gstools.field.upscaling

GStools subpackage providing upscaling routines for the spatial random field.

The following functions are provided

var_coarse_graining(model[, point_volumes]) Coarse Graning procedure to upscale the variance for uniform flow.
var_no_scaling(model, \*args, \*\*kwargs) Dummy function to bypass scaling.
gstools.field.upscaling.var_coarse_graining(model, point_volumes=0.0)[source]

Coarse Graning procedure to upscale the variance for uniform flow.

Parameters:
  • model (CovModel) – Covariance Model used for the field.
  • point_volumes (float or numpy.ndarray) – Volumes of the elements at the given points. Default: 0
Returns:

scaled_var – The upscaled variance

Return type:

float or numpy.ndarray

Notes

This procedure was presented in [Attinger03]. It applies the upscaling procedure ‘Coarse Graining’ to the Groundwater flow equation under uniform flow on a lognormal distributed conductivity field following a gaussian covariance function. A filter over a cube with a given edge-length \lambda is applied and an upscaled conductivity field is obtained. The upscaled field is again following a gaussian covariance function with scale dependent variance and length-scale:

\lambda &= V^{\frac{1}{d}} \\
\sigma^2\left(\lambda\right) &=
\sigma^2\cdot\left(
\frac{\ell^2}{\ell^2+\left(\frac{\lambda}{2}\right)^2}
\right)^{\frac{d}{2}} \\
\ell\left(\lambda\right) &=
\left(\ell^2+\left(\frac{\lambda}{2}\right)^2\right)^{\frac{1}{2}}

Therby \lambda will be calculated from the given point_volumes V by assuming a cube with the given volume.

The upscaled length scale will be ignored by this routine.

References

[Attinger03]Attinger, S. 2003, ‘’Generalized coarse graining procedures for flow in porous media’‘, Computational Geosciences, 7(4), 253–273.
gstools.field.upscaling.var_no_scaling(model, *args, **kwargs)[source]

Dummy function to bypass scaling.

Parameters:model (CovModel) – Covariance Model used for the field.
Returns:var – The model variance.
Return type:float