Principles Of Geophysics

Principles of Applied Geophysics D. S. Parasnis and C. Harrison Dwight Citation: American Journal of Physics 32, 237 (1964); doi: 10.1119/1.1970198 View online: http://dx.doi.org/10.1119/1.1970198 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/32/3?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Applying geophysical imaging algorithms to nondestructive evaluation problems J. Acoust. Soc. Am. 119, 3215 (2006); 10.1121/1.4785897 Uncertainty principle applied to the deuteron Am. J. Phys. 49, 185 (1981); 10.1119/1.12553 Geophysics Phys. Today 17, 92 (1964); 10.1063/1.3051507 Principles and Techniques of Applied Mathematics Am. J. Phys. 25, 208 (1957); 10.1119/1.1934401 Training of Physicists for Work in the Field of Applied Geophysics Am. J. Phys. 10, 185 (1942); 10.1119/1.1990368 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 23 Nov 2014 08:11:58 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 23 Nov 2014 08:11:58 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 23 Nov 2014 08:11:58
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1

PRINCIPLES OF GEOPHYSICS


GPH 201







Abdullah M. Al-Amri
Dept. of Geology & Geophysics
King Saud University, Riyadh
[email protected]
www.a-alamri.com







2


PRINCIPLES OF GEOPHYSICS ( GPH 201 )
F THE COURSE SYLLABUS O



I. FUNDAMENTAL CONSIDERATIONS ( 4 HOURS )
- Stress - Strain Relationship
- Elastic Coefficients
- Seismic Waves
- Huygen and Fermat principles
- Snell's Law in Refraction
Problem Set – 1


II. SEISMIC REFRACTION METHOD ( 4 HOURS )
- Fundamentals
- Two Horizontal Interfaces
- Dipping Interfaces
- The Non ideal Subsurface
- The Delay-Time Method
- Field Procedures & Interpretation
Problem Set - 2


III . SEISMIC REFLECTION METHOD ( 4 HOURS )
- A Single Subsurface Interface
- Analysis of Arrival Times
- Normal Move out
- Determining of Velocity & Thickness
- Dipping Interface
- Common Field Procedures
- Velocity Analysis
- Applications in Petroleum exploration
Problem Set - 3


------------------------------ FIRST MIDTERM EXAM-------------------------------

IV EARTHQUAKE SEISMOLOGY ( 3 HOURS )
- Definition and Historical review
- Classification of Earthquakes
- Earthquakes : Where and Why
- Causes of Earthquakes
- Earthquake Epicenter & Hypocenter
- Magnitude & Intensity


V ELECTRICAL METHOD ( 4 HOURS )

3

- Electrical properties of rocks
- Apparent & True resistivity
- Electrode configurations
- Electrical soundings & Profiling
- Applications in groundwater exploration
Problem Set - 4

----------------------- SECOND MIDTERM EXAM---------------------------------

VI GRAVITY PROSPECTING ( 4 HOURS )
- Fundamental principles
- Measurements
- Data reduction
- Isostasy and crustal thickness
- Interpretation & Applications
Problem Set - 5

VII MAGNETIC METHOD ( 3 HOURS )
- Basic concepts
- Description of the magnetic field
- Source of magnetic anomalies
- Interpretation & Applications
Problem Set – 6

GRADING :

First Midterm exam 10 %
Second Midterm exam 10 %
Problem Set 15 %
Lab. 25 %
Final exam. 40 %



: TEXT

An introduction to geophysical exploration (third edition) - P. Keary, M. Brooks, and I.
Hill, Blackwell Publishing, 2002. ISBN: 0-632-04929-4

Introduction to applied geophysics: Exploring the shallow subsurface - H.R. Burger, A.F.
Sheehan, and C.H. Jones, W.W. Norton and Company, 2006. ISBN: 0-393-92637-0


AMRI - ABDULLAH M. S. AL I NSTRUCTOR :
12 - OFFI CE HOURS : SAT & MON 11


4

Free Advice
Group study. is strongly encouraged. Interaction with peers and
instructors is very beneficial to your learning and digesting the material.
Make good use of it. Discuss the problems with the TA.
Ask questions. It is recommended that you put in at least 2 hours of
study for each 1 hour of lecture. It is very useful to read ahead, before
the material comes up in the classroom. Feel free to ask questions,
make comments or just express your opinions.
Think. The assigned problem sets present an opportunity for you to
think about and apply the material presented in the lectures and in some
cases to learn new material. Start them early in the week so that you
can think about them and exercise your creative potential. Note that
solving a problem does not only consist of inserting numbers into
equations and crunching out new numbers. Strive for 100% on the
assignments utilizing all the resources and opportunities available to you.
Expanding your mind. Learning is a process of expanding your mind.
This is attained by developing new connections between neurons in your
brain and strengthening selected neural pathways. Be persistent and
don‘t give up when you come to the inevitable hard sections. For some
of you that might be at the beginning, for others the resistance might
increase with time. Hang in there.
Don’t Copy. Copying the solutions to assigned problems from other
sources is not only discouraged. It is not professional, it does not reflect
well on you as a future geophyscist, and it is detrimental to your
character and professional career. It is an activity that shows disrespect
for you, for your classmates, and for your instructors. If you have
questions, ask the instructor or the TAs.
If you follow the lectures, keep up with the homework problems, and
ask questions to your classmates, TAs and the instructor, you will be on
your way to mastering the material. Persistence and motivation are the
only prerequisites to success. Make it a game to master new skills and
knowledge. Be courageous in facing new challenges.
Measured objectively, what a man can wrest from Truth by passionate
striving is utterly infinitesimal. But the striving frees us from the bonds of
the self and makes us comrades of those who are the best and the
greatest. Albert Einstein
5

INTRODUCTION
Geophysics is an interdisciplinary physical science concerned with the
nature of the earth and its environment and as such seeks to apply the
knowledge and techniques of physics, mathematics and chemistry to
understand the structure and dynamic behavior of the earth and its
environment. The required sequence of Mathematics, Physics and
Geophysics courses is designed to provide a basic structure on which to
build a program with science electives normally selected from Geology,
Astronomy, Oceanography, Mathematics, Physics and Chemistry courses.
Geophysics is the science which deals with investigating the Earth, using
the methods and techniques of Physics. The physical properties of earth
materials (rocks, air, and water masses) such as density, elasticity,
magnetization, and electrical conductivity all allow inference about those
materials to be made from measurements of the corresponding physical
fields - gravity, seismic waves, magnetic fields, and various kinds of
electrical fields. Because Geophysics incorporates the sciences of Physics,
Mathematics, Geology (and therefore Chemistry) it is a truly
multidisciplinary physical science.
The two great divisions of Geophysics conventionally are labeled as
Exploration Geophysics, and Global Geophysics. In Global
Geophysics, we study earthquakes, the main magnetic field, physical
oceanography, studies of the Earth's thermal state and meteorology
(amongst others!). In Exploration Geophysics, physical principles are
applied to the search for, and evaluation of, resources such as oil, gas,
minerals, water and building stone. Exploration geophysicists also work in
the management of resources and the associated environmental issues.
Geophysics contributes to an understanding of the internal structure and
evolution of the Earth, earthquakes, the ocean and many other physical
phenomena. There are many divisions of geophysics, including
oceanography, atmospheric physics, climatology, petroleum geophysics,
environmental geophysics, engineering geophysics and mining geophysics.
6

Geophysical Exploration Techniques

Geophysical methods are divided into two types : Active and Passive
(Natural Sources): Incorporate measurements of Passive methods
natural occurring fields or properties of the earth. Ex. SP, Magnetotelluric
(MT), Telluric, Gravity, Magnetic.
(Induced Sources) : A signal is injected into the earth Active Methods
and then measure how the earth respond to the signal. Ex. DC. Resistivity,
Seismic Refraction, IP, EM, Mise-A-LA-Masse, GPR.

Common Applications
- oil and gas exploration
- mineral exploration
- diamond exploration (kimberlites)
- hydrogeology
- geotechnical and engineering studies
- tectonic studies
- earthquake hazard assessment
- archaeology

7



8






CHAPTER 1



FUNDAMENTAL CONSIDERATIONS


- Stress - Strain Relationship
- Elastic Coefficients
- Seismic Waves
- Huygen and Fermat principles
- Snell's Law in Refraction
Problem Set – 1 ( 1-3-6-8-11)
9

THEORY OF ELASTICITY
Stress is the ratio of applied force F to the area across which it is acts.
Strain is the deformation caused in the body, and is expressed as the ratio of
change in length (or volume) to original length (or volume).
Triaxial Stress
Stresses act along three orthogonal axes, perpendicular to faces of solid, e.g.
stretching a bar:


Pressure
Forces act equally in all directions perpendicular to faces of body, e.g.
pressure on a cube in water:
11



Strain Associated with Seismic Waves Inside a uniform solid, two types of
strain can propagate as waves:

Axial Stress : Stresses act in one direction only, e.g. if sides of bar fixed:

- Change in volume of solid occurs.
- Associated with P wave propagation





11

Shear Stress : Stresses act parallel to face of solid, e.g. pushing along a
table:

- No change in volume.
- Fluids such as water and air cannot support shear stresses.
- Associated with S wave propagation.




















12

Stresses on a solid in 3 dimensions





Stress = Force applied to a body per unit area (s = F/dS).
The stress can be expressed in two sets of components:
-Normal stress (n), perpendicular to the surface of the body (c.f.
pressure) -
Shear stress (t) acting parallel to the surface
of the body For each surface one can define 3 orthogonal
components of stress. The surfaces themselves can be defined as 3
orthogonal components _ definition of 9 components of
stress (direction of the force and direction of the surface on which it acts).

For each body, it is possible to define 3 axes for which shear stresses are
zero and only the normal stresses exist (using geometrical
transformations). These axes are called the principal axes, and the
corresponding normal stresses are the principal stresses.
13


If all 3 principal stresses are equal, the body is subjected to a pressure
(lithostatic pressure in the case of solid rock).
Pressure = (sum of principal stresses)/3
Conventions for directions:
Stresses towards the interior: compression
Stresses towards the exterior: tension (extension, dilatation)
























14


Hooke’s Law

Hooke‘s Law essentially states that stress is proportional to strain.
- At low to moderate strains: Hooke‘s Law applies and a solid body is
said to behave elastically, i.e. will return to original form when stress
removed.

- At high strains: the elastic limit is exceeded and a body deforms in a
plastic or ductile manner: it is unable to return to its original shape,
being permanently strained, or damaged.
- At very high strains: a solid will fracture, e.g. in earthquake faulting.
15

Constant of proportionality is called the modulus, and is ratio of stress to
strain, e.g. Young‘s modulus in triaxial strain.

Elastic Moduli ( Constants )

(a) Young’s Modulus (E) – longitudinal strain proportional to longitudinal
stress.
E = F / A ÷ Dl / l




















( b) Bulk Modulus
(K) – describes the
change of volume
due to a change of
pressure.

K = P ÷ DV / V


16

( c ) Shear Modulus (μ) – amount of angular deformation due to the
application of a shear stress on one side of the object.
μ = t / tan θ Note: μ = 0 in liquids (no rigidity).














(d) Axial modulus (ψ) – response to longitudinal stress, similar to
Young‘s Modulus:
Y = F /A ÷ Dl / l except that strain is uniaxial – no transverse strain
associated with the application of the longitudinal stress.





17

Relationships between elastic moduli , Lamé coefficients
ì , and μ.


μ = shear modulus (as before)
ì = first Lamé coefficient (no direct physical interpretation)
Young’s Modulus: E = μ (3ì +2 μ ) ÷(ì + μ )
Bulk modulus: K =ì +2/3 μ
Poisson’s Ratio: o = ì / 2 (ì+ μ )
Lamé 1 in terms of Poisson & Young
ì =E o / (1 +o)(1 – 2o)

Poisson's ratio ( Dimensionless ratio) is the ratio of transverse contraction
strain to longitudinal extension strain in the direction of stretching force.
Tensile deformation is considered positive and compressive deformation is
considered negative. The definition of Poisson's ratio contains a minus sign
so that normal materials have a positive ratio
o = - e
trans
/ e
longitudinal


Poisson solid: material for which ì =μ , giving o = 0.25




18

SEISMIC WAVES

A. Body Waves
Seismic waves are pulses of strain energy that propagate in a solid. Two
types of seismic wave can exist inside a uniform solid:
A) P waves (Primary, Compressional, Push-Pull)
Motion of particles in the solid is in direction of wave propagation.
- P waves have highest speed.
- Volumetric change
- Sound is an example of a P wave.












19

B. S waves (Secondary, Shear, Shake)
Particle motion is in plane perpendicular to direction of propagation.
- If particle motion along a line in perpendicular plane, then S wave is
said to be plane polarised: SV in vertical plane, SH horizontal.
- No volume change
- S waves cannot exist in fluids like water or air, because the fluid is
unable to support shear stresses.






v is the poisson‘s ratio = 0 for a perfect fluid, so S-waves cannot
propagate through fluids. Poisson‘s ratio is theoretically bounded between 0
and 0.5 and for most rocks lies around 0.25, so typically V
P
/V
S
is about
1.7.

Vp /Vs ratios (hence Poisson‘s Ratio) can be characteristic of rock type or
physical property, e.g.
-Felsic rocks _ lower Poisson‘s Ratio
21

-Mafic rocks _ higher Poisson‘s Ratio
-Partial melt _ very high Poisson‘s Ratio (S wave speeds are more strongly
affected by melt than P waves)

C. Surface Waves
On the surface between a solid or liquid and air the deformation of an
element is unconstrained in the direction normal to the surface. This
changes the equation of motion for elements or particles adjacent to the
surface and gives rise to another wave type commonly called the surface
wave. Water waves are familiar and are readily seen as having a much
lower velocity than a sound wave propagating through the bulk of the
medium. Further a particle moves up and down vertically as well as in the
direction of the wave. In detail a particle traces an ellipse with a prograde
rotation as shown in the sketch below.
The surface wave on an isotropic half -space is known as a Rayleigh
wave and it is similar in form to the surface wave on a liquid half-space
except that the particle motion is retrograde. Rayleigh waves carry a large
amount of energy away from surface sources and are the noticeable, or
‗felt‘, ground motion when one is standing near a seismic source. In
exploration seismology Rayleigh wave are known as ground roll.

1) Rayleigh waves
- Propagate along the surface of Earth
- Amplitude decreases exponentially with depth.
- Near the surface the particle motion is retrograde elliptical.
- Rayleigh wave speed is slightly less than S wave: ~92% V
S
.
2. Love waves
Occur when a free surface and a deeper interface are present, and
the shear wave velocity is lower in the top layer.
- Particle motion is SH, i.e. transverse horizontal
21

- Dispersive propagation: different frequencies travel at different
velocities, but usually faster than Rayleigh waves.




The velocity of a Rayleigh wave, V
R
, is tied to the S-wave velocity and
Poisson‘s ratio (and hence to V
P
) through the solution to the following
equation (White, 1983)



For typical values of Poisson‘s ratio the Rayleigh wave velocity varies only
from 0.91 to 0.93 V
S.
.
The amplitude of the particle displacement for surface waves
decreases exponentially beneath the surface and the exponent is
proportional to the frequency as well as the elastic constants and density of
the medium. Since it is observed that velocity generally increases with
depth, Rayleigh waves of high frequency penetrate to shallow depth and
22

have low velocity whereas Rayleigh waves of low frequency penetrate to
greater depth and have high velocity. The change in velocity with frequency
is known as dispersion. For a typical surface source of Rayleigh waves the
initiating disturbance has a broad frequency spectrum so the observed
surface motion at some distance from the source is spread out in time, with
low frequencies coming first and high frequencies coming later.



Seismic velocity, attenuation and rock properties

• Rock properties that affect seismic velocity
Porosity
Lithification
Pressure
Fluid saturation
• Velocity in unconsolidated near surface soils (the weathered layer)
• Attenuation

Seismic surveys yield maps of the distribution of seismic velocities,
interfaces between rock units and, ideally, of reflection coefficients at these
interfaces. The velocities of crustal rocks vary widely as the following figure
shows.




23



Generally, the velocities depend on the elastic modulii and density
via:

These elastic constants, and densities, in turn depend on the properties
that the geologist or engineer use to characterize the rock such as porosity,
fluid saturation, texture etc. A review of the relationships between the
intrinsic rock properties and the measured velocities or reflectivities is
needed before seismic survey results can be interpreted quantitatively in
terms of lithology. Many of these relationships are empirical – velocities are
found to be related to certain rock units in a given locale by actual
laboratory measurements on core samples of the rock or soil.
It is observed from seismic surveys that velocities generally increase
with depth. Densities also increase with depth so it must be that the bulk
and shear modulii increase faster than the density. In seismic exploration
there are many empirical relationships between velocity and depth of burial
and geologic age.
24

The relationship between intrinsic rock properties such as porosity,
fracture content, fluid content and density and velocity underlie the
empirical relationships mentioned above.

Rock properties that affect seismic velocity
1) Porosity. A very rough rule due to Wyllie is the so called time
average relationship:
where φ is the porosity.
This is not based on any convincing theory but is roughly right when the
effective pressure is high and the rock is fully saturated.
2) Lithification.
Also known as cementation. The degree to which grains in a
sedimentary rock are cemented together by post depositional, usually
chemical, processes, has a strong effect on the modulii. By filling pore
space with minerals of higher density than the fluid it replaces the bulk
density is also increased. The combination of porosity reduction and
lithification causes the observed increase of velocity with depth of burial
and age.
3) Pressure.
Compressional wave velocity is strongly dependant on effective
stress. [For a rock buried in the earth the confining pressure is the pressure
of the overlying rock column, the pore water pressure may be the
25

hydrostatic pressure if there is connected porosity to the surface or it may
be greater or less than hydrostatic. The effective pressure is the difference
between the confining and pore pressure.]
In general velocity rises with increasing confining pressure and then levels
off to a ―terminal velocity‖ when the effective pressure is high. The effect is
probably due to crack closure. At low effective pressure cracks are open
and easily closed with an increase in stress (large strain for low increase in
stress—small K and low velocity). As the effective pressure increases the
cracks are all closed, K goes up and the velocity increases.
Finally even at depth, as the pore pressure increases above hydrostatic, the
effective pressure decreases as does the velocity. Overpressured zones can
be detected in a sedimentary sequence by their anomalously low velocities.
4) Fluid saturation.
From theoretical and empirical studies it is found that the compressional
wave velocity decreases with decreasing fluid saturation. As the fraction of
gas in the pores increases, K and hence velocity decreases. Less intuitive is
the fact that V
s
also decreases with an increase in gas content. The
reflection coefficient is strongly affected if one of the contacting media is
gas saturated because the impedance is lowered by both the density and
velocity decreases.


Velocity in the weathered layer

The effects of high porosity, less than 100% water saturation, lack of
cementation, low effective pressure and the low bulk modulus (due to the
ease with which native minerals can be rearranged under stress) combine
to yield very low compressional and shear wave velocities in the weathered
layer. V
p
can be as low as 200 m/sec in the unsaturated zone (vadose zone)
– less that the velocity of sound in air!
26

Attenuation
It is observed that seismic waves decrease in amplitude due to spherical
spreading and due to mechanical or other loss mechanisms in the rock
units that the wave passes through.
The attenuation for a sinusoidal propagating wave is defined formally
as the energy loss per cycle (wave length) Δ E/E where E is the energy
content of the wave.
Mathematically, the propagating wave , get an added damping
term , so the solution becomes

[We can apply this to the definition of attenuation Δ E/E by substituting A
2
for the energy at two points at distance λ (the wavelength) apart and we
find
There are many theories for explaining attenuation in rocks. Friction,
included by including a velocity term in the governing differential equation
explain laboratory measurement. Various not for the displacement does
other damping mechanisms such as viscous flow (Biot Theory) have some
success but much important work remains to be done in this area
(especially for unconsolidated material where the attenuation is very high).
Some of the theories predict attenuation as well as dispersion (the variation
of velocity with frequency).
Experimentally it is found that the a
frequency and that there is little dispersion. In fact to a good approximation
attenuation can be described by . With x in meters and f in
Hertz, a typical shale has a . So at one Hertz the amplitude falls to
A
0
/e at 10 km. But at 1000 Hz it falls to A
0
/e in 10 m. The attenuation may
be as much as 10 times greater in unconsolidated sediments.
Another important attenuation mechanism is the reduction in amplitude of
a wave by the scattering of its energy by diffraction by objects whose
27

dimensions are on the order of the wavelength. If a is an average linear
dimension of velocity inhomogeneities then the attenuation coefficient is
given approximately by:

So attenuation increases rapidly with decreasing wavelength. Consider
attenuation is an unconsolidated medium with a velocity of 250 m/sec and
a frequency of 1000 Hz. Then, λ = 0.25 m, and α = a
3
×256. The wave
would fall to 1/e of its initial amplitude when a = 157 m.
It might be reasonable to expect inhomogeneities with a
characteristic dimension on the order of 15 cm in the overburden so it is
likely that the very high attenuation observed in near surface
unconsolidated sediments is due to scattering.


Constraints on Seismic Velocity

Seismic velocities vary with mineral content, lithology, porosity, pore fluid
saturation, pore pressure, and to some extent temperature.
In igneous rocks with minimal porosity, seismic velocity increases with
increasing mafic mineral content.
In sedimentary rocks, effects of porosity and grain cementation are more
important, and seismic velocity relationships are complex.
Various empirical relationships have been estimated from either
measurements on cores or field observations:

1) P wave velocity as function of age and depth
km/s
where Z is depth in km and T is geological age in millions of years (Faust,
1951).

28


2) Time-average equation

f
and V
m
are P wave velocities of pore fluid and rock
matrix respectively (Wyllie, 1958).
- Usually V
f
≈ 1500 m/s, while V
m
depends on lithology.
- If the velocities of pore fluid and matrix known, then porosity can be
estimated from the measured P wave velocity.

29

Waves and Rays

Considering a source at point O on a homogeneous half-space, the
resulting seismic P-wave propagates such that all the points of constant
phase lie on a hemisphere centered at O. The surface of all points of the
same phase is called a wave front although the inferred meaning is that
the phase in question is associated with some identifiable first arrival of the
wave. A more rigorous definition is that the wave front is the surface of all
equal travel times from the source. A cross section of such a hemispherical
spreading wave is shown in the following cartoon for two successive time
steps, t
1
and 2t
1
.
The vector perpendicular to a wave front is defined as a ray. The ray in the
cartoon is directed along a radius from the source, but this is not always
the case. Rays are useful for describing what happens to waves when they
pass through an interface


31








Huygen’s Principle
Every point on a wavefront can be considered a secondary source of
spherical waves, and the position of the wavefront after a given time is the
envelope of these secondary wavefronts.

-
31

- Treat all the points on a wavefront as point sources that generate
secondary spherical wavefronts (‗wavelets‘).

• Given the geometry of a wavefront at time t
1
, the principle can be
used to construct the geometry of the wavefront at a later time t
2
.

• Useful for understanding reflection, refraction and diffraction of
seismic waves.


Snell’s Law

A wave incident on a boundary separating two media is reflected back into
the first medium and some of the energy is transmitted, or refracted, into
the second. The geometry of refraction and reflection is governed by
which relates the angles of incidence, reflection and refraction Snell’s Law
to the velocities of the medium.
The cartoon below illustrates the ray geometry for a P-wave incident on the
boundary between media of velocity V
1
and V
2
. The angles of incidence,
reflection and refraction, θ
1

1
‘, and θ
2
, respectively are the angles the ray
makes with the normal to the interface.

32






Snell‘s law requires that the angle of reflection is equal to the angle of
incidence.

if V
2
is less than V
1
the ray is bent towards the normal

if V
2
is greater than V
1
the ray is bent away from the normal.

If V
2
is greater than V
1
the angle of refraction is greater than the ngle
of incidence. The latter result can lead to a special condition where θ
2
=
90° . The angle of incidence for which this occurs is called the critical angle, θ
c
.
The critical angle is given by;
33

The geometry of this wave is correctly given by Snell‘s law but the nature
of a wave that propagates along the interface is not so simple. The critically
refracted wave is a disturbance that propagates along the interface with
the velocity of the lower medium. As it goes its wave front acts as a moving
source of waves that propagate back into the upper medium with velocity
V
1
along rays which are parallel to the reflected wave at the critical angle.
This phenomenon is illustrated in the following cartoon.

When a P wave is incident on a boundary, at which elastic properties
change, two reflected waves (one P, one S) and two transmitted waves
(one P, one S) are generated.
34


Angles of transmission and reflection of the S waves are less than the P
waves. Exact angles of transmission and reflection are given by:

p is known as the ray parameter.

There are two critical angles corresponding to when transmitted P and S
waves emerge at 90°.



35

Head Waves

The interaction of this wave with the interface produces secondary sources
that produce an upgoing wavefront, known as a head wave, by Huygen‘s
principle.
The ray associated with this head wave emerges from the interface at the
critical angle.


This phenomenon is the basis of the refraction surveying method.








36

Diffractions

Reflection by Huygen’s Principle

When a plane wavefront is incident on a plane boundary, each point of the
boundary acts as a secondary source. The superposition of these secondary
waves creates the reflection.

Diffraction by Huygen’s Principle
If interface truncates abruptly, then secondary waves do not cancel at the
edge, and a diffraction is observed.

- This explains how energy can propagate into shadow zones.
- A small scattering object in the subsurface such as a boulder will
produce a single diffraction.
- A finite-length interface will produce diffractions from each end, and
the interior parts of the arrivals will be opposite polarity.

37

PROBLEM SET - 1

1. A steel beam is placed vertically in the basement of a building to keep
the floor above from sagging. The load on the beam is 5.8 x10
4
N and
the length of the beam is 2.5 m, and the cross-sectional area of the beam
is 7.5 x 10
-3
m
2
. Find the vertical compression of the beam.

2. A 0.50 m long string, of cross- sectional area 1.0 x10
-6
m
2
, has a
Young‘s modulus of 2.0 x 10
9
Pa. By how much must you stretch a string
to obtain a tension of 20.0 N?

3. The upper surface of a cube of gelatin, 5.0 cm on a side, is displaced by
0.64 cm by a tangential force. If the shear modulus of the gelatin is 940 Pa,
what is the magnitude of the tangential force?

4. An anchor, made of cast iron of bulk modulus 60.0 x 10
9
Pa and a
volume of 0.230 m
3
, is lowered over the side of a ship to the bottom of the
harbor where the pressure is greater than sea level pressure by
1.75 x 10
6
Pa. Find the change in the volume of the anchor.

5. 1. AN ARKOSE HAS A DENSITY OF 2.62 G/CM, A YOUNG MODULUS OF 0.16X10
N/M , AND A POISSON'S RATIO OF 0.29 . TWELVE GEOPHONES ARE ARRANGED
ALONG A LINE AT 10 M INTERVAL. THE SHOT POINT IS LOCATED 5 M FROM THE
FIRST GEOPHONE IN THE LINE.
CONSTRUCT A GRAPH THAT ILLUSTRATES TIME OF ARRIVAL AGAINST GEOPHONE
POSITION FOR THE P-WAVE , S-WAVE AND SURFACE WAVE.


6. FROM SNELL‘S LAW, CALCULATE THE CHANGE IN DIRECTION OF A SEISMIC
WAVE WHEN IT REFRACTED FROM A SANDSTONE STRATUM ( V = 4000 M/SEC
AND 350 M THICK) INTO A LIMESTONE STRATUM (V=6000 M/SEC ) FOR EACH OF
THE FOLLOWING ANGLES OF INCIDENCE 0, 15, 25, 35, 45, AND 60.

WHAT IS THE CRITICAL ANGLE.
WHAT IS THE MINIMUM DISTANCE FOR REFRACTION ARRIVALS
WHAT IS THE CROSS- OVER DISTANCE
COMPARE THE VALUES OF X
CO
WITH THE THICKNESS OF SANDSTONE LAYER
WHAT IS THE NMO AND REFLECTION TRAVEL TIME

38




7. THE VOLUM OF ALUMINUM BLOCK WAS PLACED UNDER HYDRAULIC PRESSURE IS
0.4 M
3
A.
A. FIND THE CHANGE IN VOLUM OF THE ALUMINUM WHEN SUBJECTED TO PRESSURE
OF 2.1 X 10
7
N/M
2
. THE BULK MODULUS IS 0.85 X 1011 N/M
2
AND THE RIGIDITY
MODULUS IS 0.36 X 10
11
N/M
2
.
B. B. WHAT IS THE CUBICAL DILATATION
C. FIND YOUNG‘S MODULUS, COMPRESSIBILITY, AND POISSON‘S RATIO.




8. A 15 HZ SEISMIC WAVE TRAVELLING AT 5.5 KM/SEC PROPAGATES FOR 1500 M
THROUGH A MEDIUM WITH AN ABSORPTION COEFFICIENT OF 0.3 dB . WHAT IS
THE WAVE ATTENUATION IN dB DU E SOLELY TO ABSORPTION.



9. CALCULATE THE AMPLITUDE OF THE REFLECTED AND TRANSMITTED P- AND S-
WAVES WHERE THE INCIDENT P-WAVE STRIKE THE INTERFACE FROM A WATER
LAYER ( P-VELOCITY =2.5 K/S, 5-VELOCITY =0, DENSITY = 1.20 g/cc) AT 25
0

WHEN THE SEAFLOOR IS:

A. SOFT ( P=3K/S, S=1.5 K/S. DENSITY2.O g/cc

B. HARD ( P= 4 K/S, 5= 2.5 K/S, DENSITY =2.5 g/cc

C. REPEAT FOR AN ANGLE OF INCIDENCE 30
0




10. A rock sample is taken to the lab and is subjected to a uniaxial stress (that is, it is
stressed in only one direction with the remaining directions free). As a result of the
stress, the length of the sample increases by 3% and the width decreases by 1%.
What is the ratio of the P wave velocity to the S wave velocity in this sample?





39

11. A material has a shear modulus of 8.8×10
9
Pa, a bulk modulus of 2.35×10
10
Pa, a
density of 2200 kg/m3 and a quality factor Q = 100.
a) What are the P-wave velocity and S-wave velocity of the medium, in units of km/s
b) If a P-wave of frequency 10 Hz has a displacement amplitude of 1 μm at a distance
of 10 m from the source, what would be the wave amplitude at 100 m?



12. Near-surface fresh water in a Lake Superior has been observed to have a P-wave
velocity of 1435 meters/sec. Estimate its bulk modulus, assuming it is pure water.
2. A laboratory has determined that the Gabbro has the following properties:
Bulk modulus= 0.952130 X 10
12
dyne/cm2
Shear modulus=0.403425 X 10
12
dyne/cm2
Density=2.931 gm/cc
Determine the (a) shear wave velocity, the (b) compressional wave velocity, and
(c) Poisson's ratio for this rock.










41








CHAPTER 2






SEISMIC REFRACTION METHOD
- Fundamentals
- Two Horizontal Interfaces
- Dipping Interfaces
- The Non ideal Subsurface
- The Delay-Time Method
- Field Procedures & Interpretation
Problem Set - 2 ( 1 – 3 – 4 – 6 - 7 )
41

Seismic Refraction
A signal, similar to a sound pulse, is transmitted into the Earth. The signal
recorded at the surface can be used to infer subsurface properties. There
are two main classes of survey:
- Seismic Refraction: the signal returns to the surface by refraction
at subsurface interfaces, and is recorded at distances much greater
than depth of investigation.

- Seismic Reflection: the seismic signal is reflected back to the
surface at layer interfaces, and is recorded at distances less than
depth of investigation.

Applications
Seismic Refraction
- Rock competence for engineering applications
- Depth to Bedrock
- Groundwater exploration
42

- Correction of lateral, near-surface, variations in seismic reflection
surveys
- Crustal structure and tectonics
Seismic Reflection
- Detection of subsurface cavities
- Shallow stratigraphy
- Site surveys for offshore installations
- Hydrocarbon exploration
- Crustal structure and tectonics
Refraction surveys use the process of critical refraction to infer interface
depths and layer velocities. Critical refraction requires an increase in
velocity with depth. If not, then there is no critical; refraction: Hidden layer
problem.

- Geophones laid out in a line to record arrivals from a shot. Recording
at each geophone is a waveform called a seismogram.
- Direct signal from shot travels along top of first layer.
- Critical refraction is also recorded at distance beyond which angle of
incidence becomes critical.

43

Example
For a shallow survey, 12-24 vertical 30 Hz geophones would be laid out to
record a hammer or shotgun shot.




First Arrival Picking
In most refraction analysis, we only use the travel times of the first arrival
on each recorded seismogram.
As velocity increases at an interface, critical refraction will become first
arrival at some source-receiver offset.



First Break Picking
The onset of the first seismic wave, the first break, on each seismogram is
identified and its arrival time picked.
Example of first break picking on Strataview field monitor

44



Travel Time Curves

Analysis of seismic refraction data is primarily based on interpretation of
critical refraction travel times.
Plots of seismic arrival times vs. source-receiver offset are called travel time
curves.
Example
Travel time curves for three arrivals shown previously:
- Direct arrival from source to receiver in top layer
- Critical refraction along top of second layer
- Reflection from top of second layer
45


Critical Distance
Offset at which critical refraction first appears.
- Critical refraction has same travel time as reflection
- Angle of reflection same as critical angle
Crossover Distance
Offset at which critical refraction becomes first arrival.

Field Surveying
Usually we analyze P wave refraction data, but S wave data occasionally
recorded



Interpretation of Refraction Travel time Data

After completion of a refraction survey first arrival times are picked from
seismograms and plotted as traveltime curves
46

Interpretation objective is to infer interface depths and layer velocities
Data interpretation requires making assumption about layering in
subsurface: look at shape and number of different first arrivals.
Assumptions
- Subsurface composed of stack of layers, usually separated by plane
interfaces
- Seismic velocity is uniform in each layer
- Layer velocities increase in depth
- All ray paths are located in vertical plane, i.e. no 3-D effects with
layers dipping out of plane of profile
Analysis based on considering critical refraction ray paths through
subsurface.
[There are more sophisticated approaches to handle non-uniform velocity
and 3-D layering.]


















47

1. Horizontal Interfaces: Two Layers


For critical refraction at top of second layer, total travel time from source S
to receiver G is given by:

Hypoteneuse and horizontal side of end 90
o
-triangle are:
and respectively.
So, as two end triangles are the same:


At critical angle, Snell‘s law becomes:
Substituting for V
1
/ V
2
, and using cos
2 2

48


This equation represents a straight line of slope 1/V
2
and intercept


Interpretation of Two Layer Case

From travel times of direct arrival and critical refraction, we can find
velocities of two layers and depth to interface:
1. Velocity of layer 1 given by slope of direct arrival
2. Velocity of layer 2 given by slope of critical refraction
3. Estimate t
i
from plot and solve for Z:




49

Depth from Crossover Distance
At crossover point, traveltime of direct and refraction are equal:



Solve for Z to get:

[Depth to interface is always less than half the crossover distance]


X
critical
= Minimum distance for refraction arrival

= 2 z
1
tan
ic

51









51

2. Dipping Interfaces : Two Layer Case

When a refractor dips, the slope of the traveltime curve does not represent
the "true" layer velocity:
- shooting updip, i.e. geophones are on updip side of shot, apparent
refractor velocity is higher
- shooting downdip apparent velocity is lower
To determine both the layer velocity and the interface dip, forward and
reverse refraction profiles must be acquired.



52

Note: Travel times are equal in forward and reverse directions for
switched, reciprocal, source/receiver positions.

Geometry is same as horizontal 2-
extra time delay at D. So traveltime is:


Formulae for up/downdip times are (not proved here):



where V
u
/ V
d
and t
u
/ t
d
are the apparent refractor velocities and intercept
times.
;

Can now solve for dip, depth and velocities:
C
:
;
[V
1
is known from direct arrival, and V
u
and V
d
are estimated from the
refraction traveltime curves]
2) Can find layer 2 velocity from Snell‘s law:
53

1. Can get slant interface depth from intercept times, and convert to
vertical depth at source position:
;


3. Faulted Planar Interface ( Diffraction )



If refractor faulted, then there will be a sharp offset in the travel time
curve:

54


Can estimate throw on fault from offset in curves, i.e. difference between
two intercept times, from simple formula:





Delay Times in Refraction

For irregular travel time curves, e.g. due to bedrock topography or glacial
fill, much analysis is based on delay times.
Total Delay Time . Difference in travel time along actual ray path and
projection of ray path along refracting interface:



55

;

Total delay time is delay time at shot plus delay time at geophone:



For small dips, can assume x=x
I
and:



Refractor Depth from Delay Time

If velocities of both layers are known, then refractor depth at point A can
be calculated from delay time at point A:



Using the triangle to get lengths in terms of z:
56



Using Snell‘s law to express angles in terms of velocities:

Simplifying:

So refractor depth at A is:















57

Blind layer problem

Blind layers occur when there is a low velocity layer (LVL). Head waves
only occur for a velocity increase. Thus, there will be no refraction from
the top of the LVL and this layer will not be detected on the time-distance plot.
This is shown below.

Hidden layer problem

Hidden layers result when there is a velocity increase with layer depth, but
the head wave from the top of one layer is never the first arrival on a
time-distance plot. Head waves from a deeper layer arrive at the detectors
before the arrivals from this layer. Two factors can cause hidden layers: 1)
the layer is very thin or 2) there is only a small velocity increase at the top
of the layer. This is shown below. It is sometimes possible to recognize
hidden layers by looking for arrivals after the first arriving energy.

Layers may not be detected by first arrival analysis:
A. Velocity inversion produces no critical refraction from layer 2
B. Insufficient velocity contrast makes refraction difficult to
identify
C. Refraction from thin layer does not become first arrival
D. Geophone spacing too large to identify second refraction
58


59

PROBLEM SET - 2

Q1. Consider a case where there is a continuous velocity increase
with depth, as is commonly observed in the Earth. The propagation of
seismic waves can be investigated by assuming that the subsurface is made
up of infinitely thin layers of uniform velocity. Sketch the expected ray
paths and travel time curve.
Remember that Snell‘s Law requires that the ray parameter is constant: P
= sinθ/ v





61

2. Given the reversed refraction observations (travel time vs. distance curves) shown
below, calculate the velocities and depths to the interfaces. Calculate the dip angles of
the interfaces.





3. Given the following schematic travel-time curve, describe a subsurface structure
and/or velocity changes that may explain them.

61

4. A seismic refraction survey was conducted along an abandoned railroad grade about
2.5 miles southeast of the town of Osakis. The railroad grade is known to be resting on
Pleistocene glacial deposits. A 12 channel system was used with an sledge hammer for
an energy source, and the following first break times were picked from the traces.
Distance First break time(meters) (milliseconds)
5 14
15 23
25 30
35 36.75
45 42
55 50
65 54.5
75 61
85 65.5
95 66.5
105 70
115 73.25

A reversed profile yielded essentially identical results, so we can assume horizontal
layering. It is assumed that the first and second layers will represent the grade fill and
the glacial deposits, respectively.

Plot the travel time relationships and estimate the velocities of the first two layers (in
meter/second). What is the thickness of the grade fill?
Based on your velocity estimate for the second layer, do you think the glacial deposits
are saturated or unsaturated (below or above the water table)?
Do you have any evidence of bedrock (ie a third layer) below the glacial deposits? If so,
what is its velocity (in meters/second) and depth?


5. SUPPOSE THAT A REVERSED REFRACTION SURVEY ( USING SHOTS A AND B )
INDICATED VELOCITIES V1 = 1500 M/SEC. AND V2 = 2500 M/SEC. FROM SHOT A
AND VELOCITIES V1 = 1500 M/SEC AND V2 = 3250 M/SEC. FROM SHOT B.
FIND THE DIP OF REFRACTOR. WHAT WOULD BE THE CHANGES IN VELOCITIES IF
THE FEFRACTOR HAD A SLOPE 10 DEGREES LARGER THAN THE ONE YOU
COMPUTED.
62

6. Construct a travel time curve to a distance of 120 m for a structure with an 10 m
thick soil layer with P-wave velocity 500 m/s over a saturated layer 20 m thick with P-
wave velocity 1500 m/s over bedrock with velocity 3000 m/s. From the graph, what are
the 2 crossover distances?



7. Construct a travel time curve for the following data for both the forward and reverse
profiles. Evaluate slopes and intercept times, and use those values to determine the
subsurface structure for the first 2 layers, and then estimate the properties of the
bottom layer. Offsets are in meters and times are in milliseconds.

Offset x 5 10 15 20 25 30 35 40 45 50 55 60
Forward 8.3 16.7 25.0 33.3 41.7 50.0 58.3 66.7 72.5 75.1 77.7 80.3
Reverse 8.3 16.7 25.0 33.3 40.7 44.3 48.0 51.6 55.3 58.9 62.6 66.3



8. The following data are from a profile over a buried steep fault scarp underlying
alluvium. Use the data to determine velocities for the alluvium and bedrock and the
throw and approximate position of the buried fault step. Offsets are in meters and times
are in milliseconds.

Offset x 5 10 15 20 25 30 35 40 45 50 55 60
Forward 3.6 7.1 10.7 14.3 17.9 21.4 23.0 24.0 24.9 25.8 26.7 27.7
Reverse 3.6 7.1 10.7 14.3 17.9 21.4 25.0 28.6 30.4 25.8 26.7 27.7



63




CHAPTER 3




SEISMIC REFLECTION METHOD
- A Single Subsurface Interface
- Analysis of Arrival Times
- Normal Move out
- Determining of Velocity & Thickness
- Dipping Interface
- Common Field Procedures
- Velocity Analysis
- Applications in Petroleum exploration
Problem Set - 3
64

SEISMIC REFLECTION

1. Reflection at normal incidence

Consider two horizontal layers that have different seismic velocities (v) and
densities (ρ). A P-wave with amplitude Ai travels downward through the
upper layer and encounters the interface between the layers at 90° (normal
angle). This produces two new P-waves: a reflected P-wave that travels
upward through Layer 1 and a transmitted P-wave that enters Layer 2.
The reflection co-efficient is the ratio of the amplitudes of the reflected
and incident waves: R = Ar/Ai
Similarly, the transmission co-efficient is the ratio of the amplitudes of
the transmitted and incident waves: T = At/Ai
The amount of energy that is partitioned into transmission and reflection
depend on the angle between the incident wave and interface and on the
acoustic impedance (Z) of each layer:
Z1 = ρ1v1 and Z2 = ρ2v2

For normal incident waves, it can be shown that:

R = ρ
2
v
2
- ρ
1
v
1
/ ρ
2
v
2
+ ρ
1
v
1
= Z
2
– Z
1
/ Z
2
+ Z
1


T = 2 ρ
1
v
1
/ ρ
2
v
2
+ ρ
1
v
1
= 2Z
1
/ Z
2
+ Z
1


These are the Zoeppritz equations. There are also more complicated
forms of the Zoeppritz equations that can be used for any angle of
incidence.

65



These equations show that the reflection and transmission coefficients
depend on the difference in impedance between the two layers.
• if Z1 = Z2, there is no reflection. All energy is transmitted into the second
layer. This does not mean that ρ1=ρ2 and v1= v2! All that matters is that
ρ
1
v
1
= ρ
2
v
2
.
• R can have a value of +1 to -1. R will be negative when Z
1
> Z
2
. A
negative value means that there will be a phase change of 180° in the
phase of the reflected wave (a peak becomes a trough). This is called a
negative polarity reflection.
• T is always positive – transmitted waves have the same phase as the
incident wave. T can be larger than 1.
• Reflection coefficients for the Earth are generally less than ±0.2, with
maximum values of ±0.5. Most energy is transmitted, not reflected.


Case 1: An increase in velocity with depth

A 600 m thick layer of sandstone overlies a granite basement with a higher
velocity. A seismic wave is generated at the surface and travels vertically
downward. At the sandstonegranite interface, the incident wave is split into
a reflected wave and transmitted wave. The amplitude of the reflected and
transmitted waves (Ar and At) can be calculated from the Zoeppritz
equations. Assume that Ai = 1.0 and that there is no geometrical
66

spreading, attenuation, or scattering. Velocity and density are constant
within each layer.
First, calculate the impedance of each layer:
Z1 = ρ
1
v
1
= 2700 × 4.1 = 11,070 (kg km s
-1
m
-3
)
Z2 = ρ2v2 = 2700 × 5.6 = 15,120 (kg km s
-1
m
-3
)
The reflection and transmission co-efficients are then:
R = 0.15 , T = = 0.85

The amplitude of the two waves are:

A
r
= R x A
i
= 0.15 A
t
= T x A
i
= 0.85



Consider a seismic survey configuration where you have a seismic source
and a receiver on the ground next to each other. The receiver will record
seismic waves that travel directly between the source and receiver, as well
as seismic waves that are reflected upward to the surface. In this case,
only one reflected wave will be recorded. The time at which it arrives at the
receiver is:
Time = 2 x 600 / 4100 = 0.29s
The seismic record will look like this:
67




More than two layers:

The same Zoeppritz equations can be applied to models with more than
two layers.

Case 1: Velocity increase with depth

In this case, there will be two reflected waves that are recorded by the
seismic station (also called arrivals).
A1 is the wave that is reflected from Interface A.
A2 is the wave that is transmitted through Interface A, reflected from
Interface B, transmitted through Interface A, and then recorded at the
surface

68

.

Arrival A1:

Amplitude: given by the reflection co-efficient at interface A (RA):
Z1 = 2700 × 3.1 = 8370 Z2 = 2700 × 4.5 = 12150
Therefore: R
A
= 0.18 , Arrival time: 0.26s



Arrival A2:
To calculate the arrival time and amplitude of A2, we need to consider all
the interfaces that it has encountered between the source and receiver.

1. Transmitted through Interface A.
The amplitude of the transmitted wave at A is T
A
= 0.82


69

2. Reflected at interface B.
The reflection amplitude depends on:
• The amplitude of the incident wave. The incident wave is the one that
was transmitted through Interface A: TA = 0.82
• Reflection co-efficient from Layer 2 to Layer 3 (RB)

Z2 = 12150 Z3 = 2700 × 6.8 = 18360

R
B = 0.2


Therefore, the amplitude of the reflected wave is: TA × RB = 0.16

3. Transmitted through Interface A.
The amplitude of the transmitted wave is the product of:
• the amplitude of the reflected wave from Interface B ( = TA × RB
=0.16)
• transmission co-efficient from Layer 2 to Layer 1

Therefore the final amplitude of A2 will be: TA × RB × T’A = 0.16 ×
1.18 = 0.19
The total travel time of A2 will be: 0.13 + 0.40 + 0.13s = 0.66s
The seismic record will look like:


71

Reflection time- distance plots
Consider a source (shot point) at point A with geophones spread out along
the x-axis on either side of the shot point.


This is the equation of a hyperbola symmetric about the t axis. The travel
time plot for the direct wave arrivals and the reflected arrivals are shown in
the following plot. The first layer is 100 m thick and its velocity is 500 m/s.
The intercept of the reflected arrival on the t axis, t
i
, is the two-way zero
71

offset time and for this model is equal to 400ms. At large offsets the
hyperbola asymptotes to the direct wave with slope 1/V
1
.



In most seismic reflection surveys the geophones are placed at offsets
small compared to the depth of the reflector. Under this condition an
approximate expression can be derived via:

72

Since is less than 1, the square root can be expanded with the
binomial expansion. Keeping only the first term in the expansion the
following expression for the travel time is obtained:

This is the basic travel time equation that is used as the starting point for
the interpretation of most reflection surveys.





















73

Moveout

A useful parameter for characterizing and interpreting reflection arrivals is
the moveout, the difference in travel times to two offset distances. The
following expanded plot of one side of the hyperbola of the previous
reflection plot shows the moveout, Δt, for two small offsets.






74

Using the small offset travel time expression for x
1
and x
2
yields the
following expression for the moveout:
The normal moveout (NMO), Δt
n
, is a special term used for the moveout
when x
1
is zero. The NMO for an offset x is then:








With the value of the intercept time, t
i
, the velocity is determined via:

For a given offset the NMO decreases as the reflector depth increases
and/or as the velocity increases.
In a layered medium the velocity obtained from the NMO of a deep
reflector is an average of the intervening layer velocities. Dix (1955) found
that the root-mean-square velocity defined by:

where V
i
is the velocity in layer i and t
i
is the travel time in layer i is the best
average to use.
75

In interpretation the NMO‘s for successive reflections are used to obtain the
average velocity to each reflector. Assuming these are the V
rms
velocities
defined above then Dix (1955) showed that the velocity in the layer
bounded by the n
th
and n-1
th
layer is given by:









Dip moveout
If the interface is dipping as in the figure below the up-dip and down-dip
travel times are changed by an amount dependant on the dip angle θ. The
time-distance plot is still a hyperbola but the axis of symmetry is shifted up-
dip by 2h sinθ. (Shown by the dashed line in the figure. Note also that the
depth is still the perpendicular distance from the interface to the shot
point). The binomial expansion for the travel time for small offsets
becomes:

For geophones offset a distance x up-dip and down-dip, the dip moveout is
defined as:

For small dips when the dip moveout yields the dip via;

76

The velocity can be obtained with sufficient accuracy by averaging the
velocities obtained in the usual manner from the up-dip and down-dip
NMO‘s.

77


Common Mid-Point Gathers

There are two disadvantages to using only a shot gather for analysis:
(1) Reflections tend to have a low amplitude (generally less than 20% of
the incident wave amplitude). This means that noise in the seismic data can
obscure reflections.
(2) Each reflection occurs at a different point on the interface. The analysis
of shot gathers assumes uniform horizontal layers. If there are significant
lateral variations in structure, this assumption is no longer valid and the
analysis can result in errors.



These problems can be overcome by using shots from a number of
different points and multiple detectors. The source and detectors are
moved in between shots.
From the complete data set, a subset of traces are chosen that have a
common reflection point. The reflection point is taken to be halfway in
between the shot and the detector.
• this is valid for areas with horizontal layering
• if the reflection occurs at a dipping interface, this is an approximation, but
does not introduce large errors

78



The collection of traces with the same reflection point is called a
common mid-point gather (CMP gather) or common depth point
(CDP) gather.
79

Fresnel Zone

Tells us about the horizontal resolution on the surface of a reflector. In
designing a seismic survey, it is important to make sure that this spacing
is less than the Fresnel zone, so that the survey layout does not limit
the resolution of the data.

First Fresnel Zone
The area of a reflector that returns energy to the receiver within half a
cycle of the first reflection. The width of the first Fresnel zone, w:

( d + λ/4 )
2
= d
2
+ ( w/2)
2

W
2
= 2d λ + λ
2
/ 4





If an interface is smaller than the first Fresnel zone it appears as an
point diffractor, if it is larger it appears as an interface

Example:
30 Hz signal, 2 km depth where α = 3 km/s then λ = 0.1 km and the
width of the first Fresnel zone is 0.63 km
81

Resolution of structure


Consider a vertical step in an interface. To be detectable the step must
cause an delay of ¼ to ½ a wavelength . This means the step (h) must
be 1/8 to ¼ the wavelength (two way traveltime)

Example:
20 Hz, α = 4.8 km/s then λ = 240 m, Therefore need an offset greater
than 30 m , Shorter wavelength signal (higher frequencies) have better
resolution.



81

PROBLEM SET - 3

1. Consider a high velocity layer that overlies a low velocity
layer.




A. What are the amplitudes of the reflected and transmitted waves?
B. At what time will the reflected wave arrive back at the surface? What
is its polarity?
C. What will the seismic record at the surface look like?


82


2. Consider a case with a low velocity layer. This could
represent a gas- filled layer within high velocity rocks. What
are the first three arrivals (after the direct P- wave)?







Note : The corresponding seismic record will be:






83





3. CALCULATE THE REFLECTION AND TRANSMISSION COEFFICIENTS FOR NORMAL
INCIDENCE ON EACH BOUNDARY WITH THE FOLLOWING MATERIAL PROPERTY
CONTRASTS. WHAT PERCENT OF THE ENERGY IS REFLECTED AND
TRANSMITTED AT EACH INTERFACE.

VELOCITIES FT/S. Density


VERY STRONG REFLECTOR 11000 & 15000 NONE
GOOD REFLECTOR 14000 & 15000 NONE
WEAK REFLECTOR 8000 & 8200 NONE
SOFT OCEAN BOTTOM NONE 1.0 & 2.0
HARD OCEAN BOTTOM 5000 & 10000 1.0 & 2.5
WEATHERED OVER UNWEATH.
MATERIAL 1600 & 6000 1.6 & 2.0
SAND WITH 30% POROSITY
OVER THE SAME SAND WITH
GAS-FILLED PORES. 6000 & 9000 1.4 & 2.4


4. Seismic waves with a dominant frequency of 50 Hz travel through
sediments with a velocity of 3000 m/s.
• What is the smallest layer thickness that will be detected?
• If the deepest reflector is at a depth of 2000 m, what is the size of the
smallest horizontal feature that will be detected?
• What is the largest geophone spacing that should be used in the
survey?

84


5. Use the Dix Formula approach (RMS velocity) to determine the
velocities and thicknesses for 3 layers using the following reflection
data:

Offset x (m) 3 6 9 12 15 18 21 24 27 30 33 36
Reflection 1 21.4 25.0 30.1 36.1 42.5 49.2 56.2 63.3 70.4 77.6 84.9 92.2
Reflection 2 62.3 62.4 62.6 62.9 63.2 63.6 64.1 64.7 65.4 66.1 66.9 67.7
Reflection 3 79.4 79.5 79.6 79.9 80.1 80.5 80.9 81.3 81.8 82.4 83.0 83.7

Times are in milliseconds.
First, make a graph of t
2
-x
2
values for all 3 reflections. Note that the
slope for the first reflector is simply 1/V
1
2
, but that for the deeper
reflectors the slope corresponds to the RMS velocity. Use the first
intercept and the single-layer travel time formula to determine the
thickness of layer 1.
Then from the slopes and the intercept values, use the Dix Formula to
compute the interval velocities for layers 2 and 3.
Finally, use the fact that t
n
– t
n-1
= 2
zn
/V
n
to compute the thicknesses
for layers 2 and 3.
85







CHAPTER 4

EARTHQUAKE SEISMOLOGY

- Definition and Historical review
- Classification of Earthquakes
- Earthquakes : Where and Why
- Causes of Earthquakes
- Earthquake Epicenter & Hypocenter
- Magnitude & Intensity

86

EARTHQUAKE SEISMOLOGY
COMPOSITION AND STRUCTURE OF THE EARTH
1. Crust. The outermost rock layer, divided into continental and
oceanic crust
a. Continental Crust (averages about 35 km thick; 60 km in
mountain ranges; diagram shows range of 20-70 km) Granitic
composition
b. Oceanic Crust (5 - 12 km thick; diagram shows 7-10 km
average)
Basaltic composition. Oceanic crust has layered structure (ophiolite
complex) consisting of the following:
1. Pillow basalts,
2. " Sheeted dikes " - interconnected basaltic dikes
3. Gabbro
2. Mantle (2885 km thick). The mantle stretches from the below
the crust to 2900 km below the surface. The upper part is partially
molten and the lower part is very dense. The main mantle rock is
peridotite.
Lithosphere = outermost 100 km of Earth . Consists of the crust plus
the outermost part of the mantle. Divided into tectonic or lithospheric
plates that cover surface of Earth
Asthenosphere = low velocity zone at 100 - 250 km depth in Earth
(seismic wave velocity decreases). Rocks are at or near melting point.

87

3. Outer core (2270 km thick)
S-waves cannot pass through outer core, therefore we know
the outer core is liquid (molten).
Composition: Molten Fe (85%) with some Ni, based on
studies of composition of meteorites. Core may also contain
lighter elements such as Si, S, C, or O.
Convection in liquid outer core plus spin of solid inner core
generates Earth's magnetic field. Magnetic field is also
evidence for a dominantly iron core.
4. Inner core (1216 km radius). Solid Fe (85%) with
some Ni - based on studies of meteorites


88












89

Plate Tectonics

The Earth's lithosphere is broken up into 6 major plates and about
14 minor ones. Oceanic plates are 50-100 km thick. Continental plates
are 100-250 km thick. Tectonic plates can include both continental and
oceanic areas. Six major plates are:
1. Indian-Australian 4. Antarctic
2. Pacific 5. African
3. American (N. and S) 6. Eurasian




91

o Plate Boundaries - Tectonic plates interact in various ways
as they move across the asthenosphere, producing
volcanoes, earthquakes and mountain systems. There are 3
primary types of Tectonic Plate boundaries: Divergent
boundaries; Covergent boundaries; and Transform.

o Divergent - Plates move away from one another, creating a
tensional environment. Characterized by shallow-focus
earthquakes and volcanism. Release of pressure causes
partial melting of mantle peridotite and produces basaltic
magma. Magma rises to surface and forms new oceanic
crust. Occur in oceanic crust (oceanic ridges) and in
continental crust (rift valleys). Continental rift valleys may
eventually flood to form a new ocean basin.


91

o Convergent - Plates move toward one another, creating a
compressional environment. Characterized by deformation,
volcanism, metamorphism, mountain building, seismicity,
and important mineral deposits. Three possible kinds of
convergent boundaries:

1. Oceanic-Oceanic Boundary - One plate is
subducted, initiating andesitic ocean floor volcanism on
the other. Can eventually form an island arc volcanic
island chain with an adjacent deep ocean trench.
Characterized by a progression from shallow to deep
focus earthquakes from the trench toward the island
arc (Benioff zone). May also form a back-arc basin if
subduction rate is faster than forward motion of
overriding plate.



92

2. Oceanic-Continental Boundary - Oceanic plate is
dense and subducts under the Lighter continental
plate. Produces deep ocean trench at the edge of the
continent. About half the oceanic sediment descends
with the subducting plate; the other half is piled up
against the continent. Subducting plate and sediments
partially melt, producing andesitic or granitic magma.
Produces volcanic mountain chains on continents called
volcanic arcs and batholiths. Part of the oceanic plate
can be broken off and thrust up onto the continent
during subduction (obduction). Obduction can expose
very deep rocks (oceanic crust, sea floor sediment, and
mantle material) at the surface. Characterized by
shallow to intermediate focus earthquakes with rare
deep focus earthquakes.


93



3. Continental-Continental Boundary - Continental
crust cannot subduct, so continental rocks are piled up,
folded, and fractured into very high complex mountain
systems. Characterized by shallow-focus earthquakes,
rare intermediate-focus earthquakes. and practically no
volcanism.







94

o Transform - Plates move laterally past one another. Largely
shear stress with lithosphere being neither created nor
destroyed. Characterized by faults that parallel the direction
of plate movement, shallow-focus earthquakes, intensely
shattered rock, and no volcanic activity. Shearing motion can
produce both compressional stress and tensional stress
where a fault bends. Transform faults occur on land, connect
segments of the oceanic ridge, and provide the mechanism
by which crust can be carried to subduction zones.





95

Types of Faults


A fault is a fracture or zone of fractures between two blocks of
rock. Faults allow the blocks to move relative to each other. This
movement may occur rapidly, in the form of an earthquake - or
may occur slowly, in the form of creep. Faults may range in length
from a few millimeters to thousands of kilometers. Most faults
produce repeated displacements over geologic time. During an
earthquake, the rock on one side of the fault suddenly slips with
respect to the other. The fault surface can be horizontal or vertical
or some arbitrary angle in between.
Earth scientists use the angle of the fault with respect to the
surface (known as the dip) and the direction of slip along the fault
to classify faults. Faults that move along the direction of the dip
plane are dip-slip faults and described as either normal or reverse,
depending on their motion. Faults that move horizontally are
known as strike-slip faults and are classified as either right-lateral
or left-lateral. Faults that show both dip-slip and strike-slip motion
are known as oblique-slip fault.






96






slip fault in which the block above the - is a dip A normal fault
fault has moved downward relative to the block below. This type
of faulting occurs in response to extension and is often observed in
the Western United States Basin and Range Province and along
oceanic ridge systems.


97






ich the upper block, above slip fault in wh - is a dip A thrust fault
the fault plane, moves up and over the lower block. This type of
faulting is common in areas of compression, such as regions where
one plate is being subducted under another as in Japan and along
the Washington coast. When the dip angle is shallow, a reverse
fault is often described as a thrust fault.
98

is a fault on which the two blocks slide past slip fault - A strike
one another. These faults are identified as either right-lateral or
left lateral depending on whether the displacement of the far block
is to the right or the left when viewed from either side. The San
Andreas Fault in California is an example of a right lateral fault.




99

CAUSES OF EARTHQUAKES

An earthquake is a sudden shuddering or trembling of the earth
produced by shock waves or vibrations passing through it. Earthquakes
occur in regions of the earth that are undergoing deformation. Energy is
stored in the form of elastic strain as the region is deformed. This
process continues until the accumulated strain exceeds the strength of
the rock, and then fracture or faulting occurs. The opposite sides of the
fault rebound to a new equilibrium position, and the energy is released
in the vibrations of seismic waves and in heating and crushing of the
) focus, hypocenter ( acturing usually start from a point rock. Rock fr
close to one edge of the fault plane and propagates along the plane with
a typical velocity of some 3 Km/sec. The vertical projection of the
. er epicent hypocenter onto the earth's surface is called the




111

Types of earthquakes
There are many different types of earthquakes: tectonic, volcanic, and
explosion. The type of earthquake depends on the region where it
occurs and the geological make-up of that region.
1. tectonic earthquakes. These occur when rocks in
the earth's crust break due to geological forces created
by movement of tectonic plates.
2. volcanic earthquakes, occur in conjunction with
volcanic activity.
3. Collapse earthquakes are small earthquakes in
underground caverns and mines.
4. Explosion earthquakes result from the explosion of
nuclear and chemical devices.

EARTHQUAKES , WHY AND WHERE DO THEY OCCUR ?

It has long been recognized that earthquakes are not evenly distributed
over the earth. The eventual correlation of the earthquake pattern with
the earth's major surface features was a key to the evolution of the
. This is the most recent and broadly satisfying theory tectonics plate
explanation theory of the majority of earthquakes. The basic idea is that
the earth's outermost part ( Lithosphere ) consists of several large and
fairly stable slabs of solid and relatively rigid rock called plates. Each
plate extends to a depth of about 80 Km .

There are two major belts along which most of the world's
). earthquakes Interplate earthquakes occur (

: A large part ( 80 % ) of the seismic belt pacific - circum The 1.
energy released by all earthquakes is released along this belt. This
includes the western coasts of South and North America , Japan,
Philippines, and a strip through the East Indies and New Zealand.

111

: A high energy belt ) European - Asiatic ( Alpide The 2.
concentration ( 10 % ) can also be seen along this belt. It extends
from the Pacific belt in New Guinea through Summatra and Indonesia,
the Himalayas, and mountains and faults of the Middle East , the Alps ,
and into the Atlantic Ocean far as the Azores.



Sporadically, earthquakes also occur at rather large distances from the
, kes earthqua Intraplate respective plate margins . These so called
show a diffuse geographical distribution and there origin is still poorly
understood. It can be large and because of there unexpectedness and
infrequency can cause major disasters.


112

According to the focal depth, earthquakes are classified into
one of the three categories :


: have their foci at a depth between earthquakes focus - Shallow 1.
0 and 70 Km. and take place at oceanic ridges and transform faults as
well as at subduction zones.
: focal depth between 71 and earthquake focus - Intermediate 2.
300 Km.
: focal depth greater than 300 Km focus earthquakes - 3. Deep
Moho Most earthquakes originate within the crust. At depth beneath the
(Crust-Mantle boundary), the number falls abruptly and dies away to
zero at a depth of about 700 Km. Earthquakes along ridges usually occur
at a depth of about 10 Km or less and are of moderate size. Transform
faults generate large shocks at depth down to about 20 Km. The largest
earthquakes occur along subduction zones.

Locating the Epicenter of an Earthquake and
Measuring its Magnitude.


P waves and S waves travel at different velocities. The first P wave will
arrive at a seismic station before the first S wave. By using the
difference in their arrival times you can determine the distance between
the epicenter of any Earthquake and a seismic station using the
conversion table at the bottom of the page. Once you have determined
that distance, you can use it as the radius of a circle and can draw a
circle around the seismic station. You need two other seismic stations
doing the same thing. Where the three circles intersect is approximately
where the epicenter of the Earthquake is.

113




To determine the magnitude of an Earthquake you simply measure
the greatest amplitude of the first S wave and use the conversion
table below to find the Earthquake's magnitude.



114

Below is the conversion chart used to determine both the distance
of the epicenter of an Earthquake and the magnitude of the
earthquake.




115

SCALES OF EARTHQUAKES

Two basically different scales are used to describe the size or strength of
an earthquake and its effect :
1. Intensity : earthquake intensity represents the degree of shaking at
a particular location on the earth's surface. It indicates the local effect or
damage of the earthquake upon people, animals, buildings and objects
in the immediate environment. The intensity diminishes generally with
increasing distance from the epicenter.
One of the most widely used scales for intensity is the Modified
Mercalli Scale . The scale has the following values, ranging from I to
XII, usually written in Roman numerals :
I. Not felt
II. Felt by persons at rest
III. Felt indoor. Hanging objects swing
IV. Vibration like passing of heavy trucks. Windows rattle.
V. Felt outdoors. Sleepers awakened.
VI. Felt by all Persons walk unsteadily. Glassware broken.
VII. Difficult to stand. Hanging objects quiver.
VIII. Twisting,fall of chimneys,factory stacks,towers.
IX. General panic.Undergroundpipes broken.Frames racked.
X. Most masonry and frame structures destroyed.
XI. Rails bent greatly.Underground pipes out of surface.
XII. Damage nearly total. Objects thrown into the air.

2. Magnitude : The magnitude of an earthquake is an expression of
the actual original force or energy of the earthquake at its moment of
creation. It is measured directly by instruments, unlike the subjective
measurement of intensity.
The Richter scale is a standard for expressing magnitude of
earthquakes. Magnitude is defined as the logarithm to the base 10 of
the amplitude of the largest ground motion traced by a standard type
seismograph placed 100 Km from the earthquake's epicenter.
The Richter scale of magnitude runs from 0 through 8.9 , although
there is no lower limit or upper limit. Each unit representing a ten-fold
116

increase in amplitude of the measured waves and nearly a 30-fold
increase in energy. An earthquake of magnitude 1 can only detected by
a seismograph. The weakest earthquakes noticed by people are usually
around magnitude 2. Houses and buildings are damaged at magnitude
5. Earthquakes with a magnitude above 6 are capable of producing
serious damage. An earthquake with a magnitude of 8 and above is
considered a great earthquake.
All the magnitude scales are of the form :
M = log ( A / T ) + q ( d , h ) + a
Where :
M is the magnitude
A is the maximum amplitude of the wave
T is the period of the wave
q is a function correcting for the decrease of amplitude with
distance from the epicenter and focal depth ( h )
d is the epicentral distance
a is an empirical constant
When the surface - wave magnitude ( Ms ) and body-wave magnitude (
mb ) are calculated for an earthquake, they do not usually have the
same value.
mb = 2.94 + 0.55 Ms
Structural damage is related to ground acceleration, although building
respond differently to seismic waves of different periods. Intensity ( I )
is calibrated in terms of ground acceleration ( a ) by an approximate
relationship


117

log a = ( I / 3 ) - 2.5
The approximate relationship between the magnitude and intensity has
been estimated according to the following relationship :
M = 2 I / 3 + 1.7 log h - 1.4



Magnitude
(Richter
Scale )
Intensity
(Mecalli
Scale)
Damage Description
4 5.5 Widely felt, plaster cracked
5 7 Strong vibration, weak buildings and
chimneys damaged
6 8.5 Ordinary buildings badly damaged
7 10 Well-built buildings destroyed
8 11.5 Specially designed buildings damaged
9 12 Widespread destruction


An earthquake of Richter magnitude 5.5 turns out to have an energy of
about 10 ergs. The relation between energy and magnitude is :
log E = 5.24 + 1.44 Ms
E : is the total energy measured in joules
Ms : is the surface-wave magnitude.
An increase in magnitude Ms of 1 unit increases the amount of seismic
energy E released by a factor of about 30. In comparison , the
Hiroshima atomic bomb was approximately equivalent in terms of
energy to an earthquake of magnitude 5.3. A one megaton nuclear
explosion would release about the same amount of energy as an
earthquake of magnitude 6.5.
118


Direct effects of Earthquakes

Ground Sliding
Strong ground motion is also the primary cause of damages to the
ground and soil upon which, or in which, people must build. These
damages to the soil and ground can take a variety of forms: cracking
and fissuring and weakening, sinking, settlement and surface fault
displacement.

Ground Tilting
Sometimes, due to earthquake, there is tilting action in the ground. This
causes plain land to tilt, causing excessive stresses on buildings,
resulting in damage to buildings.

Differential Settlement
If a structure is built upon soil which is not homogeneous, then there is
differential settlement, with some part of the structure sinking more
than other. This induces excessive stresses and causes cracking.




119

Liquefaction
During an earthquake, significant damage can result due to
instability of the soil in the area affected by internal seismic waves. The
soil response depends on the mechanical characteristics of the soil
layers, the depth of the water table and the intensities and duration of
the ground shaking. If the soil consists of deposits of loose granular
materials it may be compacted by the ground vibrations induced by the
earthquake, resulting in large settlement and differential settlements of
the ground surface. This compaction of the soil may result in the
development of excess hydrostatic pore water pressures of sufficient
magnitude to cause liquefaction of the soil, resulting in settlement,
tilting and rupture of structures.

Indirect Effects of Earthquakes
Tsunamis
A tsunami is a very large sea wave that is generated by a disturbance
along the ocean floor. This disturbance can be an earthquake, a
landslide, or a volcanic eruption. A tsunami is undetectable far out
in the ocean, but once it reaches shallow water, this fast-traveling
wave grows very large. Tsunamis are very destructive, as this wall
of water can destroy everything in its path.
111

Landslides
Landslide means descent of a mass of earth and rock down a
mountain slope. Landslides may occur when water from rain and melting
snow sinks through the earth on top of a slope, seeps through cracks
and pore spaces in underlying sandstone, and encounters a layer of
slippery material, such as shale or clay, inclined toward the valley.
Earthquakes and volcanic eruptions can also cause severe, fast-moving
landslides.
Landslides that suddenly rush down a steep slope can cause great
destruction across a wide area of habitable land and sometimes cause
floods by damming up bodies of water.
Floods & Fires
The amount of damage caused by post-earthquake fire depends on the
types of building materials used, whether water lines are intact, and
whether natural gas mains have been broken. Ruptured gas mains may
lead to numerous fires, and fire fighting cannot be effective if the water
mains are not intact to transport water to the fires.
Earthquakes may also give rise to floods. Many times, large earthquakes
can cause cracking in Dams. So, to contain the increased pressure, the
authorities have to immediately release a lot of water to reduce the
reservoir pressure. This gives rise to very heavy flooding in the region,
causing great destruction.
111



CHAPTER 5


ELECTRICAL METHOD


- Electrical properties of rocks
- Apparent & True resistivity
- Electrode configurations
- Electrical soundings & Profiling
- Applications in groundwater exploration
Problem Set - 4
112

S METHOD ELECTRICAL


ELECTRICAL PROPERTIES OF ROCKS :

 Resistivity (or conductivity), which governs the amount of current
that passes when a potential difference is created.

 Electrochemical activity or polarizability, the response of certain
minerals to electrolytes in the ground, the bases for SP and IP.

 Dielectric constant or permittivity. A measure of the capacity of a
material to store charge when an electric field is applied . It
measure the polarizability of a material in an electric field = 1 +
4 π X
X : electrical susceptibility .

Electrical methods utilize direct current or Low frequency alternating
current to investigate electrical properties of the subsurface.
Electromagnetic methods use alternating electromagnetic field of high
frequencies.
Two properties are of primary concern in the Application of electrical
methods.
(1) The ability of Rocks to conduct an electrical current.
(2) The polarization which occurs when an electrical current is passed
through them (IP).

Resistivity
For a uniform wire or cube, resistance is proportional to length and
inversely proportional to cross-sectional area. Resistivity is related to
resistance but it not identical to it. The resistance R depends an length,
Area and properties of the material which we term resistivity (ohm.m) .
Constant of proportionality is called Resistivity :
113


Resistivity is the fundamental physical property of the metal in
the wire

Resistivity is measured in ohm-m


Conductivity
(S/m), equivalent to ohm
-1
m
-1
.


Classification of Materials according to Resistivities Values

114


a) Materials which lack pore spaces will show high resistivity
such as :
- massive limestone
- most igneous and metamorphic (granite, basalt)

- Materials whose pore space lacks water will show
high resistivity such as : - dry sand and gravel ,
Ice .

b) Materials whose connate water is clean (free from salinity )
will show high resistivity such as :
- clean sand or gravel , even if water saturated.

c) most other materials will show medium or low resistivity,
especially if clay is present such as :
- clay soil
- weathered rock.


The presence of clay minerals tends to decrease the Resistivity because
:
1 ) The clay minerals can combine with water .
2) The clay minerals can absorb cations in an exchangeable state on
the surface.
3) The clay minerals tend to ionize and contribute to the supply of
free ions.


115

Factors which control the Resistivity
(1) Geologic Age
(2) Salinity.
(3) Free-ion content of the connate water.
(4) Interconnection of the pore spaces (Permeability).
(5) Temperature.
(6) Porosity.
(7) Pressure
(8) Depth


Archie’s Law
Empirical relationship defining bulk resistivity of a saturated porous rock.
In sedimentary rocks, resistivity of pore fluid is probably single most
important factor controlling resistivity of whole rock.
Archie (1942) developed empirical formula for effective resistivity of
rock:



ρ
0
= bulk rock resistivity
ρ
w
= pore-water resistivity
a = empirical constant (0.6 < a < 1)
m = cementation factor (1.3 poor, unconsolidated) < m < 2.2
(good, cemented or crystalline)
φ = fractional porosity (vol liq. / vol rock)

Formation Factor:



Effects of Partial Saturation:



Sw is the volumetric saturation.
n is the saturation coefficient (1.5 < n < 2.5).
116

- Archie‘s Law ignores the effect of pore geometry, but is a
reasonable approximation in many sedimentary rocks


Current Flow in A Homogeneous Earth


1. Point current Source :
If we define a very thin shell of thickness dr we can define the
potential different dv

dv = I ( R ) = I ( ρ L / A ) = I ( ρ dr / 2π r
2
)

To determine V a t a point . We integrate the above eq. over its
distance D to to infinity :

V = I ρ / 2π D

C: current density per unit of cross sectional area :








117

2. Two current electrodes

To determine the current flow in a homogeneous, isotropic earth
when we have two current electrodes. The current must flow from
the positive (source ) to the resistive ( sink ).

The effect of the source at C1 (+) and the sink at C2 (-)

Vp
1
= i ρ / 2π r
1
+ ( - iρ / 2π r
2
)





Vp
1
= iρ / 2π { 1/ [ (d/2 + x )
2
+ Z
2
]
0.5
- 1 / [ (d/2 - x )
2
+ Z
2
]
0.5
}

118


3. Two potential Electrodes

Vp
1
= i ρ / 2π r
1
- iρ / 2π r
2


Vp
2
= i ρ / 2π r
3
- iρ / 2π r
4


Δ V = Vp
1
–Vp
2
= i ρ / 2π ( 1/r
1
– 1 / r
2
– 1 / r
3
+ 1 / r
4
)














119

ELECTRODE CONFIGURATIONS

The value of the apparent resistivity depends on the geometry of the
electrode array used (K factor)

1- Wenner Arrangement
Named after wenner (1916) .
The four electrodes A , M , N , B are equally spaced along a straight
line. The distance between adjacent electrode is called ―a‖ spacing . So
AM=MN=NB=
⅓ AB = a.


Ρ
a=
2 π a V / I

The wenner array is widely used in the western Hemisphere. This array
is sensitive to horizontal variations.

2) Schlumberger Arrangement .
This array is the most widely used in the electrical prospecting . Four
electrodes are placed along a straight line in the same order AMNB ,
but with AB ≥ 5 MN



This array is less sensitive to lateral variations and faster to use
as only the current electrodes are moved.

3. Dipole – Dipole Array .
The use of the dipole-dipole arrays has become common since the
1950‘s , Particularly in Russia. In a dipole-dipole, the distance
between the current electrode A and B (current dipole) and the
distance between the potential electrodes M and N (measuring
(
(
(
(
(
¸
(

¸

|
.
|

\
|
÷
|
.
|

\
|
× × =
MN
MN AB
I
V
a
2 2
2 2
t µ
121

dipole) are significantly smaller than the distance r , between the
centers of the two dipoles.




ρ
a = π [ ( r
2
/ a ) – r ] v/i

Or . if the separations a and b are equal and the distance between
the centers is (n+1) a then

ρ
a = n (n+1) (n+2) . π a. v/i







This array is used for deep penetration ≈ 1 km.









121

REFRACTION OF ELECTRICAL RESISTIVITY


A. Distortion of Current flow
At the boundary between two media of different resistivities the
potential remains continuous and the current lines are refracted
according to the law of tangents.


Ρ
1
tan Ө
2
= Ρ
2
tan Ө
1



If ρ2 < p1 , The current lines will be refracted away from the Normal.
The line of flow are moved downward because the lower resistivity
below the interface results in an easier path for the current within the
deeper zone.


B. Distortion of Potential

Consider a source of current I at the point S in the first layers P1 of
Semi infinite extent. The potential at any point P would be that from S
plus the amount reflected by the layer P2 as if the reflected amount
were coming from the image S
/


122





V
1
(P) = i ρ
1
/ 2π [ (1 / r
1
) + ( K / r
2
) ]


K = Reflection coefficient = ρ
2
– ρ
1
/ ρ
2
+ ρ
1

In the case where P lies in the second medium ρ2, Then transmitting
light coming from S. Since only 1 – K is transmitted through the
boundary.
The Potential in the second medium is

V
2
(P) = I ρ
2
/ 2π [ (1 / r
1
) – (K / r
1
) ]

Continuity of the potential requires that the boundary where r1 = r2
, V1(p) must be equal to V2 ( P).
At the interface r1 = r2 , V1= V2



k is electrical reflection coefficient and used in interpretation


123

The value of the dimming factor , K always lies between ± 1

If the second layer is a pure insulator
( ρ
2
= ω ) then K = + 1

If the second layer is a perfect conductor

( ρ
2
= O ) then K = - 1

When ρ
1
= ρ
2
then No electrical boundary Exists and K =
O .



SURVEY DESIGN

Two categories of field techniques exist for conventional resistivity
analysis of the subsurface. These techniques are vertical electric
sounding (VES), and Horizontal Electrical Profiling (HEP).


1- Vertical Electrical Sounding (VES) .
The object of VES is to deduce the variation of resistivity with
depth below a given point on the ground surface and to correlate
it with the available geological information in order to infer the
depths and resistivities of the layers present.


2- Horizontal Electrical profiling (HEP) .
The object of HEP is to detect lateral variations in the resistivity of
the ground, such as lithological changes, near- surface faults……
.

124


Multiple Horizontal Interfaces

For Three layers resistivities in two interface case , four possible
curve types exist.

1- Q – type ρ
1
> ρ
2
> ρ
3

2- H – Type ρ
1
> ρ
2
< ρ
3

3- K – Type ρ
1
< ρ
2
> ρ
3

4- A – Type ρ
1
< ρ
2
< ρ
3




Error

125


Applications of Resistivity Techniques

1. Bedrock Depth Determination
Both VES and CST are useful in determining bedrock depth. Bedrock
usually more resistive than overburden. HEP profiling with Wenner array
at 10 m spacing and 10 m station interval used to map bedrock highs.

2. Location of Permafrost
Permafrost represents significant difficulty to construction projects due
to excavation problems and thawing after construction.
- Ice has high resistivity of 1-120 ohm-m

3. Landfill Mapping
Resistivity increasingly used to investigate landfills:
- Leachates often conductive due to dissolved salts
- Landfills can be resistive or conductive, depends on contents
126

Limitations of Resistivity Interpretation

1- Principle of Equivalence.
If we consider three-lager curves of K (ρ
1
< ρ
2
> ρ
3
) or Q type

1
> ρ
2
> ρ
3
) we find the possible range of values for the
product T
2
= ρ
2
h
2
Turns out to be much smaller. This is called
T-equivalence. H = thickness, T : Transverse resistance it
implies that we can determine T
2
more reliably than ρ
2
and h
2

separately. If we can estimate either ρ
2
or h
2
independently we
can narrow the ambiguity. Equivalence: several models produce
the same results. Ambiguity in physics of 1D interpretation such
that different layered models basically yield the same response.
Different Scenarios: Conductive layers between two
resistors, where lateral conductance (σh) is the same.
Resistive layer between two conductors with same transverse
resistance (ρh).

2- Principle of Suppression.
This states that a thin layer may sometimes not be detectable on
the field graph within the errors of field measurements. The thin
layer will then be averaged into on overlying or underlying layer in
the interpretation. Thin layers of small resistivity contrast with
respect to background will be missed. Thin layers of greater
resistivity contrast will be detectable, but equivalence limits
resolution of boundary depths, etc. The detectibility of a layer of
given resistivity depends on its relative thickness which is defined
as the ratio of Thickness/Depth.
127

Comparison of Wenner and Schlumberger

(1) In Sch. MN ≤ 1/5 AB
Wenner MN = 1/3 AB

(2) In Sch. Sounding, MN are moved only occasionally.
In Wenner Soundings, MN and AB are moved after each
measurement.

(3) The manpower and time required for making Schlumberger
soundings are less than that required for Wenner soundings.

(4) Stray currents that are measured with long spreads effect
measurements with Wenner more easily than Sch.

(5) The effect of lateral variations in resistivity are recognized and
corrected more easily on Schlumberger than Wenner.

(6) Sch. Sounding is discontinuous resulting from enlarging MN.



Disadvantages of Wenner Array

1. All electrodes must be moved for each reading
2. Required more field time
3. More sensitive to local and near surface lateral variations
4. Interpretations are limited to simple, horizontally layered structures



Advantages of Schlumberger Array

1. Less sensitive to lateral variations in resistivity
2. Slightly faster in field operation
3. Small corrections to the field data
128


Disadvantages of Schlumberger Array

1. Interpretations are limited to simple, horizontally layered structures
2. For large current electrodes spacing, very sensitive voltmeters are
required.

Advantages of Resistivity Methods

1. Flexible
2. Relatively rapid. Field time increases with depth
3. Minimal field expenses other than personnel
4. Equipment is light and portable
5. Qualitative interpretation is straightforward
6. Respond to different material properties than do seismic and other
methods, specifically to the water content and water salinity

Disadvantages of Resistivity Methods

1- Interpretations are ambiguous, consequently, independent
geophysical and geological controls are necessary to
discriminate between valid alternative interpretation of the
resistivity data ( Principles of Suppression & Equivalence)

2- Interpretation is limited to simple structural configurations.

3- Topography and the effects of near surface resistivity variations
can mask the effects of deeper variations.

4- The depth of penetration of the method is limited by the
maximum electrical power that can be introduced into the
ground and by the practical difficulties of laying out long length
of cable. The practical depth limit of most surveys is about 1
Km.

5. Accuracy of depth determination is substantially lower than
with seismic methods or with drilling.

129

Problem Sets


1. Copper has ρ =1.7 X 10
-8
ohm.m. What is the resistance of
20 m of copper with a cross-sectional radius of 0.005 m .


2. Construct the current-flow lines beneath the interface in (a) and
(b).





3. Assume a homogeneous medium of resistivity 120 ohm-m.
Using the wenner electrode system with a 60-m spacing, assume
a current of 0.628 ampere. What is the measured potential differ-
ence? What will be the potential difference if we place the sink
(negative-current electrode) at infi ni ty?


4. Why are the electrical methods of exploration particularly suited to
hydrogeological investigations? Describe other geophysical methods
which could be used in this context, stating the reasons why they are
applicable.
131

















CHAPTER 6




GRAVITY METHOD

- Fundamental principles
- Measurements
- Data reduction
- Interpretation & Applications
Solved Problems
Problem Set - 5
131

GRAVITY METHOD

The gravity method is a nondestructive geophysical technique that
measures differences in the earth‘s gravitational field at specific
locations. It has found numerous applications in engineering and
environmental studies including locating voids and karst features, buried
stream valleys, water table levels and the determination of soil layer
thickness. The success of the gravity method depends on the different
earth materials having different bulk densities (mass) that produce
variations in the measured gravitational field. These variations can then
be interpreted by a variety of analytical and computers methods to
determine the depth, geometry and density the causes the gravity field
variations.
Gravity data in engineering and environmental applications should be
collected in a grid or along a profile with stations spacing 5 meters or
less. In addition, gravity station elevations must be determined to within
0.2 meters. Using the highly precise locations and elevations plus all
other quantifiable disturbing effects, the data are processed to remove
all these predictable effects. The most commonly used processed data
are known as Bouguer gravity anomalies, measured in mGal.
The gravity method can be a relatively easy geophysical technique to
perform and interpret. It requires only simple but precise data
processing, and for detailed studies the determination of a station‘s
elevation is the most difficult and time-consuming aspect. The technique
has good depth penetration when compared to ground penetrating
radar, high frequency electromagnetics and DC-resistivity techniques
and is not affected by the high conductivity values of near-surface clay
132

rich soils. Additionally, lateral boundaries of subsurface features can be
easily obtained especially through the measurement of the derivatives of
the gravitational field.
The main drawback is the ambiguity of the interpretation of the
anomalies. This means that a given gravity anomaly can be caused by
numerous source bodies. An accurate determination of the source
usually requires outside geophysical or geological information.
Geophysical interpretations from gravity surveys are based on the
mutual attraction experienced between two masses* as first expressed
by Isaac Newton. Newton's law of gravitation states that the mutual
attractive force between two point masses**, m1 and m2, is
proportional to one over the square of the distance between them. The
constant of proportionality is usually specified as G, the gravitational
constant. Thus, the law of gravitation





where F is the force of attraction, G is the gravitational constant
( G = 6.67 x 10
-11
m
3
kg
-1
s
-2
) and r is the distance between the two
masses, m1 and m2.
Mass is formally defined as the proportionality constant relating the
force applied to a body and the acceleration the body undergoes as
133

given by Newton's second law, usually written as F=ma. Therefore,
mass is given as m=F/a and has the units of force over acceleration.


Gravitational Acceleration

When making measurements of the earth's gravity, we usually don't
measure the gravitational force, F. Rather, we measure the gravitational
acceleration, g. The gravitational acceleration is the time rate of change
of a body's speed under the influence of the gravitational force. That is,
if you drop a rock off a cliff, it not only falls, but its speed increases as it
falls. In addition to defining the law of mutual attraction between
masses, Newton also defined the relationship between a force and an
acceleration.
Newton's second law states that force is proportional to acceleration.
The constant of proportionality is the mass of the object. Combining
Newton's second law with his law of mutual attraction, the gravitational
acceleration on the mass m2 can be shown to be equal to the mass of
attracting object, m1, over the squared distance between the center of
the two masses, r.



If an object such as a ball is dropped, it falls under the influence of
gravity in such a way that its speed increases constantly with time. That
is, the object accelerates as it falls with constant acceleration. At sea
level, the rate of acceleration is about 9.8 meters per second squared.
134

In gravity surveying, we will measure variations in the acceleration due
to the earth's gravity. Variations in this acceleration can be caused by
variations in subsurface geology. Acceleration variations due to geology,
however, tend to be much smaller than 9.8 meters per second squared.
Thus, a meter per second squared is an inconvenient system of units to
use when discussing gravity surveys.
The units typically used in describing the gravitational acceleration
variations observed in exploration gravity surveys are specified in
milliGals. A Gal is defined as a cm / sec.
2
(1 mGal=10
-3
Gal) and
microgal (1μGal = 10
-6
Gals ).
Thus, the Earth's gravitational acceleration is approximately 980 Gals.
The Gal is named after Galileo Galilei. The milliGal (mgal) is 0.001 Gal.
In milliGals, the Earth's gravitational acceleration is approximately
980,000.

135


Gravity Measurements

The instrument used to measure gravity is called a gravimeter.

Absolute gravity
This technique makes measurements of the total gravity field at a
site. There are a number of types of instruments, including free-fall
devices, the reversible pendulum, and superconducting gravimeters. The
equipment is very expensive and bulky. Lengthy observation times (24+
hrs) are required to obtain accurate readings (0.001- 0.01 mgal).
Relative gravity
In general, for interpreting gravity data, only the relative gravitational
acceleration is required. Therefore, we usually don‘t need to know the
absolute gravity at every station, just how gravity changes between
stations. The relative gravity readings can be converted into absolute
gravity if one of the survey sites is chosen to be a place where the
absolute gravity was measured previously.
a) Portable pendulum:
- based on the idea that the period of a pendulum (time taken for
the pendulum to swing back and forth) is given by:

where L is the pendulum length and g is the gravity.
- measure the period (T1) at one location and then move the
pendulum to another location and measure the period (T2). The
change in period (T2- T1) is proportional to the change in gravity
between the two locations.
- accuracy of 0.25 mgal
136

Factors Affecting Gravitational Acceleration

Factors can be subdivided into two categories: those that give rise to
temporal variations and those that give rise to spatial variations in the
gravitational acceleration.
A. Temporal Based Variations - These are changes in the
observed acceleration that are time dependent. In other words,
these factors cause variations in acceleration that would be
observed even if we didn't move our gravimeter.
Instrument Drift - Changes in the observed acceleration caused by
changes in the response of the gravimeter over time.
Tidal Affects - Changes in the observed acceleration caused by the
gravitational attraction of the sun and moon.
B. Spatial Based Variations - These are changes in the observed
acceleration that are space dependent. That is, these change the
gravitational acceleration from place to place, just like the geologic
affects, but they are not related to geology.
Latitude Variations - Changes in the observed acceleration caused by
the ellipsoidal shape and the rotation of the earth.
Elevation Variations - Changes in the observed acceleration caused
by differences in the elevations of the observation points.
Bouguer Effects - Changes in the observed acceleration caused by the
extra mass underlying observation points at higher elevations.
Topographic Effects - Changes in the observed acceleration related to
topography near the observation point.

Latitude Variations:
Two features of the earth's large-scale structure and dynamics affect our
gravity observations: its shape and its rotation.
137

Although the difference in earth radii measured at the poles and at the
equator is only 22 km (this value represents a change in earth radius of
only 0.3%), this, in conjunction with the earth's rotation, can produce a
measurable change in the gravitational acceleration with latitude.
Because this produces a spatially varying change in the gravitational
acceleration, it is possible to confuse this change with a change
produced by local geologic structure. Fortunately, it is a relatively simple
matter to correct our gravitational observations for the change in
acceleration produced by the earth's elliptical shape and rotation.
To first order*, the elliptical shape of the earth causes the gravitational
acceleration to vary with latitude because the distance between the
gravimeter and the earth's center varies with latitude. The magnitude of
the gravitational acceleration changes as one over the distance from the
center of mass of the earth to the gravimeter squared. Thus,
qualitatively, we would expect the gravitational acceleration to be
smaller at the equator than at the poles, because the surface of the
earth is farther from the earth's center at the equator than it is at the
poles.




138





The mathematical formula used to predict the components of the
gravitational acceleration produced by the earth's shape and rotation is
called the Geodetic Reference Formula of 1967. The predicted gravity is
called the normal gravity ( g
n
).
How large is this correction to our observed gravitational acceleration?
And, because we need to know the latitudes of our observation points to
make this correction, how accurately do we need to know locations? At
a latitude of 45 degrees, the gravitational acceleration varies
approximately 0.813 mgals per kilometer. Thus, to achieve an accuracy
of 0.01 mgals, we need to know the north-south location of our gravity
stations to about 12 meters.

At any latitude g
n
= 0.812 sin 2φ mgal/m


139

Elevation Variations (The Free-Air Correction )

The gravitational acceleration observed on the surface of the earth
varies at about -0.3086 mgal per meter in elevation difference. The
minus sign indicates that as the elevation increases, the observed
gravitational acceleration decreases. The magnitude of the number says
that if two gravity readings are made at the same location, but one is
done a meter above the other, the reading taken at the higher elevation
will be 0.3086 mgal less than the lower.
To apply an elevation correction to our observed gravity, we need to
know the elevation of every gravity station. If this is known, we can
correct all of the observed gravity readings to a common elevation*
(usually chosen to be sea level) by adding -0.3086 times the elevation of
the station in meters to each reading. Given the relatively large size of
the expected corrections, how accurately do we actually need to know
the station elevations?
If we require a precision of 0.01 mgals, then relative station elevations
need to be known to about 3 cm.


Variations Due to Excess Mass (Bouguer Correction)

In addition to the gravity readings differing at two stations because of
elevation differences, the readings will also contain a difference because
there is more mass below the reading taken at a higher elevation than
there is of one taken at a lower elevation. As a first-order correction for
this additional mass, we will assume that the excess mass underneath
the observation point at higher elevation, point B in the figure below,
can be approximated by a slab of uniform density and thickness.
141

Obviously, this description does not accurately describe the nature of
the mass below point B. The topography is not of uniform thickness
around point B and the density of the rocks probably varies with
location.




Corrections based on this simple slab approximation are referred to as
the Bouguer Slab Correction. It can be shown that the vertical
gravitational acceleration associated with a flat slab can be written
simply as -0.04193rh. Where the correction is given in mgals, r is the
density of the slab in gm/cm^3, and h is the elevation difference in
meters between the observation point and elevation datum. h is positive
for observation points above the datum level and negative for
observation points below the datum level. Notice that the sign of the
Bouguer Slab Correction makes sense. If an observation point is at a
141

higher elevation than the datum, there is excess mass below the
observation point that wouldn't be there if we were able to make all of
our observations at the datum elevation. Thus, our gravity reading is
larger due to the excess mass, and we would therefore have to subtract
a factor to move the observation point back down to the datum. Notice
that the sign of this correction is opposite to that used for the elevation
correction. For a density of 2.67 gm/cm
3
, the Bouguer Slab Correction
is about 0.11 mgals/m.

Variations Due to Topography ( Terrain Correction )

Although the slab correction described previously adequately describes
the gravitational variations caused by gentle topographic variations
(those that can be approximated by a slab), it does not adequately
address the gravitational variations associated with extremes in
topography near an observation point. Consider the gravitational
acceleration observed at point B shown in the figure below.


In applying the slab correction to observation point B, we remove the
effect of the mass surrounded by the blue rectangle. Note, however,
that in applying this correction in the presence of a valley to the left of
142

point B, we have accounted for too much mass because the valley
actually contains no material. Thus, a small adjustment must be added
back into our Bouguer corrected gravity to account for the mass that
was removed as part of the valley and, therefore, actually didn't exist.
The mass associated with the nearby mountain is not included in our
Bouguer correction. The presence of the mountain acts as an upward
directed gravitational acceleration. Therefore, because the mountain is
near our observation point, we observe a smaller gravitational
acceleration directed downward than we would if the mountain were not
there. Like the valley, we must add a small adjustment to our Bouguer
corrected gravity to account for the mass of the mountain. These small
adjustments are referred to as Terrain Corrections. As noted above,
Terrain Corrections are always positive in value. To compute these
corrections, we are going to need to be able to estimate the mass of the
mountain and the excess mass of the valley that was included in the
Bouguer Corrections. These masses can be computed if we know the
volume of each of these features and their average densities.
Generally, gravity data corrections are associated with each other as
follows :
Observed Gravity (gobs) - Gravity readings observed at each gravity
station after corrections have been applied for instrument drift and tides.
Latitude Correction (gn) - Correction subtracted from gobs that
accounts for the earth's elliptical shape and rotation. The gravity value
that would be observed if the earth were a perfect (no geologic or
topographic complexities), rotating ellipsoid is referred to as the normal
gravity.
143

Free Air Corrected Gravity (gfa) - The Free-Air correction accounts for
gravity variations caused by elevation differences in the observation
locations. The form of the Free-Air gravity anomaly, gfa, is given by;
gfa = gobs - gn + 0.3086h (mgal)
where h is the elevation at which the gravity station is above the
elevation datum chosen for the survey (this is usually sea level).
Bouguer Slab Corrected Gravity (gb) - The Bouguer correction is a
first-order correction to account for the excess mass underlying
observation points located at elevations higher than the elevation
datum. Conversely, it accounts for a mass deficiency at observations
points located below the elevation datum. The form of the Bouguer
gravity anomaly, gb, is given by;
gb = gobs - gn + 0.3086h - 0.04193rh (mgal)
where r is the average density of the rocks underlying the survey area.
Terrain Corrected Bouguer Gravity (gt) - The Terrain correction
accounts for variations in the observed gravitational acceleration caused
by variations in topography near each observation point. The terrain
correction is positive regardless of whether the local topography consists
of a mountain or a valley. The form of the Terrain corrected, Bouguer
gravity anomaly, gt, is given by;
gt = gobs - gn + 0.3086h - 0.04193r + TC (mgal)
where TC is the value of the computed Terrain correction. Assuming
these corrections have accurately accounted for the variations in
gravitational acceleration they were intended to account for, any
remaining variations in the gravitational acceleration associated with the
Terrain Corrected Bouguer Gravity, gt, can now be assumed to be
caused by geologic structure.
144





FREE AIR AND BOUGUER ANOMALIES
A. Free Air anomaly
The figure below shows three gravity sites in an area with a hill and a
valley.

As you go from site A to site B, you climb a hill that is 100 m high. The
corresponding change in gravity is: Δg = 0.3086 x 100 = 30.86 mgal
. The gravity at the top of the hill will be 30.86 mgal less than the
gravity at the bottom because you have moved further from the centre
of the Earth.
This variation in gravity must be removed from your data. The
correction for elevation is called the Free Air Correction. To apply this,
it is necessary to define a reference level for the survey. Any reference
level can be chosen:
- for surveys in coastal areas, sea level is often chosen as the reference
- for a survey far away from the coast, the average elevation of the
survey area could be used.
145

- in the figure, the elevation of Site A was chosen. The Free Air
Correction is written as: CFA = 0.3086 Δh, where Δh is the difference
in elevation between the site and the reference level
• If the site is above the reference level (e.g., Site B), CFA is added to
the observed gravity value.
• If the site is below the reference level (e.g., Site C), CFA is
subtracted from the observed gravity value. The resulting gravity value
is called the free air anomaly. Ag
F
= g
obs
- g
¢
+ C
F



B. Bouguer Anomaly
Consider Sites A and B. The Free Air correction will correct the gravity
observed at B for the 100 m difference in elevation. However, there is
still a difference in the amount of mass below each station. The gravity
at Site B will be affected by the gravitational pull of the 100 m thickness
of the material between it and the reference level. This ―excess‖ gravity
has to be removed in order to compare the gravity at A and B.
To first order, the difference in gravity between Site A and Site B can be
approximated by an infinite slab of uniform density and thickness. The
gravitational attraction of this layer is: g = 2πGρΔh , where Δh is the
thickness and ρ is the density of the material . The correction for the
difference in mass due to a difference elevation (Δh) is called the
Bouguer correction (CB): C = 2πGρΔh = B 0.00004193 ρ Δh (CB in
mgal). The Bouguer correction at Site B would be: 0.1119 Δh =
0.1119×100 m = 11.2 mgal.
This value must be subtracted – we want to take away the effect of
the hill. Conversely, after the gravity data at site C have been corrected
146

for elevation (CFA), they will be ―too low‖, because they were made at a
lower elevation and thus there was less mass below the station. In this
case, the Bouguer correction will add the ―missing‖ mass to the original
gravity measurement. The resulting gravity value is the Bouguer
anomaly,
Ag
B
= g
obs
- g
¢
+ C
F
- C
B
+ C
T

this is the gravity anomaly due to local geology.

Generally, measurement above reference level Add Free Air Subtract
Bouguer correction correction. Measurement below reference level
Subtract Free Air Add Bouguer correction.


Applications of the Gravity Method

 Determining Shape of the earth ( Geodesy )
 Detection of subsurface voids including caves, mine shafts
 Determining the amount of subsidence in surface collapse features
over time
 Determination of soil and glacier sediment thickness (bedrock
topography)
 Location of buried sediment valleys
 Determination of groundwater volume and changes in water table
levels over time in alluvial basins
 Mapping the volume, lateral and vertical extent of landfills
 Mapping steeply dipping contacts including faults

147

SOLVED PROBLEMS IN GRAVITY

Problem 1:

Given :
Observed gravity at base 980.30045 Gals
Observed gravity at station relative to base + 5.65 mGal
Theoretical gravity at sea level at latitude of station 980.30212 Gals
Elevation of station 100 m above sea level
Density of rock above sea level 2.0 g/cc
Terrain effect 0.15 mGals

Compute : Bouguer gravity anomaly
Free air anomaly

Ag
B
= g
obs
- g
¢
+ C
F
- C
B
+ C
T

= (980300.54 + 5.65) - (980302.12 + (0.3086 x 100) - (0.0419 x
100 x 2) + 0.15
g = 980306.19 - 980302.12 + 30.86 - 8.38 + 0.15
Ag
B
= 26.55 mGal

Ag
F
= g
obs
- g
¢
+ C
F

= 34.9 mGal
148


Problem 2 :
Given :
Observed gravity at base 980.30045 Gals
Observed gravity at station relative to base + 5.65 mGal
Theoretical gravity at sea level at latitude of station 980.30212 Gals
Elevation of station 100 m below sea level
Density of rock above sea level 2.0 g/cc
Terrain effect 0.15 mGals

Compute :
1 - Free air anomaly
2 - Bouguer gravity anomaly

Ag
F
= g
obs
- g
¢
+ C
F

= 980306.19 – 980302.12 + (0.3086 x -100)
= - 26.79 mGal

Ag
B
= g
obs
- g
¢
+ C
F
- C
B
+ C
T

= 980306.19 - 980302.12 + (0.3086 x -100) - (0.0419 x 2.0 x -
100) + 0.15
= - 18.26 mGal
149


Problem 3 :

Given :
Observed gravity relative to base + 30 mGal
Elevation of station 150 m above base
Station is 1000 m north of base
Latitude effect is 0.00025 mGal/m
Density is 1.8 g/cc
Terrain effect is 0.05 mGal

Compute : Free-air and Bouguer Anomalies


Ag
F
= g
obs
- g
¢
+ C
F

= 30 - (0.00025 x 1000) + 0.3086 x 150)
= 30 - 0.25 + 46.29
= 76.04 mGal

Ag
B
= g
obs
- g
¢
+ C
F
- C
B
+ C
T

= 76.04 - (0.0419 x 1.8 x 150) + 0.05
= 76.04 - 11.313 + 0.05
= 64.777 mGal


151


Problem 4 :

Given:
Observed gravity relative to base - 12.5 mGal
Elevation of station is 100 m below the base
Station is 2000 m south of base
Latitude effect is 0.00025 mGal/m
Density is 1.8 g/cc
Terrain effect is 0.1 mGal

Compute:
1 - Free air anomaly
2 - Bouguer gravity anomaly


Ag
F
= g
obs
- g
¢
+ C
F

= - 12.5 – (0.00025 x - 2000) + (0.3086 x - 100)
= - 12.5 + 0.5 - 30.86 = - 42.86 mGal

Ag
B
= g
obs
- g
¢
+ C
F
- C
B
+ C
T

= - 42.86 - (0.0419 x 1.8 x - 100) + 0.1
= - 42.86 +7.542+ 0.1
= - 35.218 mGal


151


Problem Sets


Q.1 What is the average gravitational acceleration at the surface of the
Earth? Mass of the Earth (ME) = 5.974 × 10
24
kg . Average radius of the
Earth (r) = 6371 km


Q. 2 What is the expected value of gravity at latitude 53.52589
0
N ?


Q.3 Surface gravity at a measuring site is 9.803244 ms
-2
, the site has
latitude 43.1
o
N and elevation 54 m. Obtain the free air gravity
anomalies.


152




CHAPTER 7


MAGNETIC METHOD

- Basic concepts
- Description of the magnetic field
- Source of magnetic anomalies
- Measurements
- Interpretation & Applications
Problem Set – 6
153

MAGNETIC METHODS


Magnetic methods are one of the most commonly used geophysical
tools. This stems from the fact that magnetic observations are obtained
relatively easily and cheaply and few corrections must be applied to the
observations. Despite these obvious advantages, like the gravitational
methods, interpretations of magnetic observations suffer from a lack of
uniqueness.

Magnetic Monopoles

Charles Coulomb, in 1785, showed that the force of attraction or
repulsion between electrically charged bodies and between magnetic
poles also obey an inverse square law like that derived for gravity by
Newton. The mathematical expression for the magnetic force
experienced between
two magnetic monopoles is given by

where µ is a constant of proportionality known as the magnetic
permeability, p1 and p2 are the strengths of the two magnetic
monopoles, and r is the distance between the two poles. the magnetic
permeability, µ, is a property of the material in which the two
monopoles, p1 and p2, are located. If they are in a vacuum, µ is called
the magnetic permeability of free space. p1 and p2 can be either
positive or negative in sign. If p1 and p2 have the same sign, the force
154

between the two monopoles is repulsive. If p1 and p2 have opposite
signs, the force between the two monopoles is attractive.
From Coulomb's expression, we know that force must be given in
Newtons,N, Permeability,mu, is defined to be a unit less constant. The
units of pole strength are defined such that if the force, F, is 1 N and the
two magnetic poles are separated by 1 m, each of the poles has a
strength of 1 Amp - m (Ampere - meters). In this case, the poles are
referred to as unit poles.



The magnetic field strength, H, is defined as the force per unit pole
strength exerted by a magnetic monopole, p1. H is nothing more than
Coulomb's expression divided by p2.


Given the units associated with force, N, and magnetic monopoles, Amp
-m, the units associated with magnetic field strength are Newtons per
Ampere-meter, N / (Amp - m). A N / (Amp - m) is referred to as a tesla
(T), named after the renowned inventor Nikola Tesla
When describing the magnetic field strength of the earth, it is more
common to use units of nanoteslas (nT), where one nanotesla is 1
billionth of a tesla. The average strength of the Earth's magnetic field is
about 50,000 nT. A nanotesla is also commonly referred to as a gamma.




155

Magnetic Induction

When a magnetic material, say iron, is placed within a magnetic field, H,
the magnetic material will produce its own magnetization. This
phenomena is called induced magnetization.
In practice, the induced magnetic field will look like it is being created by
a series of magnetic dipoles located within the magnetic material and
oriented parallel to the direction of the inducing field, H.
The strength of the magnetic field induced by the magnetic material due
to the inducing field is called the intensity of magnetization, I.



The intensity of magnetization, I, is related to the strength of the
inducing magnetic field, H, through a constant of proportionality, known
as the magnetic susceptibility.
The magnetic susceptibility ( K) is a unitless constant that is determined
by the physical properties of the magnetic material. It can take on either
positive or negative values. Positive values imply that the induced
magnetic field, I, is in the same direction as the inducing field, H.
156

Negative values imply that the induced magnetic field is in the opposite
direction as the inducing field.
Magnetic Susceptibility K is dependent on :

1- The state of magnetization.
2- Intensity of saturation magnetization.
3- Grain size.
4- Internal stress.
5- Shape
6- Mode of dispersion.

The intensity of magnetization I = M / V
M = magnetic moment = m L
V = Volume
m = pole strength
L = length

Intensity of magnetization ∞ H and has the same direction.




Mechanisms for Induced Magnetization

The nature of magnetization material is in general complex, governed by
atomic properties, and well beyond the scope of this series of notes.
Suffice it to say, there are three types of magnetic materials:
paramagnetic, diamagnetic, and ferromagnetic.
Diamagnetism - Discovered by Michael Faraday in 1846. This form of
magnetism is a fundamental property of all materials and is caused by
the alignment of magnetic moments associated with orbital electrons in
the presence of an external magnetic field. For those elements with no
unpaired electrons in their outer electron shells, this is the only form of
157

magnetism observed. The susceptibilities of diamagnetic materials are
relatively small and negative. Quartz and salt are two common
diamagnetic earth materials.
Paramagnetism - This is a form of magnetism associated with
elements that have an odd number of electrons in their outer electron
shells. Paramagnetism is associated with the alignment of electron spin
directions in the presence of an external magnetic field. It can only be
observed at relatively low temperatures. The temperature above which
paramagnetism is no longer observed is called the Curie Temperature.
The susceptibilities of paramagnetic substances are small and positive.
Ferromagnetism - This is a special case of paramagnetism in which
there is an almost perfect alignment of electron spin directions within
large portions of the material referred to as domains.
Like paramagnetism, ferromagnetism is observed only at temperatures
below the Curie temperature. There are three varieties of
ferromagnetism.
Pure Ferromagnetism - The directions of electron spin alignment
within each domain are almost all parallel to the direction of the external
inducing field. Pure ferromagnetic substances have large (approaching
1) positive susceptibilities. Ferrromagnetic minerals do not exist, but
iron, cobalt, and nickel are examples of common ferromagnetic
elements.

158



Antiferromagnetism - The directions of electron alignment within
adjacent domains are opposite and the relative abundance of domains
with each spin direction is approximately equal. The observed magnetic
intensity for the material is almost zero. Thus, the susceptibilities of
antiferromagnetic materials are almost zero. Hematite is an
antiferromagnetic material


159

Ferromagnetism - Like antiferromagnetic materials, adjacent domains
produce magnetic intensities in opposite directions. The intensities
associated with domains polarized in a direction opposite that of the
external field, however, are weaker. The observed magnetic intensity for
the entire material is in the direction of the inducing field but is much
weaker than that observed for pure ferromagnetic materials. Thus, the
susceptibilities for ferromagnetic materials are small and positive. The
most important magnetic minerals are ferromagnetic and include
magnetite, titanomagnetite, ilmenite, and pyrrhotite.



Remanent Magnetization in Rocks
- Remanent field (remains even after external field removed)
o thermoremanent
o detrital remanent
o chemical remanent
Total magnetization (J) = Remanent (Jr) + induced (Ji)
intensity of Jr is large in igneous and thermally metamorphosed Rocks.
Koenigsberger ratio. Q = Remanent (Jr) / induced (Ji)
Q > 1, Jr of sediments is smaller than Ji.

161


Magnetic Field Elements

At any point on the Earth's surface, the magnetic field, F*, has some
strength and points in some direction. The following terms are used to
describe the direction of the magnetic field.
Declination - The angle between north and the horizontal projection of
F. This value is measured positive through east and varies from 0 to 360
degrees.
Inclination - The angle between the surface of the earth and F.
Positive inclinations indicate F is pointed downward, negative inclinations
indicate F is pointed upward. Inclination varies from -90 to 90 degrees.
Magnetic Equator - The location around the surface of the Earth
where the Earth's magnetic field has an inclination of zero (the magnetic
field vector F is horizontal). This location does not correspond to the
Earth's rotational equator.
Magnetic Poles - The locations on the surface of the Earth where the
Earth's magnetic field has an inclination of either plus or minus 90
degrees (the magnetic field vector F is vertical). These locations do not
correspond to the Earth's north and south poles.


161



The total field F is resolved into its horizontal components H ( x , y ) and
it vertical components Z. The angle which F makes with its horizontal
components H is the inclination (I), and the angle between H and X
(points North) is the declination (D).
F
2
= X
2
+ y
2
+ Z
2

F
2
H
2
+ Z
2

H = F cos I
Z F sin I
X = H cos D
Z / H = tan I

Tan I = 2 tan Ø → Latitude


F at North Pole = 60,000 nT
F at South Pole = 70,000 nT
F at equator = 30.000 nT
F
pole
= 2 F
Equator



162

The Earth's Magnetic Field

Ninety percent of the Earth's magnetic field looks like a magnetic field
that would be generated from a dipolar magnetic source located at the
center of the Earth and aligned with the Earth's rotational axis. The
strength of the magnetic field at the poles is about 60,000 nT. The
remaining 10% of the magnetic field can not be explained in terms of
simple dipolar sources.
If the Earth's field were simply dipolar with the axis of the dipole
oriented along the Earth's rotational axis, all declinations would be 0
degrees (the field would always point toward the north).
The magnetic field can be broken into three separate components.
Main Field - This is the largest component of the magnetic field and is
believed to be caused by electrical currents in the Earth's fluid outer
core. For exploration work, this field acts as the inducing magnetic field.
External Magnetic Field - This is a relatively small portion of the
observed magnetic field that is generated from magnetic sources
external to the earth. This field is believed to be produced by
interactions of the Earth's ionosphere with the solar wind. Hence,
temporal variations associated with the external magnetic field are
correlated to solar activity.

Temporal Variations of the Earth's Magnetic Field

The magnetic field varies with time. When describing temporal
variations of the magnetic field, it is useful to classify these variations
into one of three types depending on their rate of occurrence and
source. Three temporal variations:
163

Secular Variations - These are long-term (changes in the field that
occur over years) variations in the main magnetic field that are
presumably caused by fluid motion in the Earth's Outer Core. Because
these variations occur slowly with respect to the time of completion of a
typical exploration magnetic survey, these variations will not complicate
data reduction efforts.
Diurnal Variations - These are variations in the magnetic field that
occur over the course of a day and are related to variations in the
Earth's external magnetic field. This variation can be on the order of 20
to 30 nT per day and should be accounted for when conducting
exploration magnetic surveys.
Magnetic Storms - Occasionally, magnetic activity in the ionosphere
will abruptly increase. The occurrence of such storms correlates with
enhanced sunspot activity. The magnetic field observed during such
times is highly irregular and unpredictable, having amplitudes as large
as 1000 nT.
Exploration magnetic surveys should not be conducted during magnetic
storms. This is because the variations in the field that they can produce
are large, rapid, and spatially varying. Therefore, it is difficult to correct
for them in acquired data.

Measuring the Earth's Magnetic Field
Magnetometers are highly accurate instruments, allowing the local
magnetic field to be measured to accuracies of 0.002%. The proton
precession, caesium vapour and gradiometer magnetometer systems are
used for commercial applications. The systems operate on broadly
similar principles utilizing proton rich fluids surrounded by an electric
164

coil. A momentary current is applied through the coil, which produces a
corresponding magnetic field that temporarily polarizes the protons.
When the current is removed, the protons realign or process into the
orientation of the Earth's magnetic field.
Gradiometers measure the magnetic field gradient rather than total
field strength, which allows the removal of background noise. Magnetic
gradient anomalies generally give a better definition of shallow buried
features such as buried tanks and drums, but are less useful for
investigating large geological features. Unlike EM surveys, the depth
penetration of magnetic surveys is not impeded by high electrical ground
conductivities associated with saline groundwater or high levels of
contamination.
Flux-gate magnetometer
- relative instrument
- can be used to find vector components, direction of field
- portable instruments usually set up to read H
Z
(vertical
component)

Proton-precession magnetometer
- simple, inexpensive, accurate, portable instrument
- measures absolute, total value of field
- 1 nT precision
- susceptible to strong magnetic gradients
It shows no appreciable instrument drift with time. One of the important
advantages of the proton precession magnetometer is its ease of use
165

and reliability. Sensor orientation need only be set to a high angle with
respect to the Earth's magnetic field. No precise leveling or orientation is
needed. The magnetic field we record with proton precession
magnetometer has two components:
The main magnetic field, or that part of the Earth's magnetic field
generated by deep (outer core) sources. The direction and size of this
component of the magnetic field at some point on the Earth's surface is
represented by the vector labeled Fe in the figure.
The anomalous magnetic field, or that part of the Earth's magnetic
field caused by magnetic induction of crustal rocks or remanent
magnetization of crustal rocks. The direction and size of this component
of the magnetic field is represented by the vector labeled Fa in the
figure. The total magnetic field we record, labeled Ft in the figure, is
nothing more than the sum of Fe and Fa.
Typically, Fe is much larger than Fa, as is shown in the figure (50,000
nT versus 100 nT). If Fe is much larger than Fa, then Ft will point almost
in the same direction as Fe regardless of the direction of Fa.That is
because the anomalous field, Fa, is so much smaller than the main field,
Fe, that the total field, Ft, will be almost parallel to the main field.

166


Similarities Between Gravity and Magnetics

- Geophysical exploration techniques that employ both gravity and
magnetics are passive (measure a naturally occurring field of the
earth.
- Collectively, the gravity and magnetic methods are often referred
to as potential methods.
- Identical physical and mathematical representations can be used
to understand magnetic and gravitational forces. For example, the
fundamental element used to define the gravitational force is the
point mass and the fundamental magnetic element is called a
magnetic monopole.
- The acquisition, reduction, and interpretation of gravity and
magnetic observations are very similar.
- Both gravity and magnetic vary in time and space and used as
reconnaissance tools in exploration.

Differences Between Gravity and Magnetics

- The fundamental parameter that controls gravity variations is rock
density and the fundamental parameter controlling the magnetic
field variations is magnetic susceptibility.
- Unlike the gravitational force, which is always attractive, the
magnetic force can be either attractive or repulsive.
- Unlike the gravitational case, single magnetic point sources
(monopoles) can never be found alone in the magnetic case.
Rather, monopoles always occur in pairs. A pair of magnetic
monopoles, referred to as a dipole, always consists of one positive
monopole and one negative monopole.
- A properly reduced gravitational field is always generated by
subsurface variations in rock density. A properly reduced magnetic
field, however, can have as its origin at least two possible sources.
It can be produced via an induced magnetization, or it can be
produced via a remnant magnetization.
- Unlike the gravitational field, which does not change significantly
with time**, the magnetic field is highly time dependent.
- gravity requires 0.1 ppm accuracy, magnetic > 10 ppm
- gravimeter is relative instrument; magnetometer is absolute
167

- densities vary from 1 to 4; susceptibility over several orders of
magnitude
- gravity anomalies smooth, regional; magnetic anomalies sharp,
local
- tides are only external gravity effect, can be corrected. Effect of
magnetic storms cannot be removed.
- gravity corrections: drift, latitude, free air, Bouguer, terrain, etc.;
magnetic corrections: ± drift, IGRF
- gravity surveys slow, expensive; magnetic costs about 1/10 of g

Applications of Rock magnetism in paleomagnetism :

a- Reversals of the earth‘s field. (most recent reversal about 20.000
y. ago., 50/50 N/R.
b- Sea floor spreading.
c- Secular variation and paleo intensity of the earth field.
d- Polar wander and continental drift.
e- Paleo climatology.
f- Magnetic dating of rocks by :
- secular variation 10
3
y.
- polarity zones 10
4
– 10
6
y.
- average paleomagnetic pole positions 10
7
– 10
9
y.
- Q ratio.
g. Tectonic movements involving rotation. Ex. Japan. By NRM.

GENARAL APPLICATIONS
o Finding buried steel tanks and waste drums
o Detecting iron and steel obstructions
o Accurately mapping archaeological features
o Locating unmarked mineshafts
o Mapping basic igneous intrusives & faults
o Evaluating the size and shape of ore bodies
168


Problem Sets

PROBLEM 1

IF THE MAGNETIC SUSCEPTIBILITY OF A SPHERICAL
PLUTON IS 0.0003 AND THE EARTH’S MAGNETIC FIELD
( B ) IS 0.0006 TESLA. THE RADIUS OF THE PLUTON IS
1 KM AND THE MAGNETIC PERMEABILITY IS 4 π X
10
-7
.

COMPUTE :
1. THE MAGNETIC FIELD STRENGTH ( H )
2. THE INTENSITY OF MAGNETIZATION ( I )
3. THE MAGNETIC MOMENT OF THE PLUTON ( M ).