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
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  1.84e+01  2.00e+05  0.00e+00  2.00e+05                         0           inf          inf
   1  1.84e+01  3.36e+02  4.22e+01  1.11e+03    2.43e+06      0      13       7.34e-04     1.79e+03
   2  1.84e+01  3.47e+01  4.56e+01  8.73e+02    1.79e+03      0      23       4.41e-04     7.87e-01   Skip BFGS
   3  1.84e+00  6.71e+00  4.86e+01  9.61e+01    5.06e+02      0      32       1.91e-04     9.68e-02   Skip BFGS
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.2447e+01 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 1.5923e-01 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 5.0585e+02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 5.0585e+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
Alpha scales: [np.float64(0.5312463186890524), 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  3.23e+01  3.53e+02  4.34e+01    2.43e+06      0      14       1.51e-04     3.67e+02
geophys. misfits: 32.3 (target 20.0 [False]) | smallness misfit: 2973.5 (target: 100.0 [False])
mref changed in  26  places
Beta cooling evaluation: progress: [32.3]; minimum progress targets: [160000.]
   2  3.15e-02  6.05e+00  5.13e+01  7.67e+00    3.67e+02      0      45       3.51e-04     1.29e-01
geophys. misfits: 6.1 (target 20.0 [True]) | smallness misfit: 1908.1 (target: 100.0 [False])
mref changed in  1  places
Beta cooling evaluation: progress: [6.1]; minimum progress targets: [25.8]
Warming alpha_pgi to favor clustering:  3.3033002338060204
   3  3.15e-02  6.18e+00  7.39e+01  8.51e+00    3.13e+00      0      50       3.00e+01     9.38e+01
geophys. misfits: 6.2 (target 20.0 [True]) | smallness misfit: 916.9 (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: [6.2]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  10.68301488348927
   4  3.15e-02  6.28e+00  1.09e+02  9.73e+00    9.40e+01      0      50       6.75e+00     6.35e+02
geophys. misfits: 6.3 (target 20.0 [True]) | smallness misfit: 383.0 (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: [6.3]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  34.00052609747121
   5  3.15e-02  6.48e+00  1.77e+02  1.21e+01    6.35e+02      0      50       1.49e-01     9.48e+01
geophys. misfits: 6.5 (target 20.0 [True]) | smallness misfit: 290.8 (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:  104.8909955019337
   6  3.15e-02  6.68e+00  2.54e+02  1.47e+01    1.03e+02      0      50       2.78e+01     2.88e+03
geophys. misfits: 6.7 (target 20.0 [True]) | smallness misfit: 139.3 (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: [6.7]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  313.91090731658386
   7  3.15e-02  8.71e+00  3.59e+02  2.00e+01    2.88e+03      0      50       7.01e-01     2.02e+03
geophys. misfits: 8.7 (target 20.0 [True]) | smallness misfit: 79.4 (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: [8.7]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  720.7844934364794
------------------------- STOP! -------------------------
1 : |fc-fOld| = 3.8498e+00 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 1.1439e-01 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 2.8766e+03 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 2.8766e+03 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      20    <= iter          =      7
------------------------- DONE! -------------------------

Estimated memory usage: 325 MB