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.

Running inversion with SimPEG v0.25.1.dev1+g9a8c46e88
Alpha scales: [np.float64(3.482494321383229), np.float64(0.0), np.float64(3.492563350370212e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers:  [0.08846538 0.91153462]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales:  [0.08846538 0.91153462]
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  1.85e+01  3.00e+05  0.00e+00  3.00e+05                         0           inf          inf
   1  1.85e+01  1.07e+03  1.71e+02  4.25e+03    1.41e+02      0      19       5.11e-04     4.62e+03
geophys. misfits: 7046.4 (target 30.0 [False]); 489.8 (target 30.0 [False]) | smallness misfit: 4172.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [7046.4  489.8]; minimum progress targets: [240000. 240000.]
   2  1.85e+01  5.49e+01  4.09e+01  8.13e+02    1.40e+02      0     100       5.58e-01     2.60e+03   Skip BFGS
geophys. misfits: 484.1 (target 30.0 [False]); 13.2 (target 30.0 [True]) | smallness misfit: 1474.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [484.1  13.2]; minimum progress targets: [5637.1  391.8]
Updating scaling for data misfits by  2.2706737663871923
New scales: [0.18057723 0.81942277]
   3  1.85e+01  4.14e+01  4.20e+01  8.19e+02    1.06e+02      0     100       1.50e-01     3.62e+02   Skip BFGS
geophys. misfits: 168.0 (target 30.0 [False]); 13.6 (target 30.0 [True]) | smallness misfit: 1130.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [168.   13.6]; minimum progress targets: [387.3  30. ]
Updating scaling for data misfits by  2.2131941537087703
New scales: [0.32783248 0.67216752]
   4  1.85e+01  3.28e+01  4.32e+01  8.33e+02    8.21e+01      0     100       4.94e-03     4.17e+00   Skip BFGS
geophys. misfits: 71.2 (target 30.0 [False]); 14.1 (target 30.0 [True]) | smallness misfit: 1046.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [71.2 14.1]; minimum progress targets: [134.4  30. ]
Updating scaling for data misfits by  2.124352536276676
New scales: [0.50886463 0.49113537]
   5  1.85e+01  2.82e+01  4.38e+01  8.40e+02    8.01e+01      0     100       1.29e+00     4.80e+02   Skip BFGS
geophys. misfits: 40.2 (target 30.0 [False]); 15.7 (target 30.0 [True]) | smallness misfit: 991.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [40.2 15.7]; minimum progress targets: [56.9 30. ]
Updating scaling for data misfits by  1.9064092024909924
New scales: [0.66389128 0.33610872]
   6  1.85e+01  2.61e+01  4.40e+01  8.42e+02    7.78e+01      0     100       3.71e-02     2.41e+01   Skip BFGS
geophys. misfits: 29.7 (target 30.0 [True]); 18.9 (target 30.0 [True]) | smallness misfit: 946.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [29.7 18.9]; minimum progress targets: [32.1 30. ]
Warming alpha_pgi to favor clustering:  1.2971579180132176
   7  1.85e+01  2.67e+01  4.46e+01  8.54e+02    4.86e+01      0     100       2.25e-02     1.54e+00
geophys. misfits: 29.1 (target 30.0 [True]); 22.1 (target 30.0 [True]) | smallness misfit: 888.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [29.1 22.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  1.5492281806981647
   8  1.85e+01  2.68e+01  4.51e+01  8.63e+02    3.92e+01      0     100       1.33e+00     7.19e+01   Skip BFGS
geophys. misfits: 27.8 (target 30.0 [True]); 25.0 (target 30.0 [True]) | smallness misfit: 848.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [27.8 25. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  1.7669119623178713
   9  1.85e+01  2.71e+01  4.55e+01  8.70e+02    5.31e+01      0     100       1.46e+00     1.30e+02   Skip BFGS
geophys. misfits: 26.8 (target 30.0 [True]); 27.7 (target 30.0 [True]) | smallness misfit: 817.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [26.8 27.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  1.9474921764941164
  10  1.85e+01  2.74e+01  4.58e+01  8.76e+02    5.59e+01      0     100       9.22e-01     1.27e+02   Skip BFGS
geophys. misfits: 26.1 (target 30.0 [True]); 29.8 (target 30.0 [True]) | smallness misfit: 792.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [26.1 29.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  2.0964681119623862
  11  1.85e+01  2.78e+01  4.61e+01  8.81e+02    5.90e+01      0     100       1.40e+00     1.89e+02   Skip BFGS
geophys. misfits: 25.8 (target 30.0 [True]); 31.9 (target 30.0 [False]) | smallness misfit: 773.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.8 31.9]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by  1.1636083316739765
New scales: [0.62928668 0.37071332]
  12  9.26e+00  1.94e+01  4.67e+01  4.52e+02    9.33e+01      0     100       1.55e+00     5.23e+02
geophys. misfits: 21.0 (target 30.0 [True]); 16.6 (target 30.0 [True]) | smallness misfit: 802.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [21.  16.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  3.3945226691292367
  13  9.26e+00  1.95e+01  4.89e+01  4.72e+02    8.34e+01      0     100       1.98e+00     1.04e+03
geophys. misfits: 18.2 (target 30.0 [True]); 21.8 (target 30.0 [True]) | smallness misfit: 696.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.2 21.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  5.137198807104712
  14  9.26e+00  2.01e+01  5.13e+01  4.95e+02    8.08e+01      0     100       1.13e-01     1.18e+02
geophys. misfits: 16.0 (target 30.0 [True]); 27.2 (target 30.0 [True]) | smallness misfit: 596.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.  27.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  7.656444438112672
  15  9.26e+00  2.33e+01  5.40e+01  5.24e+02    9.05e+01      0     100       1.78e+00     4.77e+02
geophys. misfits: 15.5 (target 30.0 [True]); 36.5 (target 30.0 [False]) | smallness misfit: 491.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.5 36.5]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by  1.9386562294117773
New scales: [0.46683945 0.53316055]
  16  4.63e+00  1.56e+01  5.54e+01  2.72e+02    9.64e+01      0     100       8.99e+00     5.23e+03
geophys. misfits: 13.7 (target 30.0 [True]); 17.3 (target 30.0 [True]) | smallness misfit: 568.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [13.7 17.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  15.045263945460675
  17  4.63e+00  2.08e+01  6.22e+01  3.09e+02    1.08e+02      0     100       1.92e-01     1.01e+03
geophys. misfits: 15.8 (target 30.0 [True]); 25.2 (target 30.0 [True]) | smallness misfit: 401.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.8 25.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  23.19758677761929
  18  4.63e+00  2.75e+01  6.69e+01  3.37e+02    9.78e+01      0     100       1.62e+00     1.73e+03
geophys. misfits: 19.0 (target 30.0 [True]); 35.0 (target 30.0 [False]) | smallness misfit: 308.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [19. 35.]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by  1.5828767692571082
New scales: [0.3561575 0.6438425]
  19  2.32e+00  1.67e+01  7.01e+01  1.79e+02    1.09e+02      0     100       7.03e-01     9.28e+02
geophys. misfits: 16.6 (target 30.0 [True]); 16.8 (target 30.0 [True]) | smallness misfit: 349.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.6 16.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  41.68419284956462
  20  2.32e+00  2.24e+01  7.97e+01  2.07e+02    1.02e+02      0     100       1.27e+01     1.27e+04
geophys. misfits: 22.8 (target 30.0 [True]); 22.2 (target 30.0 [True]) | smallness misfit: 242.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.8 22.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  55.678577260222724
  21  2.32e+00  2.62e+01  8.47e+01  2.22e+02    1.05e+02      0     100       1.51e-01     1.93e+03
geophys. misfits: 28.4 (target 30.0 [True]); 25.0 (target 30.0 [True]) | smallness misfit: 197.5 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [28.4 25. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  62.82096017694456
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.3517e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.5333e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x|    = 1.0541e+02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 1.0541e+02 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      50    <= iter          =     21
------------------------- DONE! -------------------------

Running inversion with SimPEG v0.25.1.dev1+g9a8c46e88
Alpha scales: [np.float64(0.0003453012199506967), np.float64(0.0), np.float64(3.439021894961647e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers:  [0.08846538 0.91153462]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales:  [0.08846538 0.91153462]
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  1.86e+03  3.00e+05  0.00e+00  3.00e+05                         0           inf          inf
   1  1.86e+03  6.51e+04  2.31e+01  1.08e+05    1.41e+02      0      12       9.75e-04     8.82e+03
geophys. misfits: 97689.7 (target 30.0 [False]); 61948.8 (target 30.0 [False]) | smallness misfit: 271.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [97689.7 61948.8]; minimum progress targets: [240000. 240000.]
   2  1.86e+03  8.05e+01  5.60e-01  1.12e+03    1.37e+02      0     100       6.13e-01     1.70e+04   Skip BFGS
geophys. misfits: 680.2 (target 30.0 [False]); 22.3 (target 30.0 [True]) | smallness misfit: 135.9 (target: 200.0 [True])
Beta cooling evaluation: progress: [680.2  22.3]; minimum progress targets: [78151.7 49559. ]
Updating scaling for data misfits by  1.3443785216315007
New scales: [0.11541478 0.88458522]
   3  1.86e+03  3.25e+01  1.42e-01  2.96e+02    1.16e+02      0     100       7.95e-03     1.37e+02   Skip BFGS
geophys. misfits: 109.3 (target 30.0 [False]); 22.5 (target 30.0 [True]) | smallness misfit: 53.5 (target: 200.0 [True])
Beta cooling evaluation: progress: [109.3  22.5]; minimum progress targets: [544.1  30. ]
Updating scaling for data misfits by  1.3355920492943336
New scales: [0.14839923 0.85160077]
   4  1.86e+03  3.02e+01  1.41e-01  2.92e+02    8.02e+01      0     100       5.77e-02     2.64e+01
geophys. misfits: 79.1 (target 30.0 [False]); 21.6 (target 30.0 [True]) | smallness misfit: 53.8 (target: 200.0 [True])
Beta cooling evaluation: progress: [79.1 21.6]; minimum progress targets: [87.4 30. ]
Updating scaling for data misfits by  1.3875313387610158
New scales: [0.19471088 0.80528912]
   5  1.86e+03  2.89e+01  1.42e-01  2.94e+02    6.72e+01      0     100       8.36e-01     1.06e+02
geophys. misfits: 55.5 (target 30.0 [False]); 22.4 (target 30.0 [True]) | smallness misfit: 54.7 (target: 200.0 [True])
Beta cooling evaluation: progress: [55.5 22.4]; minimum progress targets: [63.3 30. ]
Updating scaling for data misfits by  1.338008809770135
New scales: [0.24443747 0.75556253]
   6  1.86e+03  2.79e+01  1.44e-01  2.95e+02    7.58e+01      0     100       8.22e-01     1.22e+02   Skip BFGS
geophys. misfits: 42.7 (target 30.0 [False]); 23.1 (target 30.0 [True]) | smallness misfit: 55.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [42.7 23.1]; minimum progress targets: [44.4 30. ]
Updating scaling for data misfits by  1.2972994963110862
New scales: [0.29562519 0.70437481]
   7  1.86e+03  2.73e+01  1.44e-01  2.96e+02    7.54e+01      0     100       1.02e+00     1.51e+02   Skip BFGS
geophys. misfits: 35.2 (target 30.0 [False]); 24.0 (target 30.0 [True]) | smallness misfit: 55.7 (target: 200.0 [True])
Beta cooling evaluation: progress: [35.2 24. ]; minimum progress targets: [34.1 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by  1.250334518050695
New scales: [0.3441607 0.6558393]
   8  9.31e+02  1.97e+01  1.50e-01  1.59e+02    9.76e+01      0     100       5.22e+00     1.64e+03
geophys. misfits: 20.2 (target 30.0 [True]); 19.5 (target 30.0 [True]) | smallness misfit: 67.5 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [20.2 19.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  1.5136363832742856
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.8117e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 1.5161e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x|    = 9.7608e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 9.7608e+01 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      50    <= iter          =      8
------------------------- DONE! -------------------------

Running inversion with SimPEG v0.25.1.dev1+g9a8c46e88
Alpha scales: [np.float64(3.723162838586789e-05), np.float64(0.0), np.float64(3.5278165494869866e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers:  [0.08846538 0.91153462]
/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:  [0.08846538 0.91153462]
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  9.72e+05  3.00e+05  0.00e+00  3.00e+05                         0           inf          inf
   1  9.72e+05  3.81e+04  4.51e-02  8.19e+04    1.41e+02      0      23       9.30e-04     8.41e+03
geophys. misfits: 60623.6 (target 30.0 [False]); 35890.2 (target 30.0 [False])
   2  1.94e+05  3.85e+03  1.09e-01  2.50e+04    1.37e+02      0     100       1.67e-03     2.96e+01   Skip BFGS
geophys. misfits: 8483.2 (target 30.0 [False]); 3397.2 (target 30.0 [False])
   3  3.89e+04  2.40e+02  1.41e-01  5.74e+03    1.30e+02      0     100       2.90e-02     1.39e+02   Skip BFGS
geophys. misfits: 560.9 (target 30.0 [False]); 209.2 (target 30.0 [False])
   4  7.78e+03  2.41e+01  1.51e-01  1.20e+03    1.01e+02      0     100       1.41e-01     1.55e+02   Skip BFGS
geophys. misfits: 38.1 (target 30.0 [False]); 22.8 (target 30.0 [True])
Updating scaling for data misfits by  1.317612787582283
New scales: [0.11337747 0.88662253]
   5  1.56e+03  1.12e+01  1.54e-01  2.51e+02    8.72e+01      0     100       3.27e+00     9.98e+02   Skip BFGS
geophys. misfits: 10.5 (target 30.0 [True]); 11.3 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 8.7690e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 3.7130e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x|    = 8.7191e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 8.7191e+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:302: 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:309: 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:353: 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:360: 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:367: 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:412: 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:419: UserWarning:

marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.

import discretize as Mesh
import matplotlib.pyplot as plt
import matplotlib.lines as mlines
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,
    cg_maxiter=100,
    cg_rtol=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,
    cg_maxiter=100,
    cg_rtol=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,
    cg_maxiter=100,
    cg_rtol=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",
)
cs_proxy = mlines.Line2D([], [], label="True Petrophysical Distribution")

ps = axes[3].scatter(
    wires.m1 * mcluster_map,
    wires.m2 * mcluster_map,
    marker="v",
    label="Recovered model crossplot",
)
axes[3].set_title("Petrophysical Distribution")
axes[3].legend(handles=[cs_proxy, ps])
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="--",
)
axes[7].scatter(
    wires.m1 * mcluster_no_map,
    wires.m2 * mcluster_no_map,
    marker="v",
    label="Recovered model crossplot",
)
cs_modeled_proxy = mlines.Line2D(
    [], [], linestyle="--", label="Modeled Petro. Distribution"
)

axes[7].set_title("Petrophysical Distribution")
axes[7].legend(handles=[cs_proxy, cs_modeled_proxy, ps])
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")
axes[11].legend(handles=[cs_proxy, ps])
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()

Estimated memory usage: 321 MB