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.dev9+g471344c9a
Alpha scales: [np.float64(3.462290333302034), np.float64(0.0), np.float64(3.4660322501327066e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers:  [0.09375869 0.90624131]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales:  [0.09375869 0.90624131]
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  1.94e+01  3.00e+05  0.00e+00  3.00e+05                         0           inf          inf
   1  1.94e+01  5.98e+02  1.76e+02  4.01e+03    1.41e+02      0      23       2.37e-04     2.19e+03
geophys. misfits: 3067.6 (target 30.0 [False]); 342.6 (target 30.0 [False]) | smallness misfit: 3809.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [3067.6  342.6]; minimum progress targets: [240000. 240000.]
   2  1.94e+01  6.54e+01  4.05e+01  8.49e+02    1.37e+02      0     100       7.07e-01     1.68e+03   Skip BFGS
geophys. misfits: 494.0 (target 30.0 [False]); 21.1 (target 30.0 [True]) | smallness misfit: 1278.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [494.   21.1]; minimum progress targets: [2454.1  274. ]
Updating scaling for data misfits by  1.4236462396464715
New scales: [0.12837989 0.87162011]
   3  1.94e+01  5.87e+01  4.09e+01  8.51e+02    9.29e+01      0     100       3.65e-03     5.96e+00
geophys. misfits: 314.8 (target 30.0 [False]); 20.9 (target 30.0 [True]) | smallness misfit: 1068.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [314.8  20.9]; minimum progress targets: [395.2  30. ]
Updating scaling for data misfits by  1.432713294920553
New scales: [0.17425161 0.82574839]
   4  1.94e+01  5.29e+01  4.17e+01  8.61e+02    7.09e+01      0     100       3.84e-02     8.58e+00
geophys. misfits: 205.1 (target 30.0 [False]); 20.7 (target 30.0 [True]) | smallness misfit: 1003.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [205.1  20.7]; minimum progress targets: [251.8  30. ]
Updating scaling for data misfits by  1.4458005562078486
New scales: [0.23377323 0.76622677]
   5  1.94e+01  4.78e+01  4.24e+01  8.70e+02    7.13e+01      0     100       4.74e-02     1.06e+01
geophys. misfits: 136.0 (target 30.0 [False]); 20.8 (target 30.0 [True]) | smallness misfit: 953.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [136.   20.8]; minimum progress targets: [164.1  30. ]
Updating scaling for data misfits by  1.4401928929298253
New scales: [0.30526513 0.69473487]
   6  1.94e+01  4.34e+01  4.30e+01  8.76e+02    7.11e+01      0     100       3.76e-01     7.82e+01   Skip BFGS
geophys. misfits: 94.4 (target 30.0 [False]); 21.0 (target 30.0 [True]) | smallness misfit: 915.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [94.4 21. ]; minimum progress targets: [108.8  30. ]
Updating scaling for data misfits by  1.4273580815650464
New scales: [0.38543921 0.61456079]
   7  1.94e+01  3.97e+01  4.34e+01  8.81e+02    7.08e+01      0     100       3.34e-02     6.82e+00   Skip BFGS
geophys. misfits: 69.0 (target 30.0 [False]); 21.4 (target 30.0 [True]) | smallness misfit: 887.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [69.  21.4]; minimum progress targets: [75.5 30. ]
Updating scaling for data misfits by  1.4048230034777132
New scales: [0.46838896 0.53161104]
   8  1.94e+01  3.67e+01  4.38e+01  8.84e+02    6.74e+01      0     100       4.06e-02     6.26e+00   Skip BFGS
geophys. misfits: 53.5 (target 30.0 [False]); 21.9 (target 30.0 [True]) | smallness misfit: 863.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [53.5 21.9]; minimum progress targets: [55.2 30. ]
Updating scaling for data misfits by  1.3706996101542341
New scales: [0.54703755 0.45296245]
   9  1.94e+01  3.43e+01  4.40e+01  8.87e+02    6.46e+01      0     100       1.08e-01     1.34e+01   Skip BFGS
geophys. misfits: 43.9 (target 30.0 [False]); 22.7 (target 30.0 [True]) | smallness misfit: 843.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [43.9 22.7]; minimum progress targets: [42.8 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by  1.3230560507627565
New scales: [0.61506477 0.38493523]
  10  9.69e+00  1.75e+01  4.52e+01  4.55e+02    8.29e+01      0     100       1.25e+00     5.77e+02
geophys. misfits: 16.1 (target 30.0 [True]); 19.6 (target 30.0 [True]) | smallness misfit: 853.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.1 19.6]; minimum progress targets: [35.1 30. ]
Warming alpha_pgi to favor clustering:  1.69525949429319
  11  9.69e+00  1.79e+01  4.64e+01  4.67e+02    7.78e+01      0     100       8.33e-02     4.85e+01
geophys. misfits: 15.8 (target 30.0 [True]); 21.3 (target 30.0 [True]) | smallness misfit: 755.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.8 21.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  2.800453046738079
  12  9.69e+00  1.80e+01  4.81e+01  4.84e+02    6.95e+01      0     100       1.10e+00     1.36e+02
geophys. misfits: 14.4 (target 30.0 [True]); 23.6 (target 30.0 [True]) | smallness misfit: 652.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.4 23.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  4.69188843691559
  13  9.69e+00  1.86e+01  5.07e+01  5.09e+02    8.55e+01      0     100       2.06e+00     4.83e+02
geophys. misfits: 12.8 (target 30.0 [True]); 27.7 (target 30.0 [True]) | smallness misfit: 547.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [12.8 27.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  8.024748385583178
  14  9.69e+00  2.01e+01  5.42e+01  5.46e+02    1.01e+02      0     100       1.63e+00     9.59e+02
geophys. misfits: 12.8 (target 30.0 [True]); 31.8 (target 30.0 [False]) | smallness misfit: 457.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [12.8 31.8]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by  2.3451299705784807
New scales: [0.40523762 0.59476238]
  15  4.84e+00  1.74e+01  5.49e+01  2.83e+02    1.08e+02      0     100       2.08e-01     1.69e+02
geophys. misfits: 11.9 (target 30.0 [True]); 21.2 (target 30.0 [True]) | smallness misfit: 451.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [11.9 21.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  15.805285065629274
  16  4.84e+00  1.82e+01  6.22e+01  3.19e+02    9.70e+01      0     100       2.77e+00     9.90e+02
geophys. misfits: 11.4 (target 30.0 [True]); 22.7 (target 30.0 [True]) | smallness misfit: 394.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [11.4 22.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  31.162973273598524
  17  4.84e+00  1.69e+01  7.42e+01  3.76e+02    1.15e+02      0     100       1.04e+00     1.29e+03
geophys. misfits: 14.9 (target 30.0 [True]); 18.2 (target 30.0 [True]) | smallness misfit: 325.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.9 18.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  56.93188520465576
  18  4.84e+00  1.80e+01  9.23e+01  4.65e+02    1.26e+02      0     100       6.12e+00     1.13e+04
geophys. misfits: 16.3 (target 30.0 [True]); 19.2 (target 30.0 [True]) | smallness misfit: 280.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.3 19.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  96.94398026744815
  19  4.84e+00  2.39e+01  1.11e+02  5.61e+02    1.31e+02      0     100       2.73e-01     3.17e+03
geophys. misfits: 29.9 (target 30.0 [True]); 19.9 (target 30.0 [True]) | smallness misfit: 226.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [29.9 19.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  121.91347313872129
  20  4.84e+00  1.91e+01  1.13e+02  5.68e+02    1.32e+02      1     100       9.65e-01     3.56e+03
geophys. misfits: 19.8 (target 30.0 [True]); 18.6 (target 30.0 [True]) | smallness misfit: 181.8 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [19.8 18.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering:  190.45555554125082
------------------------- STOP! -------------------------
1 : |fc-fOld| = 5.5574e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 5.9302e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x|    = 1.3236e+02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 1.3236e+02 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      50    <= iter          =     20
------------------------- DONE! -------------------------

Running inversion with SimPEG v0.25.1.dev9+g471344c9a
Alpha scales: [np.float64(0.00034018480896722713), np.float64(0.0), np.float64(3.404260339291685e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers:  [0.09375869 0.90624131]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales:  [0.09375869 0.90624131]
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  1.93e+03  3.00e+05  0.00e+00  3.00e+05                         0           inf          inf
   1  1.93e+03  6.54e+04  2.27e+01  1.09e+05    1.41e+02      0      15       2.94e-04     2.72e+03
geophys. misfits: 93732.9 (target 30.0 [False]); 62453.9 (target 30.0 [False]) | smallness misfit: 248.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [93732.9 62453.9]; minimum progress targets: [240000. 240000.]
   2  1.93e+03  7.50e+01  5.52e-01  1.14e+03    1.36e+02      0     100       1.60e-01     4.32e+03   Skip BFGS
geophys. misfits: 602.4 (target 30.0 [False]); 20.5 (target 30.0 [True]) | smallness misfit: 107.8 (target: 200.0 [True])
Beta cooling evaluation: progress: [602.4  20.5]; minimum progress targets: [74986.3 49963.1]
Updating scaling for data misfits by  1.4640310811430572
New scales: [0.13154262 0.86845738]
   3  1.93e+03  2.63e+01  1.38e-01  2.92e+02    1.06e+02      0     100       1.85e-02     1.16e+02   Skip BFGS
geophys. misfits: 79.9 (target 30.0 [False]); 18.2 (target 30.0 [True]) | smallness misfit: 85.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [79.9 18.2]; minimum progress targets: [481.9  30. ]
Updating scaling for data misfits by  1.6511350349395428
New scales: [0.20005916 0.79994084]
   4  1.93e+03  2.37e+01  1.32e-01  2.78e+02    9.43e+01      0     100       2.62e-02     4.42e+01
geophys. misfits: 44.6 (target 30.0 [False]); 18.5 (target 30.0 [True]) | smallness misfit: 57.1 (target: 200.0 [True])
Beta cooling evaluation: progress: [44.6 18.5]; minimum progress targets: [63.9 30. ]
Updating scaling for data misfits by  1.625497296725418
New scales: [0.28902772 0.71097228]
   5  1.93e+03  2.16e+01  1.32e-01  2.76e+02    7.67e+01      0     100       1.13e-01     1.35e+02
geophys. misfits: 29.1 (target 30.0 [True]); 18.6 (target 30.0 [True]) | smallness misfit: 59.7 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [29.1 18.6]; minimum progress targets: [35.6 30. ]
Warming alpha_pgi to favor clustering:  1.321718629948008
------------------------- STOP! -------------------------
1 : |fc-fOld| = 5.6491e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 1.3996e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x|    = 7.6745e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 7.6745e+01 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      50    <= iter          =      5
------------------------- DONE! -------------------------

Running inversion with SimPEG v0.25.1.dev9+g471344c9a
Alpha scales: [np.float64(4.556967670769874e-05), np.float64(0.0), np.float64(3.5137225852748354e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers:  [0.09375869 0.90624131]
/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:  [0.09375869 0.90624131]
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  9.07e+05  3.00e+05  0.00e+00  3.00e+05                         0           inf          inf
   1  9.07e+05  3.58e+04  4.76e-02  7.90e+04    1.41e+02      0      23       9.53e-04     8.81e+03
geophys. misfits: 56438.2 (target 30.0 [False]); 33694.7 (target 30.0 [False])
   2  1.81e+05  3.46e+03  1.11e-01  2.36e+04    1.38e+02      0      94       9.29e-04     1.62e+01   Skip BFGS
geophys. misfits: 7092.3 (target 30.0 [False]); 3082.8 (target 30.0 [False])
   3  3.63e+04  2.26e+02  1.43e-01  5.40e+03    1.29e+02      0     100       1.45e-02     6.59e+01   Skip BFGS
geophys. misfits: 454.5 (target 30.0 [False]); 202.2 (target 30.0 [False])
   4  7.26e+03  3.01e+01  1.52e-01  1.13e+03    1.02e+02      0     100       2.50e-01     2.56e+02   Skip BFGS
geophys. misfits: 29.9 (target 30.0 [True]); 30.1 (target 30.0 [False])
Updating scaling for data misfits by  1.002649769886167
New scales: [0.09353408 0.90646592]
   5  1.45e+03  1.67e+01  1.55e-01  2.42e+02    8.22e+01      0     100       3.39e-02     1.12e+01   Skip BFGS
geophys. misfits: 5.6 (target 30.0 [True]); 17.8 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 8.6234e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.2423e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x|    = 8.2246e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 8.2246e+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