.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/00_misc/00_tpl_stable.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_00_misc_00_tpl_stable.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_00_misc_00_tpl_stable.py:


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}

.. GENERATED FROM PYTHON SOURCE LINES 43-63



.. image-sg:: /examples/00_misc/images/sphx_glr_00_tpl_stable_001.png
   :alt: Field 2D structured: (100, 100)
   :srcset: /examples/00_misc/images/sphx_glr_00_tpl_stable_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    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()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 2.151 seconds)


.. _sphx_glr_download_examples_00_misc_00_tpl_stable.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: 00_tpl_stable.ipynb <00_tpl_stable.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: 00_tpl_stable.py <00_tpl_stable.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: 00_tpl_stable.zip <00_tpl_stable.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_