def test_update_of_sparse_norms(self):
mesh = discretize.TensorMesh([8, 7, 6])
m = np.random.rand(mesh.nC)
v = np.random.rand(mesh.nC)
cell_weights = np.random.rand(mesh.nC)
reg = regularization.Sparse(mesh, cell_weights=cell_weights)
reg.norms = np.c_[2.0, 2.0, 2.0, 2.0]
self.assertTrue(
np.all(reg.norms == np.kron(np.ones((reg.regmesh.Pac.shape[1],
1)), np.c_[2.0, 2.0, 2.0,
2.0])))
self.assertTrue(np.all(reg.objfcts[0].norm == 2.0 * np.ones(mesh.nC)))
self.assertTrue(np.all(reg.objfcts[1].norm == 2.0 * np.ones(mesh.nFx)))
self.assertTrue(np.all(reg.objfcts[2].norm == 2.0 * np.ones(mesh.nFy)))
self.assertTrue(np.all(reg.objfcts[3].norm == 2.0 * np.ones(mesh.nFz)))
reg.norms = np.c_[0.0, 1.0, 1.0, 1.0]
self.assertTrue(
np.all(reg.norms == np.kron(np.ones((reg.regmesh.Pac.shape[1],
1)), np.c_[0.0, 1.0, 1.0,
1.0])))
self.assertTrue(np.all(reg.objfcts[0].norm == 0.0 * np.ones(mesh.nC)))
self.assertTrue(np.all(reg.objfcts[1].norm == 1.0 * np.ones(mesh.nFx)))
self.assertTrue(np.all(reg.objfcts[2].norm == 1.0 * np.ones(mesh.nFy)))
self.assertTrue(np.all(reg.objfcts[3].norm == 1.0 * np.ones(mesh.nFz)))
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=True,
alpha_s=1e-4,
alpha_x=1.0,
alpha_y=1.0,
alpha_z=1.0,
):
"""
Run DC 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.Tikhonov(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
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
update_Jacobi = directives.UpdatePreconditioner()
if use_sensitivity_weight:
updateSensW = directives.UpdateSensitivityWeights()
directiveList = [beta, target, updateSensW, update_Jacobi, betaest]
else:
directiveList = [beta, target, update_Jacobi, betaest]
inv = inversion.BaseInversion(invProb, directiveList=directiveList)
opt.LSshorten = 0.5
opt.remember("xc")
# Run inversion
mopt = inv.run(m0)
return mopt, invProb.dpred
def setUp(self):
mesh = discretize.TensorMesh([4, 4, 4])
# Magnetic inducing field parameter (A,I,D)
B = [50000, 90, 0]
# Create a MAGsurvey
rx = mag.Point(np.vstack([[0.25, 0.25, 0.25], [-0.25, -0.25, 0.25]]))
srcField = mag.SourceField([rx], parameters=(B[0], B[1], B[2]))
survey = mag.Survey(srcField)
# Create the forward model operator
sim = mag.Simulation3DIntegral(mesh,
survey=survey,
chiMap=maps.IdentityMap(mesh))
# Compute forward model some data
m = np.random.rand(mesh.nC)
data = sim.make_synthetic_data(m, add_noise=True)
reg = regularization.Sparse(mesh)
reg.mref = np.zeros(mesh.nC)
reg.norms = np.c_[0, 1, 1, 1]
reg.eps_p, reg.eps_q = 1e-3, 1e-3
# Data misfit function
dmis = data_misfit.L2DataMisfit(data)
dmis.W = 1.0 / data.relative_error
# Add directives to the inversion
opt = optimization.ProjectedGNCG(maxIter=2,
lower=-10.0,
upper=10.0,
maxIterCG=2)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
self.mesh = mesh
self.invProb = invProb
self.sim = sim
# 3) Optimization: the numerical approach used to solve the inverse problem
#
# Define the data misfit. Here the data misfit is the L2 norm of the weighted
# residual between the observed data and the data predicted for a given model.
# Within the data misfit, the residual between predicted and observed data are
# normalized by the data's standard deviation.
dmis = data_misfit.L2DataMisfit(data=data_object, simulation=simulation)
# Define the regularization (model objective function)
reg = regularization.Sparse(
mesh,
indActive=ind_active,
mapping=model_map,
mref=starting_model,
gradientType="total",
alpha_s=1,
alpha_x=1,
alpha_y=1,
alpha_z=1,
)
# Define sparse and blocky norms p, qx, qy, qz
reg.norms = np.c_[0, 2, 2, 2]
# Define how the optimization problem is solved. Here we will use a projected
# Gauss-Newton approach that employs the conjugate gradient solver.
opt = optimization.ProjectedGNCG(maxIter=10,
lower=0.0,
upper=1.0,
maxIterLS=20,
def setUp(self):
ndv = -100
# Create a self.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
locXYZ = np.c_[utils.mkvc(X.T), utils.mkvc(Y.T), utils.mkvc(Z.T)]
rxLoc = gravity.Point(locXYZ)
srcField = gravity.SourceField([rxLoc])
survey = gravity.Survey(srcField)
# We can now create a density 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] = 0.5
model = utils.mkvc(model)
self.model = model[actv]
# Create active map to go from reduce set to full
actvMap = maps.InjectActiveCells(self.mesh, actv, ndv)
# Create reduced identity map
idenMap = maps.IdentityMap(nP=nC)
# Create the forward model operator
sim = gravity.Simulation3DIntegral(
self.mesh,
survey=survey,
rhoMap=idenMap,
actInd=actv,
store_sensitivities="ram",
)
# Compute linear forward operator and compute some data
# computing sensitivities to ram is best using dask processes
with dask.config.set(scheduler="processes"):
data = sim.make_synthetic_data(self.model,
relative_error=0.0,
noise_floor=0.001,
add_noise=True)
print(sim.G)
# 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 = 5e-2, 1e-2
# Data misfit function
dmis = data_misfit.L2DataMisfit(simulation=sim, data=data)
# 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(dmis, reg, opt, beta=1e8)
# 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, update_Jacobi])
self.sim = sim
ax.set_aspect("equal")
plt.show()
#####################################################
# Invert on the global mesh
#
#
#
#
#
# Create reduced identity map
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 a threshold parameter empirically based on the distribution of
def run(plotIt=True):
# Define the inducing field parameter
H0 = (50000, 90, 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 = TensorMesh([hxind, hyind, hzind], "CCC")
# Get index of the center
midx = int(mesh.nCx / 2)
midy = int(mesh.nCy / 2)
# 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]
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) + 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.receivers.Point(rxLoc, components=["tmi"])
srcField = magnetics.sources.SourceField(receiver_list=[rxLoc],
parameters=H0)
survey = magnetics.survey.Survey(srcField)
# We can now create a susceptibility model and generate data
# Here a simple block in half-space
model = np.zeros((mesh.nCx, mesh.nCy, mesh.nCz))
model[(midx - 2):(midx + 2), (midy - 2):(midy + 2), -6:-2] = 0.02
model = utils.mkvc(model)
model = model[actv]
# Create active map to go from reduce set to full
actvMap = maps.InjectActiveCells(mesh, actv, -100)
# Create reduced identity map
idenMap = maps.IdentityMap(nP=nC)
# Create the forward model operator
simulation = magnetics.simulation.Simulation3DIntegral(
survey=survey,
mesh=mesh,
chiMap=idenMap,
actInd=actv,
)
# Compute linear forward operator and compute some data
d = simulation.dpred(model)
# Add noise and uncertainties
# We add some random Gaussian noise (1nT)
synthetic_data = d + np.random.randn(len(d))
wd = np.ones(len(synthetic_data)) * 1.0 # Assign flat uncertainties
data_object = data.Data(survey, dobs=synthetic_data, noise_floor=wd)
# Create a regularization
reg = regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
reg.mref = np.zeros(nC)
reg.norms = np.c_[0, 0, 0, 0]
# reg.eps_p, reg.eps_q = 1e-0, 1e-0
# Create sensitivity weights from our linear forward operator
rxLoc = survey.source_field.receiver_list[0].locations
m0 = np.ones(nC) * 1e-4 # Starting model
# Data misfit function
dmis = data_misfit.L2DataMisfit(simulation=simulation, data=data_object)
dmis.W = 1 / wd
# Add directives to the inversion
opt = optimization.ProjectedGNCG(maxIter=20,
lower=0.0,
upper=1.0,
maxIterLS=20,
maxIterCG=20,
tolCG=1e-3)
invProb = inverse_problem.BaseInvProblem(dmis, 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
IRLS = directives.Update_IRLS(f_min_change=1e-3, max_irls_iterations=40)
saveDict = directives.SaveOutputEveryIteration(save_txt=False)
update_Jacobi = directives.UpdatePreconditioner()
# Add sensitivity weights
sensitivity_weights = directives.UpdateSensitivityWeights(everyIter=False)
inv = inversion.BaseInversion(
invProb,
directiveList=[
sensitivity_weights, IRLS, betaest, update_Jacobi, saveDict
],
)
# Run the inversion
mrec = inv.run(m0)
if plotIt:
# Here is the recovered susceptibility model
ypanel = midx
zpanel = -5
m_l2 = actvMap * invProb.l2model
m_l2[m_l2 == -100] = np.nan
m_lp = actvMap * mrec
m_lp[m_lp == -100] = np.nan
m_true = actvMap * model
m_true[m_true == -100] = np.nan
# Plot the data
utils.plot_utils.plot2Ddata(rxLoc, d)
plt.figure()
# Plot L2 model
ax = plt.subplot(321)
mesh.plotSlice(
m_l2,
ax=ax,
normal="Z",
ind=zpanel,
grid=True,
clim=(model.min(), model.max()),
)
plt.plot(
([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
([mesh.vectorCCy[ypanel], mesh.vectorCCy[ypanel]]),
color="w",
)
plt.title("Plan l2-model.")
plt.gca().set_aspect("equal")
plt.ylabel("y")
ax.xaxis.set_visible(False)
plt.gca().set_aspect("equal", adjustable="box")
# Vertica section
ax = plt.subplot(322)
mesh.plotSlice(
m_l2,
ax=ax,
normal="Y",
ind=midx,
grid=True,
clim=(model.min(), model.max()),
)
plt.plot(
([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
([mesh.vectorCCz[zpanel], mesh.vectorCCz[zpanel]]),
color="w",
)
plt.title("E-W l2-model.")
plt.gca().set_aspect("equal")
ax.xaxis.set_visible(False)
plt.ylabel("z")
plt.gca().set_aspect("equal", adjustable="box")
# Plot Lp model
ax = plt.subplot(323)
mesh.plotSlice(
m_lp,
ax=ax,
normal="Z",
ind=zpanel,
grid=True,
clim=(model.min(), model.max()),
)
plt.plot(
([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
([mesh.vectorCCy[ypanel], mesh.vectorCCy[ypanel]]),
color="w",
)
plt.title("Plan lp-model.")
plt.gca().set_aspect("equal")
ax.xaxis.set_visible(False)
plt.ylabel("y")
plt.gca().set_aspect("equal", adjustable="box")
# Vertical section
ax = plt.subplot(324)
mesh.plotSlice(
m_lp,
ax=ax,
normal="Y",
ind=midx,
grid=True,
clim=(model.min(), model.max()),
)
plt.plot(
([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
([mesh.vectorCCz[zpanel], mesh.vectorCCz[zpanel]]),
color="w",
)
plt.title("E-W lp-model.")
plt.gca().set_aspect("equal")
ax.xaxis.set_visible(False)
plt.ylabel("z")
plt.gca().set_aspect("equal", adjustable="box")
# Plot True model
ax = plt.subplot(325)
mesh.plotSlice(
m_true,
ax=ax,
normal="Z",
ind=zpanel,
grid=True,
clim=(model.min(), model.max()),
)
plt.plot(
([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
([mesh.vectorCCy[ypanel], mesh.vectorCCy[ypanel]]),
color="w",
)
plt.title("Plan true model.")
plt.gca().set_aspect("equal")
plt.xlabel("x")
plt.ylabel("y")
plt.gca().set_aspect("equal", adjustable="box")
# Vertical section
ax = plt.subplot(326)
mesh.plotSlice(
m_true,
ax=ax,
normal="Y",
ind=midx,
grid=True,
clim=(model.min(), model.max()),
)
plt.plot(
([mesh.vectorCCx[0], mesh.vectorCCx[-1]]),
([mesh.vectorCCz[zpanel], mesh.vectorCCz[zpanel]]),
color="w",
)
plt.title("E-W true model.")
plt.gca().set_aspect("equal")
plt.xlabel("x")
plt.ylabel("z")
plt.gca().set_aspect("equal", adjustable="box")
# Plot convergence curves
fig, axs = plt.figure(), plt.subplot()
axs.plot(saveDict.phi_d, "k", lw=2)
axs.plot(
np.r_[IRLS.iterStart, IRLS.iterStart],
np.r_[0, np.max(saveDict.phi_d)],
"k:",
)
twin = axs.twinx()
twin.plot(saveDict.phi_m, "k--", lw=2)
axs.text(
IRLS.iterStart,
0,
"IRLS Steps",
va="bottom",
ha="center",
rotation="vertical",
size=12,
bbox={"facecolor": "white"},
)
axs.set_ylabel("$\phi_d$", size=16, rotation=0)
axs.set_xlabel("Iterations", size=14)
twin.set_ylabel("$\phi_m$", size=16, rotation=0)
idenMap = maps.IdentityMap(nP=nC)
# Create static map
simulation = magnetics.simulation.Simulation3DIntegral(
mesh=mesh,
survey=survey,
chiMap=idenMap,
actInd=surf,
store_sensitivities="ram")
wr = simulation.getJtJdiag(mstart)**0.5
wr = wr / np.max(np.abs(wr))
# 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=20,
lower=-np.inf,
upper=np.inf,
maxIterLS=20,
maxIterCG=20,
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
# Define the data misfit. Here the data misfit is the L2 norm of the weighted
# residual between the observed data and the data predicted for a given model.
# Within the data misfit, the residual between predicted and observed data are
# normalized by the data's standard deviation.
dmis = data_misfit.L2DataMisfit(data=dc_data, simulation=simulation)
# Define the regularization (model objective function). Here, 'p' defines the
# the norm of the smallness term, 'qx' defines the norm of the smoothness
# in x and 'qz' defines the norm of the smoothness in z.
regmap = maps.IdentityMap(nP=int(ind_active.sum()))
reg = regularization.Sparse(
mesh,
indActive=ind_active,
mref=starting_conductivity_model,
mapping=regmap,
gradientType="components",
alpha_s=0.01,
alpha_x=1,
alpha_y=1,
)
p = 0
qx = 1
qz = 1
reg.norms = np.c_[p, qx, qz]
# Define how the optimization problem is solved. Here we will use an inexact
# Gauss-Newton approach.
opt = optimization.InexactGaussNewton(maxIter=40)
# Here we define the inverse problem that is to be solved
#
# 1) Data Misfit: a measure of how well our recovered model explains the field data
# 2) Regularization: constraints placed on the recovered model and a priori information
# 3) Optimization: the numerical approach used to solve the inverse problem
#
# Define the data misfit. Here the data misfit is the L2 norm of the weighted
# residual between the observed data and the data predicted for a given model.
# Within the data misfit, the residual between predicted and observed data are
# normalized by the data's standard deviation.
dmis = data_misfit.L2DataMisfit(simulation=sim, data=data_obj)
# Define the regularization (model objective function). Here, 'p' defines the
# the norm of the smallness term and 'q' defines the norm of the smoothness
# term.
reg = regularization.Sparse(mesh, mapping=model_map)
reg.mref = np.zeros(nParam)
p = 0.0
q = 0.0
reg.norms = np.c_[p, q]
# Define how the optimization problem is solved.
opt = optimization.ProjectedGNCG(maxIter=100,
lower=-2.0,
upper=2.0,
maxIterLS=20,
maxIterCG=30,
tolCG=1e-4)
# Here we define the inverse problem that is to be solved
inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt)
def setUp(self):
np.random.seed(0)
# First we need to define the direction of the inducing field
# As a simple case, we pick a vertical inducing field of magnitude
# 50,000nT.
# From old convention, field orientation is given as an
# azimuth from North (positive clockwise)
# and dip from the horizontal (positive downward).
H0 = (50000.0, 90.0, 0.0)
# Create a mesh
h = [5, 5, 5]
padDist = np.ones((3, 2)) * 100
nCpad = [2, 4, 2]
# Create grid of points for topography
# Lets create a simple Gaussian topo and set the active cells
[xx, yy] = np.meshgrid(np.linspace(-200.0, 200.0, 50),
np.linspace(-200.0, 200.0, 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)
# self.mesh.finalize()
self.mesh = meshutils.mesh_builder_xyz(
xyzLoc,
h,
padding_distance=padDist,
mesh_type="TREE",
)
self.mesh = meshutils.refine_tree_xyz(
self.mesh,
topo,
method="surface",
octree_levels=nCpad,
octree_levels_padding=nCpad,
finalize=True,
)
# Define an active cells from topo
actv = utils.surface2ind_topo(self.mesh, topo)
nC = int(actv.sum())
# We can now create a susceptibility model and generate data
# Lets start with a simple block in half-space
self.model = utils.model_builder.addBlock(
self.mesh.gridCC,
np.zeros(self.mesh.nC),
np.r_[-20, -20, -15],
np.r_[20, 20, 20],
0.05,
)[actv]
# Create active map to go from reduce set to full
self.actvMap = maps.InjectActiveCells(self.mesh, actv, np.nan)
# 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="ram",
)
self.sim = sim
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.mref = np.zeros(nC)
# Data misfit function
dmis = data_misfit.L2DataMisfit(simulation=sim, data=data)
# Add directives to the inversion
opt = optimization.ProjectedGNCG(
maxIter=10,
lower=0.0,
upper=10.0,
maxIterLS=5,
maxIterCG=5,
tolCG=1e-4,
stepOffBoundsFact=1e-4,
)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt, beta=1e6)
# 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=20,
beta_tol=1e-1,
beta_search=False)
update_Jacobi = directives.UpdatePreconditioner()
sensitivity_weights = directives.UpdateSensitivityWeights()
self.inv = inversion.BaseInversion(
invProb, directiveList=[IRLS, sensitivity_weights, update_Jacobi])
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,
model_type="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_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
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 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 setUp(self):
cs = 25.0
hx = [(cs, 0, -1.3), (cs, 21), (cs, 0, 1.3)]
hy = [(cs, 0, -1.3), (cs, 21), (cs, 0, 1.3)]
hz = [(cs, 0, -1.3), (cs, 20), (cs, 0, 1.3)]
mesh = discretize.TensorMesh([hx, hy, hz], x0="CCC")
blkind0 = utils.model_builder.getIndicesSphere(
np.r_[-100.0, -100.0, -200.0], 75.0, mesh.gridCC)
blkind1 = utils.model_builder.getIndicesSphere(
np.r_[100.0, 100.0, -200.0], 75.0, mesh.gridCC)
sigma = np.ones(mesh.nC) * 1e-2
airind = mesh.gridCC[:, 2] > 0.0
sigma[airind] = 1e-8
eta = np.zeros(mesh.nC)
tau = np.ones_like(sigma) * 1.0
c = np.ones_like(sigma) * 0.5
eta[blkind0] = 0.1
eta[blkind1] = 0.1
tau[blkind0] = 0.1
tau[blkind1] = 0.01
actmapeta = maps.InjectActiveCells(mesh, ~airind, 0.0)
actmaptau = maps.InjectActiveCells(mesh, ~airind, 1.0)
actmapc = maps.InjectActiveCells(mesh, ~airind, 1.0)
x = mesh.vectorCCx[(mesh.vectorCCx > -155.0)
& (mesh.vectorCCx < 155.0)]
y = mesh.vectorCCy[(mesh.vectorCCy > -155.0)
& (mesh.vectorCCy < 155.0)]
Aloc = np.r_[-200.0, 0.0, 0.0]
Bloc = np.r_[200.0, 0.0, 0.0]
M = utils.ndgrid(x - 25.0, y, np.r_[0.0])
N = utils.ndgrid(x + 25.0, y, np.r_[0.0])
times = np.arange(10) * 1e-3 + 1e-3
rx = sip.receivers.Dipole(M, N, times)
src = sip.sources.Dipole([rx], Aloc, Bloc)
survey = sip.Survey([src])
wires = maps.Wires(("eta", actmapeta.nP), ("taui", actmaptau.nP),
("c", actmapc.nP))
problem = sip.Simulation3DNodal(
mesh,
survey=survey,
sigma=sigma,
etaMap=actmapeta * wires.eta,
tauiMap=actmaptau * wires.taui,
cMap=actmapc * wires.c,
actinds=~airind,
storeJ=False,
verbose=False,
)
problem.solver = Solver
mSynth = np.r_[eta[~airind], 1.0 / tau[~airind], c[~airind]]
dobs = problem.make_synthetic_data(mSynth, add_noise=True)
# Now set up the problem to do some minimization
dmis = data_misfit.L2DataMisfit(data=dobs, simulation=problem)
reg_eta = regularization.Sparse(mesh,
mapping=wires.eta,
indActive=~airind)
reg_taui = regularization.Sparse(mesh,
mapping=wires.taui,
indActive=~airind)
reg_c = regularization.Sparse(mesh, mapping=wires.c, indActive=~airind)
reg = reg_eta + reg_taui + reg_c
opt = optimization.InexactGaussNewton(maxIterLS=20,
maxIter=10,
tolF=1e-6,
tolX=1e-6,
tolG=1e-6,
maxIterCG=6)
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt, beta=1e4)
inv = inversion.BaseInversion(invProb)
self.inv = inv
self.reg = reg
self.p = problem
self.mesh = mesh
self.m0 = mSynth
self.survey = survey
self.dmis = dmis
self.dobs = dobs
#
# 1) Data Misfit: a measure of how well our recovered model explains the field data
# 2) Regularization: constraints placed on the recovered model and a priori information
# 3) Optimization: the numerical approach used to solve the inverse problem
#
# Define the data misfit. Here the data misfit is the L2 norm of the weighted
# residual between the observed data and the data predicted for a given model.
# Within the data misfit, the residual between predicted and observed data are
# normalized by the data's standard deviation.
dmis = data_misfit.L2DataMisfit(data=data_obj, simulation=simulation)
# Define the regularization (model objective function). Here, 'p' defines the
# the norm of the smallness term and 'q' defines the norm of the smoothness
# term.
reg = regularization.Sparse(mesh, mapping=maps.IdentityMap(nP=mesh.nC))
p = 0
qx = 0.5
qy = 0.5
reg.norms = np.c_[p, qx, qy]
# Define how the optimization problem is solved.
opt = optimization.ProjectedGNCG(maxIter=100,
lower=0.0,
upper=1e6,
maxIterLS=20,
maxIterCG=10,
tolCG=1e-4)
# Here we define the inverse problem that is to be solved
inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt)
# solution using a Cartesian coordinate system, then a sparse
# inversion in the Spherical domain.
#
# Create sensitivity weights from our linear forward operator
rxLoc = survey.source_field.receiver_list[0].locations
# This Mapping connects the regularizations for the three-component
# vector model
wires = maps.Wires(("p", nC), ("s", nC), ("t", nC))
m0 = np.ones(3 * nC) * 1e-4 # Starting model
# Create three regularizations 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=simulation, data=data_object)
dmis.W = 1.0 / data_object.standard_deviation
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_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)
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()
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_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)
# 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()
# The inverse problem is defined by 3 things:
#
# 1) Data Misfit: a measure of how well our recovered model explains the field data
# 2) Regularization: constraints placed on the recovered model and a priori information
# 3) Optimization: the numerical approach used to solve the inverse problem
#
# Define the data misfit. Here the data misfit is the L2 norm of the weighted
# residual between the observed data and the data predicted for a given model.
# Within the data misfit, the residual between predicted and observed data are
# normalized by the data's standard deviation.
dmis = data_misfit.L2DataMisfit(data=data_object, simulation=simulation)
dmis.W = utils.sdiag(1 / uncertainties)
# Define the regularization (model objective function).
reg = regularization.Sparse(mesh, indActive=ind_active, mapping=model_map)
reg.norms = np.c_[0, 2, 2, 2]
# Define how the optimization problem is solved. Here we will use a projected
# Gauss-Newton approach that employs the conjugate gradient solver.
opt = optimization.ProjectedGNCG(maxIter=100,
lower=-1.0,
upper=1.0,
maxIterLS=20,
maxIterCG=10,
tolCG=1e-3)
# Here we define the inverse problem that is to be solved
inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt)
#######################################################################
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",
)
