Note
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Straight Ray with Volume Data Misfit Term#
Based on the SEG abstract Heagy, Cockett and Oldenburg, 2014.
Heagy, L. J., Cockett, A. R., & Oldenburg, D. W. (2014, August 5). Parametrized Inversion Framework for Proppant Volume in a Hydraulically Fractured Reservoir. SEG Technical Program Expanded Abstracts 2014. Society of Exploration Geophysicists. doi:10.1190/segam2014-1639.1
This example is a simple joint inversion that consists of a
data misfit for the tomography problem
data misfit for the volume of the inclusions (uses the effective medium theory mapping)
model regularization
True Volume: 0.6240000000000001
Running inversion with SimPEG v0.25.2.dev41+g7d721d035
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 2.50e-01 8.49e+04 0.00e+00 8.49e+04 0 inf inf
1 2.50e-01 4.27e+03 9.96e-01 4.27e+03 5.90e-01 0 5 4.77e-03 1.02e+01
/home/vsts/work/1/s/simpeg/maps/_property_maps.py:1505: UserWarning: Maximum number of iterations reached
return self._sc2phaseEMTSpheroidstransform(m)
2 2.50e-01 5.17e+02 8.43e-01 5.17e+02 3.33e+01 0 5 2.00e-02 1.51e+00
3 2.50e-01 2.17e+02 7.29e-01 2.17e+02 2.89e+01 5 5 8.94e+00 8.65e+02 Skip BFGS
4 2.50e-01 7.60e+01 6.86e-01 7.62e+01 2.14e+01 2 5 6.74e-01 7.13e+01
5 2.50e-01 6.10e+01 6.79e-01 6.12e+01 1.80e+01 0 5 4.13e-01 3.07e+01 Skip BFGS
6 2.50e-01 1.48e+01 6.86e-01 1.50e+01 1.71e+01 0 5 9.55e-02 1.83e+01
7 2.50e-01 1.00e+01 6.83e-01 1.02e+01 9.53e+00 0 5 5.61e-01 2.53e+01
8 2.50e-01 5.10e+00 6.84e-01 5.27e+00 8.98e+00 0 5 2.54e-01 1.88e+01
9 2.50e-01 3.65e+00 6.86e-01 3.82e+00 6.10e+00 0 5 8.18e-01 1.57e+01
10 2.50e-01 2.45e+00 6.86e-01 2.62e+00 6.63e+00 0 5 4.93e-01 1.03e+01
11 2.50e-01 1.89e+00 6.88e-01 2.06e+00 4.71e+00 0 5 5.63e-01 8.05e+00
12 2.50e-01 1.28e+00 6.89e-01 1.45e+00 5.74e+00 0 5 4.46e-01 7.27e+00
13 2.50e-01 9.52e-01 6.89e-01 1.12e+00 4.25e+00 0 5 4.64e-01 6.15e+00
14 2.50e-01 7.86e-01 6.89e-01 9.58e-01 4.09e+00 0 5 9.69e-01 6.61e+00
15 2.50e-01 5.42e-01 6.90e-01 7.14e-01 3.42e+00 0 5 1.15e+00 8.83e+00 Skip BFGS
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.4428e-01 <= tolF*(1+|f0|) = 8.4872e+03
0 : |xc-x_last| = 3.4358e-01 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x| = 3.3723e+00 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 3.3723e+00 <= 1e3*eps = 1.0000e-02
1 : maxIter = 15 <= iter = 15
------------------------- DONE! -------------------------
Total recovered volume (no vol misfit term in inversion): 0.02066784892812048
Running inversion with SimPEG v0.25.2.dev41+g7d721d035
================================================= Projected GNCG =================================================
# beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment
-----------------------------------------------------------------------------------------------------------------
0 2.50e-01 1.20e+05 0.00e+00 1.20e+05 0 inf inf
1 2.50e-01 7.88e+03 8.50e-01 7.88e+03 2.22e+01 0 5 5.43e-02 7.98e+01
2 2.50e-01 7.87e+03 8.25e-01 7.87e+03 2.67e+01 4 5 3.37e-01 1.76e+02
3 2.50e-01 7.83e+03 8.17e-01 7.83e+03 2.63e+01 5 5 8.01e-01 3.65e+02 Skip BFGS
4 2.50e-01 7.83e+03 8.10e-01 7.83e+03 2.62e+01 6 5 4.86e-01 2.10e+02
5 2.50e-01 7.83e+03 8.09e-01 7.83e+03 2.62e+01 8 5 4.79e-01 2.33e+02
------------------------------------------------------------------
0 : ft = 7.8260e+03 <= alp*descent = 7.8259e+03
1 : maxIterLS = 10 <= iterLS = 10
------------------------- End Linesearch -------------------------
The linesearch got broken. Boo.
Total volume (vol misfit term in inversion): 0.027823988604927005
import numpy as np
import scipy.sparse as sp
import matplotlib.pyplot as plt
from simpeg.seismic import straight_ray_tomography as tomo
import discretize
from simpeg import (
maps,
utils,
regularization,
optimization,
inverse_problem,
inversion,
data_misfit,
objective_function,
)
class Volume(objective_function.BaseObjectiveFunction):
r"""
A regularization on the volume integral of the model
.. math::
\phi_v = || \int_V m dV - \text{knownVolume} ||^2
"""
def __init__(self, mesh, knownVolume=0.0, **kwargs):
self.mesh = mesh
self.knownVolume = knownVolume
super().__init__(**kwargs)
@property
def knownVolume(self):
"""known volume"""
return self._knownVolume
@knownVolume.setter
def knownVolume(self, value):
self._knownVolume = utils.validate_float("knownVolume", value, min_val=0.0)
def __call__(self, m):
return (self.estVol(m) - self.knownVolume) ** 2
def estVol(self, m):
return np.inner(self.mesh.cell_volumes, m)
def deriv(self, m):
# return (self.mesh.cell_volumes * np.inner(self.mesh.cell_volumes, m))
return (
2
* self.mesh.cell_volumes
* (self.knownVolume - np.inner(self.mesh.cell_volumes, m))
) # factor of 2 from deriv of ||estVol - knownVol||^2
def deriv2(self, m, v=None):
if v is not None:
return 2 * utils.mkvc(
self.mesh.cell_volumes * np.inner(self.mesh.cell_volumes, v)
)
else:
# TODO: this is inefficent. It is a fully dense matrix
return 2 * sp.csc_matrix(
np.outer(self.mesh.cell_volumes, self.mesh.cell_volumes)
)
def run(plotIt=True):
nC = 40
de = 1.0
h = np.ones(nC) * de / nC
M = discretize.TensorMesh([h, h])
y = np.linspace(M.cell_centers_y[0], M.cell_centers_x[-1], int(np.floor(nC / 4)))
rlocs = np.c_[0 * y + M.cell_centers_x[-1], y]
rx = tomo.Rx(rlocs)
source_list = [
tomo.Src(location=np.r_[M.cell_centers_x[0], yi], receiver_list=[rx])
for yi in y
]
# phi model
phi0 = 0
phi1 = 0.65
phitrue = utils.model_builder.create_block_in_wholespace(
M.gridCC, [0.4, 0.6], [0.6, 0.4], [phi1, phi0]
)
knownVolume = np.sum(phitrue * M.cell_volumes)
print("True Volume: {}".format(knownVolume))
# Set up true conductivity model and plot the model transform
sigma0 = np.exp(1)
sigma1 = 1e4
if plotIt:
fig, ax = plt.subplots(1, 1)
sigmaMapTest = maps.SelfConsistentEffectiveMedium(
nP=1000, sigma0=sigma0, sigma1=sigma1, rel_tol=1e-1, maxIter=150
)
testphis = np.linspace(0.0, 1.0, 1000)
sigetest = sigmaMapTest * testphis
ax.semilogy(testphis, sigetest)
ax.set_title("Model Transform")
ax.set_xlabel(r"$\varphi$")
ax.set_ylabel(r"$\sigma$")
sigmaMap = maps.SelfConsistentEffectiveMedium(M, sigma0=sigma0, sigma1=sigma1)
# scale the slowness so it is on a ~linear scale
slownessMap = maps.LogMap(M) * sigmaMap
# set up the problem and survey
survey = tomo.Survey(source_list)
problem = tomo.Simulation(M, survey=survey, slownessMap=slownessMap)
if plotIt:
_, ax = plt.subplots(1, 1)
cb = plt.colorbar(M.plot_image(phitrue, ax=ax)[0], ax=ax)
survey.plot(ax=ax)
cb.set_label(r"$\varphi$")
# get observed data
data = problem.make_synthetic_data(phitrue, relative_error=0.03, add_noise=True)
dpred = problem.dpred(np.zeros(M.nC))
# objective function pieces
reg = regularization.WeightedLeastSquares(M)
dmis = data_misfit.L2DataMisfit(simulation=problem, data=data)
dmisVol = Volume(mesh=M, knownVolume=knownVolume)
beta = 0.25
maxIter = 15
# without the volume regularization
opt = optimization.ProjectedGNCG(maxIter=maxIter, lower=0.0, upper=1.0)
opt.remember("xc")
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt, beta=beta)
inv = inversion.BaseInversion(invProb)
mopt1 = inv.run(np.zeros(M.nC) + 1e-16)
print(
"\nTotal recovered volume (no vol misfit term in inversion): "
"{}".format(dmisVol(mopt1))
)
# with the volume regularization
vol_multiplier = 9e4
reg2 = reg
dmis2 = dmis + vol_multiplier * dmisVol
opt2 = optimization.ProjectedGNCG(maxIter=maxIter, lower=0.0, upper=1.0)
opt2.remember("xc")
invProb2 = inverse_problem.BaseInvProblem(dmis2, reg2, opt2, beta=beta)
inv2 = inversion.BaseInversion(invProb2)
mopt2 = inv2.run(np.zeros(M.nC) + 1e-16)
print("\nTotal volume (vol misfit term in inversion): {}".format(dmisVol(mopt2)))
# plot results
if plotIt:
fig, ax = plt.subplots(1, 1)
ax.plot(data.dobs)
ax.plot(dpred)
ax.plot(problem.dpred(mopt1), "o")
ax.plot(problem.dpred(mopt2), "s")
ax.legend(["dobs", "dpred0", "dpred w/o Vol", "dpred with Vol"])
fig, ax = plt.subplots(1, 3, figsize=(16, 4))
im0 = M.plot_image(phitrue, ax=ax[0])[0]
im1 = M.plot_image(mopt1, ax=ax[1])[0]
im2 = M.plot_image(mopt2, ax=ax[2])[0]
for im in [im0, im1, im2]:
im.set_clim([0.0, phi1])
plt.colorbar(im0, ax=ax[0])
plt.colorbar(im1, ax=ax[1])
plt.colorbar(im2, ax=ax[2])
ax[0].set_title("true, vol: {:1.3e}".format(knownVolume))
ax[1].set_title(
"recovered(no Volume term), vol: {:1.3e} ".format(dmisVol(mopt1))
)
ax[2].set_title(
"recovered(with Volume term), vol: {:1.3e} ".format(dmisVol(mopt2))
)
plt.tight_layout()
if __name__ == "__main__":
run()
plt.show()
Total running time of the script: (0 minutes 22.946 seconds)
Estimated memory usage: 330 MB