.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "content/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_content_examples_10-pgi_plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py>`
        to download the full example code

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

.. _sphx_glr_content_examples_10-pgi_plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:


Petrophysically guided inversion: Joint linear example with nonlinear relationships
===================================================================================

We do a comparison between the classic least-squares inversion
and our formulation of a petrophysically guided inversion.
We explore it through coupling two linear problems whose respective physical
properties are linked by polynomial relationships that change between rock units.

.. GENERATED FROM PYTHON SOURCE LINES 11-432



.. image-sg:: /content/examples/10-pgi/images/sphx_glr_plot_inv_1_PGI_Linear_1D_joint_WithRelationships_001.png
   :alt: Problem 1, Problem 2, Petrophysical Distribution, Problem 1, Problem 2, Petrophysical Distribution, Problem 1, Problem 2, Petro Distribution
   :srcset: /content/examples/10-pgi/images/sphx_glr_plot_inv_1_PGI_Linear_1D_joint_WithRelationships_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    SimPEG.InvProblem will set Regularization.reference_model to m0.
    SimPEG.InvProblem will set Regularization.reference_model to m0.
    SimPEG.InvProblem will set Regularization.reference_model to m0.

                        SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
                        ***Done using the default solver Pardiso and no solver_opts.***
                    
    Alpha scales: [3.4563515102825697, 0.0, 3.4567008422694405e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09369146 0.90630854]
    <class 'SimPEG.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09369146 0.90630854]
    model has any nan: 0
    =============================== Projected GNCG ===============================
      #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment   
    -----------------------------------------------------------------------------
    x0 has any nan: 0
       0  1.89e+01  1.50e+05  0.00e+00  1.50e+05    1.41e+02      0              
    geophys. misfits: 535.7 (target 15.0 [False]); 33.8 (target 15.0 [False]) | smallness misfit: 1488.9 (target: 100.0 [False])
    Beta cooling evaluation: progress: [535.7  33.8] ; minimum progress targets: [120000. 120000.]
       1  1.89e+01  8.08e+01  2.05e+01  4.69e+02    7.42e+01      0              
    geophys. misfits: 242.5 (target 15.0 [False]); 7.5 (target 15.0 [True]) | smallness misfit: 680.8 (target: 100.0 [False])
    Beta cooling evaluation: progress: [242.5   7.5] ; minimum progress targets: [428.5  27. ]
    Updating scaling for data misfits by  2.0099452877554045
    New scales: [0.17203609 0.82796391]
       2  1.89e+01  4.79e+01  1.99e+01  4.25e+02    9.10e+01      0   Skip BFGS  
    geophys. misfits: 104.7 (target 15.0 [False]); 7.3 (target 15.0 [True]) | smallness misfit: 612.0 (target: 100.0 [False])
    Beta cooling evaluation: progress: [104.7   7.3] ; minimum progress targets: [194.  15.]
    Updating scaling for data misfits by  2.04600337958706
    New scales: [0.29830613 0.70169387]
       3  1.89e+01  3.64e+01  2.08e+01  4.30e+02    7.18e+01      0              
    geophys. misfits: 50.9 (target 15.0 [False]); 7.5 (target 15.0 [True]) | smallness misfit: 571.3 (target: 100.0 [False])
    Beta cooling evaluation: progress: [50.9  7.5] ; minimum progress targets: [83.8 15. ]
    Updating scaling for data misfits by  1.9977120474469887
    New scales: [0.459247 0.540753]
       4  1.89e+01  2.74e+01  2.14e+01  4.33e+02    7.09e+01      0   Skip BFGS  
    geophys. misfits: 30.7 (target 15.0 [False]); 8.3 (target 15.0 [True]) | smallness misfit: 540.4 (target: 100.0 [False])
    Beta cooling evaluation: progress: [30.7  8.3] ; minimum progress targets: [40.7 15. ]
    Updating scaling for data misfits by  1.8071739818330619
    New scales: [0.60548912 0.39451088]
       5  1.89e+01  2.18e+01  2.18e+01  4.34e+02    6.68e+01      0   Skip BFGS  
    geophys. misfits: 23.1 (target 15.0 [False]); 9.9 (target 15.0 [True]) | smallness misfit: 513.2 (target: 100.0 [False])
    Beta cooling evaluation: progress: [23.1  9.9] ; minimum progress targets: [24.5 15. ]
    Updating scaling for data misfits by  1.513006342027113
    New scales: [0.69898906 0.30101094]
       6  1.89e+01  1.91e+01  2.20e+01  4.35e+02    5.21e+01      0   Skip BFGS  
    geophys. misfits: 20.2 (target 15.0 [False]); 12.0 (target 15.0 [True]) | smallness misfit: 491.9 (target: 100.0 [False])
    Beta cooling evaluation: progress: [20.2 12. ] ; minimum progress targets: [18.5 15. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.2489357416430322
    New scales: [0.743603 0.256397]
       7  9.45e+00  1.81e+01  2.20e+01  2.26e+02    7.53e+01      0   Skip BFGS  
    geophys. misfits: 11.7 (target 15.0 [True]); 8.2 (target 15.0 [True]) | smallness misfit: 517.5 (target: 100.0 [False])
    Beta cooling evaluation: progress: [11.7  8.2] ; minimum progress targets: [16.2 15. ]
    Warming alpha_pgi to favor clustering:  1.558346964126058
       8  9.45e+00  1.08e+01  2.32e+01  2.30e+02    5.11e+01      0              
    geophys. misfits: 11.6 (target 15.0 [True]); 10.6 (target 15.0 [True]) | smallness misfit: 459.2 (target: 100.0 [False])
    Beta cooling evaluation: progress: [11.6 10.6] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  2.11208740112366
       9  9.45e+00  1.13e+01  2.37e+01  2.35e+02    5.57e+01      0              
    geophys. misfits: 11.5 (target 15.0 [True]); 13.1 (target 15.0 [True]) | smallness misfit: 416.6 (target: 100.0 [False])
    Beta cooling evaluation: progress: [11.5 13.1] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  2.5870703091709033
      10  9.45e+00  1.19e+01  2.40e+01  2.39e+02    3.62e+01      0   Skip BFGS  
    geophys. misfits: 11.5 (target 15.0 [True]); 15.3 (target 15.0 [False]) | smallness misfit: 386.7 (target: 100.0 [False])
    Beta cooling evaluation: progress: [11.5 15.3] ; minimum progress targets: [15. 15.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.3061581918379288
    New scales: [0.68948016 0.31051984]
      11  4.73e+00  1.27e+01  2.39e+01  1.26e+02    6.55e+01      0   Skip BFGS  
    geophys. misfits: 9.2 (target 15.0 [True]); 8.0 (target 15.0 [True]) | smallness misfit: 436.5 (target: 100.0 [False])
    Beta cooling evaluation: progress: [9.2 8. ] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  4.5484603804233865
      12  4.73e+00  8.80e+00  2.63e+01  1.33e+02    4.20e+01      0              
    geophys. misfits: 9.3 (target 15.0 [True]); 11.0 (target 15.0 [True]) | smallness misfit: 351.7 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.3 11. ] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  6.77298705312485
      13  4.73e+00  9.83e+00  2.77e+01  1.41e+02    3.81e+01      0              
    geophys. misfits: 9.3 (target 15.0 [True]); 13.9 (target 15.0 [True]) | smallness misfit: 290.2 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.3 13.9] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  9.109600670471133
      14  4.73e+00  1.07e+01  2.88e+01  1.47e+02    6.32e+01      0   Skip BFGS  
    geophys. misfits: 9.5 (target 15.0 [True]); 17.5 (target 15.0 [False]) | smallness misfit: 246.7 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.5 17.5] ; minimum progress targets: [15. 15.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.5787904164918563
    New scales: [0.58444098 0.41555902]
      15  2.36e+00  1.28e+01  2.84e+01  8.00e+01    7.68e+01      0   Skip BFGS  
    geophys. misfits: 8.6 (target 15.0 [True]); 8.5 (target 15.0 [True]) | smallness misfit: 291.7 (target: 100.0 [False])
    Beta cooling evaluation: progress: [8.6 8.5] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  16.00907106538602
      16  2.36e+00  8.55e+00  3.39e+01  8.86e+01    5.79e+01      0              
    geophys. misfits: 8.9 (target 15.0 [True]); 11.0 (target 15.0 [True]) | smallness misfit: 213.7 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 8.9 11. ] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  24.420341012994264
      17  2.36e+00  9.77e+00  3.67e+01  9.66e+01    7.67e+01      0              
    geophys. misfits: 9.3 (target 15.0 [True]); 12.1 (target 15.0 [True]) | smallness misfit: 173.1 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.3 12.1] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  34.84683728259039
      18  2.36e+00  1.05e+01  3.99e+01  1.05e+02    8.61e+01      0              
    geophys. misfits: 9.5 (target 15.0 [True]); 13.8 (target 15.0 [True]) | smallness misfit: 140.6 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.5 13.8] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  46.54759697440141
      19  2.36e+00  1.13e+01  4.27e+01  1.12e+02    8.35e+01      0              
    geophys. misfits: 10.3 (target 15.0 [True]); 17.3 (target 15.0 [False]) | smallness misfit: 112.7 (target: 100.0 [False])
    Beta cooling evaluation: progress: [10.3 17.3] ; minimum progress targets: [15. 15.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.4521282269866747
    New scales: [0.49200091 0.50799909]
      20  1.18e+00  1.39e+01  4.12e+01  6.26e+01    7.28e+01      0              
    geophys. misfits: 9.6 (target 15.0 [True]); 11.9 (target 15.0 [True]) | smallness misfit: 129.8 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.6 11.9] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  65.7457158607421
      21  1.18e+00  1.08e+01  4.87e+01  6.83e+01    7.93e+01      0              
    geophys. misfits: 9.2 (target 15.0 [True]); 8.9 (target 15.0 [True]) | smallness misfit: 123.1 (target: 100.0 [False])
    Beta cooling evaluation: progress: [9.2 8.9] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  109.09773521695293
      22  1.18e+00  9.04e+00  6.10e+01  8.11e+01    8.87e+01      0              
    geophys. misfits: 9.7 (target 15.0 [True]); 10.1 (target 15.0 [True]) | smallness misfit: 109.5 (target: 100.0 [False])
    Beta cooling evaluation: progress: [ 9.7 10.1] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  165.84176425481058
      23  1.18e+00  9.88e+00  7.23e+01  9.53e+01    9.68e+01      1              
    geophys. misfits: 9.7 (target 15.0 [True]); 9.7 (target 15.0 [True]) | smallness misfit: 100.2 (target: 100.0 [False])
    Beta cooling evaluation: progress: [9.7 9.7] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  256.2108523404665
      24  1.18e+00  9.71e+00  9.19e+01  1.18e+02    1.08e+02      0              
    geophys. misfits: 9.6 (target 15.0 [True]); 9.7 (target 15.0 [True]) | smallness misfit: 90.9 (target: 100.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [9.6 9.7] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  396.871518280846
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.5000e+04
    0 : |xc-x_last| = 4.2645e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 1.0822e+02 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 1.0822e+02 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =     25
    ------------------------- DONE! -------------------------
    SimPEG.InvProblem will set Regularization.reference_model to m0.
    SimPEG.InvProblem will set Regularization.reference_model to m0.
    SimPEG.InvProblem will set Regularization.reference_model to m0.

                        SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
                        ***Done using the default solver Pardiso and no solver_opts.***
                    
    Alpha scales: [0.0003501319300388227, 0.0, 3.4823712219446624e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09369146 0.90630854]
    <class 'SimPEG.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09369146 0.90630854]
    model has any nan: 0
    =============================== Projected GNCG ===============================
      #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment   
    -----------------------------------------------------------------------------
    x0 has any nan: 0
       0  1.90e+03  1.50e+05  0.00e+00  1.50e+05    1.41e+02      0              
    geophys. misfits: 44336.6 (target 15.0 [False]); 31031.5 (target 15.0 [False]) | smallness misfit: 137.5 (target: 100.0 [False])
    Beta cooling evaluation: progress: [44336.6 31031.5] ; minimum progress targets: [120000. 120000.]
       1  1.90e+03  3.23e+04  3.32e-01  3.29e+04    9.20e+01      0              
    geophys. misfits: 321.7 (target 15.0 [False]); 8.2 (target 15.0 [True]) | smallness misfit: 48.9 (target: 100.0 [True])
    Beta cooling evaluation: progress: [321.7   8.2] ; minimum progress targets: [35469.3 24825.2]
    Updating scaling for data misfits by  1.8383802415807917
    New scales: [0.15969651 0.84030349]
       2  1.90e+03  5.82e+01  1.36e-01  3.17e+02    1.01e+02      0   Skip BFGS  
    geophys. misfits: 36.0 (target 15.0 [False]); 8.6 (target 15.0 [True]) | smallness misfit: 25.4 (target: 100.0 [True])
    Beta cooling evaluation: progress: [36.   8.6] ; minimum progress targets: [257.4  15. ]
    Updating scaling for data misfits by  1.7390538633610058
    New scales: [0.24840321 0.75159679]
       3  1.90e+03  1.54e+01  6.88e-02  1.46e+02    8.19e+01      0   Skip BFGS  
    geophys. misfits: 22.2 (target 15.0 [False]); 7.8 (target 15.0 [True]) | smallness misfit: 34.0 (target: 100.0 [True])
    Beta cooling evaluation: progress: [22.2  7.8] ; minimum progress targets: [28.8 15. ]
    Updating scaling for data misfits by  1.916082805099921
    New scales: [0.38773008 0.61226992]
       4  1.90e+03  1.34e+01  8.71e-02  1.79e+02    8.04e+01      0              
    geophys. misfits: 16.1 (target 15.0 [False]); 8.5 (target 15.0 [True]) | smallness misfit: 31.1 (target: 100.0 [True])
    Beta cooling evaluation: progress: [16.1  8.5] ; minimum progress targets: [17.8 15. ]
    Updating scaling for data misfits by  1.7651233218695725
    New scales: [0.52781046 0.47218954]
       5  1.90e+03  1.25e+01  7.70e-02  1.59e+02    6.82e+01      0              
    geophys. misfits: 13.2 (target 15.0 [True]); 10.8 (target 15.0 [True]) | smallness misfit: 24.5 (target: 100.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [13.2 10.8] ; minimum progress targets: [15. 15.]
    Warming alpha_pgi to favor clustering:  1.2635740951830294
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.5000e+04
    0 : |xc-x_last| = 1.4628e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 6.8186e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 6.8186e+01 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =      6
    ------------------------- DONE! -------------------------
    SimPEG.InvProblem will set Regularization.reference_model to m0.
    SimPEG.InvProblem will set Regularization.reference_model to m0.

                        SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
                        ***Done using the default solver Pardiso and no solver_opts.***
                    
    Alpha scales: [3.482672955213922e-05, 0.0, 3.5050139896186884e-05, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09369146 0.90630854]
    /home/vsts/work/1/s/SimPEG/directives/directives.py:1299: UserWarning:

    There is no PGI regularization. Smallness target is turned off (TriggerSmall flag)

    Initial data misfit scales:  [0.09369146 0.90630854]
    model has any nan: 0
    =============================== Projected GNCG ===============================
      #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment   
    -----------------------------------------------------------------------------
    x0 has any nan: 0
       0  1.02e+06  1.50e+05  0.00e+00  1.50e+05    1.41e+02      0              
    geophys. misfits: 28272.7 (target 15.0 [False]); 17613.7 (target 15.0 [False])
       1  2.04e+05  1.86e+04  2.17e-02  2.30e+04    1.36e+02      0              
    geophys. misfits: 4213.1 (target 15.0 [False]); 1890.7 (target 15.0 [False])
       2  4.08e+04  2.11e+03  5.39e-02  4.31e+03    1.24e+02      0   Skip BFGS  
    geophys. misfits: 285.1 (target 15.0 [False]); 118.9 (target 15.0 [False])
       3  8.15e+03  1.34e+02  7.10e-02  7.13e+02    9.78e+01      0   Skip BFGS  
    geophys. misfits: 23.0 (target 15.0 [False]); 11.9 (target 15.0 [True])
    Updating scaling for data misfits by  1.2556154440684768
    New scales: [0.11488897 0.88511103]
       4  1.63e+03  1.32e+01  7.62e-02  1.37e+02    6.35e+01      0   Skip BFGS  
    geophys. misfits: 8.4 (target 15.0 [True]); 5.7 (target 15.0 [True])
    All targets have been reached
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.5000e+04
    0 : |xc-x_last| = 3.3061e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 6.3502e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 6.3502e+01 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =      5
    ------------------------- DONE! -------------------------
    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:301: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:308: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:346: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:353: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:360: UserWarning:

    The following kwargs were not used by contour: 'label'

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:368: UserWarning:

    The following kwargs were not used by contour: 'label'

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:402: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:409: UserWarning:

    marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.







|

.. code-block:: default


    import discretize as Mesh
    import matplotlib.pyplot as plt
    import numpy as np
    from SimPEG import (
        data_misfit,
        directives,
        inverse_problem,
        inversion,
        maps,
        optimization,
        regularization,
        simulation,
        utils,
    )

    # Random seed for reproductibility
    np.random.seed(1)
    # Mesh
    N = 100
    mesh = Mesh.TensorMesh([N])

    # Survey design parameters
    nk = 30
    jk = np.linspace(1.0, 59.0, nk)
    p = -0.25
    q = 0.25


    # Physics
    def g(k):
        return np.exp(p * jk[k] * mesh.cell_centers_x) * np.cos(
            np.pi * q * jk[k] * mesh.cell_centers_x
        )


    G = np.empty((nk, mesh.nC))

    for i in range(nk):
        G[i, :] = g(i)

    m0 = np.zeros(mesh.nC)
    m0[20:41] = np.linspace(0.0, 1.0, 21)
    m0[41:57] = np.linspace(-1, 0.0, 16)

    poly0 = maps.PolynomialPetroClusterMap(coeffyx=np.r_[0.0, -4.0, 4.0])
    poly1 = maps.PolynomialPetroClusterMap(coeffyx=np.r_[-0.0, 3.0, 6.0, 6.0])
    poly0_inverse = maps.PolynomialPetroClusterMap(coeffyx=-np.r_[0.0, -4.0, 4.0])
    poly1_inverse = maps.PolynomialPetroClusterMap(coeffyx=-np.r_[0.0, 3.0, 6.0, 6.0])
    cluster_mapping = [maps.IdentityMap(), poly0_inverse, poly1_inverse]

    m1 = np.zeros(100)
    m1[20:41] = 1.0 + (poly0 * np.vstack([m0[20:41], m1[20:41]]).T)[:, 1]
    m1[41:57] = -1.0 + (poly1 * np.vstack([m0[41:57], m1[41:57]]).T)[:, 1]

    model2d = np.vstack([m0, m1]).T
    m = utils.mkvc(model2d)

    clfmapping = utils.GaussianMixtureWithNonlinearRelationships(
        mesh=mesh,
        n_components=3,
        covariance_type="full",
        tol=1e-8,
        reg_covar=1e-3,
        max_iter=1000,
        n_init=100,
        init_params="kmeans",
        random_state=None,
        warm_start=False,
        means_init=np.array(
            [
                [0, 0],
                [m0[20:41].mean(), m1[20:41].mean()],
                [m0[41:57].mean(), m1[41:57].mean()],
            ]
        ),
        verbose=0,
        verbose_interval=10,
        cluster_mapping=cluster_mapping,
    )
    clfmapping = clfmapping.fit(model2d)

    clfnomapping = utils.WeightedGaussianMixture(
        mesh=mesh,
        n_components=3,
        covariance_type="full",
        tol=1e-8,
        reg_covar=1e-3,
        max_iter=1000,
        n_init=100,
        init_params="kmeans",
        random_state=None,
        warm_start=False,
        verbose=0,
        verbose_interval=10,
    )
    clfnomapping = clfnomapping.fit(model2d)

    wires = maps.Wires(("m1", mesh.nC), ("m2", mesh.nC))

    relatrive_error = 0.01
    noise_floor = 0.0

    prob1 = simulation.LinearSimulation(mesh, G=G, model_map=wires.m1)
    survey1 = prob1.make_synthetic_data(
        m, relative_error=relatrive_error, noise_floor=noise_floor, add_noise=True
    )

    prob2 = simulation.LinearSimulation(mesh, G=G, model_map=wires.m2)
    survey2 = prob2.make_synthetic_data(
        m, relative_error=relatrive_error, noise_floor=noise_floor, add_noise=True
    )


    dmis1 = data_misfit.L2DataMisfit(simulation=prob1, data=survey1)
    dmis2 = data_misfit.L2DataMisfit(simulation=prob2, data=survey2)
    dmis = dmis1 + dmis2
    minit = np.zeros_like(m)

    # Distance weighting
    wr1 = np.sum(prob1.G**2.0, axis=0) ** 0.5 / mesh.cell_volumes
    wr1 = wr1 / np.max(wr1)
    wr2 = np.sum(prob2.G**2.0, axis=0) ** 0.5 / mesh.cell_volumes
    wr2 = wr2 / np.max(wr2)

    reg_simple = regularization.PGI(
        mesh=mesh,
        gmmref=clfmapping,
        gmm=clfmapping,
        approx_gradient=True,
        wiresmap=wires,
        non_linear_relationships=True,
        weights_list=[wr1, wr2],
    )

    opt = optimization.ProjectedGNCG(
        maxIter=50,
        tolX=1e-6,
        maxIterCG=100,
        tolCG=1e-3,
        lower=-10,
        upper=10,
    )

    invProb = inverse_problem.BaseInvProblem(dmis, reg_simple, opt)

    # directives
    scales = directives.ScalingMultipleDataMisfits_ByEig(
        chi0_ratio=np.r_[1.0, 1.0], verbose=True, n_pw_iter=10
    )
    scaling_schedule = directives.JointScalingSchedule(verbose=True)
    alpha0_ratio = np.r_[1e6, 1e4, 1, 1]
    alphas = directives.AlphasSmoothEstimate_ByEig(
        alpha0_ratio=alpha0_ratio, n_pw_iter=10, verbose=True
    )
    beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-5, n_pw_iter=10)
    betaIt = directives.PGI_BetaAlphaSchedule(
        verbose=True,
        coolingFactor=2.0,
        progress=0.2,
    )
    targets = directives.MultiTargetMisfits(verbose=True)
    petrodir = directives.PGI_UpdateParameters(update_gmm=False)

    # Setup Inversion
    inv = inversion.BaseInversion(
        invProb,
        directiveList=[alphas, scales, beta, petrodir, targets, betaIt, scaling_schedule],
    )

    mcluster_map = inv.run(minit)

    # Inversion with no nonlinear mapping
    reg_simple_no_map = regularization.PGI(
        mesh=mesh,
        gmmref=clfnomapping,
        gmm=clfnomapping,
        approx_gradient=True,
        wiresmap=wires,
        non_linear_relationships=False,
        weights_list=[wr1, wr2],
    )

    opt = optimization.ProjectedGNCG(
        maxIter=50,
        tolX=1e-6,
        maxIterCG=100,
        tolCG=1e-3,
        lower=-10,
        upper=10,
    )

    invProb = inverse_problem.BaseInvProblem(dmis, reg_simple_no_map, opt)

    # directives
    scales = directives.ScalingMultipleDataMisfits_ByEig(
        chi0_ratio=np.r_[1.0, 1.0], verbose=True, n_pw_iter=10
    )
    scaling_schedule = directives.JointScalingSchedule(verbose=True)
    alpha0_ratio = np.r_[100.0 * np.ones(2), 1, 1]
    alphas = directives.AlphasSmoothEstimate_ByEig(
        alpha0_ratio=alpha0_ratio, n_pw_iter=10, verbose=True
    )
    beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-5, n_pw_iter=10)
    betaIt = directives.PGI_BetaAlphaSchedule(
        verbose=True,
        coolingFactor=2.0,
        progress=0.2,
    )
    targets = directives.MultiTargetMisfits(
        chiSmall=1.0, TriggerSmall=True, TriggerTheta=False, verbose=True
    )
    petrodir = directives.PGI_UpdateParameters(update_gmm=False)

    # Setup Inversion
    inv = inversion.BaseInversion(
        invProb,
        directiveList=[alphas, scales, beta, petrodir, targets, betaIt, scaling_schedule],
    )

    mcluster_no_map = inv.run(minit)

    # WeightedLeastSquares Inversion

    reg1 = regularization.WeightedLeastSquares(
        mesh, alpha_s=1.0, alpha_x=1.0, mapping=wires.m1
    )
    reg1.cell_weights = wr1
    reg2 = regularization.WeightedLeastSquares(
        mesh, alpha_s=1.0, alpha_x=1.0, mapping=wires.m2
    )
    reg2.cell_weights = wr2
    reg = reg1 + reg2

    opt = optimization.ProjectedGNCG(
        maxIter=50,
        tolX=1e-6,
        maxIterCG=100,
        tolCG=1e-3,
        lower=-10,
        upper=10,
    )

    invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)

    # directives
    alpha0_ratio = np.r_[1, 1, 1, 1]
    alphas = directives.AlphasSmoothEstimate_ByEig(
        alpha0_ratio=alpha0_ratio, n_pw_iter=10, verbose=True
    )
    scales = directives.ScalingMultipleDataMisfits_ByEig(
        chi0_ratio=np.r_[1.0, 1.0], verbose=True, n_pw_iter=10
    )
    scaling_schedule = directives.JointScalingSchedule(verbose=True)
    beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-5, n_pw_iter=10)
    beta_schedule = directives.BetaSchedule(coolingFactor=5.0, coolingRate=1)
    targets = directives.MultiTargetMisfits(
        TriggerSmall=False,
        verbose=True,
    )

    # Setup Inversion
    inv = inversion.BaseInversion(
        invProb,
        directiveList=[alphas, scales, beta, targets, beta_schedule, scaling_schedule],
    )

    mtik = inv.run(minit)


    # Final Plot
    fig, axes = plt.subplots(3, 4, figsize=(25, 15))
    axes = axes.reshape(12)
    left, width = 0.25, 0.5
    bottom, height = 0.25, 0.5
    right = left + width
    top = bottom + height

    axes[0].set_axis_off()
    axes[0].text(
        0.5 * (left + right),
        0.5 * (bottom + top),
        ("Using true nonlinear\npetrophysical relationships"),
        horizontalalignment="center",
        verticalalignment="center",
        fontsize=20,
        color="black",
        transform=axes[0].transAxes,
    )

    axes[1].plot(mesh.cell_centers_x, wires.m1 * mcluster_map, "b.-", ms=5, marker="v")
    axes[1].plot(mesh.cell_centers_x, wires.m1 * m, "k--")
    axes[1].set_title("Problem 1")
    axes[1].legend(["Recovered Model", "True Model"], loc=1)
    axes[1].set_xlabel("X")
    axes[1].set_ylabel("Property 1")

    axes[2].plot(mesh.cell_centers_x, wires.m2 * mcluster_map, "r.-", ms=5, marker="v")
    axes[2].plot(mesh.cell_centers_x, wires.m2 * m, "k--")
    axes[2].set_title("Problem 2")
    axes[2].legend(["Recovered Model", "True Model"], loc=1)
    axes[2].set_xlabel("X")
    axes[2].set_ylabel("Property 2")

    x, y = np.mgrid[-1:1:0.01, -4:2:0.01]
    pos = np.empty(x.shape + (2,))
    pos[:, :, 0] = x
    pos[:, :, 1] = y
    CS = axes[3].contour(
        x,
        y,
        np.exp(clfmapping.score_samples(pos.reshape(-1, 2)).reshape(x.shape)),
        100,
        alpha=0.25,
        cmap="viridis",
    )
    axes[3].scatter(wires.m1 * mcluster_map, wires.m2 * mcluster_map, marker="v")
    axes[3].set_title("Petrophysical Distribution")
    CS.collections[0].set_label("")
    axes[3].legend(["True Petrophysical Distribution", "Recovered model crossplot"])
    axes[3].set_xlabel("Property 1")
    axes[3].set_ylabel("Property 2")

    axes[4].set_axis_off()
    axes[4].text(
        0.5 * (left + right),
        0.5 * (bottom + top),
        ("Using a pure\nGaussian distribution"),
        horizontalalignment="center",
        verticalalignment="center",
        fontsize=20,
        color="black",
        transform=axes[4].transAxes,
    )

    axes[5].plot(mesh.cell_centers_x, wires.m1 * mcluster_no_map, "b.-", ms=5, marker="v")
    axes[5].plot(mesh.cell_centers_x, wires.m1 * m, "k--")
    axes[5].set_title("Problem 1")
    axes[5].legend(["Recovered Model", "True Model"], loc=1)
    axes[5].set_xlabel("X")
    axes[5].set_ylabel("Property 1")

    axes[6].plot(mesh.cell_centers_x, wires.m2 * mcluster_no_map, "r.-", ms=5, marker="v")
    axes[6].plot(mesh.cell_centers_x, wires.m2 * m, "k--")
    axes[6].set_title("Problem 2")
    axes[6].legend(["Recovered Model", "True Model"], loc=1)
    axes[6].set_xlabel("X")
    axes[6].set_ylabel("Property 2")

    CSF = axes[7].contour(
        x,
        y,
        np.exp(clfmapping.score_samples(pos.reshape(-1, 2)).reshape(x.shape)),
        100,
        alpha=0.5,
        label="True Petro. Distribution",
    )
    CS = axes[7].contour(
        x,
        y,
        np.exp(clfnomapping.score_samples(pos.reshape(-1, 2)).reshape(x.shape)),
        500,
        cmap="viridis",
        linestyles="--",
        label="Modeled Petro. Distribution",
    )
    axes[7].scatter(
        wires.m1 * mcluster_no_map,
        wires.m2 * mcluster_no_map,
        marker="v",
        label="Recovered model crossplot",
    )
    axes[7].set_title("Petrophysical Distribution")
    axes[7].legend()
    axes[7].set_xlabel("Property 1")
    axes[7].set_ylabel("Property 2")

    # Tikonov

    axes[8].set_axis_off()
    axes[8].text(
        0.5 * (left + right),
        0.5 * (bottom + top),
        ("Least-Squares\n~Using a single cluster"),
        horizontalalignment="center",
        verticalalignment="center",
        fontsize=20,
        color="black",
        transform=axes[8].transAxes,
    )

    axes[9].plot(mesh.cell_centers_x, wires.m1 * mtik, "b.-", ms=5, marker="v")
    axes[9].plot(mesh.cell_centers_x, wires.m1 * m, "k--")
    axes[9].set_title("Problem 1")
    axes[9].legend(["Recovered Model", "True Model"], loc=1)
    axes[9].set_xlabel("X")
    axes[9].set_ylabel("Property 1")

    axes[10].plot(mesh.cell_centers_x, wires.m2 * mtik, "r.-", ms=5, marker="v")
    axes[10].plot(mesh.cell_centers_x, wires.m2 * m, "k--")
    axes[10].set_title("Problem 2")
    axes[10].legend(["Recovered Model", "True Model"], loc=1)
    axes[10].set_xlabel("X")
    axes[10].set_ylabel("Property 2")

    CS = axes[11].contour(
        x,
        y,
        np.exp(clfmapping.score_samples(pos.reshape(-1, 2)).reshape(x.shape)),
        100,
        alpha=0.25,
        cmap="viridis",
    )
    axes[11].scatter(wires.m1 * mtik, wires.m2 * mtik, marker="v")
    axes[11].set_title("Petro Distribution")
    CS.collections[0].set_label("")
    axes[11].legend(["True Petro Distribution", "Recovered model crossplot"])
    axes[11].set_xlabel("Property 1")
    axes[11].set_ylabel("Property 2")
    plt.subplots_adjust(wspace=0.3, hspace=0.3, top=0.85)
    plt.show()


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

   **Total running time of the script:** ( 0 minutes  52.884 seconds)

**Estimated memory usage:**  23 MB


.. _sphx_glr_download_content_examples_10-pgi_plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:

.. only:: html

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




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

      :download:`Download Python source code: plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py <plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py>`

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

      :download:`Download Jupyter notebook: plot_inv_1_PGI_Linear_1D_joint_WithRelationships.ipynb <plot_inv_1_PGI_Linear_1D_joint_WithRelationships.ipynb>`


.. only:: html

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

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