def run(plotIt=True, nFreq=1):
"""
MT: 3D: Forward
===============
Forward model 3D MT data.
"""
# Make a mesh
M = simpeg.Mesh.TensorMesh([[(100,5,-1.5),(100.,10),(100,5,1.5)],[(100,5,-1.5),(100.,10),(100,5,1.5)],[(100,5,1.6),(100.,10),(100,3,2)]], x0=['C','C',-3529.5360])
# Setup the model
conds = [1e-2,1]
sig = simpeg.Utils.ModelBuilder.defineBlock(M.gridCC,[-1000,-1000,-400],[1000,1000,-200],conds)
sig[M.gridCC[:,2]>0] = 1e-8
sig[M.gridCC[:,2]<-600] = 1e-1
sigBG = np.zeros(M.nC) + conds[0]
sigBG[M.gridCC[:,2]>0] = 1e-8
## Setup the the survey object
# Receiver locations
rx_x, rx_y = np.meshgrid(np.arange(-500,501,50),np.arange(-500,501,50))
rx_loc = np.hstack((simpeg.Utils.mkvc(rx_x,2),simpeg.Utils.mkvc(rx_y,2),np.zeros((np.prod(rx_x.shape),1))))
# Make a receiver list
rxList = []
for loc in rx_loc:
# NOTE: loc has to be a (1,3) np.ndarray otherwise errors accure
for rx_orientation in ['xx','xy','yx','yy']:
rxList.append(NSEM.Rx.Point_impedance3D(simpeg.mkvc(loc,2).T,rx_orientation, 'real'))
rxList.append(NSEM.Rx.Point_impedance3D(simpeg.mkvc(loc,2).T,rx_orientation, 'imag'))
for rx_orientation in ['zx','zy']:
rxList.append(NSEM.Rx.Point_tipper3D(simpeg.mkvc(loc,2).T,rx_orientation, 'real'))
rxList.append(NSEM.Rx.Point_tipper3D(simpeg.mkvc(loc,2).T,rx_orientation, 'imag'))
# Source list
srcList =[]
for freq in np.logspace(3,-3,nFreq):
srcList.append(NSEM.Src.Planewave_xy_1Dprimary(rxList,freq))
# Survey MT
survey = NSEM.Survey(srcList)
## Setup the problem object
problem = NSEM.Problem3D_ePrimSec(M, sigmaPrimary=sigBG)
problem.pair(survey)
problem.Solver = Solver
# Calculate the data
fields = problem.fields(sig)
dataVec = survey.eval(fields)
# Make the data
mtData = NSEM.Data(survey,dataVec)
# Add plots
if plotIt:
pass
def halfSpaceProblemAnaVMDDiff(showIt=False, waveformType="STEPOFF"):
cs, ncx, ncz, npad = 20., 25, 25, 15
hx = [(cs, ncx), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.CylMesh([hx, 1, hz], '00C')
sighalf = 1e-2
siginf = np.ones(mesh.nC) * 1e-8
siginf[mesh.gridCC[:, -1] < 0.] = sighalf
eta = np.ones(mesh.nC) * 0.2
tau = np.ones(mesh.nC) * 0.005
c = np.ones(mesh.nC) * 0.7
m = np.r_[siginf, eta, tau, c]
iMap = Maps.IdentityMap(nP=int(mesh.nC))
maps = [('sigmaInf', iMap), ('eta', iMap), ('tau', iMap), ('c', iMap)]
prb = ProblemATEMIP_b(mesh, mapping=maps)
if waveformType == "GENERAL":
# timeon = np.cumsum(np.r_[np.ones(10)*1e-3, np.ones(10)*5e-4, np.ones(10)*1e-4])
timeon = np.cumsum(np.r_[np.ones(10) * 1e-3,
np.ones(10) * 5e-4,
np.ones(10) * 1e-4])
timeon -= timeon.max()
timeoff = np.cumsum(np.r_[np.ones(20) * 1e-5,
np.ones(20) * 1e-4,
np.ones(20) * 1e-3])
time = np.r_[timeon, timeoff]
current_on = np.ones_like(timeon)
current_on[[0, -1]] = 0.
current = np.r_[current_on, np.zeros_like(timeoff)]
wave = np.c_[time, current]
prb.waveformType = "GENERAL"
prb.currentwaveform(wave)
prb.t0 = time.min()
elif waveformType == "STEPOFF":
prb.timeSteps = [(1e-5, 20), (1e-4, 20), (1e-3, 10)]
offset = 20.
tobs = np.logspace(-4, -2, 21)
rx = EM.TDEM.RxTDEM(np.array([[offset, 0., 0.]]), tobs, "bz")
src = EM.TDEM.SrcTDEM_VMD_MVP([rx],
np.array([[0., 0., 0.]]),
waveformType=waveformType)
survey = EM.TDEM.SurveyTDEM([src])
prb.Solver = MumpsSolver
prb.pair(survey)
out = survey.dpred(m)
bz_ana = mu_0 * hzAnalyticDipoleT_CC(
offset, rx.times, sigmaInf=sighalf, eta=eta[0], tau=tau[0], c=c[0])
err = np.linalg.norm(bz_ana - out) / np.linalg.norm(bz_ana)
print '>> Relative error = ', err
if showIt:
plt.loglog(rx.times, abs(bz_ana), 'k')
plt.loglog(rx.times, abs(out), 'b.')
plt.show()
return err
def setFrequency(self, time=np.logspace(-7, -1, 256)):
self.Nch = self.time.size
wt = np.array([7.214369775966785e-20, 5.997984537445829e-20, 1.383536819510307e-20, 6.127201193993877e-20, 2.735622069700930e-20, 6.567948836420383e-20, 4.144963335850363e-20, 7.316414067200350e-20, 5.682375914662966e-20, 8.391977074915078e-20, 7.418756524583309e-20, 9.829637687190485e-20, 9.430643800653847e-20, 1.168146262188112e-19, 1.180370735968097e-19, 1.401723019040171e-19, 1.463726071463266e-19, 1.692722072070252e-19, 1.804796158499069e-19, 2.052560499147526e-19, 2.217507732438609e-19, 2.495469564846162e-19, 2.718603842873614e-19, 3.039069705922034e-19, 3.328334008394297e-19, 3.705052796297763e-19, 4.071277819975917e-19, 4.520053409594589e-19, 4.977334107366132e-19, 5.516707191291291e-19, 6.082931168675559e-19, 6.734956703766505e-19, 7.432489554623685e-19, 8.223651399147256e-19, 9.080210233648037e-19, 1.004250388267800e-18, 1.109225156214032e-18, 1.226448534750949e-18, 1.354938655056596e-18, 1.497875155579711e-18, 1.655024636692164e-18, 1.829422009902478e-18, 2.021527957180686e-18, 2.234394042862191e-18, 2.469158736824458e-18, 2.729043278909879e-18, 3.015882778812807e-18, 3.333221019045560e-18, 3.683642665131121e-18, 4.071174485366807e-18, 4.499238428427072e-18, 4.972519918024098e-18, 5.495403162992602e-18, 6.073431145514256e-18, 6.712116746365455e-18, 7.418091347704607e-18, 8.198210388921290e-18, 9.060466264497684e-18, 1.001332641867938e-17, 1.106647001686341e-17, 1.223031194783507e-17, 1.351661046246575e-17, 1.493814249254853e-17, 1.650922025025269e-17, 1.824549287949245e-17, 2.016440324953847e-17, 2.228509875325462e-17, 2.462885473506622e-17, 2.721908372832262e-17, 3.008174877960754e-17, 3.324546598231868e-17, 3.674192913569353e-17, 4.060610542324258e-17, 4.487669220181069e-17, 4.959641037849226e-17, 5.481251456381401e-17, 6.057719336989671e-17, 6.694815564512041e-17, 7.398915178848498e-17, 8.177066132132114e-17, 9.037055462918574e-17, 9.987491078055815e-17, 1.103788451159722e-16, 1.219874911140742e-16, 1.348170262066998e-16, 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ab = 0.7866057737580476e0
#------- Compute Frequency components reqired for transform -------#
# This is for Digital filtering and here we evalute frequency domain responses
# ritght at this bases.
# a. Generate time base
n = np.ceil(-10*np.log(time.min()/time.max()))
tbase = time.max()*np.exp(-0.1*np.arange(0, n+1))
self.wt = wt
self.ab = ab
self.n = n
self.tbase = tbase
# b. Determine required frequencies
omega_int = (ab/tbase[0])*np.exp(0.1*(np.r_[1:786+tbase.size:(786+tbase.size)*1j]-425))
# Case1: Compute frequency domain reponses right at filter coefficient values
if self.switchInterp == False:
self.frequency = omega_int/(2*np.pi)
self.Nfreq = self.frequency.size
# Case2: Compute frequency domain reponses in logarithmic then intepolate
elif self.switchInterp == True:
# This is tested decision: works well 1e-4-1e0 S/m
self.frequency = np.logspace(-3, 8, 81)
self.omega_int = omega_int
self.Nfreq = self.frequency.size
else:
raise Exception('Not implemented!!')
if self.offset is not None and np.isscalar(self.offset):
self.offset = self.offset*np.ones(self.Nfreq)
elif self.offset is not None and not np.isscalar(self.offset):
self.offset = self.offset[0]*np.ones(self.Nfreq)
def halfSpaceProblemAnaVMDDiff(showIt=False, waveformType="STEPOFF"):
cs, ncx, ncz, npad = 20., 25, 25, 15
hx = [(cs, ncx), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.CylMesh([hx, 1, hz], '00C')
prb = ProblemATEM_b(mesh)
if waveformType == "GENERAL":
timeon = np.cumsum(np.r_[np.ones(10) * 1e-3,
np.ones(10) * 5e-4,
np.ones(10) * 1e-4])
timeon -= timeon.max()
timeoff = np.cumsum(np.r_[np.ones(10) * 5e-5,
np.ones(10) * 1e-4,
np.ones(10) * 5e-4,
np.ones(10) * 1e-3,
np.ones(10) * 5e-3])
time = np.r_[timeon, timeoff]
current_on = np.ones_like(timeon)
current_on[[0, -1]] = 0.
current = np.r_[current_on, np.zeros_like(timeoff)]
wave = np.c_[time, current]
prb.waveformType = "GENERAL"
prb.currentwaveform(wave)
prb.t0 = time.min()
elif waveformType == "STEPOFF":
prb.timeSteps = [(1e-5, 10), (5e-5, 10), (1e-4, 10), (5e-4, 10),
(1e-3, 10), (5e-3, 10)]
offset = 20.
tobs = np.logspace(-4, -2, 21)
rx = EM.TDEM.RxTDEM(np.array([[offset, 0., 0.]]), tobs, "bz")
src = EM.TDEM.SrcTDEM_VMD_MVP([rx],
np.array([[0., 0., 0.]]),
waveformType=waveformType)
survey = EM.TDEM.SurveyTDEM([src])
prb.Solver = MumpsSolver
sigma = np.ones(mesh.nC) * 1e-8
active = mesh.gridCC[:, 2] < 0.
sig_half = 1e-2
sigma[active] = sig_half
prb.pair(survey)
out = survey.dpred(sigma)
bz_ana = mu_0 * hzAnalyticDipoleT(offset, rx.times, sig_half)
err = np.linalg.norm(bz_ana - out) / np.linalg.norm(bz_ana)
print '>> Relative error = ', err
if showIt:
plt.loglog(rx.times, bz_ana, 'k')
plt.loglog(rx.times, out, 'b.')
plt.show()
return err
def halfSpaceProblemAnaVMDDiff(showIt=False, waveformType="STEPOFF"):
cs, ncx, ncz, npad = 20., 25, 25, 15
hx = [(cs,ncx), (cs,npad,1.3)]
hz = [(cs,npad,-1.3), (cs,ncz), (cs,npad,1.3)]
mesh = Mesh.CylMesh([hx,1,hz], '00C')
sighalf = 1e-2
siginf = np.ones(mesh.nC)*1e-8
siginf[mesh.gridCC[:,-1]<0.] = sighalf
eta = np.ones(mesh.nC)*0.2
tau = np.ones(mesh.nC)*0.005
c = np.ones(mesh.nC)*0.7
m = np.r_[siginf, eta, tau, c]
iMap = Maps.IdentityMap(nP=int(mesh.nC))
maps = [('sigmaInf', iMap), ('eta', iMap), ('tau', iMap), ('c', iMap)]
prb = ProblemATEMIP_b(mesh, mapping = maps)
if waveformType =="GENERAL":
# timeon = np.cumsum(np.r_[np.ones(10)*1e-3, np.ones(10)*5e-4, np.ones(10)*1e-4])
timeon = np.cumsum(np.r_[np.ones(10)*1e-3, np.ones(10)*5e-4, np.ones(10)*1e-4])
timeon -= timeon.max()
timeoff = np.cumsum(np.r_[np.ones(20)*1e-5, np.ones(20)*1e-4, np.ones(20)*1e-3])
time = np.r_[timeon, timeoff]
current_on = np.ones_like(timeon)
current_on[[0,-1]] = 0.
current = np.r_[current_on, np.zeros_like(timeoff)]
wave = np.c_[time, current]
prb.waveformType = "GENERAL"
prb.currentwaveform(wave)
prb.t0 = time.min()
elif waveformType =="STEPOFF":
prb.timeSteps = [(1e-5, 20), (1e-4, 20), (1e-3, 10)]
offset = 20.
tobs = np.logspace(-4, -2, 21)
rx = EM.TDEM.RxTDEM(np.array([[offset, 0., 0.]]), tobs, "bz")
src = EM.TDEM.SrcTDEM_VMD_MVP([rx], np.array([[0., 0., 0.]]), waveformType=waveformType)
survey = EM.TDEM.SurveyTDEM([src])
prb.Solver = MumpsSolver
prb.pair(survey)
out = survey.dpred(m)
bz_ana = mu_0*hzAnalyticDipoleT_CC(offset, rx.times, sigmaInf=sighalf, eta=eta[0], tau=tau[0], c=c[0])
err = np.linalg.norm(bz_ana-out)/np.linalg.norm(bz_ana)
print '>> Relative error = ', err
if showIt:
plt.loglog(rx.times, abs(bz_ana), 'k')
plt.loglog(rx.times, abs(out), 'b.')
plt.show()
return err
def appResNorm(sigmaHalf):
nFreq = 26
m1d = Mesh.TensorMesh([[(100,5,1.5),(100.,10),(100,5,1.5)]], x0=['C'])
sigma = np.zeros(m1d.nC) + sigmaHalf
sigma[m1d.gridCC[:]>200] = 1e-8
# Calculate the analytic fields
freqs = np.logspace(4,-4,nFreq)
Z = []
for freq in freqs:
Ed, Eu, Hd, Hu = NSEM.Utils.getEHfields(m1d,sigma,freq,np.array([200]))
Z.append((Ed + Eu)/(Hd + Hu))
Zarr = np.concatenate(Z)
app_r, app_p = NSEM.Utils.appResPhs(freqs,Zarr)
return np.linalg.norm(np.abs(app_r - np.ones(nFreq)/sigmaHalf)) / np.log10(sigmaHalf)
def halfSpaceProblemAnaVMDDiff(showIt=False, waveformType="STEPOFF"): cs, ncx, ncz, npad = 20., 25, 25, 15 hx = [(cs,ncx), (cs,npad,1.3)] hz = [(cs,npad,-1.3), (cs,ncz), (cs,npad,1.3)] mesh = Mesh.CylMesh([hx,1,hz], '00C') prb = ProblemATEM_b(mesh) if waveformType =="GENERAL": timeon = np.cumsum(np.r_[np.ones(10)*1e-3, np.ones(10)*5e-4, np.ones(10)*1e-4]) timeon -= timeon.max() timeoff = np.cumsum(np.r_[np.ones(10)*5e-5, np.ones(10)*1e-4, np.ones(10)*5e-4, np.ones(10)*1e-3, np.ones(10)*5e-3]) time = np.r_[timeon, timeoff] current_on = np.ones_like(timeon) current_on[[0,-1]] = 0. current = np.r_[current_on, np.zeros_like(timeoff)] wave = np.c_[time, current] prb.waveformType = "GENERAL" prb.currentwaveform(wave) prb.t0 = time.min() elif waveformType =="STEPOFF": prb.timeSteps = [(1e-5, 10), (5e-5, 10), (1e-4, 10), (5e-4, 10), (1e-3, 10),(5e-3, 10)] offset = 20. tobs = np.logspace(-4, -2, 21) rx = EM.TDEM.RxTDEM(np.array([[offset, 0., 0.]]), tobs, "bz") src = EM.TDEM.SrcTDEM_VMD_MVP([rx], np.array([[0., 0., 0.]]), waveformType=waveformType) survey = EM.TDEM.SurveyTDEM([src]) prb.Solver = MumpsSolver sigma = np.ones(mesh.nC)*1e-8 active = mesh.gridCC[:,2]<0. sig_half = 1e-2 sigma[active] = sig_half prb.pair(survey) out = survey.dpred(sigma) bz_ana = mu_0*hzAnalyticDipoleT(offset, rx.times, sig_half) err = np.linalg.norm(bz_ana-out)/np.linalg.norm(bz_ana) print '>> Relative error = ', err if showIt: plt.loglog(rx.times, bz_ana, 'k') plt.loglog(rx.times, out, 'b.') plt.show() return err
def run(plotIt=True):
"""
1D FDEM and TDEM inversions
===========================
This example is used in the paper Heagy et al 2016 (in prep)
"""
# Set up cylindrically symmeric mesh
cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh
hx = [(cs, ncx), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.CylMesh([hx, 1, hz], '00C')
# Conductivity model
layerz = np.r_[-200., -100.]
layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1])
active = mesh.vectorCCz < 0.
sig_half = 1e-2 # Half-space conductivity
sig_air = 1e-8 # Air conductivity
sig_layer = 5e-2 # Layer conductivity
sigma = np.ones(mesh.nCz)*sig_air
sigma[active] = sig_half
sigma[layer] = sig_layer
# Mapping
actMap = Maps.InjectActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz)
mapping = Maps.ExpMap(mesh) * Maps.SurjectVertical1D(mesh) * actMap
mtrue = np.log(sigma[active])
# ----- FDEM problem & survey -----
rxlocs = Utils.ndgrid([np.r_[50.], np.r_[0], np.r_[0.]])
bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real')
bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag')
freqs = np.logspace(2, 3, 5)
srcLoc = np.array([0., 0., 0.])
print('min skin depth = ', 500./np.sqrt(freqs.max() * sig_half),
'max skin depth = ', 500./np.sqrt(freqs.min() * sig_half))
print('max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(),
'max z ', mesh.vectorCCz.max())
srcList = []
[srcList.append(FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc,
orientation='Z')) for freq in freqs]
surveyFD = FDEM.Survey(srcList)
prbFD = FDEM.Problem3D_b(mesh, mapping=mapping)
prbFD.pair(surveyFD)
std = 0.03
surveyFD.makeSyntheticData(mtrue, std)
surveyFD.eps = np.linalg.norm(surveyFD.dtrue)*1e-5
# FDEM inversion
np.random.seed(1)
dmisfit = DataMisfit.l2_DataMisfit(surveyFD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
inv = Inversion.BaseInversion(invProb, directiveList=[beta, betaest, target])
m0 = np.log(np.ones(mtrue.size)*sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbFD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptFD = inv.run(m0)
# TDEM problem
times = np.logspace(-4, np.log10(2e-3), 10)
print('min diffusion distance ', 1.28*np.sqrt(times.min()/(sig_half*mu_0)),
'max diffusion distance ', 1.28*np.sqrt(times.max()/(sig_half*mu_0)))
rx = TDEM.Rx(rxlocs, times, 'bz')
src = TDEM.Src.MagDipole([rx], waveform=TDEM.Src.StepOffWaveform(),
loc=srcLoc) # same src location as FDEM problem
surveyTD = TDEM.Survey([src])
prbTD = TDEM.Problem3D_b(mesh, mapping=mapping)
prbTD.timeSteps = [(5e-5, 10), (1e-4, 10), (5e-4, 10)]
prbTD.pair(surveyTD)
prbTD.Solver = SolverLU
std = 0.03
surveyTD.makeSyntheticData(mtrue, std)
surveyTD.std = std
surveyTD.eps = np.linalg.norm(surveyTD.dtrue)*1e-5
# TDEM inversion
dmisfit = DataMisfit.l2_DataMisfit(surveyTD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
inv = Inversion.BaseInversion(invProb, directiveList=[beta, betaest, target])
m0 = np.log(np.ones(mtrue.size)*sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbTD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptTD = inv.run(m0)
if plotIt:
import matplotlib
fig = plt.figure(figsize = (10, 8))
ax0 = plt.subplot2grid((2, 2), (0, 0), rowspan=2)
ax1 = plt.subplot2grid((2, 2), (0, 1))
ax2 = plt.subplot2grid((2, 2), (1, 1))
fs = 13 # fontsize
matplotlib.rcParams['font.size'] = fs
# Plot the model
ax0.semilogx(sigma[active], mesh.vectorCCz[active], 'k-', lw=2)
ax0.semilogx(np.exp(moptFD), mesh.vectorCCz[active], 'bo', ms=6)
ax0.semilogx(np.exp(moptTD), mesh.vectorCCz[active], 'r*', ms=10)
ax0.set_ylim(-700, 0)
ax0.set_xlim(5e-3, 1e-1)
ax0.set_xlabel('Conductivity (S/m)', fontsize=fs)
ax0.set_ylabel('Depth (m)', fontsize=fs)
ax0.grid(which='both', color='k', alpha=0.5, linestyle='-',
linewidth=0.2)
ax0.legend(['True', 'FDEM', 'TDEM'], fontsize=fs, loc=4)
# plot the data misfits - negative b/c we choose positive to be in the
# direction of primary
ax1.plot(freqs, -surveyFD.dobs[::2], 'k-', lw=2)
ax1.plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2)
dpredFD = surveyFD.dpred(moptTD)
ax1.loglog(freqs, -dpredFD[::2], 'bo', ms=6)
ax1.loglog(freqs, -dpredFD[1::2], 'b+', markeredgewidth=2., ms=10)
ax2.loglog(times, surveyTD.dobs, 'k-', lw=2)
ax2.loglog(times, surveyTD.dpred(moptTD), 'r*', ms=10)
ax2.set_xlim(times.min(), times.max())
# Labels, gridlines, etc
ax2.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.set_xlabel('Frequency (Hz)', fontsize=fs)
ax1.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.set_xlabel('Time (s)', fontsize=fs)
ax2.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.legend(("Obs", "Pred"), fontsize=fs)
ax1.legend(("Obs (real)", "Obs (imag)", "Pred (real)", "Pred (imag)"),
fontsize=fs)
ax1.set_xlim(freqs.max(), freqs.min())
ax0.set_title("(a) Recovered Models", fontsize=fs)
ax1.set_title("(b) FDEM observed vs. predicted", fontsize=fs)
ax2.set_title("(c) TDEM observed vs. predicted", fontsize=fs)
plt.tight_layout(pad=1.5)
plt.show()
def run(plotIt=True):
"""
MT: 1D: Inversion
=================
Forward model 1D MT data.
Setup and run a MT 1D inversion.
"""
## Setup the forward modeling
# Setting up 1D mesh and conductivity models to forward model data.
# Frequency
nFreq = 26
freqs = np.logspace(2,-3,nFreq)
# Set mesh parameters
ct = 10
air = simpeg.Utils.meshTensor([(ct,25,1.4)])
core = np.concatenate( ( np.kron(simpeg.Utils.meshTensor([(ct,10,-1.3)]),np.ones((5,))) , simpeg.Utils.meshTensor([(ct,5)]) ) )
bot = simpeg.Utils.meshTensor([(core[0],25,-1.4)])
x0 = -np.array([np.sum(np.concatenate((core,bot)))])
# Make the model
m1d = simpeg.Mesh.TensorMesh([np.concatenate((bot,core,air))], x0=x0)
# Setup model varibles
active = m1d.vectorCCx<0.
layer1 = (m1d.vectorCCx<-500.) & (m1d.vectorCCx>=-800.)
layer2 = (m1d.vectorCCx<-3500.) & (m1d.vectorCCx>=-5000.)
# Set the conductivity values
sig_half = 1e-2
sig_air = 1e-8
sig_layer1 = .2
sig_layer2 = .2
# Make the true model
sigma_true = np.ones(m1d.nCx)*sig_air
sigma_true[active] = sig_half
sigma_true[layer1] = sig_layer1
sigma_true[layer2] = sig_layer2
# Extract the model
m_true = np.log(sigma_true[active])
# Make the background model
sigma_0 = np.ones(m1d.nCx)*sig_air
sigma_0[active] = sig_half
m_0 = np.log(sigma_0[active])
# Set the mapping
actMap = simpeg.Maps.InjectActiveCells(m1d, active, np.log(1e-8), nC=m1d.nCx)
mappingExpAct = simpeg.Maps.ExpMap(m1d) * actMap
## Setup the layout of the survey, set the sources and the connected receivers
# Receivers
rxList = []
rxList.append(NSEM.Rx.Point_impedance1D(simpeg.mkvc(np.array([-0.5]),2).T,'real'))
rxList.append(NSEM.Rx.Point_impedance1D(simpeg.mkvc(np.array([-0.5]),2).T,'imag'))
# Source list
srcList =[]
for freq in freqs:
srcList.append(NSEM.Src.Planewave_xy_1Dprimary(rxList,freq))
# Make the survey
survey = NSEM.Survey(srcList)
survey.mtrue = m_true
## Set the problem
problem = NSEM.Problem1D_ePrimSec(m1d,sigmaPrimary=sigma_0,mapping=mappingExpAct)
problem.pair(survey)
## Forward model data
# Project the data
survey.dtrue = survey.dpred(m_true)
survey.dobs = survey.dtrue + 0.01*abs(survey.dtrue)*np.random.randn(*survey.dtrue.shape)
if plotIt:
fig = NSEM.Utils.dataUtils.plotMT1DModelData(problem,[])
fig.suptitle('Target - smooth true')
# Assign uncertainties
std = 0.05 # 5% std
survey.std = np.abs(survey.dobs*std)
# Assign the data weight
Wd = 1./survey.std
## Setup the inversion proceedure
# Define a counter
C = simpeg.Utils.Counter()
# Set the optimization
opt = simpeg.Optimization.ProjectedGNCG(maxIter = 25)
opt.counter = C
opt.lower = np.log(1e-4)
opt.upper = np.log(5)
opt.LSshorten = 0.1
opt.remember('xc')
# Data misfit
dmis = simpeg.DataMisfit.l2_DataMisfit(survey)
dmis.Wd = Wd
# Regularization - with a regularization mesh
regMesh = simpeg.Mesh.TensorMesh([m1d.hx[active]],m1d.x0)
reg = simpeg.Regularization.Tikhonov(regMesh)
reg.mrefInSmooth = True
reg.alpha_s = 1e-1
reg.alpha_x = 1.
# Inversion problem
invProb = simpeg.InvProblem.BaseInvProblem(dmis, reg, opt)
invProb.counter = C
# Beta cooling
beta = simpeg.Directives.BetaSchedule()
beta.coolingRate = 4.
beta.coolingFactor = 4.
betaest = simpeg.Directives.BetaEstimate_ByEig(beta0_ratio=100.)
betaest.beta0 = 1.
targmis = simpeg.Directives.TargetMisfit()
targmis.target = survey.nD
# Create an inversion object
inv = simpeg.Inversion.BaseInversion(invProb, directiveList=[beta,betaest,targmis])
## Run the inversion
mopt = inv.run(m_0)
if plotIt:
fig = NSEM.Utils.dataUtils.plotMT1DModelData(problem,[mopt])
fig.suptitle('Target - smooth true')
fig.axes[0].set_ylim([-10000,500])
plt.show()
def run(plotIt=True):
"""
MT: 1D: Inversion
=================
Forward model 1D MT data.
Setup and run a MT 1D inversion.
"""
## Setup the forward modeling
# Setting up 1D mesh and conductivity models to forward model data.
# Frequency
nFreq = 26
freqs = np.logspace(2, -3, nFreq)
# Set mesh parameters
ct = 10
air = simpeg.Utils.meshTensor([(ct, 25, 1.4)])
core = np.concatenate(
(np.kron(simpeg.Utils.meshTensor([(ct, 10, -1.3)]), np.ones(
(5, ))), simpeg.Utils.meshTensor([(ct, 5)])))
bot = simpeg.Utils.meshTensor([(core[0], 25, -1.4)])
x0 = -np.array([np.sum(np.concatenate((core, bot)))])
# Make the model
m1d = simpeg.Mesh.TensorMesh([np.concatenate((bot, core, air))], x0=x0)
# Setup model varibles
active = m1d.vectorCCx < 0.
layer1 = (m1d.vectorCCx < -500.) & (m1d.vectorCCx >= -800.)
layer2 = (m1d.vectorCCx < -3500.) & (m1d.vectorCCx >= -5000.)
# Set the conductivity values
sig_half = 1e-2
sig_air = 1e-8
sig_layer1 = .2
sig_layer2 = .2
# Make the true model
sigma_true = np.ones(m1d.nCx) * sig_air
sigma_true[active] = sig_half
sigma_true[layer1] = sig_layer1
sigma_true[layer2] = sig_layer2
# Extract the model
m_true = np.log(sigma_true[active])
# Make the background model
sigma_0 = np.ones(m1d.nCx) * sig_air
sigma_0[active] = sig_half
m_0 = np.log(sigma_0[active])
# Set the mapping
actMap = simpeg.Maps.InjectActiveCells(m1d,
active,
np.log(1e-8),
nC=m1d.nCx)
mappingExpAct = simpeg.Maps.ExpMap(m1d) * actMap
## Setup the layout of the survey, set the sources and the connected receivers
# Receivers
rxList = []
rxList.append(
NSEM.Rx.Point_impedance1D(simpeg.mkvc(np.array([-0.5]), 2).T, 'real'))
rxList.append(
NSEM.Rx.Point_impedance1D(simpeg.mkvc(np.array([-0.5]), 2).T, 'imag'))
# Source list
srcList = []
for freq in freqs:
srcList.append(NSEM.Src.Planewave_xy_1Dprimary(rxList, freq))
# Make the survey
survey = NSEM.Survey(srcList)
survey.mtrue = m_true
## Set the problem
problem = NSEM.Problem1D_ePrimSec(m1d,
sigmaPrimary=sigma_0,
mapping=mappingExpAct)
problem.pair(survey)
## Forward model data
# Project the data
survey.dtrue = survey.dpred(m_true)
survey.dobs = survey.dtrue + 0.01 * abs(
survey.dtrue) * np.random.randn(*survey.dtrue.shape)
if plotIt:
fig = NSEM.Utils.dataUtils.plotMT1DModelData(problem, [])
fig.suptitle('Target - smooth true')
# Assign uncertainties
std = 0.05 # 5% std
survey.std = np.abs(survey.dobs * std)
# Assign the data weight
Wd = 1. / survey.std
## Setup the inversion proceedure
# Define a counter
C = simpeg.Utils.Counter()
# Set the optimization
opt = simpeg.Optimization.ProjectedGNCG(maxIter=25)
opt.counter = C
opt.lower = np.log(1e-4)
opt.upper = np.log(5)
opt.LSshorten = 0.1
opt.remember('xc')
# Data misfit
dmis = simpeg.DataMisfit.l2_DataMisfit(survey)
dmis.Wd = Wd
# Regularization - with a regularization mesh
regMesh = simpeg.Mesh.TensorMesh([m1d.hx[active]], m1d.x0)
reg = simpeg.Regularization.Tikhonov(regMesh)
reg.mrefInSmooth = True
reg.alpha_s = 1e-1
reg.alpha_x = 1.
# Inversion problem
invProb = simpeg.InvProblem.BaseInvProblem(dmis, reg, opt)
invProb.counter = C
# Beta cooling
beta = simpeg.Directives.BetaSchedule()
beta.coolingRate = 4.
beta.coolingFactor = 4.
betaest = simpeg.Directives.BetaEstimate_ByEig(beta0_ratio=100.)
betaest.beta0 = 1.
targmis = simpeg.Directives.TargetMisfit()
targmis.target = survey.nD
# Create an inversion object
inv = simpeg.Inversion.BaseInversion(
invProb, directiveList=[beta, betaest, targmis])
## Run the inversion
mopt = inv.run(m_0)
if plotIt:
fig = NSEM.Utils.dataUtils.plotMT1DModelData(problem, [mopt])
fig.suptitle('Target - smooth true')
fig.axes[0].set_ylim([-10000, 500])
plt.show()
def run(plotIt=True):
"""
1D FDEM and TDEM inversions
===========================
This example is used in the paper Heagy et al 2016 (in prep)
"""
# Set up cylindrically symmeric mesh
cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh
hx = [(cs, ncx), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.CylMesh([hx, 1, hz], '00C')
# Conductivity model
layerz = np.r_[-200., -100.]
layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1])
active = mesh.vectorCCz < 0.
sig_half = 1e-2 # Half-space conductivity
sig_air = 1e-8 # Air conductivity
sig_layer = 5e-2 # Layer conductivity
sigma = np.ones(mesh.nCz) * sig_air
sigma[active] = sig_half
sigma[layer] = sig_layer
# Mapping
actMap = Maps.InjectActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz)
mapping = Maps.ExpMap(mesh) * Maps.SurjectVertical1D(mesh) * actMap
mtrue = np.log(sigma[active])
# ----- FDEM problem & survey -----
rxlocs = Utils.ndgrid([np.r_[50.], np.r_[0], np.r_[0.]])
bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real')
bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag')
freqs = np.logspace(2, 3, 5)
srcLoc = np.array([0., 0., 0.])
print('min skin depth = ', 500. / np.sqrt(freqs.max() * sig_half),
'max skin depth = ', 500. / np.sqrt(freqs.min() * sig_half))
print('max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(),
'max z ', mesh.vectorCCz.max())
srcList = [
FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation='Z')
for freq in freqs
]
surveyFD = FDEM.Survey(srcList)
prbFD = FDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver)
prbFD.pair(surveyFD)
std = 0.03
surveyFD.makeSyntheticData(mtrue, std)
surveyFD.eps = np.linalg.norm(surveyFD.dtrue) * 1e-5
# FDEM inversion
np.random.seed(1)
dmisfit = DataMisfit.l2_DataMisfit(surveyFD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
directiveList = [beta, betaest, target]
inv = Inversion.BaseInversion(invProb, directiveList=directiveList)
m0 = np.log(np.ones(mtrue.size) * sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbFD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptFD = inv.run(m0)
# TDEM problem
times = np.logspace(-4, np.log10(2e-3), 10)
print('min diffusion distance ',
1.28 * np.sqrt(times.min() / (sig_half * mu_0)),
'max diffusion distance ',
1.28 * np.sqrt(times.max() / (sig_half * mu_0)))
rx = TDEM.Rx.Point_b(rxlocs, times, 'z')
src = TDEM.Src.MagDipole(
[rx],
waveform=TDEM.Src.StepOffWaveform(),
loc=srcLoc # same src location as FDEM problem
)
surveyTD = TDEM.Survey([src])
prbTD = TDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver)
prbTD.timeSteps = [(5e-5, 10), (1e-4, 10), (5e-4, 10)]
prbTD.pair(surveyTD)
std = 0.03
surveyTD.makeSyntheticData(mtrue, std)
surveyTD.std = std
surveyTD.eps = np.linalg.norm(surveyTD.dtrue) * 1e-5
# TDEM inversion
dmisfit = DataMisfit.l2_DataMisfit(surveyTD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
inv = Inversion.BaseInversion(invProb, directiveList=directiveList)
m0 = np.log(np.ones(mtrue.size) * sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbTD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptTD = inv.run(m0)
if plotIt:
plt.figure(figsize=(10, 8))
ax0 = plt.subplot2grid((2, 2), (0, 0), rowspan=2)
ax1 = plt.subplot2grid((2, 2), (0, 1))
ax2 = plt.subplot2grid((2, 2), (1, 1))
fs = 13 # fontsize
matplotlib.rcParams['font.size'] = fs
# Plot the model
ax0.semilogx(sigma[active], mesh.vectorCCz[active], 'k-', lw=2)
ax0.semilogx(np.exp(moptFD), mesh.vectorCCz[active], 'bo', ms=6)
ax0.semilogx(np.exp(moptTD), mesh.vectorCCz[active], 'r*', ms=10)
ax0.set_ylim(-700, 0)
ax0.set_xlim(5e-3, 1e-1)
ax0.set_xlabel('Conductivity (S/m)', fontsize=fs)
ax0.set_ylabel('Depth (m)', fontsize=fs)
ax0.grid(which='both',
color='k',
alpha=0.5,
linestyle='-',
linewidth=0.2)
ax0.legend(['True', 'FDEM', 'TDEM'], fontsize=fs, loc=4)
# plot the data misfits - negative b/c we choose positive to be in the
# direction of primary
ax1.plot(freqs, -surveyFD.dobs[::2], 'k-', lw=2)
ax1.plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2)
dpredFD = surveyFD.dpred(moptTD)
ax1.loglog(freqs, -dpredFD[::2], 'bo', ms=6)
ax1.loglog(freqs, -dpredFD[1::2], 'b+', markeredgewidth=2., ms=10)
ax2.loglog(times, surveyTD.dobs, 'k-', lw=2)
ax2.loglog(times, surveyTD.dpred(moptTD), 'r*', ms=10)
ax2.set_xlim(times.min(), times.max())
# Labels, gridlines, etc
ax2.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.set_xlabel('Frequency (Hz)', fontsize=fs)
ax1.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.set_xlabel('Time (s)', fontsize=fs)
ax2.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.legend(("Obs", "Pred"), fontsize=fs)
ax1.legend(("Obs (real)", "Obs (imag)", "Pred (real)", "Pred (imag)"),
fontsize=fs)
ax1.set_xlim(freqs.max(), freqs.min())
ax0.set_title("(a) Recovered Models", fontsize=fs)
ax1.set_title("(b) FDEM observed vs. predicted", fontsize=fs)
ax2.set_title("(c) TDEM observed vs. predicted", fontsize=fs)
plt.tight_layout(pad=1.5)
