$\renewcommand{\div}{\nabla\cdot\,} \newcommand{\grad}{\vec \nabla} \newcommand{\curl}{{\vec \nabla}\times\,}$

# Frequency Domain Electromagnetics¶

Electromagnetic (EM) geophysical methods are used in a variety of applications from resource exploration, including for hydrocarbons and minerals, to environmental applications, such as groundwater monitoring. The primary physical property of interest in EM is electrical conductivity, which describes the ease with which electric current flows through a material.

## Background¶

Electromagnetic phenomena are governed by Maxwell’s equations. They describe the behavior of EM fields and fluxes. Electromagnetic theory for geophysical applications by Ward and Hohmann (1988) is a highly recommended resource on this topic.

### 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

\begin{align}\begin{aligned}\begin{split}F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\\end{split}\\f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d \omega\end{aligned}\end{align}

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.

## Maxwell’s Equations¶

In the frequency domain, Maxwell’s equations are given by

\begin{align}\begin{aligned}\begin{split}\curl \vec{E} + i \omega \vec{B} = \vec{S_m}\\\end{split}\\\begin{split}\curl \vec{H} - \vec{J} - i \omega \vec{D} = \vec{S_e} \\\end{split}\\\begin{split}\div \vec{B} = 0 \\\end{split}\\\div \vec{D} = \rho_f\end{aligned}\end{align}

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 ($$A/m^2$$ )

• $$\vec{S_m}$$ : magnetic source term ($$V/m^2$$ )

• $$\vec{S_e}$$ : electric source term ($$A/m^2$$ )

• $$\rho_f$$ : free charge density ($$\Omega m$$ )

### Constitutive Relations¶

The fields and fluxes are related through the constitutive relations. At each frequency, they are given by

\begin{align}\begin{aligned}\begin{split}\vec{J} = \sigma \vec{E} \\\end{split}\\\begin{split}\vec{B} = \mu \vec{H} \\\end{split}\\\vec{D} = \varepsilon \vec{E}\end{aligned}\end{align}

where:

• $$\sigma$$ : electrical conductivity ($$S/m$$)

• $$\mu$$ : magnetic permeability ($$H/m$$)

• $$\varepsilon$$ : dielectric permittivity ($$F/m$$)

$$\sigma$$, $$\mu$$, $$\varepsilon$$ 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 low-frequency geophysical applications of EM, $$\sigma$$ is the primary physical property of interest, and $$\mu$$, $$\varepsilon$$ are assumed to have their free-space values $$\mu_0 = 4\pi \times 10^{-7} H/m$$ , $$\varepsilon_0 = 1/(\mu_0 c^2) \approx 8.85 \times 10^{-12} F/m$$, where $$c$$ is the speed of light in free space.

### 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

$\begin{split}\nabla \times \vec{E} + i \omega \vec{B} = \vec{S_m} \\ \nabla \times \vec{H} - \vec{J} = \vec{S_e}\end{split}$

Geophysical methods where the quasi-static approximation, often called diffusive approximation, does not hold are high-frequency methods such as ground-penetrating radar or dielectric well-log measurements.

## 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:

$\begin{split}\nabla \times \vec{E} + i \omega \vec{B} = \vec{S}_m \\ \nabla \times \mu^{-1} \vec{B} - \sigma \vec{E} = \vec{S}_e\end{split}$

The H-J formulation is in terms of the current density and the magnetic field:

$\begin{split}\nabla \times \sigma^{-1} \vec{J} + i \omega \mu \vec{H} = \vec{S}_m \\ \nabla \times \vec{H} - \vec{J} = \vec{S}_e\end{split}$

### Discretizing¶

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 the two formulations, the discretization of the physical properties, fields and fluxes are summarized below.

Note that resistivity is the inverse of conductivity, $$\rho = \sigma^{-1}$$.

### E-B Formulation¶

$\begin{split}\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\ \mathbf{C^T} \mathbf{M^f_{\mu^{-1}}} \mathbf{b} - \mathbf{M^e_\sigma} \mathbf{e} = \mathbf{M^e} \mathbf{s_e}\end{split}$

### H-J Formulation¶

$\begin{split}\mathbf{C^T} \mathbf{M^f_\rho} \mathbf{j} + i \omega \mathbf{M^e_\mu} \mathbf{h} = \mathbf{M^e} \mathbf{s_m} \\ \mathbf{C} \mathbf{h} - \mathbf{j} = \mathbf{s_e}\end{split}$

## API¶

### FDEM Problem¶

class SimPEG.EM.FDEM.ProblemFDEM.BaseFDEMProblem(mesh, **kwargs)[source]

We start by looking at Maxwell’s equations in the electric field $$\mathbf{e}$$ and the magnetic flux density $$\mathbf{b}$$

$\begin{split}\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\ {\mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{M_{\sigma}^e} \mathbf{e} = \mathbf{s_e}}\end{split}$

if using the E-B formulation (Problem3D_e or Problem3D_b). Note that in this case, $$\mathbf{s_e}$$ is an integrated quantity.

If we write Maxwell’s equations in terms of $$\mathbf{h}$$ and current density $$\mathbf{j}$$

$\begin{split}\mathbf{C}^{\top} \mathbf{M_{\rho}^f} \mathbf{j} + i \omega \mathbf{M_{\mu}^e} \mathbf{h} = \mathbf{s_m} \\ \mathbf{C} \mathbf{h} - \mathbf{j} = \mathbf{s_e}\end{split}$

if using the H-J formulation (Problem3D_j or Problem3D_h). Note that here, $$\mathbf{s_m}$$ is an integrated quantity.

The problem performs the elimination so that we are solving the system for $$\mathbf{e},\mathbf{b},\mathbf{j}$$ or $$\mathbf{h}$$

Optional Properties:

Other Properties:

surveyPair
fieldsPair
property mu

Magnetic Permeability (H/m)

property muMap

Mapping of Magnetic Permeability (H/m) to the inversion model.

property muDeriv

Derivative of Magnetic Permeability (H/m) wrt the model.

property mui

Inverse Magnetic Permeability (m/H)

property muiMap

Mapping of Inverse Magnetic Permeability (m/H) to the inversion model.

property muiDeriv

Derivative of Inverse Magnetic Permeability (m/H) wrt the model.

fields(m=None)[source]

Solve the forward problem for the fields.

Parameters

m (numpy.ndarray) – inversion model (nP,)

Return type

numpy.ndarray

Return f

forward solution

Jvec(m, v, f=None)[source]

Sensitivity times a vector.

Parameters
Return type

numpy.ndarray

Returns

Jv (ndata,)

Jtvec(m, v, f=None)[source]

Sensitivity transpose times a vector

Parameters
Return type

numpy.ndarray

Returns

Jv (ndata,)

getSourceTerm(freq)[source]

Evaluates the sources for a given frequency and puts them in matrix form

Parameters

freq (float) – Frequency

Return type

tuple

Returns

(s_m, s_e) (nE or nF, nSrc)

class SimPEG.EM.FDEM.ProblemFDEM.Problem3D_e(mesh, **kwargs)[source]

By eliminating the magnetic flux density using

$\mathbf{b} = \frac{1}{i \omega}\left(-\mathbf{C} \mathbf{e} + \mathbf{s_m}\right)$

we can write Maxwell’s equations as a second order system in $$\mathbf{e}$$ only:

$\left(\mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{C} + i \omega \mathbf{M^e_{\sigma}} \right)\mathbf{e} = \mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} - i\omega\mathbf{M^e}\mathbf{s_e}$

which we solve for $$\mathbf{e}$$.

param discretize.base.BaseMesh mesh

mesh

Optional Properties:

Other Properties:

fieldsPair
getA(freq)[source]

System matrix

$\mathbf{A} = \mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{C} + i \omega \mathbf{M^e_{\sigma}}$
Parameters

freq (float) – Frequency

Return type

scipy.sparse.csr_matrix

Returns

A

getADeriv_sigma(freq, u, v, adjoint=False)[source]

Product of the derivative of our system matrix with respect to the conductivity model and a vector

$\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}_{\sigma}} = i \omega \frac{d \mathbf{M^e_{\sigma}}(\mathbf{u})\mathbf{v} }{d\mathbf{m}}$
Parameters
• freq (float) – frequency

• u (numpy.ndarray) – solution vector (nE,)

• v (numpy.ndarray) – vector to take prodct with (nP,) or (nD,) for adjoint

Return type

numpy.ndarray

Returns

derivative of the system matrix times a vector (nP,) or adjoint (nD,)

getADeriv_mui(freq, u, v, adjoint=False)[source]

Product of the derivative of the system matrix with respect to the permeability model and a vector.

$\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}_{\mu^{-1}} = \mathbf{C}^{ op} \frac{d \mathbf{M^f_{\mu^{-1}}}\mathbf{v}}{d\mathbf{m}}$
getADeriv(freq, u, v, adjoint=False)[source]
getRHS(freq)[source]

Right hand side for the system

$\mathbf{RHS} = \mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} - i\omega\mathbf{M_e}\mathbf{s_e}$
Parameters

freq (float) – Frequency

Return type

numpy.ndarray

Returns

RHS (nE, nSrc)

getRHSDeriv(freq, src, v, adjoint=False)[source]

Derivative of the Right-hand side with respect to the model. This includes calls to derivatives in the sources

class SimPEG.EM.FDEM.ProblemFDEM.Problem3D_b(mesh, **kwargs)[source]

We eliminate $$\mathbf{e}$$ using

$\mathbf{e} = \mathbf{M^e_{\sigma}}^{-1} \left(\mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{s_e}\right)$

and solve for $$\mathbf{b}$$ using:

$\left(\mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f} + i \omega \right)\mathbf{b} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{M^e}\mathbf{s_e}$

Note

The inverse problem will not work with full anisotropy

param discretize.base.BaseMesh mesh

mesh

Optional Properties:

Other Properties:

fieldsPair
getA(freq)[source]

System matrix

$\mathbf{A} = \mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f} + i \omega$
Parameters

freq (float) – Frequency

Return type

scipy.sparse.csr_matrix

Returns

A

getADeriv_sigma(freq, u, v, adjoint=False)[source]

Product of the derivative of our system matrix with respect to the model and a vector

$\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}} = \mathbf{C} \frac{\mathbf{M^e_{\sigma}} \mathbf{v}}{d\mathbf{m}}$
Parameters
• freq (float) – frequency

• u (numpy.ndarray) – solution vector (nF,)

• v (numpy.ndarray) – vector to take prodct with (nP,) or (nD,) for adjoint

Return type

numpy.ndarray

Returns

derivative of the system matrix times a vector (nP,) or adjoint (nD,)

getADeriv_mui(freq, u, v, adjoint=False)[source]
getADeriv(freq, u, v, adjoint=False)[source]
getRHS(freq)[source]

Right hand side for the system

$\mathbf{RHS} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{s_e}$
Parameters

freq (float) – Frequency

Return type

numpy.ndarray

Returns

RHS (nE, nSrc)

getRHSDeriv(freq, src, v, adjoint=False)[source]

Derivative of the right hand side with respect to the model

Parameters
Return type

numpy.ndarray

Returns

product of rhs deriv with a vector

class SimPEG.EM.FDEM.ProblemFDEM.Problem3D_j(mesh, **kwargs)[source]

We eliminate $$\mathbf{h}$$ using

$\mathbf{h} = \frac{1}{i \omega} \mathbf{M_{\mu}^e}^{-1} \left(-\mathbf{C}^{\top} \mathbf{M_{\rho}^f} \mathbf{j} + \mathbf{M^e} \mathbf{s_m} \right)$

and solve for $$\mathbf{j}$$ using

$\left(\mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{C}^{\top} \mathbf{M_{\rho}^f} + i \omega\right)\mathbf{j} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{M^e} \mathbf{s_m} - i\omega\mathbf{s_e}$

Note

This implementation does not yet work with full anisotropy!!

param discretize.base.BaseMesh mesh

mesh

Optional Properties:

Other Properties:

fieldsPair
getA(freq)[source]

System matrix

$\mathbf{A} = \mathbf{C} \mathbf{M^e_{\mu^{-1}}} \mathbf{C}^{\top} \mathbf{M^f_{\sigma^{-1}}} + i\omega$
Parameters

freq (float) – Frequency

Return type

scipy.sparse.csr_matrix

Returns

A

getADeriv_rho(freq, u, v, adjoint=False)[source]

Product of the derivative of our system matrix with respect to the model and a vector

In this case, we assume that electrical conductivity, $$\sigma$$ is the physical property of interest (i.e. $$\sigma$$ = model.transform). Then we want

$\frac{\mathbf{A(\sigma)} \mathbf{v}}{d \mathbf{m}} = \mathbf{C} \mathbf{M^e_{mu^{-1}}} \mathbf{C^{\top}} \frac{d \mathbf{M^f_{\sigma^{-1}}}\mathbf{v} }{d \mathbf{m}}$
Parameters
• freq (float) – frequency

• u (numpy.ndarray) – solution vector (nF,)

• v (numpy.ndarray) – vector to take prodct with (nP,) or (nD,) for adjoint

Return type

numpy.ndarray

Returns

derivative of the system matrix times a vector (nP,) or adjoint (nD,)

getADeriv_mu(freq, u, v, adjoint=False)[source]
getADeriv(freq, u, v, adjoint=False)[source]
getRHS(freq)[source]

Right hand side for the system

$\mathbf{RHS} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1}\mathbf{s_m} - i\omega \mathbf{s_e}$
Parameters

freq (float) – Frequency

Return type

numpy.ndarray

Returns

RHS (nE, nSrc)

getRHSDeriv(freq, src, v, adjoint=False)[source]

Derivative of the right hand side with respect to the model

Parameters
Return type

numpy.ndarray

Returns

product of rhs deriv with a vector

class SimPEG.EM.FDEM.ProblemFDEM.Problem3D_h(mesh, **kwargs)[source]

We eliminate $$\mathbf{j}$$ using

$\mathbf{j} = \mathbf{C} \mathbf{h} - \mathbf{s_e}$

and solve for $$\mathbf{h}$$ using

$\left(\mathbf{C}^{\top} \mathbf{M_{\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}\right) \mathbf{h} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^{\top} \mathbf{M_{\rho}^f} \mathbf{s_e}$
param discretize.base.BaseMesh mesh

mesh

Optional Properties:

Other Properties:

fieldsPair
getA(freq)[source]

System matrix

$\mathbf{A} = \mathbf{C}^{\top} \mathbf{M_{\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}$
Parameters

freq (float) – Frequency

Return type

scipy.sparse.csr_matrix

Returns

A

getADeriv_rho(freq, u, v, adjoint=False)[source]

Product of the derivative of our system matrix with respect to the model and a vector

$\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}} = \mathbf{C}^{\top}\frac{d \mathbf{M^f_{\rho}}\mathbf{v}} {d\mathbf{m}}$
Parameters
• freq (float) – frequency

• u (numpy.ndarray) – solution vector (nE,)

• v (numpy.ndarray) – vector to take prodct with (nP,) or (nD,) for adjoint

Return type

numpy.ndarray

Returns

derivative of the system matrix times a vector (nP,) or adjoint (nD,)

getADeriv_mu(freq, u, v, adjoint=False)[source]
getADeriv(freq, u, v, adjoint=False)[source]
getRHS(freq)[source]

Right hand side for the system

$\mathbf{RHS} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^{\top} \mathbf{M_{\rho}^f} \mathbf{s_e}$
Parameters

freq (float) – Frequency

Return type

numpy.ndarray

Returns

RHS (nE, nSrc)

getRHSDeriv(freq, src, v, adjoint=False)[source]

Derivative of the right hand side with respect to the model

Parameters
Return type

numpy.ndarray

Returns

product of rhs deriv with a vector

### FDEM Survey¶

class SimPEG.EM.FDEM.SurveyFDEM.Survey(srcList, **kwargs)[source]

Frequency domain electromagnetic survey

Parameters

srcList (list) – list of FDEM sources used in the survey

srcPair
rxPair
property freqs

Frequencies

property nFreq

Number of frequencies

property nSrcByFreq

Number of sources at each frequency

getSrcByFreq(freq)[source]

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

class SimPEG.EM.FDEM.SrcFDEM.BaseFDEMSrc(rxList, **kwargs)[source]

Base source class for FDEM Survey

Required Properties:

• freq (Float): frequency of the source, a float in range [0, inf]

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (Array): location of the source, a list or numpy array of <class ‘float’>, <class ‘int’> with shape (*, 3) or (3)

property freq

freq (Float): frequency of the source, a float in range [0, inf]

bPrimary(prob)[source]

Primary magnetic flux density

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

primary magnetic flux density

bPrimaryDeriv(prob, v, adjoint=False)[source]

Derivative of the primary magnetic flux density

Parameters
Return type

numpy.ndarray

Returns

primary magnetic flux density

hPrimary(prob)[source]

Primary magnetic field

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

primary magnetic field

hPrimaryDeriv(prob, v, adjoint=False)[source]

Derivative of the primary magnetic field

Parameters
Return type

numpy.ndarray

Returns

primary magnetic flux density

ePrimary(prob)[source]

Primary electric field

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

primary electric field

ePrimaryDeriv(prob, v, adjoint=False)[source]

Derivative of the primary electric field

Parameters
Return type

numpy.ndarray

Returns

primary magnetic flux density

jPrimary(prob)[source]

Primary current density

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

primary current density

jPrimaryDeriv(prob, v, adjoint=False)[source]

Derivative of the primary current density

Parameters
Return type

numpy.ndarray

Returns

primary magnetic flux density

class SimPEG.EM.FDEM.SrcFDEM.RawVec_e(rxList, freq, s_e, **kwargs)[source]

RawVec electric source. It is defined by the user provided vector s_e

param list rxList

param float freq

frequency

param numpy.ndarray s_e

electric source term

param bool integrate

Integrate the source term (multiply by Me) [False]

Required Properties:

• freq (Float): frequency of the source, a float in range [0, inf]

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (Array): location of the source, a list or numpy array of <class ‘float’>, <class ‘int’> with shape (*, 3) or (3)

s_e(prob)[source]

Electric source term

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

electric source term on mesh

class SimPEG.EM.FDEM.SrcFDEM.RawVec_m(rxList, freq, s_m, **kwargs)[source]

RawVec magnetic source. It is defined by the user provided vector s_m

param float freq

frequency

param rxList

param numpy.ndarray s_m

magnetic source term

param bool integrate

Integrate the source term (multiply by Me) [False]

Required Properties:

• freq (Float): frequency of the source, a float in range [0, inf]

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (Array): location of the source, a list or numpy array of <class ‘float’>, <class ‘int’> with shape (*, 3) or (3)

s_m(prob)[source]

Magnetic source term

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

magnetic source term on mesh

class SimPEG.EM.FDEM.SrcFDEM.RawVec(rxList, freq, s_m, s_e, **kwargs)[source]

RawVec source. It is defined by the user provided vectors s_m, s_e

param rxList

param float freq

frequency

param numpy.ndarray s_m

magnetic source term

param numpy.ndarray s_e

electric source term

param bool integrate

Integrate the source term (multiply by Me) [False]

Required Properties:

• freq (Float): frequency of the source, a float in range [0, inf]

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (Array): location of the source, a list or numpy array of <class ‘float’>, <class ‘int’> with shape (*, 3) or (3)

s_m(prob)[source]

Magnetic source term

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

magnetic source term on mesh

s_e(prob)[source]

Electric source term

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

electric source term on mesh

class SimPEG.EM.FDEM.SrcFDEM.MagDipole(rxList, freq, loc, **kwargs)[source]

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.

$\begin{split}\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\ {\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{M_{\sigma}^e} \mathbf{e} = \mathbf{s_e}}\end{split}$

We split up the fields and $$\mu^{-1}$$ into primary ($$\mathbf{P}$$) and secondary ($$\mathbf{S}$$) components

• $$\mathbf{e} = \mathbf{e^P} + \mathbf{e^S}$$

• $$\mathbf{b} = \mathbf{b^P} + \mathbf{b^S}$$

• $$\boldsymbol{\mu}^{\mathbf{-1}} = \boldsymbol{\mu}^{\mathbf{-1}^\mathbf{P}} + \boldsymbol{\mu}^{\mathbf{-1}^\mathbf{S}}$$

and define a zero-frequency primary problem, noting that the source is generated by a divergence free electric current

$\begin{split}\mathbf{C} \mathbf{e^P} = \mathbf{s_m^P} = 0 \\ {\mathbf{C}^T \mathbf{{M_{\mu^{-1}}^f}^P} \mathbf{b^P} - \mathbf{M_{\sigma}^e} \mathbf{e^P} = \mathbf{M^e} \mathbf{s_e^P}}\end{split}$

Since $$\mathbf{e^P}$$ is curl-free, divergence-free, we assume that there is no constant field background, the $$\mathbf{e^P} = 0$$, so our primary problem is

$\begin{split}\mathbf{e^P} = 0 \\ {\mathbf{C}^T \mathbf{{M_{\mu^{-1}}^f}^P} \mathbf{b^P} = \mathbf{s_e^P}}\end{split}$

Our secondary problem is then

$\begin{split}\mathbf{C} \mathbf{e^S} + i \omega \mathbf{b^S} = - i \omega \mathbf{b^P} \\ {\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b^S} - \mathbf{M_{\sigma}^e} \mathbf{e^S} = -\mathbf{C}^T \mathbf{{M_{\mu^{-1}}^f}^S} \mathbf{b^P}}\end{split}$
param list rxList

param float freq

frequency

param numpy.ndarray loc

source location (ie: np.r_[xloc,yloc,zloc])

param string orientation

‘X’, ‘Y’, ‘Z’

param float moment

magnetic dipole moment

param float mu

background magnetic permeability

Required Properties:

• freq (Float): frequency of the source (Hz), a float

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (LocationVector): location of the source, of <class ‘float’>, <class ‘int’> with shape (3), Default: [0. 0. 0.]

• moment (Float): dipole moment of the transmitter, a float in range [0.0, inf], Default: 1.0

• mu (Float): permeability of the background, a float in range [0.0, inf], Default: 1.2566370614359173e-06

• orientation (Vector3): orientation of the source, a 3D Vector of <class ‘float’> with shape (3), Default: Z

property moment

moment (Float): dipole moment of the transmitter, a float in range [0.0, inf], Default: 1.0

property mu

mu (Float): permeability of the background, a float in range [0.0, inf], Default: 1.2566370614359173e-06

property orientation

orientation (Vector3): orientation of the source, a 3D Vector of <class ‘float’> with shape (3), Default: Z

property freq

freq (Float): frequency of the source (Hz), a float

property loc

loc (LocationVector): location of the source, of <class ‘float’>, <class ‘int’> with shape (3), Default: [0. 0. 0.]

bPrimary(prob)[source]

The primary magnetic flux density from a magnetic vector potential

Parameters

prob (BaseFDEMProblem) – FDEM problem

Return type

numpy.ndarray

Returns

primary magnetic field

hPrimary(prob)[source]

The primary magnetic field from a magnetic vector potential

Parameters

prob (BaseFDEMProblem) – FDEM problem

Return type

numpy.ndarray

Returns

primary magnetic field

s_m(prob)[source]

The magnetic source term

Parameters

prob (BaseFDEMProblem) – FDEM problem

Return type

numpy.ndarray

Returns

primary magnetic field

s_e(prob)[source]

The electric source term

Parameters

prob (BaseFDEMProblem) – FDEM problem

Return type

numpy.ndarray

Returns

primary magnetic field

s_eDeriv(prob, v, adjoint=False)[source]

Derivative of electric source term with respect to the inversion model

Parameters
Return type

numpy.ndarray

Returns

product of electric source term derivative with a vector

class SimPEG.EM.FDEM.SrcFDEM.MagDipole_Bfield(rxList, freq, loc, **kwargs)[source]

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

param float freq

frequency

param numpy.ndarray loc

source location (ie: np.r_[xloc,yloc,zloc])

param string orientation

‘X’, ‘Y’, ‘Z’

param float moment

magnetic dipole moment

param float mu

background magnetic permeability

Required Properties:

• freq (Float): frequency of the source (Hz), a float

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (LocationVector): location of the source, of <class ‘float’>, <class ‘int’> with shape (3), Default: [0. 0. 0.]

• moment (Float): dipole moment of the transmitter, a float in range [0.0, inf], Default: 1.0

• mu (Float): permeability of the background, a float in range [0.0, inf], Default: 1.2566370614359173e-06

• orientation (Vector3): orientation of the source, a 3D Vector of <class ‘float’> with shape (3), Default: Z

bPrimary(prob)[source]

The primary magnetic flux density from the analytic solution for magnetic fields from a dipole

Parameters

prob (BaseFDEMProblem) – FDEM problem

Return type

numpy.ndarray

Returns

primary magnetic field

class SimPEG.EM.FDEM.SrcFDEM.CircularLoop(rxList, freq, loc, **kwargs)[source]

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

param float freq

frequency

param numpy.ndarray loc

source location (ie: np.r_[xloc,yloc,zloc])

param string orientation

‘X’, ‘Y’, ‘Z’

param float moment

magnetic dipole moment

param float mu

background magnetic permeability

Required Properties:

• current (Float): current in the loop, a float, Default: 1.0

• freq (Float): frequency of the source (Hz), a float

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (LocationVector): location of the source, of <class ‘float’>, <class ‘int’> with shape (3), Default: [0. 0. 0.]

• mu (Float): permeability of the background, a float in range [0.0, inf], Default: 1.2566370614359173e-06

• orientation (Vector3): orientation of the source, a 3D Vector of <class ‘float’> with shape (3), Default: Z

• radius (Float): radius of the loop, a float in range [0.0, inf], Default: 1.0

property radius

radius (Float): radius of the loop, a float in range [0.0, inf], Default: 1.0

property current

current (Float): current in the loop, a float, Default: 1.0

property moment

moment (Float): dipole moment of the transmitter, a float in range [0.0, inf], Default: 1.0

class SimPEG.EM.FDEM.SrcFDEM.PrimSecSigma(rxList, freq, sigBack, ePrimary, **kwargs)[source]

Required Properties:

• freq (Float): frequency of the source, a float in range [0, inf]

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (Array): location of the source, a list or numpy array of <class ‘float’>, <class ‘int’> with shape (*, 3) or (3)

s_e(prob)[source]

Electric source term

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

electric source term on mesh

s_eDeriv(prob, v, adjoint=False)[source]

Derivative of electric source term with respect to the inversion model

Parameters
Return type

numpy.ndarray

Returns

product of electric source term derivative with a vector

class SimPEG.EM.FDEM.SrcFDEM.PrimSecMappedSigma(rxList, freq, primaryProblem, primarySurvey, map2meshSecondary=None, **kwargs)[source]

Primary-Secondary Source in which a mapping is provided to put the current model onto the primary mesh. This is solved on every model update. There are a lot of layers to the derivatives here!

Required :param list rxList: Receiver List :param float freq: frequency :param BaseFDEMProblem primaryProblem: FDEM primary problem :param SurveyFDEM primarySurvey: FDEM primary survey

Optional :param Mapping map2meshSecondary: mapping current model to act as primary model on the secondary mesh

Required Properties:

• freq (Float): frequency of the source, a float in range [0, inf]

• integrate (Boolean): integrate the source term?, a boolean, Default: False

• loc (Array): location of the source, a list or numpy array of <class ‘float’>, <class ‘int’> with shape (*, 3) or (3)

ePrimary(prob, f=None)[source]

Primary electric field

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

primary electric field

ePrimaryDeriv(prob, v, adjoint=False, f=None)[source]

Derivative of the primary electric field

Parameters
Return type

numpy.ndarray

Returns

primary magnetic flux density

bPrimary(prob, f=None)[source]

Primary magnetic flux density

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

primary magnetic flux density

s_e(prob, f=None)[source]

Electric source term

Parameters

prob (BaseFDEMProblem) – FDEM Problem

Return type

numpy.ndarray

Returns

electric source term on mesh

s_eDeriv(prob, v, adjoint=False)[source]

Derivative of electric source term with respect to the inversion model

Parameters
Return type

numpy.ndarray

Returns

product of electric source term derivative with a vector

class SimPEG.EM.FDEM.RxFDEM.BaseRx(locs, orientation=None, component=None)[source]

Parameters
• locs (numpy.ndarray) – receiver locations (ie. np.r_[x,y,z])

• orientation (string) – receiver orientation ‘x’, ‘y’ or ‘z’

• component (string) – real or imaginary component ‘real’ or ‘imag’

projGLoc(f)[source]

Grid Location projection (e.g. Ex Fy …)

eval(src, mesh, f)[source]

Project fields to receivers to get data.

Parameters
Return type

numpy.ndarray

Returns

fields projected to recievers

evalDeriv(src, mesh, f, du_dm_v=None, v=None, adjoint=False)[source]

Derivative of projected fields with respect to the inversion model times a vector.

Parameters
Return type

numpy.ndarray

Returns

fields projected to recievers

class SimPEG.EM.FDEM.RxFDEM.Point_e(locs, orientation=None, component=None)[source]

Parameters
• locs (numpy.ndarray) – receiver locations (ie. np.r_[x,y,z])

• orientation (string) – receiver orientation ‘x’, ‘y’ or ‘z’

• component (string) – real or imaginary component ‘real’ or ‘imag’

class SimPEG.EM.FDEM.RxFDEM.Point_b(locs, orientation=None, component=None)[source]

Parameters
• locs (numpy.ndarray) – receiver locations (ie. np.r_[x,y,z])

• orientation (string) – receiver orientation ‘x’, ‘y’ or ‘z’

• component (string) – real or imaginary component ‘real’ or ‘imag’

class SimPEG.EM.FDEM.RxFDEM.Point_bSecondary(locs, orientation=None, component=None)[source]

Parameters
• locs (numpy.ndarray) – receiver locations (ie. np.r_[x,y,z])

• orientation (string) – receiver orientation ‘x’, ‘y’ or ‘z’

• component (string) – real or imaginary component ‘real’ or ‘imag’

class SimPEG.EM.FDEM.RxFDEM.Point_h(locs, orientation=None, component=None)[source]

Parameters
• locs (numpy.ndarray) – receiver locations (ie. np.r_[x,y,z])

• orientation (string) – receiver orientation ‘x’, ‘y’ or ‘z’

• component (string) – real or imaginary component ‘real’ or ‘imag’

class SimPEG.EM.FDEM.RxFDEM.Point_j(locs, orientation=None, component=None)[source]

Parameters
• locs (numpy.ndarray) – receiver locations (ie. np.r_[x,y,z])

• orientation (string) – receiver orientation ‘x’, ‘y’ or ‘z’

• component (string) – real or imaginary component ‘real’ or ‘imag’

### FDEM Fields¶

class SimPEG.EM.FDEM.FieldsFDEM.FieldsFDEM(mesh, survey, **kwargs)[source]

Fancy Field Storage for a FDEM survey. Only one field type is stored for each problem, the rest are computed. The fields object acts like an array and is indexed by

f = problem.fields(m)
e = f[srcList,'e']
b = f[srcList,'b']


If accessing all sources for a given field, use the :

f = problem.fields(m)
e = f[:,'e']
b = f[:,'b']


The array returned will be size (nE or nF, nSrcs $$\times$$ nFrequencies)

knownFields = {}
dtype

alias of builtins.complex

class SimPEG.EM.FDEM.FieldsFDEM.Fields3D_e(mesh, survey, **kwargs)[source]

Fields object for Problem3D_e.

Parameters
knownFields = {'eSolution': 'E'}
aliasFields = {'b': ['eSolution', 'F', '_b'], 'bPrimary': ['eSolution', 'F', '_bPrimary'], 'bSecondary': ['eSolution', 'F', '_bSecondary'], 'e': ['eSolution', 'E', '_e'], 'ePrimary': ['eSolution', 'E', '_ePrimary'], 'eSecondary': ['eSolution', 'E', '_eSecondary'], 'h': ['eSolution', 'F', '_h'], 'j': ['eSolution', 'E', '_j']}
startup()[source]
class SimPEG.EM.FDEM.FieldsFDEM.Fields3D_b(mesh, survey, **kwargs)[source]

Fields object for Problem3D_b.

Parameters
knownFields = {'bSolution': 'F'}
aliasFields = {'b': ['bSolution', 'F', '_b'], 'bPrimary': ['bSolution', 'F', '_bPrimary'], 'bSecondary': ['bSolution', 'F', '_bSecondary'], 'e': ['bSolution', 'E', '_e'], 'ePrimary': ['bSolution', 'E', '_ePrimary'], 'eSecondary': ['bSolution', 'E', '_eSecondary'], 'h': ['bSolution', 'F', '_h'], 'j': ['bSolution', 'E', '_j']}
startup()[source]
class SimPEG.EM.FDEM.FieldsFDEM.Fields3D_j(mesh, survey, **kwargs)[source]

Fields object for Problem3D_j.

Parameters
knownFields = {'jSolution': 'F'}
aliasFields = {'b': ['jSolution', 'E', '_b'], 'e': ['jSolution', 'F', '_e'], 'h': ['jSolution', 'E', '_h'], 'hPrimary': ['jSolution', 'E', '_hPrimary'], 'hSecondary': ['jSolution', 'E', '_hSecondary'], 'j': ['jSolution', 'F', '_j'], 'jPrimary': ['jSolution', 'F', '_jPrimary'], 'jSecondary': ['jSolution', 'F', '_jSecondary']}
startup()[source]
class SimPEG.EM.FDEM.FieldsFDEM.Fields3D_h(mesh, survey, **kwargs)[source]

Fields object for Problem3D_h.

Parameters
knownFields = {'hSolution': 'E'}
aliasFields = {'b': ['hSolution', 'CCV', '_b'], 'e': ['hSolution', 'CCV', '_e'], 'h': ['hSolution', 'E', '_h'], 'hPrimary': ['hSolution', 'E', '_hPrimary'], 'hSecondary': ['hSolution', 'E', '_hSecondary'], 'j': ['hSolution', 'F', '_j'], 'jPrimary': ['hSolution', 'F', '_jPrimary'], 'jSecondary': ['hSolution', 'F', '_jSecondary']}
startup()[source]
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