r"""
Truncated Power Law Variograms
------------------------------

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- :math:`\ell_{\mathrm{low}}` and
an upper length-scale :math:`\ell_{\mathrm{up}}`.

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

.. math::
   \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:

.. math::
   \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 :math:`\ell_{\mathrm{low}}=0` this results in:

.. math::
   \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}
"""
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()