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Plurigaussian Field Generation
==============================

Plurigaussian field simulations (PGS) are used to simulate correlated fields
of categorical data, e.g. lithofacies, hydrofacies, soil types, or
cementitious materials.
PGS uses one spatial random field (SRF) per dimension, e.g. two SRFs, when
working with two dimensional data. Furthermore, PGS needs a field, which
describes the categorical data and its spatial relations.
This might sound more complicated then it is, as we will see in the following
examples.

Examples
--------



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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="As a first example, we will create a two dimensional plurigaussian field (PGS). Thus, we need two spatial random fields(SRF) and on top of that, we need a field describing the categorical data and its spatial relation. We will start off by creating the two SRFs with a Gaussian variogram, which makes the fields nice and smooth. But before that, we will import all necessary libraries and define a few variables, like the number of grid cells in each dimension.">

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  .. image:: /examples/11_plurigaussian/images/thumb/sphx_glr_00_simple_thumb.png
    :alt:

  :ref:`sphx_glr_examples_11_plurigaussian_00_simple.py`

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      <div class="sphx-glr-thumbnail-title">A First and Simple Example</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In this example we want to try to understand how exactly PGS are generated and how to influence them with the categorical field. First of all, we will set everything up very similar to the first example.">

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  .. image:: /examples/11_plurigaussian/images/thumb/sphx_glr_01_pgs_thumb.png
    :alt:

  :ref:`sphx_glr_examples_11_plurigaussian_01_pgs.py`

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      <div class="sphx-glr-thumbnail-title">Understanding PGS</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In this example we will try to understand how we can influence the spatial relationships of the different categories with the lithotypes field. For simplicity, we will start very similarly to the very first example.">

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  .. image:: /examples/11_plurigaussian/images/thumb/sphx_glr_02_spatial_relations_thumb.png
    :alt:

  :ref:`sphx_glr_examples_11_plurigaussian_02_spatial_relations.py`

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      <div class="sphx-glr-thumbnail-title">Controlling Spatial Relations</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Up until now, we have only used very smooth Gaussian variograms for the underlying spatial random fields. Now, we will combine a smooth Gaussian field with a much rougher exponential field. This example should feel familiar, if you had a look at the previous examples.">

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  .. image:: /examples/11_plurigaussian/images/thumb/sphx_glr_03_correlations_thumb.png
    :alt:

  :ref:`sphx_glr_examples_11_plurigaussian_03_correlations.py`

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      <div class="sphx-glr-thumbnail-title">Understanding the Influence of Variograms</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Let&#x27;s create a 3d PGS! This will mostly feel very familiar, but the plotting will be a bit more involved.">

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  .. image:: /examples/11_plurigaussian/images/thumb/sphx_glr_04_3d_pgs_thumb.png
    :alt:

  :ref:`sphx_glr_examples_11_plurigaussian_04_3d_pgs.py`

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      <div class="sphx-glr-thumbnail-title">Creating a Three Dimensional PGS</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In case we have knowledge about values of the PGS in certain areas, we should incorporate that knowledge. We can do this by using conditional fields, which are created by combining kriged fields with SRFs. This can be done quite easily with GSTools. For more details, have a look at the kriging and conditional field examples.">

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  .. image:: /examples/11_plurigaussian/images/thumb/sphx_glr_05_conditioned_thumb.png
    :alt:

  :ref:`sphx_glr_examples_11_plurigaussian_05_conditioned.py`

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      <div class="sphx-glr-thumbnail-title">Creating conditioned PGS</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Plurigaussian fields with periodic boundaries (P-PGS) are used in various applications, including the simulation of interactions between the landsurface and the atmosphere, as well as the application of homogenisation theory to porous media, e.g. [Ricketts 2024](https://doi.org/10.1007/s11242-024-02074-z).">

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  .. image:: /examples/11_plurigaussian/images/thumb/sphx_glr_06_periodic_thumb.png
    :alt:

  :ref:`sphx_glr_examples_11_plurigaussian_06_periodic.py`

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      <div class="sphx-glr-thumbnail-title">Creating PGS with periodic boundaries</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /examples/11_plurigaussian/00_simple
   /examples/11_plurigaussian/01_pgs
   /examples/11_plurigaussian/02_spatial_relations
   /examples/11_plurigaussian/03_correlations
   /examples/11_plurigaussian/04_3d_pgs
   /examples/11_plurigaussian/05_conditioned
   /examples/11_plurigaussian/06_periodic



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

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