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.
/home/vsts/work/1/s/simpeg/optimization.py:1535: FutureWarning:
InexactCG.tolCG has been deprecated, please use InexactCG.cg_atol. It will be removed in version 0.26.0 of SimPEG.
/home/vsts/work/1/s/simpeg/optimization.py:1061: FutureWarning:
InexactCG.maxIterCG has been deprecated, please use InexactCG.cg_maxiter. It will be removed in version 0.26.0 of SimPEG.
Running inversion with SimPEG v0.24.1.dev37+gf5e5be547
Alpha scales: [np.float64(3.482392103611884), np.float64(0.0), np.float64(3.480814983191258e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09546346 0.90453654]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09546346 0.90453654]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.91e+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.91e+01 1.76e+02 1.79e+02 3.59e+03 1.40e+02 0 100 1.57e-06 1.44e+01
geophys. misfits: 1059.9 (target 30.0 [False]); 82.9 (target 30.0 [False]) | smallness misfit: 2916.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [1059.9 82.9]; minimum progress targets: [240000. 240000.]
2 1.91e+01 7.50e+01 3.99e+01 8.37e+02 7.64e+01 0 100 2.45e-02 1.84e+01 Skip BFGS
geophys. misfits: 493.9 (target 30.0 [False]); 30.8 (target 30.0 [False]) | smallness misfit: 1267.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [493.9 30.8]; minimum progress targets: [847.9 66.3]
3 1.91e+01 7.44e+01 3.99e+01 8.37e+02 2.42e+01 0 100 1.89e-01 4.58e+00
geophys. misfits: 487.5 (target 30.0 [False]); 30.8 (target 30.0 [False]) | smallness misfit: 1254.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [487.5 30.8]; minimum progress targets: [395.1 30. ]
Decreasing beta to counter data misfit decrase plateau.
4 9.54e+00 4.51e+01 4.21e+01 4.47e+02 8.34e+01 0 100 7.49e+00 2.29e+03
geophys. misfits: 200.7 (target 30.0 [False]); 28.6 (target 30.0 [True]) | smallness misfit: 1249.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [200.7 28.6]; minimum progress targets: [390. 30.]
Updating scaling for data misfits by 1.0481755163463566
New scales: [0.09960438 0.90039562]
5 9.54e+00 4.45e+01 4.22e+01 4.47e+02 8.45e+01 0 100 2.27e-02 5.41e+01 Skip BFGS
geophys. misfits: 189.2 (target 30.0 [False]); 28.5 (target 30.0 [True]) | smallness misfit: 1246.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [189.2 28.5]; minimum progress targets: [160.6 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.051583822579721
New scales: [0.10420694 0.89579306]
6 4.77e+00 3.15e+01 4.41e+01 2.42e+02 7.91e+01 0 100 1.17e+01 2.41e+03
geophys. misfits: 72.1 (target 30.0 [False]); 26.8 (target 30.0 [True]) | smallness misfit: 1363.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [72.1 26.8]; minimum progress targets: [151.4 30. ]
Updating scaling for data misfits by 1.1204465149359941
New scales: [0.11531099 0.88468901]
7 4.77e+00 3.13e+01 4.42e+01 2.42e+02 7.68e+01 0 100 9.95e-02 2.37e+02 Skip BFGS
geophys. misfits: 67.5 (target 30.0 [False]); 26.6 (target 30.0 [True]) | smallness misfit: 1366.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [67.5 26.6]; minimum progress targets: [57.7 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.1259511011925538
New scales: [0.12797587 0.87202413]
8 2.39e+00 2.53e+01 4.61e+01 1.35e+02 7.60e+01 0 100 4.33e-01 1.28e+02
geophys. misfits: 31.5 (target 30.0 [False]); 24.4 (target 30.0 [True]) | smallness misfit: 1736.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [31.5 24.4]; minimum progress targets: [54. 30.]
Updating scaling for data misfits by 1.231097889055021
New scales: [0.15302512 0.84697488]
9 2.39e+00 2.52e+01 4.60e+01 1.35e+02 7.03e+01 0 100 2.34e+00 3.20e+02
geophys. misfits: 24.8 (target 30.0 [True]); 25.2 (target 30.0 [True]) | smallness misfit: 1552.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.8 25.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.1982239889674502
10 2.39e+00 2.50e+01 4.67e+01 1.36e+02 6.37e+01 0 100 8.09e-01 2.59e+02
geophys. misfits: 22.6 (target 30.0 [True]); 25.4 (target 30.0 [True]) | smallness misfit: 1484.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.6 25.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.5002132993239634
11 2.39e+00 2.49e+01 4.77e+01 1.39e+02 6.72e+01 0 100 6.12e-01 1.59e+02
geophys. misfits: 20.2 (target 30.0 [True]); 25.8 (target 30.0 [True]) | smallness misfit: 1407.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.2 25.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.9857818862394554
12 2.39e+00 2.51e+01 4.90e+01 1.42e+02 6.39e+01 0 100 2.83e+00 4.56e+02
geophys. misfits: 18.7 (target 30.0 [True]); 26.2 (target 30.0 [True]) | smallness misfit: 1303.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.7 26.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.727073988357054
13 2.39e+00 2.56e+01 5.09e+01 1.47e+02 8.25e+01 0 100 7.07e-01 3.23e+02
geophys. misfits: 18.6 (target 30.0 [True]); 26.8 (target 30.0 [True]) | smallness misfit: 1187.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.6 26.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 3.7259976449987384
14 2.39e+00 2.60e+01 5.31e+01 1.53e+02 8.41e+01 0 100 2.28e+00 7.48e+02
geophys. misfits: 17.6 (target 30.0 [True]); 27.6 (target 30.0 [True]) | smallness misfit: 1078.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.6 27.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 5.19735754885325
15 2.39e+00 2.70e+01 5.60e+01 1.61e+02 9.21e+01 0 100 1.01e+00 7.57e+02
geophys. misfits: 19.4 (target 30.0 [True]); 28.4 (target 30.0 [True]) | smallness misfit: 963.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.4 28.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 6.75602389950755
16 2.39e+00 2.61e+01 5.84e+01 1.66e+02 9.37e+01 0 100 1.61e+00 1.23e+03
geophys. misfits: 18.0 (target 30.0 [True]); 27.5 (target 30.0 [True]) | smallness misfit: 821.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [18. 27.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 9.323429886829778
17 2.39e+00 2.62e+01 6.06e+01 1.71e+02 1.10e+02 0 100 1.24e+00 1.52e+03
geophys. misfits: 18.1 (target 30.0 [True]); 27.6 (target 30.0 [True]) | smallness misfit: 637.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.1 27.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 12.768148249815132
18 2.39e+00 2.73e+01 6.28e+01 1.77e+02 9.68e+01 0 100 1.35e+00 2.07e+03
geophys. misfits: 19.9 (target 30.0 [True]); 28.7 (target 30.0 [True]) | smallness misfit: 475.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.9 28.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 16.296788715164183
19 2.39e+00 2.83e+01 6.58e+01 1.85e+02 9.04e+01 0 100 8.80e-01 1.82e+03
geophys. misfits: 22.7 (target 30.0 [True]); 29.3 (target 30.0 [True]) | smallness misfit: 407.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.7 29.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 19.099279000327073
20 2.39e+00 2.85e+01 6.80e+01 1.91e+02 8.95e+01 0 100 6.62e-01 1.20e+03 Skip BFGS
geophys. misfits: 23.7 (target 30.0 [True]); 29.4 (target 30.0 [True]) | smallness misfit: 369.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.7 29.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 21.851326419109387
21 2.39e+00 2.93e+01 6.99e+01 1.96e+02 8.83e+01 0 100 9.16e-01 1.10e+03
geophys. misfits: 26.4 (target 30.0 [True]); 29.8 (target 30.0 [True]) | smallness misfit: 347.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [26.4 29.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 23.39967059663779
22 2.39e+00 2.96e+01 7.08e+01 1.99e+02 8.35e+01 0 100 7.25e-01 8.01e+02 Skip BFGS
geophys. misfits: 27.5 (target 30.0 [True]); 30.0 (target 30.0 [True]) | smallness misfit: 330.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [27.5 30. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 24.47286011122914
23 2.39e+00 3.01e+01 7.14e+01 2.00e+02 7.79e+01 0 100 1.65e+00 1.32e+03
geophys. misfits: 29.7 (target 30.0 [True]); 30.1 (target 30.0 [False]) | smallness misfit: 320.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [29.7 30.1]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.0109803306408114
New scales: [0.15161509 0.84838491]
24 1.19e+00 2.54e+01 7.41e+01 1.14e+02 9.10e+01 0 100 1.80e-01 2.42e+02
geophys. misfits: 14.8 (target 30.0 [True]); 27.3 (target 30.0 [True]) | smallness misfit: 352.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.8 27.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 38.255222613993766
25 1.19e+00 2.53e+01 8.43e+01 1.26e+02 7.74e+01 0 100 3.41e+00 8.77e+02
geophys. misfits: 14.8 (target 30.0 [True]); 27.2 (target 30.0 [True]) | smallness misfit: 302.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.8 27.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 60.00658983603364
26 1.19e+00 2.50e+01 9.85e+01 1.42e+02 1.01e+02 0 100 7.56e-01 6.76e+02
geophys. misfits: 15.1 (target 30.0 [True]); 26.8 (target 30.0 [True]) | smallness misfit: 285.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.1 26.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 93.40296755294365
27 1.19e+00 2.47e+01 1.18e+02 1.65e+02 1.08e+02 0 100 9.95e-01 7.86e+02
geophys. misfits: 15.8 (target 30.0 [True]); 26.3 (target 30.0 [True]) | smallness misfit: 235.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.8 26.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 142.04724826902822
28 1.19e+00 2.45e+01 1.44e+02 1.96e+02 1.16e+02 0 100 2.73e+00 2.68e+03
geophys. misfits: 13.8 (target 30.0 [True]); 26.5 (target 30.0 [True]) | smallness misfit: 231.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [13.8 26.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 234.78973740137803
29 1.19e+00 2.49e+01 1.85e+02 2.46e+02 1.28e+02 1 100 1.11e+00 3.18e+03
geophys. misfits: 15.7 (target 30.0 [True]); 26.6 (target 30.0 [True]) | smallness misfit: 203.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.7 26.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 356.1463839412645
30 1.19e+00 2.82e+01 2.24e+02 2.96e+02 1.32e+02 1 100 6.23e+00 1.65e+04
geophys. misfits: 36.3 (target 30.0 [False]); 26.8 (target 30.0 [True]) | smallness misfit: 179.6 (target: 200.0 [True])
Beta cooling evaluation: progress: [36.3 26.8]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.1199561910314613
New scales: [0.16676919 0.83323081]
31 5.97e-01 2.73e+01 2.17e+02 1.57e+02 1.19e+02 1 100 2.44e-01 1.98e+03
geophys. misfits: 27.0 (target 30.0 [True]); 27.4 (target 30.0 [True]) | smallness misfit: 169.7 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [27. 27.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 392.46716129644136
------------------------- STOP! -------------------------
1 : |fc-fOld| = 5.2717e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 3.2632e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.1856e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.1856e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 31
------------------------- DONE! -------------------------
/home/vsts/work/1/s/simpeg/optimization.py:1535: FutureWarning:
InexactCG.tolCG has been deprecated, please use InexactCG.cg_atol. It will be removed in version 0.26.0 of SimPEG.
/home/vsts/work/1/s/simpeg/optimization.py:1061: FutureWarning:
InexactCG.maxIterCG has been deprecated, please use InexactCG.cg_maxiter. It will be removed in version 0.26.0 of SimPEG.
Running inversion with SimPEG v0.24.1.dev37+gf5e5be547
Alpha scales: [np.float64(0.0004701811806052387), np.float64(0.0), np.float64(3.559609317027757e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09546346 0.90453654]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09546346 0.90453654]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.98e+03 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.98e+03 6.63e+04 2.22e+01 1.10e+05 1.41e+02 0 48 6.18e-12 5.67e-05
geophys. misfits: 90074.6 (target 30.0 [False]); 63781.2 (target 30.0 [False]) | smallness misfit: 280.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [90074.6 63781.2]; minimum progress targets: [240000. 240000.]
2 1.98e+03 9.65e+01 5.57e-01 1.20e+03 8.87e+01 0 100 1.87e-02 5.11e+02 Skip BFGS
geophys. misfits: 693.8 (target 30.0 [False]); 33.4 (target 30.0 [False]) | smallness misfit: 91.0 (target: 200.0 [True])
Beta cooling evaluation: progress: [693.8 33.4]; minimum progress targets: [72059.7 51025. ]
3 1.98e+03 4.34e+01 1.35e-01 3.10e+02 8.08e+01 0 100 2.60e-01 5.28e+02 Skip BFGS
geophys. misfits: 139.0 (target 30.0 [False]); 33.3 (target 30.0 [False]) | smallness misfit: 68.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [139. 33.3]; minimum progress targets: [555.1 30. ]
4 1.98e+03 4.14e+01 1.30e-01 2.98e+02 8.20e+01 0 100 7.83e-03 1.27e+01
geophys. misfits: 139.9 (target 30.0 [False]); 31.1 (target 30.0 [False]) | smallness misfit: 54.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [139.9 31.1]; minimum progress targets: [111.2 30. ]
Decreasing beta to counter data misfit decrase plateau.
5 9.89e+02 3.10e+01 1.36e-01 1.66e+02 9.45e+01 0 100 1.11e-01 4.18e+01
geophys. misfits: 56.5 (target 30.0 [False]); 28.3 (target 30.0 [True]) | smallness misfit: 48.6 (target: 200.0 [True])
Beta cooling evaluation: progress: [56.5 28.3]; minimum progress targets: [111.9 30. ]
Updating scaling for data misfits by 1.0606099942810125
New scales: [0.10066703 0.89933297]
6 9.89e+02 3.08e+01 1.36e-01 1.66e+02 3.77e+01 0 100 9.96e-01 4.40e+01 Skip BFGS
geophys. misfits: 51.9 (target 30.0 [False]); 28.4 (target 30.0 [True]) | smallness misfit: 48.8 (target: 200.0 [True])
Beta cooling evaluation: progress: [51.9 28.4]; minimum progress targets: [45.2 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.0563245728238762
New scales: [0.10573752 0.89426248]
7 4.95e+02 2.66e+01 1.42e-01 9.69e+01 8.06e+01 0 100 2.27e-01 3.15e+01
geophys. misfits: 24.0 (target 30.0 [True]); 26.9 (target 30.0 [True]) | smallness misfit: 51.2 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [24. 26.9]; minimum progress targets: [41.5 30. ]
Warming alpha_pgi to favor clustering: 1.1832422294450518
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.4170e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 1.9760e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.0615e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.0615e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 7
------------------------- DONE! -------------------------
/home/vsts/work/1/s/simpeg/optimization.py:1535: FutureWarning:
InexactCG.tolCG has been deprecated, please use InexactCG.cg_atol. It will be removed in version 0.26.0 of SimPEG.
/home/vsts/work/1/s/simpeg/optimization.py:1061: FutureWarning:
InexactCG.maxIterCG has been deprecated, please use InexactCG.cg_maxiter. It will be removed in version 0.26.0 of SimPEG.
Running inversion with SimPEG v0.24.1.dev37+gf5e5be547
Alpha scales: [np.float64(3.4332922983895255e-05), np.float64(0.0), np.float64(3.434957697677808e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09546346 0.90453654]
/home/vsts/work/1/s/simpeg/directives/_directives.py:334: UserWarning:
There is no PGI regularization. Smallness target is turned off (TriggerSmall flag)
Initial data misfit scales: [0.09546346 0.90453654]
================================================= 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 3.78e+04 4.26e-02 8.22e+04 1.41e+02 0 100 2.93e-04 2.68e+03
geophys. misfits: 56906.9 (target 30.0 [False]); 35835.5 (target 30.0 [False])
2 2.08e+05 4.33e+03 1.07e-01 2.65e+04 1.37e+02 0 100 6.66e-03 1.10e+02 Skip BFGS
geophys. misfits: 8525.3 (target 30.0 [False]); 3884.1 (target 30.0 [False])
3 4.16e+04 2.85e+02 1.41e-01 6.16e+03 1.31e+02 0 100 2.15e-02 1.10e+02 Skip BFGS
geophys. misfits: 571.0 (target 30.0 [False]); 254.7 (target 30.0 [False])
4 8.32e+03 4.04e+01 1.51e-01 1.30e+03 1.02e+02 0 100 6.83e+00 8.02e+03 Skip BFGS
geophys. misfits: 35.9 (target 30.0 [False]); 40.9 (target 30.0 [False])
5 1.66e+03 2.42e+01 1.55e-01 2.82e+02 9.53e+01 0 100 2.57e-03 2.06e+01 Skip BFGS
geophys. misfits: 8.3 (target 30.0 [True]); 25.9 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.0265e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 5.0672e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 9.5323e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 9.5323e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 5
------------------------- DONE! -------------------------
/home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:302: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.
/home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:309: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.
/home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:353: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.
/home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:360: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.
/home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:367: UserWarning:
The following kwargs were not used by contour: 'label'
/home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:412: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.
/home/vsts/work/1/s/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:419: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.
import discretize as Mesh
import matplotlib.pyplot as plt
import matplotlib.lines as mlines
import numpy as np
from simpeg import (
data_misfit,
directives,
inverse_problem,
inversion,
maps,
optimization,
regularization,
simulation,
utils,
)
# Random seed for reproductibility
np.random.seed(1)
# Mesh
N = 100
mesh = Mesh.TensorMesh([N])
# Survey design parameters
nk = 30
jk = np.linspace(1.0, 59.0, nk)
p = -0.25
q = 0.25
# Physics
def g(k):
return np.exp(p * jk[k] * mesh.cell_centers_x) * np.cos(
np.pi * q * jk[k] * mesh.cell_centers_x
)
G = np.empty((nk, mesh.nC))
for i in range(nk):
G[i, :] = g(i)
m0 = np.zeros(mesh.nC)
m0[20:41] = np.linspace(0.0, 1.0, 21)
m0[41:57] = np.linspace(-1, 0.0, 16)
poly0 = maps.PolynomialPetroClusterMap(coeffyx=np.r_[0.0, -4.0, 4.0])
poly1 = maps.PolynomialPetroClusterMap(coeffyx=np.r_[-0.0, 3.0, 6.0, 6.0])
poly0_inverse = maps.PolynomialPetroClusterMap(coeffyx=-np.r_[0.0, -4.0, 4.0])
poly1_inverse = maps.PolynomialPetroClusterMap(coeffyx=-np.r_[0.0, 3.0, 6.0, 6.0])
cluster_mapping = [maps.IdentityMap(), poly0_inverse, poly1_inverse]
m1 = np.zeros(100)
m1[20:41] = 1.0 + (poly0 * np.vstack([m0[20:41], m1[20:41]]).T)[:, 1]
m1[41:57] = -1.0 + (poly1 * np.vstack([m0[41:57], m1[41:57]]).T)[:, 1]
model2d = np.vstack([m0, m1]).T
m = utils.mkvc(model2d)
clfmapping = utils.GaussianMixtureWithNonlinearRelationships(
mesh=mesh,
n_components=3,
covariance_type="full",
tol=1e-8,
reg_covar=1e-3,
max_iter=1000,
n_init=100,
init_params="kmeans",
random_state=None,
warm_start=False,
means_init=np.array(
[
[0, 0],
[m0[20:41].mean(), m1[20:41].mean()],
[m0[41:57].mean(), m1[41:57].mean()],
]
),
verbose=0,
verbose_interval=10,
cluster_mapping=cluster_mapping,
)
clfmapping = clfmapping.fit(model2d)
clfnomapping = utils.WeightedGaussianMixture(
mesh=mesh,
n_components=3,
covariance_type="full",
tol=1e-8,
reg_covar=1e-3,
max_iter=1000,
n_init=100,
init_params="kmeans",
random_state=None,
warm_start=False,
verbose=0,
verbose_interval=10,
)
clfnomapping = clfnomapping.fit(model2d)
wires = maps.Wires(("m1", mesh.nC), ("m2", mesh.nC))
relatrive_error = 0.01
noise_floor = 0.0
prob1 = simulation.LinearSimulation(mesh, G=G, model_map=wires.m1)
survey1 = prob1.make_synthetic_data(
m, relative_error=relatrive_error, noise_floor=noise_floor, add_noise=True
)
prob2 = simulation.LinearSimulation(mesh, G=G, model_map=wires.m2)
survey2 = prob2.make_synthetic_data(
m, relative_error=relatrive_error, noise_floor=noise_floor, add_noise=True
)
dmis1 = data_misfit.L2DataMisfit(simulation=prob1, data=survey1)
dmis2 = data_misfit.L2DataMisfit(simulation=prob2, data=survey2)
dmis = dmis1 + dmis2
minit = np.zeros_like(m)
# Distance weighting
wr1 = np.sum(prob1.G**2.0, axis=0) ** 0.5 / mesh.cell_volumes
wr1 = wr1 / np.max(wr1)
wr2 = np.sum(prob2.G**2.0, axis=0) ** 0.5 / mesh.cell_volumes
wr2 = wr2 / np.max(wr2)
reg_simple = regularization.PGI(
mesh=mesh,
gmmref=clfmapping,
gmm=clfmapping,
approx_gradient=True,
wiresmap=wires,
non_linear_relationships=True,
weights_list=[wr1, wr2],
)
opt = optimization.ProjectedGNCG(
maxIter=50,
tolX=1e-6,
maxIterCG=100,
tolCG=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,
maxIterCG=100,
tolCG=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,
maxIterCG=100,
tolCG=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 38.440 seconds)
Estimated memory usage: 298 MB