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, cg_maxiter=50, cg_rtol=1e-3)
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 v0.25.2.dev6+gba60041a8
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  1.79e+01  2.00e+05  0.00e+00  2.00e+05                         0           inf          inf
   1  1.79e+01  3.50e+02  4.21e+01  1.10e+03    2.43e+06      0      13       7.82e-04     1.90e+03
   2  1.79e+01  3.41e+01  4.56e+01  8.51e+02    1.90e+03      0      22       7.58e-04     1.44e+00   Skip BFGS
   3  1.79e+00  6.22e+00  4.87e+01  9.35e+01    4.94e+02      0      32       3.38e-04     1.67e-01   Skip BFGS
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.2306e+01 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 1.8201e-01 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 4.9385e+02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 4.9385e+02 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      10    <= iter          =      3
------------------------- 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, cg_maxiter=50, cg_rtol=1e-3)
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 v0.25.2.dev6+gba60041a8
Alpha scales: [np.float64(0.5265216793696655), np.float64(0.0)]
<class 'simpeg.regularization.pgi.PGIsmallness'>
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  3.15e-02  2.00e+05  0.00e+00  2.00e+05                         0           inf          inf
   1  3.15e-02  3.52e+01  3.51e+02  4.63e+01    2.43e+06      0      14       2.60e-04     6.31e+02
geophys. misfits: 35.2 (target 20.0 [False]) | smallness misfit: 2958.2 (target: 100.0 [False])
mref changed in  26  places
Beta cooling evaluation: progress: [35.2]; minimum progress targets: [160000.]
   2  3.15e-02  4.37e+00  6.36e+01  6.37e+00    6.30e+02      0      44       5.99e-04     3.78e-01
geophys. misfits: 4.4 (target 20.0 [True]) | smallness misfit: 2692.8 (target: 100.0 [False])
mref changed in  1  places
Beta cooling evaluation: progress: [4.4]; minimum progress targets: [28.2]
Warming alpha_pgi to favor clustering:  4.577195453467291
   3  3.15e-02  5.03e+00  9.41e+01  7.99e+00    5.49e+00      0      50       6.95e+00     3.82e+01
geophys. misfits: 5.0 (target 20.0 [True]) | smallness misfit: 1018.1 (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: [5.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  18.20740606034946
   4  3.15e-02  5.69e+00  1.35e+02  9.96e+00    4.02e+01      0      50       3.56e+00     1.43e+02
geophys. misfits: 5.7 (target 20.0 [True]) | smallness misfit: 306.2 (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: [5.7]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  63.94834435857015
   5  3.15e-02  6.54e+00  1.96e+02  1.27e+01    1.46e+02      0      50       2.65e+00     3.87e+02
geophys. misfits: 6.5 (target 20.0 [True]) | smallness misfit: 160.2 (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: [6.5]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  195.6208940044209
   6  3.15e-02  6.89e+00  2.45e+02  1.46e+01    3.90e+02      0      50       8.74e-01     3.41e+02
geophys. misfits: 6.9 (target 20.0 [True]) | smallness misfit: 62.4 (target: 100.0 [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: [6.9]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  568.1712170924235
------------------------- STOP! -------------------------
1 : |fc-fOld| = 4.7588e+00 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 1.8644e-01 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 3.9013e+02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 3.9013e+02 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      20    <= iter          =      6
------------------------- DONE! -------------------------

Estimated memory usage: 325 MB