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 v0.26.0
Alpha scales: [np.float64(3.4711030442221444), np.float64(0.0), np.float64(3.4709346860881057e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09613795 0.90386205]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09613795 0.90386205]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 1.91e+01 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 1.91e+01 6.69e+02 1.77e+02 4.04e+03 1.41e+02 0 23 2.90e-04 2.66e+03
geophys. misfits: 3198.9 (target 30.0 [False]); 400.1 (target 30.0 [False]) | smallness misfit: 3780.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [3198.9 400.1]; minimum progress targets: [240000. 240000.]
2 1.91e+01 5.98e+01 4.05e+01 8.31e+02 1.36e+02 0 100 6.32e-02 1.77e+02 Skip BFGS
geophys. misfits: 408.2 (target 30.0 [False]); 22.7 (target 30.0 [True]) | smallness misfit: 1394.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [408.2 22.7]; minimum progress targets: [2559.1 320.1]
Updating scaling for data misfits by 1.3208303631249265
New scales: [0.12318249 0.87681751]
3 1.91e+01 5.44e+01 4.06e+01 8.28e+02 8.33e+01 0 100 2.20e-01 5.97e+01 Skip BFGS
geophys. misfits: 279.6 (target 30.0 [False]); 22.7 (target 30.0 [True]) | smallness misfit: 1186.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [279.6 22.7]; minimum progress targets: [326.6 30. ]
Updating scaling for data misfits by 1.3198216105197607
New scales: [0.15641665 0.84358335]
4 1.91e+01 4.97e+01 4.12e+01 8.35e+02 7.89e+01 0 100 9.06e-03 1.54e+00
geophys. misfits: 196.3 (target 30.0 [False]); 22.5 (target 30.0 [True]) | smallness misfit: 1133.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [196.3 22.5]; minimum progress targets: [223.7 30. ]
Updating scaling for data misfits by 1.3306308870123136
New scales: [0.19789829 0.80210171]
5 1.91e+01 4.55e+01 4.17e+01 8.41e+02 6.74e+01 0 100 2.39e-02 3.88e+00
geophys. misfits: 138.3 (target 30.0 [False]); 22.6 (target 30.0 [True]) | smallness misfit: 1089.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [138.3 22.6]; minimum progress targets: [157. 30.]
Updating scaling for data misfits by 1.3289801379282722
New scales: [0.24692682 0.75307318]
6 1.91e+01 4.17e+01 4.22e+01 8.45e+02 6.69e+01 0 100 8.37e-02 1.30e+01 Skip BFGS
geophys. misfits: 99.6 (target 30.0 [False]); 22.7 (target 30.0 [True]) | smallness misfit: 1054.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [99.6 22.7]; minimum progress targets: [110.7 30. ]
Updating scaling for data misfits by 1.3211248618434603
New scales: [0.30225412 0.69774588]
7 1.91e+01 3.84e+01 4.25e+01 8.49e+02 6.57e+01 0 100 5.19e-02 7.47e+00 Skip BFGS
geophys. misfits: 74.2 (target 30.0 [False]); 22.9 (target 30.0 [True]) | smallness misfit: 1025.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [74.2 22.9]; minimum progress targets: [79.7 30. ]
Updating scaling for data misfits by 1.307699775377887
New scales: [0.36162522 0.63837478]
8 1.91e+01 3.57e+01 4.28e+01 8.51e+02 6.42e+01 0 100 7.48e-02 9.62e+00 Skip BFGS
geophys. misfits: 57.5 (target 30.0 [False]); 23.3 (target 30.0 [True]) | smallness misfit: 1001.5 (target: 200.0 [False])
Beta cooling evaluation: progress: [57.5 23.3]; minimum progress targets: [59.4 30. ]
Updating scaling for data misfits by 1.2882878196346679
New scales: [0.42189401 0.57810599]
9 1.91e+01 3.34e+01 4.30e+01 8.53e+02 6.12e+01 0 100 1.10e-01 1.22e+01 Skip BFGS
geophys. misfits: 46.5 (target 30.0 [False]); 23.8 (target 30.0 [True]) | smallness misfit: 980.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [46.5 23.8]; minimum progress targets: [46. 30.]
Decreasing beta to counter data misfit decrase plateau.
Updating scaling for data misfits by 1.262889609620851
New scales: [0.47961115 0.52038885]
10 9.53e+00 1.88e+01 4.41e+01 4.39e+02 8.46e+01 0 100 4.21e-02 1.82e+01
geophys. misfits: 17.2 (target 30.0 [True]); 20.2 (target 30.0 [True]) | smallness misfit: 998.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [17.2 20.2]; minimum progress targets: [37.2 30. ]
Warming alpha_pgi to favor clustering: 1.6133292689386638
11 9.53e+00 1.93e+01 4.53e+01 4.51e+02 4.94e+01 0 100 1.11e+00 8.46e+01
geophys. misfits: 16.5 (target 30.0 [True]); 22.0 (target 30.0 [True]) | smallness misfit: 884.8 (target: 200.0 [False])
Beta cooling evaluation: progress: [16.5 22. ]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 2.5696536625224615
12 9.53e+00 2.01e+01 4.70e+01 4.68e+02 7.51e+01 0 100 9.75e-01 1.32e+02
geophys. misfits: 15.8 (target 30.0 [True]); 24.1 (target 30.0 [True]) | smallness misfit: 769.3 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.8 24.1]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 4.042612036190953
13 9.53e+00 2.17e+01 4.91e+01 4.90e+02 8.85e+01 0 100 7.43e-01 1.59e+02
geophys. misfits: 15.7 (target 30.0 [True]); 27.3 (target 30.0 [True]) | smallness misfit: 648.1 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.7 27.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 6.084991062664996
14 9.53e+00 2.32e+01 5.17e+01 5.16e+02 8.77e+01 0 100 2.04e+00 5.45e+02
geophys. misfits: 15.6 (target 30.0 [True]); 30.3 (target 30.0 [False]) | smallness misfit: 550.7 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.6 30.3]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.9272613638487575
New scales: [0.32350715 0.67649285]
15 4.76e+00 1.90e+01 5.26e+01 2.69e+02 8.45e+01 0 100 1.97e-01 1.41e+02
geophys. misfits: 12.8 (target 30.0 [True]); 21.9 (target 30.0 [True]) | smallness misfit: 559.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [12.8 21.9]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 11.288013230476038
16 4.76e+00 2.13e+01 5.79e+01 2.97e+02 8.36e+01 0 100 7.78e-01 1.96e+02
geophys. misfits: 15.0 (target 30.0 [True]); 24.3 (target 30.0 [True]) | smallness misfit: 433.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [15. 24.3]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 18.276910489294647
17 4.76e+00 1.87e+01 6.44e+01 3.25e+02 9.54e+01 0 100 1.36e+01 4.80e+03
geophys. misfits: 15.8 (target 30.0 [True]); 20.2 (target 30.0 [True]) | smallness misfit: 368.4 (target: 200.0 [False])
Beta cooling evaluation: progress: [15.8 20.2]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 30.96174972492287
18 4.76e+00 1.91e+01 7.27e+01 3.65e+02 1.16e+02 0 100 6.97e-01 3.37e+03
geophys. misfits: 20.7 (target 30.0 [True]); 18.4 (target 30.0 [True]) | smallness misfit: 267.2 (target: 200.0 [False])
Beta cooling evaluation: progress: [20.7 18.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 47.686327621082384
19 4.76e+00 2.17e+01 8.03e+01 4.04e+02 1.22e+02 0 100 2.26e+00 7.74e+03
geophys. misfits: 28.3 (target 30.0 [True]); 18.5 (target 30.0 [True]) | smallness misfit: 231.6 (target: 200.0 [False])
Beta cooling evaluation: progress: [28.3 18.5]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 63.95791499169522
20 4.76e+00 2.44e+01 8.63e+01 4.36e+02 1.24e+02 0 100 8.09e-01 6.30e+03
geophys. misfits: 35.4 (target 30.0 [False]); 19.2 (target 30.0 [True]) | smallness misfit: 177.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [35.4 19.2]; minimum progress targets: [30. 30.]
Decreasing beta to counter data misfit increase.
Updating scaling for data misfits by 1.5655070541208405
New scales: [0.42812851 0.57187149]
21 2.38e+00 1.68e+01 8.72e+01 2.25e+02 1.13e+02 0 100 4.27e-01 2.62e+03
geophys. misfits: 14.7 (target 30.0 [True]); 18.4 (target 30.0 [True]) | smallness misfit: 186.7 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [14.7 18.4]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 117.28288638008026
------------------------- STOP! -------------------------
1 : |fc-fOld| = 7.1005e+00 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 4.8464e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 1.1332e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.1332e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 21
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.26.0
Alpha scales: [np.float64(0.0003507500212916628), np.float64(0.0), np.float64(3.5416605434839747e-06), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09613795 0.90386205]
<class 'simpeg.regularization.pgi.PGIsmallness'>
Initial data misfit scales: [0.09613795 0.90386205]
================================================= 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 6.65e+04 2.23e+01 1.10e+05 1.41e+02 0 15 3.02e-04 2.78e+03
geophys. misfits: 92146.6 (target 30.0 [False]); 63799.5 (target 30.0 [False]) | smallness misfit: 250.9 (target: 200.0 [False])
Beta cooling evaluation: progress: [92146.6 63799.5]; minimum progress targets: [240000. 240000.]
2 1.95e+03 7.32e+01 5.50e-01 1.15e+03 1.35e+02 0 100 3.32e-02 9.02e+02 Skip BFGS
geophys. misfits: 587.2 (target 30.0 [False]); 18.5 (target 30.0 [True]) | smallness misfit: 143.2 (target: 200.0 [True])
Beta cooling evaluation: progress: [587.2 18.5]; minimum progress targets: [73717.3 51039.6]
Updating scaling for data misfits by 1.6228578595827623
New scales: [0.14720363 0.85279637]
3 1.95e+03 2.48e+01 1.33e-01 2.85e+02 1.15e+02 0 100 2.13e-02 7.25e+01
geophys. misfits: 59.1 (target 30.0 [False]); 18.9 (target 30.0 [True]) | smallness misfit: 55.7 (target: 200.0 [True])
Beta cooling evaluation: progress: [59.1 18.9]; minimum progress targets: [469.8 30. ]
Updating scaling for data misfits by 1.5881779572352337
New scales: [0.21515687 0.78484313]
4 1.95e+03 2.15e+01 1.31e-01 2.77e+02 8.28e+01 0 100 3.65e-02 3.06e+01
geophys. misfits: 34.4 (target 30.0 [False]); 17.9 (target 30.0 [True]) | smallness misfit: 61.6 (target: 200.0 [True])
Beta cooling evaluation: progress: [34.4 17.9]; minimum progress targets: [47.3 30. ]
Updating scaling for data misfits by 1.6723814659918426
New scales: [0.31434837 0.68565163]
5 1.95e+03 1.98e+01 1.30e-01 2.74e+02 8.27e+01 0 100 2.02e-02 2.14e+01
geophys. misfits: 22.2 (target 30.0 [True]); 18.7 (target 30.0 [True]) | smallness misfit: 48.1 (target: 200.0 [True])
All targets have been reached
Beta cooling evaluation: progress: [22.2 18.7]; minimum progress targets: [30. 30.]
Warming alpha_pgi to favor clustering: 1.4782229624282544
------------------------- STOP! -------------------------
1 : |fc-fOld| = 5.9485e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 1.6219e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.2711e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.2711e+01 <= 1e3*eps = 1.0000e-02
0 : maxIter = 50 <= iter = 5
------------------------- DONE! -------------------------
Running inversion with SimPEG v0.26.0
Alpha scales: [np.float64(4.05657828013591e-05), np.float64(0.0), np.float64(3.505827920158615e-05), np.float64(0.0)]
Calculating the scaling parameter.
Scale Multipliers: [0.09613795 0.90386205]
/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: [0.09613795 0.90386205]
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 9.72e+05 3.00e+05 0.00e+00 3.00e+05 0 inf inf
1 9.72e+05 3.87e+04 4.50e-02 8.24e+04 1.41e+02 0 23 9.25e-04 8.50e+03
geophys. misfits: 56768.2 (target 30.0 [False]); 36756.1 (target 30.0 [False])
2 1.94e+05 3.91e+03 1.10e-01 2.52e+04 1.37e+02 0 93 9.99e-04 1.78e+01 Skip BFGS
geophys. misfits: 7524.8 (target 30.0 [False]); 3530.7 (target 30.0 [False])
3 3.89e+04 2.59e+02 1.43e-01 5.81e+03 1.30e+02 0 100 5.10e-02 2.46e+02 Skip BFGS
geophys. misfits: 483.4 (target 30.0 [False]); 234.7 (target 30.0 [False])
4 7.77e+03 3.29e+01 1.53e-01 1.22e+03 1.01e+02 0 100 6.88e-02 7.69e+01 Skip BFGS
geophys. misfits: 30.8 (target 30.0 [False]); 33.1 (target 30.0 [False])
5 1.55e+03 1.63e+01 1.57e-01 2.60e+02 8.54e+01 0 100 1.95e-01 4.66e+01 Skip BFGS
geophys. misfits: 9.7 (target 30.0 [True]); 17.0 (target 30.0 [True])
All targets have been reached
------------------------- STOP! -------------------------
1 : |fc-fOld| = 1.0477e+01 <= tolF*(1+|f0|) = 3.0000e+04
0 : |xc-x_last| = 5.0871e-01 <= tolX*(1+|x0|) = 1.0000e-06
0 : |proj(x-g)-x| = 8.5407e+01 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 8.5407e+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 24.985 seconds)
Estimated memory usage: 331 MB