###### 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.

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: The fundamental problem, estimate the value q from the 3 samples .

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 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 describes how the sample located at position influences the estimated value throughout the nearby space. In this form 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 . 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.

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.