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.dev8+g3c268dda4
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  1.83e+01  2.00e+05  0.00e+00  2.00e+05                         0           inf          inf
   1  1.83e+01  2.95e+02  4.26e+01  1.08e+03    2.50e+06      0      13       6.89e-04     1.72e+03
   2  1.83e+01  4.02e+01  4.54e+01  8.72e+02    1.72e+03      0      22       4.23e-04     7.28e-01   Skip BFGS
   3  1.83e+00  1.23e+01  4.88e+01  1.02e+02    4.97e+02      0      38       1.92e-04     9.56e-02   Skip BFGS
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.1785e+01 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 2.6192e-01 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 4.9711e+02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 4.9711e+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.dev8+g3c268dda4
Alpha scales: [np.float64(0.5279475753353815), 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  3.24e-02  2.00e+05  0.00e+00  2.00e+05                         0           inf          inf
   1  3.24e-02  3.75e+01  3.53e+02  4.89e+01    2.50e+06      0      14       1.23e-04     3.06e+02
geophys. misfits: 37.5 (target 20.0 [False]) | smallness misfit: 3004.8 (target: 100.0 [False])
mref changed in  26  places
Beta cooling evaluation: progress: [37.5]; minimum progress targets: [160000.]
   2  3.24e-02  2.76e+00  1.24e+02  6.77e+00    3.06e+02      0      43       9.73e-04     2.98e-01   Skip BFGS
geophys. misfits: 2.8 (target 20.0 [True]) | smallness misfit: 5614.5 (target: 100.0 [False])
mref changed in  9  places
Beta cooling evaluation: progress: [2.8]; minimum progress targets: [30.]
Warming alpha_pgi to favor clustering:  7.23348751731688
   3  3.24e-02  4.39e+00  2.31e+02  1.19e+01    1.43e+01      0      50       6.54e+00     9.37e+01
geophys. misfits: 4.4 (target 20.0 [True]) | smallness misfit: 1930.0 (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: [4.4]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  32.92984385879198
   4  3.24e-02  6.42e+00  4.97e+02  2.25e+01    9.94e+01      0      50       4.18e-02     4.15e+00
geophys. misfits: 6.4 (target 20.0 [True]) | smallness misfit: 1068.9 (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: [6.4]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  102.52194444131061
   5  3.24e-02  8.05e+00  1.13e+03  4.45e+01    6.59e+01      0      50       4.96e-02     3.27e+00
geophys. misfits: 8.1 (target 20.0 [True]) | smallness misfit: 931.4 (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: [8.1]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  254.6277451235043
   6  3.24e-02  1.14e+01  2.38e+03  8.85e+01    1.34e+02      0      50       1.21e-02     1.62e+00
geophys. misfits: 11.4 (target 20.0 [True]) | smallness misfit: 865.3 (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: [11.4]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  448.4349059416474
   7  3.24e-02  1.71e+01  3.83e+03  1.41e+02    1.66e+02      0      50       3.55e-01     5.88e+01
geophys. misfits: 17.1 (target 20.0 [True]) | smallness misfit: 813.3 (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: [17.1]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  524.3684224844313
   8  3.24e-02  1.98e+01  4.36e+03  1.61e+02    8.60e+01      0      50       1.64e-01     1.41e+01   Skip BFGS
geophys. misfits: 19.8 (target 20.0 [True]) | smallness misfit: 796.1 (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: [19.8]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  529.6794660304826
   9  3.24e-02  2.00e+01  4.39e+03  1.62e+02    1.47e+01      0      50       9.12e-03     1.34e-01   Skip BFGS
geophys. misfits: 20.0 (target 20.0 [True]) | smallness misfit: 794.9 (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: [20.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  529.9340299003638
  10  3.24e-02  2.00e+01  4.40e+03  1.62e+02    2.83e-01      0      50       3.59e-02     1.02e-02   Skip BFGS
geophys. misfits: 20.0 (target 20.0 [False]) | smallness misfit: 794.9 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
------------------------- STOP! -------------------------
1 : |fc-fOld| = 7.9471e-06 <= tolF*(1+|f0|) = 2.0000e+04
1 : |xc-x_last| = 1.2111e-04 <= tolX*(1+|x0|) = 1.0000e-01
1 : |proj(x-g)-x|    = 1.0174e-02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 1.0174e-02 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      20    <= iter          =     10
------------------------- DONE! -------------------------

Estimated memory usage: 333 MB