\[\renewcommand{\div}{\nabla\cdot\,} \newcommand{\grad}{\vec \nabla} \newcommand{\curl}{{\vec \nabla}\times\,} \newcommand {\J}{{\vec J}} \renewcommand{\H}{{\vec H}} \newcommand {\E}{{\vec E}} \newcommand{\dcurl}{{\mathbf C}} \newcommand{\dgrad}{{\mathbf G}} \newcommand{\Acf}{{\mathbf A_c^f}} \newcommand{\Ace}{{\mathbf A_c^e}} \renewcommand{\S}{{\mathbf \Sigma}} \renewcommand{\Div}{{\mathbf {Div}}} \newcommand{\St}{{\mathbf \Sigma_\tau}} \newcommand{\T}{{\mathbf T}} \newcommand{\Tt}{{\mathbf T_\tau}} \newcommand{\diag}{\mathbf{diag}} \newcommand{\M}{{\mathbf M}} \newcommand{\MfMui}{{\M^f_{\mu^{-1}}}} \newcommand{\dMfMuI}{{d_m (\M^f_{\mu^{-1}})^{-1}}} \newcommand{\MeSig}{{\M^e_\sigma}} \newcommand{\MeSigInf}{{\M^e_{\sigma_\infty}}} \newcommand{\MeSigO}{{\M^e_{\sigma_0}}} \newcommand{\Me}{{\M^e}} \newcommand{\Mes}[1]{{\M^e_{#1}}} \newcommand{\Mee}{{\M^e_e}} \newcommand{\Mej}{{\M^e_j}} \newcommand{\BigO}[1]{\mathcal{O}\bigl(#1\bigr)} \newcommand{\bE}{\mathbf{E}} \newcommand{\bH}{\mathbf{H}} \newcommand{\B}{\vec{B}} \newcommand{\D}{\vec{D}} \renewcommand{\H}{\vec{H}} \newcommand{\s}{\vec{s}} \newcommand{\bfJ}{\bf{J}} \newcommand{\vecm}{\vec m} \renewcommand{\Re}{\mathsf{Re}} \renewcommand{\Im}{\mathsf{Im}} \renewcommand {\j} { {\vec j} } \newcommand {\h} { {\vec h} } \renewcommand {\b} { {\vec b} } \newcommand {\e} { {\vec e} } \newcommand {\c} { {\vec c} } \renewcommand {\d} { {\vec d} } \renewcommand {\u} { {\vec u} } \newcommand{\I}{\vec{I}}\]

Magnetics

The geomagnetic field can be ranked as the longest studied of all the geophysical properties of the earth. In addition, magnetic survey, has been used broadly in diverse realm e.g., mining, oil and gas industry and environmental engineering. Although, this geophysical application is quite common in geoscience; however, we do not have modular, well-documented and well-tested open-source codes, which perform forward and inverse problems of magnetic survey. Therefore, here we are going to build up magnetic forward and inverse modeling code based on two common methodologies for forward problem - differential equation and integral equation approaches.

First, we start with some backgrounds of magnetics, e.g., Maxwell’s equations. Based on that secondly, we use differential equation approach to solve forward problem with secondary field formulation. In order to discretzie our system here, we use finite volume approach with weak formulation. Third, we solve inverse problem through Gauss-Newton method.

Backgrounds

Maxwell’s equations for static case with out current source can be written as

\[ \begin{align}\begin{aligned}\begin{split}\nabla U = \frac{1}{\mu}\vec{B} \\\end{split}\\\nabla \cdot \vec{B} = 0\end{aligned}\end{align} \]

where \(\vec{B}\) is magnetic flux (\(T\)) and \(U\) is magnetic potential and \(\mu\) is permeability. Since we do not have any source term in above equations, boundary condition is going to be the driving force of our system as given below

\[(\vec{B}\cdot{\vec{n}})_{\partial\Omega} = B_{BC}\]

where \(\vec{n}\) means the unit normal vector on the boundary surface (\(\partial \Omega\)). By using seocondary field formulation we can rewrite above equations as

\[ \begin{align}\begin{aligned}\frac{1}{\mu}\vec{B}_s = (\frac{1}{\mu}_0-\frac{1}{\mu})\vec{B}_0+\nabla\phi_s\\\nabla \cdot \vec{B}_s = 0\\(\vec{B}_s\cdot{\vec{n}})_{\partial\Omega} = B_{sBC}\end{aligned}\end{align} \]

where \(\vec{B}_s\) is the secondary magnetic flux and \(\vec{B}_0\) is the background or primary magnetic flux. In practice, we consider our earth field, which we can get from International Geomagnetic Reference Field (IGRF) by specifying the time and location, as \(\vec{B}_0\). And based on this background fields, we compute secondary fields (\(\vec{B}_s\)). Now we introduce the susceptibility as

\[ \begin{align}\begin{aligned}\begin{split}\chi = \frac{\mu}{\mu_0} - 1 \\\end{split}\\\mu = \mu_0(1+\chi)\end{aligned}\end{align} \]

Since most materials in the earth have lower permeability than \(\mu_0\), usually \(\chi\) is greater than 0.

Note

Actually, this is an assumption, which means we are not sure exactly this is true, although we are sure, it is very rare that we can encounter those materials. Anyway, practical range of the susceptibility is \(0 < \chi < 1 \).

Since we compute secondary field based on the earth field, which can be different from different locations in the world, we can expect different anomalous responses in different locations in the earth. For instance, assume we have two susceptible spheres, which are exactly same. However, anomalous responses in Seoul and Vancouver are going to be different.

Since we can measure total fields ( \(\vec{B}\)), and usually have reasonably accurate earth field (\(\vec{B}_0\)), we can compute anomalous fields, \(\vec{B}_s\) from our observed data. If you want to download earth magnetic fields at specific location see this website (noaa).

What is our data?

In applied geophysics, which means in practice, it is common to refer to measurements as “the magnetic anomaly” and we can consider this as our observed data. For further descriptions in GPG materials for magnetic survey. Now we have the simple relation ship between “the magnetic anomaly” and the total field as

\[\triangle\vec{B} = |\hat{B}_o-\vec{B}_s|-|\hat{B}_o| \approx |\vec{B}_s|cos \theta\]

where \(\theta\) is the angle between total and anomalous fields, \(\hat{B}_o\) is the unit vector for \(\vec{B}_o\). Equivalently, we can use the vector dot product to show that the anomalous field is approximately equal to the projection of that field onto the direction of the inducing field. Using this approach we would write

\[\triangle\vec{B} = |\vec{B}_s|cos \theta = |\hat{B}_o||\vec{B}_s|cos \theta = \hat{B}_o \cdot \vec{B}_s\]

This is important because, in practice we usually use a total field magnetometer (like a proton precession or optically pumped sensor), which can measure only that part of the anomalous field which is in the direction of the earth’s main field.

Sphere in a whole space

Forward problem

Differential equation approach

\[ \begin{align}\begin{aligned}\mathbf{A}\mathbf{u} = \mathbf{rhs}\\\mathbf{A} = \Div(\MfMui)^{-1}\Div^{T}\\\mathbf{rhs} = \Div(\MfMui)^{-1}\mathbf{M}^f_{\mu_0^{-1}}\mathbf{B}_0 - \Div\mathbf{B}_0+\diag(v)\mathbf{D} \mathbf{P}_{out}^T \mathbf{B}_{sBC}\\\mathbf{B}_s = (\MfMui)^{-1}\mathbf{M}^f_{\mu_0^{-1}}\mathbf{B}_0-\mathbf{B}_0 -(\MfMui)^{-1}\Div^T \mathbf{u}\end{aligned}\end{align} \]

Mag Differential eq. approach

class SimPEG.PF.Magnetics.Problem3D_DiffSecondary(mesh, **kwargs)

Bases: SimPEG.Problem.BaseProblem

Secondary field approach using differential equations!

Optional Properties:

• model (Model): Inversion model., a numpy array of <type ‘float’>, <type ‘int’> with shape (*)
• mu (PhysicalProperty): Magnetic Permeability (H/m), a physical property, Default: 1.25663706144e-06
• muMap (Mapping): Mapping of Magnetic Permeability (H/m) to the inversion model., a SimPEG Map
• mui (PhysicalProperty): Inverse Magnetic Permeability (m/H), a physical property
• muiMap (Mapping): Mapping of Inverse Magnetic Permeability (m/H) to the inversion model., a SimPEG Map

Other Properties:

• muDeriv (Derivative): Derivative of Magnetic Permeability (H/m) wrt the model.
• muiDeriv (Derivative): Derivative of Inverse Magnetic Permeability (m/H) wrt the model.

Jtvec(*args, **kwargs)

Computing Jacobian^T multiplied by vector.

\[ \begin{align}\begin{aligned}(\frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}})^{T} = \left[ \mathbf{P}_{deriv}\frac{\partial \mathbf{\mu} } {\partial \mathbf{m} } \left[ \diag(\M^f_{\mu_{0}^{-1} } \mathbf{B}_0) \dMfMuI \ - \diag (\Div^T\mathbf{u})\dMfMuI \right ]\right]^{T}\\ - \left[\mathbf{P}_{deriv}(\MfMui)^{-1}\Div^T\frac{\delta\mathbf{u}}{\delta \mathbf{m}} \right]^{T}\end{aligned}\end{align} \]

where

\[\mathbf{P}_{derv} = \frac{\partial \mathbf{P}}{\partial\mathbf{B}}\]

Note

Here we only want to compute

\[\mathbf{J}^{T}\mathbf{v} = (\frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}})^{T} \mathbf{v}\]

Jtvec_approx(m, v, f=None)

Approximate effect of transpose of J(m) on a vector v.

Parameters:

• m (numpy.array) – model
• v (numpy.array) – vector to multiply
• f (Fields) – fields

Return type:

numpy.array

Returns:

JTv

Jvec(*args, **kwargs)

Computing Jacobian multiplied by vector

By setting our problem as

\[\mathbf{C}(\mathbf{m}, \mathbf{u}) = \mathbf{A}\mathbf{u} - \mathbf{rhs} = 0\]

And taking derivative w.r.t m

\[ \begin{align}\begin{aligned}\nabla \mathbf{C}(\mathbf{m}, \mathbf{u}) = \nabla_m \mathbf{C}(\mathbf{m}) \delta \mathbf{m} + \nabla_u \mathbf{C}(\mathbf{u}) \delta \mathbf{u} = 0\\\frac{\delta \mathbf{u}}{\delta \mathbf{m}} = - [\nabla_u \mathbf{C}(\mathbf{u})]^{-1}\nabla_m \mathbf{C}(\mathbf{m})\end{aligned}\end{align} \]

With some linear algebra we can have

\[ \begin{align}\begin{aligned}\nabla_u \mathbf{C}(\mathbf{u}) = \mathbf{A}\\\nabla_m \mathbf{C}(\mathbf{m}) = \frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} - \frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}}\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} = \frac{\partial \mathbf{\mu}}{\partial \mathbf{m}} \left[\Div \diag (\Div^T \mathbf{u}) \dMfMuI \right]\\\dMfMuI = \diag(\MfMui)^{-1}_{vec} \mathbf{Av}_{F2CC}^T\diag(\mathbf{v})\diag(\frac{1}{\mu^2})\\\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}} = \frac{\partial \mathbf{\mu}}{\partial \mathbf{m}} \left[ \Div \diag(\M^f_{\mu_{0}^{-1}}\mathbf{B}_0) \dMfMuI \right] - \diag(\mathbf{v})\mathbf{D} \mathbf{P}_{out}^T\frac{\partial B_{sBC}}{\partial \mathbf{m}}\end{aligned}\end{align} \]

In the end,

\[\frac{\delta \mathbf{u}}{\delta \mathbf{m}} = - [ \mathbf{A} ]^{-1}\left[ \frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} - \frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}} \right]\]

A little tricky point here is we are not interested in potential (u), but interested in magnetic flux (B). Thus, we need sensitivity for B. Now we take derivative of B w.r.t m and have

\[ \begin{align}\begin{aligned}\frac{\delta \mathbf{B}} {\delta \mathbf{m}} = \frac{\partial \mathbf{\mu} } {\partial \mathbf{m} } \left[ \diag(\M^f_{\mu_{0}^{-1} } \mathbf{B}_0) \dMfMuI \ - \diag (\Div^T\mathbf{u})\dMfMuI \right ]\\ - (\MfMui)^{-1}\Div^T\frac{\delta\mathbf{u}}{\delta \mathbf{m}}\end{aligned}\end{align} \]

Finally we evaluate the above, but we should remember that

Note

We only want to evalute

\[\mathbf{J}\mathbf{v} = \frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}}\mathbf{v}\]

Since forming sensitivity matrix is very expensive in that this monster is “big” and “dense” matrix!!

Jvec_approx(m, v, f=None)

Approximate effect of J(m) on a vector v

Parameters:

• m (numpy.array) – model
• v (numpy.array) – vector to multiply
• f (Fields) – fields

Return type:

numpy.array

Returns:

approxJv

MfMu0

MfMuI

MfMui

class Solver(A, **kwargs)

Bases: object

clean()

counter = None

curModel

Setting the curModel is depreciated.

Use SimPEG.Problem.model instead.

deleteTheseOnModelUpdate = []

deserialize(value, trusted=False, strict=False, assert_valid=False, **kwargs)

Creates HasProperties instance from serialized dictionary

This uses the Property deserializers to deserialize all JSON-compatible dictionary values into their corresponding Property values on a new instance of a HasProperties class. Extra keys in the dictionary that do not correspond to Properties will be ignored.

Parameters:

• value - Dictionary to deserialize new instance from.
• trusted - If True (and if the input dictionary has '__class__' keyword and this class is in the registry), the new HasProperties class will come from the dictionary. If False (the default), only the HasProperties class this method is called on will be constructed.
• strict - Requires '__class__', if present on the input dictionary, to match the deserialized instance’s class. Also disallows unused properties in the input dictionary. Default is False.
• assert_valid - Require deserialized instance to be valid. Default is False.
• Any other keyword arguments will be passed through to the Property deserializers.

equal(other)

Determine if two HasProperties instances are equivalent

Equivalence is determined by checking if all Property values on two instances are equal, using Property.equal.

fields(m)

Return magnetic potential (u) and flux (B) u: defined on the cell center [nC x 1] B: defined on the cell center [nF x 1]

After we compute u, then we update B.

\[\mathbf{B}_s = (\MfMui)^{-1}\mathbf{M}^f_{\mu_0^{-1}}\mathbf{B}_0-\mathbf{B}_0 -(\MfMui)^{-1}\Div^T \mathbf{u}\]

getA(m)

GetA creates and returns the A matrix for the Magnetics problem

The A matrix has the form:

\[\mathbf{A} = \Div(\MfMui)^{-1}\Div^{T}\]

getB0(*args, **kwargs)

To use getB0 method, SimPEG requires that the survey be specified.

getRHS(m)

\[\mathbf{rhs} = \Div(\MfMui)^{-1}\mathbf{M}^f_{\mu_0^{-1}}\mathbf{B}_0 - \Div\mathbf{B}_0+\diag(v)\mathbf{D} \mathbf{P}_{out}^T \mathbf{B}_{sBC}\]

ispaired

True if the problem is paired to a survey.

makeMassMatrices(m)

mapPair

alias of IdentityMap

mapping

Setting an unnamed mapping has been depreciated in v0.4.0. Please see the release notes for more details.

mesh = None

model

model (Model): Inversion model., a numpy array of <type ‘float’>, <type ‘int’> with shape (*)

modelPair

alias of BaseMagMap

mu

Magnetic Permeability (H/m)

muDeriv

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

muMap

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

mui

Inverse Magnetic Permeability (m/H)

muiDeriv

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

muiMap

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

needs_model

True if a model is necessary

pair(d)

Bind a survey to this problem instance using pointers.

serialize(include_class=True, save_dynamic=False, **kwargs)

Serializes a HasProperties instance to dictionary

This uses the Property serializers to serialize all Property values to a JSON-compatible dictionary. Properties that are undefined are not included. If the HasProperties instance contains a reference to itself, a properties.SelfReferenceError will be raised.

Parameters:

• include_class - If True (the default), the name of the class will also be saved to the serialized dictionary under key '__class__'
• save_dynamic - If True, dynamic properties are written to the serialized dict (default: False).
• Any other keyword arguments will be passed through to the Property serializers.

solverOpts = {}

summary()

survey

The survey object for this problem.

surveyPair

alias of BaseMagSurvey

unpair()

Unbind a survey from this problem instance.

validate()

Call all registered class validator methods

These are all methods decorated with @properties.validator. Validator methods are expected to raise a ValidationError if they fail.

class SimPEG.PF.BaseMag.BaseMagSurvey(**kwargs)

Bases: SimPEG.Survey.BaseSurvey

Base Magnetics Survey

Qfx

Qfy

Qfz

counter = None

dobs = None

dpred(m, f=None)

Create the projected data from a model. The fields, f, (if provided) will be used for the predicted data instead of recalculating the fields (which may be expensive!).

\[d_\text{pred} = P(f(m))\]

Where P is a projection of the fields onto the data space.

Note

To use survey.dpred(), SimPEG requires that a problem be bound to the survey. If a problem has not been bound, an Exception will be raised. To bind a problem to the Data object:

survey.pair(myProblem)

dtrue = None

eps = None

eval(f)

This function projects the fields onto the data space.

\[d_\text{pred} = \mathbf{P} f(m)\]

evalDeriv(f)

This function s the derivative of projects the fields onto the data space.

\[\frac{\partial d_\text{pred}}{\partial u} = \mathbf{P}\]

getSourceIndex(sources)

isSynthetic

Check if the data is synthetic.

ispaired

makeSyntheticData(m, std=0.05, f=None, force=False)

Make synthetic data given a model, and a standard deviation.

Parameters:

• m (numpy.array) – geophysical model
• std (numpy.array) – standard deviation
• u (numpy.array) – fields for the given model (if pre-calculated)
• force (bool) – force overwriting of dobs

mesh

Mesh of the paired problem.

mtrue = None

nD

Number of data

nSrc

Number of Sources

pair(p)

Bind a problem to this survey instance using pointers

prob

The geophysical problem that explains this survey, use:

survey.pair(prob)

projectFields(u)

This function projects the fields onto the data space.

Especially, here for we use total magnetic intensity (TMI) data, which is common in practice.

First we project our B on to data location

\[\mathbf{B}_{rec} = \mathbf{P} \mathbf{B}\]

then we take the dot product between B and b_0

\[\text{TMI} = \vec{B}_s \cdot \hat{B}_0\]

projectFieldsAsVector(B)

projectFieldsDeriv(*args, **kwargs)

This function projects the fields onto the data space.

\[\frac{\partial d_\text{pred}}{\partial \mathbf{B}} = \mathbf{P}\]

Especially, this function is for TMI data type

residual(m, f=None)

Parameters:

• m (numpy.array) – geophysical model
• f (numpy.array) – fields

Return type:

numpy.array

Returns:

data residual

The data residual:

\[\mu_\text{data} = \mathbf{d}_\text{pred} - \mathbf{d}_\text{obs}\]

rxLoc = None

rxType = None

setBackgroundField(Inc, Dec, Btot)

srcList

Source List

srcPair

alias of BaseSrc

std = None

unpair()

Unbind a problem from this survey instance

vnD

Vector number of data

Mag Integral eq. approach

Mag analytic solutions

SimPEG.PF.MagAnalytics.CongruousMagBC(mesh, Bo, chi)

Computing boundary condition using Congrous sphere method. This is designed for secondary field formulation.

>> Input

• mesh: Mesh class
• Bo: np.array([Box, Boy, Boz]): Primary magnetic flux
• chi: susceptibility at cell volume

\[\vec{B}(r) = \frac{\mu_0}{4\pi} \frac{m}{ \| \vec{r} - \vec{r}_0\|^3}[3\hat{m}\cdot\hat{r}-\hat{m}]\]

SimPEG.PF.MagAnalytics.IDTtoxyz(Inc, Dec, Btot)

Convert from Inclination, Declination, Total intensity of earth field to x, y, z

SimPEG.PF.MagAnalytics.MagSphereAnaFun(x, y, z, R, x0, y0, z0, mu1, mu2, H0, flag='total')

test Analytic function for Magnetics problem. The set up here is magnetic sphere in whole-space assuming that the inducing field is oriented in the x-direction.

• (x0, y0, z0)
• (x0, y0, z0 ): is the center location of sphere
• r: is the radius of the sphere

\[\mathbf{H}_0 = H_0\hat{x}\]

SimPEG.PF.MagAnalytics.MagSphereAnaFunA(x, y, z, R, xc, yc, zc, chi, Bo, flag)

Computing boundary condition using Congrous sphere method. This is designed for secondary field formulation. >> Input mesh: Mesh class Bo: np.array([Box, Boy, Boz]): Primary magnetic flux Chi: susceptibility at cell volume

\[\vec{B}(r) = \frac{\mu_0}{4\pi}\frac{m}{\| \vec{r}-\vec{r}_0\|^3}[3\hat{m}\cdot\hat{r}-\hat{m}]\]

SimPEG.PF.MagAnalytics.MagSphereFreeSpace(x, y, z, R, xc, yc, zc, chi, Bo)

Computing the induced response of magnetic sphere in free-space.

>> Input x, y, z: Observation locations R: radius of the sphere xc, yc, zc: Location of the sphere chi: Susceptibility of sphere Bo: Inducing field components [bx, by, bz]*|H0|

SimPEG.PF.MagAnalytics.spheremodel(mesh, x0, y0, z0, r)

Generate model indicies for sphere - (x0, y0, z0 ): is the center location of sphere - r: is the radius of the sphere - it returns logical indicies of cell-center model