TPL StableΒΆ

GSTools also implements truncated power law variograms, which can be represented as a superposition of scale dependant modes in form of standard variograms, which are truncated by a lower- \ell_{\mathrm{low}} and an upper length-scale \ell_{\mathrm{up}}.

This example shows the truncated power law (TPLStable) based on the Stable covariance model and is given by

\gamma_{\ell_{\mathrm{low}},\ell_{\mathrm{up}}}(r) =
\intop_{\ell_{\mathrm{low}}}^{\ell_{\mathrm{up}}}
\gamma(r,\lambda) \frac{\rm d \lambda}{\lambda}

with Stable modes on each scale:

\gamma(r,\lambda) &=
\sigma^2(\lambda)\cdot\left(1-
\exp\left[- \left(\frac{r}{\lambda}\right)^{\alpha}\right]
\right)\\
\sigma^2(\lambda) &= C\cdot\lambda^{2H}

which gives Gaussian modes for alpha=2 or Exponential modes for alpha=1.

For \ell_{\mathrm{low}}=0 this results in:

\gamma_{\ell_{\mathrm{up}}}(r) &=
\sigma^2_{\ell_{\mathrm{up}}}\cdot\left(1-
\frac{2H}{\alpha} \cdot
E_{1+\frac{2H}{\alpha}}
\left[\left(\frac{r}{\ell_{\mathrm{up}}}\right)^{\alpha}\right]
\right) \\
\sigma^2_{\ell_{\mathrm{up}}} &=
C\cdot\frac{\ell_{\mathrm{up}}^{2H}}{2H}

../../_images/sphx_glr_00_tpl_stable_001.png
import numpy as np
import gstools as gs

x = y = np.linspace(0, 100, 100)
model = gs.TPLStable(
    dim=2,  # spatial dimension
    var=1,  # variance (C is calculated internally, so variance is actually 1)
    len_low=0,  # lower truncation of the power law
    len_scale=10,  # length scale (a.k.a. range), len_up = len_low + len_scale
    nugget=0.1,  # nugget
    anis=0.5,  # anisotropy between main direction and transversal ones
    angles=np.pi / 4,  # rotation angles
    alpha=1.5,  # shape parameter from the stable model
    hurst=0.7,  # hurst coefficient from the power law
)
srf = gs.SRF(model, mean=1.0, seed=19970221)
srf.structured([x, y])
srf.plot()

Total running time of the script: ( 0 minutes 15.473 seconds)

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