.. 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


    Running inversion with SimPEG v0.22.0
    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: [3.5111974374988844, 0.0, 3.4802923651593253e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09429751 0.90570249]
    <class 'simpeg.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09429751 0.90570249]
    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+01  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 1079.1 (target 30.0 [False]); 83.7 (target 30.0 [False]) | smallness misfit: 2929.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [1079.1   83.7]; minimum progress targets: [240000. 240000.]
       1  1.91e+01  1.78e+02  4.14e+01  9.69e+02    8.21e+01      0              
    geophys. misfits: 482.7 (target 30.0 [False]); 28.2 (target 30.0 [True]) | smallness misfit: 1350.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [482.7  28.2]; minimum progress targets: [863.3  67. ]
    Updating scaling for data misfits by  1.0620202907137457
    New scales: [0.09956358 0.90043642]
       2  1.91e+01  7.35e+01  4.04e+01  8.45e+02    3.73e+01      0   Skip BFGS  
    geophys. misfits: 439.6 (target 30.0 [False]); 28.1 (target 30.0 [True]) | smallness misfit: 1319.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [439.6  28.1]; minimum progress targets: [386.2  30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.0663187291080125
    New scales: [0.1054701 0.8945299]
       3  9.56e+00  7.15e+01  4.06e+01  4.59e+02    8.41e+01      0   Skip BFGS  
    geophys. misfits: 148.9 (target 30.0 [False]); 22.7 (target 30.0 [True]) | smallness misfit: 1358.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [148.9  22.7]; minimum progress targets: [351.7  30. ]
    Updating scaling for data misfits by  1.3196861608421822
    New scales: [0.13464748 0.86535252]
       4  9.56e+00  3.97e+01  4.30e+01  4.51e+02    5.69e+01      0              
    geophys. misfits: 118.6 (target 30.0 [False]); 23.5 (target 30.0 [True]) | smallness misfit: 1290.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [118.6  23.5]; minimum progress targets: [119.1  30. ]
    Updating scaling for data misfits by  1.2791257490712276
    New scales: [0.16599247 0.83400753]
       5  9.56e+00  3.92e+01  4.33e+01  4.53e+02    6.36e+01      0   Skip BFGS  
    geophys. misfits: 81.1 (target 30.0 [False]); 23.1 (target 30.0 [True]) | smallness misfit: 1275.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [81.1 23.1]; minimum progress targets: [94.9 30. ]
    Updating scaling for data misfits by  1.298685061013985
    New scales: [0.20538886 0.79461114]
       6  9.56e+00  3.50e+01  4.39e+01  4.55e+02    5.61e+01      0              
    geophys. misfits: 63.3 (target 30.0 [False]); 23.2 (target 30.0 [True]) | smallness misfit: 1244.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [63.3 23.2]; minimum progress targets: [64.9 30. ]
    Updating scaling for data misfits by  1.2926615014373246
    New scales: [0.25044422 0.74955578]
       7  9.56e+00  3.33e+01  4.42e+01  4.56e+02    5.75e+01      0   Skip BFGS  
    geophys. misfits: 53.0 (target 30.0 [False]); 23.7 (target 30.0 [True]) | smallness misfit: 1219.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [53.  23.7]; minimum progress targets: [50.6 30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.263540875307549
    New scales: [0.2968535 0.7031465]
       8  4.78e+00  3.24e+01  4.44e+01  2.45e+02    7.61e+01      0              
    geophys. misfits: 28.9 (target 30.0 [True]); 20.9 (target 30.0 [True]) | smallness misfit: 1315.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [28.9 20.9]; minimum progress targets: [42.4 30. ]
    Warming alpha_pgi to favor clustering:  1.2344826360406898
       9  4.78e+00  2.33e+01  4.63e+01  2.45e+02    7.09e+01      0              
    geophys. misfits: 28.2 (target 30.0 [True]); 21.4 (target 30.0 [True]) | smallness misfit: 1242.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [28.2 21.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.5218665825639803
      10  4.78e+00  2.34e+01  4.69e+01  2.48e+02    6.06e+01      0              
    geophys. misfits: 26.3 (target 30.0 [True]); 21.9 (target 30.0 [True]) | smallness misfit: 1152.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [26.3 21.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.908281988237571
      11  4.78e+00  2.32e+01  4.78e+01  2.52e+02    4.72e+01      0              
    geophys. misfits: 25.4 (target 30.0 [True]); 22.5 (target 30.0 [True]) | smallness misfit: 1073.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.4 22.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  2.400424867767163
      12  4.78e+00  2.33e+01  4.89e+01  2.57e+02    3.17e+01      0              
    geophys. misfits: 24.9 (target 30.0 [True]); 23.2 (target 30.0 [True]) | smallness misfit: 987.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.9 23.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  2.9976387400152387
      13  4.78e+00  2.37e+01  5.00e+01  2.63e+02    5.81e+01      0              
    geophys. misfits: 24.5 (target 30.0 [True]); 24.1 (target 30.0 [True]) | smallness misfit: 912.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.5 24.1]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  3.6965455513930157
      14  4.78e+00  2.42e+01  5.12e+01  2.69e+02    6.56e+01      0              
    geophys. misfits: 24.9 (target 30.0 [True]); 25.1 (target 30.0 [True]) | smallness misfit: 838.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.9 25.1]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  4.4339769943829594
      15  4.78e+00  2.50e+01  5.23e+01  2.75e+02    7.87e+01      0              
    geophys. misfits: 24.5 (target 30.0 [True]); 25.7 (target 30.0 [True]) | smallness misfit: 782.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.5 25.7]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  5.305715646440647
      16  4.78e+00  2.53e+01  5.37e+01  2.82e+02    7.23e+01      0              
    geophys. misfits: 24.8 (target 30.0 [True]); 26.8 (target 30.0 [True]) | smallness misfit: 702.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.8 26.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  6.174049240549029
      17  4.78e+00  2.62e+01  5.47e+01  2.87e+02    7.78e+01      0              
    geophys. misfits: 25.0 (target 30.0 [True]); 27.5 (target 30.0 [True]) | smallness misfit: 651.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.  27.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  7.077074939554095
      18  4.78e+00  2.67e+01  5.58e+01  2.93e+02    5.73e+01      0   Skip BFGS  
    geophys. misfits: 25.4 (target 30.0 [True]); 28.2 (target 30.0 [True]) | smallness misfit: 607.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.4 28.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  7.954968826192312
      19  4.78e+00  2.73e+01  5.68e+01  2.99e+02    6.48e+01      0   Skip BFGS  
    geophys. misfits: 25.6 (target 30.0 [True]); 28.8 (target 30.0 [True]) | smallness misfit: 569.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.6 28.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  8.798348914444245
      20  4.78e+00  2.79e+01  5.77e+01  3.03e+02    6.53e+01      0   Skip BFGS  
    geophys. misfits: 26.2 (target 30.0 [True]); 29.4 (target 30.0 [True]) | smallness misfit: 538.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [26.2 29.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  9.533408118000374
      21  4.78e+00  2.84e+01  5.84e+01  3.07e+02    7.85e+01      0   Skip BFGS  
    geophys. misfits: 26.4 (target 30.0 [True]); 30.1 (target 30.0 [False]) | smallness misfit: 515.1 (target: 200.0 [False])
    Beta cooling evaluation: progress: [26.4 30.1]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.1375763963523058
    New scales: [0.27066988 0.72933012]
      22  2.39e+00  2.91e+01  5.82e+01  1.68e+02    8.85e+01      0              
    geophys. misfits: 23.7 (target 30.0 [True]); 24.9 (target 30.0 [True]) | smallness misfit: 564.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [23.7 24.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  11.784831908155274
      23  2.39e+00  2.46e+01  6.23e+01  1.73e+02    8.44e+01      0              
    geophys. misfits: 23.1 (target 30.0 [True]); 24.3 (target 30.0 [True]) | smallness misfit: 518.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [23.1 24.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  14.931368471700454
      24  2.39e+00  2.40e+01  6.58e+01  1.81e+02    9.30e+01      0              
    geophys. misfits: 21.8 (target 30.0 [True]); 23.9 (target 30.0 [True]) | smallness misfit: 491.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.8 23.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  19.650374200956737
      25  2.39e+00  2.33e+01  7.09e+01  1.93e+02    1.02e+02      0              
    geophys. misfits: 21.8 (target 30.0 [True]); 22.7 (target 30.0 [True]) | smallness misfit: 443.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.8 22.7]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  26.510066720861296
      26  2.39e+00  2.25e+01  7.74e+01  2.08e+02    1.03e+02      0              
    geophys. misfits: 21.3 (target 30.0 [True]); 22.4 (target 30.0 [True]) | smallness misfit: 415.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.3 22.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  36.37715172737612
      27  2.39e+00  2.21e+01  8.60e+01  2.28e+02    1.13e+02      0              
    geophys. misfits: 23.2 (target 30.0 [True]); 22.3 (target 30.0 [True]) | smallness misfit: 360.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [23.2 22.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  48.00309361234333
      28  2.39e+00  2.25e+01  9.42e+01  2.47e+02    1.14e+02      0              
    geophys. misfits: 22.5 (target 30.0 [True]); 22.6 (target 30.0 [True]) | smallness misfit: 351.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.5 22.6]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  63.879518397111084
      29  2.39e+00  2.26e+01  1.05e+02  2.74e+02    1.19e+02      0              
    geophys. misfits: 27.4 (target 30.0 [True]); 22.3 (target 30.0 [True]) | smallness misfit: 306.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.4 22.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  78.02898057143261
      30  2.39e+00  2.36e+01  1.12e+02  2.92e+02    1.27e+02      0              
    geophys. misfits: 24.6 (target 30.0 [True]); 23.0 (target 30.0 [True]) | smallness misfit: 278.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.6 23. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  98.43114270782824
      31  2.39e+00  2.34e+01  1.21e+02  3.13e+02    1.19e+02      0              
    geophys. misfits: 36.6 (target 30.0 [False]); 25.6 (target 30.0 [True]) | smallness misfit: 226.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [36.6 25.6]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.1732718933185409
    New scales: [0.30334275 0.69665725]
      32  1.19e+00  2.89e+01  1.16e+02  1.67e+02    1.18e+02      0              
    geophys. misfits: 27.1 (target 30.0 [True]); 25.5 (target 30.0 [True]) | smallness misfit: 213.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.1 25.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  112.27045537462944
      33  1.19e+00  2.60e+01  1.21e+02  1.71e+02    1.14e+02      0              
    geophys. misfits: 28.2 (target 30.0 [True]); 24.3 (target 30.0 [True]) | smallness misfit: 211.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [28.2 24.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  129.08852518299622
      34  1.19e+00  2.55e+01  1.28e+02  1.78e+02    1.10e+02      0              
    geophys. misfits: 29.2 (target 30.0 [True]); 25.6 (target 30.0 [True]) | smallness misfit: 194.5 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [29.2 25.6]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  142.12597606212722
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 5.0874e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 1.1035e+02 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 1.1035e+02 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =     35
    ------------------------- DONE! -------------------------

    Running inversion with SimPEG v0.22.0
    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: [0.00035290147269136, 0.0, 3.5122265027259625e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09429751 0.90570249]
    <class 'simpeg.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09429751 0.90570249]
    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: 88746.2 (target 30.0 [False]); 62261.7 (target 30.0 [False]) | smallness misfit: 272.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [88746.2 62261.7]; minimum progress targets: [240000. 240000.]
       1  1.91e+03  6.48e+04  6.57e-01  6.60e+04    9.27e+01      0              
    geophys. misfits: 642.5 (target 30.0 [False]); 24.1 (target 30.0 [True]) | smallness misfit: 112.7 (target: 200.0 [True])
    Beta cooling evaluation: progress: [642.5  24.1]; minimum progress targets: [70996.9 49809.3]
    Updating scaling for data misfits by  1.2471548335361147
    New scales: [0.11492514 0.88507486]
       2  1.91e+03  9.51e+01  3.23e-01  7.14e+02    8.44e+01      0   Skip BFGS  
    geophys. misfits: 107.4 (target 30.0 [False]); 22.3 (target 30.0 [True]) | smallness misfit: 61.3 (target: 200.0 [True])
    Beta cooling evaluation: progress: [107.4  22.3]; minimum progress targets: [514.  30.]
    Updating scaling for data misfits by  1.3427652544537596
    New scales: [0.14846894 0.85153106]
       3  1.91e+03  3.50e+01  1.53e-01  3.28e+02    1.05e+02      0   Skip BFGS  
    geophys. misfits: 76.3 (target 30.0 [False]); 21.4 (target 30.0 [True]) | smallness misfit: 46.6 (target: 200.0 [True])
    Beta cooling evaluation: progress: [76.3 21.4]; minimum progress targets: [85.9 30. ]
    Updating scaling for data misfits by  1.3995646935052177
    New scales: [0.19615537 0.80384463]
       4  1.91e+03  3.22e+01  1.27e-01  2.75e+02    7.57e+01      0              
    geophys. misfits: 56.0 (target 30.0 [False]); 22.0 (target 30.0 [True]) | smallness misfit: 47.4 (target: 200.0 [True])
    Beta cooling evaluation: progress: [56. 22.]; minimum progress targets: [61. 30.]
    Updating scaling for data misfits by  1.3642761578592137
    New scales: [0.24976333 0.75023667]
       5  1.91e+03  3.05e+01  1.29e-01  2.76e+02    7.88e+01      0   Skip BFGS  
    geophys. misfits: 44.7 (target 30.0 [False]); 22.5 (target 30.0 [True]) | smallness misfit: 47.9 (target: 200.0 [True])
    Beta cooling evaluation: progress: [44.7 22.5]; minimum progress targets: [44.8 30. ]
    Updating scaling for data misfits by  1.3358205199459254
    New scales: [0.30782035 0.69217965]
       6  1.91e+03  2.93e+01  1.30e-01  2.77e+02    7.89e+01      0   Skip BFGS  
    geophys. misfits: 38.0 (target 30.0 [False]); 23.0 (target 30.0 [True]) | smallness misfit: 48.3 (target: 200.0 [True])
    Beta cooling evaluation: progress: [38. 23.]; minimum progress targets: [35.7 30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.3017109355198204
    New scales: [0.36664205 0.63335795]
       7  9.57e+02  2.85e+01  1.30e-01  1.53e+02    9.91e+01      0   Skip BFGS  
    geophys. misfits: 26.5 (target 30.0 [True]); 20.3 (target 30.0 [True]) | smallness misfit: 50.2 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [26.5 20.3]; minimum progress targets: [30.4 30. ]
    Warming alpha_pgi to favor clustering:  1.3051451777497802
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 1.5127e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 9.9134e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 9.9134e+01 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =      8
    ------------------------- DONE! -------------------------

    Running inversion with SimPEG v0.22.0
    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: [3.494765446396261e-05, 0.0, 3.50340917582425e-05, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09429751 0.90570249]
    /home/vsts/work/1/s/simpeg/directives/directives.py:334: UserWarning:

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

    Initial data misfit scales:  [0.09429751 0.90570249]
    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  1.03e+06  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 57071.4 (target 30.0 [False]); 35672.8 (target 30.0 [False])
       1  2.07e+05  3.77e+04  4.30e-02  4.66e+04    1.38e+02      0              
    geophys. misfits: 8548.9 (target 30.0 [False]); 3856.8 (target 30.0 [False])
       2  4.14e+04  4.30e+03  1.07e-01  8.73e+03    1.30e+02      0   Skip BFGS  
    geophys. misfits: 574.4 (target 30.0 [False]); 258.8 (target 30.0 [False])
       3  8.27e+03  2.89e+02  1.41e-01  1.46e+03    1.02e+02      0   Skip BFGS  
    geophys. misfits: 45.6 (target 30.0 [False]); 37.2 (target 30.0 [False])
       4  1.65e+03  3.80e+01  1.52e-01  2.90e+02    8.81e+01      0   Skip BFGS  
    geophys. misfits: 21.2 (target 30.0 [True]); 19.8 (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| = 4.9534e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 8.8080e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 8.8080e+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 52.142 seconds)

**Estimated memory usage:**  9 MB


.. _sphx_glr_download_content_examples_10-pgi_plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:

.. only:: html

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    .. 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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