def test_update_of_sparse_norms(self):
mesh = Mesh.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., 2., 2., 2.]
self.assertTrue(
np.all(reg.norms == np.kron(np.ones((reg.regmesh.Pac.shape[1],
1)), np.c_[2., 2., 2., 2.])))
self.assertTrue(np.all(reg.objfcts[0].norm == 2. * np.ones(mesh.nC)))
self.assertTrue(np.all(reg.objfcts[1].norm == 2. * np.ones(mesh.nFx)))
self.assertTrue(np.all(reg.objfcts[2].norm == 2. * np.ones(mesh.nFy)))
self.assertTrue(np.all(reg.objfcts[3].norm == 2. * np.ones(mesh.nFz)))
reg.norms = np.c_[0., 1., 1., 1.]
self.assertTrue(
np.all(reg.norms == np.kron(np.ones((reg.regmesh.Pac.shape[1],
1)), np.c_[0., 1., 1., 1.])))
self.assertTrue(np.all(reg.objfcts[0].norm == 0. * np.ones(mesh.nC)))
self.assertTrue(np.all(reg.objfcts[1].norm == 1. * np.ones(mesh.nFx)))
self.assertTrue(np.all(reg.objfcts[2].norm == 1. * np.ones(mesh.nFy)))
self.assertTrue(np.all(reg.objfcts[3].norm == 1. * np.ones(mesh.nFz)))
def setUp(self):
mesh = Mesh.TensorMesh([4, 4, 4])
# Magnetic inducing field parameter (A,I,D)
B = [50000, 90, 0]
# Create a MAGsurvey
rx = PF.BaseMag.RxObs(
np.vstack([[0.25, 0.25, 0.25], [-0.25, -0.25, 0.25]]))
srcField = PF.BaseMag.SrcField([rx], param=(B[0], B[1], B[2]))
survey = PF.BaseMag.LinearSurvey(srcField)
# Create the forward model operator
prob = PF.Magnetics.MagneticIntegral(mesh,
chiMap=Maps.IdentityMap(mesh))
# Pair the survey and problem
survey.pair(prob)
# Compute forward model some data
m = np.random.rand(mesh.nC)
survey.makeSyntheticData(m)
reg = Regularization.Sparse(mesh)
reg.mref = np.zeros(mesh.nC)
wr = np.sum(prob.G**2., axis=0)**0.5
reg.cell_weights = wr
reg.norms = [0, 1, 1, 1]
reg.eps_p, reg.eps_q = 1e-3, 1e-3
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1. / survey.std
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=2,
lower=-10.,
upper=10.,
maxIterCG=2)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
self.mesh = mesh
self.invProb = invProb
def test_update_of_sparse_norms(self):
mesh = Mesh.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)
self.assertTrue(np.all(reg.norms == [2., 2., 2., 2.]))
self.assertTrue(reg.objfcts[0].norm == 2.)
self.assertTrue(reg.objfcts[1].norm == 2.)
self.assertTrue(reg.objfcts[2].norm == 2.)
self.assertTrue(reg.objfcts[3].norm == 2.)
reg.norms = [0., 1., 1., 1.]
self.assertTrue(np.all(reg.norms == [0., 1., 1., 1.]))
self.assertTrue(reg.objfcts[0].norm == 0.)
self.assertTrue(reg.objfcts[1].norm == 1.)
self.assertTrue(reg.objfcts[2].norm == 1.)
self.assertTrue(reg.objfcts[3].norm == 1.)
def run(plotIt=True):
# Define the inducing field parameter
H0 = (50000, 90, 0)
# Create a mesh
dx = 5.
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 = 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., 20., 20)
yr = np.linspace(-20., 20., 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.
# Create a MAGsurvey
rxLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]
rxLoc = PF.BaseMag.RxObs(rxLoc)
srcField = PF.BaseMag.SrcField([rxLoc], param=H0)
survey = PF.BaseMag.LinearSurvey(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
prob = PF.Magnetics.MagneticIntegral(mesh, chiMap=idenMap, actInd=actv)
# Pair the survey and problem
survey.pair(prob)
# Compute linear forward operator and compute some data
d = prob.fields(model)
# Add noise and uncertainties
# We add some random Gaussian noise (1nT)
data = d + np.random.randn(len(d))
wd = np.ones(len(data)) * 1. # Assign flat uncertainties
survey.dobs = data
survey.std = wd
survey.mtrue = model
# Create sensitivity weights from our linear forward operator
rxLoc = survey.srcField.rxList[0].locs
wr = np.sum(prob.G**2., axis=0)**0.5
wr = (wr / np.max(wr))
# Create a regularization
reg = Regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
reg.cell_weights = wr
reg.mref = np.zeros(nC)
reg.norms = np.c_[0, 0, 0, 0]
# reg.eps_p, reg.eps_q = 1e-0, 1e-0
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1 / wd
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=100,
lower=0.,
upper=1.,
maxIterLS=20,
maxIterCG=20,
tolCG=1e-3)
invProb = InvProblem.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-4, maxIRLSiter=40)
saveDict = Directives.SaveOutputEveryIteration(save_txt=False)
update_Jacobi = Directives.UpdatePreconditioner()
inv = Inversion.BaseInversion(
invProb, directiveList=[IRLS, betaest, update_Jacobi, saveDict])
# Run the inversion
m0 = np.ones(nC) * 1e-4 # Starting model
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.PlotUtils.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)
# Load weighting file
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1./survey.std
wr = prob.getJtJdiag(np.ones(int(nC + len(actv))), W=dmis.W)
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 = Mesh.TensorMesh([nC])
reg_m1 = Regularization.Sparse(regMesh, mapping=wires.h**o)
reg_m1.cell_weights = wires.h**o*wr*2.
reg_m1.mref = mref
# 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_[driver.lpnorms].T
reg_m2.mref = mref
reg = reg_m1 + reg_m2
opt = Optimization.ProjectedGNCG(maxIter=30, lower=driver.bounds[0],
upper=driver.bounds[1],
maxIterLS = 20, maxIterCG= 30,
tolCG = 1e-4)
# wr = np.sum((dmis.W*prob.F)**2., axis=0)**0.5
wr = np.zeros(nC)
for ii in range(survey.nD):
wr += (prob.F[ii, :] / survey.std[ii])**2.
wr = (wr / np.max(wr))
else:
wr = Mesh.TensorMesh.readModelUBC(mesh, work_dir + dsep + wgtfile)
wr = wr[actv]
wr = wr**2.
# % Create inversion objects
# Starting with regularization
reg = Regularization.Sparse(mesh,
indActive=actv,
mapping=staticCells,
gradientType='total')
reg.mref = driver.mref[dynamic]
reg.cell_weights = wr
reg.norms = driver.lpnorms
if driver.eps is not None:
reg.eps_p = driver.eps[0]
reg.eps_q = driver.eps[1]
# Optimization function
opt = Optimization.ProjectedGNCG(maxIter=100,
lower=driver.bounds[0],
upper=driver.bounds[1],
maxIterLS=50,
maxIterCG=10,
tolCG=1e-3)
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.
# %%
# Read in the input file which included all parameters at once
# (mesh, topo, model, survey, inv param, etc.)
driver = PF.GravityDriver.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
# define gravity survey locations
rxLoc = survey.srcField.rxList[0].locs
# define gravity data and errors
d = survey.dobs
wd = survey.std
# 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
prob = PF.Gravity.GravityIntegral(mesh, rhoMap=staticCells, actInd=active)
prob.solverOpts['accuracyTol'] = 1e-4
# Pair the survey and problem
survey.pair(prob)
# Apply depth weighting
wr = PF.Magnetics.get_dist_wgt(mesh, rxLoc, active, wgtexp,
np.min(mesh.hx) / 4.)
wr = wr**2.
# %% Create inversion objects
reg = Regularization.Sparse(mesh,
indActive=active,
mapping=staticCells,
gradientType='total')
reg.mref = driver.mref[dynamic]
reg.cell_weights = wr * mesh.vol[active]
reg.norms = np.c_[0., 1., 1., 1.]
# 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-3)
# Define misfit function (obs-calc)
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1. / wd
# create the default L2 inverse problem from the above objects
invProb = InvProblem.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,
maxIRLSiter=40,
beta_tol=5e-1)
# Preconditioning refreshing for each IRLS iteration
update_Jacobi = Directives.UpdatePreconditioner()
# Create combined the L2 and Lp problem
inv = Inversion.BaseInversion(invProb,
directiveList=[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
PF.Magnetics.plot_obs_2D(rxLoc, d, 'Observed Data')
# %%
# Write output model and data files and print misft 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$^3$)')
wrGlobal = wrGlobal[actvGlobal]**0.5
wrGlobal = (wrGlobal / np.max(wrGlobal))
#%% Create a regularization
actv = np.all([actv, actvMap * actvMeshGlobal], axis=0)
actvMap = Maps.InjectActiveCells(mesh, actv, 0)
actvMapAmp = Maps.InjectActiveCells(mesh, actv, -100)
nC = int(np.sum(actv))
mstart = np.ones(nC) * 1e-4
mref = np.zeros(nC)
# Create a regularization
reg = Regularization.Sparse(mesh,
indActive=actv,
mapping=Maps.IdentityMap(nP=nC))
reg.cell_weights = wrGlobal
reg.norms = np.c_[driver.lpnorms].T
reg.mref = mref
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=100,
lower=-10.,
upper=10.,
maxIterCG=20,
tolCG=1e-3)
invProb = InvProblem.BaseInvProblem(ComboMisfit, reg, opt)
betaest = Directives.BetaEstimate_ByEig()
prob = PF.Magnetics.MagneticIntegral(mesh, chiMap=idenMap, actInd=actv)
# Pair the survey and problem
survey.pair(prob)
# Create sensitivity weights from our linear forward operator
rxLoc = survey.srcField.rxList[0].locs
wr = np.zeros(prob.F.shape[1])
for ii in range(survey.nD):
wr += (prob.F[ii, :] / survey.std[ii])**2.
wr = (wr / np.max(wr))
wr = wr**0.5
# Create a regularization
reg = Regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
reg.norms = driver.lpnorms
if driver.eps is not None:
reg.eps_p = driver.eps[0]
reg.eps_q = driver.eps[1]
reg.cell_weights = wr
reg.mref = driver.mref
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1. / survey.std
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=20,
def setUp(self):
ndv = -100
# Create a mesh
dx = 5.
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)]
mesh = 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]
# Go from topo to actv cells
topo = np.c_[Utils.mkvc(xx), Utils.mkvc(yy), Utils.mkvc(zz)]
actv = Utils.surface2ind_topo(mesh, topo, 'N')
actv = np.asarray([inds for inds, elem in enumerate(actv, 1)
if elem], dtype=int) - 1
# Create active map to go from reduce space to full
actvMap = Maps.InjectActiveCells(mesh, actv, -100)
nC = len(actv)
# Create and array of observation points
xr = np.linspace(-20., 20., 20)
yr = np.linspace(-20., 20., 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.
# Create a MAGsurvey
locXYZ = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]
rxLoc = PF.BaseGrav.RxObs(locXYZ)
srcField = PF.BaseGrav.SrcField([rxLoc])
survey = PF.BaseGrav.LinearSurvey(srcField)
# We can now create a density 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] = 0.5
model = Utils.mkvc(model)
self.model = model[actv]
# Create active map to go from reduce set to full
actvMap = Maps.InjectActiveCells(mesh, actv, ndv)
# Create reduced identity map
idenMap = Maps.IdentityMap(nP=nC)
# Create the forward model operator
prob = PF.Gravity.GravityIntegral(
mesh,
rhoMap=idenMap,
actInd=actv
)
# Pair the survey and problem
survey.pair(prob)
# Compute linear forward operator and compute some data
d = prob.fields(self.model)
# Add noise and uncertainties (1nT)
data = d + np.random.randn(len(d))*0.001
wd = np.ones(len(data))*.001
survey.dobs = data
survey.std = wd
# PF.Gravity.plot_obs_2D(survey.srcField.rxList[0].locs, d=data)
# Create sensitivity weights from our linear forward operator
wr = PF.Magnetics.get_dist_wgt(mesh, locXYZ, actv, 2., 2.)
wr = wr**2.
# Create a regularization
reg = Regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
reg.cell_weights = wr
reg.norms = [0, 1, 1, 1]
reg.eps_p, reg.eps_q = 5e-2, 1e-2
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1/wd
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=100, lower=-1., upper=1.,
maxIterLS=20, maxIterCG=10,
tolCG=1e-3)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt, beta=1e+8)
# Here is where the norms are applied
IRLS = Directives.Update_IRLS(f_min_change=1e-3,
minGNiter=3)
update_Jacobi = Directives.Update_lin_PreCond(mapping=idenMap)
self.inv = Inversion.BaseInversion(invProb,
directiveList=[IRLS,
update_Jacobi])
def run_inversion(
m0, survey, actind, mesh,
std, eps,
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.,
alpha_y=1.,
alpha_z=1.,
):
"""
Run IP inversion
"""
dmisfit = DataMisfit.l2_DataMisfit(survey)
uncert = abs(survey.dobs) * std + eps
dmisfit.W = 1./uncert
# 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 = InvProblem.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
wr = (wr / np.max(wr))
wr = wr
else:
wr = Mesh.TensorMesh.readModelUBC(mesh, work_dir + dsep + wgtfile)
wr = wr[actv]
wr = wr**2.
Mesh.TensorMesh.writeModelUBC(mesh, work_dir + out_dir + 'SensWeights.den',
actvMap * (homogMap.P * wr))
idenMap = Maps.IdentityMap(nP=nC)
regMesh = Mesh.TensorMesh([nC])
# Create a regularization
reg = Regularization.Sparse(regMesh, mapping=idenMap)
reg.norms = driver.lpnorms
if driver.eps is not None:
reg.eps_p = driver.eps[0]
reg.eps_q = driver.eps[1]
reg.cell_weights = wr #driver.cell_weights*mesh.vol**0.5
reg.mref = mstart
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1. / survey.std
# Write out the predicted file and generate the forward operator
#pred = prob.fields(m0)
#
def run(N=100, plotIt=True):
np.random.seed(1)
std_noise = 1e-2
mesh = Mesh.TensorMesh([N])
m0 = np.ones(mesh.nC) * 1e-4
mref = np.zeros(mesh.nC)
nk = 20
jk = np.linspace(1., 60., nk)
p = -0.25
q = 0.25
def g(k):
return (np.exp(p * jk[k] * mesh.vectorCCx) *
np.cos(np.pi * q * jk[k] * mesh.vectorCCx))
G = np.empty((nk, mesh.nC))
for i in range(nk):
G[i, :] = g(i)
mtrue = np.zeros(mesh.nC)
mtrue[mesh.vectorCCx > 0.3] = 1.
mtrue[mesh.vectorCCx > 0.45] = -0.5
mtrue[mesh.vectorCCx > 0.6] = 0
prob = Problem.LinearProblem(mesh, G=G)
survey = Survey.LinearSurvey()
survey.pair(prob)
survey.dobs = prob.fields(mtrue) + std_noise * np.random.randn(nk)
wd = np.ones(nk) * std_noise
# Distance weighting
wr = np.sum(prob.G**2., axis=0)**0.5
wr = wr / np.max(wr)
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.Wd = 1. / wd
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e-2)
reg = Regularization.Sparse(mesh)
reg.mref = mref
reg.cell_weights = wr
reg.mref = np.zeros(mesh.nC)
opt = Optimization.ProjectedGNCG(maxIter=100,
lower=-2.,
upper=2.,
maxIterLS=20,
maxIterCG=10,
tolCG=1e-3)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
update_Jacobi = Directives.Update_lin_PreCond()
# Set the IRLS directive, penalize the lowest 25 percentile of model values
# Start with an l2-l2, then switch to lp-norms
norms = [0., 0., 2., 2.]
IRLS = Directives.Update_IRLS(norms=norms,
prctile=25,
maxIRLSiter=15,
minGNiter=3)
inv = Inversion.BaseInversion(invProb,
directiveList=[IRLS, betaest, update_Jacobi])
# Run inversion
mrec = inv.run(m0)
print("Final misfit:" + str(invProb.dmisfit.eval(mrec)))
if plotIt:
fig, axes = plt.subplots(1, 2, figsize=(12 * 1.2, 4 * 1.2))
for i in range(prob.G.shape[0]):
axes[0].plot(prob.G[i, :])
axes[0].set_title('Columns of matrix G')
axes[1].plot(mesh.vectorCCx, mtrue, 'b-')
axes[1].plot(mesh.vectorCCx, reg.l2model, 'r-')
# axes[1].legend(('True Model', 'Recovered Model'))
axes[1].set_ylim(-1.0, 1.25)
axes[1].plot(mesh.vectorCCx, mrec, 'k-', lw=2)
axes[1].legend(('True Model', 'Smooth l2-l2',
'Sparse lp: {0}, lqx: {1}'.format(*reg.norms)),
fontsize=12)
return prob, survey, mesh, mrec
wr = (wr / np.max(wr))
wr = wr**0.5
#Mesh.TensorMesh.writeModelUBC(mesh, work_dir + out_dir + 'SensWeights.sus',
# actvMap*(homogMap.P*wr))
# wr = PF.Magnetics.get_dist_wgt(mesh, rxLoc, actv, 3, 1)
# Create a block diagonal regularization
wires = Maps.Wires(('p', nC), ('s', nC), ('t', nC))
idenMap = Maps.IdentityMap(nP=nC)
regMesh = Mesh.TensorMesh([nC])
# Create a regularization
reg_p = Regularization.Sparse(regMesh, mapping=wires.p)
reg_p.cell_weights = wires.p * wr
reg_p.norms = [2, 2, 2, 2]
reg_s = Regularization.Sparse(regMesh, mapping=wires.s)
reg_s.cell_weights = wires.s * wr
reg_s.norms = [2, 2, 2, 2]
reg_t = Regularization.Sparse(regMesh, mapping=wires.t)
reg_t.cell_weights = wires.t * wr
reg_t.norms = [2, 2, 2, 2]
reg = reg_p + reg_s + reg_t
reg.mref = np.zeros(3 * nC)
# Data misfit function
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., 90., 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., 200., 50),
np.linspace(-200., 200., 50))
b = 100
A = 50
zz = A * np.exp(-0.5 * ((xx / b)**2. + (yy / b)**2.))
# 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., 100., 20)
yr = np.linspace(-100., 100., 20)
X, Y = np.meshgrid(xr, yr)
Z = A * np.exp(-0.5 * ((X / b)**2. + (Y / b)**2.)) + 5
# Create a MAGsurvey
xyzLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]
rxLoc = PF.BaseMag.RxObs(xyzLoc)
srcField = PF.BaseMag.SrcField([rxLoc], param=H0)
survey = PF.BaseMag.LinearSurvey(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.ModelBuilder.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
prob = PF.Magnetics.MagneticIntegral(self.mesh,
chiMap=idenMap,
actInd=actv)
# Pair the survey and problem
survey.pair(prob)
# Compute linear forward operator and compute some data
data = prob.fields(self.model)
# Add noise and uncertainties (1nT)
noise = np.random.randn(len(data))
data += noise
wd = np.ones(len(data)) * 1.
survey.dobs = data
survey.std = wd
# Create sensitivity weights from our linear forward operator
rxLoc = survey.srcField.rxList[0].locs
wr = prob.getJtJdiag(self.model)**0.5
wr /= np.max(wr)
# Create a regularization
reg = Regularization.Sparse(self.mesh, indActive=actv, mapping=idenMap)
reg.norms = np.c_[0, 0, 0, 0]
reg.cell_weights = wr
reg.mref = np.zeros(nC)
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1. / survey.std
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=20,
lower=0.,
upper=10.,
maxIterLS=20,
maxIterCG=20,
tolCG=1e-4,
stepOffBoundsFact=1e-4)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt, beta=1e+6)
# 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,
maxIRLSiter=20,
beta_tol=1e-1,
betaSearch=False)
update_Jacobi = Directives.UpdatePreconditioner()
# saveOuput = Directives.SaveOutputEveryIteration()
# saveModel.fileName = work_dir + out_dir + 'ModelSus'
self.inv = Inversion.BaseInversion(invProb,
directiveList=[IRLS, update_Jacobi])
wr = prob.getJtJdiag(m0, W=dmis.W)
wr = (wr / np.max(wr))
wr = wr**0.5
#Mesh.TensorMesh.writeModelUBC(mesh, work_dir + out_dir + 'SensWeights.sus',
# actvMap*(homogMap.P*wr))
# Create a block diagonal regularization
wires = Maps.Wires(('p', nC), ('s', nC), ('t', nC))
idenMap = Maps.IdentityMap(nP=nC)
regMesh = Mesh.TensorMesh([nC])
# Create a regularization
reg_p = Regularization.Sparse(regMesh, mapping=wires.p)
#reg_p.cell_weights = wires.p * wr
#reg_p.norms = [2, 2, 2, 2]
reg_s = Regularization.Sparse(regMesh, mapping=wires.s)
#reg_s.cell_weights = wires.s * wr
#reg_s.norms = [2, 2, 2, 2]
reg_t = Regularization.Sparse(regMesh, mapping=wires.t)
#reg_t.cell_weights = wires.t * wr
#reg_t.norms = [2, 2, 2, 2]
reg = reg_p + reg_s + reg_t
reg.mref = np.zeros(3 * nC)
# Add directives to the inversion
# actv = np.all([actv, actvMap*actvMeshGlobal], axis=0)
actvMap = Maps.InjectActiveCells(mesh, actv, 0) # For re-projection
actvMapAmp = Maps.InjectActiveCells(mesh, actv, -100) # For final output
nC = int(np.sum(actv))
mstart = np.ones(3 * nC) * 1e-4
# Assumes amplitude reference, distributed on 3 components
mref = np.ones(3 * nC) * (np.mean(driver.mref)**2. / 3)**0.5
# Create a block diagonal regularization
wires = Maps.Wires(('p', nC), ('s', nC), ('t', nC))
# Create a regularization
reg_p = Regularization.Sparse(mesh, indActive=actv, mapping=wires.p)
reg_p.cell_weights = (wires.p * wrGlobal)
reg_p.norms = np.c_[2, 2, 2, 2]
reg_p.mref = mref
reg_s = Regularization.Sparse(mesh, indActive=actv, mapping=wires.s)
reg_s.cell_weights = (wires.s * wrGlobal)
reg_s.norms = np.c_[2, 2, 2, 2]
reg_s.mref = mref
reg_t = Regularization.Sparse(mesh, indActive=actv, mapping=wires.t)
reg_t.cell_weights = (wires.t * wrGlobal)
reg_t.norms = np.c_[2, 2, 2, 2]
reg_t.mref = mref
# Assemble the 3-component regularizations
# Create the forward model operator
prob = PF.Magnetics.MagneticVector(mesh, chiMap=idenMap, actInd=actv)
# Pair the survey and problem
survey.pair(prob)
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1. / survey.std
wr = np.sum(prob.F**2., axis=0)**0.5
wr = (wr / np.max(wr)) * np.r_[mamp, mamp, mamp]
# Create a regularization
reg_p = Regularization.Sparse(mesh, indActive=actv, mapping=wires.prim)
reg_p.cell_weights = wires.prim * wr
reg_s = Regularization.Sparse(mesh, indActive=actv, mapping=wires.second)
reg_s.cell_weights = wires.second * wr
reg_t = Regularization.Sparse(mesh, indActive=actv, mapping=wires.third)
reg_t.cell_weights = wires.third * wr
reg = reg_p + reg_s + reg_t
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=30,
lower=-10.,
upper=10.,
maxIterCG=20,
def run(plotIt=True, survey_type="dipole-dipole", p=0., qx=2., qz=2.):
np.random.seed(1)
# Initiate I/O class for DC
IO = DC.IO()
# Obtain ABMN locations
xmin, xmax = 0., 200.
ymin, ymax = 0., 0.
zmin, zmax = 0, 0
endl = np.array([[xmin, ymin, zmin], [xmax, ymax, zmax]])
# Generate DC survey object
survey = DC.Utils.gen_DCIPsurvey(endl,
survey_type=survey_type,
dim=2,
a=10,
b=10,
n=10)
survey.getABMN_locations()
survey = IO.from_ambn_locations_to_survey(survey.a_locations,
survey.b_locations,
survey.m_locations,
survey.n_locations,
survey_type,
data_dc_type='volt')
# Obtain 2D TensorMesh
mesh, actind = IO.set_mesh()
topo, mesh1D = DC.Utils.genTopography(mesh, -10, 0, its=100)
actind = Utils.surface2ind_topo(mesh, np.c_[mesh1D.vectorCCx, topo])
survey.drapeTopo(mesh, actind, option="top")
# Build a conductivity model
blk_inds_c = Utils.ModelBuilder.getIndicesSphere(np.r_[60., -25.], 12.5,
mesh.gridCC)
blk_inds_r = Utils.ModelBuilder.getIndicesSphere(np.r_[140., -25.], 12.5,
mesh.gridCC)
layer_inds = mesh.gridCC[:, 1] > -5.
sigma = np.ones(mesh.nC) * 1. / 100.
sigma[blk_inds_c] = 1. / 10.
sigma[blk_inds_r] = 1. / 1000.
sigma[~actind] = 1. / 1e8
rho = 1. / 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.Problem2D_N(mesh,
rhoMap=mapping,
storeJ=True,
Solver=Solver,
verbose=True)
# Pair problem with survey
try:
prb.pair(survey)
except:
survey.unpair()
prb.pair(survey)
# Make synthetic DC data with 5% Gaussian noise
dtrue = survey.makeSyntheticData(mtrue, std=0.05, force=True)
IO.data_dc = dtrue
# Show apparent resisitivty pseudo-section
if plotIt:
IO.plotPseudoSection(data=survey.dobs / IO.G,
data_type='apparent_resistivity')
# Show apparent resisitivty histogram
if plotIt:
fig = plt.figure()
out = hist(survey.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.)
# Set uncertainty
# floor
eps = 10**(-3.2)
# percentage
std = 0.05
dmisfit = DataMisfit.l2_DataMisfit(survey)
uncert = abs(survey.dobs) * std + eps
dmisfit.W = 1. / 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.]
IRLS = Directives.Update_IRLS(maxIRLSiter=20,
minGNiter=1,
betaSearch=False,
fix_Jmatrix=True)
opt = Optimization.InexactGaussNewton(maxIter=40)
invProb = InvProblem.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()
## Create a regularization
# For the homogeneous model
#regMesh = Mesh.TensorMesh([nC])
#
#reg_m1 = Regularization.Sparse(regMesh, mapping=wires.h**o)
#reg_m1.cell_weights = wires.h**o*wr*2.
#if driver.eps is not None:
# reg_m1.eps_p = driver.eps[0]
# reg_m1.eps_q = driver.eps[1]
#reg_m1.norms = [2, 2, 2, 2]
#reg_m1.mref = mref
# Regularization for the voxel model
reg_m2 = Regularization.Sparse(mesh, indActive=actv, mapping=Maps.IdentityMap(nP=int(dynamic.sum())),
gradientType=gradientType)
reg_m2.cell_weights = wr
reg_m2.norms = driver.lpnorms
if driver.eps is not None:
reg_m2.eps_p = driver.eps[0]
reg_m2.eps_q = driver.eps[1]
reg_m2.mref = mref[dynamic]
reg = reg_m2
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1./survey.std
opt = Optimization.ProjectedGNCG(maxIter=30, lower=driver.bounds[0],
upper=driver.bounds[1],
maxIterLS = 20, maxIterCG= 30,
global_misfit += local_misfit
else:
global_misfit, global_weights = createLocalProb(mesh, survey,
global_weights, 0)
# Global sensitivity weights (linear)
global_weights = global_weights**0.5
global_weights = (global_weights / np.max(global_weights))
if input_dict["inversion_type"].lower() in ['grav', 'mag']:
# Create a regularization function
reg = Regularization.Sparse(mesh,
indActive=activeCells,
mapping=idenMap,
alpha_s=alphas[0],
alpha_x=alphas[1],
alpha_y=alphas[2],
alpha_z=alphas[3])
reg.norms = np.c_[model_norms].T
reg.cell_weights = global_weights
if isinstance(model_reference, str):
mref = mesh.readModelUBC(workDir + model_reference)
reg.mref = mref[activeCells]
else:
reg.mref = np.ones(nC) * model_reference[0]
if isinstance(model_start, str):
mstart = mesh.readModelUBC(workDir + model_start)
mstart = mstart[activeCells]
def run(plotIt=True):
# Create a mesh
dx = 5.
hxind = [(dx, 5, -1.3), (dx, 15), (dx, 5, 1.3)]
hyind = [(dx, 5, -1.3), (dx, 15), (dx, 5, 1.3)]
hzind = [(dx, 5, -1.3), (dx, 7), (3.5, 1), (2, 5)]
mesh = 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 actv cells
actv = Utils.surface2ind_topo(mesh, topo, 'N')
actv = np.asarray([inds for inds, elem in enumerate(actv, 1) if elem],
dtype=int) - 1
# Create active map to go from reduce space to full
actvMap = Maps.InjectActiveCells(mesh, actv, -100)
nC = len(actv)
# Create and array of observation points
xr = np.linspace(-30., 30., 20)
yr = np.linspace(-30., 30., 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] + 0.1
# Create a MAGsurvey
rxLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]
rxLoc = PF.BaseGrav.RxObs(rxLoc)
srcField = PF.BaseGrav.SrcField([rxLoc])
survey = PF.BaseGrav.LinearSurvey(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-5):(midx-1), (midy-2):(midy+2), -10:-6] = 0.5
model[(midx+1):(midx+5), (midy-2):(midy+2), -10:-6] = -0.5
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
prob = PF.Gravity.GravityIntegral(mesh, rhoMap=idenMap, actInd=actv)
# Pair the survey and problem
survey.pair(prob)
# Compute linear forward operator and compute some data
d = prob.fields(model)
# Add noise and uncertainties
# We add some random Gaussian noise (1nT)
data = d + np.random.randn(len(d))*1e-3
wd = np.ones(len(data))*1e-3 # Assign flat uncertainties
survey.dobs = data
survey.std = wd
survey.mtrue = model
# Create sensitivity weights from our linear forward operator
rxLoc = survey.srcField.rxList[0].locs
wr = np.sum(prob.G**2., axis=0)**0.5
wr = (wr/np.max(wr))
# Create a regularization
reg = Regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
reg.cell_weights = wr
reg.norms = [0, 1, 1, 1]
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = Utils.sdiag(1/wd)
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=100, lower=-1., upper=1.,
maxIterLS=20, maxIterCG=10,
tolCG=1e-3)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
betaest = Directives.BetaEstimate_ByEig()
# 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-2, minGNiter=2)
update_Jacobi = Directives.UpdatePreconditioner()
inv = Inversion.BaseInversion(invProb, directiveList=[IRLS,
betaest,
update_Jacobi])
# Run the inversion
m0 = np.ones(nC)*1e-4 # Starting model
mrec = inv.run(m0)
if plotIt:
# Here is the recovered susceptibility model
ypanel = midx
zpanel = -7
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
vmin, vmax = mrec.min(), mrec.max()
# Plot the data
PF.Gravity.plot_obs_2D(rxLoc, d=data)
plt.figure()
# Plot L2 model
ax = plt.subplot(321)
mesh.plotSlice(m_l2, ax=ax, normal='Z', ind=zpanel,
grid=True, clim=(vmin, vmax))
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=(vmin, vmax))
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=(vmin, vmax))
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=(vmin, vmax))
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=(vmin, vmax))
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=(vmin, vmax))
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')
def setUp(self):
np.random.seed(0)
H0 = (50000., 90., 0.)
# The magnetization is set along a different
# direction (induced + remanence)
M = np.array([45., 90.])
# 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. + (yy/b)**2.))
# 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., 100., 20)
yr = np.linspace(-100., 100., 20)
X, Y = np.meshgrid(xr, yr)
Z = A*np.exp(-0.5*((X/b)**2. + (Y/b)**2.)) + 5
# Create a MAGsurvey
xyzLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]
rxLoc = PF.BaseMag.RxObs(xyzLoc)
srcField = PF.BaseMag.SrcField([rxLoc], param=H0)
survey = PF.BaseMag.LinearSurvey(srcField)
# Create a mesh
h = [5, 5, 5]
padDist = np.ones((3, 2)) * 100
nCpad = [2, 4, 2]
# Get extent of points
limx = np.r_[topo[:, 0].max(), topo[:, 0].min()]
limy = np.r_[topo[:, 1].max(), topo[:, 1].min()]
limz = np.r_[topo[:, 2].max(), topo[:, 2].min()]
# Get center of the mesh
midX = np.mean(limx)
midY = np.mean(limy)
midZ = np.mean(limz)
nCx = int(limx[0]-limx[1]) / h[0]
nCy = int(limy[0]-limy[1]) / h[1]
nCz = int(limz[0]-limz[1]+int(np.min(np.r_[nCx, nCy])/3)) / h[2]
# Figure out full extent required from input
extent = np.max(np.r_[nCx * h[0] + padDist[0, :].sum(),
nCy * h[1] + padDist[1, :].sum(),
nCz * h[2] + padDist[2, :].sum()])
maxLevel = int(np.log2(extent/h[0]))+1
# Number of cells at the small octree level
nCx, nCy, nCz = 2**(maxLevel), 2**(maxLevel), 2**(maxLevel)
# Define the mesh and origin
# For now cubic cells
mesh = Mesh.TreeMesh([np.ones(nCx)*h[0],
np.ones(nCx)*h[1],
np.ones(nCx)*h[2]])
# Set origin
mesh.x0 = np.r_[
-nCx*h[0]/2.+midX,
-nCy*h[1]/2.+midY,
-nCz*h[2]/2.+midZ
]
# Refine the mesh around topography
# Get extent of points
F = NearestNDInterpolator(topo[:, :2], topo[:, 2])
zOffset = 0
# Cycle through the first 3 octree levels
for ii in range(3):
dx = mesh.hx.min()*2**ii
nCx = int((limx[0]-limx[1]) / dx)
nCy = int((limy[0]-limy[1]) / dx)
# Create a grid at the octree level in xy
CCx, CCy = np.meshgrid(
np.linspace(limx[1], limx[0], nCx),
np.linspace(limy[1], limy[0], nCy)
)
z = F(mkvc(CCx), mkvc(CCy))
# level means number of layers in current OcTree level
for level in range(int(nCpad[ii])):
mesh.insert_cells(
np.c_[
mkvc(CCx),
mkvc(CCy),
z-zOffset
], np.ones_like(z)*maxLevel-ii,
finalize=False
)
zOffset += dx
mesh.finalize()
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.matutils.dip_azimuth2cartesian(M[0], M[1])
# Get the indicies of the magnetized block
ind = Utils.ModelBuilder.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
prob = PF.Magnetics.MagneticIntegral(
mesh, chiMap=idenMap, actInd=actv,
modelType='vector'
)
# Pair the survey and problem
survey.pair(prob)
# Compute some data and add some random noise
data = prob.fields(Utils.mkvc(self.model))
std = 5 # nT
data += np.random.randn(len(data))*std
wd = np.ones(len(data))*std
# Assigne data and uncertainties to the survey
survey.dobs = data
survey.std = wd
# Create an projection matrix for plotting later
actvPlot = Maps.InjectActiveCells(mesh, actv, np.nan)
# Create sensitivity weights from our linear forward operator
rxLoc = survey.srcField.rxList[0].locs
# This Mapping connects the regularizations for the three-component
# vector model
wires = Maps.Wires(('p', nC), ('s', nC), ('t', nC))
# Create sensitivity weights from our linear forward operator
# so that all cells get equal chance to contribute to the solution
wr = np.sum(prob.G**2., axis=0)**0.5
wr = (wr/np.max(wr))
# 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_p.cell_weights = (wires.p * wr)
reg_s = Regularization.Sparse(mesh, indActive=actv, mapping=wires.s)
reg_s.mref = np.zeros(3*nC)
reg_s.cell_weights = (wires.s * wr)
reg_t = Regularization.Sparse(mesh, indActive=actv, mapping=wires.t)
reg_t.mref = np.zeros(3*nC)
reg_t.cell_weights = (wires.t * wr)
reg = reg_p + reg_s + reg_t
reg.mref = np.zeros(3*nC)
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1./survey.std
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=30, lower=-10, upper=10.,
maxIterLS=20, maxIterCG=20, tolCG=1e-4)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
# A list of directive to control the inverson
betaest = Directives.BetaEstimate_ByEig()
# 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, maxIRLSiter=0, beta_tol=5e-1
)
# Pre-conditioner
update_Jacobi = Directives.UpdatePreconditioner()
inv = Inversion.BaseInversion(invProb,
directiveList=[IRLS, update_Jacobi, betaest])
# Run the inversion
m0 = np.ones(3*nC) * 1e-4 # Starting model
mrec_MVIC = inv.run(m0)
self.mstart = Utils.matutils.cartesian2spherical(mrec_MVIC.reshape((nC, 3), order='F'))
beta = invProb.beta
dmis.prob.coordinate_system = 'spherical'
dmis.prob.model = self.mstart
# 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.] # 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. # No reference angle
reg_t.space = 'spherical'
reg_t.norms = np.c_[2., 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. # No reference angle
reg_p.space = 'spherical'
reg_p.norms = np.c_[2., 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=20,
lower=Lbound,
upper=Ubound,
maxIterLS=20,
maxIterCG=30,
tolCG=1e-3,
stepOffBoundsFact=1e-3,
)
opt.approxHinv = None
invProb = InvProblem.BaseInvProblem(dmis, reg, opt, beta=beta*10.)
# Here is where the norms are applied
IRLS = Directives.Update_IRLS(f_min_change=1e-4, maxIRLSiter=20,
minGNiter=1, beta_tol=0.5,
coolingRate=1, coolEps_q=True,
betaSearch=False)
# Special directive specific to the mag amplitude problem. The sensitivity
# weights are update between each iteration.
ProjSpherical = Directives.ProjectSphericalBounds()
update_SensWeight = Directives.UpdateSensitivityWeights()
update_Jacobi = Directives.UpdatePreconditioner()
self.inv = Inversion.BaseInversion(
invProb,
directiveList=[
ProjSpherical, IRLS, update_SensWeight, 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.survey.dobs) * 0.01 + floor
m0 = np.ones(self.mesh.nC) * m0
mref = np.ones(self.mesh.nC) * mref
if ~use_tikhonov:
reg = Regularization.Sparse(
self.mesh,
alpha_s=alpha_s,
alpha_x=alpha_x,
alpha_y=alpha_z,
mref=mref,
mapping=Maps.IdentityMap(self.mesh),
cell_weights=self.mesh.vol,
)
else:
reg = Regularization.Tikhonov(
self.mesh,
alpha_s=alpha_s,
alpha_x=alpha_x,
alpha_y=alpha_z,
mref=mref,
mapping=Maps.IdentityMap(self.mesh),
)
dmis = DataMisfit.l2_DataMisfit(self.survey)
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 = InvProblem.BaseInvProblem(dmis, reg, opt)
save = Directives.SaveOutputEveryIteration()
beta_schedule = Directives.BetaSchedule(coolingFactor=coolingFactor,
coolingRate=n_iter_per_beta)
if use_irls:
IRLS = Directives.Update_IRLS(
f_min_change=1e-4,
minGNiter=1,
silent=False,
maxIRLSiter=40,
beta_tol=5e-1,
coolEpsFact=1.3,
chifact_start=chifact,
)
if beta_start is None:
directives = [
Directives.BetaEstimate_ByEig(beta0_ratio=beta0_ratio),
IRLS,
save,
]
else:
directives = [IRLS, save]
invProb.beta = beta_start
reg.norms = np.c_[p_s, p_x, p_z, 2]
else:
target = Directives.TargetMisfit(chifact=chifact)
directives = [
Directives.BetaEstimate_ByEig(beta0_ratio=beta0_ratio),
beta_schedule,
save,
]
if use_target:
directives.append(target)
inv = Inversion.BaseInversion(invProb, directiveList=directives)
mopt = inv.run(m0)
model = opt.recall("xc")
model.append(mopt)
pred = []
for m in model:
pred.append(self.survey.dpred(m))
return model, pred, save
(activeCellsMap*model_map*global_weights)[:mesh.nC]}
)
else:
mesh.writeModelUBC(
'SensWeights.mod',
(activeCellsMap*model_map*global_weights)[:mesh.nC]
)
if not vector_property:
# Create a regularization function
reg = Regularization.Sparse(
regularization_mesh,
indActive=regularization_actv,
mapping=regularization_map,
alpha_s=alphas[0],
alpha_x=alphas[1],
alpha_y=alphas[2],
alpha_z=alphas[3]
)
reg.norms = np.c_[model_norms].T
reg.cell_weights = global_weights
reg.mref = mref
else:
# Create a regularization
reg_p = Regularization.Sparse(
mesh, indActive=activeCells,
mapping=regularization_map.p,
gradientType=gradient_type,
def setUp(self):
np.random.seed(0)
# Define the inducing field parameter
H0 = (50000, 90, 0)
# Create a mesh
dx = 5.
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)]
mesh = 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]
# Go from topo to actv cells
topo = np.c_[Utils.mkvc(xx), Utils.mkvc(yy), Utils.mkvc(zz)]
actv = Utils.surface2ind_topo(mesh, topo, 'N')
actv = np.asarray([inds for inds, elem in enumerate(actv, 1) if elem],
dtype=int) - 1
# Create active map to go from reduce space to full
actvMap = Maps.InjectActiveCells(mesh, actv, -100)
nC = len(actv)
# Create and array of observation points
xr = np.linspace(-20., 20., 20)
yr = np.linspace(-20., 20., 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.
# Create a MAGsurvey
rxLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]
rxLoc = PF.BaseMag.RxObs(rxLoc)
srcField = PF.BaseMag.SrcField([rxLoc], param=H0)
survey = PF.BaseMag.LinearSurvey(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)
self.model = model[actv]
# Create active map to go from reduce set to full
actvMap = Maps.InjectActiveCells(mesh, actv, -100)
# Creat reduced identity map
idenMap = Maps.IdentityMap(nP=nC)
# Create the forward model operator
prob = PF.Magnetics.MagneticIntegral(mesh, chiMap=idenMap, actInd=actv)
# Pair the survey and problem
survey.pair(prob)
# Compute linear forward operator and compute some data
d = prob.fields(self.model)
# Add noise and uncertainties (1nT)
data = d + np.random.randn(len(d))
wd = np.ones(len(data)) * 1.
survey.dobs = data
survey.std = wd
# Create sensitivity weights from our linear forward operator
wr = np.sum(prob.G**2., axis=0)**0.5
wr = (wr / np.max(wr))
# Create a regularization
reg = Regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
reg.cell_weights = wr
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 = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1 / wd
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=100,
lower=0.,
upper=1.,
maxIterLS=20,
maxIterCG=10,
tolCG=1e-3)
invProb = InvProblem.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()
self.inv = Inversion.BaseInversion(
invProb, directiveList=[IRLS, betaest, update_Jacobi])
actvMap = Maps.InjectActiveCells(mesh, actv, 0)
idenMap = Maps.IdentityMap(nP=int(np.sum(actv)))
# reg = Regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
# reg.norms = np.c_[driver.lpnorms].T
# if driver.eps is not None:
# reg.eps_p = driver.eps[0]
# reg.eps_q = driver.eps[1]
# reg.cell_weights = wrGlobal
# reg.mref = np.zeros(mesh.nC)[actv]
nC = actv.sum()
reg1 = Regularization.Sparse(mesh,
indActive=actv,
mapping=idenMap,
gradientType='component')
reg1.norms = np.c_[driver.lpnorms].T
# reg1.alpha_x = 1#((Dx1.max() - Dx1.min())/2)**-2.
# reg1.alpha_y = 3#((Dy1.max() - Dy1.min())/2)**-2.
# reg1.alpha_z = 4
# reg1.alpha_y = 4
# reg1.eps_p = 1e-3
# reg1.eps_q = 1e-3
reg1.cell_weights = wrGlobal
reg1.mref = np.zeros(mesh.nC)[actv]
reg1.objfcts[1].regmesh._cellDiffxStencil = Dx1
reg1.objfcts[1].regmesh._aveCC2Fx = speye(nC)
reg1.objfcts[2].regmesh._cellDiffyStencil = Dy1
def run(N=100, plotIt=True):
np.random.seed(1)
std_noise = 1e-2
mesh = Mesh.TensorMesh([N])
m0 = np.ones(mesh.nC) * 1e-4
mref = np.zeros(mesh.nC)
nk = 20
jk = np.linspace(1., 60., nk)
p = -0.25
q = 0.25
def g(k):
return (
np.exp(p*jk[k]*mesh.vectorCCx) *
np.cos(np.pi*q*jk[k]*mesh.vectorCCx)
)
G = np.empty((nk, mesh.nC))
for i in range(nk):
G[i, :] = g(i)
mtrue = np.zeros(mesh.nC)
mtrue[mesh.vectorCCx > 0.3] = 1.
mtrue[mesh.vectorCCx > 0.45] = -0.5
mtrue[mesh.vectorCCx > 0.6] = 0
prob = Problem.LinearProblem(mesh, G=G)
survey = Survey.LinearSurvey()
survey.pair(prob)
survey.dobs = prob.fields(mtrue) + std_noise * np.random.randn(nk)
wd = np.ones(nk) * std_noise
# Distance weighting
wr = np.sum(prob.getJ(m0)**2., axis=0)**0.5
wr = wr/np.max(wr)
dmis = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1./wd
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e0)
# Creat reduced identity map
idenMap = Maps.IdentityMap(nP=mesh.nC)
reg = Regularization.Sparse(mesh, mapping=idenMap)
reg.mref = mref
reg.cell_weights = wr
reg.norms = np.c_[0., 0., 2., 2.]
reg.mref = np.zeros(mesh.nC)
opt = Optimization.ProjectedGNCG(
maxIter=100, lower=-2., upper=2.,
maxIterLS=20, maxIterCG=10, tolCG=1e-3
)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
update_Jacobi = Directives.UpdatePreconditioner()
# Set the IRLS directive, penalize the lowest 25 percentile of model values
# Start with an l2-l2, then switch to lp-norms
IRLS = Directives.Update_IRLS(
maxIRLSiter=40, minGNiter=1, f_min_change=1e-4)
saveDict = Directives.SaveOutputEveryIteration(save_txt=False)
inv = Inversion.BaseInversion(
invProb,
directiveList=[IRLS, betaest, update_Jacobi, saveDict]
)
# Run inversion
mrec = inv.run(m0)
print("Final misfit:" + str(invProb.dmisfit(mrec)))
if plotIt:
fig, axes = plt.subplots(2, 2, figsize=(12*1.2, 8*1.2))
for i in range(prob.G.shape[0]):
axes[0, 0].plot(prob.G[i, :])
axes[0, 0].set_title('Columns of matrix G')
axes[0, 1].plot(mesh.vectorCCx, mtrue, 'b-')
axes[0, 1].plot(mesh.vectorCCx, invProb.l2model, 'r-')
# axes[0, 1].legend(('True Model', 'Recovered Model'))
axes[0, 1].set_ylim(-1.0, 1.25)
axes[0, 1].plot(mesh.vectorCCx, mrec, 'k-', lw=2)
axes[0, 1].legend(
(
'True Model',
'Smooth l2-l2',
'Sparse norms: {0}'.format(*reg.norms)
),
fontsize=12
)
axes[1, 1].plot(saveDict.phi_d, 'k', lw=2)
twin = axes[1, 1].twinx()
twin.plot(saveDict.phi_m, 'k--', lw=2)
axes[1, 1].plot(
np.r_[IRLS.iterStart, IRLS.iterStart],
np.r_[0, np.max(saveDict.phi_d)], 'k:'
)
axes[1, 1].text(
IRLS.iterStart, 0.,
'IRLS Start', va='bottom', ha='center',
rotation='vertical', size=12,
bbox={'facecolor': 'white'}
)
axes[1, 1].set_ylabel('$\phi_d$', size=16, rotation=0)
axes[1, 1].set_xlabel('Iterations', size=14)
axes[1, 0].axis('off')
twin.set_ylabel('$\phi_m$', size=16, rotation=0)
return prob, survey, mesh, mrec
# Create sensitivity weights from our linear forward operator
rxLoc = survey.srcField.rxList[0].locs
# This Mapping connects the regularizations for the three-component
# vector model
wires = Maps.Wires(('p', nC), ('s', nC), ('t', nC))
# Create sensitivity weights from our linear forward operator
# so that all cells get equal chance to contribute to the solution
wr = np.sum(prob.G**2., axis=0)**0.5
wr = (wr / np.max(wr))
# 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_p.cell_weights = (wires.p * wr)
reg_s = Regularization.Sparse(mesh, indActive=actv, mapping=wires.s)
reg_s.mref = np.zeros(3 * nC)
reg_s.cell_weights = (wires.s * wr)
reg_t = Regularization.Sparse(mesh, indActive=actv, mapping=wires.t)
reg_t.mref = np.zeros(3 * nC)
reg_t.cell_weights = (wires.t * wr)
reg = reg_p + reg_s + reg_t
reg.mref = np.zeros(3 * nC)
# Data misfit function
def setUp(self):
# We will assume a vertical inducing field
H0 = (50000., 90., 0.)
# The magnetization is set along a different direction (induced + remanence)
M = np.array([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. + (yy / b)**2.))
topo = np.c_[Utils.mkvc(xx), Utils.mkvc(yy), Utils.mkvc(zz)]
# Create and array of observation points
xr = np.linspace(-100., 100., 20)
yr = np.linspace(-100., 100., 20)
X, Y = np.meshgrid(xr, yr)
Z = A * np.exp(-0.5 * ((X / b)**2. + (Y / b)**2.)) + 5
# Create a MAGsurvey
rxLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]
Rx = PF.BaseMag.RxObs(rxLoc)
srcField = PF.BaseMag.SrcField([Rx], param=H0)
survey = PF.BaseMag.LinearSurvey(srcField)
# Create a mesh
h = [5, 5, 5]
padDist = np.ones((3, 2)) * 100
nCpad = [4, 4, 2]
# Get extent of points
limx = np.r_[topo[:, 0].max(), topo[:, 0].min()]
limy = np.r_[topo[:, 1].max(), topo[:, 1].min()]
limz = np.r_[topo[:, 2].max(), topo[:, 2].min()]
# Get center of the mesh
midX = np.mean(limx)
midY = np.mean(limy)
midZ = np.mean(limz)
nCx = int(limx[0] - limx[1]) / h[0]
nCy = int(limy[0] - limy[1]) / h[1]
nCz = int(limz[0] - limz[1] + int(np.min(np.r_[nCx, nCy]) / 3)) / h[2]
# Figure out full extent required from input
extent = np.max(np.r_[nCx * h[0] + padDist[0, :].sum(),
nCy * h[1] + padDist[1, :].sum(),
nCz * h[2] + padDist[2, :].sum()])
maxLevel = int(np.log2(extent / h[0])) + 1
# Number of cells at the small octree level
# For now equal in 3D
nCx, nCy, nCz = 2**(maxLevel), 2**(maxLevel), 2**(maxLevel)
# Define the mesh and origin
mesh = Mesh.TreeMesh(
[np.ones(nCx) * h[0],
np.ones(nCx) * h[1],
np.ones(nCx) * h[2]])
# Set origin
mesh.x0 = np.r_[-nCx * h[0] / 2. + midX, -nCy * h[1] / 2. + midY,
-nCz * h[2] / 2. + midZ]
# Refine the mesh around topography
# Get extent of points
F = NearestNDInterpolator(topo[:, :2], topo[:, 2])
zOffset = 0
# Cycle through the first 3 octree levels
for ii in range(3):
dx = mesh.hx.min() * 2**ii
nCx = int((limx[0] - limx[1]) / dx)
nCy = int((limy[0] - limy[1]) / dx)
# Create a grid at the octree level in xy
CCx, CCy = np.meshgrid(np.linspace(limx[1], limx[0], nCx),
np.linspace(limy[1], limy[0], nCy))
z = F(mkvc(CCx), mkvc(CCy))
# level means number of layers in current OcTree level
for level in range(int(nCpad[ii])):
mesh.insert_cells(np.c_[mkvc(CCx),
mkvc(CCy), z - zOffset],
np.ones_like(z) * maxLevel - ii,
finalize=False)
zOffset += dx
mesh.finalize()
# 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.matutils.dip_azimuth2cartesian(
np.ones(nC) * M[0],
np.ones(nC) * M[1])
# Get the indicies of the magnetized block
ind = Utils.ModelBuilder.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
self.model = model[actv]
# Create active map to go from reduce set to full
self.actvPlot = Maps.InjectActiveCells(mesh, actv, np.nan)
# Creat reduced identity map
idenMap = Maps.IdentityMap(nP=nC)
# Create the forward model operator
prob = PF.Magnetics.MagneticIntegral(mesh,
M=M_xyz,
chiMap=idenMap,
actInd=actv)
# Pair the survey and problem
survey.pair(prob)
# Compute some data and add some random noise
data = prob.fields(self.model)
# Split the data in components
nD = rxLoc.shape[0]
std = 5 # nT
data += np.random.randn(nD) * std
wd = np.ones(nD) * std
# Assigne data and uncertainties to the survey
survey.dobs = data
survey.std = wd
######################################################################
# Equivalent Source
# Get the active cells for equivalent source is the top only
surf = Utils.modelutils.surface_layer_index(mesh, topo)
# Get the layer of cells directyl below topo
nC = np.count_nonzero(surf) # Number of active cells
# 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
prob = PF.Magnetics.MagneticIntegral(mesh,
chiMap=idenMap,
actInd=surf,
parallelized=False,
equiSourceLayer=True)
prob.solverOpts['accuracyTol'] = 1e-4
# Pair the survey and problem
if survey.ispaired:
survey.unpair()
survey.pair(prob)
# Create a regularization function, in this case l2l2
reg = Regularization.Sparse(mesh,
indActive=surf,
mapping=Maps.IdentityMap(nP=nC),
scaledIRLS=False)
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 = DataMisfit.l2_DataMisfit(survey)
dmis.W = 1. / survey.std
# Create the default L2 inverse problem from the above objects
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
# Specify how the initial beta is found
betaest = Directives.BetaEstimate_ByEig()
# 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)
update_Jacobi = Directives.UpdatePreconditioner()
# Put all the parts together
inv = Inversion.BaseInversion(
invProb, directiveList=[betaest, IRLS, update_Jacobi])
# Run the equivalent source inversion
mstart = np.ones(nC) * 1e-4
mrec = inv.run(mstart)
# Won't store the sensitivity and output 'xyz' data.
prob.forwardOnly = True
prob.rx_type = 'xyz'
prob._G = None
prob.modelType = 'amplitude'
prob.model = mrec
pred = prob.fields(mrec)
bx = pred[:nD]
by = pred[nD:2 * nD]
bz = pred[2 * nD:]
bAmp = (bx**2. + by**2. + bz**2.)**0.5
# AMPLITUDE 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)
self.mstart = np.ones(nC) * 1e-4
# Create the forward model operator
prob = PF.Magnetics.MagneticIntegral(mesh,
chiMap=idenMap,
actInd=actv,
modelType='amplitude',
rx_type='xyz')
prob.model = self.mstart
# Change the survey to xyz components
surveyAmp = PF.BaseMag.LinearSurvey(survey.srcField)
# Pair the survey and problem
surveyAmp.pair(prob)
# Create a regularization function, in this case l2l2
wr = np.sum(prob.G**2., axis=0)**0.5
wr = (wr / np.max(wr))
# Re-set the observations to |B|
surveyAmp.dobs = bAmp
surveyAmp.std = wd
# Create a sparse regularization
reg = Regularization.Sparse(mesh, indActive=actv, mapping=idenMap)
reg.norms = np.c_[0, 0, 0, 0]
reg.mref = np.zeros(nC)
reg.cell_weights = wr
# Data misfit function
dmis = DataMisfit.l2_DataMisfit(surveyAmp)
dmis.W = 1. / surveyAmp.std
# Add directives to the inversion
opt = Optimization.ProjectedGNCG(maxIter=30,
lower=0.,
upper=1.,
maxIterLS=20,
maxIterCG=20,
tolCG=1e-3)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
# Here is the list of directives
betaest = Directives.BetaEstimate_ByEig()
# Specify the sparse norms
IRLS = Directives.Update_IRLS(f_min_change=1e-3,
minGNiter=1,
coolingRate=1,
betaSearch=False)
# The sensitivity weights are update between each iteration.
update_SensWeight = Directives.UpdateSensitivityWeights()
update_Jacobi = Directives.UpdatePreconditioner(threshold=1 - 3)
# Put all together
self.inv = Inversion.BaseInversion(
invProb,
directiveList=[betaest, IRLS, update_SensWeight, update_Jacobi])
self.mesh = mesh
