The covariance function in resource estimation: variograms, kernels and RBFs
A covariance function describes how strongly two samples are related as the distance between them grows, and it is the main ingredient in the weights of kriging, RBFs and SVMs.
Different methods call it different things: a variogram model in kriging, a radial basis function in implicit modelling, a kernel in support vector machines and an activation function in neural networks. Common choices are spherical, exponential and Gaussian, shaped by a range, and variography is the practical way to choose one from your data. Once the function and its range are chosen, the rest of the estimation is mostly mechanics. This is part 4 of 4.
The covariance matrix and the covariance function
In kriging, the weights come from a covariance matrix K and a vector k. The numbers in them are computed by a covariance function C that depends on the distance between sample locations. Two samples close together have a high covariance, and the value falls as they move apart.
Same idea, different names
| Method | What the covariance function is called |
|---|---|
| Kriging | Variogram model |
| Implicit modelling | Radial basis function (RBF) |
| Support vector machines | Kernel |
| Neural networks | Activation function |
Common covariance functions
The Gaussian, exponential and spherical functions are widely used, and polynomial forms appear too. Each is controlled by parameters such as the range a, the distance beyond which samples stop influencing each other, and a shape parameter. The range is usually the most important setting.
Choosing a covariance function
Variography gives a practical way of choosing a function and range from your data: you fit a model to the experimental variogram of your samples. In exploration, results are often also checked visually, by looking at whether the resulting model makes geological sense.
Anisotropy
Geological properties often vary more in some directions than others. Distances can be adjusted, by scaling them along the direction of continuity, so that the same covariance function reflects this. It works the same way for every method.
Speed
Modern RBF and machine learning methods are much faster than classical kriging, which matters when a model needs updating in near real time as new drilling arrives.
Estimation through weighted means is not that complicated. The covariance function and its range are the most important choices.
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Common questions
Covariance function FAQ
What is a covariance function?
A function that gives the expected similarity of two samples as a function of the distance between them. Kriging, RBFs, SVMs and neural networks all use one, under different names.
What is the range of a variogram?
The distance at which samples stop being related. Beyond the range, a sample has little or no influence on the estimate.
How do I choose a variogram model?
Fit a spherical, exponential or Gaussian model, or a combination, to your experimental variogram, then check the result with validation and visual inspection.