uses advanced algorithms to find the dual Kriging weights and evaluate the estimates quickly. It’s not what Leapfrog solves that is different, it’s how it does it quickly.

The reason for using the radial basis function terminology in Leapfrog was originally based on technology built for laser scanning. It was only several years after the product started that the relationship with Kriging became clear.


Chiles J-P and Delfiner P, “Geostatistics Modelling Spatial Uncertainty”, Wiley Interscience, 1999, p 186

Dowd P A, “A review of geostatistical techniques for contouring”, NATA ASI Series, vol F17, Fundamental Algorithms for Computer Graphics. Edited by R.A. Earnshaw Springer Verlag..

Galli A. and Murillo E. “Dual Kriging – Its properties and its uses in direct contouring” Geostatistics for Natural Resources Characterization, NATO ASI Series C: Mathematical and Physical Sciences vol 122 part 2, Dodrecht, Holland, pp 621-634