def setUpClass(self):
print("\n------- Testing Primary Secondary Source EB -> EB --------\n")
# receivers
self.rxlist = []
for rxtype in ["MagneticFluxDensity", "ElectricField"]:
rx = getattr(fdem.Rx, "Point{}".format(rxtype))
for orientation in ["x", "y", "z"]:
for comp in ["real", "imag"]:
self.rxlist.append(
rx(rx_locs, component=comp, orientation=orientation))
# primary
self.primarySimulation = fdem.Simulation3DMagneticFluxDensity(
meshp, sigmaMap=primaryMapping)
self.primarySimulation.solver = Solver
primarySrc = fdem.Src.MagDipole(self.rxlist,
frequency=freq,
location=src_loc)
self.primarySurvey = fdem.Survey([primarySrc])
self.secondarySrc = fdem.Src.PrimSecMappedSigma(
self.rxlist,
freq,
self.primarySimulation,
self.primarySurvey,
primaryMap2Meshs,
)
self.secondarySurvey = fdem.Survey([self.secondarySrc])
# Secondary Problem
self.secondarySimulation = fdem.Simulation3DMagneticFluxDensity(
meshs, survey=self.secondarySurvey, sigmaMap=mapping)
self.secondarySimulation.solver = Solver
# Full 3D problem to compare with
self.survey3D = fdem.Survey([primarySrc])
self.simulation3D = fdem.Simulation3DMagneticFluxDensity(
meshs, survey=self.survey3D, sigmaMap=mapping)
self.simulation3D.solver = Solver
# solve and store fields
print(" solving primary - secondary")
self.fields_primsec = self.secondarySimulation.fields(model)
print(" ... done")
self.fields_primsec = self.secondarySimulation.fields(model)
print(" solving 3D")
self.fields_3D = self.simulation3D.fields(model)
print(" ... done")
return None
def simulate(self, srcLoc, rxLoc, freqs):
bzr = fdem.receivers.PointMagneticFluxDensitySecondary(
rxLoc, orientation="z", component="real")
bzi = fdem.receivers.PointMagneticFluxDensitySecondary(
rxLoc, orientation="z", component="imag")
self.srcList = [
fdem.sources.MagDipole([bzr, bzi],
frequency=freq,
location=srcLoc,
orientation="Z") for freq in freqs
]
survey = fdem.survey.Survey(self.srcList)
sim = fdem.Simulation3DMagneticFluxDensity(self.mesh,
survey=survey,
sigmaMap=self.mapping,
mu=self.mu,
solver=Pardiso)
self.f = sim.fields(self.m)
self.sim = sim
dpred = sim.dpred(self.m, f=self.f)
self.srcLoc = srcLoc
self.rxLoc = rxLoc
return dpred
def setUp(self):
print("\nTesting Transect for analytic")
cs = 10.0
ncx, ncy, ncz = 10, 10, 10
npad = 5
freq = 1e2
hx = [(cs, npad, -1.3), (cs, ncx), (cs, npad, 1.3)]
hy = [(cs, npad, -1.3), (cs, ncy), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = discretize.TensorMesh([hx, hy, hz], "CCC")
x = np.linspace(-10, 10, 5)
XYZ = utils.ndgrid(x, np.r_[0], np.r_[0])
rxList = fdem.Rx.PointElectricField(XYZ,
orientation="x",
component="imag")
SrcList = [
fdem.Src.MagDipole([rxList],
location=np.r_[0.0, 0.0, 0.0],
frequency=freq),
fdem.Src.CircularLoop(
[rxList],
location=np.r_[0.0, 0.0, 0.0],
frequency=freq,
radius=np.sqrt(1.0 / np.pi),
),
# fdem.Src.MagDipole_Bfield(
# [rxList], loc=np.r_[0., 0., 0.],
# freq=freq
# ), # less accurate
]
survey = fdem.Survey(SrcList)
sig = 1e-1
sigma = np.ones(mesh.nC) * sig
sigma[mesh.gridCC[:, 2] > 0] = 1e-8
prb = fdem.Simulation3DMagneticFluxDensity(mesh, sigma=sigma)
prb.pair(survey)
try:
from pymatsolver import Pardiso
prb.Solver = Pardiso
except ImportError:
prb.Solver = SolverLU
self.prb = prb
self.mesh = mesh
self.sig = sig
print(" starting solve ...")
u = self.prb.fields()
print(" ... done")
self.u = u
def setUp(self):
cs = 10.0
ncx, ncy, ncz = 30.0, 30.0, 30.0
npad = 10.0
hx = [(cs, npad, -1.5), (cs, ncx), (cs, npad, 1.5)]
hy = [(cs, npad, -1.5), (cs, ncy), (cs, npad, 1.5)]
hz = [(cs, npad, -1.5), (cs, ncz), (cs, npad, 1.5)]
self.mesh = discretize.TensorMesh([hx, hy, hz], "CCC")
mapping = maps.ExpMap(self.mesh)
self.frequency = 1.0
self.prob_e = fdem.Simulation3DElectricField(self.mesh,
sigmaMap=mapping)
self.prob_b = fdem.Simulation3DMagneticFluxDensity(self.mesh,
sigmaMap=mapping)
self.prob_h = fdem.Simulation3DMagneticField(self.mesh,
sigmaMap=mapping)
self.prob_j = fdem.Simulation3DCurrentDensity(self.mesh,
sigmaMap=mapping)
loc = np.r_[0.0, 0.0, 0.0]
self.location = utils.mkvc(
self.mesh.gridCC[utils.closestPoints(self.mesh, loc, "CC"), :])
def resolve_1Dinversions(
mesh,
dobs,
src_height,
freqs,
m0,
mref,
mapping,
relative=0.08,
floor=1e-14,
rxOffset=7.86,
):
"""
Perform a single 1D inversion for a RESOLVE sounding for Horizontal
Coplanar Coil data (both real and imaginary).
:param discretize.CylMesh mesh: mesh used for the forward simulation
:param numpy.ndarray dobs: observed data
:param float src_height: height of the source above the ground
:param numpy.ndarray freqs: frequencies
:param numpy.ndarray m0: starting model
:param numpy.ndarray mref: reference model
:param maps.IdentityMap mapping: mapping that maps the model to electrical conductivity
:param float relative: percent error used to construct the data misfit term
:param float floor: noise floor used to construct the data misfit term
:param float rxOffset: offset between source and receiver.
"""
# ------------------- Forward Simulation ------------------- #
# set up the receivers
bzr = FDEM.Rx.PointMagneticFluxDensitySecondary(np.array(
[[rxOffset, 0.0, src_height]]),
orientation="z",
component="real")
bzi = FDEM.Rx.PointMagneticFluxDensity(np.array(
[[rxOffset, 0.0, src_height]]),
orientation="z",
component="imag")
# source location
srcLoc = np.array([0.0, 0.0, src_height])
srcList = [
FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation="Z")
for freq in freqs
]
# construct a forward simulation
survey = FDEM.Survey(srcList)
prb = FDEM.Simulation3DMagneticFluxDensity(mesh,
sigmaMap=mapping,
Solver=PardisoSolver)
prb.survey = survey
# ------------------- Inversion ------------------- #
# data misfit term
uncert = abs(dobs) * relative + floor
dat = data.Data(dobs=dobs, standard_deviation=uncert)
dmisfit = data_misfit.L2DataMisfit(simulation=prb, data=dat)
# regularization
regMesh = discretize.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = regularization.Simple(regMesh)
reg.mref = mref
# optimization
opt = optimization.InexactGaussNewton(maxIter=10)
# statement of the inverse problem
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
# Inversion directives and parameters
target = directives.TargetMisfit()
inv = inversion.BaseInversion(invProb, directiveList=[target])
invProb.beta = 2.0 # Fix beta in the nonlinear iterations
reg.alpha_s = 1e-3
reg.alpha_x = 1.0
prb.counter = opt.counter = utils.Counter()
opt.LSshorten = 0.5
opt.remember("xc")
# run the inversion
mopt = inv.run(m0)
return mopt, invProb.dpred, survey.dobs
def run(plotIt=True, saveFig=False):
# Set up cylindrically symmeric mesh
cs, ncx, ncz, npad = 10.0, 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 = discretize.CylMesh([hx, 1, hz], "00C")
# Conductivity model
layerz = np.r_[-200.0, -100.0]
layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1])
active = mesh.vectorCCz < 0.0
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_[50.0], np.r_[0], np.r_[0.0]])
bzr = FDEM.Rx.PointMagneticFluxDensitySecondary(rxlocs, "z", "real")
bzi = FDEM.Rx.PointMagneticFluxDensitySecondary(rxlocs, "z", "imag")
freqs = np.logspace(2, 3, 5)
srcLoc = np.array([0.0, 0.0, 0.0])
print(
"min skin depth = ",
500.0 / np.sqrt(freqs.max() * sig_half),
"max skin depth = ",
500.0 / np.sqrt(freqs.min() * sig_half),
)
print(
"max x ",
mesh.vectorCCx.max(),
"min z ",
mesh.vectorCCz.min(),
"max z ",
mesh.vectorCCz.max(),
)
source_list = [
FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation="Z") for freq in freqs
]
surveyFD = FDEM.Survey(source_list)
prbFD = FDEM.Simulation3DMagneticFluxDensity(
mesh, survey=surveyFD, sigmaMap=mapping, solver=Solver
)
rel_err = 0.03
dataFD = prbFD.make_synthetic_data(mtrue, relative_error=rel_err, add_noise=True)
dataFD.noise_floor = np.linalg.norm(dataFD.dclean) * 1e-5
# FDEM inversion
np.random.seed(1)
dmisfit = data_misfit.L2DataMisfit(simulation=prbFD, data=dataFD)
regMesh = discretize.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = regularization.Simple(regMesh)
opt = optimization.InexactGaussNewton(maxIterCG=10)
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives
beta = directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1.0, seed=518936)
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 = 1.0
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.PointMagneticFluxDensity(rxlocs, times, "z")
src = TDEM.Src.MagDipole(
[rx],
waveform=TDEM.Src.StepOffWaveform(),
location=srcLoc, # same src location as FDEM problem
)
surveyTD = TDEM.Survey([src])
prbTD = TDEM.Simulation3DMagneticFluxDensity(
mesh, survey=surveyTD, sigmaMap=mapping, solver=Solver
)
prbTD.time_steps = [(5e-5, 10), (1e-4, 10), (5e-4, 10)]
rel_err = 0.03
dataTD = prbTD.make_synthetic_data(mtrue, relative_error=rel_err, add_noise=True)
dataTD.noise_floor = np.linalg.norm(dataTD.dclean) * 1e-5
# TDEM inversion
dmisfit = data_misfit.L2DataMisfit(simulation=prbTD, data=dataTD)
regMesh = discretize.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = regularization.Simple(regMesh)
opt = optimization.InexactGaussNewton(maxIterCG=10)
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
# directives
beta = directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1.0, seed=518936)
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 = 1.0
prbTD.counter = opt.counter = utils.Counter()
opt.remember("xc")
moptTD = inv.run(m0)
# Plot the results
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
# z_true = np.repeat(mesh.vectorCCz[active][1:], 2, axis=0)
# z_true = np.r_[mesh.vectorCCz[active][0], z_true, mesh.vectorCCz[active][-1]]
activeN = mesh.vectorNz <= 0.0 + cs / 2.0
z_true = np.repeat(mesh.vectorNz[activeN][1:-1], 2, axis=0)
z_true = np.r_[mesh.vectorNz[activeN][0], z_true, mesh.vectorNz[activeN][-1]]
sigma_true = np.repeat(sigma[active], 2, axis=0)
ax0.semilogx(sigma_true, z_true, "k-", lw=2, label="True")
ax0.semilogx(
np.exp(moptFD),
mesh.vectorCCz[active],
"bo",
ms=6,
markeredgecolor="k",
markeredgewidth=0.5,
label="FDEM",
)
ax0.semilogx(
np.exp(moptTD),
mesh.vectorCCz[active],
"r*",
ms=10,
markeredgecolor="k",
markeredgewidth=0.5,
label="TDEM",
)
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(fontsize=fs, loc=4)
# plot the data misfits - negative b/c we choose positive to be in the
# direction of primary
ax1.plot(freqs, -dataFD.dobs[::2], "k-", lw=2, label="Obs (real)")
ax1.plot(freqs, -dataFD.dobs[1::2], "k--", lw=2, label="Obs (imag)")
dpredFD = prbFD.dpred(moptTD)
ax1.loglog(
freqs,
-dpredFD[::2],
"bo",
ms=6,
markeredgecolor="k",
markeredgewidth=0.5,
label="Pred (real)",
)
ax1.loglog(
freqs, -dpredFD[1::2], "b+", ms=10, markeredgewidth=2.0, label="Pred (imag)"
)
ax2.loglog(times, dataTD.dobs, "k-", lw=2, label="Obs")
ax2.loglog(
times,
prbTD.dpred(moptTD),
"r*",
ms=10,
markeredgecolor="k",
markeredgewidth=0.5,
label="Pred",
)
ax2.set_xlim(times.min() - 1e-5, times.max() + 1e-4)
# 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(fontsize=fs, loc=3)
ax1.legend(fontsize=fs, loc=3)
ax1.set_xlim(freqs.max() + 1e2, freqs.min() - 1e1)
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)
if saveFig is True:
plt.savefig("example1.png", dpi=600)
def run(plotIt=True, saveFig=False, cleanup=True):
"""
Run 1D inversions for a single sounding of the RESOLVE and SkyTEM
bookpurnong data
:param bool plotIt: show the plots?
:param bool saveFig: save the figure
:param bool cleanup: remove the downloaded results
"""
downloads, directory = download_and_unzip_data()
resolve = h5py.File(os.path.sep.join([directory, "booky_resolve.hdf5"]),
"r")
skytem = h5py.File(os.path.sep.join([directory, "booky_skytem.hdf5"]), "r")
river_path = resolve["river_path"].value
# Choose a sounding location to invert
xloc, yloc = 462100.0, 6196500.0
rxind_skytem = np.argmin(
abs(skytem["xy"][:, 0] - xloc) + abs(skytem["xy"][:, 1] - yloc))
rxind_resolve = np.argmin(
abs(resolve["xy"][:, 0] - xloc) + abs(resolve["xy"][:, 1] - yloc))
# Plot both resolve and skytem data on 2D plane
fig = plt.figure(figsize=(13, 6))
title = ["RESOLVE In-phase 400 Hz", "SkyTEM High moment 156 $\mu$s"]
ax1 = plt.subplot(121)
ax2 = plt.subplot(122)
axs = [ax1, ax2]
out_re = utils.plot2Ddata(
resolve["xy"],
resolve["data"][:, 0],
ncontour=100,
contourOpts={"cmap": "viridis"},
ax=ax1,
)
vmin, vmax = out_re[0].get_clim()
cb_re = plt.colorbar(out_re[0],
ticks=np.linspace(vmin, vmax, 3),
ax=ax1,
fraction=0.046,
pad=0.04)
temp_skytem = skytem["data"][:, 5].copy()
temp_skytem[skytem["data"][:, 5] > 7e-10] = 7e-10
out_sky = utils.plot2Ddata(
skytem["xy"][:, :2],
temp_skytem,
ncontour=100,
contourOpts={
"cmap": "viridis",
"vmax": 7e-10
},
ax=ax2,
)
vmin, vmax = out_sky[0].get_clim()
cb_sky = plt.colorbar(
out_sky[0],
ticks=np.linspace(vmin, vmax * 0.99, 3),
ax=ax2,
format="%.1e",
fraction=0.046,
pad=0.04,
)
cb_re.set_label("Bz (ppm)")
cb_sky.set_label("dB$_z$ / dt (V/A-m$^4$)")
for i, ax in enumerate(axs):
xticks = [460000, 463000]
yticks = [6195000, 6198000, 6201000]
ax.set_xticks(xticks)
ax.set_yticks(yticks)
ax.plot(xloc, yloc, "wo")
ax.plot(river_path[:, 0], river_path[:, 1], "k", lw=0.5)
ax.set_aspect("equal")
if i == 1:
ax.plot(skytem["xy"][:, 0],
skytem["xy"][:, 1],
"k.",
alpha=0.02,
ms=1)
ax.set_yticklabels([str(" ") for f in yticks])
else:
ax.plot(resolve["xy"][:, 0],
resolve["xy"][:, 1],
"k.",
alpha=0.02,
ms=1)
ax.set_yticklabels([str(f) for f in yticks])
ax.set_ylabel("Northing (m)")
ax.set_xlabel("Easting (m)")
ax.set_title(title[i])
ax.axis("equal")
# plt.tight_layout()
if saveFig is True:
fig.savefig("resolve_skytem_data.png", dpi=600)
# ------------------ Mesh ------------------ #
# Step1: Set 2D cylindrical mesh
cs, ncx, ncz, npad = 1.0, 10.0, 10.0, 20
hx = [(cs, ncx), (cs, npad, 1.3)]
npad = 12
temp = np.logspace(np.log10(1.0), np.log10(12.0), 19)
temp_pad = temp[-1] * 1.3**np.arange(npad)
hz = np.r_[temp_pad[::-1], temp[::-1], temp, temp_pad]
mesh = discretize.CylMesh([hx, 1, hz], "00C")
active = mesh.vectorCCz < 0.0
# Step2: Set a SurjectVertical1D mapping
# Note: this sets our inversion model as 1D log conductivity
# below subsurface
active = mesh.vectorCCz < 0.0
actMap = maps.InjectActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz)
mapping = maps.ExpMap(mesh) * maps.SurjectVertical1D(mesh) * actMap
sig_half = 1e-1
sig_air = 1e-8
sigma = np.ones(mesh.nCz) * sig_air
sigma[active] = sig_half
# Initial and reference model
m0 = np.log(sigma[active])
# ------------------ RESOLVE Forward Simulation ------------------ #
# Step3: Invert Resolve data
# Bird height from the surface
b_height_resolve = resolve["src_elevation"].value
src_height_resolve = b_height_resolve[rxind_resolve]
# Set Rx (In-phase and Quadrature)
rxOffset = 7.86
bzr = FDEM.Rx.PointMagneticFluxDensitySecondary(
np.array([[rxOffset, 0.0, src_height_resolve]]),
orientation="z",
component="real",
)
bzi = FDEM.Rx.PointMagneticFluxDensity(
np.array([[rxOffset, 0.0, src_height_resolve]]),
orientation="z",
component="imag",
)
# Set Source (In-phase and Quadrature)
frequency_cp = resolve["frequency_cp"].value
freqs = frequency_cp.copy()
srcLoc = np.array([0.0, 0.0, src_height_resolve])
srcList = [
FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation="Z")
for freq in freqs
]
# Set FDEM survey (In-phase and Quadrature)
survey = FDEM.Survey(srcList)
prb = FDEM.Simulation3DMagneticFluxDensity(mesh,
sigmaMap=mapping,
Solver=Solver)
prb.survey = survey
# ------------------ RESOLVE Inversion ------------------ #
# Primary field
bp = -mu_0 / (4 * np.pi * rxOffset**3)
# Observed data
cpi_inds = [0, 2, 6, 8, 10]
cpq_inds = [1, 3, 7, 9, 11]
dobs_re = (np.c_[resolve["data"][rxind_resolve, :][cpi_inds],
resolve["data"][rxind_resolve, :][cpq_inds], ].flatten() *
bp * 1e-6)
# Uncertainty
relative = np.repeat(np.r_[np.ones(3) * 0.1, np.ones(2) * 0.15], 2)
floor = 20 * abs(bp) * 1e-6
std = abs(dobs_re) * relative + floor
# Data Misfit
data_resolve = data.Data(dobs=dobs_re,
survey=survey,
standard_deviation=std)
dmisfit = data_misfit.L2DataMisfit(simulation=prb, data=data_resolve)
# Regularization
regMesh = discretize.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = regularization.Simple(regMesh, mapping=maps.IdentityMap(regMesh))
# Optimization
opt = optimization.InexactGaussNewton(maxIter=5)
# statement of the inverse problem
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
# Inversion directives and parameters
target = directives.TargetMisfit() # stop when we hit target misfit
invProb.beta = 2.0
inv = inversion.BaseInversion(invProb, directiveList=[target])
reg.alpha_s = 1e-3
reg.alpha_x = 1.0
reg.mref = m0.copy()
opt.LSshorten = 0.5
opt.remember("xc")
# run the inversion
mopt_re = inv.run(m0)
dpred_re = invProb.dpred
# ------------------ SkyTEM Forward Simulation ------------------ #
# Step4: Invert SkyTEM data
# Bird height from the surface
b_height_skytem = skytem["src_elevation"].value
src_height = b_height_skytem[rxind_skytem]
srcLoc = np.array([0.0, 0.0, src_height])
# Radius of the source loop
area = skytem["area"].value
radius = np.sqrt(area / np.pi)
rxLoc = np.array([[radius, 0.0, src_height]])
# Parameters for current waveform
t0 = skytem["t0"].value
times = skytem["times"].value
waveform_skytem = skytem["waveform"].value
offTime = t0
times_off = times - t0
# Note: we are Using theoretical VTEM waveform,
# but effectively fits SkyTEM waveform
peakTime = 1.0000000e-02
a = 3.0
dbdt_z = TDEM.Rx.PointMagneticFluxTimeDerivative(
locations=rxLoc, times=times_off[:-3] + offTime,
orientation="z") # vertical db_dt
rxList = [dbdt_z] # list of receivers
srcList = [
TDEM.Src.CircularLoop(
rxList,
loc=srcLoc,
radius=radius,
orientation="z",
waveform=TDEM.Src.VTEMWaveform(offTime=offTime,
peakTime=peakTime,
a=3.0),
)
]
# solve the problem at these times
timeSteps = [
(peakTime / 5, 5),
((offTime - peakTime) / 5, 5),
(1e-5, 5),
(5e-5, 5),
(1e-4, 10),
(5e-4, 15),
]
prob = TDEM.Simulation3DElectricField(mesh,
time_steps=timeSteps,
sigmaMap=mapping,
Solver=Solver)
survey = TDEM.Survey(srcList)
prob.survey = survey
src = srcList[0]
rx = src.receiver_list[0]
wave = []
for time in prob.times:
wave.append(src.waveform.eval(time))
wave = np.hstack(wave)
out = prob.dpred(m0)
# plot the waveform
fig = plt.figure(figsize=(5, 3))
times_off = times - t0
plt.plot(waveform_skytem[:, 0], waveform_skytem[:, 1], "k.")
plt.plot(prob.times, wave, "k-", lw=2)
plt.legend(("SkyTEM waveform", "Waveform (fit)"), fontsize=10)
for t in rx.times:
plt.plot(np.ones(2) * t, np.r_[-0.03, 0.03], "k-")
plt.ylim(-0.1, 1.1)
plt.grid(True)
plt.xlabel("Time (s)")
plt.ylabel("Normalized current")
if saveFig:
fig.savefig("skytem_waveform", dpi=200)
# Observed data
dobs_sky = skytem["data"][rxind_skytem, :-3] * area
# ------------------ SkyTEM Inversion ------------------ #
# Uncertainty
relative = 0.12
floor = 7.5e-12
std = abs(dobs_sky) * relative + floor
# Data Misfit
data_sky = data.Data(dobs=-dobs_sky, survey=survey, standard_deviation=std)
dmisfit = data_misfit.L2DataMisfit(simulation=prob, data=data_sky)
# Regularization
regMesh = discretize.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = regularization.Simple(regMesh, mapping=maps.IdentityMap(regMesh))
# Optimization
opt = optimization.InexactGaussNewton(maxIter=5)
# statement of the inverse problem
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
# Directives and Inversion Parameters
target = directives.TargetMisfit()
invProb.beta = 20.0
inv = inversion.BaseInversion(invProb, directiveList=[target])
reg.alpha_s = 1e-1
reg.alpha_x = 1.0
opt.LSshorten = 0.5
opt.remember("xc")
reg.mref = mopt_re # Use RESOLVE model as a reference model
# run the inversion
mopt_sky = inv.run(m0)
dpred_sky = invProb.dpred
# Plot the figure from the paper
plt.figure(figsize=(12, 8))
fs = 13 # fontsize
matplotlib.rcParams["font.size"] = fs
ax0 = plt.subplot2grid((2, 2), (0, 0), rowspan=2)
ax1 = plt.subplot2grid((2, 2), (0, 1))
ax2 = plt.subplot2grid((2, 2), (1, 1))
# Recovered Models
sigma_re = np.repeat(np.exp(mopt_re), 2, axis=0)
sigma_sky = np.repeat(np.exp(mopt_sky), 2, axis=0)
z = np.repeat(mesh.vectorCCz[active][1:], 2, axis=0)
z = np.r_[mesh.vectorCCz[active][0], z, mesh.vectorCCz[active][-1]]
ax0.semilogx(sigma_re, z, "k", lw=2, label="RESOLVE")
ax0.semilogx(sigma_sky, z, "b", lw=2, label="SkyTEM")
ax0.set_ylim(-50, 0)
# ax0.set_xlim(5e-4, 1e2)
ax0.grid(True)
ax0.set_ylabel("Depth (m)")
ax0.set_xlabel("Conducivity (S/m)")
ax0.legend(loc=3)
ax0.set_title("(a) Recovered Models")
# RESOLVE Data
ax1.loglog(frequency_cp,
dobs_re.reshape((5, 2))[:, 0] / bp * 1e6,
"k-",
label="Obs (real)")
ax1.loglog(
frequency_cp,
dobs_re.reshape((5, 2))[:, 1] / bp * 1e6,
"k--",
label="Obs (imag)",
)
ax1.loglog(
frequency_cp,
dpred_re.reshape((5, 2))[:, 0] / bp * 1e6,
"k+",
ms=10,
markeredgewidth=2.0,
label="Pred (real)",
)
ax1.loglog(
frequency_cp,
dpred_re.reshape((5, 2))[:, 1] / bp * 1e6,
"ko",
ms=6,
markeredgecolor="k",
markeredgewidth=0.5,
label="Pred (imag)",
)
ax1.set_title("(b) RESOLVE")
ax1.set_xlabel("Frequency (Hz)")
ax1.set_ylabel("Bz (ppm)")
ax1.grid(True)
ax1.legend(loc=3, fontsize=11)
# SkyTEM data
ax2.loglog(times_off[3:] * 1e6, dobs_sky / area, "b-", label="Obs")
ax2.loglog(
times_off[3:] * 1e6,
-dpred_sky / area,
"bo",
ms=4,
markeredgecolor="k",
markeredgewidth=0.5,
label="Pred",
)
ax2.set_xlim(times_off.min() * 1e6 * 1.2, times_off.max() * 1e6 * 1.1)
ax2.set_xlabel("Time ($\mu s$)")
ax2.set_ylabel("dBz / dt (V/A-m$^4$)")
ax2.set_title("(c) SkyTEM High-moment")
ax2.grid(True)
ax2.legend(loc=3)
a3 = plt.axes([0.86, 0.33, 0.1, 0.09], facecolor=[0.8, 0.8, 0.8, 0.6])
a3.plot(prob.times * 1e6, wave, "k-")
a3.plot(rx.times * 1e6,
np.zeros_like(rx.times),
"k|",
markeredgewidth=1,
markersize=12)
a3.set_xlim([prob.times.min() * 1e6 * 0.75, prob.times.max() * 1e6 * 1.1])
a3.set_title("(d) Waveform", fontsize=11)
a3.set_xticks([prob.times.min() * 1e6, t0 * 1e6, prob.times.max() * 1e6])
a3.set_yticks([])
# a3.set_xticklabels(['0', '2e4'])
a3.set_xticklabels(["-1e4", "0", "1e4"])
plt.tight_layout()
if saveFig:
plt.savefig("booky1D_time_freq.png", dpi=600)
if plotIt:
plt.show()
resolve.close()
skytem.close()
if cleanup:
print(os.path.split(directory)[:-1])
os.remove(
os.path.sep.join(directory.split()[:-1] +
["._bookpurnong_inversion"]))
os.remove(downloads)
shutil.rmtree(directory)
locations=rx_locs, orientation=orientation, component="imag"
)
src = FDEM.Src.MagDipole(
receiver_list=[rx_real, rx_imag],
loc=src_loc,
orientation=orientation,
freq=freq,
)
srcList.append(src)
# create the survey and problem objects for running the forward simulation
survey = FDEM.Survey(srcList)
prob = FDEM.Simulation3DMagneticFluxDensity(
mesh, survey=survey, sigmaMap=mapping, Solver=Solver
)
###############################################################################
# Set up data for inversion
# -------------------------
#
# Generate clean, synthetic data. Later we will invert the clean data, and
# assign a standard deviation of 0.05, and a floor of 1e-11.
t = time.time()
data = prob.make_synthetic_data(
m_true, relative_error=0.05, noise_floor=1e-11, add_noise=False
)
def getFDEMProblem(fdemType, comp, SrcList, freq, useMu=False, verbose=False):
cs = 10.0
ncx, ncy, ncz = 0, 0, 0
npad = 8
hx = [(cs, npad, -1.3), (cs, ncx), (cs, npad, 1.3)]
hy = [(cs, npad, -1.3), (cs, ncy), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = TensorMesh([hx, hy, hz], ["C", "C", "C"])
if useMu is True:
mapping = [("sigma", maps.ExpMap(mesh)),
("mu", maps.IdentityMap(mesh))]
else:
mapping = maps.ExpMap(mesh)
x = (
np.array([
np.linspace(-5.0 * cs, -2.0 * cs, 3),
np.linspace(5.0 * cs, 2.0 * cs, 3)
]) + cs / 4.0
) # don't sample right by the source, slightly off alignment from either staggered grid
XYZ = utils.ndgrid(x, x, np.linspace(-2.0 * cs, 2.0 * cs, 5))
Rx0 = getattr(fdem.Rx, "Point" + comp[0])
if comp[-1] == "r":
real_or_imag = "real"
elif comp[-1] == "i":
real_or_imag = "imag"
rx0 = Rx0(XYZ, comp[1], real_or_imag)
Src = []
for SrcType in SrcList:
if SrcType == "MagDipole":
Src.append(
fdem.Src.MagDipole([rx0],
frequency=freq,
location=np.r_[0.0, 0.0, 0.0]))
elif SrcType == "MagDipole_Bfield":
Src.append(
fdem.Src.MagDipole_Bfield([rx0],
frequency=freq,
location=np.r_[0.0, 0.0, 0.0]))
elif SrcType == "CircularLoop":
Src.append(
fdem.Src.CircularLoop([rx0],
frequency=freq,
location=np.r_[0.0, 0.0, 0.0]))
elif SrcType == "LineCurrent":
Src.append(
fdem.Src.LineCurrent(
[rx0],
frequency=freq,
location=np.array([[0.0, 0.0, 0.0], [20.0, 0.0, 0.0]]),
))
elif SrcType == "RawVec":
if fdemType == "e" or fdemType == "b":
S_m = np.zeros(mesh.nF)
S_e = np.zeros(mesh.nE)
S_m[utils.closestPoints(mesh, [0.0, 0.0, 0.0], "Fz") +
np.sum(mesh.vnF[:1])] = 1e-3
S_e[utils.closestPoints(mesh, [0.0, 0.0, 0.0], "Ez") +
np.sum(mesh.vnE[:1])] = 1e-3
Src.append(
fdem.Src.RawVec([rx0], freq, S_m,
mesh.getEdgeInnerProduct() * S_e))
elif fdemType == "h" or fdemType == "j":
S_m = np.zeros(mesh.nE)
S_e = np.zeros(mesh.nF)
S_m[utils.closestPoints(mesh, [0.0, 0.0, 0.0], "Ez") +
np.sum(mesh.vnE[:1])] = 1e-3
S_e[utils.closestPoints(mesh, [0.0, 0.0, 0.0], "Fz") +
np.sum(mesh.vnF[:1])] = 1e-3
Src.append(
fdem.Src.RawVec([rx0], freq,
mesh.getEdgeInnerProduct() * S_m, S_e))
if verbose:
print(" Fetching {0!s} problem".format((fdemType)))
if fdemType == "e":
survey = fdem.Survey(Src)
prb = fdem.Simulation3DElectricField(mesh, sigmaMap=mapping)
elif fdemType == "b":
survey = fdem.Survey(Src)
prb = fdem.Simulation3DMagneticFluxDensity(mesh, sigmaMap=mapping)
elif fdemType == "j":
survey = fdem.Survey(Src)
prb = fdem.Simulation3DCurrentDensity(mesh, sigmaMap=mapping)
elif fdemType == "h":
survey = fdem.Survey(Src)
prb = fdem.Simulation3DMagneticField(mesh, sigmaMap=mapping)
else:
raise NotImplementedError()
prb.pair(survey)
try:
from pymatsolver import Pardiso
prb.solver = Pardiso
except ImportError:
prb.solver = SolverLU
# prb.solver_opts = dict(check_accuracy=True)
return prb