def setUp(self):
np.random.seed(0)
H0 = (50000.0, 90.0, 0.0)
# The magnetization is set along a different
# direction (induced + remanence)
M = np.array([45.0, 90.0])
# Create grid of points for topography
# Lets create a simple Gaussian topo
# and set the active cells
[xx, yy] = np.meshgrid(np.linspace(-200, 200, 50),
np.linspace(-200, 200, 50))
b = 100
A = 50
zz = A * np.exp(-0.5 * ((xx / b)**2.0 + (yy / b)**2.0))
# We would usually load a topofile
topo = np.c_[utils.mkvc(xx), utils.mkvc(yy), utils.mkvc(zz)]
# Create and array of observation points
xr = np.linspace(-100.0, 100.0, 20)
yr = np.linspace(-100.0, 100.0, 20)
X, Y = np.meshgrid(xr, yr)
Z = A * np.exp(-0.5 * ((X / b)**2.0 + (Y / b)**2.0)) + 5
# Create a MAGsurvey
xyzLoc = np.c_[utils.mkvc(X.T), utils.mkvc(Y.T), utils.mkvc(Z.T)]
rxLoc = mag.Point(xyzLoc)
srcField = mag.SourceField([rxLoc], parameters=H0)
survey = mag.Survey(srcField)
# Create a mesh
h = [5, 5, 5]
padDist = np.ones((3, 2)) * 100
mesh = mesh_builder_xyz(xyzLoc,
h,
padding_distance=padDist,
depth_core=100,
mesh_type="tree")
mesh = refine_tree_xyz(mesh,
topo,
method="surface",
octree_levels=[4, 4],
finalize=True)
self.mesh = mesh
# Define an active cells from topo
actv = utils.surface2ind_topo(mesh, topo)
nC = int(actv.sum())
model = np.zeros((mesh.nC, 3))
# Convert the inclination declination to vector in Cartesian
M_xyz = utils.mat_utils.dip_azimuth2cartesian(M[0], M[1])
# Get the indicies of the magnetized block
ind = utils.model_builder.getIndicesBlock(
np.r_[-20, -20, -10],
np.r_[20, 20, 25],
mesh.gridCC,
)[0]
# Assign magnetization values
model[ind, :] = np.kron(np.ones((ind.shape[0], 1)), M_xyz * 0.05)
# Remove air cells
self.model = model[actv, :]
# Create active map to go from reduce set to full
self.actvMap = maps.InjectActiveCells(mesh, actv, np.nan)
# Creat reduced identity map
idenMap = maps.IdentityMap(nP=nC * 3)
# Create the forward model operator
sim = mag.Simulation3DIntegral(
self.mesh,
survey=survey,
modelType="vector",
chiMap=idenMap,
actInd=actv,
store_sensitivities="disk",
)
self.sim = sim
# Compute some data and add some random noise
data = sim.make_synthetic_data(utils.mkvc(self.model),
relative_error=0.0,
noise_floor=5.0,
add_noise=True)
# This Mapping connects the regularizations for the three-component
# vector model
wires = maps.Wires(("p", nC), ("s", nC), ("t", nC))
# Create three regularization for the different components
# of magnetization
reg_p = regularization.Sparse(mesh, indActive=actv, mapping=wires.p)
reg_p.mref = np.zeros(3 * nC)
reg_s = regularization.Sparse(mesh, indActive=actv, mapping=wires.s)
reg_s.mref = np.zeros(3 * nC)
reg_t = regularization.Sparse(mesh, indActive=actv, mapping=wires.t)
reg_t.mref = np.zeros(3 * nC)
reg = reg_p + reg_s + reg_t
reg.mref = np.zeros(3 * nC)
# Data misfit function
dmis = data_misfit.L2DataMisfit(simulation=sim, data=data)
# dmis.W = 1./survey.std
# Add directives to the inversion
opt = optimization.ProjectedGNCG(maxIter=10,
lower=-10,
upper=10.0,
maxIterLS=5,
maxIterCG=5,
tolCG=1e-4)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
# A list of directive to control the inverson
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1e1)
# Here is where the norms are applied
# Use pick a treshold parameter empirically based on the distribution of
# model parameters
IRLS = directives.Update_IRLS(f_min_change=1e-3,
max_irls_iterations=0,
beta_tol=5e-1)
# Pre-conditioner
update_Jacobi = directives.UpdatePreconditioner()
sensitivity_weights = directives.UpdateSensitivityWeights(
everyIter=False)
inv = inversion.BaseInversion(
invProb,
directiveList=[sensitivity_weights, IRLS, update_Jacobi, betaest])
# Run the inversion
m0 = np.ones(3 * nC) * 1e-4 # Starting model
mrec_MVIC = inv.run(m0)
sim.chiMap = maps.SphericalSystem(nP=nC * 3)
self.mstart = sim.chiMap.inverse(mrec_MVIC)
dmis.simulation.model = self.mstart
beta = invProb.beta
# Create a block diagonal regularization
wires = maps.Wires(("amp", nC), ("theta", nC), ("phi", nC))
# Create a Combo Regularization
# Regularize the amplitude of the vectors
reg_a = regularization.Sparse(mesh, indActive=actv, mapping=wires.amp)
reg_a.norms = np.c_[0.0, 0.0, 0.0,
0.0] # Sparse on the model and its gradients
reg_a.mref = np.zeros(3 * nC)
# Regularize the vertical angle of the vectors
reg_t = regularization.Sparse(mesh,
indActive=actv,
mapping=wires.theta)
reg_t.alpha_s = 0.0 # No reference angle
reg_t.space = "spherical"
reg_t.norms = np.c_[2.0, 0.0, 0.0, 0.0] # Only norm on gradients used
# Regularize the horizontal angle of the vectors
reg_p = regularization.Sparse(mesh, indActive=actv, mapping=wires.phi)
reg_p.alpha_s = 0.0 # No reference angle
reg_p.space = "spherical"
reg_p.norms = np.c_[2.0, 0.0, 0.0, 0.0] # Only norm on gradients used
reg = reg_a + reg_t + reg_p
reg.mref = np.zeros(3 * nC)
Lbound = np.kron(np.asarray([0, -np.inf, -np.inf]), np.ones(nC))
Ubound = np.kron(np.asarray([10, np.inf, np.inf]), np.ones(nC))
# Add directives to the inversion
opt = optimization.ProjectedGNCG(
maxIter=5,
lower=Lbound,
upper=Ubound,
maxIterLS=5,
maxIterCG=5,
tolCG=1e-3,
stepOffBoundsFact=1e-3,
)
opt.approxHinv = None
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt, beta=beta)
# Here is where the norms are applied
IRLS = directives.Update_IRLS(
f_min_change=1e-4,
max_irls_iterations=5,
minGNiter=1,
beta_tol=0.5,
coolingRate=1,
coolEps_q=True,
sphericalDomain=True,
)
# Special directive specific to the mag amplitude problem. The sensitivity
# weights are update between each iteration.
ProjSpherical = directives.ProjectSphericalBounds()
sensitivity_weights = directives.UpdateSensitivityWeights()
update_Jacobi = directives.UpdatePreconditioner()
self.inv = inversion.BaseInversion(
invProb,
directiveList=[
ProjSpherical, IRLS, sensitivity_weights, update_Jacobi
],
)
def run(plotIt=True, cleanAfterRun=True):
# Start by downloading files from the remote repository
# directory where the downloaded files are
url = "https://storage.googleapis.com/simpeg/Chile_GRAV_4_Miller/Chile_GRAV_4_Miller.tar.gz"
downloads = download(url, overwrite=True)
basePath = downloads.split(".")[0]
# unzip the tarfile
tar = tarfile.open(downloads, "r")
tar.extractall()
tar.close()
input_file = basePath + os.path.sep + "LdM_input_file.inp"
# %% User input
# Plotting parameters, max and min densities in g/cc
vmin = -0.6
vmax = 0.6
# weight exponent for default weighting
wgtexp = 3.0
# %%
# Read in the input file which included all parameters at once
# (mesh, topo, model, survey, inv param, etc.)
driver = GravityDriver_Inv(input_file)
# %%
# Now we need to create the survey and model information.
# Access the mesh and survey information
mesh = driver.mesh #
survey = driver.survey
data_object = driver.data
# [survey, data_object] = driver.survey
# define gravity survey locations
rxLoc = survey.source_field.receiver_list[0].locations
# define gravity data and errors
d = data_object.dobs
# Get the active cells
active = driver.activeCells
nC = len(active) # Number of active cells
# Create active map to go from reduce set to full
activeMap = maps.InjectActiveCells(mesh, active, -100)
# Create static map
static = driver.staticCells
dynamic = driver.dynamicCells
staticCells = maps.InjectActiveCells(None,
dynamic,
driver.m0[static],
nC=nC)
mstart = driver.m0[dynamic]
# Get index of the center
midx = int(mesh.nCx / 2)
# %%
# Now that we have a model and a survey we can build the linear system ...
# Create the forward model operator
simulation = gravity.simulation.Simulation3DIntegral(survey=survey,
mesh=mesh,
rhoMap=staticCells,
actInd=active)
# %% Create inversion objects
reg = regularization.Sparse(mesh,
indActive=active,
mapping=staticCells,
gradientType="total")
reg.mref = driver.mref[dynamic]
reg.norms = np.c_[0.0, 1.0, 1.0, 1.0]
# reg.norms = driver.lpnorms
# Specify how the optimization will proceed
opt = optimization.ProjectedGNCG(
maxIter=20,
lower=driver.bounds[0],
upper=driver.bounds[1],
maxIterLS=10,
maxIterCG=20,
tolCG=1e-4,
)
# Define misfit function (obs-calc)
dmis = data_misfit.L2DataMisfit(data=data_object, simulation=simulation)
# create the default L2 inverse problem from the above objects
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
# Specify how the initial beta is found
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1e-2)
# IRLS sets up the Lp inversion problem
# Set the eps parameter parameter in Line 11 of the
# input file based on the distribution of model (DEFAULT = 95th %ile)
IRLS = directives.Update_IRLS(f_min_change=1e-4,
max_irls_iterations=40,
coolEpsFact=1.5,
beta_tol=5e-1)
# Preconditioning refreshing for each IRLS iteration
update_Jacobi = directives.UpdatePreconditioner()
sensitivity_weights = directives.UpdateSensitivityWeights()
# Create combined the L2 and Lp problem
inv = inversion.BaseInversion(
invProb,
directiveList=[sensitivity_weights, IRLS, update_Jacobi, betaest])
# %%
# Run L2 and Lp inversion
mrec = inv.run(mstart)
if cleanAfterRun:
os.remove(downloads)
shutil.rmtree(basePath)
# %%
if plotIt:
# Plot observed data
# The sign of the data is flipped here for the change of convention
# between Cartesian coordinate system (internal SimPEG format that
# expects "positive up" gravity signal) and traditional gravity data
# conventions (positive down). For example a traditional negative
# gravity anomaly is described as "positive up" in Cartesian coordinates
# and hence the sign needs to be flipped for use in SimPEG.
plot2Ddata(rxLoc, -d)
# %%
# Write output model and data files and print misfit stats.
# reconstructing l2 model mesh with air cells and active dynamic cells
L2out = activeMap * invProb.l2model
# reconstructing lp model mesh with air cells and active dynamic cells
Lpout = activeMap * mrec
# %%
# Plot out sections and histograms of the smooth l2 model.
# The ind= parameter is the slice of the model from top down.
yslice = midx + 1
L2out[L2out == -100] = np.nan # set "air" to nan
plt.figure(figsize=(10, 7))
plt.suptitle("Smooth Inversion: Depth weight = " + str(wgtexp))
ax = plt.subplot(221)
dat1 = mesh.plotSlice(
L2out,
ax=ax,
normal="Z",
ind=-16,
clim=(vmin, vmax),
pcolorOpts={"cmap": "bwr"},
)
plt.plot(
np.array([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
np.array([mesh.vectorCCy[yslice], mesh.vectorCCy[yslice]]),
c="gray",
linestyle="--",
)
plt.scatter(rxLoc[0:, 0], rxLoc[0:, 1], color="k", s=1)
plt.title("Z: " + str(mesh.vectorCCz[-16]) + " m")
plt.xlabel("Easting (m)")
plt.ylabel("Northing (m)")
plt.gca().set_aspect("equal", adjustable="box")
cb = plt.colorbar(dat1[0],
orientation="vertical",
ticks=np.linspace(vmin, vmax, 4))
cb.set_label("Density (g/cc$^3$)")
ax = plt.subplot(222)
dat = mesh.plotSlice(
L2out,
ax=ax,
normal="Z",
ind=-27,
clim=(vmin, vmax),
pcolorOpts={"cmap": "bwr"},
)
plt.plot(
np.array([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
np.array([mesh.vectorCCy[yslice], mesh.vectorCCy[yslice]]),
c="gray",
linestyle="--",
)
plt.scatter(rxLoc[0:, 0], rxLoc[0:, 1], color="k", s=1)
plt.title("Z: " + str(mesh.vectorCCz[-27]) + " m")
plt.xlabel("Easting (m)")
plt.ylabel("Northing (m)")
plt.gca().set_aspect("equal", adjustable="box")
cb = plt.colorbar(dat1[0],
orientation="vertical",
ticks=np.linspace(vmin, vmax, 4))
cb.set_label("Density (g/cc$^3$)")
ax = plt.subplot(212)
mesh.plotSlice(
L2out,
ax=ax,
normal="Y",
ind=yslice,
clim=(vmin, vmax),
pcolorOpts={"cmap": "bwr"},
)
plt.title("Cross Section")
plt.xlabel("Easting(m)")
plt.ylabel("Elevation")
plt.gca().set_aspect("equal", adjustable="box")
cb = plt.colorbar(
dat1[0],
orientation="vertical",
ticks=np.linspace(vmin, vmax, 4),
cmap="bwr",
)
cb.set_label("Density (g/cc$^3$)")
# %%
# Make plots of Lp model
yslice = midx + 1
Lpout[Lpout == -100] = np.nan # set "air" to nan
plt.figure(figsize=(10, 7))
plt.suptitle("Compact Inversion: Depth weight = " + str(wgtexp) +
": $\epsilon_p$ = " + str(round(reg.eps_p, 1)) +
": $\epsilon_q$ = " + str(round(reg.eps_q, 2)))
ax = plt.subplot(221)
dat = mesh.plotSlice(
Lpout,
ax=ax,
normal="Z",
ind=-16,
clim=(vmin, vmax),
pcolorOpts={"cmap": "bwr"},
)
plt.plot(
np.array([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
np.array([mesh.vectorCCy[yslice], mesh.vectorCCy[yslice]]),
c="gray",
linestyle="--",
)
plt.scatter(rxLoc[0:, 0], rxLoc[0:, 1], color="k", s=1)
plt.title("Z: " + str(mesh.vectorCCz[-16]) + " m")
plt.xlabel("Easting (m)")
plt.ylabel("Northing (m)")
plt.gca().set_aspect("equal", adjustable="box")
cb = plt.colorbar(dat[0],
orientation="vertical",
ticks=np.linspace(vmin, vmax, 4))
cb.set_label("Density (g/cc$^3$)")
ax = plt.subplot(222)
dat = mesh.plotSlice(
Lpout,
ax=ax,
normal="Z",
ind=-27,
clim=(vmin, vmax),
pcolorOpts={"cmap": "bwr"},
)
plt.plot(
np.array([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
np.array([mesh.vectorCCy[yslice], mesh.vectorCCy[yslice]]),
c="gray",
linestyle="--",
)
plt.scatter(rxLoc[0:, 0], rxLoc[0:, 1], color="k", s=1)
plt.title("Z: " + str(mesh.vectorCCz[-27]) + " m")
plt.xlabel("Easting (m)")
plt.ylabel("Northing (m)")
plt.gca().set_aspect("equal", adjustable="box")
cb = plt.colorbar(dat[0],
orientation="vertical",
ticks=np.linspace(vmin, vmax, 4))
cb.set_label("Density (g/cc$^3$)")
ax = plt.subplot(212)
dat = mesh.plotSlice(
Lpout,
ax=ax,
normal="Y",
ind=yslice,
clim=(vmin, vmax),
pcolorOpts={"cmap": "bwr"},
)
plt.title("Cross Section")
plt.xlabel("Easting (m)")
plt.ylabel("Elevation (m)")
plt.gca().set_aspect("equal", adjustable="box")
cb = plt.colorbar(dat[0],
orientation="vertical",
ticks=np.linspace(vmin, vmax, 4))
cb.set_label("Density (g/cc$^3$)")
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(),
)
srcList = [
FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation="Z")
for freq in freqs
]
surveyFD = FDEM.Survey(srcList)
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(),
loc=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)
regT = regularization.Simple(mesh,
indActive=actind,
alpha_s=1e-6,
alpha_x=1.0,
alpha_y=1.0,
alpha_z=1.0)
# Optimization Scheme
opt = optimization.InexactGaussNewton(maxIter=10)
# Form the problem
opt.remember("xc")
invProb = inverse_problem.BaseInvProblem(dmis, regT, opt)
# Directives for Inversions
beta = directives.BetaEstimate_ByEig(beta0_ratio=1.0)
Target = directives.TargetMisfit()
betaSched = directives.BetaSchedule(coolingFactor=5.0, coolingRate=2)
inv = inversion.BaseInversion(invProb, directiveList=[beta, Target, betaSched])
# Run Inversion
minv = inv.run(m0)
# Final Plot
############
fig, ax = plt.subplots(2, 2, figsize=(12, 6))
ax = utils.mkvc(ax)
cyl0v = getCylinderPoints(x0, z0, r0)
cyl1v = getCylinderPoints(x1, z1, r1)
def setUp(self):
# We will assume a vertical inducing field
H0 = (50000.0, 90.0, 0.0)
# The magnetization is set along a different direction (induced + remanence)
M = np.array([45.0, 90.0])
# Block with an effective susceptibility
chi_e = 0.05
# Create grid of points for topography
# Lets create a simple Gaussian topo and set the active cells
[xx, yy] = np.meshgrid(np.linspace(-200, 200, 50),
np.linspace(-200, 200, 50))
b = 100
A = 50
zz = A * np.exp(-0.5 * ((xx / b)**2.0 + (yy / b)**2.0))
topo = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)]
# Create an array of observation points
xr = np.linspace(-100.0, 100.0, 20)
yr = np.linspace(-100.0, 100.0, 20)
X, Y = np.meshgrid(xr, yr)
Z = A * np.exp(-0.5 * ((X / b)**2.0 + (Y / b)**2.0)) + 10
# Create a MAGsurvey
rxLoc = np.c_[mkvc(X.T), mkvc(Y.T), mkvc(Z.T)]
rxList = magnetics.receivers.Point(rxLoc)
srcField = magnetics.sources.SourceField(receiver_list=[rxList],
parameters=H0)
survey = magnetics.survey.Survey(srcField)
###############################################################################
# Inversion Mesh
# Create a mesh
h = [5, 5, 5]
padDist = np.ones((3, 2)) * 100
mesh = mesh_builder_xyz(rxLoc,
h,
padding_distance=padDist,
depth_core=100,
mesh_type="tree")
mesh = refine_tree_xyz(mesh,
topo,
method="surface",
octree_levels=[4, 4],
finalize=True)
# Define an active cells from topo
actv = utils.surface2ind_topo(mesh, topo)
nC = int(actv.sum())
# Convert the inclination declination to vector in Cartesian
M_xyz = utils.mat_utils.dip_azimuth2cartesian(
np.ones(nC) * M[0],
np.ones(nC) * M[1])
# Get the indicies of the magnetized block
ind = utils.model_builder.getIndicesBlock(
np.r_[-20, -20, -10],
np.r_[20, 20, 25],
mesh.gridCC,
)[0]
# Assign magnetization value, inducing field strength will
# be applied in by the :class:`SimPEG.PF.Magnetics` problem
model = np.zeros(mesh.nC)
model[ind] = chi_e
# Remove air cells
model = model[actv]
# Creat reduced identity map
idenMap = maps.IdentityMap(nP=nC)
# Create the forward model operator
simulation = magnetics.Simulation3DIntegral(
survey=survey,
mesh=mesh,
chiMap=idenMap,
actInd=actv,
store_sensitivities="forward_only",
)
simulation.M = M_xyz
# Compute some data and add some random noise
synthetic_data = simulation.dpred(model)
# Split the data in components
nD = rxLoc.shape[0]
std = 5 # nT
synthetic_data += np.random.randn(nD) * std
wd = np.ones(nD) * std
# Assigne data and uncertainties to the survey
data_object = data.Data(survey, dobs=synthetic_data, noise_floor=wd)
######################################################################
# Equivalent Source
# Get the active cells for equivalent source is the top only
surf = utils.model_utils.surface_layer_index(mesh, topo)
nC = np.count_nonzero(surf) # Number of active cells
mstart = np.ones(nC) * 1e-4
# Create active map to go from reduce set to full
surfMap = maps.InjectActiveCells(mesh, surf, np.nan)
# Create identity map
idenMap = maps.IdentityMap(nP=nC)
# Create static map
simulation = magnetics.simulation.Simulation3DIntegral(
mesh=mesh,
survey=survey,
chiMap=idenMap,
actInd=surf,
store_sensitivities="ram",
)
simulation.model = mstart
# Create a regularization function, in this case l2l2
reg = regularization.Sparse(mesh,
indActive=surf,
mapping=maps.IdentityMap(nP=nC),
alpha_z=0)
reg.mref = np.zeros(nC)
# Specify how the optimization will proceed, set susceptibility bounds to inf
opt = optimization.ProjectedGNCG(
maxIter=10,
lower=-np.inf,
upper=np.inf,
maxIterLS=5,
maxIterCG=5,
tolCG=1e-3,
)
# Define misfit function (obs-calc)
dmis = data_misfit.L2DataMisfit(simulation=simulation,
data=data_object)
# Create the default L2 inverse problem from the above objects
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
# Specify how the initial beta is found
betaest = directives.BetaEstimate_ByEig(beta0_ratio=2)
# Target misfit to stop the inversion,
# try to fit as much as possible of the signal, we don't want to lose anything
IRLS = directives.Update_IRLS(f_min_change=1e-3,
minGNiter=1,
beta_tol=1e-1,
max_irls_iterations=5)
update_Jacobi = directives.UpdatePreconditioner()
# Put all the parts together
inv = inversion.BaseInversion(
invProb, directiveList=[betaest, IRLS, update_Jacobi])
# Run the equivalent source inversion
print("Solving for Equivalent Source")
mrec = inv.run(mstart)
########################################################
# Forward Amplitude Data
# ----------------------
#
# Now that we have an equialent source layer, we can forward model alh three
# components of the field and add them up: :math:`|B| = \sqrt{( Bx^2 + Bx^2 + Bx^2 )}`
#
rxList = magnetics.receivers.Point(rxLoc,
components=["bx", "by", "bz"])
srcField = magnetics.sources.SourceField(receiver_list=[rxList],
parameters=H0)
surveyAmp = magnetics.survey.Survey(srcField)
simulation = magnetics.simulation.Simulation3DIntegral(
mesh=mesh,
survey=surveyAmp,
chiMap=idenMap,
actInd=surf,
is_amplitude_data=True,
store_sensitivities="forward_only",
)
bAmp = simulation.fields(mrec)
######################################################################
# Amplitude Inversion
# -------------------
#
# Now that we have amplitude data, we can invert for an effective
# susceptibility. This is a non-linear inversion.
#
# Create active map to go from reduce space to full
actvMap = maps.InjectActiveCells(mesh, actv, -100)
nC = int(actv.sum())
# Create identity map
idenMap = maps.IdentityMap(nP=nC)
mstart = np.ones(nC) * 1e-4
# Create the forward model operator
simulation = magnetics.simulation.Simulation3DIntegral(
survey=surveyAmp,
mesh=mesh,
chiMap=idenMap,
actInd=actv,
is_amplitude_data=True,
)
data_obj = data.Data(survey, dobs=bAmp, noise_floor=wd)
# Create a sparse regularization
reg = regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
reg.norms = np.c_[1, 0, 0, 0]
reg.mref = np.zeros(nC)
# Data misfit function
dmis = data_misfit.L2DataMisfit(simulation=simulation, data=data_obj)
# Add directives to the inversion
opt = optimization.ProjectedGNCG(maxIter=10,
lower=0.0,
upper=1.0,
maxIterLS=5,
maxIterCG=5,
tolCG=1e-3)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
# Here is the list of directives
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1)
# Specify the sparse norms
IRLS = directives.Update_IRLS(
max_irls_iterations=5,
f_min_change=1e-3,
minGNiter=1,
coolingRate=1,
beta_search=False,
)
# Special directive specific to the mag amplitude problem. The sensitivity
# weights are update between each iteration.
update_SensWeight = directives.UpdateSensitivityWeights()
update_Jacobi = directives.UpdatePreconditioner()
# Put all together
self.inv = inversion.BaseInversion(
invProb,
directiveList=[update_SensWeight, betaest, IRLS, update_Jacobi])
self.mstart = mstart
self.model = model
self.sim = simulation
def run(plotIt=True, survey_type="dipole-dipole", p=0.0, qx=2.0, qz=2.0):
np.random.seed(1)
# Initiate I/O class for DC
IO = DC.IO()
# Obtain ABMN locations
xmin, xmax = 0.0, 200.0
ymin, ymax = 0.0, 0.0
zmin, zmax = 0, 0
endl = np.array([[xmin, ymin, zmin], [xmax, ymax, zmax]])
# Generate DC survey object
survey = gen_DCIPsurvey(endl,
survey_type=survey_type,
dim=2,
a=10,
b=10,
n=10)
survey = IO.from_ambn_locations_to_survey(
survey.locations_a,
survey.locations_b,
survey.locations_m,
survey.locations_n,
survey_type,
data_dc_type="volt",
)
# Obtain 2D TensorMesh
mesh, actind = IO.set_mesh()
topo, mesh1D = genTopography(mesh, -10, 0, its=100)
actind = utils.surface2ind_topo(mesh, np.c_[mesh1D.vectorCCx, topo])
survey.drape_electrodes_on_topography(mesh, actind, option="top")
# Build a conductivity model
blk_inds_c = utils.model_builder.getIndicesSphere(np.r_[60.0, -25.0], 12.5,
mesh.gridCC)
blk_inds_r = utils.model_builder.getIndicesSphere(np.r_[140.0, -25.0],
12.5, mesh.gridCC)
layer_inds = mesh.gridCC[:, 1] > -5.0
sigma = np.ones(mesh.nC) * 1.0 / 100.0
sigma[blk_inds_c] = 1.0 / 10.0
sigma[blk_inds_r] = 1.0 / 1000.0
sigma[~actind] = 1.0 / 1e8
rho = 1.0 / sigma
# Show the true conductivity model
if plotIt:
fig = plt.figure(figsize=(12, 3))
ax = plt.subplot(111)
temp = rho.copy()
temp[~actind] = np.nan
out = mesh.plotImage(
temp,
grid=True,
ax=ax,
gridOpts={"alpha": 0.2},
clim=(10, 1000),
pcolorOpts={
"cmap": "viridis",
"norm": colors.LogNorm()
},
)
ax.plot(survey.electrode_locations[:, 0],
survey.electrode_locations[:, 1], "k.")
ax.set_xlim(IO.grids[:, 0].min(), IO.grids[:, 0].max())
ax.set_ylim(-IO.grids[:, 1].max(), IO.grids[:, 1].min())
cb = plt.colorbar(out[0])
cb.set_label("Resistivity (ohm-m)")
ax.set_aspect("equal")
plt.show()
# Use Exponential Map: m = log(rho)
actmap = maps.InjectActiveCells(mesh,
indActive=actind,
valInactive=np.log(1e8))
mapping = maps.ExpMap(mesh) * actmap
# Generate mtrue
mtrue = np.log(rho[actind])
# Generate 2.5D DC problem
# "N" means potential is defined at nodes
prb = DC.Simulation2DNodal(mesh,
survey=survey,
rhoMap=mapping,
storeJ=True,
Solver=Solver,
verbose=True)
# Make synthetic DC data with 5% Gaussian noise
data = prb.make_synthetic_data(mtrue, relative_error=0.05, add_noise=True)
IO.data_dc = data.dobs
# Show apparent resisitivty pseudo-section
if plotIt:
IO.plotPseudoSection(data=data.dobs / IO.G,
data_type="apparent_resistivity")
# Show apparent resisitivty histogram
if plotIt:
fig = plt.figure()
out = hist(data.dobs / IO.G, bins=20)
plt.xlabel("Apparent Resisitivty ($\Omega$m)")
plt.show()
# Set initial model based upon histogram
m0 = np.ones(actmap.nP) * np.log(100.0)
# Set standard_deviation
# floor
eps = 10**(-3.2)
# percentage
relative = 0.05
dmisfit = data_misfit.L2DataMisfit(simulation=prb, data=data)
uncert = abs(data.dobs) * relative + eps
dmisfit.standard_deviation = uncert
# Map for a regularization
regmap = maps.IdentityMap(nP=int(actind.sum()))
# Related to inversion
reg = regularization.Sparse(mesh,
indActive=actind,
mapping=regmap,
gradientType="components")
# gradientType = 'components'
reg.norms = np.c_[p, qx, qz, 0.0]
IRLS = directives.Update_IRLS(max_irls_iterations=20,
minGNiter=1,
beta_search=False,
fix_Jmatrix=True)
opt = optimization.InexactGaussNewton(maxIter=40)
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
beta = directives.BetaSchedule(coolingFactor=5, coolingRate=2)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1e0)
target = directives.TargetMisfit()
update_Jacobi = directives.UpdatePreconditioner()
inv = inversion.BaseInversion(invProb, directiveList=[betaest, IRLS])
prb.counter = opt.counter = utils.Counter()
opt.LSshorten = 0.5
opt.remember("xc")
# Run inversion
mopt = inv.run(m0)
rho_est = mapping * mopt
rho_est_l2 = mapping * invProb.l2model
rho_est[~actind] = np.nan
rho_est_l2[~actind] = np.nan
rho_true = rho.copy()
rho_true[~actind] = np.nan
# show recovered conductivity
if plotIt:
vmin, vmax = rho.min(), rho.max()
fig, ax = plt.subplots(3, 1, figsize=(20, 9))
out1 = mesh.plotImage(
rho_true,
clim=(10, 1000),
pcolorOpts={
"cmap": "viridis",
"norm": colors.LogNorm()
},
ax=ax[0],
)
out2 = mesh.plotImage(
rho_est_l2,
clim=(10, 1000),
pcolorOpts={
"cmap": "viridis",
"norm": colors.LogNorm()
},
ax=ax[1],
)
out3 = mesh.plotImage(
rho_est,
clim=(10, 1000),
pcolorOpts={
"cmap": "viridis",
"norm": colors.LogNorm()
},
ax=ax[2],
)
out = [out1, out2, out3]
titles = ["True", "L2", ("L%d, Lx%d, Lz%d") % (p, qx, qz)]
for i in range(3):
ax[i].plot(survey.electrode_locations[:, 0],
survey.electrode_locations[:, 1], "kv")
ax[i].set_xlim(IO.grids[:, 0].min(), IO.grids[:, 0].max())
ax[i].set_ylim(-IO.grids[:, 1].max(), IO.grids[:, 1].min())
cb = plt.colorbar(out[i][0], ax=ax[i])
cb.set_label("Resistivity ($\Omega$m)")
ax[i].set_xlabel("Northing (m)")
ax[i].set_ylabel("Elevation (m)")
ax[i].set_aspect("equal")
ax[i].set_title(titles[i])
plt.tight_layout()
plt.show()
# `alpha_s, alpha_x, alpha_y` as input arguments when the regularization is
# created. The default reference model in the regularization is the starting
# model. To set something different, you can input an `mref` into the
# regularization.
#
# We estimate the trade-off parameter, beta, between the data
# misfit and regularization by the largest eigenvalue of the data misfit and
# the regularization. Here, we use a fixed beta, but could alternatively
# employ a beta-cooling schedule using :class:`SimPEG.directives.BetaSchedule`
dmisfit = data_misfit.L2DataMisfit(simulation=prob, data=data)
reg = regularization.Simple(inversion_mesh)
opt = optimization.InexactGaussNewton(maxIterCG=10, remember="xc")
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=0.05, n_pw_iter=1, seed=1)
target = directives.TargetMisfit()
directiveList = [betaest, target]
inv = inversion.BaseInversion(invProb, directiveList=directiveList)
print("The target misfit is {:1.2f}".format(target.target))
###############################################################################
# Run the inversion
# ------------------
#
# We start from a half-space equal to the deep conductivity.
m0 = np.log(sigma_deep) * np.ones(inversion_mesh.nC)
def run(plotIt=True, survey_type="dipole-dipole"):
np.random.seed(1)
# Initiate I/O class for DC
IO = DC.IO()
# Obtain ABMN locations
xmin, xmax = 0.0, 200.0
ymin, ymax = 0.0, 0.0
zmin, zmax = 0, 0
endl = np.array([[xmin, ymin, zmin], [xmax, ymax, zmax]])
# Generate DC survey object
survey = gen_DCIPsurvey(endl,
survey_type=survey_type,
dim=2,
a=10,
b=10,
n=10)
survey = IO.from_abmn_locations_to_survey(
survey.locations_a,
survey.locations_b,
survey.locations_m,
survey.locations_n,
survey_type,
data_dc_type="volt",
)
# Obtain 2D TensorMesh
mesh, actind = IO.set_mesh()
topo, mesh1D = genTopography(mesh, -10, 0, its=100)
actind = utils.surface2ind_topo(mesh, np.c_[mesh1D.vectorCCx, topo])
survey.drape_electrodes_on_topography(mesh, actind, option="top")
# Build a conductivity model
blk_inds_c = utils.model_builder.getIndicesSphere(np.r_[60.0, -25.0], 12.5,
mesh.gridCC)
blk_inds_r = utils.model_builder.getIndicesSphere(np.r_[140.0, -25.0],
12.5, mesh.gridCC)
layer_inds = mesh.gridCC[:, 1] > -5.0
sigma = np.ones(mesh.nC) * 1.0 / 100.0
sigma[blk_inds_c] = 1.0 / 10.0
sigma[blk_inds_r] = 1.0 / 1000.0
sigma[~actind] = 1.0 / 1e8
rho = 1.0 / sigma
# Show the true conductivity model
if plotIt:
fig = plt.figure(figsize=(12, 3))
ax = plt.subplot(111)
temp = rho.copy()
temp[~actind] = np.nan
out = mesh.plotImage(
temp,
grid=True,
ax=ax,
gridOpts={"alpha": 0.2},
clim=(10, 1000),
pcolorOpts={
"cmap": "viridis",
"norm": colors.LogNorm()
},
)
ax.plot(survey.electrode_locations[:, 0],
survey.electrode_locations[:, 1], "k.")
ax.set_xlim(IO.grids[:, 0].min(), IO.grids[:, 0].max())
ax.set_ylim(-IO.grids[:, 1].max(), IO.grids[:, 1].min())
cb = plt.colorbar(out[0])
cb.set_label("Resistivity (ohm-m)")
ax.set_aspect("equal")
plt.show()
# Use Exponential Map: m = log(rho)
actmap = maps.InjectActiveCells(mesh,
indActive=actind,
valInactive=np.log(1e8))
mapping = maps.ExpMap(mesh) * actmap
# Generate mtrue
mtrue = np.log(rho[actind])
# Generate 2.5D DC problem
# "N" means potential is defined at nodes
prb = DC.Simulation2DNodal(mesh,
survey=survey,
rhoMap=mapping,
storeJ=True,
Solver=Solver,
verbose=True)
geometric_factor = survey.set_geometric_factor(
data_type="apparent_resistivity",
survey_type="dipole-dipole",
space_type="half-space",
)
# Make synthetic DC data with 5% Gaussian noise
data = prb.make_synthetic_data(mtrue, relative_error=0.05, add_noise=True)
IO.data_dc = data.dobs
# Show apparent resisitivty pseudo-section
if plotIt:
IO.plotPseudoSection(data=data.dobs, data_type="apparent_resistivity")
# Show apparent resisitivty histogram
if plotIt:
fig = plt.figure()
out = hist(data.dobs, bins=20)
plt.xlabel("Apparent Resisitivty ($\Omega$m)")
plt.show()
# Set initial model based upon histogram
m0 = np.ones(actmap.nP) * np.log(100.0)
# Set standard_deviation
# floor (10 ohm-m)
eps = 1.0
# percentage
relative = 0.05
dmisfit = data_misfit.L2DataMisfit(simulation=prb, data=data)
uncert = abs(data.dobs) * relative + eps
dmisfit.standard_deviation = uncert
# Map for a regularization
regmap = maps.IdentityMap(nP=int(actind.sum()))
# Related to inversion
reg = regularization.Sparse(mesh, indActive=actind, mapping=regmap)
opt = optimization.InexactGaussNewton(maxIter=15)
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
beta = directives.BetaSchedule(coolingFactor=5, coolingRate=2)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1e0)
target = directives.TargetMisfit()
updateSensW = directives.UpdateSensitivityWeights()
update_Jacobi = directives.UpdatePreconditioner()
inv = inversion.BaseInversion(
invProb,
directiveList=[beta, target, updateSensW, betaest, update_Jacobi])
prb.counter = opt.counter = utils.Counter()
opt.LSshorten = 0.5
opt.remember("xc")
# Run inversion
mopt = inv.run(m0)
# Get diag(JtJ)
mask_inds = np.ones(mesh.nC, dtype=bool)
jtj = np.sqrt(updateSensW.JtJdiag[0])
jtj /= jtj.max()
temp = np.ones_like(jtj, dtype=bool)
temp[jtj > 0.005] = False
mask_inds[actind] = temp
actind_final = np.logical_and(actind, ~mask_inds)
jtj_cc = np.ones(mesh.nC) * np.nan
jtj_cc[actind] = jtj
# Show the sensitivity
if plotIt:
fig = plt.figure(figsize=(12, 3))
ax = plt.subplot(111)
temp = rho.copy()
temp[~actind] = np.nan
out = mesh.plotImage(
jtj_cc,
grid=True,
ax=ax,
gridOpts={"alpha": 0.2},
clim=(0.005, 0.5),
pcolorOpts={
"cmap": "viridis",
"norm": colors.LogNorm()
},
)
ax.plot(survey.electrode_locations[:, 0],
survey.electrode_locations[:, 1], "k.")
ax.set_xlim(IO.grids[:, 0].min(), IO.grids[:, 0].max())
ax.set_ylim(-IO.grids[:, 1].max(), IO.grids[:, 1].min())
cb = plt.colorbar(out[0])
cb.set_label("Sensitivity")
ax.set_aspect("equal")
plt.show()
# Convert obtained inversion model to resistivity
# rho = M(m), where M(.) is a mapping
rho_est = mapping * mopt
rho_est[~actind_final] = np.nan
rho_true = rho.copy()
rho_true[~actind_final] = np.nan
# show recovered conductivity
if plotIt:
vmin, vmax = rho.min(), rho.max()
fig, ax = plt.subplots(2, 1, figsize=(20, 6))
out1 = mesh.plotImage(
rho_true,
clim=(10, 1000),
pcolorOpts={
"cmap": "viridis",
"norm": colors.LogNorm()
},
ax=ax[0],
)
out2 = mesh.plotImage(
rho_est,
clim=(10, 1000),
pcolorOpts={
"cmap": "viridis",
"norm": colors.LogNorm()
},
ax=ax[1],
)
out = [out1, out2]
for i in range(2):
ax[i].plot(survey.electrode_locations[:, 0],
survey.electrode_locations[:, 1], "kv")
ax[i].set_xlim(IO.grids[:, 0].min(), IO.grids[:, 0].max())
ax[i].set_ylim(-IO.grids[:, 1].max(), IO.grids[:, 1].min())
cb = plt.colorbar(out[i][0], ax=ax[i])
cb.set_label("Resistivity ($\Omega$m)")
ax[i].set_xlabel("Northing (m)")
ax[i].set_ylabel("Elevation (m)")
ax[i].set_aspect("equal")
plt.tight_layout()
plt.show()
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.Tikhonov(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)
#########################################
# Petrophysically constrained inversion #
#########################################
# fit a Gaussian Mixture Model with n components
# on the true model to simulate the laboratory
def setUp(self):
np.random.seed(0)
# Define the inducing field parameter
H0 = (50000, 90, 0)
# Create a mesh
dx = 5.0
hxind = [(dx, 5, -1.3), (dx, 5), (dx, 5, 1.3)]
hyind = [(dx, 5, -1.3), (dx, 5), (dx, 5, 1.3)]
hzind = [(dx, 5, -1.3), (dx, 6)]
self.mesh = discretize.TensorMesh([hxind, hyind, hzind], "CCC")
# Get index of the center
midx = int(self.mesh.nCx / 2)
midy = int(self.mesh.nCy / 2)
# Lets create a simple Gaussian topo and set the active cells
[xx, yy] = np.meshgrid(self.mesh.vectorNx, self.mesh.vectorNy)
zz = -np.exp((xx**2 + yy**2) / 75**2) + self.mesh.vectorNz[-1]
# Go from topo to actv cells
topo = np.c_[utils.mkvc(xx), utils.mkvc(yy), utils.mkvc(zz)]
actv = utils.surface2ind_topo(self.mesh, topo, "N")
actv = np.where(actv)[0]
# Create active map to go from reduce space to full
self.actvMap = maps.InjectActiveCells(self.mesh, actv, -100)
nC = len(actv)
# Create and array of observation points
xr = np.linspace(-20.0, 20.0, 20)
yr = np.linspace(-20.0, 20.0, 20)
X, Y = np.meshgrid(xr, yr)
# Move the observation points 5m above the topo
Z = -np.exp((X**2 + Y**2) / 75**2) + self.mesh.vectorNz[-1] + 5.0
# Create a MAGsurvey
rxLoc = np.c_[utils.mkvc(X.T), utils.mkvc(Y.T), utils.mkvc(Z.T)]
rxLoc = mag.Point(rxLoc)
srcField = mag.SourceField([rxLoc], parameters=H0)
survey = mag.Survey(srcField)
# We can now create a susceptibility model and generate data
# Here a simple block in half-space
model = np.zeros((self.mesh.nCx, self.mesh.nCy, self.mesh.nCz))
model[(midx - 2):(midx + 2), (midy - 2):(midy + 2), -6:-2] = 0.02
model = utils.mkvc(model)
self.model = model[actv]
# Create active map to go from reduce set to full
self.actvMap = maps.InjectActiveCells(self.mesh, actv, -100)
# Creat reduced identity map
idenMap = maps.IdentityMap(nP=nC)
# Create the forward model operator
sim = mag.Simulation3DIntegral(
self.mesh,
survey=survey,
chiMap=idenMap,
actInd=actv,
store_sensitivities="disk",
)
self.sim = sim
# Compute linear forward operator and compute some data
data = sim.make_synthetic_data(self.model,
relative_error=0.0,
noise_floor=1.0,
add_noise=True)
# Create a regularization
reg = regularization.Sparse(self.mesh, indActive=actv, mapping=idenMap)
reg.norms = np.c_[0, 0, 0, 0]
reg.gradientType = "component"
# reg.eps_p, reg.eps_q = 1e-3, 1e-3
# Data misfit function
dmis = data_misfit.L2DataMisfit(simulation=sim, data=data)
# dmis.W = 1/wd
# Add directives to the inversion
opt = optimization.ProjectedGNCG(maxIter=100,
lower=0.0,
upper=1.0,
maxIterLS=20,
maxIterCG=10,
tolCG=1e-3)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
betaest = directives.BetaEstimate_ByEig()
# Here is where the norms are applied
IRLS = directives.Update_IRLS(f_min_change=1e-4, minGNiter=1)
update_Jacobi = directives.UpdatePreconditioner()
sensitivity_weights = directives.UpdateSensitivityWeights(
everyIter=False)
self.inv = inversion.BaseInversion(
invProb,
directiveList=[IRLS, sensitivity_weights, betaest, update_Jacobi])
def run(plotIt=True):
M = discretize.TensorMesh([np.ones(40)], x0="N")
M.setCellGradBC("dirichlet")
# We will use the haverkamp empirical model with parameters from Celia1990
k_fun, theta_fun = richards.empirical.haverkamp(
M,
A=1.1750e06,
gamma=4.74,
alpha=1.6110e06,
theta_s=0.287,
theta_r=0.075,
beta=3.96,
)
# Here we are making saturated hydraulic conductivity
# an exponential mapping to the model (defined below)
k_fun.KsMap = maps.ExpMap(nP=M.nC)
# Setup the boundary and initial conditions
bc = np.array([-61.5, -20.7])
h = np.zeros(M.nC) + bc[0]
prob = richards.SimulationNDCellCentered(
M,
hydraulic_conductivity=k_fun,
water_retention=theta_fun,
boundary_conditions=bc,
initial_conditions=h,
do_newton=False,
method="mixed",
debug=False,
)
prob.time_steps = [(5, 25, 1.1), (60, 40)]
# Create the survey
locs = -np.arange(2, 38, 4.0).reshape(-1, 1)
times = np.arange(30, prob.time_mesh.vectorCCx[-1], 60)
rxSat = richards.receivers.Saturation(locs, times)
survey = richards.Survey([rxSat])
prob.survey = survey
# Create a simple model for Ks
Ks = 1e-3
mtrue = np.ones(M.nC) * np.log(Ks)
mtrue[15:20] = np.log(5e-2)
mtrue[20:35] = np.log(3e-3)
mtrue[35:40] = np.log(1e-2)
m0 = np.ones(M.nC) * np.log(Ks)
# Create some synthetic data and fields
relative = 0.02 # The standard deviation for the noise
Hs = prob.fields(mtrue)
data = prob.make_synthetic_data(mtrue,
relative_error=relative,
f=Hs,
add_noise=True)
# Setup a pretty standard inversion
reg = regularization.Tikhonov(M, alpha_s=1e-1)
dmis = data_misfit.L2DataMisfit(simulation=prob, data=data)
opt = optimization.InexactGaussNewton(maxIter=20, maxIterCG=10)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
beta = directives.BetaSchedule(coolingFactor=4)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1e2)
target = directives.TargetMisfit()
dir_list = [beta, betaest, target]
inv = inversion.BaseInversion(invProb, directiveList=dir_list)
mopt = inv.run(m0)
Hs_opt = prob.fields(mopt)
if plotIt:
plt.figure(figsize=(14, 9))
ax = plt.subplot(121)
plt.semilogx(np.exp(np.c_[mopt, mtrue]), M.gridCC)
plt.xlabel("Saturated Hydraulic Conductivity, $K_s$")
plt.ylabel("Depth, cm")
plt.semilogx([10**-3.9] * len(locs), locs, "ro")
plt.legend(("$m_{rec}$", "$m_{true}$", "Data locations"), loc=4)
ax = plt.subplot(222)
mesh2d = discretize.TensorMesh([prob.time_mesh.hx / 60, prob.mesh.hx],
"0N")
sats = [theta_fun(_) for _ in Hs]
clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax)
cmap0 = matplotlib.cm.RdYlBu_r
clr[0].set_cmap(cmap0)
c = plt.colorbar(clr[0])
c.set_label("Saturation $\\theta$")
plt.xlabel("Time, minutes")
plt.ylabel("Depth, cm")
plt.title("True saturation over time")
ax = plt.subplot(224)
mesh2d = discretize.TensorMesh([prob.time_mesh.hx / 60, prob.mesh.hx],
"0N")
sats = [theta_fun(_) for _ in Hs_opt]
clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax)
cmap0 = matplotlib.cm.RdYlBu_r
clr[0].set_cmap(cmap0)
c = plt.colorbar(clr[0])
c.set_label("Saturation $\\theta$")
plt.xlabel("Time, minutes")
plt.ylabel("Depth, cm")
plt.title("Recovered saturation over time")
plt.tight_layout()
def run(plotIt=True):
H0 = (50000.0, 90.0, 0.0)
# Create a mesh
dx = 5.0
hxind = [(dx, 5, -1.3), (dx, 10), (dx, 5, 1.3)]
hyind = [(dx, 5, -1.3), (dx, 10), (dx, 5, 1.3)]
hzind = [(dx, 5, -1.3), (dx, 10)]
mesh = discretize.TensorMesh([hxind, hyind, hzind], "CCC")
# Lets create a simple Gaussian topo and set the active cells
[xx, yy] = np.meshgrid(mesh.vectorNx, mesh.vectorNy)
zz = -np.exp((xx ** 2 + yy ** 2) / 75 ** 2) + mesh.vectorNz[-1]
# We would usually load a topofile
topo = np.c_[utils.mkvc(xx), utils.mkvc(yy), utils.mkvc(zz)]
# Go from topo to array of indices of active cells
actv = utils.surface2ind_topo(mesh, topo, "N")
actv = np.where(actv)[0]
# Create and array of observation points
xr = np.linspace(-20.0, 20.0, 20)
yr = np.linspace(-20.0, 20.0, 20)
X, Y = np.meshgrid(xr, yr)
# Move the observation points 5m above the topo
Z = -np.exp((X ** 2 + Y ** 2) / 75 ** 2) + mesh.vectorNz[-1] + 5.0
# Create a MAGsurvey
rxLoc = np.c_[utils.mkvc(X.T), utils.mkvc(Y.T), utils.mkvc(Z.T)]
rxLoc = magnetics.Point(rxLoc)
srcField = magnetics.SourceField([rxLoc], parameters=H0)
survey = magnetics.Survey(srcField)
# We can now create a susceptibility model and generate data
model = np.zeros(mesh.nC)
# Change values in half the domain
model[mesh.gridCC[:, 0] < 0] = 0.01
# Add a block in half-space
model = utils.model_builder.addBlock(
mesh.gridCC, model, np.r_[-10, -10, 20], np.r_[10, 10, 40], 0.05
)
model = utils.mkvc(model)
model = model[actv]
# Create active map to go from reduce set to full
actvMap = maps.InjectActiveCells(mesh, actv, np.nan)
# Create reduced identity map
idenMap = maps.IdentityMap(nP=len(actv))
# Create the forward model operator
prob = magnetics.Simulation3DIntegral(
mesh,
survey=survey,
chiMap=idenMap,
actInd=actv,
store_sensitivities="forward_only",
)
# Compute linear forward operator and compute some data
data = prob.make_synthetic_data(
model, relative_error=0.0, noise_floor=1, add_noise=True
)
# Create a homogenous maps for the two domains
domains = [mesh.gridCC[actv, 0] < 0, mesh.gridCC[actv, 0] >= 0]
homogMap = maps.SurjectUnits(domains)
# Create a wire map for a second model space, voxel based
wires = maps.Wires(("h**o", len(domains)), ("hetero", len(actv)))
# Create Sum map
sumMap = maps.SumMap([homogMap * wires.h**o, wires.hetero])
# Create the forward model operator
prob = magnetics.Simulation3DIntegral(
mesh, survey=survey, chiMap=sumMap, actInd=actv, store_sensitivities="ram"
)
# Make depth weighting
wr = np.zeros(sumMap.shape[1])
print(prob.nC)
# print(prob.M.shape) # why does this reset nC
G = prob.G
# Take the cell number out of the scaling.
# Want to keep high sens for large volumes
scale = utils.sdiag(
np.r_[utils.mkvc(1.0 / homogMap.P.sum(axis=0)), np.ones_like(actv)]
)
for ii in range(survey.nD):
wr += (
(prob.G[ii, :] * prob.chiMap.deriv(np.ones(sumMap.shape[1]) * 1e-4) * scale)
/ data.standard_deviation[ii]
) ** 2.0
# Scale the model spaces independently
wr[wires.h**o.index] /= np.max((wires.h**o * wr))
wr[wires.hetero.index] /= np.max(wires.hetero * wr)
wr = wr ** 0.5
## Create a regularization
# For the homogeneous model
regMesh = discretize.TensorMesh([len(domains)])
reg_m1 = regularization.Sparse(regMesh, mapping=wires.h**o)
reg_m1.cell_weights = wires.h**o * wr
reg_m1.norms = np.c_[0, 2, 2, 2]
reg_m1.mref = np.zeros(sumMap.shape[1])
# Regularization for the voxel model
reg_m2 = regularization.Sparse(mesh, indActive=actv, mapping=wires.hetero)
reg_m2.cell_weights = wires.hetero * wr
reg_m2.norms = np.c_[0, 1, 1, 1]
reg_m2.mref = np.zeros(sumMap.shape[1])
reg = reg_m1 + reg_m2
# Data misfit function
dmis = data_misfit.L2DataMisfit(simulation=prob, data=data)
# Add directives to the inversion
opt = optimization.ProjectedGNCG(
maxIter=100,
lower=0.0,
upper=1.0,
maxIterLS=20,
maxIterCG=10,
tolCG=1e-3,
tolG=1e-3,
eps=1e-6,
)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
betaest = directives.BetaEstimate_ByEig()
# Here is where the norms are applied
# Use pick a threshold parameter empirically based on the distribution of
# model parameters
IRLS = directives.Update_IRLS(f_min_change=1e-3, minGNiter=1)
update_Jacobi = directives.UpdatePreconditioner()
inv = inversion.BaseInversion(invProb, directiveList=[IRLS, betaest, update_Jacobi])
# Run the inversion
m0 = np.ones(sumMap.shape[1]) * 1e-4 # Starting model
prob.model = m0
mrecSum = inv.run(m0)
if plotIt:
mesh.plot_3d_slicer(
actvMap * model,
aspect="equal",
zslice=30,
pcolorOpts={"cmap": "inferno_r"},
transparent="slider",
)
mesh.plot_3d_slicer(
actvMap * sumMap * mrecSum,
aspect="equal",
zslice=30,
pcolorOpts={"cmap": "inferno_r"},
transparent="slider",
)
invProb = inverse_problem.BaseInvProblem(dmis, reg_simple, opt)
# directives
scales = directives.ScalingMultipleDataMisfits_ByEig(chi0_ratio=np.r_[1.0,
1.0],
verbose=True,
n_pw_iter=10)
scaling_schedule = directives.JointScalingSchedule(verbose=True)
alpha0_ratio = np.r_[np.zeros(len(reg_simple.objfcts[0].objfcts)),
100.0 * np.ones(len(reg_simple.objfcts[1].objfcts)),
1.0 * np.ones(len(reg_simple.objfcts[2].objfcts)), ]
alphas = directives.AlphasSmoothEstimate_ByEig(alpha0_ratio=alpha0_ratio,
n_pw_iter=10,
verbose=True)
beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-5, n_pw_iter=10)
betaIt = directives.PGI_BetaAlphaSchedule(
verbose=True,
coolingFactor=2.0,
progress=0.2,
)
targets = directives.MultiTargetMisfits(verbose=True)
petrodir = directives.PGI_UpdateParameters(update_gmm=False)
# Setup Inversion
inv = inversion.BaseInversion(
invProb,
directiveList=[
alphas, scales, beta, petrodir, targets, betaIt, scaling_schedule
],
)
def run_inversion(
self,
maxIter=60,
m0=0.0,
mref=0.0,
percentage=5,
floor=0.1,
chifact=1,
beta0_ratio=1.0,
coolingFactor=1,
n_iter_per_beta=1,
alpha_s=1.0,
alpha_x=1.0,
alpha_z=1.0,
use_target=False,
use_tikhonov=True,
use_irls=False,
p_s=2,
p_x=2,
p_y=2,
p_z=2,
beta_start=None,
):
self.uncertainty = percentage * abs(self.data_prop.dobs) * 0.01 + floor
m0 = np.ones(self.mesh_prop.nC) * m0
mref = np.ones(self.mesh_prop.nC) * mref
if ~use_tikhonov:
reg = regularization.Sparse(
self.mesh_prop,
alpha_s=alpha_s,
alpha_x=alpha_x,
alpha_y=alpha_z,
mref=mref,
mapping=maps.IdentityMap(self.mesh_prop),
cell_weights=self.mesh_prop.vol,
)
else:
reg = regularization.Tikhonov(
self.mesh_prop,
alpha_s=alpha_s,
alpha_x=alpha_x,
alpha_y=alpha_z,
mref=mref,
mapping=maps.IdentityMap(self.mesh_prop),
)
dataObj = data.Data(self.survey_prop,
dobs=self.dobs,
noise_floor=self.uncertainty)
dmis = data_misfit.L2DataMisfit(simulation=self.simulation_prop,
data=dataObj)
dmis.W = 1.0 / self.uncertainty
opt = optimization.ProjectedGNCG(maxIter=maxIter, maxIterCG=20)
opt.lower = 0.0
opt.remember("xc")
opt.tolG = 1e-10
opt.eps = 1e-10
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
beta_schedule = directives.BetaSchedule(coolingFactor=coolingFactor,
coolingRate=n_iter_per_beta)
save = directives.SaveOutputEveryIteration()
print(chifact)
if use_irls:
IRLS = directives.Update_IRLS(
f_min_change=1e-4,
minGNiter=1,
silent=False,
max_irls_iterations=40,
beta_tol=5e-1,
coolEpsFact=1.3,
chifact_start=chifact,
)
if beta_start is None:
directives_list = [
directives.BetaEstimate_ByEig(beta0_ratio=beta0_ratio),
IRLS,
save,
]
else:
directives_list = [IRLS, save]
invProb.beta = beta_start
reg.norms = np.c_[p_s, p_x, p_z, 2]
else:
target = directives.TargetMisfit(chifact=chifact)
directives_list = [
directives.BetaEstimate_ByEig(beta0_ratio=beta0_ratio),
beta_schedule,
save,
]
if use_target:
directives_list.append(target)
inv = inversion.BaseInversion(invProb, directiveList=directives_list)
mopt = inv.run(m0)
model = opt.recall("xc")
model.append(mopt)
pred = []
for m in model:
pred.append(self.simulation_prop.dpred(m))
return model, pred, save
idenMap = maps.IdentityMap(nP=nC)
# Create a regularization
reg = regularization.Sparse(mesh, indActive=activeCells, mapping=idenMap)
m0 = np.ones(nC) * 1e-4 # Starting model
# Add directives to the inversion
opt = optimization.ProjectedGNCG(maxIter=100,
lower=-1.0,
upper=1.0,
maxIterLS=20,
maxIterCG=10,
tolCG=1e-3)
invProb = inverse_problem.BaseInvProblem(global_misfit, reg, opt)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=1e-1)
# Here is where the norms are applied
# Use pick a threshold parameter empirically based on the distribution of
# model parameters
update_IRLS = directives.Update_IRLS(
f_min_change=1e-4,
max_irls_iterations=0,
coolEpsFact=1.5,
beta_tol=1e-2,
)
saveDict = directives.SaveOutputEveryIteration(save_txt=False)
update_Jacobi = directives.UpdatePreconditioner()
sensitivity_weights = directives.UpdateSensitivityWeights(everyIter=False)
inv = inversion.BaseInversion(
invProb,
# Here we define the inverse problem that is to be solved inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt) ####################################################################### # Define Inversion Directives # --------------------------- # # Here we define any directiveas that are carried out during the inversion. This # includes the cooling schedule for the trade-off parameter (beta), stopping # criteria for the inversion and saving inversion results at each iteration. # # Defining a starting value for the trade-off parameter (beta) between the data # misfit and the regularization. starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=10) # Defining the fractional decrease in beta and the number of Gauss-Newton solves # for each beta value. beta_schedule = directives.BetaSchedule(coolingFactor=10, coolingRate=3) # Options for outputting recovered models and predicted data for each beta. save_iteration = directives.SaveOutputEveryIteration(save_txt=False) # Setting a stopping criteria for the inversion. target_misfit = directives.TargetMisfit(chifact=1) # The directives are defined as a list. directives_list = [starting_beta, beta_schedule, save_iteration, target_misfit] #####################################################################
def run_inversion(
m0,
simulation,
data,
actind,
mesh,
maxIter=15,
beta0_ratio=1e0,
coolingFactor=5,
coolingRate=2,
upper=np.inf,
lower=-np.inf,
use_sensitivity_weight=False,
alpha_s=1e-4,
alpha_x=1.0,
alpha_y=1.0,
alpha_z=1.0,
):
"""
Run IP inversion
"""
dmisfit = data_misfit.L2DataMisfit(simulation=simulation, data=data)
# Map for a regularization
regmap = maps.IdentityMap(nP=int(actind.sum()))
# Related to inversion
if use_sensitivity_weight:
reg = regularization.Sparse(mesh, indActive=actind, mapping=regmap)
reg.alpha_s = alpha_s
reg.alpha_x = alpha_x
reg.alpha_y = alpha_y
reg.alpha_z = alpha_z
else:
reg = regularization.Sparse(mesh,
indActive=actind,
mapping=regmap,
cell_weights=mesh.vol[actind])
reg.alpha_s = alpha_s
reg.alpha_x = alpha_x
reg.alpha_y = alpha_y
reg.alpha_z = alpha_z
opt = optimization.ProjectedGNCG(maxIter=maxIter, upper=upper, lower=lower)
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
beta = directives.BetaSchedule(coolingFactor=coolingFactor,
coolingRate=coolingRate)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=beta0_ratio)
target = directives.TargetMisfit()
# Need to have basice saving function
if use_sensitivity_weight:
updateSensW = directives.UpdateSensitivityWeights()
update_Jacobi = directives.UpdatePreconditioner()
directiveList = [beta, betaest, target, update_Jacobi]
else:
directiveList = [beta, betaest, target]
inv = inversion.BaseInversion(invProb, directiveList=directiveList)
opt.LSshorten = 0.5
opt.remember("xc")
# Run inversion
mopt = inv.run(m0)
return mopt, invProb.dpred
