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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.dev9+g43b0120dd
Alpha scales: [np.float64(3.491829047521848), np.float64(0.0), np.float64(3.5189375267533188e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.10096782 0.89903218]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.10096782 0.89903218]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.99e+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.99e+01 1.79e+03 1.69e+02 5.13e+03 1.41e+02 0 19 9.68e-04 9.08e+03
geophys. misfits: 8329.3 (target 30.0 [False]); 1050.9 (target 30.0 [False]) | smallness misfit: 3982.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [8329.3 1050.9]; minimum progress targets: [240000. 240000.]
2 1.99e+01 6.12e+01 4.06e+01 8.68e+02 1.40e+02 0 100 2.79e-03 2.53e+01 Skip BFGS
geophys. misfits: 475.7 (target 30.0 [False]); 14.7 (target 30.0 [True]) | smallness misfit: 1377.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [475.7 14.7]; minimum progress targets: [6663.4 840.7]
Updating scaling for data misfits by 2.0454061133079002
New scales: [0.18680272 0.81319728]
3 1.99e+01 4.82e+01 4.16e+01 8.75e+02 8.59e+01 0 100 1.40e-02 7.08e+00 Skip BFGS
geophys. misfits: 196.5 (target 30.0 [False]); 14.2 (target 30.0 [True]) | smallness misfit: 1132.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [196.5 14.2]; minimum progress targets: [380.6 30. ]
Updating scaling for data misfits by 2.119786367854586
New scales: [0.3274799 0.6725201]
4 1.99e+01 3.79e+01 4.29e+01 8.90e+02 8.21e+01 0 100 2.23e-01 1.16e+02
geophys. misfits: 85.5 (target 30.0 [False]); 14.8 (target 30.0 [True]) | smallness misfit: 1074.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [85.5 14.8]; minimum progress targets: [157.2 30. ]
Updating scaling for data misfits by 2.033251004699464
New scales: [0.49750769 0.50249231]
5 1.99e+01 3.16e+01 4.36e+01 8.98e+02 8.23e+01 0 100 3.13e-02 1.33e+01 Skip BFGS
geophys. misfits: 46.7 (target 30.0 [False]); 16.7 (target 30.0 [True]) | smallness misfit: 1017.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [46.7 16.7]; minimum progress targets: [68.4 30. ]
Updating scaling for data misfits by 1.8017555577268096
New scales: [0.64078944 0.35921056]
6 1.99e+01 2.89e+01 4.39e+01 9.01e+02 7.67e+01 0 100 5.62e-02 1.26e+01 Skip BFGS
geophys. misfits: 33.7 (target 30.0 [False]); 20.3 (target 30.0 [True]) | smallness misfit: 968.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [33.7 20.3]; minimum progress targets: [37.4 30. ]
Updating scaling for data misfits by 1.4808472768239511
New scales: [0.72539977 0.27460023]
7 1.99e+01 2.78e+01 4.40e+01 9.02e+02 6.51e+01 0 100 1.26e-01 1.37e+01 Skip BFGS
geophys. misfits: 29.0 (target 30.0 [True]); 24.7 (target 30.0 [True]) | smallness misfit: 930.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [29. 24.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.1251683149444558
8 1.99e+01 2.84e+01 4.42e+01 9.07e+02 2.51e+01 0 100 4.58e-01 1.19e+01 Skip BFGS
geophys. misfits: 28.9 (target 30.0 [True]); 27.1 (target 30.0 [True]) | smallness misfit: 904.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [28.9 27.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.2078417970919952
9 1.99e+01 2.87e+01 4.44e+01 9.11e+02 1.72e+01 0 100 6.49e-01 1.12e+01 Skip BFGS
geophys. misfits: 28.8 (target 30.0 [True]); 28.7 (target 30.0 [True]) | smallness misfit: 889.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [28.8 28.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.2607230158404406
10 1.99e+01 2.90e+01 4.45e+01 9.13e+02 1.31e+01 0 100 7.03e-01 9.18e+00 Skip BFGS
geophys. misfits: 28.7 (target 30.0 [True]); 29.8 (target 30.0 [True]) | smallness misfit: 879.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [28.7 29.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.29428625574706
11 1.99e+01 2.91e+01 4.45e+01 9.14e+02 9.35e+00 0 100 1.21e+00 1.13e+01 Skip BFGS
geophys. misfits: 28.6 (target 30.0 [True]); 30.4 (target 30.0 [False]) | smallness misfit: 873.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [28.6 30.4]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.047548086814322
New scales: [0.7160505 0.2839495]
12 9.93e+00 1.52e+01 4.55e+01 4.67e+02 9.15e+01 0 100 1.24e-02 4.93e+00
geophys. misfits: 14.4 (target 30.0 [True]); 17.1 (target 30.0 [True]) | smallness misfit: 953.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.4 17.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.4804535043827047
13 9.93e+00 1.76e+01 4.77e+01 4.91e+02 6.59e+01 0 100 1.10e-01 1.39e+01
geophys. misfits: 14.1 (target 30.0 [True]); 26.6 (target 30.0 [True]) | smallness misfit: 804.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.1 26.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.045460463842888
14 9.93e+00 2.16e+01 5.00e+01 5.18e+02 7.46e+01 0 100 3.91e-01 6.29e+01
geophys. misfits: 14.1 (target 30.0 [True]); 40.6 (target 30.0 [False]) | smallness misfit: 685.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.1 40.6]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 2.130943278543393
New scales: [0.5419982 0.4580018]
15 4.97e+00 1.30e+01 5.15e+01 2.69e+02 9.40e+01 0 100 9.42e+00 2.05e+03
geophys. misfits: 10.5 (target 30.0 [True]); 16.0 (target 30.0 [True]) | smallness misfit: 792.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [10.5 16. ]; minimum progress targets: [30. 32.4]
Warming alpha_pgi to favor clustering: 9.55046179940674
16 4.97e+00 1.79e+01 5.96e+01 3.14e+02 9.84e+01 0 100 5.24e-01 1.09e+03
geophys. misfits: 11.3 (target 30.0 [True]); 25.6 (target 30.0 [True]) | smallness misfit: 611.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [11.3 25.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 18.242355180359997
17 4.97e+00 1.71e+01 7.10e+01 3.70e+02 1.03e+02 0 100 3.66e-01 4.28e+02
geophys. misfits: 11.4 (target 30.0 [True]); 23.9 (target 30.0 [True]) | smallness misfit: 494.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [11.4 23.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 35.458940030540404
18 4.97e+00 2.28e+01 8.70e+01 4.55e+02 1.17e+02 0 100 1.78e+01 1.76e+04
geophys. misfits: 14.8 (target 30.0 [True]); 32.2 (target 30.0 [False]) | smallness misfit: 366.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.8 32.2]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 2.0278720191540045
New scales: [0.36851392 0.63148608]
19 2.48e+00 2.13e+01 8.70e+01 2.37e+02 1.08e+02 0 100 7.95e-02 9.51e+02
geophys. misfits: 15.5 (target 30.0 [True]); 24.7 (target 30.0 [True]) | smallness misfit: 343.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.5 24.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 55.78422544263968
20 2.48e+00 2.47e+01 1.02e+02 2.77e+02 1.04e+02 0 100 2.16e+00 2.25e+03
geophys. misfits: 17.4 (target 30.0 [True]); 29.0 (target 30.0 [True]) | smallness misfit: 283.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.4 29. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 77.05678672597305
21 2.48e+00 3.38e+01 1.12e+02 3.13e+02 1.08e+02 0 100 1.62e+00 3.76e+03
geophys. misfits: 24.2 (target 30.0 [True]); 39.4 (target 30.0 [False]) | smallness misfit: 243.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.2 39.4]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.2383539494813824
New scales: [0.32030283 0.67969717]
22 1.24e+00 1.98e+01 1.19e+02 1.68e+02 1.01e+02 0 100 9.83e-01 3.22e+03
geophys. misfits: 15.3 (target 30.0 [True]); 21.9 (target 30.0 [True]) | smallness misfit: 236.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.3 21.9]; minimum progress targets: [30. 31.5]
Warming alpha_pgi to favor clustering: 128.12579950834123
23 1.24e+00 2.34e+01 1.46e+02 2.04e+02 1.05e+02 0 100 1.67e+00 5.40e+03
geophys. misfits: 22.1 (target 30.0 [True]); 24.0 (target 30.0 [True]) | smallness misfit: 226.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.1 24. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 167.0954247923413
24 1.24e+00 2.34e+01 1.61e+02 2.24e+02 1.09e+02 1 100 5.31e-01 2.89e+03
geophys. misfits: 22.7 (target 30.0 [True]); 23.7 (target 30.0 [True]) | smallness misfit: 206.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.7 23.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 216.4393096032495
25 1.24e+00 2.44e+01 1.82e+02 2.50e+02 1.14e+02 1 100 2.60e-01 1.08e+03
geophys. misfits: 24.6 (target 30.0 [True]); 24.3 (target 30.0 [True]) | smallness misfit: 186.5 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [24.6 24.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 265.4455967768883
------------------------- STOP! -------------------------
1 : |fc-fOld| = 5.5080e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 3.1223e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.1362e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.1362e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 25
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev9+g43b0120dd
Alpha scales: [np.float64(0.00034866048567845284), np.float64(0.0), np.float64(3.4877913684558853e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.10096782 0.89903218]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.10096782 0.89903218]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 2.01e+03 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 2.01e+03 6.77e+04 2.17e+01 1.11e+05 1.41e+02 0 15 3.26e-04 3.06e+03
geophys. misfits: 90756.6 (target 30.0 [False]); 65086.5 (target 30.0 [False]) | smallness misfit: 249.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [90756.6 65086.5]; minimum progress targets: [240000. 240000.]
2 2.01e+03 7.95e+01 5.52e-01 1.19e+03 1.35e+02 0 100 1.60e-02 4.42e+02 Skip BFGS
geophys. misfits: 604.2 (target 30.0 [False]); 20.5 (target 30.0 [True]) | smallness misfit: 123.9 (target: 200.0 [True])
Beta cooling evaluation: progress: [604.2 20.5]; minimum progress targets: [72605.3 52069.2]
Updating scaling for data misfits by 1.4611213240493328
New scales: [0.14096321 0.85903679]
3 2.01e+03 2.88e+01 1.35e-01 3.00e+02 1.13e+02 0 100 1.49e+00 3.83e+03 Skip BFGS
geophys. misfits: 70.4 (target 30.0 [False]); 22.0 (target 30.0 [True]) | smallness misfit: 68.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [70.4 22. ]; minimum progress targets: [483.3 30. ]
Updating scaling for data misfits by 1.3630837374236162
New scales: [0.18278924 0.81721076]
4 2.01e+03 2.63e+01 1.31e-01 2.90e+02 1.05e+02 0 100 8.25e-03 3.58e+01
geophys. misfits: 48.4 (target 30.0 [False]); 21.3 (target 30.0 [True]) | smallness misfit: 54.0 (target: 200.0 [True])
Beta cooling evaluation: progress: [48.4 21.3]; minimum progress targets: [56.3 30. ]
Updating scaling for data misfits by 1.4067636808329327
New scales: [0.23934545 0.76065455]
5 2.01e+03 2.42e+01 1.32e-01 2.88e+02 7.35e+01 0 100 1.36e-02 7.34e+00
geophys. misfits: 32.8 (target 30.0 [False]); 21.5 (target 30.0 [True]) | smallness misfit: 48.6 (target: 200.0 [True])
Beta cooling evaluation: progress: [32.8 21.5]; minimum progress targets: [38.7 30. ]
Updating scaling for data misfits by 1.3933148451037627
New scales: [0.3047911 0.6952089]
6 2.01e+03 2.31e+01 1.32e-01 2.89e+02 6.80e+01 0 100 3.86e-01 4.49e+01
geophys. misfits: 23.8 (target 30.0 [True]); 22.8 (target 30.0 [True]) | smallness misfit: 49.0 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [23.8 22.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.2858204207294475
------------------------- STOP! -------------------------
1 : |fc-fOld| = 3.9784e-01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 5.8077e-02 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 6.8016e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 6.8016e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 6
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev9+g43b0120dd
Alpha scales: [np.float64(3.973700929267243e-05), np.float64(0.0), np.float64(3.484360468950745e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.10096782 0.89903218]
/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: [0.10096782 0.89903218]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.05e+06 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.05e+06 3.94e+04 4.24e-02 8.39e+04 1.41e+02 0 28 7.79e-04 7.31e+03
geophys. misfits: 52978.8 (target 30.0 [False]); 37912.3 (target 30.0 [False])
2 2.10e+05 4.34e+03 1.07e-01 2.67e+04 1.37e+02 0 100 1.99e-02 3.63e+02 Skip BFGS
geophys. misfits: 7838.1 (target 30.0 [False]); 3947.9 (target 30.0 [False])
3 4.19e+04 2.69e+02 1.41e-01 6.19e+03 1.31e+02 0 100 7.78e-02 4.01e+02 Skip BFGS
geophys. misfits: 508.2 (target 30.0 [False]); 242.0 (target 30.0 [False])
4 8.39e+03 2.36e+01 1.51e-01 1.29e+03 1.03e+02 0 100 4.44e-03 5.51e+00 Skip BFGS
geophys. misfits: 31.5 (target 30.0 [False]); 22.8 (target 30.0 [True])
Updating scaling for data misfits by 1.318599036504984
New scales: [0.12898679 0.87101321]
5 1.68e+03 1.06e+01 1.54e-01 2.69e+02 7.70e+01 0 100 8.20e-02 2.06e+01 Skip BFGS
geophys. misfits: 8.6 (target 30.0 [True]); 10.9 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 8.8322e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 2.2495e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 7.7022e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 7.7022e+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 29.520 seconds)
Estimated memory usage: 333 MB