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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 v0.25.2.dev8+g3c268dda4
Alpha scales: [np.float64(3.5449825467668394), np.float64(0.0), np.float64(3.4891094200376346e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09610193 0.90389807]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09610193 0.90389807]
================================================= 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+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.86e+01 1.33e+03 1.71e+02 4.52e+03 1.40e+02 0 21 6.46e-04 5.92e+03
geophys. misfits: 7479.1 (target 30.0 [False]); 676.1 (target 30.0 [False]) | smallness misfit: 3920.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [7479.1 676.1]; minimum progress targets: [240000. 240000.]
2 1.86e+01 7.19e+01 4.17e+01 8.49e+02 1.40e+02 0 100 4.32e-03 2.56e+01 Skip BFGS
geophys. misfits: 444.8 (target 30.0 [False]); 32.3 (target 30.0 [False]) | smallness misfit: 1578.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [444.8 32.3]; minimum progress targets: [5983.3 540.9]
3 1.86e+01 7.12e+01 4.08e+01 8.31e+02 5.50e+01 0 100 9.98e-02 1.84e+01 Skip BFGS
geophys. misfits: 436.5 (target 30.0 [False]); 32.4 (target 30.0 [False]) | smallness misfit: 1389.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [436.5 32.4]; minimum progress targets: [355.9 30. ]
Decreasing beta to counter data misfit decrase plateau.
4 9.31e+00 3.94e+01 4.31e+01 4.41e+02 8.39e+01 0 100 2.34e+00 6.84e+02
geophys. misfits: 163.1 (target 30.0 [False]); 26.3 (target 30.0 [True]) | smallness misfit: 1439.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [163.1 26.3]; minimum progress targets: [349.2 30. ]
Updating scaling for data misfits by 1.1418882428381512
New scales: [0.10826144 0.89173856]
5 9.31e+00 3.90e+01 4.33e+01 4.43e+02 6.84e+01 0 100 2.06e-02 1.39e+01 Skip BFGS
geophys. misfits: 142.1 (target 30.0 [False]); 26.5 (target 30.0 [True]) | smallness misfit: 1391.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [142.1 26.5]; minimum progress targets: [130.4 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.1331371439877118
New scales: [0.12093199 0.87906801]
6 4.66e+00 2.81e+01 4.51e+01 2.38e+02 7.44e+01 0 100 8.76e-01 1.86e+02
geophys. misfits: 61.0 (target 30.0 [False]); 23.6 (target 30.0 [True]) | smallness misfit: 1456.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [61. 23.6]; minimum progress targets: [113.7 30. ]
Updating scaling for data misfits by 1.270678415635522
New scales: [0.14879506 0.85120494]
7 4.66e+00 2.57e+01 4.56e+01 2.38e+02 6.96e+01 0 100 9.87e-02 2.40e+01 Skip BFGS
geophys. misfits: 38.0 (target 30.0 [False]); 23.5 (target 30.0 [True]) | smallness misfit: 1434.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [38. 23.5]; minimum progress targets: [48.8 30. ]
Updating scaling for data misfits by 1.2747084221849039
New scales: [0.18222195 0.81777805]
8 4.66e+00 2.52e+01 4.57e+01 2.38e+02 4.74e+01 0 100 8.62e-01 5.28e+01
geophys. misfits: 33.4 (target 30.0 [False]); 23.4 (target 30.0 [True]) | smallness misfit: 1384.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [33.4 23.4]; minimum progress targets: [30.4 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.2819027988582374
New scales: [0.2221778 0.7778222]
9 2.33e+00 1.99e+01 4.72e+01 1.30e+02 6.83e+01 0 100 2.22e-01 3.85e+01
geophys. misfits: 15.0 (target 30.0 [True]); 21.3 (target 30.0 [True]) | smallness misfit: 1551.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [15. 21.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.7021413790613629
10 2.33e+00 2.06e+01 4.89e+01 1.34e+02 3.52e+01 0 100 7.87e-01 3.20e+01
geophys. misfits: 14.7 (target 30.0 [True]); 22.2 (target 30.0 [True]) | smallness misfit: 1282.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.7 22.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.8861376981355944
11 2.33e+00 2.12e+01 5.15e+01 1.41e+02 3.76e+01 0 100 1.16e+00 4.75e+01
geophys. misfits: 13.6 (target 30.0 [True]); 23.4 (target 30.0 [True]) | smallness misfit: 1058.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [13.6 23.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 5.027219107404559
12 2.33e+00 2.22e+01 5.53e+01 1.51e+02 5.63e+01 0 100 2.41e+00 1.69e+02
geophys. misfits: 13.4 (target 30.0 [True]); 24.7 (target 30.0 [True]) | smallness misfit: 848.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [13.4 24.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 8.682470417367277
13 2.33e+00 2.38e+01 6.02e+01 1.64e+02 7.70e+01 0 100 1.22e+00 2.33e+02
geophys. misfits: 14.2 (target 30.0 [True]); 26.6 (target 30.0 [True]) | smallness misfit: 662.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.2 26.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 14.094190686039097
14 2.33e+00 2.53e+01 6.60e+01 1.79e+02 8.70e+01 0 100 1.24e+00 3.44e+02
geophys. misfits: 15.1 (target 30.0 [True]); 28.2 (target 30.0 [True]) | smallness misfit: 505.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.1 28.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 21.53388130980507
15 2.33e+00 2.79e+01 7.14e+01 1.94e+02 9.50e+01 0 100 8.24e-01 3.10e+02
geophys. misfits: 18.2 (target 30.0 [True]); 30.7 (target 30.0 [False]) | smallness misfit: 380.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.2 30.7]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.644458269673727
New scales: [0.14799284 0.85200716]
16 1.16e+00 2.37e+01 7.39e+01 1.10e+02 9.03e+01 0 100 7.58e-01 1.80e+02
geophys. misfits: 14.5 (target 30.0 [True]); 25.3 (target 30.0 [True]) | smallness misfit: 435.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.5 25.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 34.95848976218021
17 1.16e+00 2.28e+01 8.56e+01 1.22e+02 8.30e+01 0 100 9.24e-01 2.14e+02
geophys. misfits: 14.9 (target 30.0 [True]); 24.1 (target 30.0 [True]) | smallness misfit: 360.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.9 24.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 56.967326263182066
18 1.16e+00 2.28e+01 1.00e+02 1.39e+02 9.45e+01 0 100 1.08e+00 3.48e+02
geophys. misfits: 17.2 (target 30.0 [True]); 23.8 (target 30.0 [True]) | smallness misfit: 317.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.2 23.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 85.67547511748367
19 1.16e+00 2.26e+01 1.19e+02 1.61e+02 1.09e+02 0 100 1.12e+00 5.43e+02
geophys. misfits: 24.4 (target 30.0 [True]); 22.3 (target 30.0 [True]) | smallness misfit: 262.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.4 22.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 110.3132289518986
20 1.16e+00 2.48e+01 1.30e+02 1.76e+02 1.09e+02 0 100 5.81e-01 3.80e+02
geophys. misfits: 24.0 (target 30.0 [True]); 25.0 (target 30.0 [True]) | smallness misfit: 273.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [24. 25.]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 135.1002618765942
21 1.16e+00 2.47e+01 1.39e+02 1.86e+02 1.10e+02 0 100 8.21e-01 5.00e+02
geophys. misfits: 28.7 (target 30.0 [True]); 24.0 (target 30.0 [True]) | smallness misfit: 211.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [28.7 24. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 154.79994160652492
22 1.16e+00 2.85e+01 1.40e+02 1.91e+02 1.12e+02 0 100 9.94e+00 6.63e+03
geophys. misfits: 44.2 (target 30.0 [False]); 25.7 (target 30.0 [True]) | smallness misfit: 192.5 (target: 200.0 [True])
Beta cooling evaluation: progress: [44.2 25.7]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.1661018510576087
New scales: [0.1684343 0.8315657]
23 5.82e-01 2.57e+01 1.38e+02 1.06e+02 9.92e+01 0 100 2.69e-02 1.74e+02
geophys. misfits: 21.9 (target 30.0 [True]); 26.5 (target 30.0 [True]) | smallness misfit: 182.5 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [21.9 26.5]; minimum progress targets: [35.4 30. ]
Warming alpha_pgi to favor clustering: 193.67675744122803
------------------------- STOP! -------------------------
1 : |fc-fOld| = 4.2567e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 6.5801e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 9.9170e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 9.9170e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 23
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev8+g3c268dda4
Alpha scales: [np.float64(0.00035029589587952685), np.float64(0.0), np.float64(3.503472583999144e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09610193 0.90389807]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09610193 0.90389807]
================================================= 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+03 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.94e+03 6.58e+04 2.25e+01 1.10e+05 1.41e+02 0 15 2.91e-04 2.67e+03
geophys. misfits: 91897.1 (target 30.0 [False]); 63053.3 (target 30.0 [False]) | smallness misfit: 249.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [91897.1 63053.3]; minimum progress targets: [240000. 240000.]
2 1.94e+03 7.74e+01 5.48e-01 1.14e+03 1.36e+02 0 100 5.70e-02 1.55e+03 Skip BFGS
geophys. misfits: 576.1 (target 30.0 [False]); 24.4 (target 30.0 [True]) | smallness misfit: 97.7 (target: 200.0 [True])
Beta cooling evaluation: progress: [576.1 24.4]; minimum progress targets: [73517.7 50442.6]
Updating scaling for data misfits by 1.2318021034936595
New scales: [0.11579895 0.88420105]
3 1.94e+03 2.98e+01 1.39e-01 3.00e+02 1.09e+02 0 100 1.70e-01 4.29e+02 Skip BFGS
geophys. misfits: 87.2 (target 30.0 [False]); 22.3 (target 30.0 [True]) | smallness misfit: 71.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [87.2 22.3]; minimum progress targets: [460.9 30. ]
Updating scaling for data misfits by 1.3478759182480406
New scales: [0.15003851 0.84996149]
4 1.94e+03 2.79e+01 1.29e-01 2.79e+02 8.86e+01 0 100 4.47e-01 9.30e+02
geophys. misfits: 60.7 (target 30.0 [False]); 22.1 (target 30.0 [True]) | smallness misfit: 47.8 (target: 200.0 [True])
Beta cooling evaluation: progress: [60.7 22.1]; minimum progress targets: [69.8 30. ]
Updating scaling for data misfits by 1.3546993422739106
New scales: [0.1929866 0.8070134]
5 1.94e+03 2.67e+01 1.30e-01 2.80e+02 8.64e+01 0 100 9.94e-02 8.87e+01 Skip BFGS
geophys. misfits: 44.6 (target 30.0 [False]); 22.4 (target 30.0 [True]) | smallness misfit: 48.5 (target: 200.0 [True])
Beta cooling evaluation: progress: [44.6 22.4]; minimum progress targets: [48.6 30. ]
Updating scaling for data misfits by 1.339063190553036
New scales: [0.24255008 0.75744992]
6 1.94e+03 2.56e+01 1.31e-01 2.81e+02 6.89e+01 0 100 1.17e-01 1.86e+01
geophys. misfits: 34.5 (target 30.0 [False]); 22.7 (target 30.0 [True]) | smallness misfit: 48.9 (target: 200.0 [True])
Beta cooling evaluation: progress: [34.5 22.7]; minimum progress targets: [35.6 30. ]
Updating scaling for data misfits by 1.3194894094168068
New scales: [0.29702513 0.70297487]
7 1.94e+03 2.47e+01 1.32e-01 2.81e+02 6.53e+01 0 100 2.54e-01 2.46e+01 Skip BFGS
geophys. misfits: 28.2 (target 30.0 [True]); 23.2 (target 30.0 [True]) | smallness misfit: 49.3 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [28.2 23.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.1796487785374392
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.0320e-01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 1.8510e-02 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 6.5267e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 6.5267e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 7
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev8+g3c268dda4
Alpha scales: [np.float64(3.491767990037381e-05), np.float64(0.0), np.float64(4.776411268868842e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09610193 0.90389807]
/home/vsts/work/1/s/simpeg/directives/_directives.py:334: UserWarning: There is no PGI regularization. Smallness target is turned off (TriggerSmall flag)
getattr(r, ruleType)()
Initial data misfit scales: [0.09610193 0.90389807]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 8.60e+05 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 8.60e+05 3.42e+04 4.98e-02 7.70e+04 1.41e+02 0 23 9.88e-04 9.06e+03
geophys. misfits: 52839.0 (target 30.0 [False]); 32174.6 (target 30.0 [False])
2 1.72e+05 3.19e+03 1.13e-01 2.26e+04 1.37e+02 0 92 9.42e-04 1.61e+01 Skip BFGS
geophys. misfits: 6234.3 (target 30.0 [False]); 2866.0 (target 30.0 [False])
3 3.44e+04 2.22e+02 1.43e-01 5.15e+03 1.28e+02 0 100 1.52e-02 6.62e+01 Skip BFGS
geophys. misfits: 396.3 (target 30.0 [False]); 203.2 (target 30.0 [False])
4 6.88e+03 3.64e+01 1.53e-01 1.09e+03 9.98e+01 0 100 1.72e-02 1.68e+01 Skip BFGS
geophys. misfits: 32.2 (target 30.0 [False]); 36.8 (target 30.0 [False])
5 1.38e+03 2.15e+01 1.57e-01 2.37e+02 7.32e+01 0 100 9.28e-01 1.88e+02 Skip BFGS
geophys. misfits: 12.3 (target 30.0 [True]); 22.5 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 9.4605e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.8921e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 7.3204e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 7.3204e+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.
axes[1].plot(mesh.cell_centers_x, wires.m1 * mcluster_map, "b.-", ms=5, marker="v")
/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.
axes[2].plot(mesh.cell_centers_x, wires.m2 * mcluster_map, "r.-", ms=5, marker="v")
/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.
axes[5].plot(mesh.cell_centers_x, wires.m1 * mcluster_no_map, "b.-", ms=5, marker="v")
/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.
axes[6].plot(mesh.cell_centers_x, wires.m2 * mcluster_no_map, "r.-", ms=5, marker="v")
/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'
CSF = axes[7].contour(
/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.
axes[9].plot(mesh.cell_centers_x, wires.m1 * mtik, "b.-", ms=5, marker="v")
/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.
axes[10].plot(mesh.cell_centers_x, wires.m2 * mtik, "r.-", ms=5, marker="v")
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()
Total running time of the script: (0 minutes 26.705 seconds)
Estimated memory usage: 333 MB