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.dev2+gfcb9bdf36
Alpha scales: [np.float64(3.4821266225483836), np.float64(0.0), np.float64(3.4829467142566976e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09493507 0.90506493]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09493507 0.90506493]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.92e+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.92e+01 1.29e+03 1.71e+02 4.56e+03 1.40e+02 0 19 8.81e-04 8.11e+03
geophys. misfits: 7405.8 (target 30.0 [False]); 643.1 (target 30.0 [False]) | smallness misfit: 3990.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [7405.8 643.1]; minimum progress targets: [240000. 240000.]
2 1.92e+01 6.63e+01 4.10e+01 8.52e+02 1.39e+02 0 100 8.57e-02 6.98e+02 Skip BFGS
geophys. misfits: 499.8 (target 30.0 [False]); 20.9 (target 30.0 [True]) | smallness misfit: 1401.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [499.8 20.9]; minimum progress targets: [5924.6 514.5]
Updating scaling for data misfits by 1.438252350847193
New scales: [0.13108666 0.86891334]
3 1.92e+01 5.97e+01 4.11e+01 8.48e+02 1.01e+02 0 100 1.32e-01 1.32e+02 Skip BFGS
geophys. misfits: 318.0 (target 30.0 [False]); 20.7 (target 30.0 [True]) | smallness misfit: 1155.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [318. 20.7]; minimum progress targets: [399.9 30. ]
Updating scaling for data misfits by 1.4475011940978275
New scales: [0.17923398 0.82076602]
4 1.92e+01 5.39e+01 4.20e+01 8.59e+02 7.48e+01 0 100 7.14e-03 2.13e+00
geophys. misfits: 205.9 (target 30.0 [False]); 20.6 (target 30.0 [True]) | smallness misfit: 1101.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [205.9 20.6]; minimum progress targets: [254.4 30. ]
Updating scaling for data misfits by 1.4530300786661812
New scales: [0.2408738 0.7591262]
5 1.92e+01 4.88e+01 4.27e+01 8.68e+02 7.18e+01 0 100 3.37e-02 7.72e+00
geophys. misfits: 137.0 (target 30.0 [False]); 20.8 (target 30.0 [True]) | smallness misfit: 1056.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [137. 20.8]; minimum progress targets: [164.7 30. ]
Updating scaling for data misfits by 1.439936964724519
New scales: [0.31361005 0.68638995]
6 1.92e+01 4.47e+01 4.33e+01 8.75e+02 7.10e+01 0 100 2.21e-02 4.59e+00 Skip BFGS
geophys. misfits: 96.3 (target 30.0 [False]); 21.2 (target 30.0 [True]) | smallness misfit: 1020.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [96.3 21.2]; minimum progress targets: [109.6 30. ]
Updating scaling for data misfits by 1.4138536542418778
New scales: [0.39246164 0.60753836]
7 1.92e+01 4.16e+01 4.37e+01 8.79e+02 6.97e+01 0 100 1.91e-02 3.44e+00 Skip BFGS
geophys. misfits: 72.1 (target 30.0 [False]); 21.8 (target 30.0 [True]) | smallness misfit: 990.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [72.1 21.8]; minimum progress targets: [77. 30.]
Updating scaling for data misfits by 1.374132477648334
New scales: [0.47024672 0.52975328]
8 1.92e+01 3.92e+01 4.40e+01 8.83e+02 6.69e+01 0 100 3.29e-02 4.77e+00 Skip BFGS
geophys. misfits: 57.7 (target 30.0 [False]); 22.7 (target 30.0 [True]) | smallness misfit: 963.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [57.7 22.7]; minimum progress targets: [57.7 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.321154562085851
New scales: [0.53975394 0.46024606]
9 9.59e+00 2.15e+01 4.53e+01 4.56e+02 8.38e+01 0 100 3.77e-01 1.82e+02
geophys. misfits: 23.7 (target 30.0 [True]); 19.0 (target 30.0 [True]) | smallness misfit: 989.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.7 19. ]; minimum progress targets: [46.2 30. ]
Warming alpha_pgi to favor clustering: 1.423420481905699
10 9.59e+00 2.20e+01 4.61e+01 4.65e+02 6.37e+01 0 100 1.17e+00 2.15e+02
geophys. misfits: 23.4 (target 30.0 [True]); 20.4 (target 30.0 [True]) | smallness misfit: 914.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.4 20.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.9571315110743324
11 9.59e+00 2.27e+01 4.71e+01 4.74e+02 7.09e+01 0 100 1.12e+00 2.54e+02
geophys. misfits: 23.2 (target 30.0 [True]); 22.0 (target 30.0 [True]) | smallness misfit: 845.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.2 22. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.598963470123926
12 9.59e+00 2.34e+01 4.82e+01 4.86e+02 8.29e+01 0 100 1.70e+00 4.58e+02
geophys. misfits: 22.9 (target 30.0 [True]); 24.0 (target 30.0 [True]) | smallness misfit: 776.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.9 24. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 3.322739811613794
13 9.59e+00 2.46e+01 4.93e+01 4.97e+02 9.12e+01 0 100 1.68e+00 7.92e+02
geophys. misfits: 22.9 (target 30.0 [True]); 26.6 (target 30.0 [True]) | smallness misfit: 724.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.9 26.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.04855961592713
14 9.59e+00 2.53e+01 5.03e+01 5.08e+02 9.49e+01 0 100 4.00e+00 3.21e+03
geophys. misfits: 22.5 (target 30.0 [True]); 28.6 (target 30.0 [True]) | smallness misfit: 673.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.5 28.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.817445806068022
15 9.59e+00 2.69e+01 5.12e+01 5.18e+02 9.93e+01 0 100 2.28e-01 7.32e+02
geophys. misfits: 22.5 (target 30.0 [True]); 32.1 (target 30.0 [False]) | smallness misfit: 619.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.5 32.1]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.3318388624647948
New scales: [0.4682407 0.5317593]
16 4.80e+00 1.83e+01 5.25e+01 2.70e+02 9.91e+01 0 100 2.79e-01 1.82e+02
geophys. misfits: 14.5 (target 30.0 [True]); 21.5 (target 30.0 [True]) | smallness misfit: 681.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.5 21.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 8.323197201035384
17 4.80e+00 2.08e+01 5.68e+01 2.93e+02 9.57e+01 0 100 5.66e+00 1.52e+03
geophys. misfits: 15.0 (target 30.0 [True]); 25.9 (target 30.0 [True]) | smallness misfit: 535.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [15. 25.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 13.157025118220227
18 4.80e+00 1.83e+01 6.26e+01 3.18e+02 1.02e+02 0 100 1.08e+00 1.67e+03
geophys. misfits: 13.3 (target 30.0 [True]); 22.7 (target 30.0 [True]) | smallness misfit: 472.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [13.3 22.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 23.50101678269182
19 4.80e+00 1.83e+01 7.23e+01 3.65e+02 1.23e+02 0 100 8.95e+00 1.59e+04
geophys. misfits: 14.2 (target 30.0 [True]); 22.0 (target 30.0 [True]) | smallness misfit: 375.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.2 22. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 40.91918215294416
20 4.80e+00 2.32e+01 8.39e+01 4.26e+02 1.32e+02 0 100 3.18e-01 5.07e+03
geophys. misfits: 20.2 (target 30.0 [True]); 25.9 (target 30.0 [True]) | smallness misfit: 343.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.2 25.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 54.04674406648726
21 4.80e+00 2.76e+01 8.97e+01 4.58e+02 1.30e+02 0 100 7.25e-01 3.73e+03
geophys. misfits: 27.6 (target 30.0 [True]); 27.7 (target 30.0 [True]) | smallness misfit: 254.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [27.6 27.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 58.67369112656665
22 4.80e+00 2.61e+01 9.03e+01 4.59e+02 1.35e+02 0 100 6.56e-01 2.48e+03
geophys. misfits: 22.5 (target 30.0 [True]); 29.3 (target 30.0 [True]) | smallness misfit: 254.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.5 29.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 69.16656027070377
23 4.80e+00 3.10e+01 9.10e+01 4.67e+02 1.33e+02 0 100 1.04e+01 2.73e+04
geophys. misfits: 22.3 (target 30.0 [True]); 38.7 (target 30.0 [False]) | smallness misfit: 211.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.3 38.7]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.343963739787227
New scales: [0.39583934 0.60416066]
24 2.40e+00 2.83e+01 9.06e+01 2.46e+02 1.25e+02 0 100 3.09e-01 9.49e+03
geophys. misfits: 21.0 (target 30.0 [True]); 33.0 (target 30.0 [False]) | smallness misfit: 195.7 (target: 200.0 [True])
Beta cooling evaluation: progress: [21. 33.]; minimum progress targets: [30. 31.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.42549729011738
New scales: [0.31489077 0.68510923]
25 1.20e+00 2.36e+01 9.32e+01 1.35e+02 1.09e+02 0 100 4.63e-01 4.93e+03
geophys. misfits: 17.7 (target 30.0 [True]); 26.3 (target 30.0 [True]) | smallness misfit: 192.5 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [17.7 26.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 98.00180330605899
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.5629e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.0970e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.0903e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.0903e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 25
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev2+gfcb9bdf36
Alpha scales: [np.float64(0.00034350275117160276), np.float64(0.0), np.float64(3.434493896517218e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09493507 0.90506493]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09493507 0.90506493]
================================================= 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+03 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.93e+03 6.54e+04 2.26e+01 1.09e+05 1.41e+02 0 15 3.46e-04 3.18e+03
geophys. misfits: 92949.3 (target 30.0 [False]); 62530.5 (target 30.0 [False]) | smallness misfit: 249.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [92949.3 62530.5]; minimum progress targets: [240000. 240000.]
2 1.93e+03 7.81e+01 5.49e-01 1.14e+03 1.34e+02 0 100 9.17e-03 2.48e+02 Skip BFGS
geophys. misfits: 600.1 (target 30.0 [False]); 23.4 (target 30.0 [True]) | smallness misfit: 134.1 (target: 200.0 [True])
Beta cooling evaluation: progress: [600.1 23.4]; minimum progress targets: [74359.5 50024.4]
Updating scaling for data misfits by 1.282750739159099
New scales: [0.1185946 0.8814054]
3 1.93e+03 2.94e+01 1.32e-01 2.85e+02 1.00e+02 0 100 5.03e-02 1.52e+02 Skip BFGS
geophys. misfits: 85.1 (target 30.0 [False]); 21.9 (target 30.0 [True]) | smallness misfit: 52.8 (target: 200.0 [True])
Beta cooling evaluation: progress: [85.1 21.9]; minimum progress targets: [480. 30.]
Updating scaling for data misfits by 1.369789455558577
New scales: [0.15562471 0.84437529]
4 1.93e+03 2.72e+01 1.30e-01 2.79e+02 8.67e+01 0 100 5.00e-01 4.30e+02
geophys. misfits: 60.2 (target 30.0 [False]); 21.1 (target 30.0 [True]) | smallness misfit: 55.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [60.2 21.1]; minimum progress targets: [68.1 30. ]
Updating scaling for data misfits by 1.4211368418250787
New scales: [0.20756063 0.79243937]
5 1.93e+03 2.63e+01 1.31e-01 2.79e+02 8.41e+01 0 100 5.05e-02 5.30e+01
geophys. misfits: 43.4 (target 30.0 [False]); 21.8 (target 30.0 [True]) | smallness misfit: 48.5 (target: 200.0 [True])
Beta cooling evaluation: progress: [43.4 21.8]; minimum progress targets: [48.2 30. ]
Updating scaling for data misfits by 1.3758003061922723
New scales: [0.26489946 0.73510054]
6 1.93e+03 2.52e+01 1.32e-01 2.80e+02 7.55e+01 0 100 2.83e+00 3.49e+02 Skip BFGS
geophys. misfits: 33.3 (target 30.0 [False]); 22.2 (target 30.0 [True]) | smallness misfit: 49.0 (target: 200.0 [True])
Beta cooling evaluation: progress: [33.3 22.2]; minimum progress targets: [34.7 30. ]
Updating scaling for data misfits by 1.349076881545468
New scales: [0.32712079 0.67287921]
7 1.93e+03 2.42e+01 1.33e-01 2.80e+02 8.02e+01 0 100 1.86e-01 6.16e+01 Skip BFGS
geophys. misfits: 27.1 (target 30.0 [True]); 22.8 (target 30.0 [True]) | smallness misfit: 49.3 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [27.1 22.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.211152611639493
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.4105e-01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 2.3218e-02 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.0154e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.0154e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 7
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev2+gfcb9bdf36
Alpha scales: [np.float64(3.1130151906368366e-05), np.float64(0.0), np.float64(3.102638544685697e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09493507 0.90506493]
/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: [0.09493507 0.90506493]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.10e+06 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.10e+06 4.31e+04 4.00e-02 8.73e+04 1.40e+02 0 23 9.86e-04 9.08e+03
geophys. misfits: 62191.2 (target 30.0 [False]); 41116.7 (target 30.0 [False])
2 2.21e+05 4.71e+03 1.04e-01 2.78e+04 1.37e+02 0 98 7.63e-04 1.46e+01 Skip BFGS
geophys. misfits: 9263.8 (target 30.0 [False]); 4231.3 (target 30.0 [False])
3 4.42e+04 3.07e+02 1.40e-01 6.48e+03 1.31e+02 0 100 1.85e-03 9.88e+00 Skip BFGS
geophys. misfits: 630.3 (target 30.0 [False]); 272.9 (target 30.0 [False])
4 8.84e+03 3.25e+01 1.50e-01 1.36e+03 1.03e+02 0 100 3.47e-01 4.28e+02 Skip BFGS
geophys. misfits: 42.0 (target 30.0 [False]); 31.5 (target 30.0 [False])
5 1.77e+03 1.55e+01 1.54e-01 2.87e+02 8.87e+01 0 100 1.89e-02 9.40e+00 Skip BFGS
geophys. misfits: 10.5 (target 30.0 [True]); 16.0 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.0972e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.1400e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.8710e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.8710e+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,
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 25.338 seconds)
Estimated memory usage: 321 MB