.. 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: [3.466585984471504, 0.0, 3.4976674770427723e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09601038 0.90398962]
    <class 'SimPEG.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09601038 0.90398962]
    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  1.92e+01  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 1059.2 (target 30.0 [False]); 83.6 (target 30.0 [False]) | smallness misfit: 2946.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [1059.2   83.6]; minimum progress targets: [240000. 240000.]
       1  1.92e+01  1.77e+02  4.12e+01  9.69e+02    8.94e+01      0              
    geophys. misfits: 482.6 (target 30.0 [False]); 29.2 (target 30.0 [True]) | smallness misfit: 1332.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [482.6  29.2]; minimum progress targets: [847.4  66.9]
    Updating scaling for data misfits by  1.0270003248200115
    New scales: [0.09834774 0.90165226]
       2  1.92e+01  7.38e+01  4.00e+01  8.43e+02    7.76e+01      0   Skip BFGS  
    geophys. misfits: 459.7 (target 30.0 [False]); 29.1 (target 30.0 [True]) | smallness misfit: 1312.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [459.7  29.1]; minimum progress targets: [386.1  30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.0314593904410994
    New scales: [0.10112882 0.89887118]
       3  9.60e+00  7.26e+01  4.01e+01  4.58e+02    8.40e+01      0   Skip BFGS  
    geophys. misfits: 167.3 (target 30.0 [False]); 24.6 (target 30.0 [True]) | smallness misfit: 1317.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [167.3  24.6]; minimum progress targets: [367.7  30. ]
    Updating scaling for data misfits by  1.219948708502077
    New scales: [0.1206875 0.8793125]
       4  9.60e+00  4.18e+01  4.25e+01  4.50e+02    7.39e+01      0              
    geophys. misfits: 143.5 (target 30.0 [False]); 25.0 (target 30.0 [True]) | smallness misfit: 1265.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [143.5  25. ]; minimum progress targets: [133.8  30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.1996666333296457
    New scales: [0.14137794 0.85862206]
       5  4.80e+00  4.18e+01  4.27e+01  2.47e+02    7.75e+01      0   Skip BFGS  
    geophys. misfits: 57.8 (target 30.0 [False]); 22.5 (target 30.0 [True]) | smallness misfit: 1372.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [57.8 22.5]; minimum progress targets: [114.8  30. ]
    Updating scaling for data misfits by  1.3342957626919507
    New scales: [0.18012683 0.81987317]
       6  4.80e+00  2.89e+01  4.46e+01  2.43e+02    7.44e+01      0              
    geophys. misfits: 43.0 (target 30.0 [False]); 22.4 (target 30.0 [True]) | smallness misfit: 1309.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [43.  22.4]; minimum progress targets: [46.3 30. ]
    Updating scaling for data misfits by  1.3375229024814719
    New scales: [0.22711581 0.77288419]
       7  4.80e+00  2.71e+01  4.48e+01  2.42e+02    7.33e+01      0              
    geophys. misfits: 32.2 (target 30.0 [False]); 22.4 (target 30.0 [True]) | smallness misfit: 1276.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [32.2 22.4]; minimum progress targets: [34.4 30. ]
    Updating scaling for data misfits by  1.3420194632227151
    New scales: [0.28282458 0.71717542]
       8  4.80e+00  2.51e+01  4.52e+01  2.42e+02    7.54e+01      0              
    geophys. misfits: 26.0 (target 30.0 [True]); 22.4 (target 30.0 [True]) | smallness misfit: 1250.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [26.  22.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.2449219097000142
       9  4.80e+00  2.34e+01  4.61e+01  2.45e+02    6.14e+01      0              
    geophys. misfits: 25.3 (target 30.0 [True]); 22.8 (target 30.0 [True]) | smallness misfit: 1187.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.3 22.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.5573370442568115
      10  4.80e+00  2.35e+01  4.69e+01  2.49e+02    4.67e+01      0              
    geophys. misfits: 24.3 (target 30.0 [True]); 23.4 (target 30.0 [True]) | smallness misfit: 1106.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.3 23.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.95922803198821
      11  4.80e+00  2.36e+01  4.78e+01  2.53e+02    3.48e+01      0              
    geophys. misfits: 23.6 (target 30.0 [True]); 24.0 (target 30.0 [True]) | smallness misfit: 1025.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [23.6 24. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  2.467445418444921
      12  4.80e+00  2.39e+01  4.88e+01  2.58e+02    3.67e+01      0              
    geophys. misfits: 23.7 (target 30.0 [True]); 24.7 (target 30.0 [True]) | smallness misfit: 953.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [23.7 24.7]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  3.0632447674595578
      13  4.80e+00  2.44e+01  4.99e+01  2.64e+02    6.00e+01      0              
    geophys. misfits: 22.7 (target 30.0 [True]); 25.4 (target 30.0 [True]) | smallness misfit: 880.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.7 25.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  3.8375817995772943
      14  4.80e+00  2.46e+01  5.13e+01  2.71e+02    7.30e+01      0              
    geophys. misfits: 22.0 (target 30.0 [True]); 26.1 (target 30.0 [True]) | smallness misfit: 804.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.  26.1]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  4.821066325745199
      15  4.80e+00  2.49e+01  5.29e+01  2.79e+02    8.00e+01      0              
    geophys. misfits: 22.7 (target 30.0 [True]); 27.1 (target 30.0 [True]) | smallness misfit: 733.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.7 27.1]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  5.860579033410637
      16  4.80e+00  2.58e+01  5.43e+01  2.87e+02    8.09e+01      0              
    geophys. misfits: 22.2 (target 30.0 [True]); 27.8 (target 30.0 [True]) | smallness misfit: 678.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.2 27.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  7.1189553468605435
      17  4.80e+00  2.62e+01  5.61e+01  2.96e+02    8.29e+01      0              
    geophys. misfits: 22.2 (target 30.0 [True]); 28.5 (target 30.0 [True]) | smallness misfit: 615.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.2 28.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  8.541098513892216
      18  4.80e+00  2.68e+01  5.78e+01  3.04e+02    9.42e+01      0              
    geophys. misfits: 23.5 (target 30.0 [True]); 29.9 (target 30.0 [True]) | smallness misfit: 562.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [23.5 29.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  9.737412991924352
      19  4.80e+00  2.81e+01  5.89e+01  3.11e+02    8.32e+01      0              
    geophys. misfits: 25.7 (target 30.0 [True]); 30.7 (target 30.0 [False]) | smallness misfit: 514.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.7 30.7]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.1690573002451738
    New scales: [0.25224184 0.74775816]
      20  2.40e+00  2.94e+01  5.85e+01  1.70e+02    9.94e+01      0              
    geophys. misfits: 19.6 (target 30.0 [True]); 25.8 (target 30.0 [True]) | smallness misfit: 543.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [19.6 25.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  13.08986157124665
      21  2.40e+00  2.43e+01  6.40e+01  1.78e+02    8.59e+01      0              
    geophys. misfits: 18.4 (target 30.0 [True]); 25.5 (target 30.0 [True]) | smallness misfit: 490.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [18.4 25.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  18.363780573348887
      22  2.40e+00  2.37e+01  6.98e+01  1.91e+02    1.01e+02      0              
    geophys. misfits: 17.6 (target 30.0 [True]); 24.7 (target 30.0 [True]) | smallness misfit: 442.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [17.6 24.7]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  26.851793413284888
      23  2.40e+00  2.29e+01  7.80e+01  2.10e+02    1.08e+02      0              
    geophys. misfits: 17.3 (target 30.0 [True]); 23.8 (target 30.0 [True]) | smallness misfit: 391.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [17.3 23.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  40.17971519844774
      24  2.40e+00  2.22e+01  8.94e+01  2.37e+02    1.15e+02      0              
    geophys. misfits: 17.3 (target 30.0 [True]); 24.3 (target 30.0 [True]) | smallness misfit: 345.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [17.3 24.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  59.741920800674485
      25  2.40e+00  2.25e+01  1.03e+02  2.70e+02    1.18e+02      0              
    geophys. misfits: 21.1 (target 30.0 [True]); 23.0 (target 30.0 [True]) | smallness misfit: 301.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.1 23. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  81.35391606203183
      26  2.40e+00  2.26e+01  1.16e+02  3.02e+02    1.19e+02      0              
    geophys. misfits: 22.9 (target 30.0 [True]); 24.4 (target 30.0 [True]) | smallness misfit: 291.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.9 24.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  103.18660220854213
      27  2.40e+00  2.41e+01  1.29e+02  3.33e+02    1.22e+02      0              
    geophys. misfits: 25.2 (target 30.0 [True]); 23.9 (target 30.0 [True]) | smallness misfit: 225.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.2 23.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  126.07099988334166
      28  2.40e+00  2.42e+01  1.37e+02  3.54e+02    1.24e+02      0              
    geophys. misfits: 39.6 (target 30.0 [False]); 28.4 (target 30.0 [True]) | smallness misfit: 193.3 (target: 200.0 [True])
    Beta cooling evaluation: progress: [39.6 28.4]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.055587985138429
    New scales: [0.26258164 0.73741836]
      29  1.20e+00  3.14e+01  1.28e+02  1.85e+02    1.23e+02      0              
    geophys. misfits: 26.1 (target 30.0 [True]); 28.3 (target 30.0 [True]) | smallness misfit: 206.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [26.1 28.3]; minimum progress targets: [31.7 30. ]
    Warming alpha_pgi to favor clustering:  139.25290145897227
      30  1.20e+00  2.77e+01  1.32e+02  1.86e+02    1.16e+02      0              
    geophys. misfits: 25.2 (target 30.0 [True]); 26.5 (target 30.0 [True]) | smallness misfit: 193.1 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [25.2 26.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  161.7783957915216
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 7.0273e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 1.1632e+02 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 1.1632e+02 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =     31
    ------------------------- 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: [0.0003449902502617977, 0.0, 3.7949084087632503e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09601038 0.90398962]
    <class 'SimPEG.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09601038 0.90398962]
    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  1.91e+03  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 87797.2 (target 30.0 [False]); 62285.6 (target 30.0 [False]) | smallness misfit: 275.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [87797.2 62285.6]; minimum progress targets: [240000. 240000.]
       1  1.91e+03  6.47e+04  6.69e-01  6.60e+04    9.27e+01      0              
    geophys. misfits: 627.9 (target 30.0 [False]); 27.8 (target 30.0 [True]) | smallness misfit: 124.8 (target: 200.0 [True])
    Beta cooling evaluation: progress: [627.9  27.8]; minimum progress targets: [70237.8 49828.5]
    Updating scaling for data misfits by  1.0777978985257075
    New scales: [0.10271258 0.89728742]
       2  1.91e+03  8.95e+01  3.37e-01  7.34e+02    7.85e+01      0   Skip BFGS  
    geophys. misfits: 111.3 (target 30.0 [False]); 26.1 (target 30.0 [True]) | smallness misfit: 47.8 (target: 200.0 [True])
    Beta cooling evaluation: progress: [111.3  26.1]; minimum progress targets: [502.3  30. ]
    Updating scaling for data misfits by  1.149353459035614
    New scales: [0.11626944 0.88373056]
       3  1.91e+03  3.60e+01  1.32e-01  2.89e+02    8.30e+01      0   Skip BFGS  
    geophys. misfits: 98.0 (target 30.0 [False]); 24.5 (target 30.0 [True]) | smallness misfit: 51.5 (target: 200.0 [True])
    Beta cooling evaluation: progress: [98.  24.5]; minimum progress targets: [89. 30.]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.2240032601152608
    New scales: [0.13870172 0.86129828]
       4  9.57e+02  3.47e+01  1.35e-01  1.64e+02    9.86e+01      0              
    geophys. misfits: 36.3 (target 30.0 [False]); 21.9 (target 30.0 [True]) | smallness misfit: 49.3 (target: 200.0 [True])
    Beta cooling evaluation: progress: [36.3 21.9]; minimum progress targets: [78.4 30. ]
    Updating scaling for data misfits by  1.3684205634446134
    New scales: [0.18057481 0.81942519]
       5  9.57e+02  2.45e+01  1.33e-01  1.52e+02    5.30e+01      0              
    geophys. misfits: 28.9 (target 30.0 [True]); 22.1 (target 30.0 [True]) | smallness misfit: 49.7 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [28.9 22.1]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.1972365675938115
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 6.2925e-02 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 5.3045e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 5.3045e+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: [3.5188574849327566e-05, 0.0, 3.4826374354602616e-05, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09601038 0.90398962]
    /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:  [0.09601038 0.90398962]
    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  1.04e+06  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 56377.6 (target 30.0 [False]); 35775.8 (target 30.0 [False])
       1  2.08e+05  3.78e+04  4.29e-02  4.67e+04    1.38e+02      0              
    geophys. misfits: 8375.2 (target 30.0 [False]); 3857.2 (target 30.0 [False])
       2  4.15e+04  4.29e+03  1.07e-01  8.73e+03    1.31e+02      0   Skip BFGS  
    geophys. misfits: 563.0 (target 30.0 [False]); 256.2 (target 30.0 [False])
       3  8.30e+03  2.86e+02  1.41e-01  1.46e+03    1.01e+02      0   Skip BFGS  
    geophys. misfits: 42.6 (target 30.0 [False]); 38.2 (target 30.0 [False])
       4  1.66e+03  3.86e+01  1.51e-01  2.90e+02    7.91e+01      0   Skip BFGS  
    geophys. misfits: 17.6 (target 30.0 [True]); 22.1 (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| = 4.8127e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 7.9019e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 7.9019e+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 34.045 seconds)

**Estimated memory usage:**  8 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

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