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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)
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 1.86e+01 1.00e+05 0.00e+00 1.00e+05 1.26e+06 0
1 1.86e+01 2.17e+01 2.28e+01 4.46e+02 3.69e-06 0
------------------------- STOP! -------------------------
0 : |fc-fOld| = 9.9554e+04 <= tolF*(1+|f0|) = 1.0000e+04
0 : |xc-x_last| = 4.0057e+00 <= tolX*(1+|x0|) = 1.0000e-01
1 : |proj(x-g)-x| = 3.6852e-06 <= tolG = 1.0000e-01
1 : |proj(x-g)-x| = 3.6852e-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()
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: [0.5191737316982674, 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 3.29e-02 1.00e+05 0.00e+00 1.00e+05 1.26e+06 0
geophys. misfits: 5.9 (target 10.0 [True]) | smallness misfit: 1923.0 (target: 50.0 [False])
mref changed in 26 places
Beta cooling evaluation: progress: [5.9]; minimum progress targets: [80000.]
Warming alpha_pgi to favor clustering: 1.6861710127265288
1 3.29e-02 5.93e+00 4.66e+01 7.47e+00 2.89e+00 0
geophys. misfits: 6.0 (target 10.0 [True]) | smallness misfit: 928.5 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 2.8181460613013196
2 3.29e-02 5.98e+00 4.47e+01 7.46e+00 9.76e+01 0 Skip BFGS
geophys. misfits: 6.1 (target 10.0 [True]) | smallness misfit: 660.4 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.1]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 4.6472403764853265
3 3.29e-02 6.06e+00 5.25e+01 7.79e+00 9.03e+00 0 Skip BFGS
geophys. misfits: 6.2 (target 10.0 [True]) | smallness misfit: 460.0 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.2]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 7.54373961924057
4 3.29e-02 6.16e+00 6.08e+01 8.16e+00 1.75e+00 0
geophys. misfits: 6.3 (target 10.0 [True]) | smallness misfit: 311.1 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.3]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 12.03343537217359
5 3.29e-02 6.27e+00 6.91e+01 8.54e+00 3.03e+00 0
geophys. misfits: 6.4 (target 10.0 [True]) | smallness misfit: 210.7 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.4]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 18.85589407480344
6 3.29e-02 6.38e+00 7.74e+01 8.93e+00 2.08e+00 0
geophys. misfits: 6.5 (target 10.0 [True]) | smallness misfit: 146.1 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.5]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 29.015991427488434
7 3.29e-02 6.50e+00 8.62e+01 9.34e+00 2.68e+00 0
geophys. misfits: 6.6 (target 10.0 [True]) | smallness misfit: 106.3 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.6]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 43.78996576616381
8 3.29e-02 6.63e+00 9.57e+01 9.78e+00 4.88e+00 0
geophys. misfits: 6.8 (target 10.0 [True]) | smallness misfit: 81.9 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.8]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 64.69673728244304
9 3.29e-02 6.77e+00 1.06e+02 1.03e+01 2.13e+01 0
geophys. misfits: 6.9 (target 10.0 [True]) | smallness misfit: 66.4 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [6.9]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 93.33898811414642
10 3.29e-02 6.93e+00 1.19e+02 1.08e+01 4.86e+00 0
geophys. misfits: 7.1 (target 10.0 [True]) | smallness misfit: 56.0 (target: 50.0 [False])
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [7.1]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 131.11899078035825
11 3.29e-02 7.12e+00 1.32e+02 1.15e+01 5.89e+00 0
geophys. misfits: 7.3 (target 10.0 [True]) | smallness misfit: 48.6 (target: 50.0 [True])
All targets have been reached
mref changed in 0 places
Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells
Beta cooling evaluation: progress: [7.3]; minimum progress targets: [11.]
Warming alpha_pgi to favor clustering: 178.84903134969392
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.0000e+04
1 : |xc-x_last| = 2.6599e-02 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x| = 5.8875e+00 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 5.8875e+00 <= 1e3*eps = 1.0000e-02
0 : maxIter = 20 <= iter = 12
------------------------- DONE! -------------------------
Total running time of the script: (0 minutes 9.716 seconds)
Estimated memory usage: 10 MB