.. 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: [3.466585984471505, 0.0, 3.497667477042771e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.10253344 0.89746656]
    <class 'SimPEG.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.10253344 0.89746656]
    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  1.97e+01  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 976.3 (target 30.0 [False]); 75.5 (target 30.0 [False]) | smallness misfit: 2987.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [976.3  75.5]; minimum progress targets: [240000. 240000.]
       1  1.97e+01  1.68e+02  4.11e+01  9.79e+02    9.62e+01      0              
    geophys. misfits: 450.0 (target 30.0 [False]); 17.6 (target 30.0 [True]) | smallness misfit: 1375.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [450.   17.6]; minimum progress targets: [781.   60.4]
    Updating scaling for data misfits by  1.708075361874646
    New scales: [0.16328045 0.83671955]
       2  1.97e+01  8.82e+01  3.98e+01  8.73e+02    8.12e+01      0   Skip BFGS  
    geophys. misfits: 233.8 (target 30.0 [False]); 17.3 (target 30.0 [True]) | smallness misfit: 1264.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [233.8  17.3]; minimum progress targets: [360.  30.]
    Updating scaling for data misfits by  1.73480762458182
    New scales: [0.25291544 0.74708456]
       3  1.97e+01  7.20e+01  4.12e+01  8.84e+02    7.80e+01      0              
    geophys. misfits: 128.9 (target 30.0 [False]); 17.6 (target 30.0 [True]) | smallness misfit: 1189.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [128.9  17.6]; minimum progress targets: [187.  30.]
    Updating scaling for data misfits by  1.7020492337099165
    New scales: [0.36556512 0.63443488]
       4  1.97e+01  5.83e+01  4.22e+01  8.91e+02    7.70e+01      0              
    geophys. misfits: 80.8 (target 30.0 [False]); 18.7 (target 30.0 [True]) | smallness misfit: 1131.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [80.8 18.7]; minimum progress targets: [103.1  30. ]
    Updating scaling for data misfits by  1.6069709546804007
    New scales: [0.48077468 0.51922532]
       5  1.97e+01  4.85e+01  4.29e+01  8.95e+02    7.39e+01      0   Skip BFGS  
    geophys. misfits: 58.9 (target 30.0 [False]); 20.5 (target 30.0 [True]) | smallness misfit: 1083.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [58.9 20.5]; minimum progress targets: [64.6 30. ]
    Updating scaling for data misfits by  1.4662958220227225
    New scales: [0.5758598 0.4241402]
       6  1.97e+01  4.26e+01  4.33e+01  8.97e+02    6.95e+01      0   Skip BFGS  
    geophys. misfits: 48.7 (target 30.0 [False]); 22.9 (target 30.0 [True]) | smallness misfit: 1044.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [48.7 22.9]; minimum progress targets: [47.1 30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.3128511097465423
    New scales: [0.64060739 0.35939261]
       7  9.87e+00  3.94e+01  4.35e+01  4.69e+02    9.23e+01      0   Skip BFGS  
    geophys. misfits: 25.6 (target 30.0 [True]); 16.8 (target 30.0 [True]) | smallness misfit: 1100.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.6 16.8]; minimum progress targets: [39. 30.]
    Warming alpha_pgi to favor clustering:  1.4784430570260136
       8  9.87e+00  2.24e+01  4.58e+01  4.74e+02    4.17e+01      0              
    geophys. misfits: 25.4 (target 30.0 [True]); 20.2 (target 30.0 [True]) | smallness misfit: 985.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.4 20.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.9732502654200745
       9  9.87e+00  2.35e+01  4.67e+01  4.84e+02    7.70e+01      0              
    geophys. misfits: 25.2 (target 30.0 [True]); 23.5 (target 30.0 [True]) | smallness misfit: 898.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.2 23.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  2.4299925361475583
      10  9.87e+00  2.46e+01  4.74e+01  4.93e+02    7.43e+01      0   Skip BFGS  
    geophys. misfits: 25.1 (target 30.0 [True]); 26.7 (target 30.0 [True]) | smallness misfit: 832.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.1 26.7]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  2.817109952670229
      11  9.87e+00  2.57e+01  4.80e+01  4.99e+02    5.60e+01      0   Skip BFGS  
    geophys. misfits: 25.1 (target 30.0 [True]); 29.3 (target 30.0 [True]) | smallness misfit: 785.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.1 29.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  3.129877772141576
      12  9.87e+00  2.66e+01  4.84e+01  5.04e+02    3.63e+01      0   Skip BFGS  
    geophys. misfits: 25.0 (target 30.0 [True]); 31.5 (target 30.0 [False]) | smallness misfit: 752.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.  31.5]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.198529568044168
    New scales: [0.59794405 0.40205595]
      13  4.93e+00  2.76e+01  4.83e+01  2.66e+02    8.76e+01      0   Skip BFGS  
    geophys. misfits: 19.8 (target 30.0 [True]); 18.3 (target 30.0 [True]) | smallness misfit: 855.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [19.8 18.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  4.9358512209031025
      14  4.93e+00  1.92e+01  5.28e+01  2.80e+02    6.23e+01      0              
    geophys. misfits: 19.9 (target 30.0 [True]); 23.1 (target 30.0 [True]) | smallness misfit: 708.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [19.9 23.1]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  6.922849426719274
      15  4.93e+00  2.12e+01  5.52e+01  2.94e+02    6.80e+01      0              
    geophys. misfits: 20.5 (target 30.0 [True]); 29.7 (target 30.0 [True]) | smallness misfit: 600.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [20.5 29.7]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  8.557932009373166
      16  4.93e+00  2.42e+01  5.65e+01  3.03e+02    8.55e+01      0              
    geophys. misfits: 20.9 (target 30.0 [True]); 34.2 (target 30.0 [False]) | smallness misfit: 531.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [20.9 34.2]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.4360536637893107
    New scales: [0.50875087 0.49124913]
      17  2.47e+00  2.74e+01  5.60e+01  1.65e+02    9.40e+01      0   Skip BFGS  
    geophys. misfits: 18.8 (target 30.0 [True]); 18.8 (target 30.0 [True]) | smallness misfit: 633.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [18.8 18.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  13.647381256803174
      18  2.47e+00  1.88e+01  6.51e+01  1.79e+02    8.74e+01      0              
    geophys. misfits: 19.6 (target 30.0 [True]); 23.8 (target 30.0 [True]) | smallness misfit: 483.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [19.6 23.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  19.045404598769014
      19  2.47e+00  2.17e+01  6.88e+01  1.91e+02    8.65e+01      0              
    geophys. misfits: 20.5 (target 30.0 [True]); 25.7 (target 30.0 [True]) | smallness misfit: 407.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [20.5 25.7]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  25.045740275743086
      20  2.47e+00  2.31e+01  7.29e+01  2.03e+02    9.08e+01      0   Skip BFGS  
    geophys. misfits: 21.0 (target 30.0 [True]); 27.2 (target 30.0 [True]) | smallness misfit: 339.1 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.  27.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  31.722142767214407
      21  2.47e+00  2.40e+01  7.68e+01  2.13e+02    9.80e+01      0              
    geophys. misfits: 22.2 (target 30.0 [True]); 29.0 (target 30.0 [True]) | smallness misfit: 300.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.2 29. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  37.82019981845879
      22  2.47e+00  2.56e+01  7.96e+01  2.22e+02    1.01e+02      0              
    geophys. misfits: 23.1 (target 30.0 [True]); 29.2 (target 30.0 [True]) | smallness misfit: 244.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [23.1 29.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  43.95066985849938
      23  2.47e+00  2.61e+01  8.15e+01  2.27e+02    9.91e+01      0              
    geophys. misfits: 23.7 (target 30.0 [True]); 31.7 (target 30.0 [False]) | smallness misfit: 231.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [23.7 31.7]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.2665719494814376
    New scales: [0.44984254 0.55015746]
      24  1.23e+00  2.81e+01  8.06e+01  1.28e+02    1.07e+02      0              
    geophys. misfits: 21.3 (target 30.0 [True]); 19.0 (target 30.0 [True]) | smallness misfit: 257.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.3 19. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  65.70376545749437
      25  1.23e+00  2.00e+01  9.68e+01  1.39e+02    1.05e+02      0              
    geophys. misfits: 22.3 (target 30.0 [True]); 16.9 (target 30.0 [True]) | smallness misfit: 233.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.3 16.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  102.4300759001624
      26  1.23e+00  1.93e+01  1.14e+02  1.60e+02    1.10e+02      0              
    geophys. misfits: 22.6 (target 30.0 [True]); 16.1 (target 30.0 [True]) | smallness misfit: 202.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.6 16.1]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  163.69186464532837
      27  1.23e+00  1.90e+01  1.40e+02  1.92e+02    1.20e+02      0              
    geophys. misfits: 23.7 (target 30.0 [True]); 15.5 (target 30.0 [True]) | smallness misfit: 187.1 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [23.7 15.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  262.2468013455045
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 3.5166e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 1.1983e+02 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 1.1983e+02 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =     28
    ------------------------- 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: [0.00034499025026179766, 0.0, 3.7949084087632494e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.10253344 0.89746656]
    <class 'SimPEG.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.10253344 0.89746656]
    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  1.97e+03  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 84899.9 (target 30.0 [False]); 63769.0 (target 30.0 [False]) | smallness misfit: 307.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [84899.9 63769. ]; minimum progress targets: [240000. 240000.]
       1  1.97e+03  6.59e+04  7.14e-01  6.73e+04    8.95e+01      0              
    geophys. misfits: 561.9 (target 30.0 [False]); 14.5 (target 30.0 [True]) | smallness misfit: 75.2 (target: 200.0 [True])
    Beta cooling evaluation: progress: [561.9  14.5]; minimum progress targets: [67919.9 51015.2]
    Updating scaling for data misfits by  2.0742079849535013
    New scales: [0.19157515 0.80842485]
       2  1.97e+03  1.19e+02  1.99e-01  5.10e+02    1.12e+02      0   Skip BFGS  
    geophys. misfits: 52.4 (target 30.0 [False]); 14.5 (target 30.0 [True]) | smallness misfit: 50.4 (target: 200.0 [True])
    Beta cooling evaluation: progress: [52.4 14.5]; minimum progress targets: [449.5  30. ]
    Updating scaling for data misfits by  2.0657867057786627
    New scales: [0.32865019 0.67134981]
       3  1.97e+03  2.70e+01  1.41e-01  3.04e+02    8.52e+01      0   Skip BFGS  
    geophys. misfits: 32.2 (target 30.0 [False]); 15.0 (target 30.0 [True]) | smallness misfit: 59.6 (target: 200.0 [True])
    Beta cooling evaluation: progress: [32.2 15. ]; minimum progress targets: [42. 30.]
    Updating scaling for data misfits by  2.005372276424505
    New scales: [0.49538346 0.50461654]
       4  1.97e+03  2.35e+01  1.68e-01  3.54e+02    8.94e+01      0              
    geophys. misfits: 25.4 (target 30.0 [True]); 17.0 (target 30.0 [True]) | smallness misfit: 48.3 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [25.4 17. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.47406056633217
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 1.6960e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 8.9403e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 8.9403e+01 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =      5
    ------------------------- 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: [3.5188574849327586e-05, 0.0, 3.4826374354602616e-05, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.10253344 0.89746656]
    /home/vsts/work/1/s/SimPEG/directives/directives.py:332: UserWarning:

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

    Initial data misfit scales:  [0.10253344 0.89746656]
    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  1.07e+06  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 53866.9 (target 30.0 [False]); 36983.1 (target 30.0 [False])
       1  2.13e+05  3.87e+04  4.18e-02  4.76e+04    1.38e+02      0              
    geophys. misfits: 7708.0 (target 30.0 [False]); 4065.3 (target 30.0 [False])
       2  4.27e+04  4.44e+03  1.06e-01  8.95e+03    1.31e+02      0   Skip BFGS  
    geophys. misfits: 509.1 (target 30.0 [False]); 260.0 (target 30.0 [False])
       3  8.53e+03  2.86e+02  1.40e-01  1.48e+03    1.05e+02      0   Skip BFGS  
    geophys. misfits: 42.8 (target 30.0 [False]); 27.0 (target 30.0 [True])
    Updating scaling for data misfits by  1.109305984673468
    New scales: [0.11248033 0.88751967]
       4  1.71e+03  2.88e+01  1.51e-01  2.86e+02    8.56e+01      0   Skip BFGS  
    geophys. misfits: 18.4 (target 30.0 [True]); 10.7 (target 30.0 [True])
    All targets have been reached
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 4.3953e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 8.5526e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 8.5526e+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:: Python


    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, weights={"cell_weights": wr1}
    )

    reg2 = regularization.WeightedLeastSquares(
        mesh, alpha_s=1.0, alpha_x=1.0, mapping=wires.m2, weights={"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 55.403 seconds)

**Estimated memory usage:**  17 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-jupyter

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

    .. 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>`


.. only:: html

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

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