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Petrophysically guided inversion (PGI): Linear example#
We do a comparison between the classic least-squares inversion and our formulation of a petrophysically constrained inversion. We explore it through the UBC linear example.
Tikhonov Inversion##
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 = 20
jk = np.linspace(1.0, 60.0, 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)
# True model
mtrue = np.zeros(mesh.nC)
mtrue[mesh.cell_centers_x > 0.2] = 1.0
mtrue[mesh.cell_centers_x > 0.35] = 0.0
t = (mesh.cell_centers_x - 0.65) / 0.25
indx = np.abs(t) < 1
mtrue[indx] = -(((1 - t**2.0) ** 2.0)[indx])
mtrue = np.zeros(mesh.nC)
mtrue[mesh.cell_centers_x > 0.3] = 1.0
mtrue[mesh.cell_centers_x > 0.45] = -0.5
mtrue[mesh.cell_centers_x > 0.6] = 0
# simpeg problem and survey
prob = simulation.LinearSimulation(mesh, G=G, model_map=maps.IdentityMap())
std = 0.01
survey = prob.make_synthetic_data(mtrue, relative_error=std, add_noise=True)
# Setup the inverse problem
reg = regularization.WeightedLeastSquares(mesh, alpha_s=1.0, alpha_x=1.0)
dmis = data_misfit.L2DataMisfit(data=survey, simulation=prob)
opt = optimization.ProjectedGNCG(maxIter=10, maxIterCG=50, tolCG=1e-4)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
directiveslist = [
directives.BetaEstimate_ByEig(beta0_ratio=1e-5),
directives.BetaSchedule(coolingFactor=10.0, coolingRate=2),
directives.TargetMisfit(),
]
inv = inversion.BaseInversion(invProb, directiveList=directiveslist)
m0 = np.zeros_like(mtrue)
mnormal = inv.run(m0)
Running inversion with SimPEG v0.24.0
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.***
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 1.81e+01 2.00e+05 0.00e+00 2.00e+05 2.52e+06 0
1 1.81e+01 4.06e+01 4.52e+01 8.57e+02 8.74e-06 0
------------------------- STOP! -------------------------
0 : |fc-fOld| = 1.9914e+05 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 3.9666e+00 <= tolX*(1+|x0|) = 1.0000e-01
1 : |proj(x-g)-x| = 8.7369e-06 <= tolG = 1.0000e-01
1 : |proj(x-g)-x| = 8.7369e-06 <= 1e3*eps = 1.0000e-02
0 : maxIter = 10 <= iter = 1
------------------------- DONE! -------------------------
Petrophysically constrained inversion ##
# fit a Gaussian Mixture Model with n components
# on the true model to simulate the laboratory
# petrophysical measurements
n = 3
clf = utils.WeightedGaussianMixture(
mesh=mesh,
n_components=n,
covariance_type="full",
max_iter=100,
n_init=3,
reg_covar=5e-4,
)
clf.fit(mtrue.reshape(-1, 1))
# Petrophyically constrained regularization
reg = regularization.PGI(
gmmref=clf,
mesh=mesh,
alpha_pgi=1.0,
alpha_x=1.0,
)
# Optimization
opt = optimization.ProjectedGNCG(maxIter=20, maxIterCG=50, tolCG=1e-4)
opt.remember("xc")
# Setup new inverse problem
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
# directives
Alphas = directives.AlphasSmoothEstimate_ByEig(alpha0_ratio=10.0, verbose=True)
beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-8)
betaIt = directives.PGI_BetaAlphaSchedule(
verbose=True,
coolingFactor=2.0,
warmingFactor=1.0,
tolerance=0.1,
update_rate=1,
progress=0.2,
)
targets = directives.MultiTargetMisfits(verbose=True)
petrodir = directives.PGI_UpdateParameters()
addmref = directives.PGI_AddMrefInSmooth(verbose=True)
# Setup Inversion
inv = inversion.BaseInversion(
invProb, directiveList=[Alphas, beta, petrodir, targets, addmref, betaIt]
)
# Initial model same as for WeightedLeastSquares
mcluster = inv.run(m0)
# Final Plot
fig, axes = plt.subplots(1, 3, figsize=(12 * 1.2, 4 * 1.2))
for i in range(prob.G.shape[0]):
axes[0].plot(prob.G[i, :])
axes[0].set_title("Columns of matrix G")
axes[1].hist(mtrue, bins=20, linewidth=3.0, density=True, color="k")
axes[1].set_xlabel("Model value")
axes[1].set_xlabel("Occurence")
axes[1].hist(mnormal, bins=20, density=True, color="b")
axes[1].hist(mcluster, bins=20, density=True, color="r")
axes[1].legend(["Mtrue Hist.", "L2 Model Hist.", "PGI Model Hist."])
axes[2].plot(mesh.cell_centers_x, mtrue, color="black", linewidth=3)
axes[2].plot(mesh.cell_centers_x, mnormal, color="blue")
axes[2].plot(mesh.cell_centers_x, mcluster, "r-")
axes[2].plot(mesh.cell_centers_x, invProb.reg.objfcts[0].reference_model, "r--")
axes[2].legend(("True Model", "L2 Model", "PGI Model", "Learned Mref"))
axes[2].set_ylim([-2, 2])
plt.show()
Running inversion with SimPEG v0.24.0
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: [np.float64(0.5139449615465257), np.float64(0.0)]
<class 'simpeg.regularization.pgi.PGIsmallness'>
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 3.21e-02 2.00e+05 0.00e+00 2.00e+05 2.52e+06 0
geophys. misfits: 7.1 (target 20.0 [True]) | smallness misfit: 3827.8 (target: 100.0 [False])
mref changed in 24 places
Beta cooling evaluation: progress: [7.1]; minimum progress targets: [160000.]
Warming alpha_pgi to favor clustering: 2.819336279630197
1 3.21e-02 7.09e+00 1.37e+02 1.15e+01 6.55e+00 0
geophys. misfits: 7.4 (target 20.0 [True]) | smallness misfit: 1419.3 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [7.4]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 7.615139213979227
2 3.21e-02 7.40e+00 1.51e+02 1.23e+01 5.24e+00 0 Skip BFGS
geophys. misfits: 7.8 (target 20.0 [True]) | smallness misfit: 708.7 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [7.8]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 19.53960661650075
3 3.21e-02 7.79e+00 2.02e+02 1.43e+01 9.33e+00 0
geophys. misfits: 8.2 (target 20.0 [True]) | smallness misfit: 391.0 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [8.2]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 47.75167345894014
4 3.21e-02 8.18e+00 2.75e+02 1.70e+01 1.63e+01 0
geophys. misfits: 8.6 (target 20.0 [True]) | smallness misfit: 279.2 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [8.6]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 111.28011678159343
5 3.21e-02 8.58e+00 4.19e+02 2.20e+01 3.06e+01 0
geophys. misfits: 9.1 (target 20.0 [True]) | smallness misfit: 239.3 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [9.1]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 243.80382871775754
6 3.21e-02 9.13e+00 7.03e+02 3.17e+01 5.88e+01 0
geophys. misfits: 10.1 (target 20.0 [True]) | smallness misfit: 217.1 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [10.1]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 480.9066605977599
7 3.21e-02 1.01e+01 1.17e+03 4.76e+01 1.00e+02 0
geophys. misfits: 11.9 (target 20.0 [True]) | smallness misfit: 200.2 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [11.9]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 805.0776898981907
8 3.21e-02 1.19e+01 1.74e+03 6.77e+01 1.32e+02 0
geophys. misfits: 14.7 (target 20.0 [True]) | smallness misfit: 186.7 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [14.7]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1098.4620936837164
9 3.21e-02 1.47e+01 2.18e+03 8.45e+01 1.15e+02 0
geophys. misfits: 17.4 (target 20.0 [True]) | smallness misfit: 177.6 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [17.4]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1261.5291004729493
10 3.21e-02 1.74e+01 2.37e+03 9.34e+01 6.23e+01 0 Skip BFGS
geophys. misfits: 19.1 (target 20.0 [True]) | smallness misfit: 173.2 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [19.1]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1323.5699856695837
11 3.21e-02 1.91e+01 2.42e+03 9.67e+01 2.34e+01 0 Skip BFGS
geophys. misfits: 19.7 (target 20.0 [True]) | smallness misfit: 171.6 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [19.7]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1343.1059933221966
12 3.21e-02 1.97e+01 2.43e+03 9.78e+01 7.34e+00 0 Skip BFGS
geophys. misfits: 19.9 (target 20.0 [True]) | smallness misfit: 171.2 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [19.9]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1348.8511301999242
13 3.21e-02 1.99e+01 2.44e+03 9.81e+01 2.15e+00 0 Skip BFGS
geophys. misfits: 20.0 (target 20.0 [True]) | smallness misfit: 171.0 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1350.505547764556
14 3.21e-02 2.00e+01 2.44e+03 9.82e+01 6.20e-01 0 Skip BFGS
geophys. misfits: 20.0 (target 20.0 [True]) | smallness misfit: 171.0 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1350.979032278791
15 3.21e-02 2.00e+01 2.44e+03 9.82e+01 1.77e-01 0 Skip BFGS
geophys. misfits: 20.0 (target 20.0 [True]) | smallness misfit: 171.0 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1351.1143046333757
16 3.21e-02 2.00e+01 2.44e+03 9.82e+01 1.43e-01 0 Skip BFGS
geophys. misfits: 20.0 (target 20.0 [True]) | smallness misfit: 171.0 (target: 100.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering: 1351.1529310428593
17 3.21e-02 2.00e+01 2.44e+03 9.82e+01 1.45e-02 0 Skip BFGS
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.1173e-03 <= tolF*(1+|f0|) = 2.0000e+04
1 : |xc-x_last| = 4.3065e-06 <= tolX*(1+|x0|) = 1.0000e-01
1 : |proj(x-g)-x| = 1.4486e-02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 1.4486e-02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 20 <= iter = 17
------------------------- DONE! -------------------------
Total running time of the script: (0 minutes 11.307 seconds)
Estimated memory usage: 289 MB