Roy K.K. Potential theory in applied geophysics

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2.9 Equations 39

equation may be divided into two large classes, viz., linear and non–linear

equations.

A second order diﬀerential equat ion

+p(x)

+ q(x)y = r(x) (2.38)

is called a homogeneous equation if r(x) = 0.

The equation is non linear if it can be written as









y=0. (2.39)

An ordinary diﬀerential equation of nth order can be written as

n−1

(x)

n−1

+ ........+P

(x)y = r(x) (2.40)

where r, P

, P

...P

are functions of x. In a similar way this equation can

be homogeneous and inhomogeneous. Partial diﬀerential equations are those

where the functions, involved, dep end on two or more independent variables.

They are used for solving many problems of physics, viz., problems of potential

theory, electromagnetic theory, heat conduction and ﬂuid ﬂow theo ry, solid

mechanics etc. We use these partial diﬀerential equations for solving boundary

value problems of diﬀerent branches of mathematical physics.

A partial diﬀerential equation is linear if it or its partial derivative is of

ﬁrst degree independent variables. Otherwise it is nonlinear. If each term of

this typ e of equation contains dependent variables or one of its derivatives,

the equation is said to b e homogeneous. Otherwise they are inhomogeneous.

An equation of the form

∂

∂x

+2B

∂

∂x∂y

∂

∂y

=F(x, y, u,

∂u

∂x

∂u

∂y

(2.41)

is a normal form of partial diﬀerential equation. The equation is said to be

elliptic if AC−B

> 0, parabolic if AC−B

= 0 and hyperbolic if AC−B

< 0.

Here A,B,C may be function of x, y, z. As for example Laplace equation

∂

∂x

∂

∂y

∂

∂z

= 0 (2.42)

is an elliptic equation. Heat conduction equation

∂u

∂t

∂

∂x

(2.43)

is a parabolic equation and wave equation

∂

∂t

∂

∂x

(2.44)

is a hyperbolic equation.

40 2 Introductory Remarks

2.9.2 Integral Equations

The theory of Integral equations deals with the equations in which an

unknown function occurs under the integral sign. The subject was devel-

oped by V. Volterra (1860–1940), E.I. Fredhom (1866–1927) and D. Hilbert

(1862–1943).

If an external force is applied to a linear system and is described by a

function f(x) (a ≤ x ≤ b), then the result or the output is given by the system

which can be written as

f(x) =



G(x, ξ)f(ξ)dξ (2.45)

where G(x, ξ) is a Kernel function or a Green’s function which is speciﬁed

by the given system. Integral equation (IE) has a major role in geophysical

forward modeling. Here a certain class of integral equations are introduced.

Integral equations appear in a certain class of diﬀusion and potential problems

in geophysics. IE has a major success in handling three dimensional problem

both in potential (scalar and vector) and non potential problems. If b oth

the upper and lower limits of an integral are constant then the equations

are Fredhom type. If one of the limits is an ind ependent variable, then the

equations are Volterra type. An integral equation is said to be linear if all the

terms occurring in the equations are linear. These integral equations can be

of ﬁrst, second and third kind as follows :

(i) Fredhom’s integral equation of the ﬁrst kind.



K(x, t)f(t)dt = g(x) (2.46)

(ii) Fredhom’s integral equation o f the second kind



K(x, t)f(t)dt = g(x) + f(x) (2.47)

(iii) Fredhom’s integral equation of the third kind



K(x, t)f(t)dt = λf(x) (2.48)

(iv) Volterra’s integral equation of the ﬁrst kind



K(x, t)f(t)dt = g(x) (2.49)

2.10 Domain of Geophysics in Potential Theory 41

(v) Volterra’s integral equation of the second kind



K(x, t)f(t)dt = g(x) + f(x) (2.50)

(vi) Volterra’s integral equation of the third kind



K(x, t)f(t)dt = λf(x) (2.51)

In Fredhom’s integral equations, the upper and lower limits are respec-

tively ‘b’ and ‘a’ where ‘a’ and ‘b’ are constants. In Volterra’s integral equa-

tion the upper and lower limits are respectively ‘x’ and ‘a’ where x is an

independent variable and ‘a’ is a constant. Here f(t) is an unknown function

to be determined. K(x,t) and g(x) are known functions. K(x,t) is known as the

Kernel function. Equation of the third kind is a homogeneous version of the

second kind and it deﬁnes an eigen value problem. These integral equations of

second kind can be solved either iteratively or using Gauss qua drature method

of numerical integration or by series expansion method. Further discussion on

integral equation is available in Chap. 15. Eigen value is deﬁned in Chap. 17.

2.10 Domain of Geophysics in Potential Theory

Potential theory as such is a vast subject and is used by physicists, geophysi-

cists, electrical engineers, electrical communication engineers, mathematicians

working in ﬂuid dynamics, complex variables, aerodynamics, acoustics, elec-

tromagnetic wave theory, heat ﬂow, theory of gravity and magnetics etc. This

subject forms the basis of many branches of science and engineering. Only a

part of geophysics, controlled by potential theory, includes gravity, magnetic,

electrical electromagnetic heat ﬂow and ﬂuid ﬂow methods.. Seismic methods

in geophysics and nuclear geophysics are out of b ound of potential theory.

All potentials of electrochemical and electrokinetic origin strictly do not come

under potential theory.

Primary task of a geophysicist is to understand the data collected from

the ﬁeld and interprete them in the form of an acceptable geological model

within the limit of ﬁnite resolving powers of diﬀerent geophysical potential

ﬁelds For prop er execution of this task, geophysicists have to frame the earth

models and have some ideas about the nature of ﬁeld response. To have this

understanding, geophysicists solve forward problems for diﬀerent branches of

geophysics using potential theory (Scalar and Vector potentials). Therefore

potential theory, equipped with diﬀerent mathematical tools, forms the basis

for solution of forward problems. Potential theory is a must to understand

the behaviour of the geophysical data. Inverse theory comes next as a tool

42 2 Introductory Remarks

for interpretation. Advent of inverse theory revolutionized the entire proce-

dure for interpretation of geophysical data. It became possible due to very

rapid development of computer science, software technology and numerical

methods in mathematical modelling. The phenomenal developments in sci-

ence and technology in these areas during the last three decades improved

the power of vision of geophysicists to look inside the earth. In this b ook,

besides some introduction on gravitational ﬁeld, magnetostatic ﬁeld, electro-

static ﬁeld, direct current ﬂow ﬁeld and electromagnetic ﬁeld, we introduced

(i) the solution of Laplace equation and its contribution towards solving dif-

ferent type of boundary value pro blems, (ii) complex variable and its role

in so lv i n g two dimensional potential problems (iii) Green theorem and its

application in solving potential problems (iv) concept of images in p otential

theory (v) electromagnetic theory and vector potentials and their contribution

towards solving the electromagnetic boundary value problems (vi) ﬁnite ele-

ment and ﬁnite diﬀerence and integral equation methods towards solving two

and three dimensional potential problem (vii) Green’s function (viii) analyti-

cal continuation of potential ﬁelds and (ix) inversion of p otential ﬁeld data.

Scientists of other disciplines of physical sciences and technologies may

ﬁnd interest .in some of these topics.

Gravitational Potential and Field

In this chapter we included some of the preliminaries of the gravitational

potentials and ﬁelds, viz, Newton’s law of gravitational attraction, gravita-

tional ﬁelds an d potentials, universal gravitational constant and acceleration

due to gravity, the nature of earth’s gravitational ﬁeld, gravitational potentials

and ﬁelds for bodies of simpler geometries and the basic guiding equations of

the gravitational ﬁeld. A few points about the nature of the earth’s grav-

ity ﬁeld and the basic preliminaries regarding data handling are mentioned.

Advanced topics, viz, spherical harmonics Green’ theorem, Green’s equivalent

layers and analytical continuation of potential ﬁeld are given respectively in

Chaps. 7, 10 and 16.

3.1 Introductio n

Sir Isaac Newton ﬁrst published Philosophiae Naturalis Principia Mathemat-

ica in 1687. In that he gave ideas both about the law of gravitation as well as

the laws of forces. It was realised that (i) gravitational force is always a force

of attraction, (ii) It is not only a global force it is an universal force present in

the entire universe, (iii) It is one of the weak forces, (iv) It has some relation

with mass of a body, (v) principle of superposition is valid for gravitational

ﬁelds, (vi).centrepetal and centrifugal forces do exist, (vii) movement of plan-

ets round diﬀerent stars and movement of satellites round diﬀerent planets

are controlled by the gravitational force of attraction. Two important physi-

cal parameters, viz, ‘G’ and ‘g’ came in the fore front for further advancement

although mass of a body, its density, has some direct relation with the force of

gravitation was realised. Lord Cavendish experimentally determined the value

of ‘G’, the universal gravitational constant, in his laboratory by measuring the

force of attraction between two lead balls placed at a certain precise d istance

and published the value of ‘G’ in 1798 to be 6.754×10

−11

−1

−2

(MKS or

SI unit) which was reﬁned later as 6.672 ×10

−11

MKS unit. Accelaration due

to gravity was ﬁrst measured by Galileo in his famous experiment by dropping

44 3 Gravitational Poten tial and Field

objects from the leaning tower of Pisa. The numerical value of ‘g’ was found

to be around 980 cm/ sec

. In honour of Galileo the unit 1 cm/ sec

, the unit

of acceleration due to gravity was termed as ‘Gal’ or ‘gal’. It was understood

right from the b eginning that the force of gravitation is global and the gravita-

tional force is always a force of attraction and the entire network of billions of

stars, planets and satellites are controlled by the force of gravitation. Kepler’s

three law’s i.e., (i) orbit of each planet is an ellipse with sun at one of the foci

(ii) orbital radius of the earth sweeps out equal areas in equal interval of time

and (iii) the ratio of square of the planet’s period of revolution to the cube of

the semi major axis of the orbit is a constant for all the planets and could b e

explained from Newton’s law of gravitation.

Next round of researches in this area were centred around accurate evalu-

ation of the absolute value of ‘g’, the acceleration due to gravity, and G, the

universal gravitational constant. Soon it became known to the physicists that

the time period of oscillation of a simpl e pendulum, which executes simple

harmonic motion, is connected to the acceleration due to gravity g through

the relation T = 2π



l/g where l is the distance between the pivotal point

of the hinge to the centre of the mass and T is the time period of oscilla-

tion. Although expression looks very simple several stages of developments

and generation of compound pendulum was necessary for accurate measure-

ment of ‘g’. It could b e known, as soon as the value of ‘g’ is known that ‘g’

is a latitude dependent quantity and shape of the earth is nearly a spheroid

with deﬁnite ellipticity. It was known right from the beginning that mass and

density of a body are interrelated an d they have connection with the gravity

ﬁeld of the earth. Therefore both the mass and the mean density of the earth

could be known from ‘G’ and ‘g’.

Much later geophysicists came forward for accurate measurement of minute

variation of the value of ‘g’ due to minor variations of densities of rocks and

minerals. Precision of measurement has gone up to such a level that practical

unit o f measurement of variation of the gravity ﬁeld was reduced to milli-

gal (10

−3

). The minute variations of gravity is termed as ‘∆g’, the gravity

anomaly and the precision instruments are named gravimeters. Gravimeters

measure the minute variations in ‘g’ rather than their absolute values. Preci-

sion level is heading towards microgal level in 21st century. The ultra sensi-

tive gravimeters are used for geodetic survey, crustal studies and geophysical

exploration.In this chapter a brief outline of the gravitational p otential and

ﬁeld are given. Some of the topics of gravitational potentials and ﬁelds are

included in Chaps. 10 and 16.

3.2 Newt on’ s Law of Gravitation

Newton’s law of gravitation states that “Every particle in the universe attracts

every other particle with a force which is directly proportional to the product

of the masses of the particles a n d inversely proportional to the square of

3.2 Newton’s Law of Gravitation 45

the distance between them.” Therefore, Newton’s law of gravitation can be

stated as



F α

which can be written as

F=G

(3.1)

where F is the force of attraction between the two masses m

and m

placed

at a distance r from each other and G, the constant of proportionality, is the

universal gravitational constant. Here m

and m

are the two masses separated

by a distance r. This G is the force of attraction between the two particles of

unit mass separated by a unit distance. The acceleration due to gravity is ‘g’.

Its unit is cm/ sec

or Gal. The unit Gal is introduced in honour Galileo. The

practical unit is milligal = 10

−3

gal. Assuming the earth as a spherical b ody

of density ρ and radius ‘r’, the force of attraction b etween the earth and a

body of mass M is given by

= Mg (3.2)

where M

is the mass of the earth and is equal to =

πr

ρ and ρ is the mean

density of the earth (ρ =5.67 gms/cc approx). Here g is the acceleration due

to gravity and is approximately equal to g = 981 cm/ sec

. In C.G.S Unit

4πρr

15, 500, 000

(approx) (3.3)

=6.664 × 10

−8

cm/g. sec

(CGS unit)

6.664 × 10

−11

m/kg. sec

(M.K.S. Unit) (3.4)

Gisindyne–cm

/gm

that is equivalent to Newton. Cavendish ﬁrst exper-

imentally determined the value of G as mentioned.

Gravitational ﬁeld is a naturally existing conservative, solenidal (in a

source free region), irrotational, global a nd scalar potential ﬁeld of very large

dimension. Two masses separated by a great distance can experience a force

of attraction what little be the magnitude of that force. Gravitational force

exits between all the celestial bodies. It is a weak force. But the force is always

a force of attraction and the direction of the force is along the line joining the

centres of two masses m

and m

(Fig. 3.1) at P (x, y, z) and at Q (ξ, η, ζ)

where r is the distance between the two point masses. r the directional force

vector which is expressed as the distance between the two masses can b e

written as

r=



i(x− ξ)+



j(y− η)+



k(z− ζ) (3.5)

For a n unit point mass m

, gravitational ﬁeld due to a point mass is given by

Lim

→1

F = −G

.r

46 3 Gravitational Poten tial and Field

Fig. 3.1. Gravitational attraction at a point P due to a point mass m at a point Q

at a distance r from P

where r is the direction of the unit vector. Gravitational ﬁeld is a scalar

potential ﬁeld i.e., the potential is a scalar and the ﬁeld is a vector. They

are related by the relation



E=−gradφ. Many authors use negative sign to

express,



F=−G

r (3.6)

As mentioned already gravitational ﬁeld, is always an irrotational ﬁeld i.e.

curl g=0.

Absolute values of densities or their variations in crust mantle silicates,

originated due to thermal processes inside the ear th as well as due to tec-

tonic evolution, are responsible generation of global gravity map. Purpose of

investigation ﬁxes the scale of measurement.

3.3 Gravity Field at a Point due to Number

of Point Sources

Suppose the masses m

, m

and m

are distributed in a space and their

distances from the point of observation P are respectively g iven by r

, r

etc (Fig. 3.2). Let the gravitational attraction at the point P for the masses

be g

, g

and g

respectively. Since this will be a vector ﬁeld, the total

ﬁeld can be determined by resolving the components of forces along the three

co-ordinates axes.

Let ξ

, η

and ζ

be the co-ordinates of the ith particle and m

be its mass.

The force acting at the point P due to m

. The components of the force

along the x, y and z directions are r espectively given by

x =

x −ξ

(3.7)

y − η

(3.8)

z − ζ

. (3.9)

3.4 Gravitational Field for a Large Body 47

Fig. 3.2. Gravitational attraction at a point P due to masses m

, m

and m

Therefore, the total gravitational ﬁeld components g

, g

using principle

of superposition are given by



i=1

(x −ξ

)

, g



i=1

(y −η

)

and



i=1

(z − ζ

)

. (3.10)

Here



. (3.11)

That gives the expression for gravitational force of attraction at a p oint due

to n number of isolated discrete point masses.

3.4 Gravitational Field for a Large Body

Let us assume an arbitrary shaped large solid three dimensional body of den-

sity ρ. Let us take a point of observation P at coordinates (x, y, z) and an

elementary mass at Q having coordinates (ξ, η, ζ) within the body (Fig. 3.3).

The elementary volume dν =dξdηdζ. We can divide the body into number of

elementary volumes. The volume density of source is Lt

∆v→0

ρ =

∆m

∆v

where ρ

is the density of the body. For a small volume dv, its mass is ρdv. The ﬁeld

due to the entire mass along the x, y, and z directions are given by





ρ (x − ξ)dξdηdζ



(x − ξ)

+(y− η)

+(z− ζ)



3/2

(3.12)

48 3 Gravitational Poten tial and Field

Fig. 3.3. Gravitational attraction at the point P due to a large solid mass





ρ (y − η)dξdηdζ



(x −ξ)

+(y− η)

+(z− ζ)



3/2

(3.13)





ρ (z −ζ)dξdηdζ



(x −ξ)

+(y− η)

+(z− ζ)



3/2

(3.14)

3.5 Gravitational Field due to a Line Source

Let AB is a line source of mass having linear density λ where λ =Lt

∆l→O

∆m

∆l

(Fig. 3.4). The line is put in the xy plane such that the centre of the line is

at the orig i n . Let the line be divided into a number of segments dξ.Thusthe

ﬁeld components along the x direction is given by

Fig. 3.4. Gravitational ﬁeld at an external point P due to a line source of ﬁnite

length