def test_updateMultipliers(self):
nP = 10
m = np.random.rand(nP)
W1 = utils.sdiag(np.random.rand(nP))
W2 = utils.sdiag(np.random.rand(nP))
phi1 = objective_function.L2ObjectiveFunction(W=W1)
phi2 = objective_function.L2ObjectiveFunction(W=W2)
phi = phi1 + phi2
self.assertTrue(phi(m) == phi1(m) + phi2(m))
phi.multipliers[0] = utils.Zero()
self.assertTrue(phi(m) == phi2(m))
phi.multipliers[0] = 1.0
phi.multipliers[1] = utils.Zero()
self.assertTrue(len(phi.objfcts) == 2)
self.assertTrue(len(phi.multipliers) == 2)
self.assertTrue(len(phi) == 2)
self.assertTrue(phi(m) == phi1(m))
def solve_2D_J(rho1, rho2, h, A, B):
ex, ez, V = solve_2D_E(rho1, rho2, h, A, B)
sigma = 1.0 / rho2 * np.ones(mesh.nC)
sigma[mesh.gridCC[:, 1] >= -h] = 1.0 / rho1 # since the model is 2D
return utils.sdiag(sigma) * ex, utils.sdiag(sigma) * ez, V
def M(self):
"""
M: ndarray
Magnetization matrix
"""
if getattr(self, "_M", None) is None:
if self.modelType == "vector":
self._M = sp.identity(self.nC) * self.survey.source_field.parameters[0]
else:
mag = mat_utils.dip_azimuth2cartesian(
np.ones(self.nC) * self.survey.source_field.parameters[1],
np.ones(self.nC) * self.survey.source_field.parameters[2],
)
self._M = sp.vstack(
(
sdiag(mag[:, 0] * self.survey.source_field.parameters[0]),
sdiag(mag[:, 1] * self.survey.source_field.parameters[0]),
sdiag(mag[:, 2] * self.survey.source_field.parameters[0]),
)
)
return self._M
def solve_2D_potentials(rho1, rho2, h, A, B):
"""
Here we solve the 2D DC problem for potentials (using SimPEG Mesh Class)
"""
sigma = 1.0 / rho2 * np.ones(mesh.nC)
sigma[mesh.gridCC[:, 1] >= -h] = 1.0 / rho1 # since the model is 2D
q = np.zeros(mesh.nC)
a = utils.closestPoints(mesh, A[:2])
q[a] = 1.0 / mesh.vol[a]
if B is not None:
b = utils.closestPoints(mesh, B[:2])
q[b] = -1.0 / mesh.vol[b]
# Use a Neumann Boundary Condition (so pole source is reasonable)
fxm, fxp, fym, fyp = mesh.faceBoundaryInd
n_xm = fxm.sum()
n_xp = fxp.sum()
n_ym = fym.sum()
n_yp = fyp.sum()
xBC_xm = np.zeros(n_xm) # 0.5*a_xm
xBC_xp = np.zeros(n_xp) # 0.5*a_xp/b_xp
yBC_xm = np.ones(n_xm) # 0.5*(1.-b_xm)
yBC_xp = np.ones(n_xp) # 0.5*(1.-1./b_xp)
xBC_ym = np.zeros(n_ym) # 0.5*a_ym
xBC_yp = np.zeros(n_yp) # 0.5*a_yp/b_yp
yBC_ym = np.ones(n_ym) # 0.5*(1.-b_ym)
yBC_yp = np.ones(n_yp) # 0.5*(1.-1./b_yp)
sortindsfx = np.argsort(np.r_[np.arange(mesh.nFx)[fxm],
np.arange(mesh.nFx)[fxp]])
sortindsfy = np.argsort(np.r_[np.arange(mesh.nFy)[fym],
np.arange(mesh.nFy)[fyp]])
xBC_x = np.r_[xBC_xm, xBC_xp][sortindsfx]
xBC_y = np.r_[xBC_ym, xBC_yp][sortindsfy]
yBC_x = np.r_[yBC_xm, yBC_xp][sortindsfx]
yBC_y = np.r_[yBC_ym, yBC_yp][sortindsfy]
x_BC = np.r_[xBC_x, xBC_y]
y_BC = np.r_[yBC_x, yBC_y]
V = utils.sdiag(mesh.vol)
Div = V * mesh.faceDiv
P_BC, B = mesh.getBCProjWF_simple()
M = B * mesh.aveCC2F
Grad = Div.T - P_BC * utils.sdiag(y_BC) * M
A = (Div * utils.sdiag(1.0 / (mesh.dim * mesh.aveF2CC.T * (1.0 / sigma))) *
Grad)
A[0, 0] = A[0, 0] + 1. # Because Neumann
Ainv = Pardiso(A)
V = Ainv * q
return V
def M(self, M):
"""
Create magnetization matrix from unit vector orientation
:parameter
M: array (3*nC,) or (nC, 3)
"""
if self.modelType == "vector":
self._M = sdiag(mkvc(M) * self.survey.source_field.parameters[0])
else:
M = M.reshape((-1, 3))
self._M = sp.vstack((
sdiag(M[:, 0] * self.survey.source_field.parameters[0]),
sdiag(M[:, 1] * self.survey.source_field.parameters[0]),
sdiag(M[:, 2] * self.survey.source_field.parameters[0]),
))
def BiotSavartFun(mesh, r_pts, component="z"):
"""
Compute systematrix G using Biot-Savart Law
G = np.vstack((G1,G2,G3..,Gnpts)
.. math::
"""
if r_pts.ndim == 1:
npts = 1
else:
npts = r_pts.shape[0]
e = np.ones((mesh.nC, 1))
o = np.zeros((mesh.nC, 1))
const = mu_0 / 4 / np.pi
G = np.zeros((npts, mesh.nC * 3))
for i in range(npts):
if npts == 1:
r_rx = np.repeat(utils.mkvc(r_pts).reshape([1, -1]),
mesh.nC,
axis=0)
else:
r_rx = np.repeat(r_pts[i, :].reshape([1, -1]), mesh.nC, axis=0)
r_CC = mesh.gridCC
r = r_rx - r_CC
r_abs = np.sqrt((r**2).sum(axis=1))
rxind = r_abs == 0.0
# r_abs[rxind] = mesh.vol.min()**(1./3.)*0.5
r_abs[rxind] = 1e20
Sx = const * utils.sdiag(mesh.vol * r[:, 0] / r_abs**3)
Sy = const * utils.sdiag(mesh.vol * r[:, 1] / r_abs**3)
Sz = const * utils.sdiag(mesh.vol * r[:, 2] / r_abs**3)
# G_temp = sp.vstack((sp.hstack(( o.T, e.T*Sz, -e.T*Sy)), \
# sp.hstack((-e.T*Sz, o.T, e.T*Sx)), \
# sp.hstack((-e.T*Sy, e.T*Sx, o.T ))))
if component == "x":
G_temp = np.hstack((o.T, e.T * Sz, -e.T * Sy))
elif component == "y":
G_temp = np.hstack((-e.T * Sz, o.T, e.T * Sx))
elif component == "z":
G_temp = np.hstack((e.T * Sy, -e.T * Sx, o.T))
G[i, :] = G_temp
return G
def solve_2D_potentials(rho1, rho2, h, A, B):
"""
Here we solve the 2D DC problem for potentials (using SimPEG Mesg Class)
"""
sigma = 1.0 / rho2 * np.ones(mesh.nC)
sigma[mesh.gridCC[:, 1] >= -h] = 1.0 / rho1 # since the model is 2D
q = np.zeros(mesh.nC)
a = utils.closestPoints(mesh, A[:2])
b = utils.closestPoints(mesh, B[:2])
q[a] = 1.0 / mesh.vol[a]
q[b] = -1.0 / mesh.vol[b]
# q = q * 1./mesh.vol
A = (
mesh.cellGrad.T
* utils.sdiag(1.0 / (mesh.dim * mesh.aveF2CC.T * (1.0 / sigma)))
* mesh.cellGrad
)
Ainv = Pardiso(A)
V = Ainv * q
return V
def test_basic(self):
expMap = maps.ExpMap(discretize.TensorMesh((3,)))
assert expMap.nP == 3
for Example in [SimpleExample, ShortcutExample]:
PM = Example(sigmaMap=expMap)
assert PM.sigmaMap is not None
assert PM.sigmaMap is expMap
# There is currently no model, so sigma, which is mapped, fails
self.assertRaises(AttributeError, getattr, PM, "sigma")
PM.model = np.r_[1.0, 2.0, 3.0]
assert np.all(PM.sigma == np.exp(np.r_[1.0, 2.0, 3.0]))
# PM = pickle.loads(pickle.dumps(PM))
# PM = maps.ExpMap.deserialize(PM.serialize())
assert np.all(
PM.sigmaDeriv.todense()
== utils.sdiag(np.exp(np.r_[1.0, 2.0, 3.0])).todense()
)
# If we set sigma, we should delete the mapping
PM.sigma = np.r_[1.0, 2.0, 3.0]
assert np.all(PM.sigma == np.r_[1.0, 2.0, 3.0])
# PM = pickle.loads(pickle.dumps(PM))
assert PM.sigmaMap is None
assert PM.sigmaDeriv == 0
del PM.model
# sigma is not changed
assert np.all(PM.sigma == np.r_[1.0, 2.0, 3.0])
def getJtJdiag(self, m, W=None):
"""
Return the diagonal of JtJ
"""
self.model = m
if W is None:
W = np.ones(self.survey.nD)
else:
W = W.diagonal() ** 2
if getattr(self, "_gtg_diagonal", None) is None:
diag = np.zeros(self.G.shape[1])
if not self.is_amplitude_data:
for i in range(len(W)):
diag += W[i] * (self.G[i] * self.G[i])
else:
fieldDeriv = self.fieldDeriv
Gx = self.G[::3]
Gy = self.G[1::3]
Gz = self.G[2::3]
for i in range(len(W)):
row = (
fieldDeriv[0, i] * Gx[i]
+ fieldDeriv[1, i] * Gy[i]
+ fieldDeriv[2, i] * Gz[i]
)
diag += W[i] * (row * row)
self._gtg_diagonal = diag
else:
diag = self._gtg_diagonal
return mkvc((sdiag(np.sqrt(diag)) @ self.chiDeriv).power(2).sum(axis=0))
def makeMassMatrices(self, m):
mu = self.muMap * m
self._MfMui = self.mesh.getFaceInnerProduct(1.0 / mu) / self.mesh.dim
# self._MfMui = self.mesh.getFaceInnerProduct(1./mu)
# TODO: this will break if tensor mu
self._MfMuI = sdiag(1.0 / self._MfMui.diagonal())
self._MfMu0 = self.mesh.getFaceInnerProduct(1.0 / mu_0) / self.mesh.dim
def test_early_exits(self):
nP = 10
m = np.random.rand(nP)
v = np.random.rand(nP)
W1 = utils.sdiag(np.random.rand(nP))
phi1 = objective_function.L2ObjectiveFunction(W=W1)
phi2 = Error_if_Hit_ObjFct()
objfct = phi1 + 0 * phi2
self.assertTrue(len(objfct) == 2)
self.assertTrue(np.all(objfct.multipliers == np.r_[1, 0]))
self.assertTrue(objfct(m) == phi1(m))
self.assertTrue(np.all(objfct.deriv(m) == phi1.deriv(m)))
self.assertTrue(np.all(objfct.deriv2(m, v) == phi1.deriv2(m, v)))
objfct.multipliers[1] = utils.Zero()
self.assertTrue(len(objfct) == 2)
self.assertTrue(np.all(objfct.multipliers == np.r_[1, 0]))
self.assertTrue(objfct(m) == phi1(m))
self.assertTrue(np.all(objfct.deriv(m) == phi1.deriv(m)))
self.assertTrue(np.all(objfct.deriv2(m, v) == phi1.deriv2(m, v)))
def __init__(self, mesh, **kwargs):
super().__init__(mesh, **kwargs)
Pbc, Pin, self._Pout = self.mesh.getBCProjWF("neumann", discretization="CC")
Dface = self.mesh.faceDiv
Mc = sdiag(self.mesh.vol)
self._Div = Mc * Dface * Pin.T * Pin
def test_sum_fail(self):
nP1 = 10
nP2 = 30
phi1 = objective_function.L2ObjectiveFunction(
W=utils.sdiag(np.random.rand(nP1))
)
phi2 = objective_function.L2ObjectiveFunction(
W=utils.sdiag(np.random.rand(nP2))
)
with self.assertRaises(Exception):
phi = phi1 + phi2
with self.assertRaises(Exception):
phi = phi1 + 100 * phi2
def getRHS(self, m):
""""""
Mc = utils.sdiag(self.mesh.vol)
self.model = m
rho = self.rho
return Mc * rho
def CasingMagDipoleDeriv_z(z):
obsloc = np.vstack([xobs, yobs, z]).T
f = Casing._getCasingHertzMagDipole(srcloc, obsloc, freq, sigma, a, b, mu)
g = utils.sdiag(
Casing._getCasingHertzMagDipoleDeriv_z(srcloc, obsloc, freq, sigma, a,
b, mu))
return f, g
def test_NewtonRoot(self):
fun = (lambda x, return_g=True: np.sin(x)
if not return_g else (np.sin(x), sdiag(np.cos(x))))
x = np.array([np.pi - 0.3, np.pi + 0.1, 0])
xopt = optimization.NewtonRoot(comments=False).root(fun, x)
x_true = np.array([np.pi, np.pi, 0])
print("Newton Root Finding")
print("xopt: ", xopt)
print("x_true: ", x_true)
self.assertTrue(np.linalg.norm(xopt - x_true, 2) < TOL, True)
def getFields(self, bType="b", ifreq=0):
src = self.srcList[ifreq]
Pfx = self.mesh.getInterpolationMat(self.mesh.gridCC[self.activeCC, :],
locType="Fx")
Pfz = self.mesh.getInterpolationMat(self.mesh.gridCC[self.activeCC, :],
locType="Fz")
Ey = self.mesh.aveE2CC * self.f[src, "e"]
Jy = utils.sdiag(self.sim.sigma) * Ey
self.Ey = utils.mkvc(self.mirrorArray(Ey[self.activeCC],
direction="y"))
self.Jy = utils.mkvc(self.mirrorArray(Jy[self.activeCC],
direction="y"))
self.Bx = utils.mkvc(
self.mirrorArray(Pfx * self.f[src, bType], direction="x"))
self.Bz = utils.mkvc(
self.mirrorArray(Pfz * self.f[src, bType], direction="z"))
def test_scalarmul(self):
scalar = 10.0
nP = 100
objfct_a = objective_function.L2ObjectiveFunction(
W=utils.sdiag(np.random.randn(nP))
)
objfct_b = scalar * objfct_a
m = np.random.rand(nP)
objfct_c = objfct_a + objfct_b
self.assertTrue(scalar * objfct_a(m) == objfct_b(m))
self.assertTrue(objfct_b.test())
self.assertTrue(objfct_c(m) == objfct_a(m) + objfct_b(m))
self.assertTrue(len(objfct_c.objfcts) == 2)
self.assertTrue(len(objfct_c.multipliers) == 2)
self.assertTrue(len(objfct_c) == 2)
def getJtJdiag(self, m, W=None):
"""
Return the diagonal of JtJ
"""
self.model = m
if W is None:
W = np.ones(self.survey.nD)
else:
W = W.diagonal() ** 2
if getattr(self, "_gtg_diagonal", None) is None:
diag = np.zeros(self.G.shape[1])
for i in range(len(W)):
diag += W[i] * (self.G[i] * self.G[i])
self._gtg_diagonal = diag
else:
diag = self._gtg_diagonal
return mkvc((sdiag(np.sqrt(diag)) @ self.rhoDeriv).power(2).sum(axis=0))
def test_mappings_and_cell_weights(self):
mesh = discretize.TensorMesh([8, 7, 6])
m = np.random.rand(2 * mesh.nC)
v = np.random.rand(2 * mesh.nC)
cell_weights = np.random.rand(mesh.nC)
wires = maps.Wires(("sigma", mesh.nC), ("mu", mesh.nC))
reg = regularization.SimpleSmall(mesh,
mapping=wires.sigma,
cell_weights=cell_weights)
objfct = objective_function.L2ObjectiveFunction(W=utils.sdiag(
np.sqrt(cell_weights)),
mapping=wires.sigma)
self.assertTrue(reg(m) == objfct(m))
self.assertTrue(np.all(reg.deriv(m) == objfct.deriv(m)))
self.assertTrue(np.all(reg.deriv2(m, v=v) == objfct.deriv2(m, v=v)))
def test_2sum(self):
nP = 80
alpha1 = 100
alpha2 = 200
phi1 = (
objective_function.L2ObjectiveFunction(W=utils.sdiag(np.random.rand(nP)))
+ alpha1 * objective_function.L2ObjectiveFunction()
)
phi2 = objective_function.L2ObjectiveFunction() + alpha2 * phi1
self.assertTrue(phi2.test(eps=EPS))
self.assertTrue(len(phi1.multipliers) == 2)
self.assertTrue(len(phi2.multipliers) == 2)
self.assertTrue(len(phi1.objfcts) == 2)
self.assertTrue(len(phi2.objfcts) == 2)
self.assertTrue(len(phi2) == 2)
self.assertTrue(len(phi1) == 2)
self.assertTrue(len(phi2) == 2)
self.assertTrue(np.all(phi1.multipliers == np.r_[1.0, alpha1]))
self.assertTrue(np.all(phi2.multipliers == np.r_[1.0, alpha2]))
def getFields(self, itime):
src = self.srcList[0]
Ey = self.mesh.aveE2CC * self.f[src, "e", itime]
Jy = utils.sdiag(self.sim.sigma) * Ey
self.Ey = utils.mkvc(self.mirrorArray(Ey[self.activeCC],
direction="y"))
self.Jy = utils.mkvc(self.mirrorArray(Jy[self.activeCC],
direction="y"))
self.Bx = utils.mkvc(
self.mirrorArray(self.Pfx * self.f[src, "b", itime],
direction="x"))
self.Bz = utils.mkvc(
self.mirrorArray(self.Pfz * self.f[src, "b", itime],
direction="z"))
self.dBxdt = utils.mkvc(
self.mirrorArray(-self.Pfx * self.mesh.edgeCurl *
self.f[src, "e", itime],
direction="x"))
self.dBzdt = utils.mkvc(
self.mirrorArray(-self.Pfz * self.mesh.edgeCurl *
self.f[src, "e", itime],
direction="z"))
# Define the Inverse Problem
# --------------------------
#
# 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
def Jtvec(self, m, v, u=None):
"""
Computing Jacobian^T multiplied by vector.
.. math ::
(\\frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}})^{T} = \left[ \mathbf{P}_{deriv}\\frac{\partial \mathbf{\mu} } {\partial \mathbf{m} }
\left[
\diag(\M^f_{\mu_{0}^{-1} } \mathbf{B}_0) \dMfMuI \\
- \diag (\Div^T\mathbf{u})\dMfMuI
\\right ]\\right]^{T}
- \left[\mathbf{P}_{deriv}(\MfMui)^{-1}\Div^T\\frac{\delta\mathbf{u}}{\delta \mathbf{m}} \\right]^{T}
where
.. math ::
\mathbf{P}_{derv} = \\frac{\partial \mathbf{P}}{\partial\mathbf{B}}
.. note ::
Here we only want to compute
.. math ::
\mathbf{J}^{T}\mathbf{v} = (\\frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}})^{T} \mathbf{v}
"""
if u is None:
u = self.fields(m)
B, u = u["B"], u["u"]
mu = self.mapping * (m)
dmu_dm = self.mapping.deriv(m)
# dchidmu = sdiag(1 / mu_0 * np.ones(self.mesh.nC))
vol = self.mesh.vol
Div = self._Div
Dface = self.mesh.faceDiv
P = self.survey.projectFieldsDeriv(B) # Projection matrix
B0 = self.getB0()
MfMuIvec = 1 / self.MfMui.diagonal()
dMfMuI = sdiag(MfMuIvec ** 2) * self.mesh.aveF2CC.T * sdiag(vol * 1.0 / mu ** 2)
# A = self._Div*self.MfMuI*self._Div.T
# RHS = Div*MfMuI*MfMu0*B0 - Div*B0 + Mc*Dface*Pout.T*Bbc
# C(m,u) = A*m-rhs
# dudm = -(dCdu)^(-1)dCdm
dCdu = self.getA(m)
s = Div * (self.MfMuI.T * (P.T * v))
Ainv = self.solver(dCdu.T, **self.solver_opts)
sol = Ainv * s
Ainv.clean()
# dCdm_A = Div * ( sdiag( Div.T * u )* dMfMuI *dmu_dm )
# dCdm_Atsol = ( dMfMuI.T*( sdiag( Div.T * u ) * (Div.T * dmu_dm)) ) * sol
dCdm_Atsol = (dmu_dm.T * dMfMuI.T * (sdiag(Div.T * u) * Div.T)) * sol
# dCdm_RHS1 = Div * (sdiag( self.MfMu0*B0 ) * dMfMuI)
# dCdm_RHS1tsol = (dMfMuI.T*( sdiag( self.MfMu0*B0 ) ) * Div.T * dmu_dm) * sol
dCdm_RHS1tsol = (dmu_dm.T * dMfMuI.T * (sdiag(self.MfMu0 * B0)) * Div.T) * sol
# temp1 = (Dface*(self._Pout.T*self.Bbc_const*self.Bbc))
# temp1sol = (Dface.T * (sdiag(vol) * sol))
# temp2 = self.Bbc_const * (self._Pout.T * self.Bbc).T
# dCdm_RHS2v = (sdiag(vol)*temp1)*np.inner(vol, dchidmu*dmu_dm*v)
# dCdm_RHS2tsol = (dmu_dm.T * dchidmu.T * vol) * np.inner(temp2, temp1sol)
# dCdm_RHSv = dCdm_RHS1*(dmu_dm*v) + dCdm_RHS2v
# temporary fix
# dCdm_RHStsol = dCdm_RHS1tsol - dCdm_RHS2tsol
dCdm_RHStsol = dCdm_RHS1tsol
# dCdm_RHSv = dCdm_RHS1*(dmu_dm*v) + dCdm_RHS2v
# dCdm_v = dCdm_A*v - dCdm_RHSv
Ctv = dCdm_Atsol - dCdm_RHStsol
# B = self.MfMuI*self.MfMu0*B0-B0-self.MfMuI*self._Div.T*u
# dBdm = d\mudm*dBd\mu
# dPBdm^T*v = Atemp^T*P^T*v - Btemp^T*P^T*v - Ctv
Atemp = sdiag(self.MfMu0 * B0) * (dMfMuI * (dmu_dm))
Btemp = sdiag(Div.T * u) * (dMfMuI * (dmu_dm))
Jtv = Atemp.T * (P.T * v) - Btemp.T * (P.T * v) - Ctv
return mkvc(Jtv)
def Jvec(self, m, v, u=None):
"""
Computing Jacobian multiplied by vector
By setting our problem as
.. math ::
\mathbf{C}(\mathbf{m}, \mathbf{u}) = \mathbf{A}\mathbf{u} - \mathbf{rhs} = 0
And taking derivative w.r.t m
.. math ::
\\nabla \mathbf{C}(\mathbf{m}, \mathbf{u}) = \\nabla_m \mathbf{C}(\mathbf{m}) \delta \mathbf{m} +
\\nabla_u \mathbf{C}(\mathbf{u}) \delta \mathbf{u} = 0
\\frac{\delta \mathbf{u}}{\delta \mathbf{m}} = - [\\nabla_u \mathbf{C}(\mathbf{u})]^{-1}\\nabla_m \mathbf{C}(\mathbf{m})
With some linear algebra we can have
.. math ::
\\nabla_u \mathbf{C}(\mathbf{u}) = \mathbf{A}
\\nabla_m \mathbf{C}(\mathbf{m}) =
\\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} - \\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}}
.. math ::
\\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} =
\\frac{\partial \mathbf{\mu}}{\partial \mathbf{m}} \left[\Div \diag (\Div^T \mathbf{u}) \dMfMuI \\right]
\dMfMuI = \diag(\MfMui)^{-1}_{vec} \mathbf{Av}_{F2CC}^T\diag(\mathbf{v})\diag(\\frac{1}{\mu^2})
\\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}} = \\frac{\partial \mathbf{\mu}}{\partial \mathbf{m}} \left[
\Div \diag(\M^f_{\mu_{0}^{-1}}\mathbf{B}_0) \dMfMuI \\right] - \diag(\mathbf{v})\mathbf{D} \mathbf{P}_{out}^T\\frac{\partial B_{sBC}}{\partial \mathbf{m}}
In the end,
.. math ::
\\frac{\delta \mathbf{u}}{\delta \mathbf{m}} =
- [ \mathbf{A} ]^{-1}\left[ \\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u}
- \\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}} \\right]
A little tricky point here is we are not interested in potential (u), but interested in magnetic flux (B).
Thus, we need sensitivity for B. Now we take derivative of B w.r.t m and have
.. math ::
\\frac{\delta \mathbf{B}} {\delta \mathbf{m}} = \\frac{\partial \mathbf{\mu} } {\partial \mathbf{m} }
\left[
\diag(\M^f_{\mu_{0}^{-1} } \mathbf{B}_0) \dMfMuI \\
- \diag (\Div^T\mathbf{u})\dMfMuI
\\right ]
- (\MfMui)^{-1}\Div^T\\frac{\delta\mathbf{u}}{\delta \mathbf{m}}
Finally we evaluate the above, but we should remember that
.. note ::
We only want to evalute
.. math ::
\mathbf{J}\mathbf{v} = \\frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}}\mathbf{v}
Since forming sensitivity matrix is very expensive in that this monster is "big" and "dense" matrix!!
"""
if u is None:
u = self.fields(m)
B, u = u["B"], u["u"]
mu = self.muMap * (m)
dmu_dm = self.muDeriv
# dchidmu = sdiag(1 / mu_0 * np.ones(self.mesh.nC))
vol = self.mesh.vol
Div = self._Div
P = self.survey.projectFieldsDeriv(B) # Projection matrix
B0 = self.getB0()
MfMuIvec = 1 / self.MfMui.diagonal()
dMfMuI = sdiag(MfMuIvec ** 2) * self.mesh.aveF2CC.T * sdiag(vol * 1.0 / mu ** 2)
# A = self._Div*self.MfMuI*self._Div.T
# RHS = Div*MfMuI*MfMu0*B0 - Div*B0 + Mc*Dface*Pout.T*Bbc
# C(m,u) = A*m-rhs
# dudm = -(dCdu)^(-1)dCdm
dCdu = self.getA(m) # = A
dCdm_A = Div * (sdiag(Div.T * u) * dMfMuI * dmu_dm)
dCdm_RHS1 = Div * (sdiag(self.MfMu0 * B0) * dMfMuI)
# temp1 = (Dface * (self._Pout.T * self.Bbc_const * self.Bbc))
# dCdm_RHS2v = (sdiag(vol) * temp1) * \
# np.inner(vol, dchidmu * dmu_dm * v)
# dCdm_RHSv = dCdm_RHS1*(dmu_dm*v) + dCdm_RHS2v
dCdm_RHSv = dCdm_RHS1 * (dmu_dm * v)
dCdm_v = dCdm_A * v - dCdm_RHSv
Ainv = self.solver(dCdu, **self.solver_opts)
sol = Ainv * dCdm_v
dudm = -sol
dBdmv = (
sdiag(self.MfMu0 * B0) * (dMfMuI * (dmu_dm * v))
- sdiag(Div.T * u) * (dMfMuI * (dmu_dm * v))
- self.MfMuI * (Div.T * (dudm))
)
Ainv.clean()
return mkvc(P * dBdmv)
def run(runIt=False, plotIt=True, saveIt=False, saveFig=False, cleanup=True):
"""
Run the bookpurnong 1D stitched RESOLVE inversions.
:param bool runIt: re-run the inversions? Default downloads and plots saved results
:param bool plotIt: show the plots?
:param bool saveIt: save the re-inverted results?
:param bool saveFig: save the figure
:param bool cleanup: remove the downloaded results
"""
# download the data
downloads, directory = download_and_unzip_data()
# Load resolve data
resolve = h5py.File(os.path.sep.join([directory, "booky_resolve.hdf5"]),
"r")
river_path = resolve["river_path"][()] # River path
nSounding = resolve["data"].shape[0] # the # of soundings
# Bird height from surface
b_height_resolve = resolve["src_elevation"][()]
# fetch the frequencies we are considering
cpi_inds = [0, 2, 6, 8, 10] # Indices for HCP in-phase
cpq_inds = [1, 3, 7, 9, 11] # Indices for HCP quadrature
frequency_cp = resolve["frequency_cp"][()]
# build a mesh
cs, ncx, ncz, npad = 1.0, 10.0, 10.0, 20
hx = [(cs, ncx), (cs, npad, 1.3)]
npad = 12
temp = np.logspace(np.log10(1.0), np.log10(12.0), 19)
temp_pad = temp[-1] * 1.3**np.arange(npad)
hz = np.r_[temp_pad[::-1], temp[::-1], temp, temp_pad]
mesh = discretize.CylMesh([hx, 1, hz], "00C")
active = mesh.vectorCCz < 0.0
# survey parameters
rxOffset = 7.86 # tx-rx separation
bp = -mu_0 / (4 * np.pi * rxOffset**3) # primary magnetic field
# re-run the inversion
if runIt:
# set up the mappings - we are inverting for 1D log conductivity
# below the earth's surface.
actMap = maps.InjectActiveCells(mesh,
active,
np.log(1e-8),
nC=mesh.nCz)
mapping = maps.ExpMap(mesh) * maps.SurjectVertical1D(mesh) * actMap
# build starting and reference model
sig_half = 1e-1
sig_air = 1e-8
sigma = np.ones(mesh.nCz) * sig_air
sigma[active] = sig_half
m0 = np.log(1e-1) * np.ones(active.sum()) # starting model
mref = np.log(1e-1) * np.ones(active.sum()) # reference model
# initalize empty lists for storing inversion results
mopt_re = [] # recovered model
dpred_re = [] # predicted data
dobs_re = [] # observed data
# downsample the data for the inversion
nskip = 40
# set up a noise model
# 10% for the 3 lowest frequencies, 15% for the two highest
relative = np.repeat(np.r_[np.ones(3) * 0.1, np.ones(2) * 0.15], 2)
floor = abs(20 * bp * 1e-6) # floor of 20ppm
# loop over the soundings and invert each
for rxind in range(nSounding):
# convert data from ppm to magnetic field (A/m^2)
dobs = (np.c_[resolve["data"][rxind, :][cpi_inds].astype(float),
resolve["data"][
rxind, :][cpq_inds].astype(float), ].flatten() *
bp * 1e-6)
# perform the inversion
src_height = b_height_resolve[rxind].astype(float)
mopt, dpred, dobs = resolve_1Dinversions(
mesh,
dobs,
src_height,
frequency_cp,
m0,
mref,
mapping,
relative=relative,
floor=floor,
)
# add results to our list
mopt_re.append(mopt)
dpred_re.append(dpred)
dobs_re.append(dobs)
# save results
mopt_re = np.vstack(mopt_re)
dpred_re = np.vstack(dpred_re)
dobs_re = np.vstack(dobs_re)
if saveIt:
np.save("mopt_re_final", mopt_re)
np.save("dobs_re_final", dobs_re)
np.save("dpred_re_final", dpred_re)
mopt_re = resolve["mopt"][()]
dobs_re = resolve["dobs"][()]
dpred_re = resolve["dpred"][()]
sigma = np.exp(mopt_re)
indz = -7 # depth index
# so that we can visually compare with literature (eg Viezzoli, 2010)
cmap = "jet"
# dummy figure for colobar
fig = plt.figure()
out = plt.scatter(np.ones(3),
np.ones(3),
c=np.linspace(-2, 1, 3),
cmap=cmap)
plt.close(fig)
# plot from the paper
fs = 13 # fontsize
# matplotlib.rcParams['font.size'] = fs
plt.figure(figsize=(13, 7))
ax0 = plt.subplot2grid((2, 3), (0, 0), rowspan=2, colspan=2)
ax1 = plt.subplot2grid((2, 3), (0, 2))
ax2 = plt.subplot2grid((2, 3), (1, 2))
# titles of plots
title = [
("(a) Recovered model, %.1f m depth") %
(-mesh.vectorCCz[active][indz]),
"(b) Obs (Real 400 Hz)",
"(c) Pred (Real 400 Hz)",
]
temp = sigma[:, indz]
tree = cKDTree(list(zip(resolve["xy"][:, 0], resolve["xy"][:, 1])))
d, d_inds = tree.query(list(zip(resolve["xy"][:, 0], resolve["xy"][:, 1])),
k=20)
w = 1.0 / (d + 100.0)**2.0
w = utils.sdiag(1.0 / np.sum(w, axis=1)) * (w)
xy = resolve["xy"]
temp = (temp.flatten()[d_inds] * w).sum(axis=1)
utils.plot2Ddata(
xy,
temp,
ncontour=100,
scale="log",
dataloc=False,
contourOpts={
"cmap": cmap,
"vmin": 1e-2,
"vmax": 1e1
},
ax=ax0,
)
ax0.plot(resolve["xy"][:, 0], resolve["xy"][:, 1], "k.", alpha=0.02, ms=1)
cb = plt.colorbar(out,
ax=ax0,
ticks=np.linspace(-2, 1, 4),
format="$10^{%.1f}$")
cb.set_ticklabels(["0.01", "0.1", "1", "10"])
cb.set_label("Conductivity (S/m)")
ax0.plot(river_path[:, 0], river_path[:, 1], "k-", lw=0.5)
# plot observed and predicted data
freq_ind = 0
axs = [ax1, ax2]
temp_dobs = dobs_re[:, freq_ind].copy()
ax1.plot(river_path[:, 0], river_path[:, 1], "k-", lw=0.5)
inf = temp_dobs / abs(bp) * 1e6
print(inf.min(), inf.max())
out = utils.plot2Ddata(
resolve["xy"][()],
temp_dobs / abs(bp) * 1e6,
ncontour=100,
scale="log",
dataloc=False,
ax=ax1,
contourOpts={"cmap": "viridis"},
)
vmin, vmax = out[0].get_clim()
print(vmin, vmax)
cb = plt.colorbar(
out[0],
ticks=np.logspace(np.log10(vmin), np.log10(vmax), 3),
ax=ax1,
format="%.1e",
fraction=0.046,
pad=0.04,
)
cb.set_label("Bz (ppm)")
temp_dpred = dpred_re[:, freq_ind].copy()
# temp_dpred[mask_:_data] = np.nan
ax2.plot(river_path[:, 0], river_path[:, 1], "k-", lw=0.5)
utils.plot2Ddata(
resolve["xy"][()],
temp_dpred / abs(bp) * 1e6,
ncontour=100,
scale="log",
dataloc=False,
contourOpts={
"vmin": vmin,
"vmax": vmax,
"cmap": "viridis"
},
ax=ax2,
)
cb = plt.colorbar(
out[0],
ticks=np.logspace(np.log10(vmin), np.log10(vmax), 3),
ax=ax2,
format="%.1e",
fraction=0.046,
pad=0.04,
)
cb.set_label("Bz (ppm)")
for i, ax in enumerate([ax0, ax1, ax2]):
xticks = [460000, 463000]
yticks = [6195000, 6198000, 6201000]
xloc, yloc = 462100.0, 6196500.0
ax.set_xticks(xticks)
ax.set_yticks(yticks)
# ax.plot(xloc, yloc, 'wo')
ax.plot(river_path[:, 0], river_path[:, 1], "k", lw=0.5)
ax.set_aspect("equal")
ax.plot(resolve["xy"][:, 0],
resolve["xy"][:, 1],
"k.",
alpha=0.02,
ms=1)
ax.set_yticklabels([str(f) for f in yticks])
ax.set_ylabel("Northing (m)")
ax.set_xlabel("Easting (m)")
ax.set_title(title[i])
plt.tight_layout()
if plotIt:
plt.show()
if saveFig is True:
fig.savefig("obspred_resolve.png", dpi=200)
resolve.close()
if cleanup:
os.remove(downloads)
shutil.rmtree(directory)
def test_CylMeshEBDipoles(self, plotIt=plotIt):
print("Testing CylMesh Electric and Magnetic Dipoles in a wholespace-"
" Analytic: J-formulation")
sigmaback = 1.0
mur = 2.0
freq = 1.0
skdpth = 500.0 / np.sqrt(sigmaback * freq)
csx, ncx, npadx = 5, 50, 25
csz, ncz, npadz = 5, 50, 25
hx = utils.meshTensor([(csx, ncx), (csx, npadx, 1.3)])
hz = utils.meshTensor([(csz, npadz, -1.3), (csz, ncz),
(csz, npadz, 1.3)])
# define the cylindrical mesh
mesh = discretize.CylMesh([hx, 1, hz], [0.0, 0.0, -hz.sum() / 2])
if plotIt:
mesh.plotGrid()
# make sure mesh is big enough
self.assertTrue(mesh.hz.sum() > skdpth * 2.0)
self.assertTrue(mesh.hx.sum() > skdpth * 2.0)
# set up source
# test electric dipole
src_loc = np.r_[0.0, 0.0, 0.0]
s_ind = utils.closestPoints(mesh, src_loc, "Fz") + mesh.nFx
de = np.zeros(mesh.nF, dtype=complex)
de[s_ind] = 1.0 / csz
de_p = [fdem.Src.RawVec_e([], freq, de / mesh.area)]
dm_p = [fdem.Src.MagDipole([], freq, src_loc)]
# Pair the problem and survey
surveye = fdem.Survey(de_p)
surveym = fdem.Survey(dm_p)
prbe = fdem.Simulation3DMagneticField(mesh,
survey=surveye,
sigma=sigmaback,
mu=mur * mu_0)
prbm = fdem.Simulation3DElectricField(mesh,
survey=surveym,
sigma=sigmaback,
mu=mur * mu_0)
# solve
fieldsBackE = prbe.fields()
fieldsBackM = prbm.fields()
rlim = [20.0, 500.0]
# lookAtTx = de_p
r = mesh.vectorCCx[np.argmin(np.abs(mesh.vectorCCx - rlim[0])):np.
argmin(np.abs(mesh.vectorCCx - rlim[1]))]
z = 100.0
# where we choose to measure
XYZ = utils.ndgrid(r, np.r_[0.0], np.r_[z])
Pf = mesh.getInterpolationMat(XYZ, "CC")
Zero = sp.csr_matrix(Pf.shape)
Pfx, Pfz = sp.hstack([Pf, Zero]), sp.hstack([Zero, Pf])
jn = fieldsBackE[de_p, "j"]
bn = fieldsBackM[dm_p, "b"]
SigmaBack = sigmaback * np.ones((mesh.nC))
Rho = utils.sdiag(1.0 / SigmaBack)
Rho = sp.block_diag([Rho, Rho])
en = Rho * mesh.aveF2CCV * jn
bn = mesh.aveF2CCV * bn
ex, ez = Pfx * en, Pfz * en
bx, bz = Pfx * bn, Pfz * bn
# get analytic solution
exa, eya, eza = analytics.FDEM.ElectricDipoleWholeSpace(XYZ,
src_loc,
sigmaback,
freq,
"Z",
mu_r=mur)
exa = utils.mkvc(exa, 2)
eya = utils.mkvc(eya, 2)
eza = utils.mkvc(eza, 2)
bxa, bya, bza = analytics.FDEM.MagneticDipoleWholeSpace(XYZ,
src_loc,
sigmaback,
freq,
"Z",
mu_r=mur)
bxa = utils.mkvc(bxa, 2)
bya = utils.mkvc(bya, 2)
bza = utils.mkvc(bza, 2)
print(
" comp, anayltic, numeric, num - ana, (num - ana)/ana"
)
print(
" ex:",
np.linalg.norm(exa),
np.linalg.norm(ex),
np.linalg.norm(exa - ex),
np.linalg.norm(exa - ex) / np.linalg.norm(exa),
)
print(
" ez:",
np.linalg.norm(eza),
np.linalg.norm(ez),
np.linalg.norm(eza - ez),
np.linalg.norm(eza - ez) / np.linalg.norm(eza),
)
print(
" bx:",
np.linalg.norm(bxa),
np.linalg.norm(bx),
np.linalg.norm(bxa - bx),
np.linalg.norm(bxa - bx) / np.linalg.norm(bxa),
)
print(
" bz:",
np.linalg.norm(bza),
np.linalg.norm(bz),
np.linalg.norm(bza - bz),
np.linalg.norm(bza - bz) / np.linalg.norm(bza),
)
if plotIt is True:
# Edipole
plt.subplot(221)
plt.plot(r, ex.real, "o", r, exa.real, linewidth=2)
plt.grid(which="both")
plt.title("Ex Real")
plt.xlabel("r (m)")
plt.subplot(222)
plt.plot(r, ex.imag, "o", r, exa.imag, linewidth=2)
plt.grid(which="both")
plt.title("Ex Imag")
plt.legend(["Num", "Ana"], bbox_to_anchor=(1.5, 0.5))
plt.xlabel("r (m)")
plt.subplot(223)
plt.plot(r, ez.real, "o", r, eza.real, linewidth=2)
plt.grid(which="both")
plt.title("Ez Real")
plt.xlabel("r (m)")
plt.subplot(224)
plt.plot(r, ez.imag, "o", r, eza.imag, linewidth=2)
plt.grid(which="both")
plt.title("Ez Imag")
plt.xlabel("r (m)")
plt.tight_layout()
# Bdipole
plt.subplot(221)
plt.plot(r, bx.real, "o", r, bxa.real, linewidth=2)
plt.grid(which="both")
plt.title("Bx Real")
plt.xlabel("r (m)")
plt.subplot(222)
plt.plot(r, bx.imag, "o", r, bxa.imag, linewidth=2)
plt.grid(which="both")
plt.title("Bx Imag")
plt.legend(["Num", "Ana"], bbox_to_anchor=(1.5, 0.5))
plt.xlabel("r (m)")
plt.subplot(223)
plt.plot(r, bz.real, "o", r, bza.real, linewidth=2)
plt.grid(which="both")
plt.title("Bz Real")
plt.xlabel("r (m)")
plt.subplot(224)
plt.plot(r, bz.imag, "o", r, bza.imag, linewidth=2)
plt.grid(which="both")
plt.title("Bz Imag")
plt.xlabel("r (m)")
plt.tight_layout()
self.assertTrue(
np.linalg.norm(exa - ex) / np.linalg.norm(exa) < tol_EBdipole)
self.assertTrue(
np.linalg.norm(eza - ez) / np.linalg.norm(eza) < tol_EBdipole)
self.assertTrue(
np.linalg.norm(bxa - bx) / np.linalg.norm(bxa) < tol_EBdipole)
self.assertTrue(
np.linalg.norm(bza - bz) / np.linalg.norm(bza) < tol_EBdipole)
def getEBJcore(src0):
B0 = np.r_[Pfx * f[src0, "b"], Pfy * f[src0, "b"], Pfz * f[src0, "b"]]
E0 = np.r_[Pex * f[src0, "e"], Pey * f[src0, "e"], Pez * f[src0, "e"]]
J0 = utils.sdiag(np.r_[sigma_core, sigma_core, sigma_core]) * E0
return E0, B0, J0
def simpleFail(x):
return np.sin(x), -sdiag(np.cos(x))
def simpleFunction(x):
return np.sin(x), lambda xi: sdiag(np.cos(x)) * xi