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.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using the default solver Pardiso and no solver_opts.***
Alpha scales: [3.466585984471505, 0.0, 3.497667477042771e-06, 0.0]
Calculating the scaling parameter.
Scale Multipliers: [0.09996043 0.90003957]
<class 'SimPEG.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09996043 0.90003957]
model has any nan: 0
=============================== Projected GNCG ===============================
# beta phi_d phi_m f |proj(x-g)-x| LS Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
0 1.93e+01 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 991.5 (target 30.0 [False]); 75.8 (target 30.0 [False]) | smallness misfit: 2954.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [991.5 75.8]; minimum progress targets: [240000. 240000.]
1 1.93e+01 1.67e+02 4.17e+01 9.71e+02 7.90e+01 0
geophys. misfits: 460.4 (target 30.0 [False]); 22.0 (target 30.0 [True]) | smallness misfit: 1328.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [460.4 22. ]; minimum progress targets: [793.2 60.7]
Updating scaling for data misfits by 1.3632428281069984
New scales: [0.13149575 0.86850425]
2 1.93e+01 7.96e+01 4.03e+01 8.57e+02 7.49e+01 0 Skip BFGS
geophys. misfits: 307.1 (target 30.0 [False]); 21.8 (target 30.0 [True]) | smallness misfit: 1262.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [307.1 21.8]; minimum progress targets: [368.3 30. ]
Updating scaling for data misfits by 1.3745508045649883
New scales: [0.1722633 0.8277367]
3 1.93e+01 7.10e+01 4.12e+01 8.65e+02 6.96e+01 0 Skip BFGS
geophys. misfits: 206.4 (target 30.0 [False]); 21.8 (target 30.0 [True]) | smallness misfit: 1217.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [206.4 21.8]; minimum progress targets: [245.7 30. ]
Updating scaling for data misfits by 1.374927306128513
New scales: [0.22248035 0.77751965]
4 1.93e+01 6.29e+01 4.20e+01 8.72e+02 6.99e+01 0
geophys. misfits: 141.7 (target 30.0 [False]); 22.0 (target 30.0 [True]) | smallness misfit: 1184.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [141.7 22. ]; minimum progress targets: [165.1 30. ]
Updating scaling for data misfits by 1.3665305949015345
New scales: [0.28110339 0.71889661]
5 1.93e+01 5.56e+01 4.26e+01 8.77e+02 7.03e+01 0 Skip BFGS
geophys. misfits: 100.8 (target 30.0 [False]); 22.2 (target 30.0 [True]) | smallness misfit: 1158.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [100.8 22.2]; minimum progress targets: [113.4 30. ]
Updating scaling for data misfits by 1.3502522954868221
New scales: [0.34553966 0.65446034]
6 1.93e+01 4.94e+01 4.31e+01 8.81e+02 6.83e+01 0 Skip BFGS
geophys. misfits: 75.0 (target 30.0 [False]); 22.6 (target 30.0 [True]) | smallness misfit: 1136.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [75. 22.6]; minimum progress targets: [80.6 30. ]
Updating scaling for data misfits by 1.3256197409819215
New scales: [0.41172873 0.58827127]
7 1.93e+01 4.42e+01 4.35e+01 8.84e+02 6.65e+01 0 Skip BFGS
geophys. misfits: 58.9 (target 30.0 [False]); 23.2 (target 30.0 [True]) | smallness misfit: 1116.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [58.9 23.2]; minimum progress targets: [60. 30.]
Updating scaling for data misfits by 1.29179095191996
New scales: [0.47482284 0.52517716]
8 1.93e+01 4.01e+01 4.38e+01 8.85e+02 6.42e+01 0 Skip BFGS
geophys. misfits: 48.6 (target 30.0 [False]); 24.0 (target 30.0 [True]) | smallness misfit: 1098.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [48.6 24. ]; minimum progress targets: [47.1 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.2498550712172474
New scales: [0.53052044 0.46947956]
9 9.64e+00 3.71e+01 4.40e+01 4.62e+02 8.61e+01 0 Skip BFGS
geophys. misfits: 19.2 (target 30.0 [True]); 20.5 (target 30.0 [True]) | smallness misfit: 1127.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.2 20.5]; minimum progress targets: [38.9 30. ]
Warming alpha_pgi to favor clustering: 1.5120952340729237
10 9.64e+00 1.98e+01 4.65e+01 4.68e+02 5.90e+01 0
geophys. misfits: 18.9 (target 30.0 [True]); 22.1 (target 30.0 [True]) | smallness misfit: 1025.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.9 22.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.2273881909462587
11 9.64e+00 2.04e+01 4.80e+01 4.83e+02 6.00e+01 0
geophys. misfits: 18.5 (target 30.0 [True]); 24.5 (target 30.0 [True]) | smallness misfit: 922.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.5 24.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 3.1707062568707705
12 9.64e+00 2.13e+01 4.97e+01 5.01e+02 7.04e+01 0
geophys. misfits: 18.3 (target 30.0 [True]); 28.0 (target 30.0 [True]) | smallness misfit: 819.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.3 28. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.294087348358102
13 9.64e+00 2.29e+01 5.15e+01 5.20e+02 8.23e+01 0
geophys. misfits: 18.3 (target 30.0 [True]); 32.4 (target 30.0 [False]) | smallness misfit: 734.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.3 32.4]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.6436251619718456
New scales: [0.40741294 0.59258706]
14 4.82e+00 2.67e+01 5.12e+01 2.74e+02 9.48e+01 0
geophys. misfits: 13.5 (target 30.0 [True]); 21.8 (target 30.0 [True]) | smallness misfit: 774.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [13.5 21.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 7.740352239154019
15 4.82e+00 1.84e+01 5.82e+01 2.99e+02 7.65e+01 0
geophys. misfits: 14.0 (target 30.0 [True]); 24.4 (target 30.0 [True]) | smallness misfit: 636.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [14. 24.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 13.052272139095345
16 4.82e+00 2.02e+01 6.49e+01 3.33e+02 9.68e+01 0
geophys. misfits: 15.5 (target 30.0 [True]); 24.5 (target 30.0 [True]) | smallness misfit: 517.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.5 24.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 20.601095892307253
17 4.82e+00 2.08e+01 7.26e+01 3.71e+02 1.12e+02 0
geophys. misfits: 17.8 (target 30.0 [True]); 33.2 (target 30.0 [False]) | smallness misfit: 410.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.8 33.2]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.6853625289462562
New scales: [0.28973915 0.71026085]
18 2.41e+00 2.87e+01 6.97e+01 1.97e+02 1.02e+02 0
geophys. misfits: 16.4 (target 30.0 [True]); 23.1 (target 30.0 [True]) | smallness misfit: 400.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.4 23.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 32.241744820349545
19 2.41e+00 2.11e+01 8.18e+01 2.18e+02 9.05e+01 0
geophys. misfits: 17.8 (target 30.0 [True]); 22.6 (target 30.0 [True]) | smallness misfit: 362.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.8 22.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 48.50155108532288
20 2.41e+00 2.12e+01 9.44e+01 2.49e+02 1.14e+02 0
geophys. misfits: 22.0 (target 30.0 [True]); 20.6 (target 30.0 [True]) | smallness misfit: 315.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [22. 20.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 68.45961472984136
21 2.41e+00 2.10e+01 1.08e+02 2.81e+02 1.17e+02 0
geophys. misfits: 25.9 (target 30.0 [True]); 19.5 (target 30.0 [True]) | smallness misfit: 268.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.9 19.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 92.31185039875561
22 2.41e+00 2.13e+01 1.20e+02 3.11e+02 1.19e+02 0
geophys. misfits: 28.8 (target 30.0 [True]); 19.4 (target 30.0 [True]) | smallness misfit: 245.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [28.8 19.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 119.38299610376257
23 2.41e+00 2.21e+01 1.32e+02 3.41e+02 1.28e+02 0
geophys. misfits: 32.5 (target 30.0 [False]); 20.7 (target 30.0 [True]) | smallness misfit: 211.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [32.5 20.7]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.4520752529313796
New scales: [0.37199739 0.62800261]
24 1.21e+00 2.51e+01 1.28e+02 1.80e+02 1.24e+02 0
geophys. misfits: 16.6 (target 30.0 [True]); 18.9 (target 30.0 [True]) | smallness misfit: 207.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.6 18.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 202.6104295221019
25 1.21e+00 1.81e+01 1.70e+02 2.22e+02 1.19e+02 0
geophys. misfits: 21.6 (target 30.0 [True]); 21.7 (target 30.0 [True]) | smallness misfit: 192.6 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [21.6 21.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 280.67086134385863
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.7182e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.1898e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.1898e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 26
------------------------- DONE! -------------------------
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using the default solver Pardiso and no solver_opts.***
Alpha scales: [0.00034499025026179766, 0.0, 3.7949084087632494e-06, 0.0]
Calculating the scaling parameter.
Scale Multipliers: [0.09996043 0.90003957]
<class 'SimPEG.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09996043 0.90003957]
model has any nan: 0
=============================== Projected GNCG ===============================
# beta phi_d phi_m f |proj(x-g)-x| LS Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
0 1.92e+03 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 86414.9 (target 30.0 [False]); 62820.2 (target 30.0 [False]) | smallness misfit: 274.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [86414.9 62820.2]; minimum progress targets: [240000. 240000.]
1 1.92e+03 6.52e+04 6.61e-01 6.65e+04 9.27e+01 0
geophys. misfits: 599.1 (target 30.0 [False]); 24.0 (target 30.0 [True]) | smallness misfit: 121.1 (target: 200.0 [True])
Beta cooling evaluation: progress: [599.1 24. ]; minimum progress targets: [69131.9 50256.2]
Updating scaling for data misfits by 1.2507779926458842
New scales: [0.12197076 0.87802924]
2 1.92e+03 9.41e+01 3.25e-01 7.20e+02 9.64e+01 0 Skip BFGS
geophys. misfits: 86.8 (target 30.0 [False]); 24.1 (target 30.0 [True]) | smallness misfit: 56.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [86.8 24.1]; minimum progress targets: [479.2 30. ]
Updating scaling for data misfits by 1.2451935395338916
New scales: [0.14746699 0.85253301]
3 1.92e+03 3.33e+01 1.47e-01 3.17e+02 8.47e+01 0 Skip BFGS
geophys. misfits: 64.2 (target 30.0 [False]); 21.7 (target 30.0 [True]) | smallness misfit: 54.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [64.2 21.7]; minimum progress targets: [69.4 30. ]
Updating scaling for data misfits by 1.3834625249315229
New scales: [0.19309586 0.80690414]
4 1.92e+03 2.99e+01 1.40e-01 2.99e+02 8.65e+01 0
geophys. misfits: 43.4 (target 30.0 [False]); 22.3 (target 30.0 [True]) | smallness misfit: 47.8 (target: 200.0 [True])
Beta cooling evaluation: progress: [43.4 22.3]; minimum progress targets: [51.4 30. ]
Updating scaling for data misfits by 1.344890151839822
New scales: [0.24347786 0.75652214]
5 1.92e+03 2.74e+01 1.30e-01 2.77e+02 6.43e+01 0
geophys. misfits: 31.7 (target 30.0 [False]); 23.0 (target 30.0 [True]) | smallness misfit: 48.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [31.7 23. ]; minimum progress targets: [34.7 30. ]
Updating scaling for data misfits by 1.305767025342401
New scales: [0.2958966 0.7041034]
6 1.92e+03 2.56e+01 1.31e-01 2.78e+02 6.53e+01 0 Skip BFGS
geophys. misfits: 25.0 (target 30.0 [True]); 23.6 (target 30.0 [True]) | smallness misfit: 48.6 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [25. 23.6]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.2367863827143057
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 3.4587e-02 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 6.5294e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 6.5294e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 7
------------------------- DONE! -------------------------
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using the default solver Pardiso and no solver_opts.***
Alpha scales: [3.5188574849327586e-05, 0.0, 3.4826374354602616e-05, 0.0]
Calculating the scaling parameter.
Scale Multipliers: [0.09996043 0.90003957]
/home/ssoler/simpeg/SimPEG/directives/directives.py:332: UserWarning:
There is no PGI regularization. Smallness target is turned off (TriggerSmall flag)
Initial data misfit scales: [0.09996043 0.90003957]
model has any nan: 0
=============================== Projected GNCG ===============================
# beta phi_d phi_m f |proj(x-g)-x| LS Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
0 1.04e+06 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 55179.1 (target 30.0 [False]); 36188.8 (target 30.0 [False])
1 2.08e+05 3.81e+04 4.28e-02 4.70e+04 1.38e+02 0
geophys. misfits: 8012.6 (target 30.0 [False]); 3915.1 (target 30.0 [False])
2 4.17e+04 4.32e+03 1.07e-01 8.79e+03 1.31e+02 0 Skip BFGS
geophys. misfits: 521.4 (target 30.0 [False]); 249.6 (target 30.0 [False])
3 8.34e+03 2.77e+02 1.42e-01 1.46e+03 1.03e+02 0 Skip BFGS
geophys. misfits: 33.8 (target 30.0 [False]); 32.4 (target 30.0 [False])
4 1.67e+03 3.25e+01 1.52e-01 2.85e+02 8.43e+01 0 Skip BFGS
geophys. misfits: 10.6 (target 30.0 [True]); 18.2 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.3829e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.4218e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.4218e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 5
------------------------- DONE! -------------------------
/home/ssoler/simpeg/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:301: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.
/home/ssoler/simpeg/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:308: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "r.-" (-> marker='.'). The keyword argument will take precedence.
/home/ssoler/simpeg/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:346: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.
/home/ssoler/simpeg/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 "r.-" (-> marker='.'). The keyword argument will take precedence.
/home/ssoler/simpeg/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:360: UserWarning:
The following kwargs were not used by contour: 'label'
/home/ssoler/simpeg/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:368: UserWarning:
The following kwargs were not used by contour: 'label'
/home/ssoler/simpeg/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:402: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "b.-" (-> marker='.'). The keyword argument will take precedence.
/home/ssoler/simpeg/examples/10-pgi/plot_inv_1_PGI_Linear_1D_joint_WithRelationships.py:409: 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 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",
)
axes[3].scatter(wires.m1 * mcluster_map, wires.m2 * mcluster_map, marker="v")
axes[3].set_title("Petrophysical Distribution")
CS.collections[0].set_label("")
axes[3].legend(["True Petrophysical Distribution", "Recovered model crossplot"])
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="--",
label="Modeled Petro. Distribution",
)
axes[7].scatter(
wires.m1 * mcluster_no_map,
wires.m2 * mcluster_no_map,
marker="v",
label="Recovered model crossplot",
)
axes[7].set_title("Petrophysical Distribution")
axes[7].legend()
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")
CS.collections[0].set_label("")
axes[11].legend(["True Petro Distribution", "Recovered model crossplot"])
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 36.963 seconds)
Estimated memory usage: 10 MB