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.dev14+g41727cb54
Alpha scales: [np.float64(3.4653841234053897), np.float64(0.0), np.float64(3.4650317226504908e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09629391 0.90370609]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09629391 0.90370609]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.95e+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.95e+01 1.38e+03 1.70e+02 4.70e+03 1.41e+02 0 19 7.60e-04 7.04e+03
geophys. misfits: 7763.8 (target 30.0 [False]); 696.8 (target 30.0 [False]) | smallness misfit: 3968.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [7763.8 696.8]; minimum progress targets: [240000. 240000.]
2 1.95e+01 6.98e+01 4.05e+01 8.60e+02 1.40e+02 0 100 1.00e-02 7.06e+01 Skip BFGS
geophys. misfits: 468.9 (target 30.0 [False]); 27.3 (target 30.0 [True]) | smallness misfit: 1416.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [468.9 27.3]; minimum progress targets: [6211.1 557.4]
Updating scaling for data misfits by 1.0979251213967502
New scales: [0.10473588 0.89526412]
3 1.95e+01 6.79e+01 3.99e+01 8.46e+02 7.62e+01 0 100 6.98e+00 1.25e+03 Skip BFGS
geophys. misfits: 417.6 (target 30.0 [False]); 26.9 (target 30.0 [True]) | smallness misfit: 1237.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [417.6 26.9]; minimum progress targets: [375.1 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.1133355719787592
New scales: [0.11523827 0.88476173]
4 9.74e+00 3.37e+01 4.24e+01 4.47e+02 8.71e+01 0 100 3.35e-02 4.75e+01
geophys. misfits: 128.9 (target 30.0 [False]); 21.3 (target 30.0 [True]) | smallness misfit: 1286.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [128.9 21.3]; minimum progress targets: [334.1 30. ]
Updating scaling for data misfits by 1.4074893600146723
New scales: [0.15492177 0.84507823]
5 9.74e+00 3.26e+01 4.29e+01 4.51e+02 7.30e+01 0 100 1.09e+00 1.37e+02 Skip BFGS
geophys. misfits: 91.8 (target 30.0 [False]); 21.8 (target 30.0 [True]) | smallness misfit: 1223.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [91.8 21.8]; minimum progress targets: [103.1 30. ]
Updating scaling for data misfits by 1.3792461042199953
New scales: [0.20181775 0.79818225]
6 9.74e+00 3.07e+01 4.34e+01 4.53e+02 7.69e+01 0 100 1.19e+00 2.55e+02 Skip BFGS
geophys. misfits: 64.4 (target 30.0 [False]); 22.2 (target 30.0 [True]) | smallness misfit: 1186.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [64.4 22.2]; minimum progress targets: [73.4 30. ]
Updating scaling for data misfits by 1.3537904289973477
New scales: [0.25501086 0.74498914]
7 9.74e+00 2.76e+01 4.38e+01 4.55e+02 7.33e+01 0 100 7.69e-01 2.63e+02
geophys. misfits: 42.5 (target 30.0 [False]); 22.5 (target 30.0 [True]) | smallness misfit: 1160.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [42.5 22.5]; minimum progress targets: [51.5 30. ]
Updating scaling for data misfits by 1.3313283369307218
New scales: [0.31305264 0.68694736]
8 9.74e+00 2.73e+01 4.39e+01 4.55e+02 7.45e+01 0 100 7.80e-01 2.68e+02
geophys. misfits: 35.8 (target 30.0 [False]); 23.4 (target 30.0 [True]) | smallness misfit: 1095.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [35.8 23.4]; minimum progress targets: [34. 30.]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.27968465083424
New scales: [0.36835681 0.63164319]
9 4.87e+00 1.81e+01 4.52e+01 2.38e+02 8.61e+01 0 100 1.60e+00 6.96e+02
geophys. misfits: 14.2 (target 30.0 [True]); 20.3 (target 30.0 [True]) | smallness misfit: 1230.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.2 20.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.792572125704074
10 4.87e+00 1.87e+01 4.68e+01 2.47e+02 9.19e+01 0 100 2.46e-01 1.71e+02
geophys. misfits: 12.3 (target 30.0 [True]); 22.5 (target 30.0 [True]) | smallness misfit: 1029.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [12.3 22.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 3.3842114924407114
11 4.87e+00 1.98e+01 4.98e+01 2.63e+02 7.50e+01 0 100 3.95e-01 6.93e+01
geophys. misfits: 11.1 (target 30.0 [True]); 24.9 (target 30.0 [True]) | smallness misfit: 842.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [11.1 24.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 6.608110769694341
12 4.87e+00 2.15e+01 5.50e+01 2.90e+02 7.66e+01 0 100 7.39e+00 1.18e+03
geophys. misfits: 9.6 (target 30.0 [True]); 28.4 (target 30.0 [True]) | smallness misfit: 676.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [ 9.6 28.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 13.845339443324315
13 4.87e+00 2.47e+01 6.43e+01 3.38e+02 1.09e+02 0 100 7.57e-01 9.35e+02
geophys. misfits: 11.7 (target 30.0 [True]); 32.3 (target 30.0 [False]) | smallness misfit: 547.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [11.7 32.3]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 2.5746895787220536
New scales: [0.18467315 0.81532685]
14 2.44e+00 2.14e+01 6.57e+01 1.81e+02 1.07e+02 0 100 4.55e-01 2.28e+02
geophys. misfits: 11.5 (target 30.0 [True]); 23.6 (target 30.0 [True]) | smallness misfit: 554.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [11.5 23.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 26.859947102771752
15 2.44e+00 2.54e+01 7.92e+01 2.18e+02 9.89e+01 0 100 2.40e+01 8.93e+03
geophys. misfits: 17.7 (target 30.0 [True]); 27.1 (target 30.0 [True]) | smallness misfit: 466.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.7 27.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 37.62557354682475
16 2.44e+00 2.25e+01 8.96e+01 2.41e+02 1.07e+02 0 100 1.55e-01 1.38e+03
geophys. misfits: 18.2 (target 30.0 [True]); 23.5 (target 30.0 [True]) | smallness misfit: 388.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.2 23.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 55.10272304929867
17 2.44e+00 2.47e+01 1.00e+02 2.69e+02 1.14e+02 0 100 6.25e+00 9.38e+03
geophys. misfits: 37.9 (target 30.0 [False]); 21.7 (target 30.0 [True]) | smallness misfit: 315.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [37.9 21.7]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.3839672714260103
New scales: [0.23865867 0.76134133]
18 1.22e+00 2.03e+01 1.01e+02 1.43e+02 1.18e+02 0 100 3.29e-01 3.98e+03
geophys. misfits: 14.5 (target 30.0 [True]); 22.2 (target 30.0 [True]) | smallness misfit: 295.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [14.5 22.2]; minimum progress targets: [30.3 30. ]
Warming alpha_pgi to favor clustering: 94.1390243160845
19 1.22e+00 2.33e+01 1.22e+02 1.71e+02 1.11e+02 0 100 5.84e-01 2.34e+03
geophys. misfits: 17.6 (target 30.0 [True]); 25.0 (target 30.0 [True]) | smallness misfit: 230.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.6 25. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 136.51157538622294
20 1.22e+00 2.35e+01 1.44e+02 1.99e+02 1.07e+02 0 100 4.00e+00 9.53e+03
geophys. misfits: 16.1 (target 30.0 [True]); 25.8 (target 30.0 [True]) | smallness misfit: 212.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.1 25.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 206.27847976595226
21 1.22e+00 2.80e+01 1.76e+02 2.42e+02 1.15e+02 0 100 2.17e+00 2.07e+04
geophys. misfits: 20.1 (target 30.0 [True]); 30.5 (target 30.0 [False]) | smallness misfit: 196.7 (target: 200.0 [True])
Beta cooling evaluation: progress: [20.1 30.5]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.4961524359505196
New scales: [0.17322459 0.82677541]
22 6.09e-01 2.35e+01 1.82e+02 1.34e+02 1.15e+02 0 100 5.19e-01 7.82e+03
geophys. misfits: 14.0 (target 30.0 [True]); 25.5 (target 30.0 [True]) | smallness misfit: 206.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [14. 25.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 342.26905241826756
23 6.09e-01 2.28e+01 2.49e+02 1.75e+02 1.18e+02 1 100 1.55e+00 1.22e+04
geophys. misfits: 14.5 (target 30.0 [True]); 24.6 (target 30.0 [True]) | smallness misfit: 198.7 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [14.5 24.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 563.1147482949501
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.1495e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.7468e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.1830e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.1830e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 23
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev14+g41727cb54
Alpha scales: [np.float64(0.0003240473506795812), np.float64(0.0), np.float64(3.1949944469179706e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09629391 0.90370609]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09629391 0.90370609]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.95e+03 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.95e+03 6.61e+04 2.25e+01 1.10e+05 1.41e+02 0 15 2.93e-04 2.71e+03
geophys. misfits: 92253.0 (target 30.0 [False]); 63282.3 (target 30.0 [False]) | smallness misfit: 248.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [92253. 63282.3]; minimum progress targets: [240000. 240000.]
2 1.95e+03 7.69e+01 5.49e-01 1.15e+03 1.35e+02 0 100 1.80e+00 4.88e+04 Skip BFGS
geophys. misfits: 575.1 (target 30.0 [False]); 23.8 (target 30.0 [True]) | smallness misfit: 131.1 (target: 200.0 [True])
Beta cooling evaluation: progress: [575.1 23.8]; minimum progress targets: [73802.4 50625.8]
Updating scaling for data misfits by 1.2601319908852104
New scales: [0.11837778 0.88162222]
3 1.95e+03 3.15e+01 1.31e-01 2.86e+02 1.17e+02 0 69 9.82e-04 5.92e+01 Skip BFGS
geophys. misfits: 68.2 (target 30.0 [False]); 26.6 (target 30.0 [True]) | smallness misfit: 61.4 (target: 200.0 [True])
Beta cooling evaluation: progress: [68.2 26.6]; minimum progress targets: [460.1 30. ]
Updating scaling for data misfits by 1.1267530187269181
New scales: [0.13141073 0.86858927]
4 1.95e+03 2.69e+01 1.28e-01 2.76e+02 5.96e+01 0 100 1.63e-01 1.52e+02
geophys. misfits: 64.4 (target 30.0 [False]); 21.2 (target 30.0 [True]) | smallness misfit: 64.8 (target: 200.0 [True])
Beta cooling evaluation: progress: [64.4 21.2]; minimum progress targets: [54.5 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.4167173044070154
New scales: [0.17650618 0.82349382]
5 9.74e+02 1.94e+01 1.30e-01 1.46e+02 1.01e+02 0 100 6.16e-01 4.34e+02
geophys. misfits: 17.0 (target 30.0 [True]); 19.9 (target 30.0 [True]) | smallness misfit: 48.5 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [17. 19.9]; minimum progress targets: [51.6 30. ]
Warming alpha_pgi to favor clustering: 1.6362872792265468
------------------------- STOP! -------------------------
1 : |fc-fOld| = 4.3684e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 2.7752e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.0095e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.0095e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 5
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.25.2.dev14+g41727cb54
Alpha scales: [np.float64(3.301295070635529e-05), np.float64(0.0), np.float64(3.292429864225739e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09629391 0.90370609]
/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: [0.09629391 0.90370609]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.13e+06 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.13e+06 4.17e+04 3.96e-02 8.66e+04 1.41e+02 0 36 4.69e-04 4.35e+03
geophys. misfits: 66398.1 (target 30.0 [False]); 39086.8 (target 30.0 [False])
2 2.27e+05 4.93e+03 1.04e-01 2.85e+04 1.37e+02 0 100 2.26e-01 4.04e+03 Skip BFGS
geophys. misfits: 9423.2 (target 30.0 [False]); 4446.0 (target 30.0 [False])
3 4.54e+04 3.26e+02 1.40e-01 6.67e+03 1.32e+02 0 100 4.41e-02 2.97e+02 Skip BFGS
geophys. misfits: 634.6 (target 30.0 [False]); 293.6 (target 30.0 [False])
4 9.07e+03 3.59e+01 1.51e-01 1.41e+03 1.05e+02 0 100 1.96e-02 2.55e+01 Skip BFGS
geophys. misfits: 36.5 (target 30.0 [False]); 35.9 (target 30.0 [False])
5 1.81e+03 1.69e+01 1.55e-01 2.98e+02 7.93e+01 0 100 4.37e-02 1.15e+01 Skip BFGS
geophys. misfits: 6.7 (target 30.0 [True]); 18.0 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.2209e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.4603e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 7.9339e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 7.9339e+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 24.530 seconds)
Estimated memory usage: 332 MB