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The Covariance Model

One of the fundamental 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 γ\gamma, of a spatial random field. In GSTools, we use the following formulation for an isotropic and stationary field:



  • r r is the lag distance
  • \ell is the main correlation length
  • s s is a scaling factor for unit conversion or normalization
  • σ2 \sigma^2 is the variance
  • n n is the nugget (subscale variance)
  • cor(h) \mathrm{cor}(h) is the normalized correlation function depending on the non-dimensional distance h=sr h=s\cdot\frac{r}{\ell}

Depending on the normalized correlation function, all covariance models in GSTools are providing the following functions:

  • ρ(r)=cor(sr) \rho(r)=\mathrm{cor}\left(s\cdot\frac{r}{\ell}\right) is the so called correlation function
  • C(r)=σ2ρ(r) C(r)=\sigma^2\cdot\rho(r) is the so called covariance function, which gives the name for our GSTools class

.. note::

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

  • x0 x_0 (main direction)
  • x1 x_1 (1. transversal direction)
  • x2 x_2 (2. transversal direction)
  • ...

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

Provided Covariance Models

CovModel list

Taken from Müller et al. (2022).

  1. Müller, S., Schüler, L., Zech, A., & Heße, F. (2022). GSTools v1.3: a toolbox for geostatistical modelling in Python. Geoscientific Model Development, 15(7), 3161–3182. 10.5194/gmd-15-3161-2022