Note
Go to the end to download the full example code.
Petrophysically guided inversion: Joint linear example with nonlinear relationships#
We do a comparison between the classic least-squares inversion and our formulation of a petrophysically guided inversion. We explore it through coupling two linear problems whose respective physical properties are linked by polynomial relationships that change between rock units.
Running inversion with SimPEG v0.22.2.dev13+g048ef809f
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: [3.4811348413305074, 0.0, 3.914991224545892e-06, 0.0]
Calculating the scaling parameter.
Scale Multipliers: [0.09423011 0.90576989]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09423011 0.90576989]
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 1.87e+01 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 1052.3 (target 30.0 [False]); 61.9 (target 30.0 [False]) | smallness misfit: 3008.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [1052.3 61.9]; minimum progress targets: [240000. 240000.]
1 1.87e+01 1.55e+02 4.14e+01 9.31e+02 8.71e+01 0
geophys. misfits: 496.2 (target 30.0 [False]); 14.2 (target 30.0 [True]) | smallness misfit: 1352.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [496.2 14.2]; minimum progress targets: [841.8 49.5]
Updating scaling for data misfits by 2.1160662904718928
New scales: [0.18042264 0.81957736]
2 1.87e+01 1.01e+02 4.00e+01 8.50e+02 8.65e+01 0 Skip BFGS
geophys. misfits: 204.2 (target 30.0 [False]); 14.4 (target 30.0 [True]) | smallness misfit: 1215.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [204.2 14.4]; minimum progress targets: [397. 30.]
Updating scaling for data misfits by 2.0859225680651106
New scales: [0.31469168 0.68530832]
3 1.87e+01 7.41e+01 4.19e+01 8.59e+02 8.13e+01 0
geophys. misfits: 100.5 (target 30.0 [False]); 14.9 (target 30.0 [True]) | smallness misfit: 1142.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [100.5 14.9]; minimum progress targets: [163.4 30. ]
Updating scaling for data misfits by 2.0120144568174263
New scales: [0.48022556 0.51977444]
4 1.87e+01 5.60e+01 4.32e+01 8.65e+02 8.12e+01 0 Skip BFGS
geophys. misfits: 62.6 (target 30.0 [False]); 16.6 (target 30.0 [True]) | smallness misfit: 1086.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [62.6 16.6]; minimum progress targets: [80.4 30. ]
Updating scaling for data misfits by 1.8038101515905047
New scales: [0.62498512 0.37501488]
5 1.87e+01 4.53e+01 4.39e+01 8.68e+02 8.25e+01 0 Skip BFGS
geophys. misfits: 48.4 (target 30.0 [False]); 19.9 (target 30.0 [True]) | smallness misfit: 1038.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [48.4 19.9]; minimum progress targets: [50.1 30. ]
Updating scaling for data misfits by 1.5078788758847963
New scales: [0.71534073 0.28465927]
6 1.87e+01 4.03e+01 4.43e+01 8.69e+02 7.09e+01 0 Skip BFGS
geophys. misfits: 43.0 (target 30.0 [False]); 24.1 (target 30.0 [True]) | smallness misfit: 1001.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [43. 24.1]; minimum progress targets: [38.7 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.244377047318195
New scales: [0.7576982 0.2423018]
7 9.36e+00 3.84e+01 4.44e+01 4.54e+02 8.23e+01 0 Skip BFGS
geophys. misfits: 25.6 (target 30.0 [True]); 16.5 (target 30.0 [True]) | smallness misfit: 1026.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.6 16.5]; minimum progress targets: [34.4 30. ]
Warming alpha_pgi to favor clustering: 1.492567793376275
8 9.36e+00 2.34e+01 4.66e+01 4.60e+02 6.02e+01 0
geophys. misfits: 25.3 (target 30.0 [True]); 20.4 (target 30.0 [True]) | smallness misfit: 936.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.3 20.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.9803558861450226
9 9.36e+00 2.41e+01 4.75e+01 4.69e+02 4.27e+01 0
geophys. misfits: 24.9 (target 30.0 [True]); 24.3 (target 30.0 [True]) | smallness misfit: 868.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.9 24.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.4150382338273
10 9.36e+00 2.48e+01 4.83e+01 4.77e+02 7.43e+01 0 Skip BFGS
geophys. misfits: 24.7 (target 30.0 [True]); 28.3 (target 30.0 [True]) | smallness misfit: 816.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.7 28.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.7461326207880123
11 9.36e+00 2.56e+01 4.88e+01 4.82e+02 3.51e+01 0 Skip BFGS
geophys. misfits: 24.7 (target 30.0 [True]); 31.5 (target 30.0 [False]) | smallness misfit: 781.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [24.7 31.5]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.2161566886975606
New scales: [0.71998869 0.28001131]
12 4.68e+00 2.66e+01 4.87e+01 2.54e+02 8.64e+01 0 Skip BFGS
geophys. misfits: 19.4 (target 30.0 [True]); 16.3 (target 30.0 [True]) | smallness misfit: 849.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.4 16.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.645406479683851
13 4.68e+00 1.86e+01 5.35e+01 2.69e+02 5.68e+01 0
geophys. misfits: 19.4 (target 30.0 [True]); 21.7 (target 30.0 [True]) | smallness misfit: 707.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.4 21.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 6.802775386436049
14 4.68e+00 2.00e+01 5.63e+01 2.83e+02 5.50e+01 0
geophys. misfits: 19.3 (target 30.0 [True]); 27.8 (target 30.0 [True]) | smallness misfit: 601.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.3 27.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 8.968556795229663
15 4.68e+00 2.17e+01 5.86e+01 2.96e+02 8.63e+01 0 Skip BFGS
geophys. misfits: 19.4 (target 30.0 [True]); 34.8 (target 30.0 [False]) | smallness misfit: 524.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.4 34.8]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.546833965058424
New scales: [0.6243833 0.3756167]
16 2.34e+00 2.52e+01 5.80e+01 1.61e+02 8.76e+01 0 Skip BFGS
geophys. misfits: 17.3 (target 30.0 [True]); 16.1 (target 30.0 [True]) | smallness misfit: 603.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.3 16.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 16.12538084604493
17 2.34e+00 1.69e+01 6.99e+01 1.80e+02 7.13e+01 0
geophys. misfits: 17.2 (target 30.0 [True]); 20.9 (target 30.0 [True]) | smallness misfit: 447.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.2 20.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 25.644088067959622
18 2.34e+00 1.86e+01 7.75e+01 2.00e+02 8.67e+01 0
geophys. misfits: 17.8 (target 30.0 [True]); 23.3 (target 30.0 [True]) | smallness misfit: 368.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.8 23.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 38.11179674968368
19 2.34e+00 1.99e+01 8.61e+01 2.21e+02 9.52e+01 0
geophys. misfits: 17.5 (target 30.0 [True]); 27.0 (target 30.0 [True]) | smallness misfit: 303.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.5 27. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 53.83285050948876
20 2.34e+00 2.11e+01 9.53e+01 2.44e+02 9.94e+01 0
geophys. misfits: 19.8 (target 30.0 [True]); 34.1 (target 30.0 [False]) | smallness misfit: 228.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.8 34.1]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.5171183525361407
New scales: [0.52282975 0.47717025]
21 1.17e+00 2.66e+01 9.08e+01 1.33e+02 1.03e+02 0
geophys. misfits: 18.7 (target 30.0 [True]); 26.4 (target 30.0 [True]) | smallness misfit: 256.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.7 26.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 73.74493592258051
22 1.17e+00 2.24e+01 1.05e+02 1.45e+02 8.26e+01 0
geophys. misfits: 17.9 (target 30.0 [True]); 19.5 (target 30.0 [True]) | smallness misfit: 241.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.9 19.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 118.50129133397314
23 1.17e+00 1.87e+01 1.30e+02 1.71e+02 1.02e+02 0
geophys. misfits: 19.2 (target 30.0 [True]); 21.9 (target 30.0 [True]) | smallness misfit: 210.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.2 21.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 173.66285878709874
24 1.17e+00 2.05e+01 1.54e+02 2.01e+02 1.11e+02 0
geophys. misfits: 19.1 (target 30.0 [True]); 22.5 (target 30.0 [True]) | smallness misfit: 198.1 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [19.1 22.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 252.22849634813764
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 3.5955e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.1054e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.1054e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 25
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.22.2.dev13+g048ef809f
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: [0.0003503488753989945, 0.0, 3.4886036364489414e-06, 0.0]
Calculating the scaling parameter.
Scale Multipliers: [0.09423011 0.90576989]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09423011 0.90576989]
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 1.89e+03 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 87847.1 (target 30.0 [False]); 61313.9 (target 30.0 [False]) | smallness misfit: 275.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [87847.1 61313.9]; minimum progress targets: [240000. 240000.]
1 1.89e+03 6.38e+04 6.69e-01 6.51e+04 9.27e+01 0
geophys. misfits: 651.5 (target 30.0 [False]); 21.4 (target 30.0 [True]) | smallness misfit: 107.0 (target: 200.0 [True])
Beta cooling evaluation: progress: [651.5 21.4]; minimum progress targets: [70277.7 49051.1]
Updating scaling for data misfits by 1.4015138772506277
New scales: [0.12725032 0.87274968]
2 1.89e+03 1.02e+02 2.91e-01 6.50e+02 9.05e+01 0 Skip BFGS
geophys. misfits: 98.2 (target 30.0 [False]); 22.1 (target 30.0 [True]) | smallness misfit: 64.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [98.2 22.1]; minimum progress targets: [521.2 30. ]
Updating scaling for data misfits by 1.356734067145079
New scales: [0.16514803 0.83485197]
3 1.89e+03 3.47e+01 1.61e-01 3.38e+02 9.36e+01 0 Skip BFGS
geophys. misfits: 72.4 (target 30.0 [False]); 21.2 (target 30.0 [True]) | smallness misfit: 48.5 (target: 200.0 [True])
Beta cooling evaluation: progress: [72.4 21.2]; minimum progress targets: [78.5 30. ]
Updating scaling for data misfits by 1.4131455866467326
New scales: [0.21847184 0.78152816]
4 1.89e+03 3.24e+01 1.31e-01 2.80e+02 6.97e+01 0
geophys. misfits: 54.2 (target 30.0 [False]); 21.7 (target 30.0 [True]) | smallness misfit: 49.3 (target: 200.0 [True])
Beta cooling evaluation: progress: [54.2 21.7]; minimum progress targets: [57.9 30. ]
Updating scaling for data misfits by 1.383299946992473
New scales: [0.27886026 0.72113974]
5 1.89e+03 3.08e+01 1.33e-01 2.81e+02 6.72e+01 0 Skip BFGS
geophys. misfits: 44.3 (target 30.0 [False]); 22.4 (target 30.0 [True]) | smallness misfit: 49.7 (target: 200.0 [True])
Beta cooling evaluation: progress: [44.3 22.4]; minimum progress targets: [43.4 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.3383226355241724
New scales: [0.34103056 0.65896944]
6 9.43e+02 2.99e+01 1.34e-01 1.56e+02 1.01e+02 0 Skip BFGS
geophys. misfits: 28.3 (target 30.0 [True]); 18.3 (target 30.0 [True]) | smallness misfit: 52.2 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [28.3 18.3]; minimum progress targets: [35.5 30. ]
Warming alpha_pgi to favor clustering: 1.3505351012806603
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 2.1926e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.0115e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.0115e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 7
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.22.2.dev13+g048ef809f
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: [4.574820883253992e-05, 0.0, 3.502354431922508e-05, 0.0]
Calculating the scaling parameter.
Scale Multipliers: [0.09423011 0.90576989]
/home/vsts/work/1/s/simpeg/directives/directives.py:339: UserWarning:
There is no PGI regularization. Smallness target is turned off (TriggerSmall flag)
Initial data misfit scales: [0.09423011 0.90576989]
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 8.88e+05 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 50077.2 (target 30.0 [False]); 29993.3 (target 30.0 [False])
1 1.78e+05 3.19e+04 4.89e-02 4.06e+04 1.37e+02 0
geophys. misfits: 6830.5 (target 30.0 [False]); 2928.5 (target 30.0 [False])
2 3.55e+04 3.30e+03 1.12e-01 7.27e+03 1.30e+02 0 Skip BFGS
geophys. misfits: 447.9 (target 30.0 [False]); 175.2 (target 30.0 [False])
3 7.10e+03 2.01e+02 1.42e-01 1.21e+03 1.04e+02 0 Skip BFGS
geophys. misfits: 39.1 (target 30.0 [False]); 20.2 (target 30.0 [True])
Updating scaling for data misfits by 1.4841625813715522
New scales: [0.13375073 0.86624927]
4 1.42e+03 2.27e+01 1.51e-01 2.37e+02 7.56e+01 0 Skip BFGS
geophys. misfits: 16.7 (target 30.0 [True]); 10.2 (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| = 3.9073e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 7.5551e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 7.5551e+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:301: 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:308: 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:329: MatplotlibDeprecationWarning:
The collections attribute was deprecated in Matplotlib 3.8 and will be removed in 3.10.
/home/vsts/work/1/s/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/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 "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:360: 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:368: 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:402: 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:409: 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:426: MatplotlibDeprecationWarning:
The collections attribute was deprecated in Matplotlib 3.8 and will be removed in 3.10.
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 48.976 seconds)
Estimated memory usage: 234 MB