the electromagnetic (em) method magnetotelluric (mt)

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ERTH2020 Introduction to Geophysics

The Electromagnetic (EM) MethodMagnetotelluric (MT)

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Magnetotelluric

combination of magnetic and telluric* methods

(Latin ‘tellūs’ ‘earth’ “Earth current”)

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Magnetotelluric

…other scientists Tikhonov (1950) and Rikitake (1951), Kato & Kikuchi (1950).

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Induction

I

• DC Resistivity

I

• Induced Polarisation

I

• Inductive EM

R

CL

Equivalent Circuits

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DC / IP

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Magnetotelluric (Passive EM)

𝐻 𝑧

𝐻 𝑦𝐻 𝑥

𝐸𝑥

𝐸𝑦

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Goal

𝜕2

𝜕𝑧 2𝐅−iω μσ𝐅=0

→𝑝=√ 2ωμσ ≈500√𝑇 𝜌𝑎

Skin Depth (Penetration Depth)

1D diffusion equation

1.

2.

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DC Resistivity Induced Polarisation Passive EM Active EM

Method

Direct Electrical Connection (galvanic) No direct electrical connection (inductive)

Injected DC current via electrodesInduced primary magnetic field via natural EM fields

Induced primary magnetic field via loop

Measured

Electrical potential

Decay of electrical potential

Ratio of E and H fields

Secondary magnetic field

(or its decay)

Resistivity Resistivity & Chargeability Conductivity Conductivity

Overview

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Contents

• Introductiono Maxwell Equationso Inductiono Sourceso Example

• EM theoryo Divergence & Curlo Diffusion equationo 1D Magnetotellurico Skin Deptho Apparent Resistivity & Phase

• 2D MT Introductiono Example

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Electromagnetic Induction

Ampere’s Law (1826)

electric current density (A/m2)magnetic field intensity (A/m)

Faraday’s Law (1831)

magnetic induction (Wb/m2 or T)magnetic field intensity (V/m)

(magneto) quasi-static approximation , i.e. separation of electrical charges occur sufficiently slowly that the system can be taken to be in equilibrium at all times

e.g. http://farside.ph.utexas.edu/teaching/302l/lectures/node70.htmlhttp://farside.ph.utexas.edu/teaching/302l/lectures/node85.html

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Simpson F. and Bahr K, 2005, p.18

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Primary field


Plane Wave Source

𝐻 𝑥

𝐸𝑦

Faraday’s Law

Ampere’s Law

Ohm’s Law𝐻 𝑥

𝐸𝑦

𝐸𝑦

𝐻 𝑥

45∘

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Magnetotelluric

Sources

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Magnetotelluric


Sources

Power spectrum: signal's power (energy per unit time) falling within given frequency bins

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Magnetotelluric


Applications

• Mineral exploration

• Hydrocarbon exploration (oil/gas)

• Deep crustal studies

• Geothermal studies

• Groundwater monitoring

• Earthquake monitoring

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Magnetotelluric

Hill et al., 2009

Example 2D-MT resistivity model

• White and red dots show the locations of the magnetotelluric measurements; measurement sites shown in red were used for 2D inversion.

• The east–west line (red) shows the profile onto which these measurements were projected. The coloured area shows the region of high conductances. (=conductivity X thickness)

• The green-to-orange transition corresponds to a conductance of 3000 Siemens.

• Locations of MT measurement sites, Mount St Helens and nearby Cascades volcanoes.

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Magnetotelluric

Hill et al., 2009

the conductivity anomalies are caused by the presence of partial melt

Example 2D-MT resistivity model after inversion

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EM Theory

𝛻×𝐅𝛻𝑈curlgradient

(𝜕𝑥𝜕 𝑦𝜕𝑧)×(𝐹 𝑥

𝐹 𝑦𝐹 𝑧

)(𝜕𝑥𝑈𝜕 𝑦𝑈𝜕𝑧𝑈 )

(𝜕𝒚 𝐹 𝒛−𝜕𝒛 𝐹 𝒚𝜕𝒛 𝐹 𝒙−𝜕 𝒙𝐹 𝒛𝜕𝒙 𝐹 𝒚−𝜕 𝒚 𝐹 𝒙

)𝜕𝑥𝑈 +𝜕𝑦𝑈 +𝜕𝑧𝑈

𝛻 ∙𝐅divergence

(𝜕𝑥𝜕 𝑦𝜕𝑧) ∙(

𝐹 𝑥

𝐹 𝑦𝐹 𝑧

)𝜕𝒙𝐹 𝒙+𝜕𝒚 𝐹 𝒚+𝜕𝒛 𝐹 𝒛

(vector) (scalar) (vector)

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Divergence (Interpretation)

The divergence measures how much a vector field ``spreads out'' or diverges from a given point, here (0,0):• Left: divergence > 0 since the vector field is ‘spreading out’• Centre: divergence = 0 everywhere since the vectors are not spreading out. • Right: divergence < 0 since the vectors are coming closer together

instead of spreading out.

is the extent to which the vector field flow behaves like a source or a sink at a given point. (If the divergence is nonzero at some point then there must be a source or sink at that position)

http://citadel.sjfc.edu/faculty/kgreen/vector/block2/del_op/node5.html

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Curl (Interpretation)

The curl of a vector field measures the tendency for the vector field to “swirl around”. (For example, let the vector field represents the velocity vectors of water in a lake. If the vector field swirls around, then when we stick a paddle wheel into the water, it will tend to spin.) • Left: curl > 0 (right-hand-rule thumb is up+)• Centre: curl = 0 everywhere since the field has no ‘swirling’. • Right: curl 0 since the vectors produce a torque on a test paddle

wheel.

describes the infinitesimal rotation of a vector field ( p.s. The name "curl" was first suggested by James Clerk Maxwell in 1871)

http://citadel.sjfc.edu/faculty/kgreen/vector/block2/del_op/node5.html & Wikipedia (Curl)

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EM Theory

(Faraday)

𝛻×𝐄=− 𝜕𝐁𝜕𝑡

𝛻×𝐇= 𝐉(Ampere)

Time-Domain Maxwell Equations (magneto-quasi-static)

Note the use of the constitutive relations:

𝐁=μ𝐇 𝐉=𝜎𝐄𝐃=ε𝐄→ 1μ 𝛻×𝐄=− 𝜕𝐇 𝜕𝑡

→ 1𝜎 𝛻×𝐇=𝐄

first order, coupled PDEs

Also note that generally

μ=μ (𝑥 , 𝑦 . 𝑧 ) 𝜎=𝜎 (𝑥 , 𝑦 .𝑧 )

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EM Theory

(Faraday)

(Ampere)


1μ 𝛻×𝐄=− 𝜕𝐇

𝜕𝑡

1𝜎 𝛻×𝐇=𝐄

Second order, uncoupled PDEs

to uncouple, take the curl

→𝛻× 1μ 𝛻×𝐄=− 𝜕 𝜕𝑡 (𝛻×𝐇 )

→𝛻× 1𝜎 𝛻×𝐇=(𝛻×𝐄 )

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EM Theory



→𝛻× 1μ 𝛻×𝐄=−𝜎 𝜕𝐄𝜕𝑡

→𝛻× 1𝜎 𝛻×𝐇=−μ 𝜕𝐇 𝜕𝑡

𝐄 (𝑡 )=𝐄0𝑒𝑖 𝜔𝑡

𝐇 (𝑡 )=𝐇0𝑒𝑖𝜔𝑡

Plane wave source sinusoidal time variation

where the angular frequency and the imaginary unit

• Complex numbers arise e.g. from equations such as .

• Generally complex numbers have a real and imaginary part and are written as where is the real part and the imaginary part.

• Complex numbers can also be written as

• Compact way to describe waves

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EM Theory



→𝛻× 1μ 𝛻×𝐄=−𝜎 𝜕𝐄𝜕𝑡 =− 𝑖𝜎 𝜔𝐄

→𝛻× 1𝜎 𝛻×𝐇=−μ 𝜕𝐇 𝜕𝑡 =−𝑖 μ𝜔𝐇

𝐄 (𝑡 )=𝐄0𝑒𝑖 𝜔𝑡

𝐇 (𝑡 )=𝐇0𝑒𝑖𝜔𝑡

Plane wave source sinusoidal time variation

where the angular frequency and the imaginary unit

• Complex numbers arise e.g. from equations such as .

• Generally complex numbers have a real and imaginary part and are written as where is the real part and the imaginary part.

• Complex numbers can also be written as

• Compact way to describe waves

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EM Theory

Frequency Domain Diffusion Equations


General equations for inductive EM

→𝛻× 1μ 𝛻×𝐄+𝑖𝜔𝜎𝐄=0

→𝛻× 1𝜎 𝛻×𝐇+𝑖 𝜔 μ𝐇=0

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EM Theory

1D solution

Diffusion Equations (Frequency Domain)

𝛻×𝛻×𝐅=𝛻 (𝛻 ∙𝐅 )− (𝛻 ∙𝛻 )𝐅with vector identity

→𝛻 (𝛻 ∙𝐄 )⏞¿ 0

− (𝛻 ∙𝛻 )𝐄=−iωμσ𝐄

→𝛻 (𝛻 ∙𝐇 )⏟¿ 0

− (𝛻 ∙𝛻 )𝐇=−iω μσ𝐇

→𝛻2𝐄−iωμσ𝐄=0

→𝛻2𝐇−iω μσ𝐇=0

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EM Theory

1D solution

𝛻 ∙𝐄=𝟎 𝛻 ∙𝐇=𝟎Divergence of Ampere’s law

→𝛻 ∙𝛻×𝐄=−𝛻 ∙ 𝜕𝐁𝜕𝑡 =− 𝜕𝜕𝑡 (𝛻 ∙𝐁)=0

→𝛻 ∙𝐁=0 (Gauss law for magnetism, i.e. no magnetic monopoles)

Divergence of Faraday’s law

→𝛻 ∙𝛻×𝐇=𝛻 ∙ 𝐉=𝛻 ∙ (σ𝐄 )=0𝛻 ∙ (σ𝐄)=σ 𝛻 ∙𝐄+𝐄 ∙𝛻 σ=0→σ𝛻 ∙𝐄=−𝐄 ∙𝛻σ

𝛻 σ=0→𝛻 ∙𝐄=0

via Cartesian coordinates

Proof

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𝑧<0

𝑧>0

EM Theory

1D solution

→𝛻2𝐅−iω μσ𝐅=0⇔𝐅=𝐅1𝑒𝑖ωt −𝑞𝑧+𝐅2𝑒𝑖ωt+𝑞𝑧

General solution for second-order PDE:

decreases in amplitude with z

increases in amplitude with z unphysical


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EM Theory

1D solution

𝐅=𝐅1𝑒𝑖ωt −𝑞𝑧

Taking the second derivative with respect to z


𝜕2

𝜕𝑧 2𝐅=𝑞2𝐅1𝑒𝑖ωt −𝑞𝑧=𝑞2𝐅↔

𝜕2

𝜕 𝑧 2𝐅−iωμσ𝐅=0

→𝑞=√ 𝑖ωμσ=√𝑖√ω μσ= (1+ 𝑖 ) √ωμσ /2=√ωμσ /2+ 𝑖√ωμσ /2Real part Imaginary part

→𝑝=1ℜ𝔢 (𝑞 )

=√ 2ωμσ


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EM Theory

1D solution

𝐅=𝐅1𝑒𝑖ωt −𝑞𝑧

Taking the second derivative with respect to z


𝜕2

𝜕𝑧 2𝐅=𝑞2𝐅1𝑒𝑖ωt −𝑞𝑧=𝑞2𝐅↔

𝜕2

𝜕 𝑧 2𝐅− iωμσ𝐅=0

→𝑞=√ 𝑖ωμσ=√𝑖√ω μσ= (1+ 𝑖 ) √ωμσ /2=√ωμσ /2+ 𝑖√ωμσ /2Real part Imaginary part

→𝑝=1ℜ𝔢 (𝑞 )

=√ 2ωμσ


For angular frequency for a half-space with conductivity

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EM Theory

1D solution 𝜎=( piecewise ) constant ,𝜇≡constant𝜇→𝜇0=4𝜋 ∙10−7

Simpson F. and Bahr K, 2005, p.22 & http://userpage.fu-berlin.de/~mtag/MT-principles.html

→𝑝=1ℜ𝔢 (𝑞 )

=√ 2ωμσ Skin Depth (Penetration Depth)

≈ 107

4 0

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EM Theory

1D solution


𝑞=√ωμσ /2+√𝑖ωμσ /2

Real part Imaginary part

The inverse of q is the Schmucker-Weidelt Transfer Function

𝐶=1𝑞=

𝑝2 +𝑖 𝑝2

and𝑝=1ℜ𝔢 (𝑞 )

=√2/ωμσ

..has dimensions of length but is complex

The Transfer Function C establishes a linear relationship between the physical properties that are measured in the field.

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EM Theory

1D solution


Schmucker-Weidelt Transfer Function

𝐶=1𝑞=

𝑝2 +𝑖 𝑝2with𝑝=√2/ωμσ

𝐸𝑥=𝐸1𝑥𝑒𝑖ωt −𝑞𝑧→

𝜕𝐸𝑥

𝜕 𝑧 =−𝑞𝐸𝑥

We had with the general solution earlier

Therefore

(𝛻×𝐄 )𝑦=𝜕𝐸𝑥

𝜕 𝑧 =− 𝑖ωμ𝐻 𝑦

However Faraday’s law is

−𝑖ωμ𝐻 𝑦=−𝑞 𝐸𝑥→𝐶= 1𝑞= 1𝑖ωμ

𝐸𝑥

𝐻 𝑦=− 1

𝑖ωμ𝐸𝑦

𝐻𝑥

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EM Theory

1D solution


Schmucker-Weidelt Transfer Function

𝐶= 1𝑞= 1𝑖ωμ

𝐸𝑥

𝐻 𝑦=− 1

𝑖ωμ𝐸 𝑦

𝐻 𝑥

• is calculated from measured and fields (or and ) .• from the apparent resistivity can be calculated:

with q=√𝑖ωμσ→|𝑞|2=ωμσ→σ=|𝑞|2

ωμor ρ= 1

|𝑞|2ωμ

→ρ=|𝐶|2ωμapparent resistivity

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EM Theory

Apparent Resistivity and Phase


𝜙=tan−1(ℑ𝔪𝐶ℜ𝔢𝐶 )phase

𝜌𝑎=|𝐶|2ω μapparent resistivity

The phase is the lag between the E and H field and together with apparent resistivity one of the most important parameters in MT

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EM Theory

Apparent Resistivity and Phase


For a homogeneous half space:

• diagnostic of substrata in which resistivity increases with depth

• diagnostic of substrata in which resistivity decreases with depth

𝜌𝑎=|𝐶|2ω μ 𝜙= tan−1(ℑ𝔪𝐶ℜ𝔢𝐶 ) 𝐶=𝑝2 +𝑖 𝑝2

with𝑝=√2/ωμσ

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EM Theory


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2D-MT Introduction


For this 2-D case, EM fields can be decoupled into two independent modes: • E-fields parallel to strike with induced B-fields perpendicular to strike and in

the vertical plane (E-polarisation or TE mode).• B-fields parallel to strike with induced E-fields perpendicular to strike and in

the vertical plane (B-polarisation or TM mode).

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2D-MT Introduction


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Numerical Modelling in 2D

2D solution

TE-mode (E-Polarisation)

𝛻 ∙ (𝛻𝐸𝒙 )−𝑖𝜔𝜇𝜎 𝐸𝑥=0

Numerical schemes, e.g.:• Finite Differences • Finite Elements

Escript Finite Element Solver (Geocomp UQ)

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Dirichlet boundary conditions via a single analytical 1D solution applied Left and Right; Top & Bottom via interpolation

σ = 10-14 S/m

σ = 0.1 S/m

σ = 0.01 S/m


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Electric Field (Imaginary) Electric Field (Real)


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Apparent Resistivity at selected station (all frequencies)


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σ = 0.4 S/mσ = 0.001 S/m

σ = 10-14 S/m

σ = 0.2 S/m

σ = 0.1 S/mσ = 0.04 S/m

# Zones = 71389

# Nodes = 36343


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Real Part Imaginary Part


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Apparent Resistivity

f = 1 Hz


r = 2.5 Ωmr = 1000 Ωm

r = 10 Ωmr = 25 ΩmSkin-depth

r = 2 Ωm

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References

Simpson F. and Bahr K.: “Practical magnetotellurics”, 2005, Cambridge University Press

Cagniard, L. (1953) Basic theory of the magneto-telluric method of geophysical prospecting, Geophysics, 18, 605–635

Hill G J., Caldwell T.G, Heise W., Chertkoff D.G., Bibby H.M., Burgess M.K., Cull J.P., Cas R.A.F.: "Distribution of melt beneath Mount St Helens and Mount Adams inferred from magnetotelluric data", Nature Geosci., 2009, V2, pp.785

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Unused slides

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EM Theory

(Faraday)

→𝛻 ∙𝛻×𝐄=−𝛻 ∙ 𝜕𝐁𝜕𝑡 =− 𝜕𝜕𝑡 (𝛻 ∙𝐁)=0

𝛻×𝐄=− 𝜕𝐁𝜕𝑡

→𝛻 ∙𝐁=0 (Gauss law for magnetism)

via Cartesian coordinates

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𝛻×𝐇= 𝐉+ 𝜕𝐃𝜕𝑡(Ampere)

EM Theory

→𝛻 ∙ 𝐉=− 𝜕𝜕𝑡 (𝛻 ∙𝐃 )

(Gauss law)

→𝛻 ∙ 𝐉+𝛻 ∙ 𝜕𝐃𝜕 t =𝛻 ∙ 𝐉+ 𝜕𝜕𝑡 (𝛻 ∙𝐃 )=0

however, the rate of change of the charge density ρ equals the divergence of the current density J Continuity equation

→𝛻 ∙ 𝐉=− 𝜕𝜕𝑡 (𝛻 ∙𝐃 )=− 𝜕

𝜕𝑡 ρ →𝛻 ∙𝐃=𝜌

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2D-MT Introduction


(Faraday)

(Ampere) 𝛻×𝐇=(𝜕𝒚 𝐻𝒛−𝜕𝒛 𝐸𝒚𝜕𝒛𝐻 𝒙−𝜕 𝒙𝐸𝒛𝜕𝒙𝐻 𝒚−𝜕 𝒚𝐸𝒙

)=(𝜕𝒚𝐻 𝒛−𝜕𝒛𝐻 𝒚𝜕𝒛𝐻 𝒙−𝜕𝒚𝐻 𝒙

)=σ (𝐸𝒙𝐸𝒚𝐸𝒛

)

𝛻×𝐄=(𝜕𝒚 𝐸𝒛−𝜕𝒛 𝐸𝒚𝜕𝒛 𝐸𝒙 −𝜕𝒙𝐸𝒛𝜕𝒙𝐸𝒚 −𝜕𝒚𝐸𝒙

)=(𝜕𝒚 𝐸𝒛−𝜕𝒛 𝐸𝒚𝜕𝒛 𝐸𝒙−𝜕𝒚𝐸𝒙

)=−𝑖𝜔𝜇 (𝐻𝒙𝐻 𝒚𝐻𝒛

)

TE-mode (E-Polarisation) TM-mode (B-Polarisation)

σ 𝐸 𝒙=𝜕𝒚𝐻 𝒛−𝜕𝒛𝐻 𝒚

𝜕𝒛 𝐸𝒙=−𝑖 𝜔𝜇𝐻 𝒚

𝜕𝒚 𝐸𝒙=𝑖𝜔𝜇𝐻 𝒛

−𝑖 𝜔𝜇𝐻 𝒙=𝜕𝒚𝐸𝒛−𝜕𝒛 𝐸𝒚

𝜕𝒛𝐻 𝒙=σ 𝐸𝒚

𝜕𝒚 𝐻𝒙=−σ 𝐸𝒛

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2D solution

σ 𝐸 𝒙=𝜕𝒚𝐻 𝒛−𝜕𝒛𝐻 𝒚

𝜕𝒛 𝐸𝒙=−𝑖 𝜔𝜇𝐻 𝒚

𝜕𝒚 𝐸𝒙=𝑖𝜔𝜇𝐻 𝒛

TE-mode (E-Polarisation)

𝜕𝒛 𝜕𝒛 𝐸𝒙=−𝑖𝜔𝜇𝜕𝒛𝐻 𝒚

𝜕𝒚𝜕 𝒚𝐸𝒙=𝑖𝜔𝜇𝜕 𝒚𝐻 𝒛

𝜕𝒚𝜕 𝒚𝐸𝒙+𝜕𝒛𝜕𝒛 𝐸𝒙=𝑖 𝜔𝜇 (𝜕𝑦𝐻 𝑧−𝜕𝑧 𝐻 𝑦 )=𝑖𝜔𝜇𝜎 𝐸𝑥

𝛻 ∙ (𝛻𝐸𝒙 )−𝑖𝜔𝜇𝜎 𝐸𝑥=0 Scalar PDE of

the electromagnetic (em) method magnetotelluric (mt)

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