C : Theory of electromagnetic (EM) field propagation in the Earth

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18 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013

1
C : Theory of electromagnetic (EM) field propagation in the Earth


In the previous section we saw that the resistivity of the crust and upper mantle varies by
orders of magnitude. The resistivity depends on the composition and temperature and is
especially sensitive to the presence of high conductivity phases such as melt, water and
graphite. It is relatively simple to measure the resistivity of a rock sample in the lab.

Resistivity can also be measured in situ through well logging. However this approach is
restricted to the rocks with a few meters of a borehole. For a review of well-logging, see
Ellis and Singer, Well logging for Earth Scientists, Springer, 2008

Electrical methods were introduced in GEOP325 where DC (direct current) methods
were introduced and give measurements away from boreholes. These methods use electric
current with zero frequency to measure the resistivity (conductivity), so the physics is
relative simple. However, their depth of exploration is limited as it is related to the
spacing of the electrodes used in the survey.

To measure the resistivity of most of the crust and mantle we must use electromagnetic
geophysics. This requires an understanding of how EM signals propagate in the Earth. We
will develop this understanding in the following sections, beginning with Maxwell’s
equations are developing solutions for the EM signals that travel in the Earth. This
requires that we also consider the other properties that characterize the subsurface, and
which influence the way that EM signals propagate (permittivity and permeability).


C1 Maxwell’s equations


Quantities describing the electromagnetic field

E = electric field strength (V/m)

D = displacement, electric flux density

J = electric current density (A/ m
2
)

B = magnetic flux density

H = magnetic field strength

P = polarization

Q = electric charge density

Quantities describing the properties of the Earth

σ = electrical conductivity

ε = dielectric permittivity

μ = magnetic permeability
Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013

2
Constitutive relations
These quantities are linked through the constitutive relations as follows

J = σ E; B = μ H; D = ε E + P

In these equations the terms on the right (E, H ) can be thought as inputs. The equations
then predict the response (output) based on the Earth properties.

Earth properties in more detail

(a) Electrical Conductivity (σ) : Measures the ability of a material to conduct
electric current (see 424B for details)

(b) Magnetic permeability (μ) : On the atomic scale, atoms can behave as magnetic
dipoles. These dipoles interact with an applied magnetic field to give a
characteristic flux density in the material. Usually consider the relative permeability,
μ
r
, which is defined as:

μ = μ
0
μ
r

where μ
0
is the permeability of free space ( μ
0
= 4 π x 10
7−
H/m). Can also
consider the magnetic susceptibility

μ
r
= 1+k

There are two types of behaviour.

Diamagnetism : The dipoles align anti-parallel to an applied magnetic field (H).
Magnetic flux density represents the combination of the applied magnetic field and
the effect of the material. All materials exhibit this property, and it can be
considered as EM induction in a circuit that comprises the electrons orbiting the
nucleus of the atom.

Magnetic susceptibility : k < 0
Relative permeability: μ
r
< 1

Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013

3
Paramagnetism : In this case the atom has a net magnetic moment due to the
configuration of unpaired electrons in the outer shell. The atoms will align parallel to an
applied magnetic field and reinforce the applied magnetic field. The paramagnetic
response will be much stronger that the diamagnetic response.

Magnetic susceptibility : k > 0
Relative permeability: μ
r
> 1

Variations in μ
r
are usually encountered only in major ore bodies. Thus for the rest
of this class, we will assume that μ = μ
0
(unless there is a good reason).






(c) Dielectric permittivity (ε)



On the atomic scale, some
molecules have a overall electric
dipole moment (e.g. H
2
O). These
molecules will align with an
applied electric field (E) to generate
a displacement (D).

D = ε E

Usually express the permittivity as a relative permittivity (ε
r
) that is defined as:

ε = ε
r
ε
0

where ε
0
= 8.85 x 10
12−
F/m and is defined as the permittivity of free space.

A material that lacks polar molecules will have ε
r
= 1. Water is a polar molecule and
has ε
r
= 80 and the quantity of water in soil or rock will have a major effect on the
overall permittivity (see 424G1 on ground penetrating radar for more detail). In fact
Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013

4
the velocity of a radar wave in soil depends primarily on the water content. The
velocity can be used to remotely sense the water content.

Note that permittivity is only important when displacement current must be
considered. We will see that at the low frequencies used in EM geophysics, that
displacement current is rarely significant so we will assume that ε = ε
0

However in GPR the frequencies are high enough that displacement current is
significant.


Summary
In general, σ, μ and ε all vary with position in the earth, and also with direction. In the
most general case they should be expressed as tensors. However to simplify the analysis
in this class, we will start with the assumption that they are scalars and that they vary
only with position in the Earth.




Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013

5
Maxwell’s equations


(1) Coulombs Law

.∇
E = ρ/ε
0


S
E. dS = ρ/ε
0

Quantifies how electric fields are generated by
electric charges.


(2) Continuous magnetic flux

.∇
B = 0

S
B.dS = 0
Based on the observation that magnetic flux
is continuous, and that magnetic monopoles
do not exist in isolation.






(3) Ampere’s Law

∧∇
B = μ J

B . dl = μ

S
J.ds
Constant current > Magnetic field

Based on the observation that an electric
current, J, generates a magnetic field, B.
This was quantified by Ampère through a
study of the forces between current carrying
conductors.

Note that this includes only conduction
current on right hand side








(4) Faraday’s Law

∧∇
E = -
t∂
∂B


E.dl = -
t




S
B.dS
Changing magnetic field > electric field

Equation (4) : After the discovery of
Ampère’s Law, the reverse effect was
looked for. i.e. a magnetic field producing
an electric current. First quantified by
Michael Faraday in 1831 as magneto-
electric induction. The negative sign comes
from Lenz’s Law (simply conservation of
energy). Faraday’s Law states that the
induced e.m.f. (voltage) is proportional to
the rate of change of magnetic flux.

V =

E.dl














Disp
l



















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∧∇
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Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013

7
C2: Boundary conditions


In this section, the goal is to understand how EM signals will travel in the Earth. We will
solve Maxwell’s equations for each region in the area we are investigating. To connect
these solutions, we will need to link them at the boundaries between regions with
different properties (conductivity, permeability, permittivity). In this section we will
develop the boundary conditions needed to do this.

Consider the horizontal interface between two half-spaces, with conductivity σ
1
and σ
2

Assume that both half-spaces have the same permeability (μ) and permittivity (ε).

1
x
E
2
x
E
= horizontal electric fields

1
z
E
2
z
E
= vertical electric fields

1
y
B
2
y
B
= horizontal magnetic field

1
z
B
2
z
B
= vertical magnetic fields



(1) Electric field parallel to interface




Consider a rectangular loop ‘L’ of length
Δx
and height
Δz
. The loop encloses the surface,
‘S’.

From Faradays Law
∫∫


−= dSB
t
dlE
L
x
..

xExEdlE
xx
L
x
Δ−Δ=

21
.

As
0→Δz

0.→

S
dSB

Thus
0
21
=Δ−Δ xExE
xx


and
21
xx
EE =



E
parallel to interface is continuous
Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


8
(2) Magnetic field parallel to interface




Consider a rectangular loop ‘L’ of length
Δy
and height
Δz
. The loop encloses a surface,
‘S’.

From Ampere’s Law
∫∫
=
SL
dSJdlB
..μ
yByBdlB
yy
L
Δ−Δ=

21
.


As
0→Δz
, 0.→

S
dSJ

Thus
0
21
=Δ−Δ yByB
yy

and
21
yy
BB =


B
parallel to interface is continuous




(3) Electric field normal to the interface





Consider a small cylinder with surface area ‘
S
’ on the top and bottom. As electric current
flows across the boundary, a steady state is established with equal amounts of charge
entering and leaving the cylinder. Thus:

21
zz
SJSJ =
;
21
zz
JJ =
;
2
2
1
1
zz
EE σσ =



E
normal to interface is discontinuous

Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


9
This shows that the electric field normal to the interface is
discontinuous
. This is due to
a layer of electric charge on the interface. The
surface charge density
can be calculated
as follows:


=
S
q
dSE
ε
.


where
q
is the
total electric charge
inside the cylinder


−=−=
S
zzz
SESESEdSE )1(.
2
1
121
σ
σ


Thus the surface charge density (Q
s
) can be defined as
ε
σ
σ
)1(
2
1
1
−==
zS
E
S
q
Q



(4) Magnetic field normal to the interface







=
S
dSB
0.

=−=
S
zz
SBSBdSB
0.
21

21
zz
BB =


B
normal to interface is continuous













Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


10

C3 : Diffusion of EM fields in a conductive medium


C3.1 General solution of Maxwell’s equations



Consider an electromagnetic wave that is travelling through a region characterized by
conductivity (σ), magnetic permeability (µ) and dielectric permittivity (ε). The local
electric charge density is Q
(x,y,z)


The electric and magnetic fields vary in space and time as
E
(
x,y,z,t
) and
B

(x,y,z,t
) and are
described by Maxwell’s equations :

׏ ר ۰ ൌ
μ
۸ ൅
μ
ε
ப۳
ப୲
(1)

׏ ר ۳ ൌ െ
ப۰
ப୲
(2)

The ultimate goal is to eliminate
B
and obtain an equation for
E

The constitutive relation gives us
۸ ൌ ߪ۳
with Amperes Law becoming

׏ ר ۰ ൌ
μσ
۳ ൅
μ
ε
ப۳
ப୲
(3)

Note that the first term on the right hand side represents
conduction current
, while the
second represents
displacement current
. Taking the curl of Faradays Law give us
 
׏ ר

׏ ר ۳

ൌ െ

ப୲
ሺ׏ ר ۰ሻ    ሺ4ሻ 

Substituting (3) into (4) and using the vector identity

׏ ר

׏ ר ۳

ൌ ׏

׏.۳

െ׏

۳
(5)


we can show that

Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


11
׏

׏.۳

െ׏

۳ ൌ െ

ப୲

μσ
۳ ൅
μ
ε
ப۳
ப୲

(6)

Now if it assumed that the Earth properties (conductivity, permeability, permittivity) do
not vary with time, then we can write.

׏

׏.۳

െ׏

۳ ൌ െ
μσ
ப۳
ப୲

μ
ε


۳
ப୲

(7)

Coulombs Law states that
 
׏.۳ ൌ


           
(8)
 

Assumption 1 :
Assume there are no free electric charges (Q =0). Note that this will not
be true if electric current crosses boundaries between regions of differing resistivity.
This requires that:

 
׏.۳ ൌ 0   ሺ9ሻ


which simplifies (7) to give

׏

۳ ൌ
μσ
ப۳
ப୲

μ
ε


۳
ப୲

(10)

We now have a second order differential equation for
E
(
x,y,z,t
) where the time-variation
can be assumed to be completely general. This represents the time-domain.


Limiting cases

(1)

If the conduction current term is much larger than the displacement current term,
then (10) simplifies to a diffusion equation
 
׏

۳ ൌ µσ
ப۳
ப୲
(11)
 

(2)

However if the displacement current term is much larger than the conduction
current term, then (10) simplifies to a wave equation.
׏

۳ ൌ
μ
ε


۳
ப୲

(12)
Comparison with the standard wave equation shows that the wave velocity is

ܿ ൌ


ఓఌ
(13)

Which for free space values of µ and ε gives c = 3 x 10
8
m/s, showing that this is a
radio wave moving at the speed of light.



Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


12
C3.2 Transform to the frequency domain


Need to determine which of these limiting cases will apply for EM signals travelling in
the Earth.

Assumption 2
: Analysis is simplified if analysis is transformed into the frequency
domain i.e. we assume that the electric and magnetic fields have a
harmonic time
variation
at an angular frequency
ω.
The angular frequency and frequency are related as
߱ ൌ 2ߨ݂
. This allows variables to be separated as

۳ሺݔ,ݕ,ݖ,ݐሻ ൌ ۳


ݔ,ݕ,ݖ

݁
ି௜ఠ௧
(14)

where i


െ1
. Substitution of (14) into (10) gives
 
׏

۳

ൌ െi߱
μσ
۳

൅߱

μ
ε۳

       
(15)
 

Note that the first term on the right hand side represents
conduction current
, while the
second represents
displacement current
. To understand which term will dominate,
consider some numerical values for two common types of geophysical exploration.

The ratio of displacement current to conduction current can be written as
ܴ ൌ ߱ߝ/ߪ


Magnetotellurics
: This exploration method uses a low frequency EM
signal at f = 1 Hz in a region where σ = 0.01 S/m, so assuming a free space
value of dielectric permittivity, this gives R = 5.56 x 10
-9
showing that
conduction current is dominant and the signal will propagate by
diffusion
.

Ground-penetrating radar (GPR)
: A 1 GHz signal travels in glacial ice
with σ = 10
-5
S/m giving R = 5561. Displacement current is dominant and
the signal travels as a EM wave.

Assumption 3
: This shows that for all practical EM applications in the Earth,
displacement current can be ignored
. Thus we will use the pre-Maxwell form of
Ampere’s Law, with the major contribution by JCM ignored!




C3.3 Plane, polarized EM signal travelling vertically in the Earth



Next will consider a
simplified geometry
of an EM signal travelling vertically in the
Earth. When the displacement current is ignored, can write:

׏

۳

ൌ െi߱
μσ
۳


(16)


This can be expressed in component form (less concisely) as





డ௫






డ௬






డ௭

൅݅߱ߤߪܧ

ൌ 0
(17a)
Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


13




డ௫






డ௬






డ௭

൅݅߱ߤߪܧ

ൌ 0
(17b)




డ௫






డ௬






డ௭

൅݅߱ߤߪܧ

ൌ 0
(17c)

where
۳

ൌ ቌ
ܧ

ܧ

ܧ





Assumption 4
: If it assumed that the electric field is
polarized
in the x-direction, then
can write
۳

ൌ ൭
ܧ

0
0

and equations (17b) and (17c) can be ignored.
Thus we need to solve the second-order partial differential equation:





డ௫






డ௬






డ௭

൅݅߱ߤߪܧ

ൌ 0
(18)


Assumption 5
: Now if the wave is assumed to be
planar
, then it will not vary in the x
and y directions, which can be expressed as
߲/߲ݔ ൌ 0    
and
   ߲/߲ݕ ൌ 0
. This further
simplifies (18) to an ordinary differential equation





ௗ௭

൅݅߱ߤߪܧ

ൌ 0
(19)

We can seek a trial solution of the form
ܧ


ݖ

ൌ ܣ݁
௞௭
(20)

where A and k are constants to be determined. Substitution into (19) gives
 
݇

ܣ ൅݅߱ߤߪܣ ൌ 0        
(21)

which gives
݇ ൌ

െ݅߱ߤߪ
 
(22)

Since there are two possible solutions,

݇ ൌ േሺ1 െ݅ሻ

ఠఓఙ

 
(23)

we need to write a general form of the solution as:

ܧ


ݖ

ൌ ܣ

݁
ሺଵି௜ሻට
ഘഋ഑


൅ܣ

݁
ିሺଵି௜ሻට
ഘഋ഑


(24)

which can be expanded as
ܧ


ݖ

ൌ ܣ

݁

ഘഋ഑


݁
ି௜ට
ഘഋ഑


൅ܣ

݁
ିට
ഘഋ഑


݁
௜ට
ഘഋ഑


(25)

Note that each term consists of an
exponential
function and an
oscillatory
function.
Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


14

We need to determine
ܣ

and
ܣ

by applying appropriate boundary conditions.


Boundary condition 1

This solution must remain bounded as
ݖ ՜∞
. This is only possible if
ܣ

= 0.


Boundary condition 2

Can specify that at
ݖ ൌ 0 

ܧ


0

ൌ  ܧ


, which requires
ܣ

ൌ  ܧ




Thus the solution can be written as :
ܧ


ݖ

ൌ ܧ


݁
ିට
ഘഋ഑


݁
௜ට
ഘഋ഑


(26)

This expression has both
real
and
imaginary
parts, both of which are solutions.

Considering the real part, we can write:
ܧ


ݖ

ൌ ܧ


݁
ିට
ഘഋ഑


cos ൬

ఠఓఙ

ݖ൰
(27)


C3.4 Skin depth


Consider the
modulus
of the electric field
|
ܧ


ݖ
ሻ|

|
ܧ


|
݁
ିට
ഘഋ഑


(28)

This decays
monotonically
as z increases. The depth at which
|ܧݔ ሺݖሻ |
has decreased
from the value at
ݖ ൌ 0
by a factor of
1/e
is defined as the
skin depth
(δ)

|



௭ୀఋ
ሻ|
|



|



ൌ  ݁
ିට
ഘഋ഑


(29)

which requires

 

ఠఓఙ

ߜ ൌ 1
(30)
This can be arranged to give
ߜ ൌ


ఠఓఙ
(31)

Substituting for
ߤ ൌ ߤ

ൌ 4ߨ10
ି଻
gives an expression for the skin depth in
metres
as

ߜ ൌ
ହ଴ଷ

ఙ௙
(32)

This is the
most important equation
derived in Geophysics 424



Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


15
Attenuation of a plane EM wave



Solid line : Real (in phase component)
Dashed line : Imaginary (out of phase component)



Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


16





What three things are incorrect in this figure?

You should be able to answer after section D has been completed.

(1)


(2)


(3)


Geophysics 424 – January 2013
© Martyn Unsworth, University of Alberta, 2013


17
Final note on electrical and magnetic Earth properties

The electrical and magnetic properties of the Earth are described by
three
parameters:

Electrical conductivity (resistivity)
Magnetic permeability
Dielectric permittivity

Variations in the
conductivity
are usually much larger than variations in the other two
parameters, often orders of magnitude. This is why our analysis of MT focuses on
conductivity variations.

Variations in subsurface
magnetic permeability
will influence how EM signals travel in
the Earth. If these variations are ignored, our solutions to Maxwell’s’ equations will be
approximate.

What about variations in
dielectric permittivity
? At the low frequencies used in MT, we
showed that displacement current is much smaller than conduction current. The only term
where permittivity was found was in the displacement current term. Thus at the low
frequencies used in MT we are completely
insensitive
to variations in permittivity. No
errors are caused by assuming the permittivity has the free space value at low frequency.