Note
Go to the end to download the full example code.
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.dev41+g7d721d035
Alpha scales: [np.float64(3.428251980035151), np.float64(0.0), np.float64(3.4278237288720794e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09041516 0.90958484]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09041516 0.90958484]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.90e+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.90e+01 1.11e+03 1.71e+02 4.36e+03 1.41e+02 0 20 4.71e-04 4.28e+03
geophys. misfits: 7003.2 (target 30.0 [False]); 527.8 (target 30.0 [False]) | smallness misfit: 4037.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [7003.2 527.8]; minimum progress targets: [240000. 240000.]
2 1.90e+01 6.90e+01 4.08e+01 8.44e+02 1.40e+02 0 100 5.56e-03 2.41e+01 Skip BFGS
geophys. misfits: 501.3 (target 30.0 [False]); 26.0 (target 30.0 [True]) | smallness misfit: 1526.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [501.3 26. ]; minimum progress targets: [5602.6 422.3]
Updating scaling for data misfits by 1.1537750229438315
New scales: [0.10288824 0.89711176]
3 1.90e+01 6.58e+01 4.03e+01 8.30e+02 6.52e+01 0 100 3.24e-02 6.20e+00 Skip BFGS
geophys. misfits: 411.3 (target 30.0 [False]); 26.2 (target 30.0 [True]) | smallness misfit: 1293.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [411.3 26.2]; minimum progress targets: [401. 30.]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.147166455249313
New scales: [0.11626942 0.88373058]
4 9.49e+00 3.26e+01 4.29e+01 4.39e+02 8.48e+01 0 100 4.30e-02 1.58e+01
geophys. misfits: 138.8 (target 30.0 [False]); 18.6 (target 30.0 [True]) | smallness misfit: 1410.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [138.8 18.6]; minimum progress targets: [329. 30.]
Updating scaling for data misfits by 1.6104580927455672
New scales: [0.17483748 0.82516252]
5 9.49e+00 3.20e+01 4.35e+01 4.45e+02 6.92e+01 0 100 2.27e-01 3.72e+01 Skip BFGS
geophys. misfits: 90.4 (target 30.0 [False]); 19.6 (target 30.0 [True]) | smallness misfit: 1288.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [90.4 19.6]; minimum progress targets: [111.1 30. ]
Updating scaling for data misfits by 1.5318095267854863
New scales: [0.24503436 0.75496564]
6 9.49e+00 3.10e+01 4.40e+01 4.49e+02 6.91e+01 0 100 2.52e+00 4.47e+02 Skip BFGS
geophys. misfits: 63.3 (target 30.0 [False]); 20.6 (target 30.0 [True]) | smallness misfit: 1202.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [63.3 20.6]; minimum progress targets: [72.4 30. ]
Updating scaling for data misfits by 1.4570329541943086
New scales: [0.32106722 0.67893278]
7 9.49e+00 2.94e+01 4.44e+01 4.51e+02 8.49e+01 0 100 1.29e+00 5.55e+02 Skip BFGS
geophys. misfits: 46.5 (target 30.0 [False]); 21.4 (target 30.0 [True]) | smallness misfit: 1140.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [46.5 21.4]; minimum progress targets: [50.6 30. ]
Updating scaling for data misfits by 1.4027171505511487
New scales: [0.39880172 0.60119828]
8 9.49e+00 2.94e+01 4.45e+01 4.52e+02 8.67e+01 0 100 8.71e-02 6.00e+01
geophys. misfits: 40.3 (target 30.0 [False]); 22.2 (target 30.0 [True]) | smallness misfit: 1104.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [40.3 22.2]; minimum progress targets: [37.2 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.3499332613816515
New scales: [0.47242668 0.52757332]
9 4.75e+00 2.14e+01 4.58e+01 2.39e+02 7.80e+01 0 100 1.71e-01 5.12e+01
geophys. misfits: 24.9 (target 30.0 [True]); 18.2 (target 30.0 [True]) | smallness misfit: 1247.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.9 18.2]; minimum progress targets: [32.2 30. ]
Warming alpha_pgi to favor clustering: 1.4256506707490755
10 4.75e+00 2.24e+01 4.66e+01 2.44e+02 4.56e+01 0 100 1.12e+00 5.77e+01
geophys. misfits: 24.9 (target 30.0 [True]); 20.2 (target 30.0 [True]) | smallness misfit: 1067.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.9 20.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.9160903313537365
11 4.75e+00 2.28e+01 4.75e+01 2.48e+02 5.08e+01 0 100 1.58e+00 9.02e+01 Skip BFGS
geophys. misfits: 23.9 (target 30.0 [True]); 21.7 (target 30.0 [True]) | smallness misfit: 956.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.9 21.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.5245820633036913
12 4.75e+00 2.36e+01 4.86e+01 2.54e+02 5.82e+01 0 100 1.60e-01 1.50e+01
geophys. misfits: 23.9 (target 30.0 [True]); 23.2 (target 30.0 [True]) | smallness misfit: 868.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.9 23.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 3.2133651821936042
13 4.75e+00 2.41e+01 4.97e+01 2.60e+02 2.95e+01 0 100 5.90e-01 2.06e+01
geophys. misfits: 23.7 (target 30.0 [True]); 24.5 (target 30.0 [True]) | smallness misfit: 794.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.7 24.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.006487688384849
14 4.75e+00 2.47e+01 5.08e+01 2.66e+02 4.06e+01 0 100 9.22e-01 4.36e+01
geophys. misfits: 23.8 (target 30.0 [True]); 25.6 (target 30.0 [True]) | smallness misfit: 734.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.8 25.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.872936701510822
15 4.75e+00 2.53e+01 5.20e+01 2.72e+02 5.62e+01 0 100 2.54e+00 1.76e+02
geophys. misfits: 23.9 (target 30.0 [True]); 26.6 (target 30.0 [True]) | smallness misfit: 676.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.9 26.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 5.808325860281779
16 4.75e+00 2.57e+01 5.33e+01 2.79e+02 7.35e+01 0 100 7.54e-01 1.38e+02
geophys. misfits: 23.8 (target 30.0 [True]); 27.5 (target 30.0 [True]) | smallness misfit: 625.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.8 27.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 6.834220064432398
17 4.75e+00 2.66e+01 5.44e+01 2.85e+02 7.32e+01 0 100 5.66e-01 8.67e+01
geophys. misfits: 24.6 (target 30.0 [True]); 28.4 (target 30.0 [True]) | smallness misfit: 579.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.6 28.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 7.783033944851308
18 4.75e+00 2.67e+01 5.55e+01 2.90e+02 6.58e+01 0 100 1.90e+00 2.05e+02
geophys. misfits: 24.2 (target 30.0 [True]); 28.9 (target 30.0 [True]) | smallness misfit: 538.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.2 28.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 8.859100177541267
19 4.75e+00 2.73e+01 5.66e+01 2.96e+02 8.37e+01 0 100 1.43e+00 3.09e+02
geophys. misfits: 24.3 (target 30.0 [True]); 29.9 (target 30.0 [True]) | smallness misfit: 498.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.3 29.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 9.915295211002007
20 4.75e+00 2.89e+01 5.72e+01 3.01e+02 8.80e+01 0 100 1.21e+00 3.87e+02
geophys. misfits: 26.4 (target 30.0 [True]); 31.2 (target 30.0 [False]) | smallness misfit: 450.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [26.4 31.2]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.1356309694360949
New scales: [0.44087943 0.55912057]
21 2.37e+00 2.39e+01 5.87e+01 1.63e+02 9.33e+01 0 100 6.78e-01 3.01e+02
geophys. misfits: 22.7 (target 30.0 [True]); 24.9 (target 30.0 [True]) | smallness misfit: 509.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.7 24.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 12.53816102728801
22 2.37e+00 2.32e+01 6.16e+01 1.69e+02 8.14e+01 0 100 4.67e-01 1.41e+02
geophys. misfits: 21.3 (target 30.0 [True]); 24.7 (target 30.0 [True]) | smallness misfit: 454.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [21.3 24.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 16.459906582360404
23 2.37e+00 2.26e+01 6.54e+01 1.78e+02 7.98e+01 0 100 7.98e-01 1.40e+02
geophys. misfits: 20.9 (target 30.0 [True]); 24.0 (target 30.0 [True]) | smallness misfit: 417.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.9 24. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 22.119147815513635
24 2.37e+00 2.21e+01 7.04e+01 1.89e+02 9.27e+01 0 100 2.17e+00 4.78e+02
geophys. misfits: 20.3 (target 30.0 [True]); 23.4 (target 30.0 [True]) | smallness misfit: 368.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.3 23.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 30.48699299330211
25 2.37e+00 2.21e+01 7.66e+01 2.04e+02 1.05e+02 0 100 8.42e-01 4.62e+02
geophys. misfits: 20.1 (target 30.0 [True]); 23.7 (target 30.0 [True]) | smallness misfit: 335.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.1 23.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 42.06103380252559
26 2.37e+00 2.24e+01 8.42e+01 2.22e+02 1.13e+02 0 100 1.15e+00 6.86e+02
geophys. misfits: 19.8 (target 30.0 [True]); 24.4 (target 30.0 [True]) | smallness misfit: 284.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.8 24.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 57.716535610672516
27 2.37e+00 2.36e+01 9.28e+01 2.44e+02 1.18e+02 0 100 4.36e+00 3.71e+03
geophys. misfits: 21.0 (target 30.0 [True]); 25.7 (target 30.0 [True]) | smallness misfit: 270.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [21. 25.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 74.85841475555637
28 2.37e+00 2.41e+01 1.02e+02 2.65e+02 1.19e+02 0 100 1.20e+00 4.51e+03
geophys. misfits: 22.5 (target 30.0 [True]); 25.4 (target 30.0 [True]) | smallness misfit: 215.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.5 25.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 94.1212250793135
29 2.37e+00 2.79e+01 1.05e+02 2.76e+02 1.25e+02 0 100 7.31e-01 3.35e+03
geophys. misfits: 25.8 (target 30.0 [True]); 29.5 (target 30.0 [True]) | smallness misfit: 189.2 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [25.8 29.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 102.61421289835057
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.1608e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.8441e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.2536e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.2536e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 29
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev41+g7d721d035
Alpha scales: [np.float64(0.00034928024314171574), np.float64(0.0), np.float64(3.4882957394587844e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09041516 0.90958484]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09041516 0.90958484]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.89e+03 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.89e+03 6.42e+04 2.32e+01 1.08e+05 1.41e+02 0 15 3.00e-04 2.72e+03
geophys. misfits: 94916.6 (target 30.0 [False]); 61155.3 (target 30.0 [False]) | smallness misfit: 251.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [94916.6 61155.3]; minimum progress targets: [240000. 240000.]
2 1.89e+03 8.26e+01 5.51e-01 1.12e+03 1.36e+02 0 100 4.83e-03 1.29e+02 Skip BFGS
geophys. misfits: 636.9 (target 30.0 [False]); 27.5 (target 30.0 [True]) | smallness misfit: 141.7 (target: 200.0 [True])
Beta cooling evaluation: progress: [636.9 27.5]; minimum progress targets: [75933.3 48924.3]
Updating scaling for data misfits by 1.0899353106657237
New scales: [0.09775181 0.90224819]
3 1.89e+03 3.38e+01 1.31e-01 2.82e+02 1.03e+02 0 100 1.36e-02 3.92e+01
geophys. misfits: 119.9 (target 30.0 [False]); 24.4 (target 30.0 [True]) | smallness misfit: 69.4 (target: 200.0 [True])
Beta cooling evaluation: progress: [119.9 24.4]; minimum progress targets: [509.6 30. ]
Updating scaling for data misfits by 1.2278042359708572
New scales: [0.11740566 0.88259434]
4 1.89e+03 3.36e+01 1.26e-01 2.72e+02 6.60e+01 0 100 1.08e+00 1.61e+03
geophys. misfits: 100.1 (target 30.0 [False]); 24.8 (target 30.0 [True]) | smallness misfit: 46.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [100.1 24.8]; minimum progress targets: [95.9 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.2100927101039762
New scales: [0.13865174 0.86134826]
5 9.45e+02 2.64e+01 1.32e-01 1.51e+02 1.03e+02 0 100 1.48e-01 2.78e+02
geophys. misfits: 44.3 (target 30.0 [False]); 23.5 (target 30.0 [True]) | smallness misfit: 48.8 (target: 200.0 [True])
Beta cooling evaluation: progress: [44.3 23.5]; minimum progress targets: [80.1 30. ]
Updating scaling for data misfits by 1.2745549891128587
New scales: [0.1702387 0.8297613]
6 9.45e+02 2.61e+01 1.33e-01 1.52e+02 8.59e+01 0 100 7.73e-01 2.17e+02 Skip BFGS
geophys. misfits: 38.3 (target 30.0 [False]); 23.6 (target 30.0 [True]) | smallness misfit: 49.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [38.3 23.6]; minimum progress targets: [35.4 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.2689199364229402
New scales: [0.20656272 0.79343728]
7 4.73e+02 2.36e+01 1.37e-01 8.82e+01 9.73e+01 0 100 2.47e-01 7.58e+01
geophys. misfits: 26.7 (target 30.0 [True]); 22.8 (target 30.0 [True]) | smallness misfit: 51.1 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [26.7 22.8]; minimum progress targets: [30.6 30. ]
Warming alpha_pgi to favor clustering: 1.2202956011897852
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.1478e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 6.7895e-02 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 9.7284e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 9.7284e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 7
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev41+g7d721d035
Alpha scales: [np.float64(3.486809571975142e-05), np.float64(0.0), np.float64(3.48703327050052e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09041516 0.90958484]
/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: [0.09041516 0.90958484]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.02e+06 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.02e+06 3.99e+04 4.31e-02 8.39e+04 1.41e+02 0 24 9.20e-04 8.36e+03
geophys. misfits: 61686.3 (target 30.0 [False]); 37781.7 (target 30.0 [False])
2 2.04e+05 4.17e+03 1.07e-01 2.60e+04 1.37e+02 0 99 8.29e-04 1.50e+01 Skip BFGS
geophys. misfits: 8955.4 (target 30.0 [False]); 3695.8 (target 30.0 [False])
3 4.08e+04 2.86e+02 1.41e-01 6.02e+03 1.30e+02 0 100 7.71e-03 3.86e+01 Skip BFGS
geophys. misfits: 615.2 (target 30.0 [False]); 253.5 (target 30.0 [False])
4 8.16e+03 3.39e+01 1.52e-01 1.27e+03 1.03e+02 0 100 1.70e-01 1.94e+02 Skip BFGS
geophys. misfits: 49.9 (target 30.0 [False]); 32.3 (target 30.0 [False])
5 1.63e+03 1.15e+01 1.57e-01 2.67e+02 9.19e+01 0 100 1.04e-02 3.26e+00 Skip BFGS
geophys. misfits: 19.7 (target 30.0 [True]); 10.6 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.4036e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 7.1792e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 9.1940e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 9.1940e+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: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 28.084 seconds)
Estimated memory usage: 329 MB