Transcript ERTH2020_(Magnetotelluric) - Exploration Geophysics at the

ERTH2020
Introduction to Geophysics
The Electromagnetic (EM) Method
Magnetotelluric (MT)
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Magnetotelluric
combination of magnetic and telluric* methods
(Latin ‘tellūs’ = ‘earth’  “Earth current”)
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Magnetotelluric
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…other scientists Tikhonov (1950) and Rikitake (1951), Kato & Kikuchi (1950).
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Induction
Equivalent Circuits
I
I
•
I
DC Resistivity
•
Induced Polarisation
C
L
R
•
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Inductive EM
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DC / IP
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Magnetotelluric (Passive EM)
𝐻𝑧
𝐸𝑥
𝐸𝑦
𝐻𝑥
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𝐻𝑦
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Goal
𝜕2
𝐅 − iωμσ𝐅 = 0
2
𝜕𝑧
1.
1D diffusion equation
2.
→
𝑝=
2
≈ 500 𝑇𝜌𝑎
ωμσ
Skin Depth (Penetration Depth)
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Overview
DC Resistivity
Induced Polarisation
Method
Direct Electrical Connection (galvanic)
Passive EM
Active EM
No direct electrical connection (inductive)
Induced primary
magnetic field via loop
Electrical
potential
Decay of electrical
potential
Ratio of E and H
fields
Secondary magnetic
field
Resistivity
Resistivity &
Chargeability
Conductivity
Conductivity
Measured
Injected DC current via electrodes
Induced primary
magnetic field via
natural EM fields
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(or its decay)
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Contents
• Introduction
o
o
o
o
Maxwell Equations
Induction
Sources
Example
• EM theory
o
o
o
o
o
Divergence & Curl
Diffusion equation
1D Magnetotelluric
Skin Depth
Apparent Resistivity & Phase
• 2D MT Introduction
o Example
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Electromagnetic Induction
Ampere’s Law (1826)
𝛻 × 𝐇 = J + 𝜕t 𝐃
Faraday’s Law (1831)
𝛻 × 𝐄 = −𝜕t 𝐁
(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
J electric current density (A/m2)
H magnetic field intensity (A/m)
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e.g.
B magnetic induction (Wb/m2 or T)
E magnetic field intensity (V/m)
http://farside.ph.utexas.edu/teaching/302l/lectures/node70.html
http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html
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Electromagnetic Induction
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Simpson F. and Bahr K, 2005, p.18
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Electromagnetic Induction
Plane Wave
Source
𝐸𝑦
45∘
𝐻𝑥
𝐸𝑦
𝐻𝑥
Faraday’s Law
𝛻 × 𝐄 = −𝜕t 𝐁
Primary field
Ampere’s Law
𝛻×𝐇=J
𝐸𝑦
Ohm’s Law
𝐻𝑥
J = σ∙𝐄
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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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Simpson F. and Bahr K, 2005, p.3
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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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Simpson F. and Bahr K, 2005, p.3
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Magnetotelluric
Example 2D-MT resistivity model
• Locations of MT measurement
sites, Mount St Helens and nearby
Cascades volcanoes.
• 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.
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Hill et al., 2009
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Magnetotelluric
Example 2D-MT resistivity model after inversion
the conductivity anomalies are caused by the presence of partial melt
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Hill et al., 2009
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EM Theory
gradient
divergence
𝛻𝑈
𝛻∙𝐅
curl
𝛻×𝐅
𝜕𝑥 𝑈
𝜕𝑦 𝑈
𝜕𝑧 𝑈
𝜕𝑥
𝐹𝑥
𝜕𝑦 ∙ 𝐹𝑦
𝐹𝑧
𝜕𝑧
𝜕𝑥
𝐹𝑥
𝜕𝑦 × 𝐹𝑦
𝐹𝑧
𝜕𝑧
𝜕𝒙 𝐹𝒙 + 𝜕𝒚 𝐹𝒚 + 𝜕𝒛 𝐹𝒛
𝜕𝒚 𝐹𝒛 − 𝜕𝒛 𝐹𝒚
𝜕𝒛 𝐹𝒙 − 𝜕𝒙 𝐹𝒛
𝜕𝒙 𝐹𝒚 − 𝜕𝒚 𝐹𝒙
𝜕𝑥 𝑈 𝑖 + 𝜕𝑦 𝑈 𝑗 + 𝜕𝑧 𝑈 𝑘
(vector)
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(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)
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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)
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http://citadel.sjfc.edu/faculty/kgreen/vector/block2/del_op/node5.html & Wikipedia (Curl)
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EM Theory
Time-Domain Maxwell Equations (magneto-quasi-static)
(Faraday)
𝜕𝐁
𝛻×𝐄=−
𝜕𝑡
1
𝜕𝐇
→ 𝛻×𝐄=−
μ
𝜕𝑡
(Ampere)
𝛻×𝐇=𝐉
Note the use of the constitutive relations:
𝐁 = μ𝐇
𝐃 = ε𝐄
𝐉 = 𝜎𝐄
Also note that generally
1
→ 𝛻×𝐇=𝐄
𝜎
μ = μ 𝑥, 𝑦. 𝑧
𝜎 = 𝜎 𝑥, 𝑦. 𝑧
 first order, coupled PDEs
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EM Theory
Time-Domain Maxwell Equations (magneto-quasi-static)
 to uncouple, take the curl
(Faraday)
1
𝜕𝐇
𝛻×𝐄 =−
μ
𝜕𝑡
1
𝜕
→𝛻× 𝛻×𝐄=−
𝛻×𝐇
μ
𝜕𝑡
(Ampere)
1
𝛻×𝐇 =𝐄
𝜎
1
→𝛻× 𝛻×𝐇= 𝛻×𝐄
𝜎
 Second order, uncoupled PDEs
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EM Theory
Time-Domain Maxwell Equations (magneto-quasi-static)
 Second order, uncoupled PDEs
1
𝜕𝐄
→ 𝛻 × 𝛻 × 𝐄 = −𝜎
μ
𝜕𝑡
Plane wave source  sinusoidal time variation
𝐄 𝑡 = 𝐄0 𝑒 𝑖𝜔𝑡
1
𝜕𝐇
→ 𝛻 × 𝛻 × 𝐇 = −μ
𝜎
𝜕𝑡
𝐇 𝑡 = 𝐇0 𝑒 𝑖𝜔𝑡
where 𝜔 = 2𝜋𝑓 the angular frequency and
𝑖 = −1 the imaginary unit
•
Complex numbers arise e.g. from equations such as
𝑥 2 + 1 = 0 → 𝑥 2 = −1 → 𝑥 = −1 ≡ 𝑖.
•
Complex numbers can also
be written as 𝑧 = 𝑒 𝑖𝜃
•
Generally complex numbers have a real and
imaginary part and are written as 𝑧 = 𝑥 + 𝑖𝑦 where
𝑥 is the real part and 𝑦 the imaginary part.
•
𝑧 = 𝑒 𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃
•
Compact way to describe
waves
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EM Theory
Time-Domain Maxwell Equations (magneto-quasi-static)
 Second order, uncoupled PDEs
Plane wave source  sinusoidal time variation
1
𝜕𝐄
→ 𝛻 × 𝛻 × 𝐄 = −𝜎
= −𝑖𝜎𝜔𝐄
μ
𝜕𝑡
𝐄 𝑡 = 𝐄0 𝑒 𝑖𝜔𝑡
1
𝜕𝐇
→ 𝛻 × 𝛻 × 𝐇 = −μ
= −𝑖μ𝜔𝐇
𝜎
𝜕𝑡
𝐇 𝑡 = 𝐇0 𝑒 𝑖𝜔𝑡
where 𝜔 = 2𝜋𝑓 the angular frequency and
𝑖 = −1 the imaginary unit
•
Complex numbers arise e.g. from equations such as
𝑥 2 + 1 = 0 → 𝑥 2 = −1 → 𝑥 = −1 ≡ 𝑖.
•
Complex numbers can also
be written as 𝑧 = 𝑒 𝑖𝜃
•
Generally complex numbers have a real and
imaginary part and are written as 𝑧 = 𝑥 + 𝑖𝑦 where
𝑥 is the real part and 𝑦 the imaginary part.
•
𝑧 = 𝑒 𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃
•
Compact way to describe
waves
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EM Theory
Frequency Domain Diffusion Equations
 Second order, uncoupled PDEs
1
→ 𝛻 × 𝛻 × 𝐄 + 𝑖𝜔𝜎𝐄 = 0
μ
1
→ 𝛻 × 𝛻 × 𝐇 + 𝑖𝜔μ𝐇 = 0
𝜎
 General equations for inductive EM
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EM Theory
1D solution 
𝜎 = (piecewise) constant, 𝜇 ≡ constant
with vector identity 𝛻 × 𝛻 × 𝐅 = 𝛻 𝛻 ∙ 𝐅 − 𝛻 ∙ 𝛻 𝐅
=0
→ 𝛻 𝛻 ∙ 𝐄 − 𝛻 ∙ 𝛻 𝐄 = −iωμσ𝐄
→ 𝛻 2 𝐄 − iωμσ𝐄 = 0
→ 𝛻 𝛻 ∙ 𝐇 − 𝛻 ∙ 𝛻 𝐇 = −iωμσ𝐇
→ 𝛻 2 𝐇 − iωμσ𝐇 = 0
=0
 Diffusion Equations (Frequency Domain)
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EM Theory
1D solution 
𝜎 = (piecewise) constant, 𝜇 ≡ constant
𝛻∙𝐄=𝟎
𝛻∙𝐇=𝟎
Divergence of Faraday’s law
Divergence of Ampere’s law
→ 𝛻 ∙ 𝛻 × 𝐄 = −𝛻 ∙
→ 𝛻 ∙ 𝛻 × 𝐇 = 𝛻 ∙ 𝐉 = 𝛻 ∙ σ𝐄 = 0
𝛻 ∙ σ𝐄 = σ𝛻 ∙ 𝐄 + 𝐄 ∙ 𝛻σ = 0
→𝛻∙𝐁=0
𝜕𝐁
𝜕
=−
𝛻∙𝐁 =0
𝜕𝑡
𝜕𝑡
(Gauss law for magnetism,
i.e. no magnetic monopoles)
→ σ𝛻 ∙ 𝐄 = −𝐄 ∙ 𝛻σ
→𝛻∙𝐄=0
𝛻σ = 0
Proof 𝛻 ∙ 𝛻 × 𝐅 = 0
via Cartesian coordinates
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𝜕𝒙
𝜕𝒙
𝐹𝒙
𝛻 ∙ 𝛻 × 𝐅 = 𝜕𝒚 ∙ 𝜕𝒚 × 𝐹𝒚 =
𝐹𝒛
𝜕𝒛
𝜕𝒛
= 𝜕𝒙 𝜕𝒚 𝐹𝒛 − 𝜕𝒙 𝜕𝒛 𝐹𝒚 + 𝜕𝒚 𝜕𝒛 𝐹𝒙 − 𝜕𝒚 𝜕𝒙 𝐹𝒛
𝜕𝒚 𝐹𝒛 − 𝜕𝒛 𝐹𝒚
𝜕𝒙
𝜕𝒚 ∙ 𝜕𝒛 𝐹𝒙 − 𝜕𝒙 𝐹𝒛
𝜕𝒙 𝐹𝒚 − 𝜕𝒚 𝐹𝒙
𝜕𝒛
+ 𝜕𝒛 𝜕𝒙 𝐹𝒚 − 𝜕𝒛 𝜕𝒚 𝐹𝒙 = 0
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EM Theory
1D solution 
𝜎 = (piecewise) constant, 𝜇 ≡ constant
→ 𝛻 2 𝐅 − iωμσ𝐅 = 0 ⇔
𝜕2
𝐅 − iωμσ𝐅 = 0
𝜕𝑧 2
General solution for second-order PDE:
𝐅 = 𝐅1 𝑒 𝑖ωt−𝑞𝑧 + 𝐅2 𝑒 𝑖ωt+𝑞𝑧
decreases in
amplitude with z
𝑧<0
increases in
amplitude with z
 unphysical
𝑧>0
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Simpson F. and Bahr K, 2005, p.21
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EM Theory
1D solution 
𝜎 = (piecewise) constant, 𝜇 ≡ constant
𝐅 = 𝐅1 𝑒 𝑖ωt−𝑞𝑧
Taking the second derivative with respect to z
𝜕2
𝐅 = 𝑞2 𝐅1 𝑒 𝑖ωt−𝑞𝑧 = 𝑞2 𝐅
2
𝜕𝑧
→
𝑞=
𝑖ωμσ = 𝑖 ωμσ = 1 + 𝑖
↔
𝜕2
𝐅 − iωμσ𝐅 = 0
𝜕𝑧 2
ωμσ 2 =
ωμσ/2 + 𝑖 ωμσ/2
Real part
→
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1
𝑝=
=
ℜ𝔢 𝑞
2
ωμσ
Imaginary part
Skin Depth (Penetration Depth)
Simpson F. and Bahr K, 2005, p.22
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EM Theory
1D solution 
𝜎 = (piecewise) constant, 𝜇 ≡ constant
𝐅 = 𝐅1 𝑒 𝑖ωt−𝑞𝑧
Taking the second derivative with respect to z
𝜕2
𝐅 = 𝑞2 𝐅1 𝑒 𝑖ωt−𝑞𝑧 = 𝑞2 𝐅
2
𝜕𝑧
→
𝑞=
𝑖ωμσ = 𝑖 ωμσ = 1 + 𝑖
↔
𝜕2
𝐅 − iωμσ𝐅 = 0
𝜕𝑧 2
ωμσ 2 =
ωμσ/2 + 𝑖 ωμσ/2
Real part
→
1
𝑝=
=
ℜ𝔢 𝑞
2
ωμσ
Imaginary part
Skin Depth (Penetration Depth)
For angular frequency 𝜔 for a half-space with conductivity 𝜎
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Simpson F. and Bahr K, 2005, p.22
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EM Theory
1D solution 
1
𝑝=
=
ℜ𝔢 𝑞
→
𝑝=
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𝜇 ≡ constant𝜇 → 𝜇0 = 4𝜋 ∙ 10−7
𝜎 = (piecewise) constant,
2
ωμσ
2
=
4𝜋 ∙ 10−7 ωσ
Skin Depth (Penetration Depth)
107 2 𝑇𝜌
=
4𝜋 ∙ 2𝜋
107
≈
40
107
𝑇𝜌 ≈ 500 𝑇𝜌
4𝜋 2
Simpson F. and Bahr K, 2005, p.22 & http://userpage.fu-berlin.de/~mtag/MT-principles.html
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EM Theory
1D solution 
𝜎 = (piecewise) constant, 𝜇 ≡ constant
Real part
𝑞=
Imaginary part
ωμσ/2 + 𝑖ωμσ/2
and
1
𝑝=
=
ℜ𝔢 𝑞
2/ωμσ
The inverse of q is the Schmucker-Weidelt Transfer Function
1 𝑝
𝑝
𝐶 = = +𝑖
𝑞 2
2
..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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Simpson F. and Bahr K, 2005, p.22
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EM Theory
1D solution 
𝜎 = (piecewise) constant, 𝜇 ≡ constant
Schmucker-Weidelt Transfer Function
1 𝑝
𝑝
𝐶 = = +𝑖
with 𝑝 = 2/ωμσ
𝑞 2
2
We had with the general solution earlier
𝐸𝑥 = 𝐸1𝑥 𝑒 𝑖ωt−𝑞𝑧 →
𝜕𝐸𝑥
= −𝑞𝐸𝑥
𝜕𝑧
However Faraday’s law is
𝛻×𝐄
Therefore
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𝑦
𝜕𝐸𝑥
=
= −𝑖ωμ𝐻𝑦
𝜕𝑧
−𝑖ωμ𝐻𝑦 = −𝑞𝐸𝑥 → 𝐶 =
1
1 𝐸𝑥
1 𝐸𝑦
=
=−
𝑞 𝑖ωμ 𝐻𝑦
𝑖ωμ 𝐻𝑥
Simpson F. and Bahr K, 2005, p.22
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EM Theory
1D solution 
𝜎 = (piecewise) constant, 𝜇 ≡ constant
Schmucker-Weidelt Transfer Function
1
1 𝐸𝑥
1 𝐸𝑦
𝐶= =
=−
𝑞 𝑖ωμ 𝐻𝑦
𝑖ωμ 𝐻𝑥
• 𝐶 is calculated from measured 𝐸𝑥 and fields 𝐻𝑦 (or 𝐸𝑦 and 𝐻𝑥 ) .
• from 𝐶 the apparent resistivity can be calculated:
with q =
𝑖ωμσ → 𝑞
apparent resistivity
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2
𝑞2
1
= ωμσ → σ =
or ρ = 2 ωμ
ωμ
𝑞
→ ρ = 𝐶 2 ωμ
Simpson F. and Bahr K, 2005, p.22
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EM Theory
Apparent Resistivity and Phase
apparent resistivity
phase
𝜌𝑎 = 𝐶 2 ωμ
𝜙=
tan−1
ℑ𝔪 𝐶
ℜ𝔢 𝐶
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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Simpson F. and Bahr K, 2005, p.22
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EM Theory
Apparent Resistivity and Phase
𝜌𝑎 = 𝐶
2 ωμ
−1
𝜙 = tan
𝐶=
ℑ𝔪 𝐶
ℜ𝔢 𝐶
with
𝑝
𝑝
+𝑖
2
2
𝑝=
2/ωμσ
For a homogeneous half space:
ℑ𝔪 𝐶 = ℜ𝔢 𝐶 → 𝜙 = tan−1 1 = 45°
• 𝜙 < 45° → diagnostic of substrata in which resistivity increases with depth
• 𝜙 > 45° → diagnostic of substrata in which resistivity decreases with depth
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Simpson F. and Bahr K, 2005, p.26
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EM Theory
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Simpson F. and Bahr K, 2005, p.27
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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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Simpson F. and Bahr K, 2005, p.27
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2D-MT Introduction
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Simpson F. and Bahr K, 2005, p.30
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Numerical Modelling in 2D
2D solution 
𝜎 = 𝜎(𝑥, 𝑧), 𝜇 ≡ constant
𝐸𝒙 → TE-mode (E-Polarisation)
𝛻 ∙ 𝛻𝐸𝒙 − 𝑖𝜔𝜇𝜎𝐸𝑥 = 0
Numerical schemes, e.g.:
• Finite Differences
• Finite Elements
Escript Finite Element Solver
(Geocomp UQ)
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Numerical Modelling in 2D
σ = 10-14 S/m
σ = 0.1 S/m
σ = 0.01 S/m
Dirichlet boundary conditions via a single analytical 1D solution
applied Left and Right; Top & Bottom via interpolation
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Numerical Modelling in 2D
Electric Field (Real)
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Electric Field (Imaginary)
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Numerical Modelling in 2D
Apparent Resistivity at selected station (all frequencies)
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Numerical Modelling in 2D
σ = 10-14 S/m
σ = 0.04 S/m
σ = 0.1 S/m
σ = 0.4 S/m
σ = 0.001 S/m
# Zones = 71389
σ = 0.2 S/m
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# Nodes = 36343
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Numerical Modelling in 2D
Real Part 𝐸𝑥
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Imaginary Part 𝐸𝑥
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Numerical Modelling in 2D
Apparent
Resistivity
f = 1 Hz
r = 25 Ωm
Skin-depth
𝑝 ≃ 500 𝑇𝜌𝑎 ≃ 500 12 ≃ 1700
r = 10 Ωm
r = 2.5 Ωm
r = 1000 Ωm
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
ERTH2020
47
EM Theory
(Faraday)
𝜕𝐁
𝛻×𝐄=−
𝜕𝑡
→ 𝛻 ∙ 𝛻 × 𝐄 = −𝛻 ∙
→𝛻∙𝐁 = 0
𝜕𝐁
𝜕
=−
𝛻∙𝐁 =0
𝜕𝑡
𝜕𝑡
(Gauss law for magnetism)
via Cartesian coordinates
𝜕𝒚 𝐸𝒛 − 𝜕𝒛 𝐸𝒚
𝜕𝒙
𝜕𝒙
𝜕𝒙
𝐸𝒙
𝛻 ∙ 𝛻 × 𝐄 = 𝜕𝒚 ∙ 𝜕𝒚 × 𝐸𝒚 = 𝜕𝒚 ∙ 𝜕𝒛 𝐸𝒙 − 𝜕𝒙 𝐸𝒛 =
𝜕𝒙 𝐸𝒚 − 𝜕𝒚 𝐸𝒙
𝐸𝒛
𝜕𝒛
𝜕𝒛
𝜕𝒛
= 𝜕𝒙 𝜕𝒚 𝐸𝒛 − 𝜕𝒙 𝜕𝒛 𝐸𝒚 + 𝜕𝒚 𝜕𝒛 𝐸𝒙 − 𝜕𝒚 𝜕𝒙 𝐸𝒛 + 𝜕𝒛 𝜕𝒙 𝐸𝒚 − 𝜕𝒛 𝜕𝒚 𝐸𝒙 = 0
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EM Theory
𝜕𝐃
𝛻×𝐇=𝐉+
𝜕𝑡
𝜕𝐃
𝜕
→𝛻∙𝐉+𝛻∙
=𝛻∙𝐉+
𝛻∙𝐃 =0
𝜕t
𝜕𝑡
𝜕
→𝛻∙𝐉=−
𝛻∙𝐃
𝜕𝑡
(Ampere)
however, the rate of change of the charge density ρ equals the
divergence of the current density J  Continuity equation
𝜕
𝜕
→𝛻∙𝐉=−
𝛻∙𝐃 =− ρ
𝜕𝑡
𝜕𝑡
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→𝛻∙𝐃=𝜌
(Gauss law)
49
2D-MT Introduction
(Faraday)
𝜕𝒚 𝐸𝒛 − 𝜕𝒛 𝐸𝒚
𝛻 × 𝐄 = 𝜕𝒛 𝐸𝒙 − 𝜕𝒙 𝐸𝒛
𝜕𝒙 𝐸𝒚 − 𝜕𝒚 𝐸𝒙
𝜕𝒚 𝐸𝒛 − 𝜕𝒛 𝐸𝒚
𝐻𝒙
𝜕𝒛 𝐸𝒙
=
= −𝑖𝜔𝜇 𝐻𝒚
−𝜕𝒚 𝐸𝒙
𝐻𝒛
(Ampere)
𝜕𝒚 𝐻𝒛 − 𝜕𝒛 𝐸𝒚
𝛻 × 𝐇 = 𝜕𝒛 𝐻𝒙 − 𝜕𝒙 𝐸𝒛
𝜕𝒙 𝐻𝒚 − 𝜕𝒚 𝐸𝒙
𝜕𝒚 𝐻𝒛 − 𝜕𝒛 𝐻𝒚
𝜕𝒛 𝐻𝒙
=
−𝜕𝒚 𝐻𝒙
𝐸𝒙 → TE-mode (E-Polarisation)
𝐻𝒙 → TM-mode (B-Polarisation)
𝜕𝒛 𝐸𝒙 = −𝑖𝜔𝜇 𝐻𝒚
𝜕𝒛 𝐻𝒙 = σ𝐸𝒚
𝜕𝒚 𝐸𝒙 = 𝑖𝜔𝜇 𝐻𝒛
𝜕𝒚 𝐻𝒙 = −σ𝐸𝒛
σ𝐸𝒙 = 𝜕𝒚 𝐻𝒛 − 𝜕𝒛 𝐻𝒚
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𝐸𝒙
= σ 𝐸𝒚
𝐸𝒛
−𝑖𝜔𝜇𝐻𝒙 = 𝜕𝒚 𝐸𝒛 − 𝜕𝒛 𝐸𝒚
Simpson F. and Bahr K, 2005, p.28
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Numerical Modelling in 2D
2D solution 
𝜎 = 𝜎(𝑥, 𝑧), 𝜇 ≡ constant
𝐸𝒙 → TE-mode (E-Polarisation)
𝜕𝒛 𝐸𝒙 = −𝑖𝜔𝜇 𝐻𝒚
𝜕𝒚 𝐸𝒙 = 𝑖𝜔𝜇 𝐻𝒛
σ𝐸𝒙 = 𝜕𝒚 𝐻𝒛 − 𝜕𝒛 𝐻𝒚
𝜕𝒛 𝜕𝒛 𝐸𝒙 = −𝑖𝜔𝜇 𝜕𝒛 𝐻𝒚
𝜕𝒚 𝜕𝒚 𝐸𝒙 = 𝑖𝜔𝜇𝜕𝒚 𝐻𝒛
𝜕𝒚 𝜕𝒚 𝐸𝒙 + 𝜕𝒛 𝜕𝒛 𝐸𝒙 = 𝑖𝜔𝜇 𝜕𝑦 𝐻𝑧 − 𝜕𝑧 𝐻𝑦 = 𝑖𝜔𝜇𝜎𝐸𝑥
𝛻 ∙ 𝛻𝐸𝒙 − 𝑖𝜔𝜇𝜎𝐸𝑥 = 0
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Scalar PDE of 𝐸𝑥
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