.. 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 v0.23.0
    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: [np.float64(3.5076401722247157), np.float64(0.0), np.float64(3.508490434878856e-06), np.float64(0.0)]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09326744 0.90673256]
    <class 'simpeg.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09326744 0.90673256]
    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  1.87e+01  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 1096.0 (target 30.0 [False]); 70.7 (target 30.0 [False]) | smallness misfit: 2986.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [1096.    70.7]; minimum progress targets: [240000. 240000.]
       1  1.87e+01  1.66e+02  4.16e+01  9.44e+02    8.82e+01      0              
    geophys. misfits: 510.0 (target 30.0 [False]); 18.9 (target 30.0 [True]) | smallness misfit: 1373.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [510.   18.9]; minimum progress targets: [876.8  56.5]
    Updating scaling for data misfits by  1.5859329981033554
    New scales: [0.14025139 0.85974861]
       2  1.87e+01  8.78e+01  4.05e+01  8.44e+02    8.95e+01      0   Skip BFGS  
    geophys. misfits: 291.8 (target 30.0 [False]); 18.6 (target 30.0 [True]) | smallness misfit: 1269.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [291.8  18.6]; minimum progress targets: [408.  30.]
    Updating scaling for data misfits by  1.6128264996374915
    New scales: [0.208298 0.791702]
       3  1.87e+01  7.55e+01  4.18e+01  8.56e+02    7.51e+01      0   Skip BFGS  
    geophys. misfits: 172.5 (target 30.0 [False]); 18.7 (target 30.0 [True]) | smallness misfit: 1200.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [172.5  18.7]; minimum progress targets: [233.5  30. ]
    Updating scaling for data misfits by  1.6048541987288436
    New scales: [0.29688358 0.70311642]
       4  1.87e+01  6.43e+01  4.28e+01  8.65e+02    7.47e+01      0              
    geophys. misfits: 111.6 (target 30.0 [False]); 19.2 (target 30.0 [True]) | smallness misfit: 1148.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [111.6  19.2]; minimum progress targets: [138.  30.]
    Updating scaling for data misfits by  1.5624977847730535
    New scales: [0.39749904 0.60250096]
       5  1.87e+01  5.59e+01  4.36e+01  8.71e+02    8.06e+01      0   Skip BFGS  
    geophys. misfits: 80.1 (target 30.0 [False]); 20.2 (target 30.0 [True]) | smallness misfit: 1106.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [80.1 20.2]; minimum progress targets: [89.3 30. ]
    Updating scaling for data misfits by  1.4865180376781295
    New scales: [0.49513509 0.50486491]
       6  1.87e+01  4.99e+01  4.41e+01  8.75e+02    8.43e+01      0   Skip BFGS  
    geophys. misfits: 63.7 (target 30.0 [False]); 21.7 (target 30.0 [True]) | smallness misfit: 1066.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [63.7 21.7]; minimum progress targets: [64.1 30. ]
    Updating scaling for data misfits by  1.3800352190916845
    New scales: [0.5750899 0.4249101]
       7  1.87e+01  4.58e+01  4.45e+01  8.78e+02    8.33e+01      0   Skip BFGS  
    geophys. misfits: 55.1 (target 30.0 [False]); 24.1 (target 30.0 [True]) | smallness misfit: 1029.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [55.1 24.1]; minimum progress targets: [50.9 30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.2469048359310606
    New scales: [0.62792216 0.37207784]
       8  9.35e+00  4.36e+01  4.47e+01  4.61e+02    9.42e+01      0   Skip BFGS  
    geophys. misfits: 30.1 (target 30.0 [False]); 17.5 (target 30.0 [True]) | smallness misfit: 1105.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [30.1 17.5]; minimum progress targets: [44.1 30. ]
    Updating scaling for data misfits by  1.7174836259931823
    New scales: [0.74348727 0.25651273]
       9  9.35e+00  2.68e+01  4.59e+01  4.56e+02    7.63e+01      0              
    geophys. misfits: 27.5 (target 30.0 [True]); 20.4 (target 30.0 [True]) | smallness misfit: 1044.1 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.5 20.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.2786296516513311
      10  9.35e+00  2.57e+01  4.67e+01  4.62e+02    7.63e+01      0   Skip BFGS  
    geophys. misfits: 27.4 (target 30.0 [True]); 23.0 (target 30.0 [True]) | smallness misfit: 973.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.4 23. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.5327952864032601
      11  9.35e+00  2.63e+01  4.71e+01  4.67e+02    7.60e+01      0              
    geophys. misfits: 27.3 (target 30.0 [True]); 24.9 (target 30.0 [True]) | smallness misfit: 927.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.3 24.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.7667402301739434
      12  9.35e+00  2.67e+01  4.75e+01  4.71e+02    7.71e+01      0   Skip BFGS  
    geophys. misfits: 27.2 (target 30.0 [True]); 27.2 (target 30.0 [True]) | smallness misfit: 884.1 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.2 27.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.9511303520110128
      13  9.35e+00  2.72e+01  4.78e+01  4.75e+02    7.54e+01      0              
    geophys. misfits: 27.4 (target 30.0 [True]); 29.5 (target 30.0 [True]) | smallness misfit: 856.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.4 29.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  2.063663142327159
      14  9.35e+00  2.79e+01  4.80e+01  4.76e+02    7.55e+01      0   Skip BFGS  
    geophys. misfits: 27.1 (target 30.0 [True]); 30.6 (target 30.0 [False]) | smallness misfit: 841.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.1 30.6]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.1050286910055975
    New scales: [0.72398232 0.27601768]
      15  4.68e+00  2.81e+01  4.80e+01  2.52e+02    9.47e+01      0              
    geophys. misfits: 21.5 (target 30.0 [True]); 19.4 (target 30.0 [True]) | smallness misfit: 944.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.5 19.4]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  3.0324546470037728
      16  4.68e+00  2.09e+01  5.10e+01  2.59e+02    8.39e+01      0              
    geophys. misfits: 21.5 (target 30.0 [True]); 22.3 (target 30.0 [True]) | smallness misfit: 837.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.5 22.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  4.149483792621317
      17  4.68e+00  2.18e+01  5.27e+01  2.68e+02    5.66e+01      0              
    geophys. misfits: 21.5 (target 30.0 [True]); 23.9 (target 30.0 [True]) | smallness misfit: 766.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.5 23.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  5.4994530769824985
      18  4.68e+00  2.22e+01  5.49e+01  2.79e+02    8.52e+01      0   Skip BFGS  
    geophys. misfits: 21.6 (target 30.0 [True]); 25.9 (target 30.0 [True]) | smallness misfit: 690.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.6 25.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  6.997932378586193
      19  4.68e+00  2.28e+01  5.69e+01  2.89e+02    9.58e+01      0              
    geophys. misfits: 21.8 (target 30.0 [True]); 29.8 (target 30.0 [True]) | smallness misfit: 648.9 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.8 29.8]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  8.345249132782294
      20  4.68e+00  2.40e+01  5.84e+01  2.97e+02    9.18e+01      0              
    geophys. misfits: 22.2 (target 30.0 [True]); 29.2 (target 30.0 [True]) | smallness misfit: 578.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.2 29.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  9.909691752145934
      21  4.68e+00  2.42e+01  6.01e+01  3.05e+02    9.66e+01      0              
    geophys. misfits: 22.9 (target 30.0 [True]); 35.7 (target 30.0 [False]) | smallness misfit: 487.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.9 35.7]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.3078380663123905
    New scales: [0.66728403 0.33271597]
      22  2.34e+00  2.72e+01  5.92e+01  1.66e+02    1.07e+02      0   Skip BFGS  
    geophys. misfits: 20.3 (target 30.0 [True]); 23.5 (target 30.0 [True]) | smallness misfit: 553.6 (target: 200.0 [False])
    Beta cooling evaluation: progress: [20.3 23.5]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  13.643321046096773
      23  2.34e+00  2.14e+01  6.54e+01  1.74e+02    7.84e+01      0              
    geophys. misfits: 20.6 (target 30.0 [True]); 21.0 (target 30.0 [True]) | smallness misfit: 509.5 (target: 200.0 [False])
    Beta cooling evaluation: progress: [20.6 21. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  19.667094190799734
      24  2.34e+00  2.07e+01  7.20e+01  1.89e+02    1.00e+02      0              
    geophys. misfits: 20.6 (target 30.0 [True]); 19.2 (target 30.0 [True]) | smallness misfit: 468.0 (target: 200.0 [False])
    Beta cooling evaluation: progress: [20.6 19.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  29.74263976375323
      25  2.34e+00  2.01e+01  8.22e+01  2.12e+02    1.14e+02      0              
    geophys. misfits: 20.6 (target 30.0 [True]); 18.2 (target 30.0 [True]) | smallness misfit: 426.3 (target: 200.0 [False])
    Beta cooling evaluation: progress: [20.6 18.2]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  46.136205946409
      26  2.34e+00  1.98e+01  9.69e+01  2.46e+02    1.24e+02      0              
    geophys. misfits: 21.6 (target 30.0 [True]); 18.0 (target 30.0 [True]) | smallness misfit: 384.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [21.6 18. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  70.45051841767439
      27  2.34e+00  2.04e+01  1.16e+02  2.92e+02    1.29e+02      0              
    geophys. misfits: 22.5 (target 30.0 [True]); 19.0 (target 30.0 [True]) | smallness misfit: 342.8 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.5 19. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  102.63896119771891
      28  2.34e+00  2.13e+01  1.38e+02  3.44e+02    1.31e+02      0              
    geophys. misfits: 22.4 (target 30.0 [True]); 25.0 (target 30.0 [True]) | smallness misfit: 332.4 (target: 200.0 [False])
    Beta cooling evaluation: progress: [22.4 25. ]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  130.30166640202273
      29  2.34e+00  2.33e+01  1.57e+02  3.89e+02    1.31e+02      0              
    geophys. misfits: 27.5 (target 30.0 [True]); 28.6 (target 30.0 [True]) | smallness misfit: 272.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [27.5 28.6]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  139.33865961698652
      30  2.34e+00  2.79e+01  1.57e+02  3.95e+02    1.34e+02      0              
    geophys. misfits: 25.4 (target 30.0 [True]); 27.1 (target 30.0 [True]) | smallness misfit: 260.1 (target: 200.0 [False])
    Beta cooling evaluation: progress: [25.4 27.1]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  159.4082639968159
      31  2.34e+00  2.60e+01  1.61e+02  4.02e+02    1.31e+02      1              
    geophys. misfits: 24.3 (target 30.0 [True]); 25.9 (target 30.0 [True]) | smallness misfit: 231.7 (target: 200.0 [False])
    Beta cooling evaluation: progress: [24.3 25.9]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  190.80460809699628
      32  2.34e+00  2.48e+01  1.68e+02  4.17e+02    1.32e+02      1              
    geophys. misfits: 28.7 (target 30.0 [True]); 50.5 (target 30.0 [False]) | smallness misfit: 174.4 (target: 200.0 [True])
    Beta cooling evaluation: progress: [28.7 50.5]; minimum progress targets: [30. 30.]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.0459707133398461
    New scales: [0.65723155 0.34276845]
      33  1.17e+00  3.62e+01  1.55e+02  2.17e+02    1.25e+02      0              
    geophys. misfits: 23.8 (target 30.0 [True]); 59.5 (target 30.0 [False]) | smallness misfit: 153.9 (target: 200.0 [True])
    Beta cooling evaluation: progress: [23.8 59.5]; minimum progress targets: [30.  40.4]
    Decreasing beta to counter data misfit increase.
    Updating scaling for data misfits by  1.2596773070109566
    New scales: [0.60351331 0.39648669]
      34  5.84e-01  3.80e+01  1.43e+02  1.22e+02    1.10e+02      0              
    geophys. misfits: 22.2 (target 30.0 [True]); 25.6 (target 30.0 [True]) | smallness misfit: 178.8 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [22.2 25.6]; minimum progress targets: [30.  47.6]
    Warming alpha_pgi to favor clustering:  240.72513045838488
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 7.9996e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 1.0991e+02 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 1.0991e+02 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =     35
    ------------------------- DONE! -------------------------

    Running inversion with SimPEG v0.23.0
    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: [np.float64(0.00039124967802256705), np.float64(0.0), np.float64(3.4991079985601646e-06), np.float64(0.0)]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09326744 0.90673256]
    <class 'simpeg.regularization.pgi.PGIsmallness'>
    Initial data misfit scales:  [0.09326744 0.90673256]
    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  1.93e+03  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 89877.2 (target 30.0 [False]); 62289.4 (target 30.0 [False]) | smallness misfit: 294.2 (target: 200.0 [False])
    Beta cooling evaluation: progress: [89877.2 62289.4]; minimum progress targets: [240000. 240000.]
       1  1.93e+03  6.49e+04  6.93e-01  6.62e+04    9.06e+01      0              
    geophys. misfits: 675.2 (target 30.0 [False]); 20.1 (target 30.0 [True]) | smallness misfit: 96.5 (target: 200.0 [True])
    Beta cooling evaluation: progress: [675.2  20.1]; minimum progress targets: [71901.8 49831.5]
    Updating scaling for data misfits by  1.493484799701024
    New scales: [0.13316447 0.86683553]
       2  1.93e+03  1.07e+02  2.53e-01  5.95e+02    9.57e+01      0   Skip BFGS  
    geophys. misfits: 94.0 (target 30.0 [False]); 18.9 (target 30.0 [True]) | smallness misfit: 51.3 (target: 200.0 [True])
    Beta cooling evaluation: progress: [94.  18.9]; minimum progress targets: [540.1  30. ]
    Updating scaling for data misfits by  1.5900801673882476
    New scales: [0.19631611 0.80368389]
       3  1.93e+03  3.36e+01  1.46e-01  3.16e+02    1.06e+02      0   Skip BFGS  
    geophys. misfits: 59.7 (target 30.0 [False]); 18.9 (target 30.0 [True]) | smallness misfit: 47.9 (target: 200.0 [True])
    Beta cooling evaluation: progress: [59.7 18.9]; minimum progress targets: [75.2 30. ]
    Updating scaling for data misfits by  1.588985131008552
    New scales: [0.27961255 0.72038745]
       4  1.93e+03  3.03e+01  1.31e-01  2.84e+02    7.87e+01      0              
    geophys. misfits: 44.2 (target 30.0 [False]); 19.2 (target 30.0 [True]) | smallness misfit: 48.7 (target: 200.0 [True])
    Beta cooling evaluation: progress: [44.2 19.2]; minimum progress targets: [47.8 30. ]
    Updating scaling for data misfits by  1.5626108355934907
    New scales: [0.3775345 0.6224655]
       5  1.93e+03  2.86e+01  1.33e-01  2.85e+02    8.45e+01      0   Skip BFGS  
    geophys. misfits: 36.3 (target 30.0 [False]); 20.4 (target 30.0 [True]) | smallness misfit: 49.1 (target: 200.0 [True])
    Beta cooling evaluation: progress: [36.3 20.4]; minimum progress targets: [35.4 30. ]
    Decreasing beta to counter data misfit decrase plateau.
    Updating scaling for data misfits by  1.4725735801347535
    New scales: [0.47177637 0.52822363]
       6  9.64e+02  2.79e+01  1.34e-01  1.57e+02    1.05e+02      0              
    geophys. misfits: 25.9 (target 30.0 [True]); 17.3 (target 30.0 [True]) | smallness misfit: 51.1 (target: 200.0 [True])
    All targets have been reached
    Beta cooling evaluation: progress: [25.9 17.3]; minimum progress targets: [30. 30.]
    Warming alpha_pgi to favor clustering:  1.4442529544627405
    ------------------------- STOP! -------------------------
    1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
    0 : |xc-x_last| = 1.5816e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 1.0514e+02 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 1.0514e+02 <= 1e3*eps       = 1.0000e-02
    0 : maxIter   =      50    <= iter          =      7
    ------------------------- DONE! -------------------------

    Running inversion with SimPEG v0.23.0
    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: [np.float64(3.510762811221696e-05), np.float64(0.0), np.float64(4.420996595587482e-05), np.float64(0.0)]
    Calculating the scaling parameter.
    Scale Multipliers:  [0.09326744 0.90673256]
    /home/vsts/work/1/s/simpeg/directives/directives.py:339: UserWarning:

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

    Initial data misfit scales:  [0.09326744 0.90673256]
    model has any nan: 0
    =============================== Projected GNCG ===============================
      #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment   
    -----------------------------------------------------------------------------
    x0 has any nan: 0
       0  1.02e+06  3.00e+05  0.00e+00  3.00e+05    1.41e+02      0              
    geophys. misfits: 56827.8 (target 30.0 [False]); 34909.5 (target 30.0 [False])
       1  2.03e+05  3.70e+04  4.35e-02  4.58e+04    1.38e+02      0              
    geophys. misfits: 8507.7 (target 30.0 [False]); 3707.9 (target 30.0 [False])
       2  4.07e+04  4.16e+03  1.07e-01  8.53e+03    1.30e+02      0   Skip BFGS  
    geophys. misfits: 582.4 (target 30.0 [False]); 239.5 (target 30.0 [False])
       3  8.14e+03  2.71e+02  1.41e-01  1.42e+03    1.04e+02      0   Skip BFGS  
    geophys. misfits: 50.8 (target 30.0 [False]); 27.3 (target 30.0 [True])
    Updating scaling for data misfits by  1.0986468594984395
    New scales: [0.10153382 0.89846618]
       4  1.63e+03  2.97e+01  1.52e-01  2.76e+02    8.03e+01      0   Skip BFGS  
    geophys. misfits: 20.7 (target 30.0 [True]); 12.6 (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| = 4.3399e-01 <= tolX*(1+|x0|) = 1.0000e-06
    0 : |proj(x-g)-x|    = 8.0315e+01 <= tolG          = 1.0000e-01
    0 : |proj(x-g)-x|    = 8.0315e+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:329: MatplotlibDeprecationWarning:

    The collections attribute was deprecated in Matplotlib 3.8 and will be removed in 3.10.

    /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.

    /home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:426: MatplotlibDeprecationWarning:

    The collections attribute was deprecated in Matplotlib 3.8 and will be removed in 3.10.







|

.. 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 56.858 seconds)

**Estimated memory usage:**  288 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>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_inv_1_PGI_Linear_1D_joint_WithRelationships.zip <plot_inv_1_PGI_Linear_1D_joint_WithRelationships.zip>`


.. only:: html

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

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