Note
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Straight Ray with Volume Data Misfit TermΒΆ
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
import numpy as np
import scipy.sparse as sp
import properties
import matplotlib.pyplot as plt
from SimPEG.SEIS import StraightRay
from SimPEG import (
Mesh, Maps, Utils, Regularization, Optimization,
InvProblem, Inversion, DataMisfit, Directives, ObjectiveFunction
)
class Volume(ObjectiveFunction.BaseObjectiveFunction):
"""
A regularization on the volume integral of the model
.. math::
\phi_v = \frac{1}{2}|| \int_V m dV - \text{knownVolume} ||^2
"""
knownVolume = properties.Float("known volume", default=0., min=0.)
def __init__(self, mesh, **kwargs):
self.mesh = mesh
super(Volume, self).__init__(**kwargs)
def __call__(self, m):
return 0.5*(self.estVol(m) - self.knownVolume)**2
def estVol(self, m):
return np.inner(self.mesh.vol, m)
def deriv(self, m):
# return (self.mesh.vol * np.inner(self.mesh.vol, m))
return self.mesh.vol * (self.knownVolume - np.inner(self.mesh.vol, m))
def deriv2(self, m, v=None):
if v is not None:
return Utils.mkvc(self.mesh.vol * np.inner(self.mesh.vol, v))
else:
# TODO: this is inefficent. It is a fully dense matrix
return sp.csc_matrix(np.outer(self.mesh.vol, self.mesh.vol))
def run(plotIt=True):
nC = 40
de = 1.
h = np.ones(nC)*de/nC
M = Mesh.TensorMesh([h, h])
y = np.linspace(M.vectorCCy[0], M.vectorCCx[-1], int(np.floor(nC/4)))
rlocs = np.c_[0*y+M.vectorCCx[-1], y]
rx = StraightRay.Rx(rlocs, None)
srcList = [
StraightRay.Src(loc=np.r_[M.vectorCCx[0], yi], rxList=[rx]) for yi in y
]
# phi model
phi0 = 0
phi1 = 0.65
phitrue = Utils.ModelBuilder.defineBlock(
M.gridCC, [0.4, 0.6], [0.6, 0.4], [phi1, phi0]
)
knownVolume = np.sum(phitrue*M.vol)
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., 1., 1000)
sigetest = sigmaMapTest * testphis
ax.semilogy(testphis, sigetest)
ax.set_title('Model Transform')
ax.set_xlabel('$\\varphi$')
ax.set_ylabel('$\sigma$')
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 true sig model and log model dobs
sigtrue = sigmaMap * phitrue
# modt = Model.BaseModel(M);
slownesstrue = slownessMap * phitrue # true model (m = log(sigma))
# set up the problem and survey
survey = StraightRay.Survey(srcList)
problem = StraightRay.Problem(M, slownessMap=slownessMap)
problem.pair(survey)
if plotIt:
fig, ax = plt.subplots(1, 1)
cb = plt.colorbar(M.plotImage(phitrue, ax=ax)[0], ax=ax)
survey.plot(ax=ax)
cb.set_label('$\\varphi$')
# get observed data
dobs = survey.makeSyntheticData(phitrue, std=0.03, force=True)
dpred = survey.dpred(np.zeros(M.nC))
# objective function pieces
reg = Regularization.Tikhonov(M)
dmis = DataMisfit.l2_DataMisfit(survey)
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 = InvProblem.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 = InvProblem.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(dobs)
ax.plot(dpred)
ax.plot(survey.dpred(mopt1), 'o')
ax.plot(survey.dpred(mopt2), 's')
ax.legend(['dobs', 'dpred0', 'dpred w/o Vol', 'dpred with Vol'])
fig, ax = plt.subplots(1, 3, figsize=(16, 4))
cb0 = plt.colorbar(M.plotImage(phitrue, ax=ax[0])[0], ax=ax[0])
cb1 = plt.colorbar(M.plotImage(mopt1, ax=ax[1])[0], ax=ax[1])
cb2 = plt.colorbar(M.plotImage(mopt2, ax=ax[2])[0], ax=ax[2])
for cb in [cb0, cb1, cb2]:
cb.set_clim([0., phi1])
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 0.000 seconds)