Tutorial 2: The Covariance Model

One of the core-features of GSTools is the powerful CovModel class, which allows you to easily define arbitrary covariance models by yourself. The resulting models provide a bunch of nice features to explore the covariance models.

A covariance model is used to characterize the semi-variogram, denoted by \gamma, of a spatial random field. In GSTools, we use the following form for an isotropic and stationary field:

\gamma\left(r\right)=
\sigma^2\cdot\left(1-\rho\left(r\right)\right)+n

Where:

  • \rho(r) is the so called correlation function depending on the distance r
  • \sigma^2 is the variance
  • n is the nugget (subscale variance)

Note

We are not limited to isotropic models. GSTools supports anisotropy ratios for length scales in orthogonal transversal directions like:

  • x (main direction)
  • y (1. transversal direction)
  • z (2. transversal direction)

These main directions can also be rotated. Just have a look at the corresponding examples.

Provided Covariance Models

The following standard covariance models are provided by GSTools

Gaussian([dim, var, len_scale, nugget, …]) The Gaussian covariance model.
Exponential([dim, var, len_scale, nugget, …]) The Exponential covariance model.
Matern([dim, var, len_scale, nugget, anis, …]) The Matérn covariance model.
Stable([dim, var, len_scale, nugget, anis, …]) The stable covariance model.
Rational([dim, var, len_scale, nugget, …]) The rational quadratic covariance model.
Linear([dim, var, len_scale, nugget, anis, …]) The bounded linear covariance model.
Circular([dim, var, len_scale, nugget, …]) The circular covariance model.
Spherical([dim, var, len_scale, nugget, …]) The Spherical covariance model.
Intersection([dim, var, len_scale, nugget, …]) The Intersection covariance model.

As a special feature, we also provide truncated power law (TPL) covariance models

TPLGaussian([dim, var, len_scale, nugget, …]) Truncated-Power-Law with Gaussian modes.
TPLExponential([dim, var, len_scale, …]) Truncated-Power-Law with Exponential modes.
TPLStable([dim, var, len_scale, nugget, …]) Truncated-Power-Law with Stable modes.