simpeg.electromagnetics.frequency_domain.sources.MagDipole#

class simpeg.electromagnetics.frequency_domain.sources.MagDipole(receiver_list, frequency, location=None, moment=1.0, orientation='z', mu=1.25663706127e-06, **kwargs)[source]#

Bases: BaseFDEMSrc

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 simulation, 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

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

To obtain $mathbf{b^P}$, we compute it by taking the curl of the vector potential due to a point dipole. This is provided by geoana.em.static.MagneticDipoleWholeSpace.vector_potential(). Specifically,

\[\vec{B}^P = \nabla \times \vec{A}\]

Parameters:

receiver_listlist of simpeg.electromagnetics.frequency_domain.receivers.BaseRx: A list of FDEM receivers
frequencyfloat: Source frequency
location(dim) numpy.ndarray, default: numpy.r_[0., 0., 0.]: Source location.
momentfloat: Magnetic dipole moment amplitude
orientation{‘z’, x’, ‘y’} or (dim) numpy.ndarray: Orientation of the dipole.
mufloat: Background magnetic permeability

Attributes

`frequency`	Source frequency
`integrate`	Integrated source term
`location`	Location of the dipole
`moment`	Amplitude of the dipole moment of the magnetic dipole ($A/m^2$)
`mu`	Magnetic permeability in H/m
`nD`	Number of data associated with the source.
`orientation`	Orientation of the dipole as a normalized vector
`receiver_list`	List of receivers associated with the source
`uid`	Universal unique identifier
`vnD`	Vector number of data.

Methods

`bPrimary`(simulation)	Compute primary magnetic flux density.
`bPrimaryDeriv`(simulation, v[, adjoint])	Compute derivative of primary magnetic flux density times a vector
`ePrimary`(simulation)	Compute primary electric field
`ePrimaryDeriv`(simulation, v[, adjoint])	Compute derivative of primary electric field times a vector
`eval`(simulation)	Return magnetic and electric source terms
`evalDeriv`(simulation[, v, adjoint])	Return derivative of the magnetic and electric source terms with respect to the model.
`get_receiver_indices`(receivers)	Get indices for a subset of receivers within the source's receivers list.
`hPrimary`(simulation)	Compute primary magnetic field.
`hPrimaryDeriv`(simulation, v[, adjoint])	Compute derivative of primary magnetic field times a vector
`jPrimary`(simulation)	Compute primary current density
`jPrimaryDeriv`(simulation, v[, adjoint])	Compute derivative of primary current density times a vector
`s_e`(simulation)	Electric source term (s_e)
`s_eDeriv`(simulation, v[, adjoint])	Derivative of electric source term with respect to the inversion model
`s_m`(simulation)	Magnetic source term (s_m)
`s_mDeriv`(simulation, v[, adjoint])	Derivative of magnetic source term with respect to the inversion model