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 v0.22.0
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  1.80e+01  2.00e+05  0.00e+00  2.00e+05    2.43e+06      0
   1  1.80e+01  3.51e+01  4.60e+01  8.63e+02    7.46e-06      0
------------------------- STOP! -------------------------
0 : |fc-fOld| = 1.9914e+05 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 4.0059e+00 <= tolX*(1+|x0|) = 1.0000e-01
1 : |proj(x-g)-x|    = 7.4629e-06 <= tolG          = 1.0000e-01
1 : |proj(x-g)-x|    = 7.4629e-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 v0.22.0
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: [0.5292980696187559, 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    2.43e+06      0
geophys. misfits: 3.6 (target 20.0 [True]) | smallness misfit: 3478.7 (target: 100.0 [False])
mref changed in  26  places
Beta cooling evaluation: progress: [3.6]; minimum progress targets: [160000.]
Warming alpha_pgi to favor clustering:  5.507417862014973
   1  3.21e-02  3.63e+00  2.20e+02  1.07e+01    1.01e+01      0
geophys. misfits: 4.0 (target 20.0 [True]) | smallness misfit: 687.2 (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: [4.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  27.79485229833153
   2  3.21e-02  3.96e+00  2.47e+02  1.19e+01    2.16e+01      0   Skip BFGS
geophys. misfits: 4.4 (target 20.0 [True]) | smallness misfit: 170.6 (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: [4.4]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  126.0299604153538
   3  3.21e-02  4.41e+00  3.13e+02  1.45e+01    3.47e+02      0
geophys. misfits: 5.0 (target 20.0 [True]) | smallness misfit: 85.0 (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: [5.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  506.17589392199756
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 1.4864e-01 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 3.4699e+02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 3.4699e+02 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      20    <= iter          =      4
------------------------- DONE! -------------------------

Estimated memory usage: 12 MB