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 v0.26.1.dev2+g3562108fd
Alpha scales: [np.float64(3.327161705501232), np.float64(0.0), np.float64(3.3335439594155046e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09431717 0.90568283]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09431717 0.90568283]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 2.05e+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 2.05e+01 1.29e+03 1.69e+02 4.76e+03 1.41e+02 0 20 6.66e-04 6.20e+03
geophys. misfits: 7466.3 (target 30.0 [False]); 652.0 (target 30.0 [False]) | smallness misfit: 3968.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [7466.3 652. ]; minimum progress targets: [240000. 240000.]
2 2.05e+01 6.43e+01 3.85e+01 8.56e+02 1.39e+02 0 100 4.24e-01 2.64e+03 Skip BFGS
geophys. misfits: 499.9 (target 30.0 [False]); 19.0 (target 30.0 [True]) | smallness misfit: 1356.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [499.9 19. ]; minimum progress targets: [5973. 521.6]
Updating scaling for data misfits by 1.5803559774710303
New scales: [0.14131924 0.85868076]
3 2.05e+01 5.63e+01 3.88e+01 8.54e+02 8.16e+01 0 100 2.45e-03 9.73e+00 Skip BFGS
geophys. misfits: 285.5 (target 30.0 [False]); 18.5 (target 30.0 [True]) | smallness misfit: 1118.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [285.5 18.5]; minimum progress targets: [399.9 30. ]
Updating scaling for data misfits by 1.6178931040854063
New scales: [0.21027791 0.78972209]
4 2.05e+01 4.87e+01 3.99e+01 8.67e+02 7.58e+01 0 100 7.23e-03 2.29e+00
geophys. misfits: 162.0 (target 30.0 [False]); 18.5 (target 30.0 [True]) | smallness misfit: 1065.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [162. 18.5]; minimum progress targets: [228.4 30. ]
Updating scaling for data misfits by 1.6177153885132058
New scales: [0.30106403 0.69893597]
5 2.05e+01 4.28e+01 4.06e+01 8.77e+02 7.66e+01 0 100 5.14e-01 1.53e+02 Skip BFGS
geophys. misfits: 98.0 (target 30.0 [False]); 19.0 (target 30.0 [True]) | smallness misfit: 1021.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [98. 19.]; minimum progress targets: [129.6 30. ]
Updating scaling for data misfits by 1.5806076027388878
New scales: [0.40505965 0.59494035]
6 2.05e+01 3.85e+01 4.11e+01 8.83e+02 7.60e+01 0 100 8.04e-01 2.55e+02 Skip BFGS
geophys. misfits: 65.9 (target 30.0 [False]); 19.8 (target 30.0 [True]) | smallness misfit: 986.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [65.9 19.8]; minimum progress targets: [78.4 30. ]
Updating scaling for data misfits by 1.5123713659880702
New scales: [0.50731249 0.49268751]
7 2.05e+01 3.57e+01 4.15e+01 8.87e+02 7.38e+01 0 100 1.09e+00 4.00e+02 Skip BFGS
geophys. misfits: 49.7 (target 30.0 [False]); 21.2 (target 30.0 [True]) | smallness misfit: 955.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [49.7 21.2]; minimum progress targets: [52.7 30. ]
Updating scaling for data misfits by 1.4155081477227704
New scales: [0.59308674 0.40691326]
8 2.05e+01 3.39e+01 4.16e+01 8.89e+02 7.09e+01 0 100 7.94e-01 3.80e+02 Skip BFGS
geophys. misfits: 41.4 (target 30.0 [False]); 22.9 (target 30.0 [True]) | smallness misfit: 929.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [41.4 22.9]; minimum progress targets: [39.8 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.307450240521018
New scales: [0.6558421 0.3441579]
9 1.03e+01 1.89e+01 4.27e+01 4.57e+02 8.62e+01 0 100 3.24e-01 1.99e+02
geophys. misfits: 19.4 (target 30.0 [True]); 17.9 (target 30.0 [True]) | smallness misfit: 962.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.4 17.9]; minimum progress targets: [33.1 30. ]
Warming alpha_pgi to favor clustering: 1.6105451614958417
10 1.03e+01 1.97e+01 4.38e+01 4.70e+02 5.96e+01 0 100 5.90e-01 1.23e+02
geophys. misfits: 19.0 (target 30.0 [True]); 21.1 (target 30.0 [True]) | smallness misfit: 864.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [19. 21.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.415798204397411
11 1.03e+01 2.11e+01 4.52e+01 4.85e+02 6.24e+01 0 100 1.37e+00 2.03e+02
geophys. misfits: 19.0 (target 30.0 [True]); 25.3 (target 30.0 [True]) | smallness misfit: 781.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [19. 25.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 3.344577537809849
12 1.03e+01 2.30e+01 4.66e+01 5.02e+02 7.67e+01 0 100 2.67e+00 5.98e+02
geophys. misfits: 19.0 (target 30.0 [True]); 30.6 (target 30.0 [False]) | smallness misfit: 709.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [19. 30.6]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.5809190549653167
New scales: [0.54656793 0.45343207]
13 5.13e+00 1.56e+01 4.77e+01 2.61e+02 9.21e+01 0 100 9.29e-01 5.06e+02
geophys. misfits: 13.7 (target 30.0 [True]); 17.8 (target 30.0 [True]) | smallness misfit: 776.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [13.7 17.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 6.478243729691913
14 5.13e+00 1.77e+01 5.25e+01 2.87e+02 8.60e+01 0 100 2.18e+00 1.16e+03
geophys. misfits: 14.0 (target 30.0 [True]); 22.2 (target 30.0 [True]) | smallness misfit: 641.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [14. 22.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 11.321943668887103
15 5.13e+00 2.03e+01 5.87e+01 3.22e+02 9.51e+01 0 100 1.62e+00 1.92e+03
geophys. misfits: 14.9 (target 30.0 [True]); 26.8 (target 30.0 [True]) | smallness misfit: 531.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.9 26.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 17.739840284747885
16 5.13e+00 2.65e+01 6.43e+01 3.56e+02 1.08e+02 0 100 1.40e+00 2.72e+03
geophys. misfits: 16.2 (target 30.0 [True]); 38.9 (target 30.0 [False]) | smallness misfit: 408.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.2 38.9]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.8487554778635598
New scales: [0.39467579 0.60532421]
17 2.57e+00 1.69e+01 6.72e+01 1.89e+02 1.01e+02 0 100 7.30e-01 1.49e+03
geophys. misfits: 13.6 (target 30.0 [True]); 19.1 (target 30.0 [True]) | smallness misfit: 432.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [13.6 19.1]; minimum progress targets: [30. 31.1]
Warming alpha_pgi to favor clustering: 33.54997887770393
18 2.57e+00 2.01e+01 8.07e+01 2.27e+02 1.00e+02 0 100 1.71e+00 2.59e+03
geophys. misfits: 15.7 (target 30.0 [True]); 23.0 (target 30.0 [True]) | smallness misfit: 331.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.7 23. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 53.98493428933941
19 2.57e+00 1.80e+01 9.64e+01 2.65e+02 1.06e+02 0 100 3.04e-01 8.00e+02
geophys. misfits: 18.0 (target 30.0 [True]); 18.0 (target 30.0 [True]) | smallness misfit: 287.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [18. 18.]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 90.12784111527974
20 2.57e+00 1.77e+01 1.21e+02 3.28e+02 1.22e+02 0 100 2.42e+00 2.64e+03
geophys. misfits: 22.2 (target 30.0 [True]); 14.7 (target 30.0 [True]) | smallness misfit: 239.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.2 14.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 152.94264364227837
21 2.57e+00 2.29e+01 1.48e+02 4.03e+02 1.30e+02 1 100 1.25e+00 3.71e+03
geophys. misfits: 25.3 (target 30.0 [True]); 21.3 (target 30.0 [True]) | smallness misfit: 220.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.3 21.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 198.10652976054138
22 2.57e+00 2.73e+01 1.68e+02 4.58e+02 1.34e+02 0 100 1.84e+00 4.47e+03
geophys. misfits: 31.9 (target 30.0 [False]); 24.2 (target 30.0 [True]) | smallness misfit: 188.1 (target: 200.0 [True])
Beta cooling evaluation: progress: [31.9 24.2]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.2382691279389555
New scales: [0.44670694 0.55329306]
23 1.28e+00 2.79e+01 1.60e+02 2.34e+02 1.23e+02 1 100 1.05e+00 5.12e+03
geophys. misfits: 27.4 (target 30.0 [True]); 28.2 (target 30.0 [True]) | smallness misfit: 185.0 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [27.4 28.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 213.58960522840803
------------------------- STOP! -------------------------
1 : |fc-fOld| = 9.3134e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 5.4337e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.2323e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.2323e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 23
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.26.1.dev2+g3562108fd
Alpha scales: [np.float64(0.00034666573599637736), np.float64(0.0), np.float64(3.470135979546567e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09431717 0.90568283]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09431717 0.90568283]
================================================= 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 6.84e+04 2.14e+01 1.11e+05 1.41e+02 0 12 9.92e-04 9.24e+03
geophys. misfits: 96763.8 (target 30.0 [False]); 65448.5 (target 30.0 [False]) | smallness misfit: 284.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [96763.8 65448.5]; minimum progress targets: [240000. 240000.]
2 1.98e+03 7.91e+01 5.55e-01 1.18e+03 1.37e+02 0 100 1.16e-01 3.30e+03 Skip BFGS
geophys. misfits: 670.6 (target 30.0 [False]); 17.5 (target 30.0 [True]) | smallness misfit: 117.0 (target: 200.0 [True])
Beta cooling evaluation: progress: [670.6 17.5]; minimum progress targets: [77411. 52358.8]
Updating scaling for data misfits by 1.711480526709208
New scales: [0.15127098 0.84872902]
3 1.98e+03 2.37e+01 1.39e-01 2.99e+02 1.12e+02 0 100 1.13e-02 4.85e+01 Skip BFGS
geophys. misfits: 73.6 (target 30.0 [False]); 14.8 (target 30.0 [True]) | smallness misfit: 52.9 (target: 200.0 [True])
Beta cooling evaluation: progress: [73.6 14.8]; minimum progress targets: [536.5 30. ]
Updating scaling for data misfits by 2.0269236096564427
New scales: [0.26538832 0.73461168]
4 1.98e+03 2.03e+01 1.42e-01 3.02e+02 1.02e+02 0 100 4.55e-02 1.63e+01 Skip BFGS
geophys. misfits: 34.4 (target 30.0 [False]); 15.2 (target 30.0 [True]) | smallness misfit: 54.6 (target: 200.0 [True])
Beta cooling evaluation: progress: [34.4 15.2]; minimum progress targets: [58.9 30. ]
Updating scaling for data misfits by 1.9739268313414065
New scales: [0.41626548 0.58373452]
5 1.98e+03 1.99e+01 1.42e-01 3.01e+02 8.80e+01 0 100 4.99e-02 1.62e+01
geophys. misfits: 22.3 (target 30.0 [True]); 18.1 (target 30.0 [True]) | smallness misfit: 54.1 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [22.3 18.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.500871103073776
------------------------- STOP! -------------------------
1 : |fc-fOld| = 4.6311e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 1.2609e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.7996e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.7996e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 5
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.26.1.dev2+g3562108fd
Alpha scales: [np.float64(3.5075776029559405e-05), np.float64(0.0), np.float64(3.5003924873987546e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09431717 0.90568283]
/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: [0.09431717 0.90568283]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.09e+06 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.09e+06 4.26e+04 4.07e-02 8.68e+04 1.41e+02 0 24 9.17e-04 8.54e+03
geophys. misfits: 61402.0 (target 30.0 [False]); 40588.2 (target 30.0 [False])
2 2.17e+05 4.60e+03 1.05e-01 2.75e+04 1.37e+02 0 100 5.47e-02 1.02e+03 Skip BFGS
geophys. misfits: 9044.4 (target 30.0 [False]); 4140.2 (target 30.0 [False])
3 4.34e+04 2.93e+02 1.40e-01 6.39e+03 1.31e+02 0 100 1.62e-02 8.71e+01 Skip BFGS
geophys. misfits: 606.9 (target 30.0 [False]); 260.0 (target 30.0 [False])
4 8.69e+03 2.85e+01 1.51e-01 1.34e+03 1.04e+02 0 100 9.21e-01 1.12e+03 Skip BFGS
geophys. misfits: 38.5 (target 30.0 [False]); 27.4 (target 30.0 [True])
Updating scaling for data misfits by 1.0931644431262788
New scales: [0.10220609 0.89779391]
5 1.74e+03 1.17e+01 1.54e-01 2.80e+02 9.26e+01 0 100 9.54e-01 1.08e+03 Skip BFGS
geophys. misfits: 12.0 (target 30.0 [True]); 11.6 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.0953e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.2870e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 9.2572e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 9.2572e+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 24.900 seconds)
Estimated memory usage: 343 MB