Note
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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.dev23+g390d5f500
Alpha scales: [np.float64(3.4700404765874278), np.float64(0.0), np.float64(3.4776412391370852e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09474081 0.90525919]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09474081 0.90525919]
================================================= 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+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.93e+01 1.23e+03 1.71e+02 4.54e+03 1.40e+02 0 20 6.55e-04 6.02e+03
geophys. misfits: 6887.1 (target 30.0 [False]); 640.1 (target 30.0 [False]) | smallness misfit: 4022.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [6887.1 640.1]; minimum progress targets: [240000. 240000.]
2 1.93e+01 6.98e+01 4.05e+01 8.53e+02 1.39e+02 0 100 1.21e+00 7.26e+03 Skip BFGS
geophys. misfits: 484.6 (target 30.0 [False]); 26.4 (target 30.0 [True]) | smallness misfit: 1491.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [484.6 26.4]; minimum progress targets: [5509.7 512. ]
Updating scaling for data misfits by 1.1351912675794176
New scales: [0.10618886 0.89381114]
3 1.93e+01 6.65e+01 4.01e+01 8.41e+02 7.69e+01 0 72 8.99e-04 7.32e+00 Skip BFGS
geophys. misfits: 409.1 (target 30.0 [False]); 25.8 (target 30.0 [True]) | smallness misfit: 1321.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [409.1 25.8]; minimum progress targets: [387.7 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.1610117359494485
New scales: [0.12121404 0.87878596]
4 9.66e+00 3.30e+01 4.27e+01 4.45e+02 8.63e+01 0 100 1.52e-02 5.87e+00
geophys. misfits: 128.8 (target 30.0 [False]); 19.8 (target 30.0 [True]) | smallness misfit: 1354.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [128.8 19.8]; minimum progress targets: [327.3 30. ]
Updating scaling for data misfits by 1.5149211093097112
New scales: [0.17284168 0.82715832]
5 9.66e+00 3.23e+01 4.32e+01 4.50e+02 6.63e+01 0 100 6.02e+00 8.66e+02 Skip BFGS
geophys. misfits: 89.6 (target 30.0 [False]); 20.4 (target 30.0 [True]) | smallness misfit: 1279.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [89.6 20.4]; minimum progress targets: [103.1 30. ]
Updating scaling for data misfits by 1.4725688097377314
New scales: [0.2353019 0.7646981]
6 9.66e+00 3.08e+01 4.37e+01 4.53e+02 7.36e+01 0 100 1.01e+00 1.21e+03 Skip BFGS
geophys. misfits: 62.8 (target 30.0 [False]); 21.0 (target 30.0 [True]) | smallness misfit: 1231.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [62.8 21. ]; minimum progress targets: [71.7 30. ]
Updating scaling for data misfits by 1.4283661856120973
New scales: [0.30532221 0.69467779]
7 9.66e+00 2.78e+01 4.42e+01 4.55e+02 7.35e+01 0 100 2.50e-01 3.95e+02 Skip BFGS
geophys. misfits: 42.6 (target 30.0 [False]); 21.3 (target 30.0 [True]) | smallness misfit: 1199.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [42.6 21.3]; minimum progress targets: [50.3 30. ]
Updating scaling for data misfits by 1.4059481073631332
New scales: [0.381929 0.618071]
8 9.66e+00 2.82e+01 4.43e+01 4.56e+02 7.41e+01 0 100 2.79e+00 1.42e+03
geophys. misfits: 37.5 (target 30.0 [False]); 22.5 (target 30.0 [True]) | smallness misfit: 1130.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [37.5 22.5]; minimum progress targets: [34.1 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.3343123823824994
New scales: [0.45191098 0.54808902]
9 4.83e+00 1.97e+01 4.55e+01 2.39e+02 8.23e+01 0 100 7.72e-01 1.33e+03
geophys. misfits: 20.4 (target 30.0 [True]); 19.1 (target 30.0 [True]) | smallness misfit: 1242.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.4 19.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.5229065965001332
10 4.83e+00 2.05e+01 4.66e+01 2.45e+02 8.54e+01 0 100 9.53e-01 1.27e+03
geophys. misfits: 19.9 (target 30.0 [True]); 21.0 (target 30.0 [True]) | smallness misfit: 1061.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.9 21. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.23375029740058
11 4.83e+00 2.07e+01 4.81e+01 2.53e+02 6.71e+01 0 100 5.87e-01 7.46e+02 Skip BFGS
geophys. misfits: 18.9 (target 30.0 [True]); 22.1 (target 30.0 [True]) | smallness misfit: 956.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.9 22.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 3.2889781427123124
12 4.83e+00 2.14e+01 5.00e+01 2.63e+02 6.12e+01 0 100 2.87e-01 2.14e+02
geophys. misfits: 18.3 (target 30.0 [True]); 24.0 (target 30.0 [True]) | smallness misfit: 860.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.3 24. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.753358202208385
13 4.83e+00 2.21e+01 5.25e+01 2.76e+02 6.26e+01 0 100 1.93e+00 4.34e+02
geophys. misfits: 17.5 (target 30.0 [True]); 25.9 (target 30.0 [True]) | smallness misfit: 758.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.5 25.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 6.82326000608938
14 4.83e+00 2.35e+01 5.56e+01 2.92e+02 7.56e+01 0 100 2.65e+00 1.18e+03
geophys. misfits: 17.9 (target 30.0 [True]); 28.0 (target 30.0 [True]) | smallness misfit: 675.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.9 28. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 9.371322422656338
15 4.83e+00 2.50e+01 5.89e+01 3.10e+02 8.81e+01 0 100 1.60e+00 1.91e+03
geophys. misfits: 16.8 (target 30.0 [True]); 31.7 (target 30.0 [False]) | smallness misfit: 595.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.8 31.7]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.7872969944117632
New scales: [0.31568857 0.68431143]
16 2.41e+00 1.98e+01 6.07e+01 1.66e+02 8.86e+01 0 100 3.46e-01 4.61e+02
geophys. misfits: 15.8 (target 30.0 [True]); 21.7 (target 30.0 [True]) | smallness misfit: 659.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.8 21.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 15.382184232746768
17 2.41e+00 2.17e+01 6.79e+01 1.86e+02 8.34e+01 0 100 2.82e+00 1.37e+03
geophys. misfits: 16.9 (target 30.0 [True]); 23.9 (target 30.0 [True]) | smallness misfit: 537.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.9 23.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 23.27325057431955
18 2.41e+00 2.17e+01 7.65e+01 2.06e+02 9.93e+01 0 100 1.31e+00 1.82e+03
geophys. misfits: 18.5 (target 30.0 [True]); 23.2 (target 30.0 [True]) | smallness misfit: 452.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.5 23.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 33.89082048789787
19 2.41e+00 2.69e+01 8.39e+01 2.30e+02 1.05e+02 0 100 1.07e+00 1.96e+03
geophys. misfits: 23.2 (target 30.0 [True]); 28.7 (target 30.0 [True]) | smallness misfit: 360.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.2 28.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 39.66596976897929
20 2.41e+00 3.02e+01 8.67e+01 2.40e+02 9.98e+01 0 100 9.46e-01 1.86e+03
geophys. misfits: 22.7 (target 30.0 [True]); 33.7 (target 30.0 [False]) | smallness misfit: 330.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.7 33.7]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.318943061993772
New scales: [0.25913147 0.74086853]
21 1.21e+00 2.23e+01 9.12e+01 1.32e+02 1.01e+02 0 100 7.57e-01 1.29e+03
geophys. misfits: 18.8 (target 30.0 [True]); 23.5 (target 30.0 [True]) | smallness misfit: 364.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.8 23.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 56.97409719941942
22 1.21e+00 2.09e+01 1.06e+02 1.48e+02 9.13e+01 0 100 8.50e-01 1.10e+03
geophys. misfits: 18.1 (target 30.0 [True]); 21.8 (target 30.0 [True]) | smallness misfit: 318.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.1 21.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 86.2553879032618
23 1.21e+00 2.13e+01 1.25e+02 1.72e+02 1.05e+02 0 100 2.20e+00 2.48e+03
geophys. misfits: 21.0 (target 30.0 [True]); 21.4 (target 30.0 [True]) | smallness misfit: 292.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [21. 21.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 122.20925527863776
24 1.21e+00 2.07e+01 1.45e+02 1.96e+02 1.14e+02 1 100 1.21e+00 3.04e+03
geophys. misfits: 19.9 (target 30.0 [True]); 21.0 (target 30.0 [True]) | smallness misfit: 255.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.9 21. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 179.52362241873712
25 1.21e+00 2.10e+01 1.76e+02 2.34e+02 1.18e+02 0 100 4.10e-01 9.49e+02
geophys. misfits: 24.5 (target 30.0 [True]); 19.8 (target 30.0 [True]) | smallness misfit: 228.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.5 19.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 245.98291541903055
26 1.21e+00 3.05e+01 2.04e+02 2.77e+02 1.22e+02 0 100 1.86e+00 2.22e+03
geophys. misfits: 51.2 (target 30.0 [False]); 23.2 (target 30.0 [True]) | smallness misfit: 215.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [51.2 23.2]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.2923955555925977
New scales: [0.31131256 0.68868744]
27 6.04e-01 2.33e+01 2.13e+02 1.52e+02 1.12e+02 0 100 8.59e-01 1.97e+03
geophys. misfits: 19.5 (target 30.0 [True]); 25.0 (target 30.0 [True]) | smallness misfit: 218.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.5 25. ]; minimum progress targets: [41. 30.]
Warming alpha_pgi to favor clustering: 336.64142282223116
28 6.04e-01 2.23e+01 2.48e+02 1.72e+02 1.18e+02 1 100 2.10e+00 4.28e+03
geophys. misfits: 17.5 (target 30.0 [True]); 24.5 (target 30.0 [True]) | smallness misfit: 202.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.5 24.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 494.4792026580972
29 6.04e-01 2.63e+01 3.10e+02 2.13e+02 1.22e+02 0 100 5.63e-01 1.01e+03
geophys. misfits: 25.3 (target 30.0 [True]); 26.8 (target 30.0 [True]) | smallness misfit: 173.4 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [25.3 26.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 570.1034060654368
------------------------- STOP! -------------------------
1 : |fc-fOld| = 5.3266e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 6.0193e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.2218e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.2218e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 29
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev23+g390d5f500
Alpha scales: [np.float64(0.00034969266069668223), np.float64(0.0), np.float64(3.493888005438109e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09474081 0.90525919]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09474081 0.90525919]
================================================= 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 6.59e+04 2.25e+01 1.10e+05 1.41e+02 0 15 4.06e-04 3.74e+03
geophys. misfits: 92525.0 (target 30.0 [False]); 63101.9 (target 30.0 [False]) | smallness misfit: 245.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [92525. 63101.9]; minimum progress targets: [240000. 240000.]
2 1.94e+03 7.31e+01 5.48e-01 1.14e+03 1.35e+02 0 100 2.88e-02 7.84e+02 Skip BFGS
geophys. misfits: 578.0 (target 30.0 [False]); 20.3 (target 30.0 [True]) | smallness misfit: 104.0 (target: 200.0 [True])
Beta cooling evaluation: progress: [578. 20.3]; minimum progress targets: [74020. 50481.5]
Updating scaling for data misfits by 1.4774471695185354
New scales: [0.13391698 0.86608302]
3 1.94e+03 2.55e+01 1.36e-01 2.90e+02 1.18e+02 0 100 2.27e+00 5.82e+03 Skip BFGS
geophys. misfits: 79.0 (target 30.0 [False]); 17.2 (target 30.0 [True]) | smallness misfit: 61.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [79. 17.2]; minimum progress targets: [462.4 30. ]
Updating scaling for data misfits by 1.741832174045479
New scales: [0.21218192 0.78781808]
4 1.94e+03 2.35e+01 1.33e-01 2.81e+02 9.45e+01 0 100 5.86e-02 5.49e+02
geophys. misfits: 46.1 (target 30.0 [False]); 17.4 (target 30.0 [True]) | smallness misfit: 73.6 (target: 200.0 [True])
Beta cooling evaluation: progress: [46.1 17.4]; minimum progress targets: [63.2 30. ]
Updating scaling for data misfits by 1.7231697148478857
New scales: [0.316986 0.683014]
5 1.94e+03 2.19e+01 1.30e-01 2.75e+02 8.46e+01 0 100 5.16e-02 8.37e+01
geophys. misfits: 30.8 (target 30.0 [False]); 17.8 (target 30.0 [True]) | smallness misfit: 57.6 (target: 200.0 [True])
Beta cooling evaluation: progress: [30.8 17.8]; minimum progress targets: [36.8 30. ]
Updating scaling for data misfits by 1.6883187418243892
New scales: [0.43931944 0.56068056]
6 1.94e+03 2.20e+01 1.29e-01 2.73e+02 8.71e+01 0 100 9.01e-02 6.20e+01
geophys. misfits: 24.0 (target 30.0 [True]); 20.4 (target 30.0 [True]) | smallness misfit: 47.7 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [24. 20.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.3611633600523336
------------------------- STOP! -------------------------
1 : |fc-fOld| = 3.3394e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 1.1108e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.7149e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.7149e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 6
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev23+g390d5f500
Alpha scales: [np.float64(3.410383120208843e-05), np.float64(0.0), np.float64(3.701966407904588e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09474081 0.90525919]
/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: [0.09474081 0.90525919]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.04e+06 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.04e+06 4.11e+04 4.23e-02 8.52e+04 1.40e+02 0 23 9.16e-04 8.42e+03
geophys. misfits: 59486.5 (target 30.0 [False]); 39202.6 (target 30.0 [False])
2 2.08e+05 4.33e+03 1.07e-01 2.66e+04 1.37e+02 0 100 1.08e-02 1.97e+02 Skip BFGS
geophys. misfits: 8374.9 (target 30.0 [False]); 3912.0 (target 30.0 [False])
3 4.17e+04 2.89e+02 1.41e-01 6.17e+03 1.31e+02 0 100 1.73e-03 8.85e+00 Skip BFGS
geophys. misfits: 554.2 (target 30.0 [False]); 261.6 (target 30.0 [False])
4 8.34e+03 3.37e+01 1.52e-01 1.30e+03 1.03e+02 0 100 4.38e-02 5.12e+01 Skip BFGS
geophys. misfits: 40.9 (target 30.0 [False]); 33.0 (target 30.0 [False])
5 1.67e+03 1.53e+01 1.56e-01 2.75e+02 8.19e+01 0 100 2.98e+00 7.38e+02 Skip BFGS
geophys. misfits: 15.4 (target 30.0 [True]); 15.3 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.1702e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 5.1717e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.1893e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.1893e+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 27.507 seconds)
Estimated memory usage: 332 MB