# 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  1.88e+01  2.00e+05  0.00e+00  2.00e+05    2.54e+06      0
1  1.88e+01  3.26e+01  4.48e+01  8.75e+02    7.22e-06      0
------------------------- STOP! -------------------------
0 : |fc-fOld| = 1.9912e+05 <= tolF*(1+|f0|) = 2.0000e+04
0 : |xc-x_last| = 3.9721e+00 <= tolX*(1+|x0|) = 1.0000e-01
1 : |proj(x-g)-x|    = 7.2206e-06 <= tolG          = 1.0000e-01
1 : |proj(x-g)-x|    = 7.2206e-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()

# 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: [0.5285451004836603, 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  3.19e-02  2.00e+05  0.00e+00  2.00e+05    2.54e+06      0
geophys. misfits: 1.1 (target 20.0 [True]) | smallness misfit: 4478.1 (target: 100.0 [False])
mref changed in  25  places
Beta cooling evaluation: progress: [1.1]; minimum progress targets: [160000.]
Warming alpha_pgi to favor clustering:  17.653803190131757
1  3.19e-02  1.13e+00  8.20e+02  2.73e+01    3.31e+01      0
geophys. misfits: 3.1 (target 20.0 [True]) | smallness misfit: 508.8 (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: [3.1]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  112.86369168671682
2  3.19e-02  3.13e+00  6.59e+02  2.42e+01    6.42e+01      0
geophys. misfits: 5.6 (target 20.0 [True]) | smallness misfit: 226.0 (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: [5.6]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  403.1655530904333
3  3.19e-02  5.60e+00  1.03e+03  3.83e+01    1.09e+03      0
geophys. misfits: 9.0 (target 20.0 [True]) | smallness misfit: 153.7 (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: [9.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  893.0790288772854
4  3.19e-02  9.03e+00  1.50e+03  5.70e+01    1.60e+03      0
geophys. misfits: 13.0 (target 20.0 [True]) | smallness misfit: 131.9 (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: [13.]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  1373.9742690730213
5  3.19e-02  1.30e+01  1.94e+03  7.51e+01    2.22e+03      0
geophys. misfits: 16.6 (target 20.0 [True]) | smallness misfit: 121.5 (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: [16.6]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  1655.030687445534
6  3.19e-02  1.66e+01  2.14e+03  8.50e+01    4.40e+03      0   Skip BFGS
geophys. misfits: 18.9 (target 20.0 [True]) | smallness misfit: 116.7 (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: [18.9]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  1751.4632591325244
7  3.19e-02  1.89e+01  2.18e+03  8.84e+01    5.39e+03      0   Skip BFGS
geophys. misfits: 19.7 (target 20.0 [True]) | smallness misfit: 115.2 (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: [19.7]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  1779.8004135388192
8  3.19e-02  1.97e+01  2.18e+03  8.94e+01    5.58e+03      0   Skip BFGS
geophys. misfits: 19.9 (target 20.0 [True]) | smallness misfit: 114.9 (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: [19.9]; minimum progress targets: [22.]
Warming alpha_pgi to favor clustering:  1789.6868816062495
9  3.19e-02  1.99e+01  2.19e+03  8.97e+01    5.49e+03      0   Skip BFGS
geophys. misfits: 20.0 (target 20.0 [True]) | smallness misfit: 114.7 (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:  1792.799252719421
10  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.49e+03      0   Skip BFGS
geophys. misfits: 20.0 (target 20.0 [True]) | smallness misfit: 114.7 (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:  1793.4614508512916
11  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.49e+03      0   Skip BFGS
geophys. misfits: 20.0 (target 20.0 [True]) | smallness misfit: 114.7 (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:  1793.8515547317995
12  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.50e+03      0
geophys. misfits: 20.0 (target 20.0 [False]) | smallness misfit: 114.7 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
13  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.50e+03      0
geophys. misfits: 20.0 (target 20.0 [False]) | smallness misfit: 114.7 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
14  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.50e+03      0
geophys. misfits: 20.0 (target 20.0 [False]) | smallness misfit: 114.7 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
15  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.50e+03      0
geophys. misfits: 20.0 (target 20.0 [False]) | smallness misfit: 114.7 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
16  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.50e+03      0
geophys. misfits: 20.0 (target 20.0 [False]) | smallness misfit: 114.7 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
17  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.49e+03      0
geophys. misfits: 20.0 (target 20.0 [False]) | smallness misfit: 114.7 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.]; minimum progress targets: [22.]
18  3.19e-02  2.00e+01  2.19e+03  8.99e+01    5.50e+03      0
geophys. misfits: 20.4 (target 20.0 [False]) | smallness misfit: 112.4 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.4]; minimum progress targets: [22.]
19  3.19e-02  2.04e+01  2.15e+03  8.90e+01    1.27e+03      0   Skip BFGS
geophys. misfits: 20.4 (target 20.0 [False]) | smallness misfit: 112.4 (target: 100.0 [False])
mref changed in  0  places
Beta cooling evaluation: progress: [20.4]; minimum progress targets: [22.]
20  3.19e-02  2.04e+01  2.15e+03  8.90e+01    6.12e+01      0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 8.7337e-03 <= tolF*(1+|f0|) = 2.0000e+04
1 : |xc-x_last| = 2.4168e-03 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 6.1162e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 6.1162e+01 <= 1e3*eps       = 1.0000e-02
1 : maxIter   =      20    <= iter          =     20
------------------------- DONE! -------------------------
```

Total running time of the script: (0 minutes 18.023 seconds)

Estimated memory usage: 11 MB

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