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.. "examples/02_cov_model/03_spectral_methods.py"
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    .. note::
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        :ref:`Go to the end <sphx_glr_download_examples_02_cov_model_03_spectral_methods.py>`
        to download the full example code.

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.. _sphx_glr_examples_02_cov_model_03_spectral_methods.py:


Spectral methods
================

The spectrum of a covariance model is given by:

.. math:: S(\mathbf{k}) = \left(\frac{1}{2\pi}\right)^n
    \int C(\Vert\mathbf{r}\Vert) e^{i b\mathbf{k}\cdot\mathbf{r}} d^n\mathbf{r}

Since the covariance function :math:`C(r)` is radially symmetric, we can
calculate this by the
`hankel-transformation <https://en.wikipedia.org/wiki/Hankel_transform>`_:

.. math:: S(k) = \left(\frac{1}{2\pi}\right)^n \cdot
    \frac{(2\pi)^{n/2}}{(bk)^{n/2-1}}
    \int_0^\infty r^{n/2-1} C(r) J_{n/2-1}(bkr) r dr

Where :math:`k=\left\Vert\mathbf{k}\right\Vert`.

Depending on the spectrum, the spectral-density is defined by:

.. math:: \tilde{S}(k) = \frac{S(k)}{\sigma^2}

You can access these methods by:

.. GENERATED FROM PYTHON SOURCE LINES 26-33

.. code-block:: Python


    import gstools as gs

    model = gs.Gaussian(dim=3, var=2.0, len_scale=10)
    ax = model.plot("spectrum")
    model.plot("spectral_density", ax=ax)




.. image-sg:: /examples/02_cov_model/images/sphx_glr_03_spectral_methods_001.png
   :alt: 03 spectral methods
   :srcset: /examples/02_cov_model/images/sphx_glr_03_spectral_methods_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 34-48

.. note::
   The spectral-density is given by the radius of the input phase. But it is
   **not** a probability density function for the radius of the phase.
   To obtain the pdf for the phase-radius, you can use the methods
   :any:`CovModel.spectral_rad_pdf`
   or :any:`CovModel.ln_spectral_rad_pdf` for the logarithm.

   The user can also provide a cdf (cumulative distribution function) by
   defining a method called ``spectral_rad_cdf``
   and/or a ppf (percent-point function)
   by ``spectral_rad_ppf``.

   The attributes :any:`CovModel.has_cdf`
   and :any:`CovModel.has_ppf` will check for that.


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

   **Total running time of the script:** (0 minutes 0.056 seconds)


.. _sphx_glr_download_examples_02_cov_model_03_spectral_methods.py:

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      :download:`Download zipped: 03_spectral_methods.zip <03_spectral_methods.zip>`


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