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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: [0.10253344 0.89746656]
<class 'SimPEG.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.10253344 0.89746656]
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 1.97e+01 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 976.3 (target 30.0 [False]); 75.5 (target 30.0 [False]) | smallness misfit: 2987.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [976.3 75.5]; minimum progress targets: [240000. 240000.]
1 1.97e+01 1.68e+02 4.11e+01 9.79e+02 9.62e+01 0
geophys. misfits: 450.0 (target 30.0 [False]); 17.6 (target 30.0 [True]) | smallness misfit: 1375.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [450. 17.6]; minimum progress targets: [781. 60.4]
Updating scaling for data misfits by 1.708075361874646
New scales: [0.16328045 0.83671955]
2 1.97e+01 8.82e+01 3.98e+01 8.73e+02 8.12e+01 0 Skip BFGS
geophys. misfits: 233.8 (target 30.0 [False]); 17.3 (target 30.0 [True]) | smallness misfit: 1264.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [233.8 17.3]; minimum progress targets: [360. 30.]
Updating scaling for data misfits by 1.73480762458182
New scales: [0.25291544 0.74708456]
3 1.97e+01 7.20e+01 4.12e+01 8.84e+02 7.80e+01 0
geophys. misfits: 128.9 (target 30.0 [False]); 17.6 (target 30.0 [True]) | smallness misfit: 1189.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [128.9 17.6]; minimum progress targets: [187. 30.]
Updating scaling for data misfits by 1.7020492337099165
New scales: [0.36556512 0.63443488]
4 1.97e+01 5.83e+01 4.22e+01 8.91e+02 7.70e+01 0
geophys. misfits: 80.8 (target 30.0 [False]); 18.7 (target 30.0 [True]) | smallness misfit: 1131.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [80.8 18.7]; minimum progress targets: [103.1 30. ]
Updating scaling for data misfits by 1.6069709546804007
New scales: [0.48077468 0.51922532]
5 1.97e+01 4.85e+01 4.29e+01 8.95e+02 7.39e+01 0 Skip BFGS
geophys. misfits: 58.9 (target 30.0 [False]); 20.5 (target 30.0 [True]) | smallness misfit: 1083.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [58.9 20.5]; minimum progress targets: [64.6 30. ]
Updating scaling for data misfits by 1.4662958220227225
New scales: [0.5758598 0.4241402]
6 1.97e+01 4.26e+01 4.33e+01 8.97e+02 6.95e+01 0 Skip BFGS
geophys. misfits: 48.7 (target 30.0 [False]); 22.9 (target 30.0 [True]) | smallness misfit: 1044.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [48.7 22.9]; minimum progress targets: [47.1 30. ]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.3128511097465423
New scales: [0.64060739 0.35939261]
7 9.87e+00 3.94e+01 4.35e+01 4.69e+02 9.23e+01 0 Skip BFGS
geophys. misfits: 25.6 (target 30.0 [True]); 16.8 (target 30.0 [True]) | smallness misfit: 1100.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.6 16.8]; minimum progress targets: [39. 30.]
Warming alpha_pgi to favor clustering: 1.4784430570260136
8 9.87e+00 2.24e+01 4.58e+01 4.74e+02 4.17e+01 0
geophys. misfits: 25.4 (target 30.0 [True]); 20.2 (target 30.0 [True]) | smallness misfit: 985.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.4 20.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.9732502654200745
9 9.87e+00 2.35e+01 4.67e+01 4.84e+02 7.70e+01 0
geophys. misfits: 25.2 (target 30.0 [True]); 23.5 (target 30.0 [True]) | smallness misfit: 898.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.2 23.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.4299925361475583
10 9.87e+00 2.46e+01 4.74e+01 4.93e+02 7.43e+01 0 Skip BFGS
geophys. misfits: 25.1 (target 30.0 [True]); 26.7 (target 30.0 [True]) | smallness misfit: 832.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.1 26.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.817109952670229
11 9.87e+00 2.57e+01 4.80e+01 4.99e+02 5.60e+01 0 Skip BFGS
geophys. misfits: 25.1 (target 30.0 [True]); 29.3 (target 30.0 [True]) | smallness misfit: 785.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [25.1 29.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 3.129877772141576
12 9.87e+00 2.66e+01 4.84e+01 5.04e+02 3.63e+01 0 Skip BFGS
geophys. misfits: 25.0 (target 30.0 [True]); 31.5 (target 30.0 [False]) | smallness misfit: 752.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [25. 31.5]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.198529568044168
New scales: [0.59794405 0.40205595]
13 4.93e+00 2.76e+01 4.83e+01 2.66e+02 8.76e+01 0 Skip BFGS
geophys. misfits: 19.8 (target 30.0 [True]); 18.3 (target 30.0 [True]) | smallness misfit: 855.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.8 18.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.9358512209031025
14 4.93e+00 1.92e+01 5.28e+01 2.80e+02 6.23e+01 0
geophys. misfits: 19.9 (target 30.0 [True]); 23.1 (target 30.0 [True]) | smallness misfit: 708.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.9 23.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 6.922849426719274
15 4.93e+00 2.12e+01 5.52e+01 2.94e+02 6.80e+01 0
geophys. misfits: 20.5 (target 30.0 [True]); 29.7 (target 30.0 [True]) | smallness misfit: 600.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.5 29.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 8.557932009373166
16 4.93e+00 2.42e+01 5.65e+01 3.03e+02 8.55e+01 0
geophys. misfits: 20.9 (target 30.0 [True]); 34.2 (target 30.0 [False]) | smallness misfit: 531.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.9 34.2]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.4360536637893107
New scales: [0.50875087 0.49124913]
17 2.47e+00 2.74e+01 5.60e+01 1.65e+02 9.40e+01 0 Skip BFGS
geophys. misfits: 18.8 (target 30.0 [True]); 18.8 (target 30.0 [True]) | smallness misfit: 633.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [18.8 18.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 13.647381256803174
18 2.47e+00 1.88e+01 6.51e+01 1.79e+02 8.74e+01 0
geophys. misfits: 19.6 (target 30.0 [True]); 23.8 (target 30.0 [True]) | smallness misfit: 483.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [19.6 23.8]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 19.045404598769014
19 2.47e+00 2.17e+01 6.88e+01 1.91e+02 8.65e+01 0
geophys. misfits: 20.5 (target 30.0 [True]); 25.7 (target 30.0 [True]) | smallness misfit: 407.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.5 25.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 25.045740275743086
20 2.47e+00 2.31e+01 7.29e+01 2.03e+02 9.08e+01 0 Skip BFGS
geophys. misfits: 21.0 (target 30.0 [True]); 27.2 (target 30.0 [True]) | smallness misfit: 339.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [21. 27.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 31.722142767214407
21 2.47e+00 2.40e+01 7.68e+01 2.13e+02 9.80e+01 0
geophys. misfits: 22.2 (target 30.0 [True]); 29.0 (target 30.0 [True]) | smallness misfit: 300.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.2 29. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 37.82019981845879
22 2.47e+00 2.56e+01 7.96e+01 2.22e+02 1.01e+02 0
geophys. misfits: 23.1 (target 30.0 [True]); 29.2 (target 30.0 [True]) | smallness misfit: 244.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.1 29.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 43.95066985849938
23 2.47e+00 2.61e+01 8.15e+01 2.27e+02 9.91e+01 0
geophys. misfits: 23.7 (target 30.0 [True]); 31.7 (target 30.0 [False]) | smallness misfit: 231.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [23.7 31.7]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.2665719494814376
New scales: [0.44984254 0.55015746]
24 1.23e+00 2.81e+01 8.06e+01 1.28e+02 1.07e+02 0
geophys. misfits: 21.3 (target 30.0 [True]); 19.0 (target 30.0 [True]) | smallness misfit: 257.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [21.3 19. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 65.70376545749437
25 1.23e+00 2.00e+01 9.68e+01 1.39e+02 1.05e+02 0
geophys. misfits: 22.3 (target 30.0 [True]); 16.9 (target 30.0 [True]) | smallness misfit: 233.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.3 16.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 102.4300759001624
26 1.23e+00 1.93e+01 1.14e+02 1.60e+02 1.10e+02 0
geophys. misfits: 22.6 (target 30.0 [True]); 16.1 (target 30.0 [True]) | smallness misfit: 202.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [22.6 16.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 163.69186464532837
27 1.23e+00 1.90e+01 1.40e+02 1.92e+02 1.20e+02 0
geophys. misfits: 23.7 (target 30.0 [True]); 15.5 (target 30.0 [True]) | smallness misfit: 187.1 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [23.7 15.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 262.2468013455045
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 3.5166e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.1983e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.1983e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 28
------------------------- 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: [0.10253344 0.89746656]
<class 'SimPEG.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.10253344 0.89746656]
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 1.97e+03 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 84899.9 (target 30.0 [False]); 63769.0 (target 30.0 [False]) | smallness misfit: 307.0 (target: 200.0 [False])
Beta cooling evaluation: progress: [84899.9 63769. ]; minimum progress targets: [240000. 240000.]
1 1.97e+03 6.59e+04 7.14e-01 6.73e+04 8.95e+01 0
geophys. misfits: 561.9 (target 30.0 [False]); 14.5 (target 30.0 [True]) | smallness misfit: 75.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [561.9 14.5]; minimum progress targets: [67919.9 51015.2]
Updating scaling for data misfits by 2.0742079849535013
New scales: [0.19157515 0.80842485]
2 1.97e+03 1.19e+02 1.99e-01 5.10e+02 1.12e+02 0 Skip BFGS
geophys. misfits: 52.4 (target 30.0 [False]); 14.5 (target 30.0 [True]) | smallness misfit: 50.4 (target: 200.0 [True])
Beta cooling evaluation: progress: [52.4 14.5]; minimum progress targets: [449.5 30. ]
Updating scaling for data misfits by 2.0657867057786627
New scales: [0.32865019 0.67134981]
3 1.97e+03 2.70e+01 1.41e-01 3.04e+02 8.52e+01 0 Skip BFGS
geophys. misfits: 32.2 (target 30.0 [False]); 15.0 (target 30.0 [True]) | smallness misfit: 59.6 (target: 200.0 [True])
Beta cooling evaluation: progress: [32.2 15. ]; minimum progress targets: [42. 30.]
Updating scaling for data misfits by 2.005372276424505
New scales: [0.49538346 0.50461654]
4 1.97e+03 2.35e+01 1.68e-01 3.54e+02 8.94e+01 0
geophys. misfits: 25.4 (target 30.0 [True]); 17.0 (target 30.0 [True]) | smallness misfit: 48.3 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [25.4 17. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.47406056633217
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 1.6960e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.9403e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.9403e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 5
------------------------- 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: [0.10253344 0.89746656]
/home/vsts/work/1/s/SimPEG/directives/directives.py:332: UserWarning:
There is no PGI regularization. Smallness target is turned off (TriggerSmall flag)
Initial data misfit scales: [0.10253344 0.89746656]
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 1.07e+06 3.00e+05 0.00e+00 3.00e+05 1.41e+02 0
geophys. misfits: 53866.9 (target 30.0 [False]); 36983.1 (target 30.0 [False])
1 2.13e+05 3.87e+04 4.18e-02 4.76e+04 1.38e+02 0
geophys. misfits: 7708.0 (target 30.0 [False]); 4065.3 (target 30.0 [False])
2 4.27e+04 4.44e+03 1.06e-01 8.95e+03 1.31e+02 0 Skip BFGS
geophys. misfits: 509.1 (target 30.0 [False]); 260.0 (target 30.0 [False])
3 8.53e+03 2.86e+02 1.40e-01 1.48e+03 1.05e+02 0 Skip BFGS
geophys. misfits: 42.8 (target 30.0 [False]); 27.0 (target 30.0 [True])
Updating scaling for data misfits by 1.109305984673468
New scales: [0.11248033 0.88751967]
4 1.71e+03 2.88e+01 1.51e-01 2.86e+02 8.56e+01 0 Skip BFGS
geophys. misfits: 18.4 (target 30.0 [True]); 10.7 (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| = 4.3953e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.5526e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.5526e+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: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.
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 55.403 seconds)
Estimated memory usage: 17 MB