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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 βtellΕ«sβ βearthβ βEarth currentβ)
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Magnetotelluric
β¦other scientists Tikhonov (1950) and Rikitake (1951), Kato & Kikuchi (1950).
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Induction
I
β’ DC Resistivity
I
β’ Induced Polarisation
I
β’ Inductive EM
R
CL
Equivalent Circuits
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DC / IP
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Magnetotelluric (Passive EM)
π» π§
π» π¦π» π₯
πΈπ₯
πΈπ¦
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Goal
π2
ππ§ 2π βiΟ ΞΌΟπ =0
βπ=β 2ΟΞΌΟ β500βπ ππ
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
β’ Introductiono Maxwell Equationso Inductiono Sourceso Example
β’ EM theoryo Divergence & Curlo Diffusion equationo 1D Magnetotellurico Skin Deptho Apparent Resistivity & Phase
β’ 2D MT Introductiono Example
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Electromagnetic Induction
Ampereβs Law (1826)
electric current density (A/m2)magnetic field intensity (A/m)
Faradayβs 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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Electromagnetic Induction
Simpson F. and Bahr K, 2005, p.18
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Primary field
Electromagnetic Induction
Plane Wave Source
π» π₯
πΈπ¦
Faradayβs Law
Ampereβs Law
Ohmβs Lawπ» π₯
πΈπ¦
πΈπ¦
π» π₯
45β
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Magnetotelluric
Sources
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Magnetotelluric
Simpson F. and Bahr K, 2005, p.3
Sources
Power spectrum: signal's power (energy per unit time) falling within given frequency bins
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Magnetotelluric
Simpson F. and Bahr K, 2005, p.3
Applications
β’ Mineral exploration
β’ Hydrocarbon exploration (oil/gas)
β’ Deep crustal studies
β’ Geothermal studies
β’ Groundwater monitoring
β’ Earthquake monitoring
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Magnetotelluric
Hill et al., 2009
Example 2D-MT resistivity model
β’ White and red dots show the locations of the magnetotelluric measurements; measurement sites shown in red were used for 2D inversion.
β’ The eastβwest line (red) shows the profile onto which these measurements were projected. The coloured area shows the region of high conductances. (=conductivity X thickness)
β’ The green-to-orange transition corresponds to a conductance of 3000 Siemens.
β’ 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
π»Γπ π»πcurlgradient
(ππ₯π π¦ππ§)Γ(πΉ π₯
πΉ π¦πΉ π§
)(ππ₯ππ π¦πππ§π )
(ππ πΉ πβππ πΉ πππ πΉ πβπ ππΉ πππ πΉ πβπ π πΉ π
)ππ₯π +ππ¦π +ππ§π
π» βπ divergence
(ππ₯π π¦ππ§) β(
πΉ π₯
πΉ π¦πΉ π§
)πππΉ π+ππ πΉ π+ππ πΉ π
(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):β’ Left: divergence > 0 since the vector field is βspreading outββ’ Centre: divergence = 0 everywhere since the vectors are not spreading out. β’ 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 βswirl aroundβ. (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.) β’ Left: curl > 0 (right-hand-rule thumb is up+)β’ Centre: curl = 0 everywhere since the field has no βswirlingβ. β’ 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)
π»Γπ=β ππππ‘
π»Γπ= π(Ampere)
Time-Domain Maxwell Equations (magneto-quasi-static)
Note the use of the constitutive relations:
π=ΞΌπ π=πππ=Ξ΅πβ 1ΞΌ π»Γπ=β ππ ππ‘
β 1π π»Γπ=π
first order, coupled PDEs
Also note that generally
ΞΌ=ΞΌ (π₯ , π¦ . π§ ) π=π (π₯ , π¦ .π§ )
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EM Theory
(Faraday)
(Ampere)
Time-Domain Maxwell Equations (magneto-quasi-static)
1ΞΌ π»Γπ=β ππ
ππ‘
1π π»Γπ=π
Second order, uncoupled PDEs
to uncouple, take the curl
βπ»Γ 1ΞΌ π»Γπ=β π ππ‘ (π»Γπ )
βπ»Γ 1π π»Γπ=(π»Γπ )
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EM Theory
Time-Domain Maxwell Equations (magneto-quasi-static)
Second order, uncoupled PDEs
βπ»Γ 1ΞΌ π»Γπ=βπ ππππ‘
βπ»Γ 1π π»Γπ=βΞΌ ππ ππ‘
π (π‘ )=π0ππ ππ‘
π (π‘ )=π0ππππ‘
Plane wave source sinusoidal time variation
where the angular frequency and the imaginary unit
β’ Complex numbers arise e.g. from equations such as .
β’ Generally complex numbers have a real and imaginary part and are written as where is the real part and the imaginary part.
β’ Complex numbers can also be written as
β’ Compact way to describe waves
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EM Theory
Time-Domain Maxwell Equations (magneto-quasi-static)
Second order, uncoupled PDEs
βπ»Γ 1ΞΌ π»Γπ=βπ ππππ‘ =β ππ ππ
βπ»Γ 1π π»Γπ=βΞΌ ππ ππ‘ =βπ ΞΌππ
π (π‘ )=π0ππ ππ‘
π (π‘ )=π0ππππ‘
Plane wave source sinusoidal time variation
where the angular frequency and the imaginary unit
β’ Complex numbers arise e.g. from equations such as .
β’ Generally complex numbers have a real and imaginary part and are written as where is the real part and the imaginary part.
β’ Complex numbers can also be written as
β’ Compact way to describe waves
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EM Theory
Frequency Domain Diffusion Equations
Second order, uncoupled PDEs
General equations for inductive EM
βπ»Γ 1ΞΌ π»Γπ+ππππ=0
βπ»Γ 1π π»Γπ+π π ΞΌπ=0
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EM Theory
1D solution
Diffusion Equations (Frequency Domain)
π»Γπ»Γπ =π» (π» βπ )β (π» βπ» )π with vector identity
βπ» (π» βπ )βΒΏ 0
β (π» βπ» )π=βiΟΞΌΟπ
βπ» (π» βπ )βΒΏ 0
β (π» βπ» )π=βiΟ ΞΌΟπ
βπ»2πβiΟΞΌΟπ=0
βπ»2πβiΟ ΞΌΟπ=0
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EM Theory
1D solution
π» βπ=π π» βπ=πDivergence of Ampereβs law
βπ» βπ»Γπ=βπ» β ππππ‘ =β πππ‘ (π» βπ)=0
βπ» βπ=0 (Gauss law for magnetism, i.e. no magnetic monopoles)
Divergence of Faradayβs law
βπ» βπ»Γπ=π» β π=π» β (Οπ )=0π» β (Οπ)=Ο π» βπ+π βπ» Ο=0βΟπ» βπ=βπ βπ»Ο
π» Ο=0βπ» βπ=0
via Cartesian coordinates
Proof
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π§<0
π§>0
EM Theory
1D solution
βπ»2π βiΟ ΞΌΟπ =0βπ =π 1ππΟt βππ§+π 2ππΟt+ππ§
General solution for second-order PDE:
decreases in amplitude with z
increases in amplitude with z unphysical
Simpson F. and Bahr K, 2005, p.21
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EM Theory
1D solution
π =π 1ππΟt βππ§
Taking the second derivative with respect to z
Simpson F. and Bahr K, 2005, p.22
π2
ππ§ 2π =π2π 1ππΟt βππ§=π2π β
π2
π π§ 2π βiΟΞΌΟπ =0
βπ=β πΟΞΌΟ=βπβΟ ΞΌΟ= (1+ π ) βΟΞΌΟ /2=βΟΞΌΟ /2+ πβΟΞΌΟ /2Real part Imaginary part
βπ=1βπ’ (π )
=β 2ΟΞΌΟ
Skin Depth (Penetration Depth)
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EM Theory
1D solution
π =π 1ππΟt βππ§
Taking the second derivative with respect to z
Simpson F. and Bahr K, 2005, p.22
π2
ππ§ 2π =π2π 1ππΟt βππ§=π2π β
π2
π π§ 2π β iΟΞΌΟπ =0
βπ=β πΟΞΌΟ=βπβΟ ΞΌΟ= (1+ π ) βΟΞΌΟ /2=βΟΞΌΟ /2+ πβΟΞΌΟ /2Real part Imaginary part
βπ=1βπ’ (π )
=β 2ΟΞΌΟ
Skin Depth (Penetration Depth)
For angular frequency for a half-space with conductivity
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EM Theory
1D solution π=( piecewise ) constant ,πβ‘constantπβπ0=4π β10β7
Simpson F. and Bahr K, 2005, p.22 & http://userpage.fu-berlin.de/~mtag/MT-principles.html
βπ=1βπ’ (π )
=β 2ΟΞΌΟ Skin Depth (Penetration Depth)
β 107
4 0
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EM Theory
1D solution
Simpson F. and Bahr K, 2005, p.22
π=βΟΞΌΟ /2+βπΟΞΌΟ /2
Real part Imaginary part
The inverse of q is the Schmucker-Weidelt Transfer Function
πΆ=1π=
π2 +π π2
andπ=1βπ’ (π )
=β2/ΟΞΌΟ
..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
Simpson F. and Bahr K, 2005, p.22
Schmucker-Weidelt Transfer Function
πΆ=1π=
π2 +π π2withπ=β2/ΟΞΌΟ
πΈπ₯=πΈ1π₯ππΟt βππ§β
ππΈπ₯
π π§ =βππΈπ₯
We had with the general solution earlier
Therefore
(π»Γπ )π¦=ππΈπ₯
π π§ =β πΟΞΌπ» π¦
However Faradayβs law is
βπΟΞΌπ» π¦=βπ πΈπ₯βπΆ= 1π= 1πΟΞΌ
πΈπ₯
π» π¦=β 1
πΟΞΌπΈπ¦
π»π₯
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EM Theory
1D solution
Simpson F. and Bahr K, 2005, p.22
Schmucker-Weidelt Transfer Function
πΆ= 1π= 1πΟΞΌ
πΈπ₯
π» π¦=β 1
πΟΞΌπΈ π¦
π» π₯
β’ is calculated from measured and fields (or and ) .β’ from the apparent resistivity can be calculated:
with q=βπΟΞΌΟβ|π|2=ΟΞΌΟβΟ=|π|2
ΟΞΌor Ο= 1
|π|2ΟΞΌ
βΟ=|πΆ|2ΟΞΌapparent resistivity
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EM Theory
Apparent Resistivity and Phase
Simpson F. and Bahr K, 2005, p.22
π=tanβ1(βπͺπΆβπ’πΆ )phase
ππ=|πΆ|2Ο ΞΌapparent 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
Simpson F. and Bahr K, 2005, p.26
For a homogeneous half space:
β’ diagnostic of substrata in which resistivity increases with depth
β’ diagnostic of substrata in which resistivity decreases with depth
ππ=|πΆ|2Ο ΞΌ π= tanβ1(βπͺπΆβπ’πΆ ) πΆ=π2 +π π2
withπ=β2/ΟΞΌΟ
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EM Theory
Simpson F. and Bahr K, 2005, p.27
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2D-MT Introduction
Simpson F. and Bahr K, 2005, p.27
For this 2-D case, EM fields can be decoupled into two independent modes: β’ E-fields parallel to strike with induced B-fields perpendicular to strike and in
the vertical plane (E-polarisation or TE mode).β’ 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
Simpson F. and Bahr K, 2005, p.30
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Numerical Modelling in 2D
2D solution
TE-mode (E-Polarisation)
π» β (π»πΈπ )βππππ πΈπ₯=0
Numerical schemes, e.g.:β’ Finite Differences β’ 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
Ο = 10-14 S/m
Ο = 0.1 S/m
Ο = 0.01 S/m
Numerical Modelling in 2D
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Electric Field (Imaginary) Electric Field (Real)
Numerical Modelling in 2D
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Apparent Resistivity at selected station (all frequencies)
Numerical Modelling in 2D
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Ο = 0.4 S/mΟ = 0.001 S/m
Ο = 10-14 S/m
Ο = 0.2 S/m
Ο = 0.1 S/mΟ = 0.04 S/m
# Zones = 71389
# Nodes = 36343
Numerical Modelling in 2D
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Real Part Imaginary Part
Numerical Modelling in 2D
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Apparent Resistivity
f = 1 Hz
Numerical Modelling in 2D
r = 2.5 Ξ©mr = 1000 Ξ©m
r = 10 Ξ©mr = 25 Ξ©mSkin-depth
r = 2 Ξ©m
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References
Simpson F. and Bahr K.: βPractical magnetotelluricsβ, 2005, Cambridge University Press
Cagniard, L. (1953) Basic theory of the magneto-telluric method of geophysical prospecting, Geophysics, 18, 605β635
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)
βπ» βπ»Γπ=βπ» β ππππ‘ =β πππ‘ (π» βπ)=0
π»Γπ=β ππππ‘
βπ» βπ=0 (Gauss law for magnetism)
via Cartesian coordinates
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π»Γπ= π+ ππππ‘(Ampere)
EM Theory
βπ» β π=β πππ‘ (π» βπ )
(Gauss law)
βπ» β π+π» β πππ t =π» β π+ πππ‘ (π» βπ )=0
however, the rate of change of the charge density Ο equals the divergence of the current density J Continuity equation
βπ» β π=β πππ‘ (π» βπ )=β π
ππ‘ Ο βπ» βπ=π
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2D-MT Introduction
Simpson F. and Bahr K, 2005, p.28
(Faraday)
(Ampere) π»Γπ=(ππ π»πβππ πΈππππ» πβπ ππΈππππ» πβπ ππΈπ
)=(πππ» πβπππ» ππππ» πβπππ» π
)=Ο (πΈππΈππΈπ
)
π»Γπ=(ππ πΈπβππ πΈπππ πΈπ βπππΈππππΈπ βπππΈπ
)=(ππ πΈπβππ πΈπππ πΈπβπππΈπ
)=βπππ (π»ππ» ππ»π
)
TE-mode (E-Polarisation) TM-mode (B-Polarisation)
Ο πΈ π=πππ» πβπππ» π
ππ πΈπ=βπ πππ» π
ππ πΈπ=ππππ» π
βπ πππ» π=πππΈπβππ πΈπ
πππ» π=Ο πΈπ
ππ π»π=βΟ πΈπ
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Numerical Modelling in 2D
2D solution
Ο πΈ π=πππ» πβπππ» π
ππ πΈπ=βπ πππ» π
ππ πΈπ=ππππ» π
TE-mode (E-Polarisation)
ππ ππ πΈπ=βππππππ» π
πππ ππΈπ=ππππ ππ» π
πππ ππΈπ+ππππ πΈπ=π ππ (ππ¦π» π§βππ§ π» π¦ )=ππππ πΈπ₯
π» β (π»πΈπ )βππππ πΈπ₯=0 Scalar PDE of