Tutorials

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:

γ(r)=σ2(1cor(sr))+n\gamma\left(r\right)=\sigma^2\cdot\left(1-\mathrm{cor}\left(s\cdot\frac{r}{\ell}\right)\right)+n

Where:

  • 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 .

References
  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