.. 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 v0.22.1
    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: [3.49503918967184, 0.0, 3.6131602071908994e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09850237 0.90149763]
    <class 'simpeg.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09850237 0.90149763]
    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  1.95e+01  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 1037.7 (target 30.0 [False]); 81.3 (target 30.0 [False]) | smallness misfit: 2942.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [1037.7   81.3]; minimum progress targets: [240000. 240000.]
       1  1.95e+01  1.75e+02  4.15e+01  9.86e+02    8.68e+01      0              
    geophys. misfits: 474.9 (target 30.0 [False]); 25.3 (target 30.0 [True]) | smallness misfit: 1334.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [474.9  25.3]; minimum progress targets: [830.2  65. ]
    Updating scaling for data misfits by  1.1861041074440497
    New scales: [0.11473085 0.88526915]
       2  1.95e+01  7.69e+01  4.02e+01  8.63e+02    6.07e+01      0   Skip BFGS  
    geophys. misfits: 377.2 (target 30.0 [False]); 25.1 (target 30.0 [True]) | smallness misfit: 1288.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [377.2  25.1]; minimum progress targets: [379.9  30. ]
    Updating scaling for data misfits by  1.1947439955698549
    New scales: [0.13407827 0.86592173]
       3  1.95e+01  7.23e+01  4.07e+01  8.69e+02    6.61e+01      0   Skip BFGS  
    geophys. misfits: 300.7 (target 30.0 [False]); 25.1 (target 30.0 [True]) | smallness misfit: 1252.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [300.7  25.1]; minimum progress targets: [301.7  30. ]
    Updating scaling for data misfits by  1.197372183297028
    New scales: [0.15640266 0.84359734]
       4  1.95e+01  6.82e+01  4.12e+01  8.74e+02    5.85e+01      0   Skip BFGS  
    geophys. misfits: 240.9 (target 30.0 [False]); 25.1 (target 30.0 [True]) | smallness misfit: 1224.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [240.9  25.1]; minimum progress targets: [240.6  30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.1962573769740719
    New scales: [0.18152586 0.81847414]
       5  9.77e+00  6.43e+01  4.17e+01  4.71e+02    8.76e+01      0              
    geophys. misfits: 73.7 (target 30.0 [False]); 21.5 (target 30.0 [True]) | smallness misfit: 1231.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [73.7 21.5]; minimum progress targets: [192.8  30. ]
    Updating scaling for data misfits by  1.394424574604439
    New scales: [0.23621177 0.76378823]
       6  9.77e+00  3.38e+01  4.38e+01  4.62e+02    6.63e+01      0              
    geophys. misfits: 53.3 (target 30.0 [False]); 21.6 (target 30.0 [True]) | smallness misfit: 1197.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [53.3 21.6]; minimum progress targets: [59. 30.]
    Updating scaling for data misfits by  1.3878836708388158
    New scales: [0.30031847 0.69968153]
       7  9.77e+00  3.11e+01  4.42e+01  4.63e+02    6.62e+01      0   Skip BFGS  
    geophys. misfits: 41.0 (target 30.0 [False]); 21.8 (target 30.0 [True]) | smallness misfit: 1170.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [41.  21.8]; minimum progress targets: [42.6 30. ]
    Updating scaling for data misfits by  1.3765406908399755
    New scales: [0.37140171 0.62859829]
       8  9.77e+00  2.89e+01  4.45e+01  4.64e+02    5.68e+01      0   Skip BFGS  
    geophys. misfits: 33.4 (target 30.0 [False]); 22.1 (target 30.0 [True]) | smallness misfit: 1143.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [33.4 22.1]; minimum progress targets: [32.8 30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.357874025735914
    New scales: [0.44514965 0.55485035]
       9  4.89e+00  2.71e+01  4.48e+01  2.46e+02    7.45e+01      0   Skip BFGS  
    geophys. misfits: 18.7 (target 30.0 [True]); 20.0 (target 30.0 [True]) | smallness misfit: 1192.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [18.7 20. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.5522062066452478
      10  4.89e+00  1.94e+01  4.72e+01  2.50e+02    7.72e+01      0              
    geophys. misfits: 18.3 (target 30.0 [True]); 21.2 (target 30.0 [True]) | smallness misfit: 1058.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [18.3 21.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  2.37124802951631
      11  4.89e+00  1.99e+01  4.89e+01  2.59e+02    5.77e+01      0              
    geophys. misfits: 17.7 (target 30.0 [True]); 22.6 (target 30.0 [True]) | smallness misfit: 939.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [17.7 22.6]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  3.5783204919103775
      12  4.89e+00  2.04e+01  5.12e+01  2.71e+02    6.79e+01      0              
    geophys. misfits: 17.5 (target 30.0 [True]); 24.3 (target 30.0 [True]) | smallness misfit: 828.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [17.5 24.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  5.267298370992695
      13  4.89e+00  2.13e+01  5.40e+01  2.85e+02    7.41e+01      0              
    geophys. misfits: 17.0 (target 30.0 [True]); 26.2 (target 30.0 [True]) | smallness misfit: 736.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [17.  26.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  7.652794277100461
      14  4.89e+00  2.21e+01  5.76e+01  3.04e+02    9.33e+01      0              
    geophys. misfits: 17.3 (target 30.0 [True]); 28.9 (target 30.0 [True]) | smallness misfit: 644.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [17.3 28.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  10.606773691903875
      15  4.89e+00  2.37e+01  6.12e+01  3.23e+02    9.25e+01      0              
    geophys. misfits: 17.8 (target 30.0 [True]); 32.7 (target 30.0 [False]) | smallness misfit: 554.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [17.8 32.7]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.6862696339324812
    New scales: [0.32239071 0.67760929]
      16  2.44e+00  2.79e+01  6.03e+01  1.75e+02    1.02e+02      0              
    geophys. misfits: 15.7 (target 30.0 [True]); 23.5 (target 30.0 [True]) | smallness misfit: 600.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [15.7 23.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  16.904836392612
      17  2.44e+00  2.10e+01  7.03e+01  1.93e+02    9.47e+01      0              
    geophys. misfits: 16.4 (target 30.0 [True]); 25.4 (target 30.0 [True]) | smallness misfit: 486.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [16.4 25.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  25.414556384964417
      18  2.44e+00  2.25e+01  7.81e+01  2.13e+02    1.02e+02      0              
    geophys. misfits: 21.8 (target 30.0 [True]); 22.3 (target 30.0 [True]) | smallness misfit: 419.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.8 22.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  34.573459035756194
      19  2.44e+00  2.22e+01  8.42e+01  2.28e+02    1.09e+02      0              
    geophys. misfits: 19.6 (target 30.0 [True]); 21.4 (target 30.0 [True]) | smallness misfit: 341.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [19.6 21.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  50.69791210176808
      20  2.44e+00  2.08e+01  9.46e+01  2.52e+02    1.18e+02      0              
    geophys. misfits: 25.7 (target 30.0 [True]); 27.4 (target 30.0 [True]) | smallness misfit: 249.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.7 27.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  57.39230776509479
      21  2.44e+00  2.68e+01  9.25e+01  2.53e+02    1.08e+02      0              
    geophys. misfits: 33.8 (target 30.0 [False]); 31.5 (target 30.0 [False]) | smallness misfit: 243.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [33.8 31.5]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
      22  1.22e+00  3.22e+01  9.02e+01  1.42e+02    1.03e+02      0              
    geophys. misfits: 21.8 (target 30.0 [True]); 23.7 (target 30.0 [True]) | smallness misfit: 254.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.8 23.7]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  75.83211639696805
      23  1.22e+00  2.31e+01  1.04e+02  1.50e+02    9.58e+01      0              
    geophys. misfits: 24.1 (target 30.0 [True]); 23.8 (target 30.0 [True]) | smallness misfit: 206.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.1 23.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  94.83677821047651
      24  1.22e+00  2.39e+01  1.10e+02  1.59e+02    1.02e+02      0              
    geophys. misfits: 25.5 (target 30.0 [True]); 25.2 (target 30.0 [True]) | smallness misfit: 207.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.5 25.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  112.24697235709137
      25  1.22e+00  2.53e+01  1.17e+02  1.68e+02    1.07e+02      0              
    geophys. misfits: 25.8 (target 30.0 [True]); 26.2 (target 30.0 [True]) | smallness misfit: 187.2 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [25.8 26.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  129.68758391780855
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 4.9887e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 1.0719e+02 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 1.0719e+02 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =     26
    ------------------------- DONE! -------------------------

    Running inversion with SimPEG v0.22.1
    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: [0.0003502883080097992, 0.0, 3.486404571820042e-06, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09850237 0.90149763]
    <class 'simpeg.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09850237 0.90149763]
    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: 87810.7 (target 30.0 [False]); 63893.7 (target 30.0 [False]) | smallness misfit: 282.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [87810.7 63893.7]; minimum progress targets: [240000. 240000.]
       1  1.97e+03  6.62e+04  6.73e-01  6.76e+04    9.16e+01      0              
    geophys. misfits: 624.0 (target 30.0 [False]); 22.5 (target 30.0 [True]) | smallness misfit: 101.2 (target: 200.0 [True])
    Beta cooling evaluation: progress: [624.   22.5]; minimum progress targets: [70248.6 51115. ]
    Updating scaling for data misfits by  1.335112655453233
    New scales: [0.12730936 0.87269064]
       2  1.97e+03  9.90e+01  2.76e-01  6.45e+02    8.89e+01      0   Skip BFGS  
    geophys. misfits: 93.5 (target 30.0 [False]); 21.9 (target 30.0 [True]) | smallness misfit: 50.9 (target: 200.0 [True])
    Beta cooling evaluation: progress: [93.5 21.9]; minimum progress targets: [499.2  30. ]
    Updating scaling for data misfits by  1.3726883101291
    New scales: [0.16684005 0.83315995]
       3  1.97e+03  3.38e+01  1.36e-01  3.03e+02    9.63e+01      0   Skip BFGS  
    geophys. misfits: 64.0 (target 30.0 [False]); 21.0 (target 30.0 [True]) | smallness misfit: 47.3 (target: 200.0 [True])
    Beta cooling evaluation: progress: [64. 21.]; minimum progress targets: [74.8 30. ]
    Updating scaling for data misfits by  1.4268414202187247
    New scales: [0.22222847 0.77777153]
       4  1.97e+03  3.06e+01  1.29e-01  2.84e+02    7.00e+01      0              
    geophys. misfits: 45.2 (target 30.0 [False]); 21.7 (target 30.0 [True]) | smallness misfit: 48.0 (target: 200.0 [True])
    Beta cooling evaluation: progress: [45.2 21.7]; minimum progress targets: [51.2 30. ]
    Updating scaling for data misfits by  1.3823203086624076
    New scales: [0.28313507 0.71686493]
       5  1.97e+03  2.84e+01  1.30e-01  2.85e+02    8.14e+01      0   Skip BFGS  
    geophys. misfits: 35.7 (target 30.0 [False]); 22.6 (target 30.0 [True]) | smallness misfit: 48.5 (target: 200.0 [True])
    Beta cooling evaluation: progress: [35.7 22.6]; minimum progress targets: [36.2 30. ]
    Updating scaling for data misfits by  1.3264641324828763
    New scales: [0.34379075 0.65620925]
       6  1.97e+03  2.71e+01  1.31e-01  2.85e+02    6.30e+01      0   Skip BFGS  
    geophys. misfits: 30.4 (target 30.0 [False]); 23.5 (target 30.0 [True]) | smallness misfit: 48.7 (target: 200.0 [True])
    Beta cooling evaluation: progress: [30.4 23.5]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.2784008935309614
    New scales: [0.40111137 0.59888863]
       7  9.87e+02  2.63e+01  1.31e-01  1.56e+02    9.90e+01      0   Skip BFGS  
    geophys. misfits: 20.8 (target 30.0 [True]); 19.8 (target 30.0 [True]) | smallness misfit: 50.6 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [20.8 19.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.477871975281782
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 1.7688e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 9.8972e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 9.8972e+01 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =      8
    ------------------------- DONE! -------------------------

    Running inversion with SimPEG v0.22.1
    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: [3.136682718424762e-05, 0.0, 3.1417153861777845e-05, 0.0]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09850237 0.90149763]
    /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:  [0.09850237 0.90149763]
    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  1.12e+06  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 58639.3 (target 30.0 [False]); 38893.0 (target 30.0 [False])
       1  2.25e+05  4.08e+04  4.00e-02  4.98e+04    1.38e+02      0              
    geophys. misfits: 8958.4 (target 30.0 [False]); 4393.2 (target 30.0 [False])
       2  4.49e+04  4.84e+03  1.04e-01  9.53e+03    1.31e+02      0   Skip BFGS  
    geophys. misfits: 603.1 (target 30.0 [False]); 286.4 (target 30.0 [False])
       3  8.99e+03  3.18e+02  1.40e-01  1.57e+03    1.03e+02      0   Skip BFGS  
    geophys. misfits: 42.3 (target 30.0 [False]); 36.9 (target 30.0 [False])
       4  1.80e+03  3.74e+01  1.51e-01  3.08e+02    8.76e+01      0   Skip BFGS  
    geophys. misfits: 15.4 (target 30.0 [True]); 18.7 (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| = 5.4077e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 8.7562e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 8.7562e+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 34.766 seconds)

**Estimated memory usage:**  9 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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