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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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Electromagnetic Induction
Simpson F. and Bahr K, 2005, p.18
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Primary field
Electromagnetic Induction
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
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
• 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)
Time-Domain Maxwell Equations (magneto-quasi-static)
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
Time-Domain Maxwell Equations (magneto-quasi-static)
Second order, uncoupled PDEs
→𝛻× 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
Time-Domain Maxwell Equations (magneto-quasi-static)
Second order, uncoupled PDEs
→𝛻× 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
Second order, uncoupled PDEs
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
Simpson F. and Bahr K, 2005, p.21
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EM Theory
1D solution
𝐅=𝐅1𝑒𝑖ωt −𝑞𝑧
Taking the second derivative with respect to z
Simpson F. and Bahr K, 2005, p.22
𝜕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
Skin Depth (Penetration Depth)
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EM Theory
1D solution
𝐅=𝐅1𝑒𝑖ωt −𝑞𝑧
Taking the second derivative with respect to z
Simpson F. and Bahr K, 2005, p.22
𝜕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
Skin Depth (Penetration Depth)
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
Simpson F. and Bahr K, 2005, p.22
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
Simpson F. and Bahr K, 2005, p.22
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
Simpson F. and Bahr K, 2005, p.22
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
Simpson F. and Bahr K, 2005, p.22
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
Simpson F. and Bahr K, 2005, p.26
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
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: • 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
Simpson F. and Bahr K, 2005, p.30
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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
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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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
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 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
Simpson F. and Bahr K, 2005, p.28
(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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Numerical Modelling in 2D
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