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Merge branch 'master' of https://github.com/simpeg/simpeg into Examples

Browse Source 
Conflicts:
	SimPEG/Examples/__init__.py
This commit is contained in:
seogi_macbook
2016-02-10 23:38:51 -08:00
parent 433457f649 8ba9815137
commit 774d612c18
41 changed files with 3197 additions and 1573 deletions
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.travis.yml
+1 -1
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@@ -33,7 +33,7 @@ before_install:
# Install packages
install:
- conda install --yes pip python=$TRAVIS_PYTHON_VERSION numpy scipy matplotlib cython ipython nose
- conda install --yes pip python=$TRAVIS_PYTHON_VERSION numpy scipy matplotlib cython ipython nose vtk
- pip install nose-cov python-coveralls
- git clone https://github.com/rowanc1/pymatsolver.git
SimPEG/DataMisfit.py
+11 -17
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@@ -59,20 +59,6 @@ class BaseDataMisfit(object):
"""
raise NotImplementedError('This method should be overwritten.')
# TODO: implement target misfit as a property, or possibly as an inversion directive.
# def target(self, forward):
# """target(forward)
# Target for data misfit. By default this is the number of data,
# which satisfies the Discrepancy Principle.
# :rtype: float
# :return: data misfit target
# """
# prob, survey = self.splitForward(forward)
# return survey.nD
class l2_DataMisfit(BaseDataMisfit):
@@ -103,10 +89,18 @@ class l2_DataMisfit(BaseDataMisfit):
"""
if getattr(self, '_Wd', None) is None:
print 'SimPEG.l2_DataMisfit is creating default weightings for Wd.'
survey = self.survey
eps = np.linalg.norm(Utils.mkvc(survey.dobs),2)*1e-5
self._Wd = Utils.sdiag(1/(abs(survey.dobs)*survey.std+eps))
if getattr(survey,'std', None) is None:
print 'SimPEG.DataMisfit.l2_DataMisfit assigning default std of 5%'
survey.std = 0.05
if getattr(survey, 'eps', None) is None:
print 'SimPEG.DataMisfit.l2_DataMisfit assigning default eps of 1e-5 * ||dobs||'
survey.eps = np.linalg.norm(Utils.mkvc(survey.dobs),2)*1e-5
self._Wd = Utils.sdiag(1/(abs(survey.dobs)*survey.std+survey.eps))
return self._Wd
@Wd.setter
SimPEG/Directives.py
+30
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@@ -206,6 +206,36 @@ class SaveOutputEveryIteration(_SaveEveryIteration):
f.write(' %3d %1.4e %1.4e %1.4e %1.4e\n'%(self.opt.iter, self.invProb.beta, self.invProb.phi_d, self.invProb.phi_m, self.opt.f))
f.close()
class SaveOutputDictEveryIteration(_SaveEveryIteration):
"""SaveOutputDictEveryIteration"""
def initialize(self):
print "SimPEG.SaveOutputDictEveryIteration will save your inversion progress as dictionary: '###-%s.npz'"%self.fileName
def endIter(self):
# Save the data.
ms = self.reg.Ws * ( self.reg.mapping * (self.invProb.curModel - self.reg.mref) )
phi_ms = 0.5*ms.dot(ms)
if self.reg.smoothModel == True:
mref = self.reg.mref
else:
mref = 0
mx = self.reg.Wx * ( self.reg.mapping * (self.invProb.curModel - mref) )
phi_mx = 0.5 * mx.dot(mx)
if self.prob.mesh.dim==2:
my = self.reg.Wy * ( self.reg.mapping * (self.invProb.curModel - mref) )
phi_my = 0.5 * my.dot(my)
else:
phi_my = 'NaN'
if self.prob.mesh.dim==3:
mz = self.reg.Wz * ( self.reg.mapping * (self.invProb.curModel - mref) )
phi_mz = 0.5 * mz.dot(mz)
else:
phi_mz = 'NaN'
# Save the file as a npz
np.savez('{:03d}-{:s}'.format(self.opt.iter,self.fileName), iter=self.opt.iter, beta=self.invProb.beta, phi_d=self.invProb.phi_d, phi_m=self.invProb.phi_m, phi_ms=phi_ms, phi_mx=phi_mx, phi_my=phi_my, phi_mz=phi_mz,f=self.opt.f, m=self.invProb.curModel,dpred=self.invProb.dpred)
SimPEG/EM/FDEM/FDEM.py
+248 -113
View File
@@ -15,18 +15,20 @@ class BaseFDEMProblem(BaseEMProblem):
.. math ::
\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\\\
{\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{M_{\sigma}^e} \mathbf{e} = \mathbf{M^e} \mathbf{s_e}}
{\mathbf{C}^{\\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{M_{\sigma}^e} \mathbf{e} = \mathbf{s_e}}
if using the E-B formulation (:code:`Problem_e`
or :code:`Problem_b`) or the magnetic field
or :code:`Problem_b`). Note that in this case, :math:`\mathbf{s_e}` is an integrated quantity.
If we write Maxwell's equations in terms of
\\\(\\\mathbf{h}\\\) and current density \\\(\\\mathbf{j}\\\)
.. math ::
\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{j} + i \omega \mathbf{M_{\mu}^e} \mathbf{h} = \mathbf{M^e} \mathbf{s_m} \\\\
\mathbf{C}^{\\top} \mathbf{M_{\\rho}^f} \mathbf{j} + i \omega \mathbf{M_{\mu}^e} \mathbf{h} = \mathbf{s_m} \\\\
\mathbf{C} \mathbf{h} - \mathbf{j} = \mathbf{s_e}
if using the H-J formulation (:code:`Problem_j` or :code:`Problem_h`).
if using the H-J formulation (:code:`Problem_j` or :code:`Problem_h`). Note that here, :math:`\mathbf{s_m}` is an integrated quantity.
The problem performs the elimination so that we are solving the system for \\\(\\\mathbf{e},\\\mathbf{b},\\\mathbf{j} \\\) or \\\(\\\mathbf{h}\\\)
"""
@@ -36,7 +38,11 @@ class BaseFDEMProblem(BaseEMProblem):
def fields(self, m=None):
"""
Solve the forward problem for the fields.
Solve the forward problem for the fields.
:param numpy.array m: inversion model (nP,)
:rtype numpy.array:
:return F: forward solution
"""
self.curModel = m
@@ -50,16 +56,22 @@ class BaseFDEMProblem(BaseEMProblem):
Srcs = self.survey.getSrcByFreq(freq)
ftype = self._fieldType + 'Solution'
F[Srcs, ftype] = sol
Ainv.clean()
return F
def Jvec(self, m, v, f=None):
def Jvec(self, m, v, u=None):
"""
Sensitivity times a vector
Sensitivity times a vector.
:param numpy.array m: inversion model (nP,)
:param numpy.array v: vector which we take sensitivity product with (nP,)
:param SimPEG.EM.FDEM.Fields u: fields object
:rtype numpy.array:
:return: Jv (ndata,)
"""
if f is None:
f = self.fields(m)
if u is None:
u = self.fields(m)
self.curModel = m
@@ -71,33 +83,41 @@ class BaseFDEMProblem(BaseEMProblem):
for src in self.survey.getSrcByFreq(freq):
ftype = self._fieldType + 'Solution'
u_src = f[src, ftype]
u_src = u[src, ftype]
dA_dm = self.getADeriv_m(freq, u_src, v)
dRHS_dm = self.getRHSDeriv_m(freq, src, v)
du_dm = Ainv * ( - dA_dm + dRHS_dm )
for rx in src.rxList:
df_duFun = getattr(f, '_%sDeriv_u'%rx.projField, None)
df_duFun = getattr(u, '_%sDeriv_u'%rx.projField, None)
df_dudu_dm = df_duFun(src, du_dm, adjoint=False)
df_dmFun = getattr(f, '_%sDeriv_m'%rx.projField, None)
df_dmFun = getattr(u, '_%sDeriv_m'%rx.projField, None)
df_dm = df_dmFun(src, v, adjoint=False)
Df_Dm = np.array(df_dudu_dm + df_dm,dtype=complex)
P = lambda v: rx.projectFieldsDeriv(src, self.mesh, f, v) # wrt u, also have wrt m
P = lambda v: rx.projectFieldsDeriv(src, self.mesh, u, v) # wrt u, also have wrt m
Jv[src, rx] = P(Df_Dm)
Ainv.clean()
return Utils.mkvc(Jv)
def Jtvec(self, m, v, f=None):
def Jtvec(self, m, v, u=None):
"""
Sensitivity transpose times a vector
Sensitivity transpose times a vector
:param numpy.array m: inversion model (nP,)
:param numpy.array v: vector which we take adjoint product with (nP,)
:param SimPEG.EM.FDEM.Fields u: fields object
:rtype numpy.array:
:return: Jv (ndata,)
"""
if f is None:
f = self.fields(m)
if u is None:
u = self.fields(m)
self.curModel = m
@@ -113,12 +133,12 @@ class BaseFDEMProblem(BaseEMProblem):
for src in self.survey.getSrcByFreq(freq):
ftype = self._fieldType + 'Solution'
u_src = f[src, ftype]
u_src = u[src, ftype]
for rx in src.rxList:
PTv = rx.projectFieldsDeriv(src, self.mesh, f, v[src, rx], adjoint=True) # wrt u, need possibility wrt m
PTv = rx.projectFieldsDeriv(src, self.mesh, u, v[src, rx], adjoint=True) # wrt u, need possibility wrt m
df_duTFun = getattr(f, '_%sDeriv_u'%rx.projField, None)
df_duTFun = getattr(u, '_%sDeriv_u'%rx.projField, None)
df_duT = df_duTFun(src, PTv, adjoint=True)
ATinvdf_duT = ATinv * df_duT
@@ -127,11 +147,12 @@ class BaseFDEMProblem(BaseEMProblem):
dRHS_dmT = self.getRHSDeriv_m(freq,src, ATinvdf_duT, adjoint=True)
du_dmT = -dA_dmT + dRHS_dmT
df_dmFun = getattr(f, '_%sDeriv_m'%rx.projField, None)
df_dmFun = getattr(u, '_%sDeriv_m'%rx.projField, None)
dfT_dm = df_dmFun(src, PTv, adjoint=True)
du_dmT += dfT_dm
# TODO: this should be taken care of by the reciever
real_or_imag = rx.projComp
if real_or_imag is 'real':
Jtv += np.array(du_dmT,dtype=complex).real
@@ -139,16 +160,18 @@ class BaseFDEMProblem(BaseEMProblem):
Jtv += - np.array(du_dmT,dtype=complex).real
else:
raise Exception('Must be real or imag')
ATinv.clean()
return Jtv
return Utils.mkvc(Jtv)
def getSourceTerm(self, freq):
"""
Evaluates the sources for a given frequency and puts them in matrix form
Evaluates the sources for a given frequency and puts them in matrix form
:param float freq: Frequency
:rtype: numpy.ndarray (nE or nF, nSrc)
:return: S_m, S_e
:param float freq: Frequency
:rtype: (numpy.ndarray, numpy.ndarray)
:return: S_m, S_e (nE or nF, nSrc)
"""
Srcs = self.survey.getSrcByFreq(freq)
if self._eqLocs is 'FE':
@@ -172,20 +195,22 @@ class BaseFDEMProblem(BaseEMProblem):
class Problem_e(BaseFDEMProblem):
"""
By eliminating the magnetic flux density using
.. math ::
\mathbf{b} = \\frac{1}{i \omega}\\left(-\mathbf{C} \mathbf{e} + \mathbf{s_m}\\right)
we can write Maxwell's equations as a second order system in \\\(\\\mathbf{e}\\\) only:
By eliminating the magnetic flux density using
.. math ::
\\left(\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{C}+ i \omega \mathbf{M^e_{\sigma}} \\right)\mathbf{e} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{M^e}\mathbf{s_e}
\mathbf{b} = \\frac{1}{i \omega}\\left(-\mathbf{C} \mathbf{e} + \mathbf{s_m}\\right)
which we solve for \\\(\\\mathbf{e}\\\).
we can write Maxwell's equations as a second order system in \\\(\\\mathbf{e}\\\) only:
.. math ::
\\left(\mathbf{C}^{\\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{C}+ i \omega \mathbf{M^e_{\sigma}} \\right)\mathbf{e} = \mathbf{C}^{\\top} \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{M^e}\mathbf{s_e}
which we solve for :math:`\mathbf{e}`.
:param SimPEG.Mesh mesh: mesh
"""
_fieldType = 'e'
@@ -197,13 +222,16 @@ class Problem_e(BaseFDEMProblem):
def getA(self, freq):
"""
.. math ::
\mathbf{A} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{C} + i \omega \mathbf{M^e_{\sigma}}
System matrix
.. math ::
\mathbf{A} = \mathbf{C}^{\\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{C} + i \omega \mathbf{M^e_{\sigma}}
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
"""
MfMui = self.MfMui
MeSigma = self.MeSigma
C = self.mesh.edgeCurl
@@ -212,6 +240,20 @@ class Problem_e(BaseFDEMProblem):
def getADeriv_m(self, freq, u, v, adjoint=False):
"""
Product of the derivative of our system matrix with respect to the model and a vector
.. math ::
\\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}} = i \omega \\frac{d \mathbf{M^e_{\sigma}}\mathbf{v} }{d\mathbf{m}}
:param float freq: frequency
:param numpy.ndarray u: solution vector (nE,)
:param numpy.ndarray v: vector to take prodct with (nP,) or (nD,) for adjoint
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: derivative of the system matrix times a vector (nP,) or adjoint (nD,)
"""
dsig_dm = self.curModel.sigmaDeriv
dMe_dsig = self.MeSigmaDeriv(u)
@@ -222,26 +264,37 @@ class Problem_e(BaseFDEMProblem):
def getRHS(self, freq):
"""
.. math ::
\mathbf{RHS} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{M_e}\mathbf{s_e}
Right hand side for the system
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)
:return: RHS
.. math ::
\mathbf{RHS} = \mathbf{C}^{\\top} \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{M_e}\mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray
:return: RHS (nE, nSrc)
"""
S_m, S_e = self.getSourceTerm(freq)
C = self.mesh.edgeCurl
MfMui = self.MfMui
RHS = C.T * (MfMui * S_m) -1j * omega(freq) * S_e
return RHS
return C.T * (MfMui * S_m) -1j * omega(freq) * S_e
def getRHSDeriv_m(self, freq, src, v, adjoint=False):
"""
Derivative of the right hand side with respect to the model
:param float freq: frequency
:param SimPEG.EM.FDEM.Src src: FDEM source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of rhs deriv with a vector
"""
C = self.mesh.edgeCurl
MfMui = self.MfMui
S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint)
S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint=adjoint)
if adjoint:
dRHS = MfMui * (C * v)
@@ -253,20 +306,22 @@ class Problem_e(BaseFDEMProblem):
class Problem_b(BaseFDEMProblem):
"""
We eliminate \\\(\\\mathbf{e}\\\) using
We eliminate :math:`\mathbf{e}` using
.. math ::
.. math ::
\mathbf{e} = \mathbf{M^e_{\sigma}}^{-1} \\left(\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{s_e}\\right)
\mathbf{e} = \mathbf{M^e_{\sigma}}^{-1} \\left(\mathbf{C}^{\\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{s_e}\\right)
and solve for \\\(\\\mathbf{b}\\\) using:
and solve for :math:`\mathbf{b}` using:
.. math ::
.. math ::
\\left(\mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} + i \omega \\right)\mathbf{b} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{M^e}\mathbf{s_e}
\\left(\mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^{\\top} \mathbf{M_{\mu^{-1}}^f} + i \omega \\right)\mathbf{b} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{M^e}\mathbf{s_e}
.. note ::
The inverse problem will not work with full anisotropy
.. note ::
The inverse problem will not work with full anisotropy
:param SimPEG.Mesh mesh: mesh
"""
_fieldType = 'b'
@@ -278,12 +333,14 @@ class Problem_b(BaseFDEMProblem):
def getA(self, freq):
"""
.. math ::
\mathbf{A} = \mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} + i \omega
System matrix
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
.. math ::
\mathbf{A} = \mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^{\\top} \mathbf{M_{\mu^{-1}}^f} + i \omega
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
"""
MfMui = self.MfMui
@@ -299,6 +356,20 @@ class Problem_b(BaseFDEMProblem):
def getADeriv_m(self, freq, u, v, adjoint=False):
"""
Product of the derivative of our system matrix with respect to the model and a vector
.. math ::
\\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}} = \mathbf{C} \\frac{\mathbf{M^e_{\sigma}} \mathbf{v}}{d\mathbf{m}}
:param float freq: frequency
:param numpy.ndarray u: solution vector (nF,)
:param numpy.ndarray v: vector to take prodct with (nP,) or (nD,) for adjoint
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: derivative of the system matrix times a vector (nP,) or adjoint (nD,)
"""
MfMui = self.MfMui
C = self.mesh.edgeCurl
MeSigmaIDeriv = self.MeSigmaIDeriv
@@ -318,12 +389,14 @@ class Problem_b(BaseFDEMProblem):
def getRHS(self, freq):
"""
.. math ::
\mathbf{RHS} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{s_e}
Right hand side for the system
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)
:return: RHS
.. math ::
\mathbf{RHS} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray
:return: RHS (nE, nSrc)
"""
S_m, S_e = self.getSourceTerm(freq)
@@ -339,6 +412,17 @@ class Problem_b(BaseFDEMProblem):
return RHS
def getRHSDeriv_m(self, freq, src, v, adjoint=False):
"""
Derivative of the right hand side with respect to the model
:param float freq: frequency
:param SimPEG.EM.FDEM.Src src: FDEM source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of rhs deriv with a vector
"""
C = self.mesh.edgeCurl
S_m, S_e = src.eval(self)
MfMui = self.MfMui
@@ -347,7 +431,7 @@ class Problem_b(BaseFDEMProblem):
v = self.MfMui * v
MeSigmaIDeriv = self.MeSigmaIDeriv(S_e)
S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint)
S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint=adjoint)
if not adjoint:
RHSderiv = C * (MeSigmaIDeriv * v)
@@ -370,21 +454,22 @@ class Problem_b(BaseFDEMProblem):
class Problem_j(BaseFDEMProblem):
"""
We eliminate \\\(\\\mathbf{h}\\\) using
We eliminate \\\(\\\mathbf{h}\\\) using
.. math ::
.. math ::
\mathbf{h} = \\frac{1}{i \omega} \mathbf{M_{\mu}^e}^{-1} \\left(-\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{j} + \mathbf{M^e} \mathbf{s_m} \\right)
\mathbf{h} = \\frac{1}{i \omega} \mathbf{M_{\mu}^e}^{-1} \\left(-\mathbf{C}^{\\top} \mathbf{M_{\\rho}^f} \mathbf{j} + \mathbf{M^e} \mathbf{s_m} \\right)
and solve for \\\(\\\mathbf{j}\\\) using
and solve for \\\(\\\mathbf{j}\\\) using
.. math ::
.. math ::
\\left(\mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{C}^T \mathbf{M_{\\rho}^f} + i \omega\\right)\mathbf{j} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{M^e} \mathbf{s_m} -i\omega\mathbf{s_e}
\\left(\mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{C}^{\\top} \mathbf{M_{\\rho}^f} + i \omega\\right)\mathbf{j} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{M^e} \mathbf{s_m} -i\omega\mathbf{s_e}
.. note::
This implementation does not yet work with full anisotropy!!
.. note::
This implementation does not yet work with full anisotropy!!
:param SimPEG.Mesh mesh: mesh
"""
_fieldType = 'j'
@@ -396,12 +481,14 @@ class Problem_j(BaseFDEMProblem):
def getA(self, freq):
"""
.. math ::
\\mathbf{A} = \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C}^T \\mathbf{M^f_{\\sigma^{-1}}} + i\\omega
System matrix
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
.. math ::
\\mathbf{A} = \\mathbf{C} \\mathbf{M^e_{\\mu^{-1}}} \\mathbf{C}^{\\top} \\mathbf{M^f_{\\sigma^{-1}}} + i\\omega
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
"""
MeMuI = self.MeMuI
@@ -418,12 +505,20 @@ class Problem_j(BaseFDEMProblem):
def getADeriv_m(self, freq, u, v, adjoint=False):
"""
In this case, we assume that electrical conductivity, \\\(\\\sigma\\\) is the physical property of interest (i.e. \\\(\\\sigma\\\) = model.transform). Then we want
Product of the derivative of our system matrix with respect to the model and a vector
.. math ::
In this case, we assume that electrical conductivity, :math:`\sigma` is the physical property of interest (i.e. :math:`\sigma` = model.transform). Then we want
\\frac{\mathbf{A(\sigma)} \mathbf{v}}{d \\mathbf{m}} &= \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{m}}
&= \\mathbf{C} \\mathbf{M^e_{mu}^{-1}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{\\sigma^{-1}}} \\frac{d \\mathbf{\\sigma^{-1}}}{d \\mathbf{\\sigma}} \\frac{d \\mathbf{\\sigma}}{d \\mathbf{m}}
.. math ::
\\frac{\mathbf{A(\sigma)} \mathbf{v}}{d \mathbf{m}} = \mathbf{C} \mathbf{M^e_{mu^{-1}}} \mathbf{C^{\\top}} \\frac{d \mathbf{M^f_{\sigma^{-1}}}\mathbf{v} }{d \mathbf{m}}
:param float freq: frequency
:param numpy.ndarray u: solution vector (nF,)
:param numpy.ndarray v: vector to take prodct with (nP,) or (nD,) for adjoint
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: derivative of the system matrix times a vector (nP,) or adjoint (nD,)
"""
MeMuI = self.MeMuI
@@ -443,12 +538,15 @@ class Problem_j(BaseFDEMProblem):
def getRHS(self, freq):
"""
.. math ::
Right hand side for the system
\mathbf{RHS} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1}\mathbf{s_m} -i\omega \mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)
:return: RHS
.. math ::
\mathbf{RHS} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1}\mathbf{s_m} -i\omega \mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)
:return: RHS
"""
S_m, S_e = self.getSourceTerm(freq)
@@ -463,9 +561,20 @@ class Problem_j(BaseFDEMProblem):
return RHS
def getRHSDeriv_m(self, freq, src, v, adjoint=False):
"""
Derivative of the right hand side with respect to the model
:param float freq: frequency
:param SimPEG.EM.FDEM.Src src: FDEM source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of rhs deriv with a vector
"""
C = self.mesh.edgeCurl
MeMuI = self.MeMuI
S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint)
S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint=adjoint)
if adjoint:
if self._makeASymmetric:
@@ -486,18 +595,19 @@ class Problem_j(BaseFDEMProblem):
class Problem_h(BaseFDEMProblem):
"""
We eliminate \\\(\\\mathbf{j}\\\) using
We eliminate \\\(\\\mathbf{j}\\\) using
.. math ::
.. math ::
\mathbf{j} = \mathbf{C} \mathbf{h} - \mathbf{s_e}
\mathbf{j} = \mathbf{C} \mathbf{h} - \mathbf{s_e}
and solve for \\\(\\\mathbf{h}\\\) using
and solve for \\\(\\\mathbf{h}\\\) using
.. math ::
.. math ::
\\left(\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}\\right) \mathbf{h} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{s_e}
\\left(\mathbf{C}^{\\top} \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}\\right) \mathbf{h} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^{\\top} \mathbf{M_{\\rho}^f} \mathbf{s_e}
:param SimPEG.Mesh mesh: mesh
"""
_fieldType = 'h'
@@ -509,13 +619,14 @@ class Problem_h(BaseFDEMProblem):
def getA(self, freq):
"""
.. math ::
System matrix
\mathbf{A} = \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}
.. math::
\mathbf{A} = \mathbf{C}^{\\top} \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
"""
MeMu = self.MeMu
@@ -525,6 +636,19 @@ class Problem_h(BaseFDEMProblem):
return C.T * (MfRho * C) + 1j*omega(freq)*MeMu
def getADeriv_m(self, freq, u, v, adjoint=False):
"""
Product of the derivative of our system matrix with respect to the model and a vector
.. math::
\\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}} = \mathbf{C}^{\\top}\\frac{d \mathbf{M^f_{\\rho}}\mathbf{v} }{d\mathbf{m}}
:param float freq: frequency
:param numpy.ndarray u: solution vector (nE,)
:param numpy.ndarray v: vector to take prodct with (nP,) or (nD,) for adjoint
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: derivative of the system matrix times a vector (nP,) or adjoint (nD,)
"""
MeMu = self.MeMu
C = self.mesh.edgeCurl
@@ -536,24 +660,35 @@ class Problem_h(BaseFDEMProblem):
def getRHS(self, freq):
"""
.. math ::
Right hand side for the system
\mathbf{RHS} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{s_e}
.. math ::
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)
:return: RHS
\mathbf{RHS} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^{\\top} \mathbf{M_{\\rho}^f} \mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray
:return: RHS (nE, nSrc)
"""
S_m, S_e = self.getSourceTerm(freq)
C = self.mesh.edgeCurl
MfRho = self.MfRho
RHS = S_m + C.T * ( MfRho * S_e )
return RHS
return S_m + C.T * ( MfRho * S_e )
def getRHSDeriv_m(self, freq, src, v, adjoint=False):
"""
Derivative of the right hand side with respect to the model
:param float freq: frequency
:param SimPEG.EM.FDEM.Src src: FDEM source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of rhs deriv with a vector
"""
_, S_e = src.eval(self)
C = self.mesh.edgeCurl
MfRho = self.MfRho
@@ -564,7 +699,7 @@ class Problem_h(BaseFDEMProblem):
elif adjoint:
RHSDeriv = MfRhoDeriv.T * (C * v)
S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint)
S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint=adjoint)
return RHSDeriv + S_mDeriv(v) + C.T * (MfRho * S_eDeriv(v))
SimPEG/EM/FDEM/FieldsFDEM.py
+512 -9
View File
@@ -7,11 +7,39 @@ from SimPEG.Utils import Zero, Identity
class Fields(SimPEG.Problem.Fields):
"""Fancy Field Storage for a FDEM survey."""
"""
Fancy Field Storage for a FDEM survey. Only one field type is stored for
each problem, the rest are computed. The fields obejct acts like an array and is indexed by
.. code-block:: python
f = problem.fields(m)
e = f[srcList,'e']
b = f[srcList,'b']
If accessing all sources for a given field, use the :code:`:`
.. code-block:: python
f = problem.fields(m)
e = f[:,'e']
b = f[:,'b']
The array returned will be size (nE or nF, nSrcs :math:`\\times` nFrequencies)
"""
knownFields = {}
dtype = complex
class Fields_e(Fields):
"""
Fields object for Problem_e.
:param Mesh mesh: mesh
:param Survey survey: survey
"""
knownFields = {'eSolution':'E'}
aliasFields = {
'e' : ['eSolution','E','_e'],
@@ -30,6 +58,15 @@ class Fields_e(Fields):
self._edgeCurl = self.survey.prob.mesh.edgeCurl
def _ePrimary(self, eSolution, srcList):
"""
Primary electric field from source
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary electric field as defined by the sources
"""
ePrimary = np.zeros_like(eSolution)
for i, src in enumerate(srcList):
ep = src.ePrimary(self.prob)
@@ -37,19 +74,67 @@ class Fields_e(Fields):
return ePrimary
def _eSecondary(self, eSolution, srcList):
"""
Secondary electric field is the thing we solved for
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary electric field
"""
return eSolution
def _e(self, eSolution, srcList):
"""
Total electric field is sum of primary and secondary
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: total electric field
"""
return self._ePrimary(eSolution,srcList) + self._eSecondary(eSolution,srcList)
def _eDeriv_u(self, src, v, adjoint = False):
"""
Derivative of the total electric field with respect to the thing we
solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the electric field with respect to the field we solved for with a vector
"""
return Identity()*v
def _eDeriv_m(self, src, v, adjoint = False):
"""
Derivative of the total electric field with respect to the inversion model. Here, we assume that the primary does not depend on the model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: SimPEG.Utils.Zero
:return: product of the electric field derivative with respect to the inversion model with a vector
"""
# assuming primary does not depend on the model
return Zero()
def _bPrimary(self, eSolution, srcList):
"""
Primary magnetic flux density from source
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary magnetic flux density as defined by the sources
"""
bPrimary = np.zeros([self._edgeCurl.shape[0],eSolution.shape[1]],dtype = complex)
for i, src in enumerate(srcList):
bp = src.bPrimary(self.prob)
@@ -57,6 +142,15 @@ class Fields_e(Fields):
return bPrimary
def _bSecondary(self, eSolution, srcList):
"""
Secondary magnetic flux density from eSolution
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary magnetic flux density
"""
C = self._edgeCurl
b = (C * eSolution)
for i, src in enumerate(srcList):
@@ -66,29 +160,84 @@ class Fields_e(Fields):
return b
def _bSecondaryDeriv_u(self, src, v, adjoint = False):
"""
Derivative of the secondary magnetic flux density with respect to the thing we solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the secondary magnetic flux density with respect to the field we solved for with a vector
"""
C = self._edgeCurl
if adjoint:
return - 1./(1j*omega(src.freq)) * (C.T * v)
return - 1./(1j*omega(src.freq)) * (C * v)
def _bSecondaryDeriv_m(self, src, v, adjoint = False):
S_mDeriv, _ = src.evalDeriv(self.prob, adjoint)
S_mDeriv = S_mDeriv(v)
"""
Derivative of the secondary magnetic flux density with respect to the inversion model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the secondary magnetic flux density derivative with respect to the inversion model with a vector
"""
S_mDeriv, _ = src.evalDeriv(self.prob, v, adjoint)
return 1./(1j * omega(src.freq)) * S_mDeriv
def _b(self, eSolution, srcList):
"""
Total magnetic flux density is sum of primary and secondary
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: total magnetic flux density
"""
return self._bPrimary(eSolution, srcList) + self._bSecondary(eSolution, srcList)
def _bDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the total magnetic flux density with respect to the thing we solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the magnetic flux density with respect to the field we solved for with a vector
"""
# Primary does not depend on u
return self._bSecondaryDeriv_u(src, v, adjoint)
def _bDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the total magnetic flux density with respect to the inversion model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: SimPEG.Utils.Zero
:return: product of the magnetic flux density derivative with respect to the inversion model with a vector
"""
# Assuming the primary does not depend on the model
return self._bSecondaryDeriv_m(src, v, adjoint)
class Fields_b(Fields):
"""
Fields object for Problem_b.
:param Mesh mesh: mesh
:param Survey survey: survey
"""
knownFields = {'bSolution':'F'}
aliasFields = {
'b' : ['bSolution','F','_b'],
@@ -111,6 +260,15 @@ class Fields_b(Fields):
self._Me = self.survey.prob.Me
def _bPrimary(self, bSolution, srcList):
"""
Primary magnetic flux density from source
:param numpy.ndarray bSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary electric field as defined by the sources
"""
bPrimary = np.zeros_like(bSolution)
for i, src in enumerate(srcList):
bp = src.bPrimary(self.prob)
@@ -118,19 +276,66 @@ class Fields_b(Fields):
return bPrimary
def _bSecondary(self, bSolution, srcList):
"""
Secondary magnetic flux density is the thing we solved for
:param numpy.ndarray bSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary magnetic flux density
"""
return bSolution
def _b(self, bSolution, srcList):
"""
Total magnetic flux density is sum of primary and secondary
:param numpy.ndarray bSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: total magnetic flux density
"""
return self._bPrimary(bSolution, srcList) + self._bSecondary(bSolution, srcList)
def _bDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the total magnetic flux density with respect to the thing we
solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the magnetic flux density with respect to the field we solved for with a vector
"""
return Identity()*v
def _bDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the total magnetic flux density with respect to the inversion model. Here, we assume that the primary does not depend on the model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: SimPEG.Utils.Zero
:return: product of the magnetic flux density derivative with respect to the inversion model with a vector
"""
# assuming primary does not depend on the model
return Zero()
def _ePrimary(self, bSolution, srcList):
"""
Primary electric field from source
:param numpy.ndarray bSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary electric field as defined by the sources
"""
ePrimary = np.zeros([self._edgeCurl.shape[1],bSolution.shape[1]],dtype = complex)
for i,src in enumerate(srcList):
ep = src.ePrimary(self.prob)
@@ -138,6 +343,15 @@ class Fields_b(Fields):
return ePrimary
def _eSecondary(self, bSolution, srcList):
"""
Secondary electric field from bSolution
:param numpy.ndarray bSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary electric field
"""
e = self._MeSigmaI * ( self._edgeCurl.T * ( self._MfMui * bSolution))
for i,src in enumerate(srcList):
_,S_e = src.eval(self.prob)
@@ -145,12 +359,32 @@ class Fields_b(Fields):
return e
def _eSecondaryDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the secondary electric field with respect to the thing we solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the secondary electric field with respect to the field we solved for with a vector
"""
if not adjoint:
return self._MeSigmaI * ( self._edgeCurl.T * ( self._MfMui * v) )
else:
return self._MfMui.T * (self._edgeCurl * (self._MeSigmaI.T * v))
def _eSecondaryDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the secondary electric field with respect to the inversion model
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the secondary electric field with respect to the model with a vector
"""
bSolution = self[[src],'bSolution']
_,S_e = src.eval(self.prob)
Me = self._Me
@@ -166,25 +400,60 @@ class Fields_b(Fields):
elif adjoint:
de_dm = self._MeSigmaIDeriv(w).T * v
_, S_eDeriv = src.evalDeriv(self.prob, adjoint)
Se_Deriv = S_eDeriv(v)
_, S_eDeriv = src.evalDeriv(self.prob, v, adjoint)
de_dm = de_dm - self._MeSigmaI * Se_Deriv
de_dm = de_dm - self._MeSigmaI * S_eDeriv
return de_dm
def _e(self, bSolution, srcList):
"""
Total electric field is sum of primary and secondary
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: total electric field
"""
return self._ePrimary(bSolution, srcList) + self._eSecondary(bSolution, srcList)
def _eDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the total electric field with respect to the thing we solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the electric field with respect to the field we solved for with a vector
"""
return self._eSecondaryDeriv_u(src, v, adjoint)
def _eDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the total electric field density with respect to the inversion model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the electric field derivative with respect to the inversion model with a vector
"""
# assuming primary doesn't depend on model
return self._eSecondaryDeriv_m(src, v, adjoint)
class Fields_j(Fields):
"""
Fields object for Problem_j.
:param Mesh mesh: mesh
:param Survey survey: survey
"""
knownFields = {'jSolution':'F'}
aliasFields = {
'j' : ['jSolution','F','_j'],
@@ -207,6 +476,15 @@ class Fields_j(Fields):
self._Me = self.survey.prob.Me
def _jPrimary(self, jSolution, srcList):
"""
Primary current density from source
:param numpy.ndarray jSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary current density as defined by the sources
"""
jPrimary = np.zeros_like(jSolution,dtype = complex)
for i, src in enumerate(srcList):
jp = src.jPrimary(self.prob)
@@ -214,19 +492,66 @@ class Fields_j(Fields):
return jPrimary
def _jSecondary(self, jSolution, srcList):
"""
Secondary current density is the thing we solved for
:param numpy.ndarray jSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary current density
"""
return jSolution
def _j(self, jSolution, srcList):
"""
Total current density is sum of primary and secondary
:param numpy.ndarray jSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: total current density
"""
return self._jPrimary(jSolution, srcList) + self._jSecondary(jSolution, srcList)
def _jDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the total current density with respect to the thing we
solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the current density with respect to the field we solved for with a vector
"""
return Identity()*v
def _jDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the total current density with respect to the inversion model. Here, we assume that the primary does not depend on the model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: SimPEG.Utils.Zero
:return: product of the current density derivative with respect to the inversion model with a vector
"""
# assuming primary does not depend on the model
return Zero()
def _hPrimary(self, jSolution, srcList):
"""
Primary magnetic field from source
:param numpy.ndarray hSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary magnetic field as defined by the sources
"""
hPrimary = np.zeros([self._edgeCurl.shape[1],jSolution.shape[1]],dtype = complex)
for i, src in enumerate(srcList):
hp = src.hPrimary(self.prob)
@@ -234,6 +559,15 @@ class Fields_j(Fields):
return hPrimary
def _hSecondary(self, jSolution, srcList):
"""
Secondary magnetic field from bSolution
:param numpy.ndarray jSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary magnetic field
"""
h = self._MeMuI * (self._edgeCurl.T * (self._MfRho * jSolution) )
for i, src in enumerate(srcList):
h[:,i] *= -1./(1j*omega(src.freq))
@@ -242,12 +576,32 @@ class Fields_j(Fields):
return h
def _hSecondaryDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the secondary magnetic field with respect to the thing we solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the secondary magnetic field with respect to the field we solved for with a vector
"""
if not adjoint:
return -1./(1j*omega(src.freq)) * self._MeMuI * (self._edgeCurl.T * (self._MfRho * v) )
elif adjoint:
return -1./(1j*omega(src.freq)) * self._MfRho.T * (self._edgeCurl * ( self._MeMuI.T * v))
def _hSecondaryDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the secondary magnetic field with respect to the inversion model
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the secondary magnetic field with respect to the model with a vector
"""
jSolution = self[[src],'jSolution']
MeMuI = self._MeMuI
C = self._edgeCurl
@@ -260,7 +614,7 @@ class Fields_j(Fields):
elif adjoint:
hDeriv_m = -1./(1j*omega(src.freq)) * MfRhoDeriv(jSolution).T * ( C * (MeMuI.T * v ) )
S_mDeriv,_ = src.evalDeriv(self.prob, adjoint)
S_mDeriv,_ = src.evalDeriv(self.prob, adjoint = adjoint)
if not adjoint:
S_mDeriv = S_mDeriv(v)
@@ -272,17 +626,53 @@ class Fields_j(Fields):
def _h(self, jSolution, srcList):
"""
Total magnetic field is sum of primary and secondary
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: total magnetic field
"""
return self._hPrimary(jSolution, srcList) + self._hSecondary(jSolution, srcList)
def _hDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the total magnetic field with respect to the thing we solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the magnetic field with respect to the field we solved for with a vector
"""
return self._hSecondaryDeriv_u(src, v, adjoint)
def _hDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the total magnetic field density with respect to the inversion model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the magnetic field derivative with respect to the inversion model with a vector
"""
# assuming the primary doesn't depend on the model
return self._hSecondaryDeriv_m(src, v, adjoint)
class Fields_h(Fields):
"""
Fields object for Problem_h.
:param Mesh mesh: mesh
:param Survey survey: survey
"""
knownFields = {'hSolution':'E'}
aliasFields = {
'h' : ['hSolution','E','_h'],
@@ -303,6 +693,15 @@ class Fields_h(Fields):
self._MfRho = self.survey.prob.MfRho
def _hPrimary(self, hSolution, srcList):
"""
Primary magnetic field from source
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary magnetic field as defined by the sources
"""
hPrimary = np.zeros_like(hSolution,dtype = complex)
for i, src in enumerate(srcList):
hp = src.hPrimary(self.prob)
@@ -310,19 +709,67 @@ class Fields_h(Fields):
return hPrimary
def _hSecondary(self, hSolution, srcList):
"""
Secondary magnetic field is the thing we solved for
:param numpy.ndarray hSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary magnetic field
"""
return hSolution
def _h(self, hSolution, srcList):
"""
Total magnetic field is sum of primary and secondary
:param numpy.ndarray hSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: total magnetic field
"""
return self._hPrimary(hSolution, srcList) + self._hSecondary(hSolution, srcList)
def _hDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the total magnetic field with respect to the thing we
solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the magnetic field with respect to the field we solved for with a vector
"""
return Identity()*v
def _hDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the total magnetic field with respect to the inversion model. Here, we assume that the primary does not depend on the model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: SimPEG.Utils.Zero
:return: product of the magnetic field derivative with respect to the inversion model with a vector
"""
# assuming primary does not depend on the model
return Zero()
def _jPrimary(self, hSolution, srcList):
"""
Primary current density from source
:param numpy.ndarray hSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary current density as defined by the sources
"""
jPrimary = np.zeros([self._edgeCurl.shape[0], hSolution.shape[1]], dtype = complex)
for i, src in enumerate(srcList):
jp = src.jPrimary(self.prob)
@@ -330,6 +777,15 @@ class Fields_h(Fields):
return jPrimary
def _jSecondary(self, hSolution, srcList):
"""
Secondary current density from eSolution
:param numpy.ndarray hSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary current density
"""
j = self._edgeCurl*hSolution
for i, src in enumerate(srcList):
_,S_e = src.eval(self.prob)
@@ -337,22 +793,69 @@ class Fields_h(Fields):
return j
def _jSecondaryDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the secondary current density with respect to the thing we solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the secondary current density with respect to the field we solved for with a vector
"""
if not adjoint:
return self._edgeCurl*v
elif adjoint:
return self._edgeCurl.T*v
def _jSecondaryDeriv_m(self, src, v, adjoint=False):
_,S_eDeriv = src.evalDeriv(self.prob, adjoint)
S_eDeriv = S_eDeriv(v)
"""
Derivative of the secondary current density with respect to the inversion model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the secondary current density derivative with respect to the inversion model with a vector
"""
_,S_eDeriv = src.evalDeriv(self.prob, v, adjoint)
return -S_eDeriv
def _j(self, hSolution, srcList):
"""
Total current density is sum of primary and secondary
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: total current density
"""
return self._jPrimary(hSolution, srcList) + self._jSecondary(hSolution, srcList)
def _jDeriv_u(self, src, v, adjoint=False):
"""
Derivative of the total current density with respect to the thing we solved for
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of the derivative of the current density with respect to the field we solved for with a vector
"""
return self._jSecondaryDeriv_u(src,v,adjoint)
def _jDeriv_m(self, src, v, adjoint=False):
"""
Derivative of the total current density with respect to the inversion model.
:param SimPEG.EM.FDEM.Src src: source
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: SimPEG.Utils.Zero
:return: product of the current density with respect to the inversion model with a vector
"""
# assuming the primary does not depend on the model
return self._jSecondaryDeriv_m(src,v,adjoint)
SimPEG/EM/FDEM/SrcFDEM.py
+274 -19
View File
@@ -1,55 +1,141 @@
from SimPEG import Survey, Problem, Utils, np, sp
from scipy.constants import mu_0
from SimPEG.EM.Utils import *
from SimPEG.Utils import Zero
# from SurveyFDEM import Rx
from SimPEG.Utils import Zero
class BaseSrc(Survey.BaseSrc):
"""
Base source class for FDEM Survey
"""
freq = None
# rxPair = Rx
# rxPair = RxFDEM
integrate = True
def eval(self, prob):
"""
Evaluate the source terms.
- :math:`S_m` : magnetic source term
- :math:`S_e` : electric source term
:param Problem prob: FDEM Problem
:rtype: (numpy.ndarray, numpy.ndarray)
:return: tuple with magnetic source term and electric source term
"""
S_m = self.S_m(prob)
S_e = self.S_e(prob)
return S_m, S_e
def evalDeriv(self, prob, v, adjoint=False):
return lambda v: self.S_mDeriv(prob,v,adjoint), lambda v: self.S_eDeriv(prob,v,adjoint)
def evalDeriv(self, prob, v=None, adjoint=False):
"""
Derivatives of the source terms with respect to the inversion model
- :code:`S_mDeriv` : derivative of the magnetic source term
- :code:`S_eDeriv` : derivative of the electric source term
:param Problem prob: FDEM Problem
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: (numpy.ndarray, numpy.ndarray)
:return: tuple with magnetic source term and electric source term derivatives times a vector
"""
if v is not None:
return self.S_mDeriv(prob,v,adjoint), self.S_eDeriv(prob,v,adjoint)
else:
return lambda v: self.S_mDeriv(prob,v,adjoint), lambda v: self.S_eDeriv(prob,v,adjoint)
def bPrimary(self, prob):
"""
Primary magnetic flux density
:param Problem prob: FDEM Problem
:rtype: numpy.ndarray
:return: primary magnetic flux density
"""
return Zero()
def hPrimary(self, prob):
"""
Primary magnetic field
:param Problem prob: FDEM Problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
return Zero()
def ePrimary(self, prob):
"""
Primary electric field
:param Problem prob: FDEM Problem
:rtype: numpy.ndarray
:return: primary electric field
"""
return Zero()
def jPrimary(self, prob):
"""
Primary current density
:param Problem prob: FDEM Problem
:rtype: numpy.ndarray
:return: primary current density
"""
return Zero()
def S_m(self, prob):
"""
Magnetic source term
:param Problem prob: FDEM Problem
:rtype: numpy.ndarray
:return: magnetic source term on mesh
"""
return Zero()
def S_e(self, prob):
"""
Electric source term
:param Problem prob: FDEM Problem
:rtype: numpy.ndarray
:return: electric source term on mesh
"""
return Zero()
def S_mDeriv(self, prob, v, adjoint = False):
"""
Derivative of magnetic source term with respect to the inversion model
:param Problem prob: FDEM Problem
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of magnetic source term derivative with a vector
"""
return Zero()
def S_eDeriv(self, prob, v, adjoint = False):
"""
Derivative of electric source term with respect to the inversion model
:param Problem prob: FDEM Problem
:param numpy.ndarray v: vector to take product with
:param bool adjoint: adjoint?
:rtype: numpy.ndarray
:return: product of electric source term derivative with a vector
"""
return Zero()
class RawVec_e(BaseSrc):
"""
RawVec electric source. It is defined by the user provided vector S_e
RawVec electric source. It is defined by the user provided vector S_e
:param numpy.array S_e: electric source term
:param float freq: frequency
:param rxList: receiver list
:param list rxList: receiver list
:param float freq: frequency
:param numpy.array S_e: electric source term
"""
def __init__(self, rxList, freq, S_e): #, ePrimary=None, bPrimary=None, hPrimary=None, jPrimary=None):
@@ -58,16 +144,17 @@ class RawVec_e(BaseSrc):
BaseSrc.__init__(self, rxList)
def S_e(self, prob):
return self._S_e
class RawVec_m(BaseSrc):
"""
RawVec magnetic source. It is defined by the user provided vector S_m
RawVec magnetic source. It is defined by the user provided vector S_m
:param numpy.array S_m: magnetic source term
:param float freq: frequency
:param rxList: receiver list
:param float freq: frequency
:param rxList: receiver list
:param numpy.array S_m: magnetic source term
"""
def __init__(self, rxList, freq, S_m, integrate = True): #ePrimary=Zero(), bPrimary=Zero(), hPrimary=Zero(), jPrimary=Zero()):
@@ -78,17 +165,24 @@ class RawVec_m(BaseSrc):
BaseSrc.__init__(self, rxList)
def S_m(self, prob):
"""
Magnetic source term
:param Problem prob: FDEM Problem
:rtype: numpy.ndarray
:return: magnetic source term on mesh
"""
return self._S_m
class RawVec(BaseSrc):
"""
RawVec source. It is defined by the user provided vectors S_m, S_e
RawVec source. It is defined by the user provided vectors S_m, S_e
:param numpy.array S_m: magnetic source term
:param numpy.array S_e: electric source term
:param float freq: frequency
:param rxList: receiver list
:param rxList: receiver list
:param float freq: frequency
:param numpy.array S_m: magnetic source term
:param numpy.array S_e: electric source term
"""
def __init__(self, rxList, freq, S_m, S_e, integrate = True):
self._S_m = np.array(S_m,dtype=complex)
@@ -109,6 +203,51 @@ class RawVec(BaseSrc):
class MagDipole(BaseSrc):
"""
Point magnetic dipole source calculated by taking the curl of a magnetic
vector potential. By taking the discrete curl, we ensure that the magnetic
flux density is divergence free (no magnetic monopoles!).
This approach uses a primary-secondary in frequency. Here we show the
derivation for E-B formulation noting that similar steps are followed for
the H-J formulation.
.. math::
\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\\\
{\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{M_{\sigma}^e} \mathbf{e} = \mathbf{s_e}}
We split up the fields and :math:`\mu^{-1}` into primary (:math:`\mathbf{P}`) and secondary (:math:`\mathbf{S}`) components
- :math:`\mathbf{e} = \mathbf{e^P} + \mathbf{e^S}`
- :math:`\mathbf{b} = \mathbf{b^P} + \mathbf{b^S}`
- :math:`\\boldsymbol{\mu}^{\mathbf{-1}} = \\boldsymbol{\mu}^{\mathbf{-1}^\mathbf{P}} + \\boldsymbol{\mu}^{\mathbf{-1}^\mathbf{S}}`
and define a zero-frequency primary problem, noting that the source is
generated by a divergence free electric current
.. math::
\mathbf{C} \mathbf{e^P} = \mathbf{s_m^P} = 0 \\\\
{\mathbf{C}^T \mathbf{{M_{\mu^{-1}}^f}^P} \mathbf{b^P} - \mathbf{M_{\sigma}^e} \mathbf{e^P} = \mathbf{M^e} \mathbf{s_e^P}}
Since :math:`\mathbf{e^P}` is curl-free, divergence-free, we assume that there is no constant field background, the :math:`\mathbf{e^P} = 0`, so our primary problem is
.. math::
\mathbf{e^P} = 0 \\\\
{\mathbf{C}^T \mathbf{{M_{\mu^{-1}}^f}^P} \mathbf{b^P} = \mathbf{s_e^P}}
Our secondary problem is then
.. math::
\mathbf{C} \mathbf{e^S} + i \omega \mathbf{b^S} = - i \omega \mathbf{b^P} \\\\
{\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b^S} - \mathbf{M_{\sigma}^e} \mathbf{e^S} = -\mathbf{C}^T \mathbf{{M_{\mu^{-1}}^f}^S} \mathbf{b^P}}
:param list rxList: receiver list
:param float freq: frequency
:param numpy.ndarray loc: source location (ie: :code:`np.r_[xloc,yloc,zloc]`)
:param string orientation: 'X', 'Y', 'Z'
:param float moment: magnetic dipole moment
:param float mu: background magnetic permeability
"""
#TODO: right now, orientation doesn't actually do anything! The methods in SrcUtils should take care of that
def __init__(self, rxList, freq, loc, orientation='Z', moment=1., mu = mu_0):
@@ -121,6 +260,13 @@ class MagDipole(BaseSrc):
BaseSrc.__init__(self, rxList)
def bPrimary(self, prob):
"""
The primary magnetic flux density from a magnetic vector potential
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
eqLocs = prob._eqLocs
if eqLocs is 'FE':
@@ -152,14 +298,37 @@ class MagDipole(BaseSrc):
return C*a
def hPrimary(self, prob):
"""
The primary magnetic field from a magnetic vector potential
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
b = self.bPrimary(prob)
return h_from_b(prob,b)
def S_m(self, prob):
"""
The magnetic source term
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
b_p = self.bPrimary(prob)
return -1j*omega(self.freq)*b_p
def S_e(self, prob):
"""
The electric source term
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
if all(np.r_[self.mu] == np.r_[prob.curModel.mu]):
return Zero()
else:
@@ -179,6 +348,21 @@ class MagDipole(BaseSrc):
class MagDipole_Bfield(BaseSrc):
"""
Point magnetic dipole source calculated with the analytic solution for the
fields from a magnetic dipole. No discrete curl is taken, so the magnetic
flux density may not be strictly divergence free.
This approach uses a primary-secondary in frequency in the same fashion as the MagDipole.
:param list rxList: receiver list
:param float freq: frequency
:param numpy.ndarray loc: source location (ie: :code:`np.r_[xloc,yloc,zloc]`)
:param string orientation: 'X', 'Y', 'Z'
:param float moment: magnetic dipole moment
:param float mu: background magnetic permeability
"""
#TODO: right now, orientation doesn't actually do anything! The methods in SrcUtils should take care of that
#TODO: neither does moment
def __init__(self, rxList, freq, loc, orientation='Z', moment=1., mu = mu_0):
@@ -190,6 +374,14 @@ class MagDipole_Bfield(BaseSrc):
BaseSrc.__init__(self, rxList)
def bPrimary(self, prob):
"""
The primary magnetic flux density from the analytic solution for magnetic fields from a dipole
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
eqLocs = prob._eqLocs
if eqLocs is 'FE':
@@ -221,14 +413,35 @@ class MagDipole_Bfield(BaseSrc):
return b
def hPrimary(self, prob):
"""
The primary magnetic field from a magnetic vector potential
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
b = self.bPrimary(prob)
return h_from_b(prob, b)
def S_m(self, prob):
"""
The magnetic source term
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
b = self.bPrimary(prob)
return -1j*omega(self.freq)*b
def S_e(self, prob):
"""
The electric source term
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
if all(np.r_[self.mu] == np.r_[prob.curModel.mu]):
return Zero()
else:
@@ -247,6 +460,20 @@ class MagDipole_Bfield(BaseSrc):
class CircularLoop(BaseSrc):
"""
Circular loop magnetic source calculated by taking the curl of a magnetic
vector potential. By taking the discrete curl, we ensure that the magnetic
flux density is divergence free (no magnetic monopoles!).
This approach uses a primary-secondary in frequency in the same fashion as the MagDipole.
:param list rxList: receiver list
:param float freq: frequency
:param numpy.ndarray loc: source location (ie: :code:`np.r_[xloc,yloc,zloc]`)
:param string orientation: 'X', 'Y', 'Z'
:param float moment: magnetic dipole moment
:param float mu: background magnetic permeability
"""
#TODO: right now, orientation doesn't actually do anything! The methods in SrcUtils should take care of that
def __init__(self, rxList, freq, loc, orientation='Z', radius = 1., mu=mu_0):
@@ -259,6 +486,13 @@ class CircularLoop(BaseSrc):
BaseSrc.__init__(self, rxList)
def bPrimary(self, prob):
"""
The primary magnetic flux density from a magnetic vector potential
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
eqLocs = prob._eqLocs
if eqLocs is 'FE':
@@ -289,14 +523,35 @@ class CircularLoop(BaseSrc):
return C*a
def hPrimary(self, prob):
"""
The primary magnetic field from a magnetic vector potential
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
b = self.bPrimary(prob)
return 1./self.mu*b
def S_m(self, prob):
"""
The magnetic source term
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
b = self.bPrimary(prob)
return -1j*omega(self.freq)*b
def S_e(self, prob):
"""
The electric source term
:param Problem prob: FDEM problem
:rtype: numpy.ndarray
:return: primary magnetic field
"""
if all(np.r_[self.mu] == np.r_[prob.curModel.mu]):
return Zero()
else:
SimPEG/EM/FDEM/SurveyFDEM.py
+42 -2
View File
@@ -10,6 +10,12 @@ import SrcFDEM as Src
####################################################
class Rx(SimPEG.Survey.BaseRx):
"""
Frequency domain receivers
:param numpy.ndarray locs: receiver locations (ie. :code:`np.r_[x,y,z]`)
:param string rxType: reciever type from knownRxTypes
"""
knownRxTypes = {
'exr':['e', 'Ex', 'real'],
@@ -61,6 +67,15 @@ class Rx(SimPEG.Survey.BaseRx):
return self.knownRxTypes[self.rxType][2]
def projectFields(self, src, mesh, u):
"""
Project fields to recievers to get data.
:param Source src: FDEM source
:param Mesh mesh: mesh used
:param Fields u: fields object
:rtype: numpy.ndarray
:return: fields projected to recievers
"""
P = self.getP(mesh)
u_part_complex = u[src, self.projField]
# get the real or imag component
@@ -69,6 +84,16 @@ class Rx(SimPEG.Survey.BaseRx):
return P*u_part
def projectFieldsDeriv(self, src, mesh, u, v, adjoint=False):
"""
Derivative of projected fields with respect to the inversion model times a vector.
:param Source src: FDEM source
:param Mesh mesh: mesh used
:param Fields u: fields object
:param numpy.ndarray v: vector to multiply
:rtype: numpy.ndarray
:return: fields projected to recievers
"""
P = self.getP(mesh)
if not adjoint:
@@ -95,10 +120,13 @@ class Rx(SimPEG.Survey.BaseRx):
class Survey(SimPEG.Survey.BaseSurvey):
"""
docstring for SurveyFDEM
Frequency domain electromagnetic survey
:param list srcList: list of FDEM sources used in the survey
"""
srcPair = Src.BaseSrc
rxPaair = Rx
def __init__(self, srcList, **kwargs):
# Sort these by frequency
@@ -126,6 +154,7 @@ class Survey(SimPEG.Survey.BaseSurvey):
@property
def nSrcByFreq(self):
"""Number of sources at each frequency"""
if getattr(self, '_nSrcByFreq', None) is None:
self._nSrcByFreq = {}
for freq in self.freqs:
@@ -133,11 +162,22 @@ class Survey(SimPEG.Survey.BaseSurvey):
return self._nSrcByFreq
def getSrcByFreq(self, freq):
"""Returns the sources associated with a specific frequency."""
"""
Returns the sources associated with a specific frequency.
:param float freq: frequency for which we look up sources
:rtype: dictionary
:return: sources at the sepcified frequency
"""
assert freq in self._freqDict, "The requested frequency is not in this survey."
return self._freqDict[freq]
def projectFields(self, u):
"""
Project fields to receiver locations
:param Fields u: fields object
:rtype: numpy.ndarray
:return: data
"""
data = SimPEG.Survey.Data(self)
for src in self.srcList:
for rx in src.rxList:
SimPEG/EM/TDEM/BaseTDEM.py
+9 -1
View File
@@ -37,13 +37,21 @@ class BaseTDEMProblem(BaseTimeProblem, BaseEMProblem):
_FieldsForward_pair = FieldsTDEM #: used for the forward calculation only
waveformType = "STEPOFF"
current = None
def currentwaveform(self, wave):
self._timeSteps = np.diff(wave[:,0])
self.current = wave[:,1]
self.waveformType = "GENERAL"
def fields(self, m):
if self.verbose: print '%s\nCalculating fields(m)\n%s'%('*'*50,'*'*50)
self.curModel = m
# Create a fields storage object
F = self._FieldsForward_pair(self.mesh, self.survey)
for src in self.survey.srcList:
# Set the initial conditions
# Set the initial conditions
F[src,:,0] = src.getInitialFields(self.mesh)
F = self.forward(m, self.getRHS, F=F)
if self.verbose: print '%s\nDone calculating fields(m)\n%s'%('*'*50,'*'*50)
SimPEG/EM/__init__.py
+1 -1
View File
@@ -1,6 +1,6 @@
# from EM import *
import TDEM
import FDEM
import Base
import Analytics
import Utils
from scipy.constants import mu_0, epsilon_0
SimPEG/Examples/EM_FDEM_1D_Inversion.py
+116
View File
@@ -0,0 +1,116 @@
from SimPEG import *
import SimPEG.EM as EM
from SimPEG.EM import mu_0
def run(plotIt=True):
"""
EM: FDEM: 1D: Inversion
=======================
Here we will create and run a FDEM 1D inversion.
"""
cs, ncx, ncz, npad = 5., 25, 15, 15
hx = [(cs,ncx), (cs,npad,1.3)]
hz = [(cs,npad,-1.3), (cs,ncz), (cs,npad,1.3)]
mesh = Mesh.CylMesh([hx,1,hz], '00C')
layerz = -100.
active = mesh.vectorCCz<0.
layer = (mesh.vectorCCz<0.) & (mesh.vectorCCz>=layerz)
actMap = Maps.ActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz)
mapping = Maps.ExpMap(mesh) * Maps.Vertical1DMap(mesh) * actMap
sig_half = 2e-2
sig_air = 1e-8
sig_layer = 1e-2
sigma = np.ones(mesh.nCz)*sig_air
sigma[active] = sig_half
sigma[layer] = sig_layer
mtrue = np.log(sigma[active])
if plotIt:
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1,1, figsize = (3, 6))
plt.semilogx(sigma[active], mesh.vectorCCz[active])
ax.set_ylim(-500, 0)
ax.set_xlim(1e-3, 1e-1)
ax.set_xlabel('Conductivity (S/m)', fontsize = 14)
ax.set_ylabel('Depth (m)', fontsize = 14)
ax.grid(color='k', alpha=0.5, linestyle='dashed', linewidth=0.5)
rxOffset=10.
bzi = EM.FDEM.Rx(np.array([[rxOffset, 0., 1e-3]]), 'bzi')
freqs = np.logspace(1,3,10)
srcLoc = np.array([0., 0., 10.])
srcList = []
[srcList.append(EM.FDEM.Src.MagDipole([bzi],freq, srcLoc,orientation='Z')) for freq in freqs]
survey = EM.FDEM.Survey(srcList)
prb = EM.FDEM.Problem_b(mesh, mapping=mapping)
try:
from pymatsolver import MumpsSolver
prb.Solver = MumpsSolver
except ImportError, e:
prb.Solver = SolverLU
prb.pair(survey)
std = 0.05
survey.makeSyntheticData(mtrue, std)
survey.std = std
survey.eps = np.linalg.norm(survey.dtrue)*1e-5
if plotIt:
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1,1, figsize = (6, 6))
ax.semilogx(freqs,survey.dtrue[:freqs.size], 'b.-')
ax.semilogx(freqs,survey.dobs[:freqs.size], 'r.-')
ax.legend(('Noisefree', '$d^{obs}$'), fontsize = 16)
ax.set_xlabel('Time (s)', fontsize = 14)
ax.set_ylabel('$B_z$ (T)', fontsize = 16)
ax.set_xlabel('Time (s)', fontsize = 14)
ax.grid(color='k', alpha=0.5, linestyle='dashed', linewidth=0.5)
dmisfit = DataMisfit.l2_DataMisfit(survey)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Tikhonov(regMesh)
opt = Optimization.InexactGaussNewton(maxIter = 6)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Create an inversion object
beta = Directives.BetaSchedule(coolingFactor=5, coolingRate=2)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e0)
inv = Inversion.BaseInversion(invProb, directiveList=[beta,betaest])
m0 = np.log(np.ones(mtrue.size)*sig_half)
reg.alpha_s = 1e-3
reg.alpha_x = 1.
prb.counter = opt.counter = Utils.Counter()
opt.LSshorten = 0.5
opt.remember('xc')
mopt = inv.run(m0)
if plotIt:
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1,1, figsize = (3, 6))
plt.semilogx(sigma[active], mesh.vectorCCz[active])
plt.semilogx(np.exp(mopt), mesh.vectorCCz[active])
ax.set_ylim(-500, 0)
ax.set_xlim(1e-3, 1e-1)
ax.set_xlabel('Conductivity (S/m)', fontsize = 14)
ax.set_ylabel('Depth (m)', fontsize = 14)
ax.grid(color='k', alpha=0.5, linestyle='dashed', linewidth=0.5)
plt.legend(['$\sigma_{true}$', '$\sigma_{pred}$'],loc='best')
plt.show()
if __name__ == '__main__':
run()
SimPEG/Examples/EM_FDEM_SusEffects.py
+145
View File
@@ -0,0 +1,145 @@
from SimPEG import *
from SimPEG import EM
from pymatsolver import MumpsSolver
from scipy.constants import mu_0
def run(plotIt=True):
"""
FDEM: Effects of susceptibility
===============================
When airborne freqeuncy domain EM (AFEM) survey is flown over
the earth including significantly susceptible bodies (magnetite-rich rocks),
negative data is often observed in the real part of the lowest frequency
(e.g. Dighem system 900 Hz). This phenomenon mostly based upon magnetization
occurs due to a susceptible body when the magnetic field applied.
To clarify what is happening in the earth when we are exciting the earth with
a loop source in the frequency domain we run three forward modelling:
- F[:math:`\sigma`, :math:`\mu`]: Anomalous conductivity and susceptibility
- F[:math:`\sigma`, :math:`\mu_0`]: Anomalous conductivity
- F[:math:`\sigma_{air}`, :math:`\mu_0`]: primary field
We plot vector magnetic fields in the earth. For secondary fields we provide
F[:math:`\sigma`, :math:`\mu`]-F[:math:`\sigma`, :math:`\mu_0`]. Following
figure show only real part, since that is our interest.
"""
# Generate Cylindrical mesh
cs, ncx, ncz, npad = 5, 25, 24, 20.
hx = [(cs,ncx), (cs,npad,1.3)]
hz = [(cs,npad,-1.3), (cs,ncz), (cs,npad,1.3)]
mesh = Mesh.CylMesh([hx,1,hz], '00C')
sighalf = 1e-3
sigma = np.ones(mesh.nC)*1e-8
sigmahomo = sigma.copy()
mu = np.ones(mesh.nC)*mu_0
sigma[mesh.gridCC[:,-1]<0.] = sighalf
blkind = np.logical_and(mesh.gridCC[:,0]<30., (mesh.gridCC[:,2]<0)&(mesh.gridCC[:,2]>-150)&(mesh.gridCC[:,2]<-50))
sigma[blkind] = 1e-1
mu[blkind] = mu_0*1.1
offset = 0.
frequency = np.r_[10., 100., 1000.]
rx0 = EM.FDEM.Rx(np.array([[8., 0., 30.]]), 'bzr')
rx1 = EM.FDEM.Rx(np.array([[8., 0., 30.]]), 'bzi')
srcLists = []
nfreq = frequency.size
for ifreq in range(nfreq):
src = EM.FDEM.Src.CircularLoop([rx0, rx1], frequency[ifreq], np.array([[0., 0., 30.]]), radius=5.)
srcLists.append(src)
survey = EM.FDEM.Survey(srcLists)
iMap = Maps.IdentityMap(nP=int(mesh.nC))
# Use PhysPropMap
maps = [('sigma', iMap), ('mu', iMap)]
prob = EM.FDEM.Problem_b(mesh, mapping=maps)
prob.Solver = MumpsSolver
survey.pair(prob)
m = np.r_[sigma, mu]
survey0 = EM.FDEM.Survey(srcLists)
prob0 = EM.FDEM.Problem_b(mesh, mapping=maps)
prob0.Solver = MumpsSolver
survey0.pair(prob0)
m = np.r_[sigma, mu]
m0 = np.r_[sigma, np.ones(mesh.nC)*mu_0]
m00 = np.r_[np.ones(mesh.nC)*1e-8, np.ones(mesh.nC)*mu_0]
# Anomalous conductivity and susceptibility
F = prob.fields(m)
# Only anomalous conductivity
F0 = prob.fields(m0)
# Primary field
F00 = prob.fields(m00)
if plotIt:
import matplotlib.pyplot as plt
def vizfields(ifreq=0, primsec="secondary",realimag="real"):
titles = ["F[$\sigma$, $\mu$]", "F[$\sigma$, $\mu_0$]", "F[$\sigma$, $\mu$]-F[$\sigma$, $\mu_0$]"]
actind = np.logical_and(mesh.gridCC[:,0]<200., (mesh.gridCC[:,2]>-400)&(mesh.gridCC[:,2]<200))
if primsec=="secondary":
bCCprim = (mesh.aveF2CCV*F00[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')
bCC = (mesh.aveF2CCV*F[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')-bCCprim
bCC0 = (mesh.aveF2CCV*F0[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')-bCCprim
elif primsec=="primary":
bCC = (mesh.aveF2CCV*F[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')
bCC0 = (mesh.aveF2CCV*F0[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')
XYZ = mesh.gridCC[actind,:]
X = XYZ[:,0].reshape((31,43), order='F')
Z = XYZ[:,2].reshape((31,43), order='F')
bx = bCC[actind,0].reshape((31,43), order='F')
bz = bCC[actind,1].reshape((31,43), order='F')
bx0 = bCC0[actind,0].reshape((31,43), order='F')
bz0 = bCC0[actind,1].reshape((31,43), order='F')
bxsec = (bCC[actind,0]-bCC0[actind,0]).reshape((31,43), order='F')
bzsec = (bCC[actind,1]-bCC0[actind,1]).reshape((31,43), order='F')
absbreal = np.sqrt(bx.real**2+bz.real**2)
absbimag = np.sqrt(bx.imag**2+bz.imag**2)
absb0real = np.sqrt(bx0.real**2+bz0.real**2)
absb0imag = np.sqrt(bx0.imag**2+bz0.imag**2)
absbrealsec = np.sqrt(bxsec.real**2+bzsec.real**2)
absbimagsec = np.sqrt(bxsec.imag**2+bzsec.imag**2)
fig = plt.figure(figsize=(15,5))
ax1 = plt.subplot(131)
ax2 = plt.subplot(132)
ax3 = plt.subplot(133)
typefield="real"
scale=20
if realimag=="real":
ax1.contourf(X, Z,np.log10(absbreal), 100)
ax1.quiver(X, Z,bx.real/absbreal,bz.real/absbreal,scale=scale,width=0.005, alpha = 0.5)
ax2.contourf(X, Z,np.log10(absb0real), 100)
ax2.quiver(X, Z,bx0.real/absb0real,bz0.real/absb0real,scale=scale,width=0.005, alpha = 0.5)
ax3.contourf(X, Z,np.log10(absbrealsec), 100)
ax3.quiver(X, Z,bxsec.real/absbrealsec,bzsec.real/absbrealsec,scale=scale,width=0.005, alpha = 0.5)
elif realimag=="imag":
ax1.contourf(X, Z,np.log10(absbimag), 100)
ax1.quiver(X, Z,bx.imag/absbimag,bz.imag/absbimag,scale=scale,width=0.005, alpha = 0.5)
ax2.contourf(X, Z,np.log10(absb0imag), 100)
ax2.quiver(X, Z,bx0.imag/absb0imag,bz0.imag/absb0imag,scale=scale,width=0.005, alpha = 0.5)
ax3.contourf(X, Z,np.log10(absbimagsec), 100)
ax3.quiver(X, Z,bxsec.imag/absbimagsec,bzsec.imag/absbimagsec,scale=scale,width=0.005, alpha = 0.5)
ax = [ax1, ax2, ax3]
ax3.text(30, 50, ("Frequency=%5.2f Hz")%(frequency[ifreq]), color="k", fontsize=18)
ax2.text(30, 50, primsec, color="k", fontsize=18)
for i, axtemp in enumerate(ax):
axtemp.plot(np.r_[0, 29.75], np.r_[-50, -50], 'w', lw=3)
axtemp.plot(np.r_[29.5, 29.5], np.r_[-50, -142.5], 'w', lw=3)
axtemp.plot(np.r_[0, 29.5], np.r_[-142.5, -142.5], 'w', lw=3)
axtemp.plot(np.r_[0, 100.], np.r_[0, 0], 'w', lw=3)
axtemp.set_ylim(-200, 100.)
axtemp.set_xlim(10, 100.)
axtemp.set_title(titles[i])
plt.show()
vizfields(1, primsec="primary", realimag="real")
vizfields(1, primsec="secondary", realimag="real")
if __name__ == '__main__':
run()
SimPEG/Examples/EM_TDEM_1D_Inversion.py
+8 -9
View File
@@ -1,6 +1,6 @@
from SimPEG import *
import SimPEG.EM as EM
from scipy.constants import mu_0
from SimPEG.EM import mu_0
def run(plotIt=True):
@@ -50,20 +50,18 @@ def run(plotIt=True):
prb.Solver = SolverLU
prb.timeSteps = [(1e-06, 20),(1e-05, 20), (0.0001, 20)]
prb.pair(survey)
dtrue = survey.dpred(mtrue)
survey.dtrue = dtrue
# create observed data
std = 0.05
noise = std*abs(survey.dtrue)*np.random.randn(*survey.dtrue.shape)
survey.dobs = survey.dtrue+noise
survey.std = survey.dobs*0 + std
survey.Wd = 1/(abs(survey.dobs)*std)
survey.dobs = survey.makeSyntheticData(mtrue,std)
survey.std = std
survey.eps = 1e-5*np.linalg.norm(survey.dobs)
if plotIt:
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1,1, figsize = (10, 6))
ax.loglog(rx.times, dtrue, 'b.-')
ax.loglog(rx.times, survey.dtrue, 'b.-')
ax.loglog(rx.times, survey.dobs, 'r.-')
ax.legend(('Noisefree', '$d^{obs}$'), fontsize = 16)
ax.set_xlabel('Time (s)', fontsize = 14)
@@ -76,6 +74,7 @@ def run(plotIt=True):
reg = Regularization.Tikhonov(regMesh)
opt = Optimization.InexactGaussNewton(maxIter = 5)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Create an inversion object
beta = Directives.BetaSchedule(coolingFactor=5, coolingRate=2)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e0)
SimPEG/InvProblem.py
+2 -2
View File
@@ -66,8 +66,8 @@ class BaseInvProblem(object):
self.curModel = m0
print """SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using same solver as the problem***"""
self.opt.bfgsH0 = self.prob.Solver(self.reg.eval2Deriv(self.curModel))
***Done using same Solver and solverOpts as the problem***"""
self.opt.bfgsH0 = self.prob.Solver(self.reg.eval2Deriv(self.curModel), **self.prob.solverOpts)
@property
def warmstart(self):
SimPEG/Maps.py
+54 -45
View File
@@ -10,21 +10,25 @@ class IdentityMap(object):
SimPEG Map
"""
__metaclass__ = Utils.SimPEGMetaClass
mesh = None #: A SimPEG Mesh
def __init__(self, mesh, **kwargs):
def __init__(self, mesh=None, nP=None, **kwargs):
Utils.setKwargs(self, **kwargs)
if nP is not None:
assert type(nP) in [int, long], ' Number of parameters must be an integer.'
self.mesh = mesh
self._nP = nP
@property
def nP(self):
"""
:rtype: int
:return: number of parameters in the model
:return: number of parameters that the mapping accepts
"""
if self._nP is not None:
return self._nP
if self.mesh is None:
return '*'
return self.mesh.nC
@@ -32,11 +36,15 @@ class IdentityMap(object):
@property
def shape(self):
"""
The default shape is (mesh.nC, nP).
The default shape is (mesh.nC, nP) if the mesh is defined.
If this is a meshless mapping (i.e. nP is defined independently)
the shape will be the the shape (nP,nP).
:rtype: (int,int)
:return: shape of the operator as a tuple
"""
if self._nP is not None:
return (self.nP, self.nP)
if self.mesh is None:
return ('*', self.nP)
return (self.mesh.nC, self.nP)
@@ -118,6 +126,7 @@ class IdentityMap(object):
def __str__(self):
return "%s(%s,%s)" % (self.__class__.__name__, self.shape[0], self.shape[1])
class ComboMap(IdentityMap):
"""Combination of various maps."""
@@ -475,7 +484,7 @@ class ActiveCells(IdentityMap):
else:
self.valInactive = valInactive.copy()
self.valInactive[self.indActive] = 0
inds = np.nonzero(self.indActive)[0]
self.P = sp.csr_matrix((np.ones(inds.size),(inds, range(inds.size))), shape=(self.nC, self.nP))
@@ -708,7 +717,7 @@ class PolyMap(IdentityMap):
Parameterize the model space using a polynomials in a wholespace.
..math::
y = \mathbf{V} c
Define the model as:
@@ -752,10 +761,10 @@ class PolyMap(IdentityMap):
else:
raise(Exception("Input for normal = X or Y or Z"))
#3D
elif self.mesh.dim == 3:
elif self.mesh.dim == 3:
X = self.mesh.gridCC[:,0]
Y = self.mesh.gridCC[:,1]
Z = self.mesh.gridCC[:,2]
Y = self.mesh.gridCC[:,1]
Z = self.mesh.gridCC[:,2]
if self.normal =='X':
f = polynomial.polyval2d(Y, Z, c.reshape((self.order[0]+1,self.order[1]+1))) - X
elif self.normal =='Y':
@@ -766,43 +775,43 @@ class PolyMap(IdentityMap):
raise(Exception("Input for normal = X or Y or Z"))
else:
raise(Exception("Only supports 2D"))
return sig1+(sig2-sig1)*(np.arctan(alpha*f)/np.pi+0.5)
def deriv(self, m):
alpha = self.slope
sig1,sig2, c = m[0],m[1],m[2:]
if self.logSigma:
sig1, sig2 = np.exp(sig1), np.exp(sig2)
#2D
if self.mesh.dim == 2:
if self.mesh.dim == 2:
X = self.mesh.gridCC[:,0]
Y = self.mesh.gridCC[:,1]
if self.normal =='X':
f = polynomial.polyval(Y, c) - X
V = polynomial.polyvander(Y, len(c)-1)
V = polynomial.polyvander(Y, len(c)-1)
elif self.normal =='Y':
f = polynomial.polyval(X, c) - Y
V = polynomial.polyvander(X, len(c)-1)
V = polynomial.polyvander(X, len(c)-1)
else:
raise(Exception("Input for normal = X or Y or Z"))
raise(Exception("Input for normal = X or Y or Z"))
#3D
elif self.mesh.dim == 3:
elif self.mesh.dim == 3:
X = self.mesh.gridCC[:,0]
Y = self.mesh.gridCC[:,1]
Z = self.mesh.gridCC[:,2]
if self.normal =='X':
f = polynomial.polyval2d(Y, Z, c.reshape((self.order[0]+1,self.order[1]+1))) - X
V = polynomial.polyvander2d(Y, Z, self.order)
V = polynomial.polyvander2d(Y, Z, self.order)
elif self.normal =='Y':
f = polynomial.polyval2d(X, Z, c.reshape((self.order[0]+1,self.order[1]+1))) - Y
V = polynomial.polyvander2d(X, Z, self.order)
V = polynomial.polyvander2d(X, Z, self.order)
elif self.normal =='Z':
f = polynomial.polyval2d(X, Y, c.reshape((self.order[0]+1,self.order[1]+1))) - Z
V = polynomial.polyvander2d(X, Y, self.order)
V = polynomial.polyvander2d(X, Y, self.order)
else:
raise(Exception("Input for normal = X or Y or Z"))
@@ -815,16 +824,16 @@ class PolyMap(IdentityMap):
g3 = Utils.sdiag(alpha*(sig2-sig1)/(1.+(alpha*f)**2)/np.pi)*V
return sp.csr_matrix(np.c_[g1,g2,g3])
return sp.csr_matrix(np.c_[g1,g2,g3])
class SplineMap(IdentityMap):
"""SplineMap
Parameterize the boundary of two geological units using a spline interpolation
Parameterize the boundary of two geological units using a spline interpolation
..math::
g = f(x)-y
Define the model as:
@@ -849,7 +858,7 @@ class SplineMap(IdentityMap):
def nP(self):
if self.mesh.dim == 2:
return np.size(self.pts)+2
elif self.mesh.dim == 3:
elif self.mesh.dim == 3:
return np.size(self.pts)*2+2
else:
raise(Exception("Only supports 2D and 3D"))
@@ -866,28 +875,28 @@ class SplineMap(IdentityMap):
X = self.mesh.gridCC[:,0]
Y = self.mesh.gridCC[:,1]
self.spl = UnivariateSpline(self.pts, c, k=self.order, s=0)
if self.normal =='X':
if self.normal =='X':
f = self.spl(Y) - X
elif self.normal =='Y':
f = self.spl(X) - Y
else:
raise(Exception("Input for normal = X or Y or Z"))
# 3D:
# Comments:
# 3D:
# Comments:
# Make two spline functions and link them using linear interpolation.
# This is not quite direct extension of 2D to 3D case
# Using 2D interpolation is possible
elif self.mesh.dim == 3:
elif self.mesh.dim == 3:
X = self.mesh.gridCC[:,0]
Y = self.mesh.gridCC[:,1]
Y = self.mesh.gridCC[:,1]
Z = self.mesh.gridCC[:,2]
npts = np.size(self.pts)
npts = np.size(self.pts)
if np.mod(c.size, 2):
raise(Exception("Put even points!"))
self.spl = {"splb":UnivariateSpline(self.pts, c[:npts], k=self.order, s=0),
"splt":UnivariateSpline(self.pts, c[npts:], k=self.order, s=0)}
@@ -902,7 +911,7 @@ class SplineMap(IdentityMap):
raise(Exception("Input for normal = X or Y or Z"))
else:
raise(Exception("Only supports 2D and 3D"))
return sig1+(sig2-sig1)*(np.arctan(alpha*f)/np.pi+0.5)
@@ -912,7 +921,7 @@ class SplineMap(IdentityMap):
if self.logSigma:
sig1, sig2 = np.exp(sig1), np.exp(sig2)
#2D
if self.mesh.dim == 2:
if self.mesh.dim == 2:
X = self.mesh.gridCC[:,0]
Y = self.mesh.gridCC[:,1]
@@ -921,9 +930,9 @@ class SplineMap(IdentityMap):
elif self.normal =='Y':
f = self.spl(X) - Y
else:
raise(Exception("Input for normal = X or Y or Z"))
raise(Exception("Input for normal = X or Y or Z"))
#3D
elif self.mesh.dim == 3:
elif self.mesh.dim == 3:
X = self.mesh.gridCC[:,0]
Y = self.mesh.gridCC[:,1]
Z = self.mesh.gridCC[:,2]
@@ -931,7 +940,7 @@ class SplineMap(IdentityMap):
zb = self.ptsv[0]
zt = self.ptsv[1]
flines = (self.spl["splt"](Y)-self.spl["splb"](Y))*(Z-zb)/(zt-zb) + self.spl["splb"](Y)
f = flines - X
f = flines - X
# elif self.normal =='Y':
# elif self.normal =='Z':
else:
@@ -944,7 +953,7 @@ class SplineMap(IdentityMap):
g1 = -(np.arctan(alpha*f)/np.pi + 0.5) + 1.0
g2 = (np.arctan(alpha*f)/np.pi + 0.5)
if self.mesh.dim ==2:
g3 = np.zeros((self.mesh.nC, self.npts))
if self.normal =='Y':
@@ -958,7 +967,7 @@ class SplineMap(IdentityMap):
cb = c.copy()
dy = self.mesh.hy[ind]*1.5
ca[i] = ctemp+dy
cb[i] = ctemp-dy
cb[i] = ctemp-dy
spla = UnivariateSpline(self.pts, ca, k=self.order, s=0)
splb = UnivariateSpline(self.pts, cb, k=self.order, s=0)
fderiv = (spla(X)-splb(X))/(2*dy)
@@ -968,7 +977,7 @@ class SplineMap(IdentityMap):
g3 = np.zeros((self.mesh.nC, self.npts*2))
if self.normal =='X':
# Here we use perturbation to compute sensitivity
for i in range(self.npts*2):
for i in range(self.npts*2):
ctemp = c[i]
ind = np.argmin(abs(self.mesh.vectorCCy-ctemp))
ca = c.copy()
@@ -982,20 +991,20 @@ class SplineMap(IdentityMap):
splbb = UnivariateSpline(self.pts, cb[:self.npts], k=self.order, s=0)
flinesa = (self.spl["splt"](Y)-splba(Y))*(Z-zb)/(zt-zb) + splba(Y) - X
flinesb = (self.spl["splt"](Y)-splbb(Y))*(Z-zb)/(zt-zb) + splbb(Y) - X
#treat top boundary
#treat top boundary
else:
splta = UnivariateSpline(self.pts, ca[self.npts:], k=self.order, s=0)
spltb = UnivariateSpline(self.pts, ca[self.npts:], k=self.order, s=0)
flinesa = (self.spl["splt"](Y)-splta(Y))*(Z-zb)/(zt-zb) + splta(Y) - X
flinesb = (self.spl["splt"](Y)-spltb(Y))*(Z-zb)/(zt-zb) + spltb(Y) - X
fderiv = (flinesa-flinesb)/(2*dy)
flinesb = (self.spl["splt"](Y)-spltb(Y))*(Z-zb)/(zt-zb) + spltb(Y) - X
fderiv = (flinesa-flinesb)/(2*dy)
g3[:,i] = Utils.sdiag(alpha*(sig2-sig1)/(1.+(alpha*f)**2)/np.pi)*fderiv
else :
raise(Exception("Not Implemented for Y and Z, your turn :)"))
return sp.csr_matrix(np.c_[g1,g2,g3])
return sp.csr_matrix(np.c_[g1,g2,g3])
SimPEG/Mesh/MeshIO.py
+416
View File
@@ -0,0 +1,416 @@
import numpy as np, os
from SimPEG import Utils
class TensorMeshIO(object):
@classmethod
def readUBC(TensorMesh, fileName):
"""
Read UBC GIF 3DTensor mesh and generate 3D Tensor mesh in simpegTD
Input:
:param fileName, path to the UBC GIF mesh file
Output:
:param SimPEG TensorMesh object
"""
# Interal function to read cell size lines for the UBC mesh files.
def readCellLine(line):
for seg in line.split():
if '*' in seg:
st = seg
sp = seg.split('*')
re = np.array(sp[0],dtype=int)*(' ' + sp[1])
line = line.replace(st,re.strip())
return np.array(line.split(),dtype=float)
# Read the file as line strings, remove lines with comment = !
msh = np.genfromtxt(fileName,delimiter='\n',dtype=np.str,comments='!')
# Fist line is the size of the model
sizeM = np.array(msh[0].split(),dtype=float)
# Second line is the South-West-Top corner coordinates.
x0 = np.array(msh[1].split(),dtype=float)
# Read the cell sizes
h1 = readCellLine(msh[2])
h2 = readCellLine(msh[3])
h3temp = readCellLine(msh[4])
h3 = h3temp[::-1] # Invert the indexing of the vector to start from the bottom.
# Adjust the reference point to the bottom south west corner
x0[2] = x0[2] - np.sum(h3)
# Make the mesh
tensMsh = TensorMesh([h1,h2,h3],x0)
return tensMsh
@classmethod
def readVTK(TensorMesh, fileName):
"""
Read VTK Rectilinear (vtr xml file) and return SimPEG Tensor mesh and model
Input:
:param vtrFileName, path to the vtr model file to write to
Output:
:return SimPEG TensorMesh object
:return SimPEG model dictionary
"""
# Import
from vtk import vtkXMLRectilinearGridReader as vtrFileReader
from vtk.util.numpy_support import vtk_to_numpy
# Read the file
vtrReader = vtrFileReader()
vtrReader.SetFileName(fileName)
vtrReader.Update()
vtrGrid = vtrReader.GetOutput()
# Sort information
hx = np.abs(np.diff(vtk_to_numpy(vtrGrid.GetXCoordinates())))
xR = vtk_to_numpy(vtrGrid.GetXCoordinates())[0]
hy = np.abs(np.diff(vtk_to_numpy(vtrGrid.GetYCoordinates())))
yR = vtk_to_numpy(vtrGrid.GetYCoordinates())[0]
zD = np.diff(vtk_to_numpy(vtrGrid.GetZCoordinates()))
# Check the direction of hz
if np.all(zD < 0):
hz = np.abs(zD[::-1])
zR = vtk_to_numpy(vtrGrid.GetZCoordinates())[-1]
else:
hz = np.abs(zD)
zR = vtk_to_numpy(vtrGrid.GetZCoordinates())[0]
x0 = np.array([xR,yR,zR])
# Make the SimPEG object
tensMsh = TensorMesh([hx,hy,hz],x0)
# Grap the models
models = {}
for i in np.arange(vtrGrid.GetCellData().GetNumberOfArrays()):
modelName = vtrGrid.GetCellData().GetArrayName(i)
if np.all(zD < 0):
modFlip = vtk_to_numpy(vtrGrid.GetCellData().GetArray(i))
tM = tensMsh.r(modFlip,'CC','CC','M')
modArr = tensMsh.r(tM[:,:,::-1],'CC','CC','V')
else:
modArr = vtk_to_numpy(vtrGrid.GetCellData().GetArray(i))
models[modelName] = modArr
# Return the data
return tensMsh, models
def writeVTK(mesh, fileName, models=None):
"""
Makes and saves a VTK rectilinear file (vtr) for a simpeg Tensor mesh and model.
Input:
:param str, path to the output vtk file
:param mesh, SimPEG TensorMesh object - mesh to be transfer to VTK
:param models, dictionary of numpy.array - Name('s) and array('s). Match number of cells
"""
# Import
from vtk import vtkRectilinearGrid as rectGrid, vtkXMLRectilinearGridWriter as rectWriter, VTK_VERSION
from vtk.util.numpy_support import numpy_to_vtk
# Deal with dimensionalities
if mesh.dim >= 1:
vX = mesh.vectorNx
xD = mesh.nNx
yD,zD = 1,1
vY, vZ = np.array([0,0])
if mesh.dim >= 2:
vY = mesh.vectorNy
yD = mesh.nNy
if mesh.dim == 3:
vZ = mesh.vectorNz
zD = mesh.nNz
# Use rectilinear VTK grid.
# Assign the spatial information.
vtkObj = rectGrid()
vtkObj.SetDimensions(xD,yD,zD)
vtkObj.SetXCoordinates(numpy_to_vtk(vX,deep=1))
vtkObj.SetYCoordinates(numpy_to_vtk(vY,deep=1))
vtkObj.SetZCoordinates(numpy_to_vtk(vZ,deep=1))
# Assign the model('s) to the object
if models is not None:
for item in models.iteritems():
# Convert numpy array
vtkDoubleArr = numpy_to_vtk(item[1],deep=1)
vtkDoubleArr.SetName(item[0])
vtkObj.GetCellData().AddArray(vtkDoubleArr)
# Set the active scalar
vtkObj.GetCellData().SetActiveScalars(models.keys()[0])
# vtkObj.Update()
# Check the extension of the fileName
ext = os.path.splitext(fileName)[1]
if ext is '':
fileName = fileName + '.vtr'
elif ext not in '.vtr':
raise IOError('{:s} is an incorrect extension, has to be .vtr')
# Write the file.
vtrWriteFilter = rectWriter()
if float(VTK_VERSION.split('.')[0]) >=6:
vtrWriteFilter.SetInputData(vtkObj)
else:
vtuWriteFilter.SetInput(vtuObj)
vtrWriteFilter.SetFileName(fileName)
vtrWriteFilter.Update()
def readModelUBC(mesh, fileName):
"""
Read UBC 3DTensor mesh model and generate 3D Tensor mesh model in simpeg
Input:
:param fileName, path to the UBC GIF mesh file to read
:param mesh, TensorMesh object, mesh that coresponds to the model
Output:
:return numpy array, model with TensorMesh ordered
"""
f = open(fileName, 'r')
model = np.array(map(float, f.readlines()))
f.close()
model = np.reshape(model, (mesh.nCz, mesh.nCx, mesh.nCy), order = 'F')
model = model[::-1,:,:]
model = np.transpose(model, (1, 2, 0))
model = Utils.mkvc(model)
return model
def writeModelUBC(mesh, fileName, model):
"""
Writes a model associated with a SimPEG TensorMesh
to a UBC-GIF format model file.
:param str fileName: File to write to
:param simpeg.Mesh.TensorMesh mesh: The mesh
:param numpy.ndarray model: The model
"""
# Reshape model to a matrix
modelMat = mesh.r(model,'CC','CC','M')
# Transpose the axes
modelMatT = modelMat.transpose((2,0,1))
# Flip z to positive down
modelMatTR = Utils.mkvc(modelMatT[::-1,:,:])
np.savetxt(fileName, modelMatTR.ravel())
def writeUBC(mesh, fileName, models=None):
"""
Writes a SimPEG TensorMesh to a UBC-GIF format mesh file.
:param str fileName: File to write to
:param simpeg.Mesh.TensorMesh mesh: The mesh
"""
assert mesh.dim == 3
s = ''
s += '%i %i %i\n' %tuple(mesh.vnC)
origin = mesh.x0 + np.array([0,0,mesh.hz.sum()]) # Have to it in the same operation or use mesh.x0.copy(), otherwise the mesh.x0 is updated.
origin.dtype = float
s += '%.2f %.2f %.2f\n' %tuple(origin)
s += ('%.2f '*mesh.nCx+'\n')%tuple(mesh.hx)
s += ('%.2f '*mesh.nCy+'\n')%tuple(mesh.hy)
s += ('%.2f '*mesh.nCz+'\n')%tuple(mesh.hz[::-1])
f = open(fileName, 'w')
f.write(s)
f.close()
if models is None: return
assert type(models) is dict, 'models must be a dict'
for key in models:
assert type(key) is str, 'The dict key is a file name'
mesh.writeModelUBC(key, models[key])
class TreeMeshIO(object):
def writeUBC(mesh, fileName, models=None):
"""
Write UBC ocTree mesh and model files from a simpeg ocTree mesh and model.
:param str fileName: File to write to
:param simpeg.Mesh.TreeMesh mesh: The mesh
:param dictionary models: The models in a dictionary, where the keys is the name of the of the model file
"""
# Calculate information to write in the file.
# Number of cells in the underlying mesh
nCunderMesh = np.array([h.size for h in mesh.h],dtype=np.int64)
# The top-south-west most corner of the mesh
tswCorn = mesh.x0 + np.array([0,0,np.sum(mesh.h[2])])
# Smallest cell size
smallCell = np.array([h.min() for h in mesh.h])
# Number of cells
nrCells = mesh.nC
## Extract iformation about the cells.
# cell pointers
cellPointers = np.array([c._pointer for c in mesh])
# cell with
cellW = np.array([ mesh._levelWidth(i) for i in cellPointers[:,-1] ])
# Need to shift the pointers to work with UBC indexing
# UBC Octree indexes always the top-left-close (top-south-west) corner first and orders the cells in z(top-down),x,y vs x,y,z(bottom-up).
# Shift index up by 1
ubcCellPt = cellPointers[:,0:-1].copy() + np.array([1.,1.,1.])
# Need reindex the z index to be from the top-left-close corner and to be from the global top.
ubcCellPt[:,2] = ( nCunderMesh[-1] + 2) - (ubcCellPt[:,2] + cellW)
# Reorder the ubcCellPt
ubcReorder = np.argsort(ubcCellPt.view(','.join(3*['float'])),axis=0,order=['f2','f1','f0'])[:,0]
# Make a array with the pointers and the withs, that are order in the ubc ordering
indArr = np.concatenate((ubcCellPt[ubcReorder,:],cellW[ubcReorder].reshape((-1,1)) ),axis=1)
## Write the UBC octree mesh file
with open(fileName,'w') as mshOut:
mshOut.write('{:.0f} {:.0f} {:.0f}\n'.format(nCunderMesh[0],nCunderMesh[1],nCunderMesh[2]))
mshOut.write('{:.4f} {:.4f} {:.4f}\n'.format(tswCorn[0],tswCorn[1],tswCorn[2]))
mshOut.write('{:.3f} {:.3f} {:.3f}\n'.format(smallCell[0],smallCell[1],smallCell[2]))
mshOut.write('{:.0f} \n'.format(nrCells))
np.savetxt(mshOut,indArr,fmt='%i')
## Print the models
# Assign the model('s) to the object
if models is not None:
# indUBCvector = np.argsort(cX0[np.argsort(np.concatenate((cX0[:,0:2],cX0[:,2:3].max() - cX0[:,2:3]),axis=1).view(','.join(3*['float'])),axis=0,order=('f2','f1','f0'))[:,0]].view(','.join(3*['float'])),axis=0,order=('f2','f1','f0'))[:,0]
for item in models.iteritems():
# Save the data
np.savetxt(item[0],item[1][ubcReorder],fmt='%3.5e')
@classmethod
def readUBC(TreeMesh, meshFile):
"""
Read UBC 3D OcTree mesh and/or modelFiles
Input:
:param str meshFile: path to the UBC GIF OcTree mesh file to read
Output:
:return SimPEG.Mesh.TreeMesh mesh: The octree mesh
:return list of ndarray's: models as a list of numpy array's
"""
## Read the file lines
fileLines = np.genfromtxt(meshFile,dtype=str,delimiter='\n')
# Extract the data
nCunderMesh = np.array(fileLines[0].split(),dtype=float)
# I think this is the case?
if np.unique(nCunderMesh).size >1:
raise Exception('SimPEG TreeMeshes have the same number of cell in all directions')
tswCorn = np.array(fileLines[1].split(),dtype=float)
smallCell = np.array(fileLines[2].split(),dtype=float)
nrCells = np.array(fileLines[3].split(),dtype=float)
# Read the index array
indArr = np.genfromtxt(fileLines[4::],dtype=np.int)
## Calculate simpeg parameters
h1,h2,h3 = [np.ones(nr)*sz for nr,sz in zip(nCunderMesh,smallCell)]
x0 = tswCorn - np.array([0,0,np.sum(h3)])
# Need to convert the index array to a points list that complies with SimPEG TreeMesh.
# Shift to start at 0
simpegCellPt = indArr[:,0:-1].copy()
simpegCellPt[:,2] = ( nCunderMesh[-1] + 2) - (simpegCellPt[:,2] + indArr[:,3])
# Need reindex the z index to be from the bottom-left-close corner and to be from the global bottom.
simpegCellPt = simpegCellPt - np.array([1.,1.,1.])
# Calculate the cell level
simpegLevel = np.log2(np.min(nCunderMesh)) - np.log2(indArr[:,3])
# Make a pointer matrix
simpegPointers = np.concatenate((simpegCellPt,simpegLevel.reshape((-1,1))),axis=1)
## Make the tree mesh
mesh = TreeMesh([h1,h2,h3],x0)
mesh._cells = set([mesh._index(p) for p in simpegPointers.tolist()])
# Figure out the reordering
mesh._simpegReorderUBC = np.argsort(np.array([mesh._index(i) for i in simpegPointers.tolist()]))
# mesh._simpegReorderUBC = np.argsort((np.array([[1,1,1,-1]])*simpegPointers).view(','.join(4*['float'])),axis=0,order=['f3','f2','f1','f0'])[:,0]
return mesh
def readModelUBC(mesh, fileName):
"""
Read UBC OcTree model and get vector
Input:
:param fileName, path to the UBC GIF model file to read
Output:
:return numpy array, OcTree model
"""
if type(fileName) is list:
out = {}
for f in fileName:
out[f] = mesh.readModelUBC(f)
return out
assert hasattr(mesh, '_simpegReorderUBC'), 'The file must have been loaded from a UBC format.'
assert mesh.dim == 3
modList = []
modArr = np.loadtxt(fileName)
if len(modArr.shape) == 1:
modList.append(modArr[mesh._simpegReorderUBC])
else:
modList.append(modArr[mesh._simpegReorderUBC,:])
return modList
def writeVTK(mesh, fileName, models=None):
"""
Function to write a VTU file from a SimPEG TreeMesh and model.
"""
import vtk
from vtk import vtkXMLUnstructuredGridWriter as Writer, VTK_VERSION
from vtk.util.numpy_support import numpy_to_vtk, numpy_to_vtkIdTypeArray
if str(type(mesh)).split()[-1][1:-2] not in 'SimPEG.Mesh.TreeMesh.TreeMesh':
raise IOError('mesh is not a SimPEG TreeMesh.')
# Make the data parts for the vtu object
# Points
mesh.number()
ptsMat = mesh._gridN + mesh.x0
vtkPts = vtk.vtkPoints()
vtkPts.SetData(numpy_to_vtk(ptsMat,deep=True))
# Cells
cellConn = np.array([c.nodes for c in mesh],dtype=np.int64)
cellsMat = np.concatenate((np.ones((cellConn.shape[0],1),dtype=np.int64)*cellConn.shape[1],cellConn),axis=1).ravel()
cellsArr = vtk.vtkCellArray()
cellsArr.SetNumberOfCells(cellConn.shape[0])
cellsArr.SetCells(cellConn.shape[0],numpy_to_vtkIdTypeArray(cellsMat,deep=True))
# Make the object
vtuObj = vtk.vtkUnstructuredGrid()
vtuObj.SetPoints(vtkPts)
vtuObj.SetCells(vtk.VTK_VOXEL,cellsArr)
# Add the level of refinement as a cell array
cellSides = np.array([np.array(vtuObj.GetCell(i).GetBounds()).reshape((3,2)).dot(np.array([-1, 1])) for i in np.arange(vtuObj.GetNumberOfCells())])
uniqueLevel, indLevel = np.unique(np.prod(cellSides,axis=1),return_inverse=True)
refineLevelArr = numpy_to_vtk(indLevel.max() - indLevel,deep=1)
refineLevelArr.SetName('octreeLevel')
vtuObj.GetCellData().AddArray(refineLevelArr)
# Assign the model('s) to the object
if models is not None:
for item in models.iteritems():
# Convert numpy array
vtkDoubleArr = numpy_to_vtk(item[1],deep=1)
vtkDoubleArr.SetName(item[0])
vtuObj.GetCellData().AddArray(vtkDoubleArr)
# Make the writer
vtuWriteFilter = Writer()
if float(VTK_VERSION.split('.')[0]) >=6:
vtuWriteFilter.SetInputData(vtuObj)
else:
vtuWriteFilter.SetInput(vtuObj)
vtuWriteFilter.SetFileName(fileName)
# Write the file
vtuWriteFilter.Update()
SimPEG/Mesh/TensorMesh.py
+559 -558
View File
File diff suppressed because it is too large Load Diff
SimPEG/Mesh/TreeMesh.py
+59 -123
View File
@@ -100,11 +100,12 @@ except Exception, e:
from InnerProducts import InnerProducts
from TensorMesh import TensorMesh, BaseTensorMesh
from MeshIO import TreeMeshIO
import time
MAX_BITS = 20
class TreeMesh(BaseTensorMesh, InnerProducts):
class TreeMesh(BaseTensorMesh, InnerProducts, TreeMeshIO):
_meshType = 'TREE'
@@ -564,15 +565,18 @@ class TreeMesh(BaseTensorMesh, InnerProducts):
return [p - (p % mod) for p in pointer[:-1]] + [pointer[-1]-1]
def _cellN(self, p):
"""Node location [x,y(,z)] of a single cell, closest to origin, given a pointer."""
p = self._asPointer(p)
return [hi[:p[ii]].sum() for ii, hi in enumerate(self.h)]
def _cellH(self, p):
"""Widths of a single cell given a pointer."""
p = self._asPointer(p)
w = self._levelWidth(p[-1])
return [hi[p[ii]:p[ii]+w].sum() for ii, hi in enumerate(self.h)]
def _cellC(self, p):
"""Cell center of a single cell (without origin correction), given a pointer."""
return (np.array(self._cellH(p))/2.0 + self._cellN(p)).tolist()
def _levelWidth(self, level):
@@ -827,8 +831,10 @@ class TreeMesh(BaseTensorMesh, InnerProducts):
def _numberCells(self, force=False):
if not self.__dirtyCells__ and not force: return
self._cc2i = dict()
self._i2cc = dict()
for ii, c in enumerate(sorted(self._cells)):
self._cc2i[c] = ii
self._i2cc[ii] = c
self.__dirtyCells__ = False
def _numberNodes(self, force=False):
@@ -1704,9 +1710,9 @@ class TreeMesh(BaseTensorMesh, InnerProducts):
"Construct the averaging operator on cell faces to cell centers."
if getattr(self, '_aveF2CC', None) is None:
if self.dim == 2:
self._aveF2CC = 1./self.dim*sp.hstack([self.aveFx2CC, self.aveFy2CC])
self._aveF2CC = 1./self.dim*sp.hstack([self.aveFx2CC, self.aveFy2CC]).tocsr()
elif self.dim == 3:
self._aveF2CC = 1./self.dim*sp.hstack([self.aveFx2CC, self.aveFy2CC, self.aveFz2CC])
self._aveF2CC = 1./self.dim*sp.hstack([self.aveFx2CC, self.aveFy2CC, self.aveFz2CC]).tocsr()
return self._aveF2CC
@property
@@ -1714,9 +1720,9 @@ class TreeMesh(BaseTensorMesh, InnerProducts):
"Construct the averaging operator on cell faces to cell centers."
if getattr(self, '_aveF2CCV', None) is None:
if self.dim == 2:
self._aveF2CCV = sp.block_diag([self.aveFx2CC, self.aveFy2CC])
self._aveF2CCV = sp.block_diag([self.aveFx2CC, self.aveFy2CC]).tocsr()
elif self.dim == 3:
self._aveF2CCV = sp.block_diag([self.aveFx2CC, self.aveFy2CC, self.aveFz2CC])
self._aveF2CCV = sp.block_diag([self.aveFx2CC, self.aveFy2CC, self.aveFz2CC]).tocsr()
return self._aveF2CCV
@property
@@ -2218,6 +2224,25 @@ class TreeMesh(BaseTensorMesh, InnerProducts):
if showIt: plt.show()
return tuple(out)
def __len__(self): return self.nC
def __getitem__(self, key):
if isinstance( key, slice ) :
#Get the start, stop, and step from the slice
return [self[ii] for ii in xrange(*key.indices(len(self)))]
elif isinstance( key, int ) :
if key < 0 : #Handle negative indices
key += len( self )
if key >= len( self ) :
raise IndexError, "The index (%d) is out of range."%key
self._numberCells() # no-op if numbered
index = self._i2cc[key]
pointer = self._asPointer(index)
return Cell(self, index, pointer)
else:
raise TypeError, "Invalid argument type."
class Cell(object):
def __init__(self, mesh, index, pointer):
@@ -2225,6 +2250,35 @@ class Cell(object):
self._index = index
self._pointer = pointer
@property
def nodes(self):
"""The node index in _gridN (this may include hanging nodes)."""
M = self.mesh
M._numberNodes()
p = self._pointer
i = self._index
w = M._levelWidth(p[-1])
if M.dim == 2:
n = [
i,
M._index([ p[0] + w, p[1] , p[2]]),
M._index([ p[0] , p[1]+ w, p[2]]),
M._index([ p[0] + w, p[1]+ w, p[2]]),
]
elif self.dim == 3:
n = [
i,
M._index([ p[0] + w, p[1] , p[2] ,p[3]]),
M._index([ p[0] , p[1] + w, p[2] ,p[3]]),
M._index([ p[0] + w, p[1] + w, p[2] ,p[3]]),
M._index([ p[0] , p[1] , p[2] + w,p[3]]),
M._index([ p[0] + w, p[1] , p[2] + w,p[3]]),
M._index([ p[0] , p[1] + w, p[2] + w,p[3]]),
M._index([ p[0] + w, p[1] + w, p[2] + w,p[3]]),
]
return [M._n2i[_] for _ in n]
@property
def center(self):
if getattr(self, '_center', None) is None:
@@ -2282,121 +2336,3 @@ class NotBalancedException(TreeException):
pass
class CellLookUpException(TreeException):
pass
if __name__ == '__main__':
import matplotlib.pyplot as plt
import matplotlib
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.colors as colors
import matplotlib.cm as cmx
def topo(x):
return np.sin(x*(2.*np.pi))*0.3 + 0.5
def function(cell):
r = cell.center - np.array([0.5]*len(cell.center))
dist = np.sqrt(r.dot(r))
# dist2 = np.abs(cell.center[-1] - topo(cell.center[0]))
# dist = min([dist1,dist2])
# if dist < 0.05:
# return 5
if dist < 0.1:
return 5
if dist < 0.2:
return 4
if dist < 0.4:
return 3
return 2
# T = TreeMesh([[(1,128)],[(1,128)],[(1,128)]],levels=7)
# T = TreeMesh([128,128,128])
# T = TreeMesh([64,64],levels=6)
T = TreeMesh([4,4,4])
# T = TreeMesh([[(1,128)],[(1,128)]],levels=7)
# T.refine(lambda xc:2, balance=False)
# T._index([0,0,0])
# T._pointer(0)
# tic = time.time()
T.refine(function)#, balance=False)
# print time.time() - tic
# print T.nC
T.plotSlice(np.log(T.vol))#np.random.rand(T.nC))
plt.show()
blah
# T.plotImage(np.arange(len(T.vol)),showIt=True)
# print T.getFaceInnerProduct()
# print T.gridFz
# T._refineCell([8,0,1])
# T._refineCell([8,0,2])
# T._refineCell([12,0,2])
# T._refineCell([8,4,2])
# T._refineCell([6,0,3])
# T._refineCell([8,8,1])
# T._refineCell([0,0,0,1])
# T.__dirty__ = True
# print T.gridFx.shape[0], T.nFx
ax = plt.subplot(211)
ax.spy(T.edgeCurl)
# print Mesh.TensorMesh([2,2,2]).edgeCurl.todense()
# print T.edgeCurl.todense()
# print Mesh.TensorMesh([2,2,2]).edgeCurl.todense() - T.edgeCurl.todense()
# print T.gridEy - Mesh.TensorMesh([2,2,2]).gridEy
# print T.edge
# T.plotGrid(ax=ax)
# R = deflationMatrix(T._facesX, T._hangingFx, T._fx2i)
# print R
ax = plt.subplot(212)#, projection='3d')
ax.spy(Mesh.TensorMesh([2,2,2]).edgeCurl)
# ax = plt.subplot(313)
# ax.spy(T.faceDiv[:,:T.nFx] * R)
# T.balance()
# T.plotGrid(ax=ax)
# cx = T._getNextCell([0,0,1],direction=0,positive=True)
# print cx
# # print [T._asPointer(_) for _ in cx]
# cx = T._getNextCell([8,0,3],direction=0,positive=False)
# print T._asPointer(cx)
# cx = T._getNextCell([8,8,1],direction=1,positive=False)
# print cx, #[T._asPointer(_) for _ in cx]
# cm = T._getNextCell([64,80,4],direction=0,positive=False)
# cy = T._getNextCell([64,80,4],direction=1,positive=True)
# cp = T._getNextCell([64,80,4],direction=1,positive=False)
# ax.plot( T._cellN([4,0,1])[0],T._cellN([4,0,1])[1], 'yd')
# ax.plot( T._cellN(cx)[0],T._cellN(cx)[1], 'ys')
# ax.plot( T._cellN(cm)[0],T._cellN(cm)[1], 'ys')
# ax.plot( T._cellN(cy)[0],T._cellN(cy)[1], 'ys')
# ax.plot( T._cellN(cp[0])[0],T._cellN(cp[0])[1], 'ys')
# ax.plot( T._cellN(cp[1])[0],T._cellN(cp[1])[1], 'ys')
# print T.nN
plt.show()
SimPEG/Optimization.py
+14
View File
@@ -990,4 +990,18 @@ class ProjectedGNCG(BFGS, Minimize, Remember):
cgFlag = 1
# End CG Iterations
# Take a gradient step on the active cells if exist
if temp != self.xc.size:
rhs_a = (Active) * -self.g
dm_i = max( abs( delx ) )
dm_a = max( abs(rhs_a) )
delx = delx + rhs_a * dm_i / dm_a /10.
# Only keep gradients going in the right direction on the active set
indx = ((self.xc<=self.lower) & (delx < 0)) | ((self.xc>=self.upper) & (delx > 0))
delx[indx] = 0.
return delx
SimPEG/Problem.py
+2 -10
View File
@@ -32,8 +32,8 @@ class BaseProblem(object):
val._assertMatchesPair(self.mapPair)
self._mapping = val
else:
self._mapping = self.PropMap(val)
self._mapping = self.PropMap(val)
def __init__(self, mesh, mapping=None, **kwargs):
Utils.setKwargs(self, **kwargs)
assert isinstance(mesh, Mesh.BaseMesh), "mesh must be a SimPEG.Mesh object."
@@ -158,9 +158,6 @@ class BaseProblem(object):
class BaseTimeProblem(BaseProblem):
"""Sets up that basic needs of a time domain problem."""
waveformType = "STEPOFF"
current = None
@property
def timeSteps(self):
@@ -187,11 +184,6 @@ class BaseTimeProblem(BaseProblem):
self._timeSteps = Utils.meshTensor(value)
del self.timeMesh
def currentwaveform(self, wave):
self._timeSteps = np.diff(wave[:,0])
self.current = wave[:,1]
self.waveformType = "GENERAL"
@property
def nT(self):
"Number of time steps."
SimPEG/Regularization.py
+44 -5
View File
@@ -20,12 +20,13 @@ class BaseRegularization(object):
mesh = None #: A SimPEG.Mesh instance.
mref = None #: Reference model.
def __init__(self, mesh, mapping=None, **kwargs):
def __init__(self, mesh, mapping=None, indActive=None, **kwargs):
Utils.setKwargs(self, **kwargs)
self.mesh = mesh
assert isinstance(mesh, Mesh.BaseMesh), "mesh must be a SimPEG.Mesh object."
self.mapping = mapping or self.mapPair(mesh)
self.mapping._assertMatchesPair(self.mapPair)
self.indActive = indActive
@property
def parent(self):
@@ -112,8 +113,6 @@ class BaseRegularization(object):
return mD.T * ( self.W.T * ( self.W * ( mD * v) ) )
class Tikhonov(BaseRegularization):
"""
"""
@@ -126,14 +125,18 @@ class Tikhonov(BaseRegularization):
alpha_yy = Utils.dependentProperty('_alpha_yy', 0.0, ['_W', '_Wyy'], "Weight for the second derivative in the y direction")
alpha_zz = Utils.dependentProperty('_alpha_zz', 0.0, ['_W', '_Wzz'], "Weight for the second derivative in the z direction")
def __init__(self, mesh, mapping=None, **kwargs):
def __init__(self, mesh, mapping=None, indActive = None, **kwargs):
BaseRegularization.__init__(self, mesh, mapping=mapping, **kwargs)
self.indActive = indActive
@property
def Ws(self):
"""Regularization matrix Ws"""
if getattr(self,'_Ws', None) is None:
self._Ws = Utils.sdiag((self.mesh.vol*self.alpha_s)**0.5)
self._Ws = Utils.sdiag((self.mesh.vol*self.alpha_s)**0.5)
if self.indActive is not None:
Pac = Utils.speye(self.mesh.nC)[:,self.indActive]
self._Ws = Pac.T * self._Ws * Pac
return self._Ws
@property
@@ -142,6 +145,13 @@ class Tikhonov(BaseRegularization):
if getattr(self, '_Wx', None) is None:
Ave_x_vol = self.mesh.aveF2CC[:,:self.mesh.nFx].T*self.mesh.vol
self._Wx = Utils.sdiag((Ave_x_vol*self.alpha_x)**0.5)*self.mesh.cellGradx
if self.indActive is not None:
indActive_Fx = (self.mesh.aveFx2CC.T * self.indActive) == 1
Pac = Utils.speye(self.mesh.nC)[:,self.indActive]
Pafx = Utils.speye(self.mesh.nFx)[:,indActive_Fx]
self._Wx = Pafx.T*self._Wx*Pac
return self._Wx
@property
@@ -150,6 +160,13 @@ class Tikhonov(BaseRegularization):
if getattr(self, '_Wy', None) is None:
Ave_y_vol = self.mesh.aveF2CC[:,self.mesh.nFx:np.sum(self.mesh.vnF[:2])].T*self.mesh.vol
self._Wy = Utils.sdiag((Ave_y_vol*self.alpha_y)**0.5)*self.mesh.cellGrady
if self.indActive is not None:
indActive_Fy = (self.mesh.aveFy2CC.T * self.indActive) == 1
Pac = Utils.speye(self.mesh.nC)[:,self.indActive]
Pafy = Utils.speye(self.mesh.nFy)[:,indActive_Fy]
self._Wy = Pafy.T*self._Wy*Pac
return self._Wy
@property
@@ -158,6 +175,13 @@ class Tikhonov(BaseRegularization):
if getattr(self, '_Wz', None) is None:
Ave_z_vol = self.mesh.aveF2CC[:,np.sum(self.mesh.vnF[:2]):].T*self.mesh.vol
self._Wz = Utils.sdiag((Ave_z_vol*self.alpha_z)**0.5)*self.mesh.cellGradz
if self.indActive is not None:
indActive_Fz = (self.mesh.aveFz2CC.T * self.indActive) == 1
Pac = Utils.speye(self.mesh.nC)[:,self.indActive]
Pafz = Utils.speye(self.mesh.nFz)[:,indActive_Fz]
self._Wz = Pafz.T*self._Wz*Pac
return self._Wz
@property
@@ -165,6 +189,11 @@ class Tikhonov(BaseRegularization):
"""Regularization matrix Wxx"""
if getattr(self, '_Wxx', None) is None:
self._Wxx = Utils.sdiag((self.mesh.vol*self.alpha_xx)**0.5)*self.mesh.faceDivx*self.mesh.cellGradx
if self.indActive is not None:
Pac = Utils.speye(self.mesh.nC)[:,self.indActive]
self._Wxx = Pac.T*self._Wxx*Pac
return self._Wxx
@property
@@ -172,6 +201,11 @@ class Tikhonov(BaseRegularization):
"""Regularization matrix Wyy"""
if getattr(self, '_Wyy', None) is None:
self._Wyy = Utils.sdiag((self.mesh.vol*self.alpha_yy)**0.5)*self.mesh.faceDivy*self.mesh.cellGrady
if self.indActive is not None:
Pac = Utils.speye(self.mesh.nC)[:,self.indActive]
self._Wyy = Pac.T*self._Wyy*Pac
return self._Wyy
@property
@@ -179,6 +213,11 @@ class Tikhonov(BaseRegularization):
"""Regularization matrix Wzz"""
if getattr(self, '_Wzz', None) is None:
self._Wzz = Utils.sdiag((self.mesh.vol*self.alpha_zz)**0.5)*self.mesh.faceDivz*self.mesh.cellGradz
if self.indActive is not None:
Pac = Utils.speye(self.mesh.nC)[:,self.indActive]
self._Wzz = Pac.T*self._Wzz*Pac
return self._Wzz
@property
SimPEG/Survey.py
+1
View File
@@ -205,6 +205,7 @@ class BaseSurvey(object):
__metaclass__ = Utils.SimPEGMetaClass
std = None #: Estimated Standard Deviations
eps = None #: Estimated Noise Floor
dobs = None #: Observed data
dtrue = None #: True data, if data is synthetic
mtrue = None #: True model, if data is synthetic
SimPEG/Utils/SolverUtils.py
+17 -4
View File
@@ -26,7 +26,14 @@ def SolverWrapD(fun, factorize=True, checkAccuracy=True, accuracyTol=1e-6):
def __init__(self, A, **kwargs):
self.A = A.tocsc()
self.checkAccuracy = kwargs.get("checkAccuracy", checkAccuracy)
if kwargs.has_key("checkAccuracy"): del kwargs["checkAccuracy"]
self.accuracyTol = kwargs.get("accuracyTol", accuracyTol)
if kwargs.has_key("accuracyTol"): del kwargs["accuracyTol"]
self.kwargs = kwargs
if factorize:
self.solver = fun(self.A, **kwargs)
@@ -57,8 +64,8 @@ def SolverWrapD(fun, factorize=True, checkAccuracy=True, accuracyTol=1e-6):
else:
X[:,i] = fun(self.A, b[:,i], **self.kwargs)
if checkAccuracy:
_checkAccuracy(self.A, b, X, accuracyTol)
if self.checkAccuracy:
_checkAccuracy(self.A, b, X, self.accuracyTol)
return X
def clean(self):
@@ -81,6 +88,12 @@ def SolverWrapI(fun, checkAccuracy=True, accuracyTol=1e-5):
def __init__(self, A, **kwargs):
self.A = A
self.checkAccuracy = kwargs.get("checkAccuracy", checkAccuracy)
if kwargs.has_key("checkAccuracy"): del kwargs["checkAccuracy"]
self.accuracyTol = kwargs.get("accuracyTol", accuracyTol)
if kwargs.has_key("accuracyTol"): del kwargs["accuracyTol"]
self.kwargs = kwargs
def __mul__(self, b):
@@ -108,8 +121,8 @@ def SolverWrapI(fun, checkAccuracy=True, accuracyTol=1e-5):
else:
X[:,i] = out
if checkAccuracy:
_checkAccuracy(self.A, b, X, accuracyTol)
if self.checkAccuracy:
_checkAccuracy(self.A, b, X, self.accuracyTol)
return X
def clean(self):
SimPEG/Utils/__init__.py
+1 -1
View File
@@ -1,6 +1,6 @@
from matutils import *
from codeutils import *
from meshutils import exampleLrmGrid, meshTensor, closestPoints, readUBCTensorMesh, writeUBCTensorMesh, writeUBCTensorModel, readVTRFile, writeVTRFile
from meshutils import *
from curvutils import volTetra, faceInfo, indexCube
from interputils import interpmat
from CounterUtils import *
SimPEG/Utils/codeutils.py
+1 -1
View File
@@ -17,7 +17,7 @@ def memProfileWrapper(towrap, *funNames):
For example::
foo_mem = memProfile(foo,'my_func')
foo_mem = memProfileWrapper(foo,['my_func'])
fooi = foo_mem()
for i in range(5):
fooi.my_func()
SimPEG/Utils/matutils.py
+12 -1
View File
@@ -2,7 +2,6 @@ import numpy as np
import scipy.sparse as sp
from codeutils import isScalar
def mkvc(x, numDims=1):
"""Creates a vector with the number of dimension specified
@@ -26,6 +25,9 @@ def mkvc(x, numDims=1):
if hasattr(x, 'tovec'):
x = x.tovec()
if isinstance(x, Zero):
return x
assert isinstance(x, np.ndarray), "Vector must be a numpy array"
if numDims == 1:
@@ -37,6 +39,9 @@ def mkvc(x, numDims=1):
def sdiag(h):
"""Sparse diagonal matrix"""
if isinstance(h, Zero):
return Zero()
return sp.spdiags(mkvc(h), 0, h.size, h.size, format="csr")
def sdInv(M):
@@ -417,6 +422,12 @@ class Zero(object):
def __ge__(self, v):return 0 >= v
def __gt__(self, v):return 0 > v
@property
def transpose(self): return Zero()
@property
def T(self): return Zero()
class Identity(object):
_positive = True
def __init__(self, positive=True):
SimPEG/Utils/meshutils.py
-217
View File
@@ -102,223 +102,6 @@ def closestPoints(mesh, pts, gridLoc='CC'):
return nodeInds
def readUBCTensorMesh(fileName):
"""
Read UBC GIF 3DTensor mesh and generate 3D Tensor mesh in simpegTD
Input:
:param fileName, path to the UBC GIF mesh file
Output:
:param SimPEG TensorMesh object
:return
"""
# Interal function to read cell size lines for the UBC mesh files.
def readCellLine(line):
for seg in line.split():
if '*' in seg:
st = seg
sp = seg.split('*')
re = np.array(sp[0],dtype=int)*(' ' + sp[1])
line = line.replace(st,re.strip())
return np.array(line.split(),dtype=float)
# Read the file as line strings, remove lines with comment = !
msh = np.genfromtxt(fileName,delimiter='\n',dtype=np.str,comments='!')
# Fist line is the size of the model
sizeM = np.array(msh[0].split(),dtype=float)
# Second line is the South-West-Top corner coordinates.
x0 = np.array(msh[1].split(),dtype=float)
# Read the cell sizes
h1 = readCellLine(msh[2])
h2 = readCellLine(msh[3])
h3temp = readCellLine(msh[4])
h3 = h3temp[::-1] # Invert the indexing of the vector to start from the bottom.
# Adjust the reference point to the bottom south west corner
x0[2] = x0[2] - np.sum(h3)
# Make the mesh
from SimPEG import Mesh
tensMsh = Mesh.TensorMesh([h1,h2,h3],x0)
return tensMsh
def readUBCTensorModel(fileName, mesh):
"""
Read UBC 3DTensor mesh model and generate 3D Tensor mesh model in simpeg
Input:
:param fileName, path to the UBC GIF mesh file to read
:param mesh, TensorMesh object, mesh that coresponds to the model
Output:
:return numpy array, model with TensorMesh ordered
"""
f = open(fileName, 'r')
model = np.array(map(float, f.readlines()))
f.close()
model = np.reshape(model, (mesh.nCz, mesh.nCx, mesh.nCy), order = 'F')
model = model[::-1,:,:]
model = np.transpose(model, (1, 2, 0))
model = mkvc(model)
return model
def writeUBCTensorMesh(fileName, mesh):
"""
Writes a SimPEG TensorMesh to a UBC-GIF format mesh file.
:param str fileName: File to write to
:param simpeg.Mesh.TensorMesh mesh: The mesh
"""
assert mesh.dim == 3
s = ''
s += '%i %i %i\n' %tuple(mesh.vnC)
origin = mesh.x0 + np.array([0,0,mesh.hz.sum()]) # Have to it in the same operation or use mesh.x0.copy(), otherwise the mesh.x0 is updated.
origin.dtype = float
s += '%.2f %.2f %.2f\n' %tuple(origin)
s += ('%.2f '*mesh.nCx+'\n')%tuple(mesh.hx)
s += ('%.2f '*mesh.nCy+'\n')%tuple(mesh.hy)
s += ('%.2f '*mesh.nCz+'\n')%tuple(mesh.hz[::-1])
f = open(fileName, 'w')
f.write(s)
f.close()
def writeUBCTensorModel(fileName, mesh, model):
"""
Writes a model associated with a SimPEG TensorMesh
to a UBC-GIF format model file.
:param str fileName: File to write to
:param simpeg.Mesh.TensorMesh mesh: The mesh
:param numpy.ndarray model: The model
"""
# Reshape model to a matrix
modelMat = mesh.r(model,'CC','CC','M')
# Transpose the axes
modelMatT = modelMat.transpose((2,0,1))
# Flip z to positive down
modelMatTR = mkvc(modelMatT[::-1,:,:])
np.savetxt(fileName, modelMatTR.ravel())
def readVTRFile(fileName):
"""
Read VTK Rectilinear (vtr xml file) and return SimPEG Tensor mesh and model
Input:
:param vtrFileName, path to the vtr model file to write to
Output:
:return SimPEG TensorMesh object
:return SimPEG model dictionary
"""
# Import
from vtk import vtkXMLRectilinearGridReader as vtrFileReader
from vtk.util.numpy_support import vtk_to_numpy
# Read the file
vtrReader = vtrFileReader()
vtrReader.SetFileName(fileName)
vtrReader.Update()
vtrGrid = vtrReader.GetOutput()
# Sort information
hx = np.abs(np.diff(vtk_to_numpy(vtrGrid.GetXCoordinates())))
xR = vtk_to_numpy(vtrGrid.GetXCoordinates())[0]
hy = np.abs(np.diff(vtk_to_numpy(vtrGrid.GetYCoordinates())))
yR = vtk_to_numpy(vtrGrid.GetYCoordinates())[0]
zD = np.diff(vtk_to_numpy(vtrGrid.GetZCoordinates()))
# Check the direction of hz
if np.all(zD < 0):
hz = np.abs(zD[::-1])
zR = vtk_to_numpy(vtrGrid.GetZCoordinates())[-1]
else:
hz = np.abs(zD)
zR = vtk_to_numpy(vtrGrid.GetZCoordinates())[0]
x0 = np.array([xR,yR,zR])
# Make the SimPEG object
from SimPEG import Mesh
tensMsh = Mesh.TensorMesh([hx,hy,hz],x0)
# Grap the models
modelDict = {}
for i in np.arange(vtrGrid.GetCellData().GetNumberOfArrays()):
modelName = vtrGrid.GetCellData().GetArrayName(i)
if np.all(zD < 0):
modFlip = vtk_to_numpy(vtrGrid.GetCellData().GetArray(i))
tM = tensMsh.r(modFlip,'CC','CC','M')
modArr = tensMsh.r(tM[:,:,::-1],'CC','CC','V')
else:
modArr = vtk_to_numpy(vtrGrid.GetCellData().GetArray(i))
modelDict[modelName] = modArr
# Return the data
return tensMsh, modelDict
def writeVTRFile(fileName,mesh,model=None):
"""
Makes and saves a VTK rectilinear file (vtr) for a simpeg Tensor mesh and model.
Input:
:param str, path to the output vtk file
:param mesh, SimPEG TensorMesh object - mesh to be transfer to VTK
:param model, dictionary of numpy.array - Name('s) and array('s). Match number of cells
"""
# Import
from vtk import vtkRectilinearGrid as rectGrid, vtkXMLRectilinearGridWriter as rectWriter
from vtk.util.numpy_support import numpy_to_vtk
# Deal with dimensionalities
if mesh.dim >= 1:
vX = mesh.vectorNx
xD = mesh.nNx
yD,zD = 1,1
vY, vZ = np.array([0,0])
if mesh.dim >= 2:
vY = mesh.vectorNy
yD = mesh.nNy
if mesh.dim == 3:
vZ = mesh.vectorNz
zD = mesh.nNz
# Use rectilinear VTK grid.
# Assign the spatial information.
vtkObj = rectGrid()
vtkObj.SetDimensions(xD,yD,zD)
vtkObj.SetXCoordinates(numpy_to_vtk(vX,deep=1))
vtkObj.SetYCoordinates(numpy_to_vtk(vY,deep=1))
vtkObj.SetZCoordinates(numpy_to_vtk(vZ,deep=1))
# Assign the model('s) to the object
for item in model.iteritems():
# Convert numpy array
vtkDoubleArr = numpy_to_vtk(item[1],deep=1)
vtkDoubleArr.SetName(item[0])
vtkObj.GetCellData().AddArray(vtkDoubleArr)
# Set the active scalar
vtkObj.GetCellData().SetActiveScalars(model.keys()[0])
vtkObj.Update()
# Check the extension of the fileName
ext = os.path.splitext(fileName)[1]
if ext is '':
fileName = fileName + '.vtr'
elif ext not in '.vtr':
raise IOError('{:s} is an incorrect extension, has to be .vtr')
# Write the file.
vtrWriteFilter = rectWriter()
vtrWriteFilter.SetInput(vtkObj)
vtrWriteFilter.SetFileName(fileName)
vtrWriteFilter.Update()
def ExtractCoreMesh(xyzlim, mesh, meshType='tensor'):
"""
Extracts Core Mesh from Global mesh
docs/em/api_FDEM.rst
+62 -42
View File
@@ -19,14 +19,14 @@ Electromagnetic phenomena are governed by Maxwell's equations. They describe the
Fourier Transform Convention
----------------------------
In order to examine Maxwell's equations in the frequency domain, we must first define our choice of harmonic time-dependence by choosing a Fourier transform convention. We use the \\(e^{i \\omega t} \\) convention, so we define our Fourier Transform pair as
In order to examine Maxwell's equations in the frequency domain, we must first define our choice of harmonic time-dependence by choosing a Fourier transform convention. We use the :math:`e^{i \omega t}` convention, so we define our Fourier Transform pair as
.. math ::
F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\
F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\
f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d \omega
f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d \omega
where \\(\\omega\\) is angular frequency, \\(t\\) is time, \\(F(\\omega)\\) is the function defined in the frequency domain and \\(f(t)\\) is the function defined in the time domain.
where :math:`\omega` is angular frequency, :math:`t` is time, :math:`F(\omega)` is the function defined in the frequency domain and :math:`f(t)` is the function defined in the time domain.
Maxwell's Equations
@@ -34,44 +34,46 @@ Maxwell's Equations
In the frequency domain, Maxwell's equations are given by
.. math ::
\curl \vec{E} = - i \omega \vec{B} \\
\curl \vec{E} + i \omega \vec{B} = \vec{S_m}\\
\curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{S} \\
\curl \vec{H} - \vec{J} - i \omega \vec{D} = \vec{S_e} \\
\div \vec{B} = 0 \\
\div \vec{B} = 0 \\
\div \vec{D} = \rho_f
\div \vec{D} = \rho_f
where:
- \\(\\vec{E}\\) : electric field (\\(V/m\\))
- \\(\\vec{H}\\) : magnetic field (\\(A/m\\))
- \\(\\vec{B}\\) : magnetic flux density (\\(Wb/m^2\\))
- \\(\\vec{D}\\) : electric displacement / electric flux density (\\(C/m^2\\))
- \\(\\vec{J}\\) : electric current density (\\(A/m^2\\))
- \\(\\rho_f\\) : free charge density
- :math:`\vec{E}` : electric field (:math:`V/m` )
- :math:`\vec{H}` : magnetic field (:math:`A/m` )
- :math:`\vec{B}` : magnetic flux density (:math:`Wb/m^2` )
- :math:`\vec{D}` : electric displacement / electric flux density (:math:`C/m^2` )
- :math:`\vec{J}` : electric current density (:math:`A/m^2` )
- :math:`\vec{S_m}` : magnetic source term (:math:`V/m^2` )
- :math:`\vec{S_e}` : electric source term (:math:`A/m^2` )
- :math:`\rho_f` : free charge density (:math:`\Omega m` )
The source term is \\(\\vec{S}\\)
Constitutive Relations
----------------------
The fields and fluxes are related through the constitutive relations. At each frequency, they are given by
.. math ::
\vec{J} = \sigma \vec{E} \\
\vec{J} = \sigma \vec{E} \\
\vec{B} = \mu \vec{H} \\
\vec{B} = \mu \vec{H} \\
\vec{D} = \varepsilon \vec{E}
\vec{D} = \varepsilon \vec{E}
where:
- \\(\\sigma\\) : electrical conductivity \\(S/m\\)
- \\(\\mu\\) : magnetic permeability \\(H/m\\)
- \\(\\varepsilon\\) : dielectric permittivity \\(F/m\\)
- :math:`\sigma` : electrical conductivity (:math:`S/m`)
- :math:`\mu` : magnetic permeability (:math:`H/m`)
- :math:`\varepsilon` : dielectric permittivity (:math:`F/m`)
\\(\\sigma\\), \\(\\mu\\), \\(\\varepsilon\\) are physical properties which depend on the material. \\(\\sigma\\) describes how easily electric current passes through a material, \\(\\mu\\) describes how easily a material is magnetized, and \\(\\varepsilon\\) describes how easily a material is electrically polarized. In most geophysical applications of EM, \\(\\sigma\\) is the the primary physical property of interest, and \\(\\mu\\), \\(\\varepsilon\\) are assumed to have their free-space values \\(\\mu_0 = 4\\pi \\times 10^{-7} H/m \\), \\(\\varepsilon_0 = 8.85 \\times 10^{-12} F/m\\)
:math:`\sigma`, :math:`\mu`, :math:`\varepsilon` are physical properties which depend on the material. :math:`\sigma` describes how easily electric current passes through a material, :math:`\mu` describes how easily a material is magnetized, and :math:`\varepsilon` describes how easily a material is electrically polarized. In most geophysical applications of EM, :math:`\sigma` is the the primary physical property of interest, and :math:`\mu`, :math:`\varepsilon` are assumed to have their free-space values :math:`\mu_0 = 4\pi \times 10^{-7} H/m` , :math:`\varepsilon_0 = 8.85 \times 10^{-12} F/m`
Quasi-static Approximation
@@ -80,8 +82,8 @@ Quasi-static Approximation
For the frequency range typical of most geophysical surveys, the contribution of the electric displacement is negligible compared to the electric current density. In this case, we use the Quasi-static approximation and assume that this term can be neglected, giving
.. math ::
\nabla \times \vec{E} = -i \omega \vec{B} \\
\nabla \times \vec{H} = \vec{J} + \vec{S}
\nabla \times \vec{E} + i \omega \vec{B} = \vec{S_m} \\
\nabla \times \vec{H} - \vec{J} = \vec{S_e}
Implementation in SimPEG.EM
@@ -90,14 +92,14 @@ Implementation in SimPEG.EM
We consider two formulations in SimPEG.EM, both first-order and both in terms of one field and one flux. We allow for the definition of magnetic and electric sources (see for example: Ward and Hohmann, starting on page 144). The E-B formulation is in terms of the electric field and the magnetic flux:
.. math ::
\nabla \times \vec{E} + i \omega \vec{B} = \vec{S}_m \\
\nabla \times \mu^{-1} \vec{B} - \sigma \vec{E} = \vec{S}_e
\nabla \times \vec{E} + i \omega \vec{B} = \vec{S}_m \\
\nabla \times \mu^{-1} \vec{B} - \sigma \vec{E} = \vec{S}_e
The H-J formulation is in terms of the current density and the magnetic field:
.. math ::
\nabla \times \sigma^{-1} \vec{J} + i \omega \mu \vec{H} = \vec{S}_m \\
\nabla \times \vec{H} - \vec{J} = \vec{S}_e
\nabla \times \sigma^{-1} \vec{J} + i \omega \mu \vec{H} = \vec{S}_m \\
\nabla \times \vec{H} - \vec{J} = \vec{S}_e
Discretizing
@@ -106,34 +108,34 @@ For both formulations, we use a finite volume discretization
and discretize fields on cell edges, fluxes on cell faces and
physical properties in cell centers. This is particularly
important when using symmetry to reduce the dimensionality of a problem
(for instance on a 2D CylMesh, there are \\(r\\), \\(z\\) faces and \\(\\theta\\) edges)
(for instance on a 2D CylMesh, there are :math:`r`, :math:`z` faces and :math:`\theta` edges)
.. figure:: ../images/finitevolrealestate.png
:align: center
:scale: 60 %
:align: center
:scale: 60 %
For the two formulations, the discretization of the physical properties, fields and fluxes are summarized below.
.. figure:: ../images/ebjhdiscretizations.png
:align: center
:scale: 60 %
:align: center
:scale: 60 %
Note that resistivity is the inverse of conductivity, \\(\\rho = \\sigma^{-1}\\).
Note that resistivity is the inverse of conductivity, :math:`\rho = \sigma^{-1}`.
E-B Formulation:
****************
E-B Formulation
---------------
.. math ::
\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\
\mathbf{C^T} \mathbf{M^f_{\mu^{-1}}} \mathbf{b} - \mathbf{M^e_\sigma} \mathbf{e} = \mathbf{M^e} \mathbf{s_e}
\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\
\mathbf{C^T} \mathbf{M^f_{\mu^{-1}}} \mathbf{b} - \mathbf{M^e_\sigma} \mathbf{e} = \mathbf{M^e} \mathbf{s_e}
H-J Formulation:
****************
H-J Formulation
---------------
.. math ::
\mathbf{C^T} \mathbf{M^f_\rho} \mathbf{j} + i \omega \mathbf{M^e_\mu} \mathbf{h} = \mathbf{M^e} \mathbf{s_m} \\
\mathbf{C} \mathbf{h} - \mathbf{j} = \mathbf{s_e}
\mathbf{C^T} \mathbf{M^f_\rho} \mathbf{j} + i \omega \mathbf{M^e_\mu} \mathbf{h} = \mathbf{M^e} \mathbf{s_m} \\
\mathbf{C} \mathbf{h} - \mathbf{j} = \mathbf{s_e}
.. Forward Problem
@@ -144,6 +146,10 @@ H-J Formulation:
API
===
FDEM Problem
------------
.. automodule:: SimPEG.EM.FDEM.FDEM
:show-inheritance:
:members:
@@ -157,3 +163,17 @@ FDEM Survey
:show-inheritance:
:members:
:undoc-members:
.. automodule:: SimPEG.EM.FDEM.SrcFDEM
:show-inheritance:
:members:
:undoc-members:
FDEM Fields
-----------
.. automodule:: SimPEG.EM.FDEM.FieldsFDEM
:show-inheritance:
:members:
:undoc-members:
docs/em/api_TDEM.rst
+299
View File
@@ -48,6 +48,305 @@
\newcommand{\I}{\vec{I}}
Time Domain Electromagnetics
****************************
.. _api_TDEM_derivation:
Time-Domain EM Derivation
=========================
The following shows the derivation for the TDEM problem. We use the b-formulation below.
(More to come soon..!)
Sensitivity Calculation
-----------------------
.. math::
\begin{align}
\dcurl \e^{(t+1)} + \frac{\b^{(t+1)} - \b^{(t)}}{\delta t} = 0 \\
\dcurl^\top \MfMui \b^{(t+1)} - \MeSig \e^{(t+1)} = \Me \j_s^{(t+1)}
\end{align}
Using Gauss-Newton to solve the inverse problem requires the ability to calculate the product of the
Jacobian and a vector, as well as the transpose of the Jacobian times a vector.
The above system can be rewritten as:
.. math::
\begin{align}
\mathbf{A} \u^{(t+1)} + \mathbf{B} \u^{(t)}= \s^{(t+1)}
\end{align}
where
.. math::
\begin{align}
\mathbf{A} =
\left[
\begin{array}{cc}
\frac{1}{\delta t} \MfMui & \MfMui\dcurl \\
\dcurl^\top \MfMui & -\MeSig
\end{array}
\right] \\
\mathbf{B} =
\left[
\begin{array}{cc}
-\frac{1}{\delta t} \MfMui & 0 \\
0 & 0
\end{array}
\right] \\
\u^{(k)} = \left[
\begin{array}{c}
\b^{(k)}\\
\e^{(k)}
\end{array}
\right] \\
\s^{(k)} = \left[
\begin{array}{c}
0\\
\Me \j^{(k)}_s
\end{array}
\right]
\end{align}
.. note::
Here we have multiplied through by \\(\\MfMui\\) to make A and B symmetric!
The entire time dependent system can be written in a single matrix expression
.. math::
\begin{align}
\hat{\mathbf{A}} \hat{u} = \hat{s}
\end{align}
where
.. math::
\begin{align}
\mathbf{\hat{A}} = \left[
\begin{array}{cccc}
A & 0 & & \\
B & A & & \\
& \ddots & \ddots & \\
& & B & A
\end{array}
\right] \\
\hat{u} = \left[
\begin{array}{c}
\u^{(1)} \\
\u^{(2)} \\
\vdots \\
\u^{(N)}
\end{array} \right]\\
\hat{s} = \left[
\begin{array}{c}
\s^{(1)} - \mathbf{B} \u^{(0)} \\
\s^{(2)} \\
\vdots \\
\s^{(N)}
\end{array}
\right]
\end{align}
For the fields \\(\\u\\), the measured data is given by
.. math::
\begin{align}
\vec{d} = \mathbf{Q} \u
\end{align}
The sensitivity matrix **J** is then defined as
.. math::
\begin{align}
\mathbf{J} = \mathbf{Q} \frac{\partial \u}{\partial \sigma}
\end{align}
Defining the function \\(\\c(m,\\u)\\) to be
.. math::
\begin{align}
\vec{c}(m,\u) = \hat{\mathbf{A}} \vec{u} - \vec{q} = \vec{0}
\end{align}
then
.. math::
\begin{align}
\frac{\partial \vec{c}}{\partial m} \partial m
+ \frac{\partial \vec{c}}{\partial \u} \partial \vec{u} = 0
\end{align}
or
.. math::
\begin{align}
\frac{\partial \vec{u}}{\partial m} = -\left(\frac{\partial \vec{c}}{\partial \u} \right)^{-1} \frac{\partial \vec{c}}{\partial m}
\end{align}
Differentiating, we find that
.. math::
\begin{align}
\frac{\partial \vec{c}}{\partial \hat{u}} = \hat{\mathbf{A}}
\end{align}
and
.. math::
\begin{align}
\frac{\partial \vec{c}}{\partial \sigma} = \mathbf{G}_\sigma =
\left[
\begin{array}{c}
g_\sigma^{(1)}\\
g_\sigma^{(2)}\\
\vdots \\
g_\sigma^{(N)}
\end{array}
\right]
\end{align}
with
.. math::
\begin{align}
g_\sigma^{(n)} =
\left[
\begin{array}{c}
\mathbf{0} \\
- \diag{\e^{(n)}} \Ace \diag{\vec{V}}
\end{array}
\right]
\end{align}
Implementing **J** times a vector
---------------------------------
Multiplying **J** onto a vector can be broken into three steps
* Compute \\(\\vec{p} = \\mathbf{G}m\\)
* Solve \\(\\hat{\\mathbf{A}} \\vec{y} = \\vec{p}\\)
* Compute \\(\\vec{w} = -\\mathbf{Q} \\vec{y}\\)
.. math::
\begin{align}
\vec{p}^{(n)} = \left[
\begin{array}{c}
\vec{p}_b^{(n)} \\
\vec{p}_e^{(n)}
\end{array}
\right] \\
\vec{p}_b^{(n)} = 0 \\
\vec{p}_e^{(n)} = - \diag{\e^{(n)}} \Ace \diag{V} m
\end{align}
For all time steps:
.. math::
\begin{align}
\frac{1}{\delta t} \MfMui\vec{y}_{b}^{(t+1)} + \MfMui\dcurl \vec{y}_{e}^{(t+1)}
- \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(t)}
= \vec{p}_b^{(t+1)} \\
\dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig \vec{y}_e^{(t+1)} = \vec{p}_e^{(t+1)}
\end{align}
and
.. math::
\begin{align}
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t+1)} =
\frac{1}{\delta t} \MfMui \vec{y}_b^{(t)}
+ \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+1)} + \vec{p}_b^{(t+1)} \\
\vec{y}_e^{(t+1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig^{-1} \vec{p}_e^{(t+1)}
\end{align}
.. note::
For the first time step, \\\(t=0\\\), the term: \\\(\\frac{1}{\\delta t} \\MfMui \\vec{y}_b^{(0)}\\\) is zero.
Implementing **J** transpose times a vector
-------------------------------------------
Multiplying \\(\\mathbf{J}^\\top\\) onto a vector can be broken into three steps
* Compute \\(\\vec{p} = \\mathbf{Q}^\\top \\vec{v}\\)
* Solve \\(\\hat{\\mathbf{A}}^\\top \\vec{y} = \\vec{p}\\)
* Compute \\(\\vec{w} = -\\mathbf{G}^\\top y\\)
.. math::
\mathbf{\hat{A}}^\top = \left[
\begin{array}{cccc}
A & B & & \\
& \ddots & \ddots & \\
& & A & B \\
& & 0 & A
\end{array}
\right]
For the all time-steps (going backwards in time):
.. math::
A \vec{y}^{(t)} + B \vec{y}^{(t+1)} = \vec{p}^{(t)}
.. math::
\begin{align}
\frac{1}{\delta t} \MfMui\vec{y}_{b}^{(t)} + \MfMui\dcurl \vec{y}_{e}^{(t)}
- \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(t+1)}
= \vec{p}_b^{(t)} \\
\dcurl^\top \MfMui \vec{y}_b^{(t)} - \MeSig \vec{y}_e^{(t)} = \vec{p}_e^{(t)}
\end{align}
and
.. math::
\begin{align}
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t)} =
\frac{1}{\delta t} \MfMui \vec{y}_b^{(t+1)}
+ \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t)} + \vec{p}_b^{(t)} \\
\vec{y}_e^{(t)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t)} - \MeSig^{-1} \vec{p}_e^{(t)}
\end{align}
.. note::
For the last time step, \\\(t=N\\\), the term: \\\(\\frac{1}{\\delta t} \\MfMui \\vec{y}_b^{(N+1)}\\\) is zero.
TDEM - B formulation
====================
docs/em/api_TDEM_derivation.rst
-341
View File
@@ -1,341 +0,0 @@
.. _api_TDEM_derivation:
.. math::
\renewcommand{\div}{\nabla\cdot\,}
\newcommand{\grad}{\vec \nabla}
\newcommand{\curl}{{\vec \nabla}\times\,}
\newcommand {\J}{{\vec J}}
\renewcommand{\H}{{\vec H}}
\newcommand {\E}{{\vec E}}
\newcommand{\dcurl}{{\mathbf C}}
\newcommand{\dgrad}{{\mathbf G}}
\newcommand{\Acf}{{\mathbf A_c^f}}
\newcommand{\Ace}{{\mathbf A_c^e}}
\renewcommand{\S}{{\mathbf \Sigma}}
\newcommand{\St}{{\mathbf \Sigma_\tau}}
\newcommand{\T}{{\mathbf T}}
\newcommand{\Tt}{{\mathbf T_\tau}}
\newcommand{\diag}[1]{\,{\sf diag}\left( #1 \right)}
\newcommand{\M}{{\mathbf M}}
\newcommand{\MfMui}{{\M^f_{\mu^{-1}}}}
\newcommand{\MeSig}{{\M^e_\sigma}}
\newcommand{\MeSigInf}{{\M^e_{\sigma_\infty}}}
\newcommand{\MeSigO}{{\M^e_{\sigma_0}}}
\newcommand{\Me}{{\M^e}}
\newcommand{\Mes}[1]{{\M^e_{#1}}}
\newcommand{\Mee}{{\M^e_e}}
\newcommand{\Mej}{{\M^e_j}}
\newcommand{\BigO}[1]{\mathcal{O}\bigl(#1\bigr)}
\newcommand{\bE}{\mathbf{E}}
\newcommand{\bH}{\mathbf{H}}
\newcommand{\B}{\vec{B}}
\newcommand{\D}{\vec{D}}
\renewcommand{\H}{\vec{H}}
\newcommand{\s}{\vec{s}}
\newcommand{\bfJ}{\bf{J}}
\newcommand{\vecm}{\vec m}
\renewcommand{\Re}{\mathsf{Re}}
\renewcommand{\Im}{\mathsf{Im}}
\renewcommand {\j} { {\vec j} }
\newcommand {\h} { {\vec h} }
\renewcommand {\b} { {\vec b} }
\newcommand {\e} { {\vec e} }
\newcommand {\c} { {\vec c} }
\renewcommand {\d} { {\vec d} }
\renewcommand {\u} { {\vec u} }
\newcommand{\I}{\vec{I}}
Time-Domain EM Derivation
*************************
The following shows the derivation for the TDEM problem. We use the b-formulation below.
(More to come soon..!)
Sensitivity Calculation
=======================
.. math::
\begin{align}
\dcurl \e^{(t+1)} + \frac{\b^{(t+1)} - \b^{(t)}}{\delta t} = 0 \\
\dcurl^\top \MfMui \b^{(t+1)} - \MeSig \e^{(t+1)} = \Me \j_s^{(t+1)}
\end{align}
Using Gauss-Newton to solve the inverse problem requires the ability to calculate the product of the
Jacobian and a vector, as well as the transpose of the Jacobian times a vector.
The above system can be rewritten as:
.. math::
\begin{align}
\mathbf{A} \u^{(t+1)} + \mathbf{B} \u^{(t)}= \s^{(t+1)}
\end{align}
where
.. math::
\begin{align}
\mathbf{A} =
\left[
\begin{array}{cc}
\frac{1}{\delta t} \MfMui & \MfMui\dcurl \\
\dcurl^\top \MfMui & -\MeSig
\end{array}
\right] \\
\mathbf{B} =
\left[
\begin{array}{cc}
-\frac{1}{\delta t} \MfMui & 0 \\
0 & 0
\end{array}
\right] \\
\u^{(k)} = \left[
\begin{array}{c}
\b^{(k)}\\
\e^{(k)}
\end{array}
\right] \\
\s^{(k)} = \left[
\begin{array}{c}
0\\
\Me \j^{(k)}_s
\end{array}
\right]
\end{align}
.. note::
Here we have multiplied through by \\(\\MfMui\\) to make A and B symmetric!
The entire time dependent system can be written in a single matrix expression
.. math::
\begin{align}
\hat{\mathbf{A}} \hat{u} = \hat{s}
\end{align}
where
.. math::
\begin{align}
\mathbf{\hat{A}} = \left[
\begin{array}{cccc}
A & 0 & & \\
B & A & & \\
& \ddots & \ddots & \\
& & B & A
\end{array}
\right] \\
\hat{u} = \left[
\begin{array}{c}
\u^{(1)} \\
\u^{(2)} \\
\vdots \\
\u^{(N)}
\end{array} \right]\\
\hat{s} = \left[
\begin{array}{c}
\s^{(1)} - \mathbf{B} \u^{(0)} \\
\s^{(2)} \\
\vdots \\
\s^{(N)}
\end{array}
\right]
\end{align}
For the fields \\(\\u\\), the measured data is given by
.. math::
\begin{align}
\vec{d} = \mathbf{Q} \u
\end{align}
The sensitivity matrix **J** is then defined as
.. math::
\begin{align}
\mathbf{J} = \mathbf{Q} \frac{\partial \u}{\partial \sigma}
\end{align}
Defining the function \\(\\c(m,\\u)\\) to be
.. math::
\begin{align}
\vec{c}(m,\u) = \hat{\mathbf{A}} \vec{u} - \vec{q} = \vec{0}
\end{align}
then
.. math::
\begin{align}
\frac{\partial \vec{c}}{\partial m} \partial m
+ \frac{\partial \vec{c}}{\partial \u} \partial \vec{u} = 0
\end{align}
or
.. math::
\begin{align}
\frac{\partial \vec{u}}{\partial m} = -\left(\frac{\partial \vec{c}}{\partial \u} \right)^{-1} \frac{\partial \vec{c}}{\partial m}
\end{align}
Differentiating, we find that
.. math::
\begin{align}
\frac{\partial \vec{c}}{\partial \hat{u}} = \hat{\mathbf{A}}
\end{align}
and
.. math::
\begin{align}
\frac{\partial \vec{c}}{\partial \sigma} = \mathbf{G}_\sigma =
\left[
\begin{array}{c}
g_\sigma^{(1)}\\
g_\sigma^{(2)}\\
\vdots \\
g_\sigma^{(N)}
\end{array}
\right]
\end{align}
with
.. math::
\begin{align}
g_\sigma^{(n)} =
\left[
\begin{array}{c}
\mathbf{0} \\
- \diag{\e^{(n)}} \Ace \diag{\vec{V}}
\end{array}
\right]
\end{align}
Implementing **J** times a vector
=================================
Multiplying **J** onto a vector can be broken into three steps
* Compute \\(\\vec{p} = \\mathbf{G}m\\)
* Solve \\(\\hat{\\mathbf{A}} \\vec{y} = \\vec{p}\\)
* Compute \\(\\vec{w} = -\\mathbf{Q} \\vec{y}\\)
.. math::
\begin{align}
\vec{p}^{(n)} = \left[
\begin{array}{c}
\vec{p}_b^{(n)} \\
\vec{p}_e^{(n)}
\end{array}
\right] \\
\vec{p}_b^{(n)} = 0 \\
\vec{p}_e^{(n)} = - \diag{\e^{(n)}} \Ace \diag{V} m
\end{align}
For all time steps:
.. math::
\begin{align}
\frac{1}{\delta t} \MfMui\vec{y}_{b}^{(t+1)} + \MfMui\dcurl \vec{y}_{e}^{(t+1)}
- \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(t)}
= \vec{p}_b^{(t+1)} \\
\dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig \vec{y}_e^{(t+1)} = \vec{p}_e^{(t+1)}
\end{align}
and
.. math::
\begin{align}
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t+1)} =
\frac{1}{\delta t} \MfMui \vec{y}_b^{(t)}
+ \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+1)} + \vec{p}_b^{(t+1)} \\
\vec{y}_e^{(t+1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig^{-1} \vec{p}_e^{(t+1)}
\end{align}
.. note::
For the first time step, \\\(t=0\\\), the term: \\\(\\frac{1}{\\delta t} \\MfMui \\vec{y}_b^{(0)}\\\) is zero.
Implementing **J** transpose times a vector
===========================================
Multiplying \\(\\mathbf{J}^\\top\\) onto a vector can be broken into three steps
* Compute \\(\\vec{p} = \\mathbf{Q}^\\top \\vec{v}\\)
* Solve \\(\\hat{\\mathbf{A}}^\\top \\vec{y} = \\vec{p}\\)
* Compute \\(\\vec{w} = -\\mathbf{G}^\\top y\\)
.. math::
\mathbf{\hat{A}}^\top = \left[
\begin{array}{cccc}
A & B & & \\
& \ddots & \ddots & \\
& & A & B \\
& & 0 & A
\end{array}
\right]
For the all time-steps (going backwards in time):
.. math::
A \vec{y}^{(t)} + B \vec{y}^{(t+1)} = \vec{p}^{(t)}
.. math::
\begin{align}
\frac{1}{\delta t} \MfMui\vec{y}_{b}^{(t)} + \MfMui\dcurl \vec{y}_{e}^{(t)}
- \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(t+1)}
= \vec{p}_b^{(t)} \\
\dcurl^\top \MfMui \vec{y}_b^{(t)} - \MeSig \vec{y}_e^{(t)} = \vec{p}_e^{(t)}
\end{align}
and
.. math::
\begin{align}
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t)} =
\frac{1}{\delta t} \MfMui \vec{y}_b^{(t+1)}
+ \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t)} + \vec{p}_b^{(t)} \\
\vec{y}_e^{(t)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t)} - \MeSig^{-1} \vec{p}_e^{(t)}
\end{align}
.. note::
For the last time step, \\\(t=N\\\), the term: \\\(\\frac{1}{\\delta t} \\MfMui \\vec{y}_b^{(N+1)}\\\) is zero.
docs/em/api_Utils.rst
+10 -9
View File
@@ -4,6 +4,16 @@ simpegEM Utilities
SimPEG for EM provides a few EM specific utility codes,
sources, and analytic functions.
Utilities for Electromagnetics
==============================
.. automodule:: SimPEG.EM.Utils
:show-inheritance:
:members:
:undoc-members:
:inherited-members:
Analytic Functions - Time
=========================
@@ -22,12 +32,3 @@ Analytic Functions - Frequency
:members:
:undoc-members:
:inherited-members:
Sources
=======
.. autoclass:: SimPEG.EM.FDEM.SrcFDEM.MagDipole
:show-inheritance:
:members:
:undoc-members:
docs/em/index.rst
+9 -27
View File
@@ -3,42 +3,24 @@ Electromagnetics
================
`SimPEG.EM` uses SimPEG as the framework for the forward and inverse
electromagnetics geophysical problems.
electromagnetics geophysical problems.
Time Domian Electromagnetics
----------------------------
.. toctree::
:maxdepth: 2
api_TDEM_derivation
To solve for predicted data, we follow the framework shown below. The model is
what we invert for. This is mapped to a physical property on the simulation
mesh. A source which is used to excite the system is specified. Having a model
and a source, we can solve Maxwell's equations for fields. We sample these
fields with recievers to give us predicted data.
Code for Time Domian Electromagnetics
-------------------------------------
.. image:: ../images/simpegEM_noMath.png
:scale: 50%
.. toctree::
:maxdepth: 2
api_TDEM
Frequency Domian Electromagnetics
---------------------------------
.. toctree::
:maxdepth: 2
api_FDEM
Utility Codes
-------------
.. toctree::
:maxdepth: 2
api_TDEM
api_Utils
docs/examples/EM_FDEM_1D_Inversion.rst
+26
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@@ -0,0 +1,26 @@
.. _examples_EM_FDEM_1D_Inversion:
.. --------------------------------- ..
.. ..
.. THIS FILE IS AUTO GENEREATED ..
.. ..
.. SimPEG/Examples/__init__.py ..
.. ..
.. --------------------------------- ..
EM: FDEM: 1D: Inversion
=======================
Here we will create and run a FDEM 1D inversion.
.. plot::
from SimPEG import Examples
Examples.EM_FDEM_1D_Inversion.run()
.. literalinclude:: ../../SimPEG/Examples/EM_FDEM_1D_Inversion.py
:language: python
:linenos:
docs/flow/index.rst
+1 -1
View File
@@ -41,7 +41,7 @@ Here we reproduce the results from Celia et al. (1990):
Richards
========
.. automodule:: simpegFLOW.Richards.Empirical
.. automodule:: SimPEG.FLOW.Richards.Empirical
:show-inheritance:
:members:
:undoc-members:
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tests/base/test_regularization.py
+63 -12
View File
@@ -4,11 +4,17 @@ from SimPEG import *
from scipy.sparse.linalg import dsolve
import inspect
TOL = 1e-20
class RegularizationTests(unittest.TestCase):
def setUp(self):
self.mesh2 = Mesh.TensorMesh([3, 2])
hx, hy, hz = np.random.rand(10), np.random.rand(9), np.random.rand(8)
hx, hy, hz = hx/hx.sum(), hy/hy.sum(), hz/hz.sum()
mesh1 = Mesh.TensorMesh([hx])
mesh2 = Mesh.TensorMesh([hx, hy])
mesh3 = Mesh.TensorMesh([hx, hy, hz])
self.meshlist = [mesh1,mesh2, mesh3]
def test_regularization(self):
for R in dir(Regularization):
@@ -16,18 +22,63 @@ class RegularizationTests(unittest.TestCase):
if not inspect.isclass(r): continue
if not issubclass(r, Regularization.BaseRegularization):
continue
# if 'Regularization' not in R: continue
mapping = r.mapPair(self.mesh2)
reg = r(self.mesh2, mapping=mapping)
m = np.random.rand(mapping.nP)
reg.mref = m[:]*np.mean(m)
print 'Check:', R
passed = Tests.checkDerivative(lambda m : [reg.eval(m), reg.evalDeriv(m)], m, plotIt=False)
self.assertTrue(passed)
print 'Check 2 Deriv:', R
passed = Tests.checkDerivative(lambda m : [reg.evalDeriv(m), reg.eval2Deriv(m)], m, plotIt=False)
self.assertTrue(passed)
for i, mesh in enumerate(self.meshlist):
print 'Testing %iD'%mesh.dim
mapping = r.mapPair(mesh)
reg = r(mesh, mapping=mapping)
m = np.random.rand(mapping.nP)
reg.mref = np.ones_like(m)*np.mean(m)
print 'Check: phi_m (mref) = %f' %reg.eval(reg.mref)
passed = reg.eval(reg.mref) < TOL
self.assertTrue(passed)
print 'Check:', R
passed = Tests.checkDerivative(lambda m : [reg.eval(m), reg.evalDeriv(m)], m, plotIt=False)
self.assertTrue(passed)
print 'Check 2 Deriv:', R
passed = Tests.checkDerivative(lambda m : [reg.evalDeriv(m), reg.eval2Deriv(m)], m, plotIt=False)
self.assertTrue(passed)
def test_regularization_ActiveCells(self):
for R in dir(Regularization):
r = getattr(Regularization, R)
if not inspect.isclass(r): continue
if not issubclass(r, Regularization.BaseRegularization):
continue
for i, mesh in enumerate(self.meshlist):
print 'Testing Active Cells %iD'%(mesh.dim)
if mesh.dim == 1:
indAct = Utils.mkvc(mesh.gridCC <= 0.8)
elif mesh.dim == 2:
indAct = Utils.mkvc(mesh.gridCC[:,-1] <= 2*np.sin(2*np.pi*mesh.gridCC[:,0])+0.5)
elif mesh.dim == 3:
indAct = Utils.mkvc(mesh.gridCC[:,-1] <= 2*np.sin(2*np.pi*mesh.gridCC[:,0])+0.5 * 2*np.sin(2*np.pi*mesh.gridCC[:,1])+0.5)
mapping = Maps.IdentityMap(nP=indAct.nonzero()[0].size)
reg = r(mesh, mapping=mapping, indActive=indAct)
m = np.random.rand(mesh.nC)[indAct]
reg.mref = np.ones_like(m)*np.mean(m)
print 'Check: phi_m (mref) = %f' %reg.eval(reg.mref)
passed = reg.eval(reg.mref) < TOL
self.assertTrue(passed)
print 'Check:', R
passed = Tests.checkDerivative(lambda m : [reg.eval(m), reg.evalDeriv(m)], m, plotIt=False)
self.assertTrue(passed)
print 'Check 2 Deriv:', R
passed = Tests.checkDerivative(lambda m : [reg.evalDeriv(m), reg.eval2Deriv(m)], m, plotIt=False)
self.assertTrue(passed)
if __name__ == '__main__':
tests/examples/test_examples.py
+21 -1
View File
@@ -1,8 +1,28 @@
import unittest
import sys
import os
from SimPEG import Examples
import numpy as np
class compareInitFiles(unittest.TestCase):
def test_compareInitFiles(self):
print 'Checking that __init__.py up-to-date in SimPEG/Examples'
fName = os.path.abspath(__file__)
ExamplesDir = os.path.sep.join(fName.split(os.path.sep)[:-3] + ['SimPEG', 'Examples'])
files = os.listdir(ExamplesDir)
pyfiles = []
[pyfiles.append(py.rstrip('.py')) for py in files if py.endswith('.py') and py != '__init__.py']
setdiff = set(pyfiles) - set(Examples.__examples__)
print ' Any missing files? ', setdiff
didpass = (setdiff == set())
self.assertTrue(didpass, "Examples not up to date, run 'python __init__.py' from SimPEG/Examples to update")
def get(test):
def test_func(self):
print '\nTesting %s.run(plotIt=False)\n'%test
@@ -10,11 +30,11 @@ def get(test):
self.assertTrue(True)
return test_func
attrs = dict()
for test in Examples.__examples__:
attrs['test_'+test] = get(test)
TestExamples = type('TestExamples', (unittest.TestCase,), attrs)
if __name__ == '__main__':
unittest.main()
tests/mesh/test_MeshIO.py
+100
View File
@@ -0,0 +1,100 @@
import numpy as np
import unittest, os
import SimPEG as simpeg
from SimPEG.Mesh import TensorMesh, TreeMesh
class TestTensorMeshIO(unittest.TestCase):
def setUp(self):
h = np.ones(16)
mesh = TensorMesh([h,2*h,3*h])
self.mesh = mesh
def test_UBCfiles(self):
mesh = self.mesh
# Make a vector
vec = np.arange(mesh.nC)
# Write and read
mesh.writeUBC('temp.msh', {'arange.txt':vec})
meshUBC = TensorMesh.readUBC('temp.msh')
vecUBC = meshUBC.readModelUBC('arange.txt')
# The mesh
assert mesh.__str__() == meshUBC.__str__()
assert np.sum(mesh.gridCC - meshUBC.gridCC) == 0
assert np.sum(vec - vecUBC) == 0
assert np.all(np.array(mesh.h) - np.array(meshUBC.h) == 0)
vecUBC = mesh.readModelUBC('arange.txt')
assert np.sum(vec - vecUBC) == 0
mesh.writeModelUBC('arange2.txt', vec + 1)
vec2UBC = mesh.readModelUBC('arange2.txt')
assert np.sum(vec + 1 - vec2UBC) == 0
print 'IO of UBC tensor mesh files is working'
os.remove('temp.msh')
os.remove('arange.txt')
os.remove('arange2.txt')
def test_VTKfiles(self):
mesh = self.mesh
vec = np.arange(mesh.nC)
mesh.writeVTK('temp.vtr', {'arange.txt':vec})
meshVTR, models = TensorMesh.readVTK('temp.vtr')
assert mesh.__str__() == meshVTR.__str__()
assert np.all(np.array(mesh.h) - np.array(meshVTR.h) == 0)
assert 'arange.txt' in models
vecVTK = models['arange.txt']
assert np.sum(vec - vecVTK) == 0
print 'IO of VTR tensor mesh files is working'
os.remove('temp.vtr')
class TestOcTreeMeshIO(unittest.TestCase):
def setUp(self):
h = np.ones(16)
mesh = TreeMesh([h,2*h,3*h])
mesh.refine(3)
mesh._refineCell([0,0,0,3])
mesh._refineCell([0,2,0,3])
self.mesh = mesh
def test_UBCfiles(self):
mesh = self.mesh
# Make a vector
vec = np.arange(mesh.nC)
# Write and read
mesh.writeUBC('temp.msh', {'arange.txt':vec})
meshUBC = TreeMesh.readUBC('temp.msh')
vecUBC = meshUBC.readModelUBC('arange.txt')
# The mesh
assert mesh.__str__() == meshUBC.__str__()
assert np.sum(mesh.gridCC - meshUBC.gridCC) == 0
assert np.sum(vec - vecUBC) == 0
assert np.all(np.array(mesh.h) - np.array(meshUBC.h) == 0)
print 'IO of UBC octree files is working'
os.remove('temp.msh')
os.remove('arange.txt')
def test_VTUfiles(self):
mesh = self.mesh
vec = np.arange(mesh.nC)
mesh.writeVTK('temp.vtu',{'arange':vec})
print 'Writing of VTU files is working'
os.remove('temp.vtu')
if __name__ == '__main__':
unittest.main()
tests/mesh/test_TreeMesh.py
+21
View File
@@ -26,6 +26,27 @@ class TestSimpleQuadTree(unittest.TestCase):
assert np.allclose(np.r_[M._areaFxFull, M._areaFyFull], M._deflationMatrix('F') * M.area)
def test_getitem(self):
M = Mesh.TreeMesh([4,4])
M.refine(1)
assert M.nC == 4
assert len(M) == M.nC
assert np.allclose(M[0].center, [0.25,0.25])
actual = [[0,0],[0.5,0],[0,0.5],[0.5,0.5]]
for i, n in enumerate(M[0].nodes):
assert np.allclose(M._gridN[n,:], actual[i])
def test_getitem3D(self):
M = Mesh.TreeMesh([4,4,4])
M.refine(1)
assert M.nC == 8
assert len(M) == M.nC
assert np.allclose(M[0].center, [0.25,0.25,0.25])
actual = [[0,0,0],[0.5,0,0],[0,0.5,0],[0.5,0.5,0],
[0,0,0.5],[0.5,0,0.5],[0,0.5,0.5],[0.5,0.5,0.5]]
for i, n in enumerate(M[0].nodes):
assert np.allclose(M._gridN[n,:], actual[i])
def test_refine(self):
M = Mesh.TreeMesh([4,4,4])
M.refine(1)
tests/utils/test_Zero.py
+6 -1
View File
@@ -1,5 +1,5 @@
import unittest
from SimPEG.Utils import Zero, Identity, sdiag
from SimPEG.Utils import Zero, Identity, sdiag, mkvc
from SimPEG import np, sp
class Tests(unittest.TestCase):
@@ -29,6 +29,11 @@ class Tests(unittest.TestCase):
assert a == 1
self.assertRaises(ZeroDivisionError, lambda:3/z)
assert mkvc(z) == 0
assert sdiag(z)*a == 0
assert z.T == 0
assert z.transpose == 0
def test_mat_zero(self):
z = Zero()
S = sdiag(np.r_[2,3])