Roy K.K. Potential theory in applied geophysics

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Electromagnetic Theory (Vector Potentials)

In this chapter basics of electromagnetics are introduced. Nature of a wavelet,

characteristics of waves, its amplitude and phase and their characteristic

equations, elliptic polarization, mutual inductance and their complex natur e,

nature of wave propagation, attenuation of waves, skin depth, propagation

constant, Maxwell’s electromagnetic equations, Helmholtz electromagnetic

wave equations, Hertz and Fitzerald vector potentials, boundary con d i tions

in electromagnetics and poynting vector are discussed.

12.1 Introducti on

In this chapter and in the next we shall discuss a few important aspects of

electromagnetic theory or the theory of time varying electromagnetic ﬁelds.

The treatments are based on vector potentials. Therefore the mathematical

treatments are essentially based on Maxwell’s electromagnetic equations and

Helmholtz electromagnetic wave equations. The treatments have typical geo-

physical orientation.

Magnetic ﬁeld originates due to ﬂow of charge or ﬂow of current. When

this current becomes time varying in a source, electromagnetic ﬁeld originates

and electromagnetic waves start propagating. These electromagnetic waves

are transverse waves, i.e., the direction o f vibration of a particle in a medium,

through which the wave passes, is at right angles to the direction of propaga-

tion (Fig. 12.1).

Electroma gnetic waves are similar to light waves and shear elastic waves

so far as the nature of propagation is concerned. Electromagnetic waves can

travel through any medium like air, gas, vacuum but not through a perfect

conductor. Through vacuum EM waves travel at a speed of light. In the very

broad spectrum of electromagnetic waves starting from gamma rays to radio

waves and beyond, visible light waves occupy only a narrow band VIBGYOR.

Velocity of electromagnetic waves get signiﬁcantly retarded by several order

350 12 Electromagnetic Theory (Vector P otentials)

Fig. 12.1. Transverse nature of electromagnetic wavelets orthogonal to the direction

of propagation

of magnitude when it travels through a medium of ﬁnite electrical conductiv-

ity. Even then the velocity of electromagnetic waves remain signiﬁcantly more

than velocity of sound and elastic longitudinal P waves. Any wire loop or a

linear conductor carrying alternating or time varying current can become a

source of electromagnetic waves. As early as 1912 Faraday has demonstrated

through his famous experiment on electromagnetic induction that when a bar

magnet approaches a coil (circular or any other shape), current ﬂows throug h

the coil and this current deﬂects a galvanometer (Chap. 5). Magnitude of

induced e.m.f. is prop or tional to the rate of change of number of lines of forces.

Magnitude of current depends on speed of forward and backward movement

to and from a coil. That is how electricity and magnetism are attached to each

other both in magnetostatics and in time varying electromagnetic ﬁelds. So

whenever time varying magnetic ﬁeld is cut by a conductor, i.e. a body of ﬁnite

electrical conductivity, induced currents or eddy currents are generated within

these conductors. All the important terminologies, viz. wavelength, velocity,

frequency, time period, amplitude, phase, wave number are used to charac-

terise an electromagnetic or time varying ﬁeld. Magnitude of these induced

currents o r eddy currents are very much dependent upon the frequency of the

electromagnetic signals and the conductivity of the medium through which

an electromagnetic wave travels. So higher the conductivity and/or frequency

higher will be the eddy currents and higher will be the attenuation of EM sig-

nals according to the Lenz’s law of electromagnetic indu ction. So electromag-

netic waves travel through vacuum almost unattenuated. it travels through air

with very little attenuation but gets attenuated signiﬁcantly when it travels

through a co n ductor. According to Lenz’s l aw, the eddy current opposes the

12.1 Introduction 351

incoming electromagnetic waves. So higher the conductivity and/or frequency

of the em signal, stronger will be the eddy currents, stronger will be the force

of opposition, greater will be the attenuation.

Electromagnetic waves are used in many walks of life such as information

technology, space technology, communication technology, defence, sp orts, nav-

igation etc. The details are beyond the scope of this book.

Scientists and engineers of ma ny disciplines viz. electrical communication,

electrical p ower, aerospace engineering, physics, geophysics etc use electro-

magnetic waves. The role of electromagnetic waves in geophysics is very much

special in the sense that geophysicists deal with the earth, a spherical solid

body of radius nearly 6370 km and the upper atmosphere upto the height of

magnetosphere/ magnetopause level, 3 to 6 times the radius of the earth, in

the outer space and the space above. Electrical Communication Engineers and

space technologists deal with chips, microchips, semi conductor s, dielectrics or

machines for very long distance propagation, reﬂection, refraction, diﬀraction

and scattering of electromagnetic waves.

Geophysicists probably use the widest frequency band of the electromag-

netic waves. Their necessity forced them to do so. To study the earth from

electrical conductivity point of view, one needs very large depth of penetration

of the electro magnetic waves. Paciﬁc or Atlantic Ocean with 4 to 5 km of saline

water cover, absorbs all the high frequency electromagnetic signals propa-

gated from the upper atmosphere towards the earth. So the starting frequency

of the signal is cycles/12 hours over ocean surface. Cycles/day cycles/week,

cycles/month, cycles/year, cycles/11 year are the frequency ranges for o cean

bottom electromagnetic studies. To have larger depth of penetration of elec-

tromagnetic waves geop hysicists went toward lower and lower frequencies or

longer and longer periods. In exploration geophysics, the requirement for

depth of penetration is considerably less for shallow ground water and mineral

exploration problems. For oil exploration problem maximum depth of inves-

tigation should be of the order of 5 to 6 kms from the surface and that too in

sedimentary rocks. For solid earth geophysics, when we try to reach the centre

of the earth, one needs very very long perio d magnetic signals available only

from the p ermanent geomagnetic observatories. It was observed that resistiv-

ity of sedimentary rock is of the order of 1 to 30 ohm-m. Therefore 4 to 5 kms

of sedimentary ro cks as in Cambay basin of western India, Bengal basin in

eastern India can considerab ly absorb the high frequency signals and atten-

uate the longer period signals. Hence for deeper probing, one should avoid

soft rocks and should try to take measurements over hard rock exposures.

Scientists search for granite windows to see beyond Moharovicic discontinu-

ity, lithosphere – asthenosphere boundary, olivine-spinel transition zone and

beyond. Highly resistive granites allow the high frequency signals to penetrate

deep inside.

There are other problems b eyond upper crustal level. Depth of penetration

is controlled by skin depth (discussed later). Shallow magma chambers in a

high heat ﬂow areas, lower crustal conductors originated due to accumulation

352 12 Electromagnetic Theory (Vector P otentials)

of ﬂuids, presence of continuous phase grain boundary graphites will absorb

the relatively higher frequency signals. Electrical conductivity of the earth’s

crust mantle silicates have very strong dependence on temperature. Since tem-

perature is increasing with depth and it reaches about 1200

◦

C to 1300

◦

Cat

the lithosphere asthenosphere boundary, an insulator at room temperature can

become a conductor at 1300

◦

C/1400

◦

C. Conducting asthenosphere absorbs

electromagnetic signals. Longer and longer perio d electromagnetic signals

with poorer and poorer resolving power enter deep inside the earth. There-

fore, magnetovariational sounding (MVS), magnetotelluric sounding (MTS)

and long period geomagnetic depth sounding (GDS) are used to study deep

inside the earth. Fortunately the energy level of the earth’s natural electro-

magnetic ﬁeld is very high (of the order of 10

ergs) and the energy level

increases with p eriod of the em signals coming from the magnetosphere level

and ab ove. This naturally gifted property of the em signals coming from the

outer space with a very wide band of frequencies make the magnetotelluric

(MTS ), magnetovariational(MVS) and geomagnetic depth(GDS) sounding

possible.

For deeper probing, these earth’s natural electromagnetic ﬁelds are used.

These natural signals which originate due to interaction of the solar ﬂares

with the magnetosphere, tidal oscillation of the ionosphere in the magne-

tosphere, ring current, diurnal and seasonal variations of the time varying

extraterrestrial electromagnetic ﬁelds, thunderstorm activities in between the

earth ionosphere waveguide have a very wide range of frequencies. Due to

phenomenal advancement in instrumentation, and development of softwa res

for data processing and interpretation, it is now possible to collect data

over a very wide band of frequencies specially for very very low frequencies.

Four to ﬁve decades of research in instrumentation made these measurements

possible.

For shallow electromag netics, used for groundwater and mineral explo-

ration, man made low power units having frequencies in the range of 300 C/s

to 5000 C/s are available. Signals within the frequency range of 20,000 Hz are

used for induction logging. Signals of GigaHertz and MegaHertz frequencies

are used in electromagnetic propagation to ol (EPT) in b orehole geophysics.

EPT is used to demarcate the oil-water contact at the borehole wall. Here

half to one inch of penetration of EM signal is good enough. Therefore, the

frequency could be raised to the level which are used by communication engi-

neers. One naturally gifted properly which help ed the geophysicists is the

signiﬁcant diﬀerence in dielectric constant of water (80) and hydrocarb ons

(5). At very high frequencies, displacement current dominates over the con-

duction current. Since displacement current components have direct relation

with the dielectric constant, it become possible to detect the oil-water con-

tact in a borehole within the 1 to 2 inches of depth of p enetration along the

borehole wall. Inside the coal mines also one uses MHz to GHz signals to

estimate the thickness of the coal columns. A promising very high frequency

tool came up during the last two decades is known as ‘Ground Penetrating

12.1 Introduction 353

Radar’(GPR). Its operating frequency is in gegahertz range(GHz) and its

resolving p ower is comparable to that o f seismic P waves. It can see o nly

5 to 10 meters from the gro un d surface. Successive reduction in frequency

increases the depth of investigation. It can map very ﬁner details of the shal-

low sub surface. Besides these three uses of the high frequency signa l s, most of

the geophysical EM jobs are in the audiofrequency ranges where displacement

currents are negligible.

Conduction current and diﬀusion of electromagnetic waves through the air

and earth are important for geophysicists. Most of the geophysical work are

done in the conduction zone where the electrical conductivity rather than the

dielectric constant control the electromagnetic wave propagation. Since elec-

tromagnetic waves can travel through the air; even an antenna in the air can

excite the ground. That brought the air borne electromagnetics, an important

branch of electromagnetics, in geophysics. Air borne geoelectromagnetics is a

very big branch of geophysics, where both time domain and frequency domain

measurements are possible and are done. The other name of time domain elec-

tromagnetics is electromagnetic transients. It is a very important branch of

geophysics for probing upto the upper crust of the earth. Fortunately the time

domain decay of an em signal is much faster than the time domain decay of

induced polarization signals. That is why it became possible to isolate the

em transients from IP in time domain measurements. Since the behaviour of

the induced polarization signal of electro chemical origin has some similarity

with the presence of capacitance in a resistive circuit, variation of impedances

and their phases of these capacitors with frequency are studied in frequency

domain and in spectral induced polarization, measurement of IP phase or

variation of decay during discharging of a capacitor in a time domain IP are

studied.

For direct current ﬂow one needs galvanic contact of current electrodes with

the ground and potentials are measured th rough a pair of potential electrodes.

In electromagnetics transmitting and receiving di po les are used for one type

of measurement. Current can be sent through a large loop insulated from the

ground or a grounded loop or line source and the response can be measured

in receiver loo ps. For earth’s natural electromagnetic ﬁeld, the magnetic sig-

nals a re received in induction coils or in magnetometers (ﬂux gate, proton

procession, SQUIDS etc.) and electrical signals are measured by two pa i rs

of non pola risable electrodes planted on the ground. A coil carrying current

is termed as magnetic dipole. When alternating current passes through it,

the coil becomes an oscillating magnetic dipole(Chap. 5 and 13). The mea-

sured signals in electr omagnetics are complex quantities. Therefore real and

imaginary components or amplitude and phase are measured. EM signals

are measured in many diﬀerent ways. The design and nature of transmitters

and receivers are pur pose an d problem dependent. Only some basic theories,

needed for the graduate students to start with, are discussed in this chapter

and Chap. 13.

354 12 Electromagnetic Theory (Vector P otentials)

12.2 Elementary Wavelet

In a continuous medium, any disturbance in the form of an electrical pulse or

sound pulse will continue to propagate waves. These waves are functions o f

space and time and can be expressed as

φ =f(x, t). (12.1)

This disturbance will move forward or backward with a certain velocity c. φ

changes to

φ =f(x− ct) for a forward moving wave and (12.2)

φ = f(x + ct) for a backward moving wave (12.3)

where t is time.

Any time varying and propagating harmonic signal can be expressed as

φ =acosωxorasinωx. (12.4)

Here a is the amplitude and ω (= 2πn, n is the frequency of the time varying

signal) is the angular frequency and x is the displacement. Many diﬀerent

ways, the same (12.4) can be written. The same expression for a time varying

ﬁeld can be written as

φ =acosω(x − ct). (12.5)

For a harmonic wave equation, we can write

x=x+

2π

(12.6)

where 2π = 360

◦

. Figure 12.2 shows a harmonic wave where wavelength

λ, amplitude a, phases 0

◦

, 90

◦

, 180

◦

, 270

◦

and 360

◦

are shown. In the phase

domain plot, the vector Ex

comes back to the original position after rotation

for an angle 2π. Amplitude of the wave is the length of the vector Ex

and

it is the radius of the circle presented in Fig. 12.3. Figure 12.2 shows the two

signals of same amplitude and frequency but diﬀerence in phase. Phase is an

angle and it is measured either in degrees or in radian or milliradian.The angle

shown between the vectors Ex

and Ex

is the phase angle and i t is the phase

diﬀerence between two sinusoidal signals shown in Fig. 12.2. For a particular

signal phase repeats after the lapse of time period T.

Therefore (12.5) can be written as

φ =acosω



2π

− ct



. (12.7)

Frequency n (ω =2πn) of a time varying signal is the numb er o f wavelets of

length λ cross a particular point in the space domain per second. Here λ is the

wave length and T is the time period of the signal and n =

i.e., frequency

12.2 Elementary Wavelet 355

Fig. 12.2. Two sine wavelets of same frequency and same amplitude with a dieﬀer-

ence of phase

and time period have reciprocal relation. ωT=2π,or,T=

2π

i.e., after a

complete time period T, the phase will rotate by 2π and the wave will repeat

itself. That is why (12.5) could be written as (12.7) using the relation (12.6).

Equation (12.7) can be written as

φ =acos[ω(x −ct) + 2π] (12.8)

=acosω (x − ct).

After traveling a distance

2π

, the wave proﬁle repeat itself.

Since c = nλ; λ =

; ωT=2π and T =

, therefore n =

2π

. Hence λ =

2πc

= ω,and

φ =acos

2πc

(x −ct) . (12.9)

Wave number k =

,and

=T=

, therefore,

ω (x − ct) =

2πc



− t



2πc



nλ

− t



2πc



− t



=2πc



−



(12.10)

Fig. 12.3. Representation of two electric vector with a phase diﬀerence; projection

of these vectors on the abscissa executes simple harmonic motion

356 12 Electromagnetic Theory (Vector P otentials)

Fig. 12.4. Primary electromagnetic ﬁeld due to a transmitting magnetic dipole;

generation of eddy currents and secondary ﬁeld in the presence of a conductor

⇒ φ =acos2πc



−



(12.11)

⇒ acos2πc



− nt



(12.12)

⇒ acos2πc(kx −nt). (12.13)

If we insert a phase component, we get

φ =acos[2πc(kx − nt)+ ∈] (12.14)

where ∈ is the change in phase.

From (12.5) we get, after diﬀerentiating twice,

∂

∂x

∂

∂t

(12.15)

This is the one dimensional wave equation. For two and three dimensional

cases the expressions for the waves are respectively g iven by

φ =f(lx+my− ct) (12.16)

φ =f(lx+my+pz−ct) (12.17)

where l, m and p a re direction cosines.

12.3 Elliptic Polarisation of Electromagnetic Waves

When two electromagnetic waves diﬀering in amplitude and phase interact,

they get elliptically po l arised. These sources of the electromagnetic waves can

be two independent sources or it may be due to interaction of primary and

secondary ﬁelds. When an electromagnetic wave propagates from a source, it

12.3 Elliptic Polarisation of Electromagnetic Waves 357

generates a primary ﬁeld. When these time varying signals start propagat-

ing through a conductor, eddy currents are generated within it due to rate

of change of number of lines of forces in any section. These eddy currents

start generating electromagnetic waves of diﬀerent amplitude and phase but

same frequency. This is a secondary ﬁeld (Fig. 12.4 ). These two primary and

secondary signals interact and get elliptically polarized. Let the two signals be

cos ωt (12.18)

cos (ωt − φ) (12.19)

where E

, E

are the amplitudes and φ is the phase diﬀerence between the

two signals E

and E

. From (12.18) and (12.19), we can write

cos φ +

1 −

sin φ



−

cos φ





1 −



sin

⇒

cos

ϕ −

cos ϕ

sin

= 1 (12.20)

⇒ LE

+NE

− 2ME

= 1 (12.21)

when M

− LN < 1. Then (12.21) becomes an equation of an ellipse. The

major and minor axes of the ellipse are respectively given by

L+N−



+(L− N)

(12.22)

L+N+



+(L− N)

. (12.23)

Here

cos φ

sin

(12.24)

sin

(12.25)

sin

. (12.26)

− LN =

cos

sin

−

sin



cos

ϕ −1



. (12.27)

358 12 Electromagnetic Theory (Vector P otentials)

Fig. 12.5. Elliptic polarisation due to interaction of two electromagnetic waves

having diﬀerent amplitude and phase

Since cos

φ < 1, therefore M

− LN < 1, therefore the ﬁeld is elliptically

polarised (Fig. 12.5). Th e angle of tilt of the ellipse is given by

tan 2ψ =

L −N

. (12.28)

Hence when an alternating current is sent through the ground, an elliptically

polarised ﬁeld is established. It has no longer constant direction and magni-

tude but traces ellipse in the x y/xz/yz plane. This property generated the dip

angle or tilt angle method of electromagnetic prospecting in g eophysics.

12.4 Mutual Inductance

The Fig. 12.6 shows an arrangement of two coils for developing the induced

e.m.f.

The magnetic ﬁeld inside the longer solenoid is uniform and has the magnitude



4π

(12.29)

where N

is the number of turns and l is its length in coil 1; I

is the cur-

rent ﬂowing through it. Cross sectional area of the coil 1 is S; the ﬂux is its

magnitude time S. If the coil 2 has N

turns, this ﬂux links the coil N

times.

Therefore, emf in coil 2 is given by

∂



∂t

(12.30)

Equation (12.30) can b e written as

4π

. (12.31)