ADA179487

0
ANALYTICAL STUDIES OF BODY WAVE
PROPAGATION AND ATTENUATION
DTIC
ELECTE
by
Ignacio Sanchez-Salinero, Jose M. Roesset
and Kenneth H. Stokoe, II
a report on research
sponsored by
United States Air Force
Office of Scientific Research
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UNIVERSITY OF TEXAS AT AUSTIN
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AIR FORCE OFFICE
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AFOSR/NA
AFOSR-83-0062
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LLING AFB, DC
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11. TITLE
,nciud.
Security Clasificaton) ANALYT ICAL STUDIESOF
BODY WAVE PROPAGATION AND ATTENUATION
61102F
WORK UNIT
NO.
TASK
NO.
NO.
2307
Cl
12. PERSONAL AUTHOR(S)
IGNACIO SANCHEZ-SALINERO, JOSE M. ROESSET AND KENNETH H. STOKOE, II
13.. TYPE OF REPORT
13b. TIME COVERED
2 1 83
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FROM
TECHNICAL
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11.
DATE OF REPORT (Yr.. Mo.. Day)
15. PAGE COUNT
SEPTEMBER, 1986
TO6/15/86
296
16. SUPPLEMENTARY NOTATION
17.
"
•
COSATI CODES
FIELO
GROUP
I.
SUB.
SUBJECT TERMS (Continue on reverse if necesewy and Identify by block number)
ANALYTICAL STUDY, ATTENUATION, BODY WAVES, COMPRESSION WAVES
CROSS CORRELATION, DAMPED WAVE VELOCITY, DISPERSION, FARFIELD WAVES,. FREQUENCY-DOMAIN ANALYSIS, GREEN'S FUNCTIONS,
OR.
16. ABSTRACT (Continue on reverseifnecesary and Identifyby block number)
(con
In situ seismic methods are becoming widely used as a means of nondestructively
evaluating the elastic properties of geotechnical systems. The crosshole and downhole
seismic methods are most often used. Elastic constants are calculated from the records of
body waves (longitudinal and transverse) traveling through the media. Measurements are made
by generating a seismic disturbance at one point and measuring the time required for the
disturbance to travel to one or more seismic receivers. Several simplifying assumptions are
made in the traditional analysis of seismic measurements for engineering purposes. These
assumptions include assuming plane wave propagation, measurement of only far-field waves.
and independence of the measurements on the source-receiver configuration and on the amount
of material damping. An analytical study of the effects and the validity of the different
assumptions is presented. A review of existing techniques to evaluate seismic data based on
time domain analysis is performed, and new techniques based on correlation and spectral
analyses are presented. Emphasis is placed on determination of wave velocities and
(continued on reverse
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DR. SPENCER T. WU
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18. (cont.) HYSTERETIC DAMPING, LINE SOURCES, LONGITUDINAL MOTION, MATERIAL DAMPING,
NEAR-FIELD EFFECTS, NEAR-FIELD WAVES, PLANE WAVE PROPAGATION, POINT SOURCES, SHEAR
WAVES, SEISMIC WAVES, SPECTRAL ANALYSIS, SPHERICAL WAVE FRONT, SOURCE-RECEIVER
CONFIGURATION, TIME-DOMAIN ANALYSIS, TRANSVERSE MOTION, VELOCITY, WHOLE SPACE
19. (cont.)
which elastic properties and material damping can be
from "i.'
,:,.
~~~calcqltqd.f parameters
~ attenuation
r
It is found that, for the range of distances and frequencies typically used in
engineering applications, body wave fronts generated by point sources cannot be
considered plane and that near-field effects associated with spherical wave fronts
can be very important. The near-field effects are caused by coupling between waves
which exhibit the same particle motion but which propagate at different velocities
ard attenuate at different rates. To minimize the detrimental effects of near-field
waves in those methods based on spectral analysis techniques, it is recommended that,
in the field setup, the ratio of distances from the source to the second and first
receivers be of the order of two or greater. For typical setups in which the
distance from the source to the first receiver is equal to the distance between the
receivers, near-field effects can be neglected if'measurements are made at
frequencies such that the distance from the source to the closest receiver is at
least one wavelength. Additional recommendations regarding body wave techniques are
presented.
I.,.,
I
*
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d 7- 0 44 2
AFOSR - TM.
ANALYTICAL STUDIES OF BODY WAVE PROPAGATION AND ATTENUATION
Approved for public reesDo;'
distribut ion unlimitod-
by
Ignacio Sinchez-Salinero, Josd M. Roesset and Kenneth H. Stokoe, II
Pit
a report on research
sponsored by
United States Air Force
Office of Scientific Research
Bolling Air Force Base
(AFSC)
AIR FORCE OFFICE OF SCIENTIFIC RESEARC1
DTIC
TO
NOTICE '-F TRANSMTTAL
and is
, This technicai repor hs been, reviewed
t.ic reteae iAW AFR 190-12
asp.'o.:VJ
bistriout ":j
;
..
MAT IHZ. J. tKEP.P-R-"
Chief, Technical information Division
11.
September, 1986
%
Geotechnical Engineering Report GR86-15
Geotechnical Engineering Center
Civil Engineering Department
The University of Texas at Austin
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ACKNOWLEDGEMENTS
The authors wish to express their gratitude to:
-
Linda M. Iverson for her assistance in typing this report, and to
The United States Air Force Office of Scientific Research (AFOSR),
Boiling Air Force Base, Washington, D.C., for supporting this research
under grant AFOSR 83-0062, Major John J. Allen was the initial project
manager, after which Lt. Col. Dale Hokanson became the project manager.
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ABSTRACT
In situ seismic methods arepJbecominjwidely used as a means of
nondestructively evaluating the elastic properties of geotechnical systems.
The crosshole and downhole seismic methods are most often used. -The elastic
constants are calculated from the records of body waves (longitudinal and
transverse waves) traveling through the media.
Measurements are made by
generating a seismic disturbance at one point and measuring the time required
for the disturbance to travel to one or more seismic receivers.
Several
simplifying assumptions are made in the traditional analysis of seismic
measurements for engineering purposes.
These 4sumptions)include assuming
plane wave propagation, measurement of only far-field waves, and independence
of the measurements on the source-receiver configuration and on the amount of
material damping.
An analytical study of the effects and the validity of the
different assumptions is presented.
A review of existing techniques to
evaluate seismic data based on time domain analysis is performed, and new
techniques based on correlation and spectral analyses are presented.
Emphasis is placed on determination of wave velocities and attenuation
parameters from which elastic properties and material damping can be
calculated.
,It is found that, for the range of distances and frequencies typically
used in engineering applications; body wave fronts generated by point sources
cannot be considered plane 3 and i*tnear-field effects associated with
spherical wave fronts can be very important.
The near-field effects are
caused by coupling between waves which exhibit the same particle motion but
which propagate at different velocities and attenuate at different rates.
To
minimize the detrimental effects of near-field waves in those methods based
on spectral analysis techniques, it is recommended that, in the field setup,
the ratio of distances from the source to the second and first receivers be
of the order of two or greater.
For typical setups in which the distance
from the source to the first receiver is equal to the distance between the
receivers,
near-field
effects
ca
be
neglected
if
measurements
are
made
- S
at
frequencies such that the distance from the source to the closest receiver is
at least one wavelength.
Additional recommendations regarding body wave
techniques are presented.
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TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS .......................................................... iii,
ABSTRACT ................................................................... iv
LIST OF FIGURES ........................................................... vii
LIST OF SYMBOLS ........................................................... xxi
CHAPTER ONE
INTRODUCTION ................................................................ 1
1.1 Seismic methods in civil engineering ................................... 1
1.1.1 Uses and benefits of seismic methods ............................ 2
1.2 Seismic methods based on body waves .................................... 6
1.3 Organization and objectives of this report ............................. 8
CHAPTER TWO
ANALYTICAL FORMULATION FOR BODY WAVES IN A FULL SPACE ...................... 13
2.1 Introduction .......................................................... 13
2.2 Theoretical background ................................................ 14
2.2.1 Two-dimensional antiplane motion ............................... 17
2.2.2 Two-dimensional in-plane motions ............................... 19
2.2.3 Three-dimensional motions ...................................... 20
2.3 Time and frequency domain parameters .................................. 22
2.4 Summary ............................................................... 31
CHAPTER THREE
CHARACTERISTICS OF BODY WAVE SIGNATURES .................................... 33
3.1 Introduction .......................................................... 33
3.2 Analysis of time records ...........
............................. 33
3.2.1 Medium with no material damping ................................ 34
3.2.2 Medium with material damping ................................... 43
3.2.3 Effect of Poisson's ratio ...................................... 44
3.3 Near-field effects ............
................................. 50
3.4 Polarity reversals upon reversing the impulse ......................... 52
3.5 Wave amplitude decay .................................................. 57
3.6 Summary ............................................................... 76
CHAPTER FOUR
TECHNIQUES OF EVALUATING BODY WAVE VELOCITIES .............................. 79
4.1 Introduction .................................................. 79
4.2 Wave velocities from direct times of arrival .......................... 79
4.3 Wave velocities from interval times of arrival ........................ 80
4.4 Wave velocities from cross-correlation records ........................ 92
4.4.1 Cross-correlation function ..................................... 92
4.4.2 Analysis of cross-correlation records .......................... 94
4.5 Wave velocities using spectral analysis techniques .................... 106
4.5.1 Theoretical background ........................................ 106
4.5.2 Dispersion curves from cross spectrum function ................ 110
4.5.3 Dispersion curves from transfer functions ..................... 129
4.6 Summary .............................................................. 137
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CHAPTER FIVE
EVALUATION OF BODY WAVE ATTENUATION ........................... ........... 141
5.1 Introduction ......................................................... 141
5.2 Theoretical background ............................................. 141
5.3 Material damping calculated from seismic records ..................... 144
5.4 Curves of apparent damping ratio ..................................... 149
5.5 Summary .............................................................. 160
CHAPTER SIX
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ................................. 173
6.1 Introduction ......................................................... 173
6.2 Body wave characteristics ............................................ 174
6.3 Body wave velocities ............................................... 176
6.4 Wave attenuation and material damping ................................ 177
APPENDIX A
TIME DOMAIN RECORDS ....................................................... 179
APPENDIX B
RELATIONSHIP BETWEEN ELASTIC AND DAMPED BODY WAVE VELOCITIES AND MODULI.. .209
APPENDIX C
CROSS-CORRELATION RECORDS ................................................. 215
APPENDIX D
BODY WAVE DISPERSION CURVES ............................................... 237
BIBLIOGRAPHY .............................................................. 269
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LIST OF FIGURES
Page
Figure
1.1
Crosshole seismic method (after Stokoe and Hoar, 1978) ............ 9
1.2
Downhole seismic method (After Stokoe and Hoar, 1978) ............ 10
2.1
Particle motion for antiplane and inplane loading ................ 18
2.2
Particle motion in three-dimensional case ........................ 21
2.3
Loading function and fast Fourier transform parameters ........... 23
2.4
Displacement, velocity and acceleration records for
two-dimensional longtudinal-motion ............................... 26
2.5
Displacement, velocity and acceleration records for
two-dimensional in-plane shear-motion ............................ 27
2.6
Displacement, velocity and acceleration records for
two-dimensional antiplane shear-motion ........................... 28
2.7
Displacement, velocity and acceleration records for
three-dimensional longitudinal-motion ............................ 29
2.8
Displacement, velocity and acceleration records for
three-dimensional shear-motion ................................... 30
3.1
Two-dimensional in-plane longitudinal motion. Effect of
different damping ratios ......................................... 35
3.2
Two-dimensional in-plane shear motion. Effect of different
damping ratios ................................................... 37
3.3
Two-dimensional anitplane shear motion. Effect of
different damping ratios ......................................... 39
3.4
Three-dimansional longitudinal motion. Effect of different
damping ratios ................................................... 41
3.5
Three-dimensional shear motion. Effect of different
damping ratios ................................................... 42
3.6
Two-dimensional in-plane longitudinal motion. Effect of
Poisson's ratio .................................................. 45
3.7
Two-dimensional in-plane shear motion. Effect of
Poisson's ratio .................................................. 46
3.8
Two-dimensional antiplane shear motion. Effect of
Poisson's ratio .................................................. 47
vii
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viii
Figure
Page
3.9
Three-dimensional longitudinal motion.
Effect of
Poisson's ratio .................................................. 48
3.10
Three-dimensional shear motion.
Effect of Poisson's ratio ....... 49
3.11
Longitudinal- and transverse-motion records as monitored
by a perfect vertical receiver ................................... 54
3.12
Transverse-motion records produced by reversed impulses as
monitored by a perfect vertical receiver ......................... 55
3.13
"Transverse-motion" records produced by reversed impulses
as monitored by a receiver with a 30 degree inclination .......... 56
3.14
Wave amplitude decay due to radiational damping.
Two-dimensional in-plane longitudinal motion ..................... 60
3.15
Wave amplitude decay due to radiational damping.
Two-dimensional in-plane shear motion ............................ 61
3.16
Wave amplitude decay due to radiational damping.
Two-dimensional antiplane shear motion ........................... 62
3.17
Wave amplitude decay due to radiational damping.
Three-dimensional longitudinal motion ............................ 63
3.18
Wave amplitude decay due to radiational damping.
Three-dimensional shear motion ................................... 64
3.19
Wave amplitude decay due to radiational damping and two
percent material damping. Two-dimensional in-plane
longitudinal motion
........................................ 66
3.20
Wave amplitude decay dLe to radiational damping and two
percent material damping. Two-dimensional in-plane
shear motion ..................................................... 67
3.21
Wave amplitude decay due to radiational damping and two
percent material damping. Two-dimensional antiplane shear
motion ........................................................... 68
3.22
Wave amplitude decay due to radiational damping and two
percent material damping. Three-dimensional longitudinal
motion ........................................................... 69
3.23
Wave amplitude decay due to radiational damping and two
percent material damping. Three-dimensional shear motion ........ 70
3.24
Wave amplitude decay due to radiational damping and five
percent material damping. Two-dimensional in-plane
longitudinal motion .............................................. 71
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3.25
Wave amplitude decay due to radiational damping and five
percent material damping. Two-dimensional in-plane
shear motion ..................................................... 72
3.26
Wave amplitude decay due to radiational damping and five
percent material damping. Two-dimensional antiplane shear
motion ........................................................... 73
3.27
Wave amplitude decay due to radiational damping and five
percent material damping. Three-dimensional longitudinal
motion ........................................................... 74
3.28
Wave amplitude decay due to radiational damping and five
percent material damping. Three-dimensional shear motion ........ 75
4.1
Two-dimensional in-plane longitudinal motion at two
different points in a medium with no damping .................. 81
4.2
Two-dimensional in-plane shear motion at two different
points in a medium with no damping ............................... 82
4.3
Two-dimensional antiplane shear motion at two different
points in a medium with no damping. Illustration of
interval travel time determination ............................... 83
4.4
Three-dimensional longitudinal motion at two different
points in a medium with no damping ............................... 84
4.5
Three-dimensional shear motion at two different points
in a medium with no damping ...................................... 85
4.6
Two-dimensional in-plane longitudinal motion at two
different points in a medium with five percent damping ........... 86
4.7
Two-dimensional in-plane shear motion at two different
points in a medium with five percent damping ..................... 87
4.8
Two-dimensional antiplane shear motion at two different
points in a medium with five percent damping. Illustration
of interval travel time determination ............................ 88
4.9
Three-dimensional longitudinal motion at two different
points in a medium with five percent damping ..................... 89
4.10
Three-dimensional shear motion at two different points
in a medium with five percent damping ............................ 90
4.11
Cross-correlation function of two-dimensional P-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 0%................................................. 91
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4.12
Cross-correlation function of two-dimensional SV-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 0% ................................................. 96
4.13
Cross-correlation function of two-dimensional SH-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 0% .......... ...................................... 97
4.14
Cross-correlation function of three-dimensional P-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 0% ................................................. 98
4.15
Cross-correlation function of three-dimensional S-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 0% ................................................. 99
4.16
Cross-correlation function of two-dimensional P-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 0% ................................................ 100
4.17
Cross-correlation function of two-dimensional SV-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 0% ................................................ 101
4.18
Cross-correlation function of two-dimensional SH-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 0% ................................................ 102
4.19
Cross-correlation function of three-dimensional P-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 0% ................................................ 103
4.20
Cross-correlation function of three-dimensional S-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 0% ................................................ 104
4.21
Dispersion curves for two-dimensional in-plane longitudinal
motion in a medium with no damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 111
4.22
Dispersion curves for two-dimensional in-plane longitudinal
motion in a medium with no damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 112
4.23
Dispersion curves for two-dimensional in-plane shear
motion in a medium with no damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 113
4.24
Dispersion curves for two-dimensional in-plane shear
motion in a medium with no damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 114
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Figure
Page
4.25
Dispersion curves for two-dimensional antiplane shear
motion in a medium with no damping and Poisson s ratio = 0.25.
Small spacing between receivers ................................. 115
4.26
Dispersion curves for two-dimensional antiplane shear
motion in a medium with no damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 116
4.27
Dispersion curves for three-dimensional longitudinal
motion in a medium with no damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 117
4.28
Dispersion curves for three-dimensional longitudinal
motion in a medium with no damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 118
4.29
Dispersion curves for three-dimensional shear
motion in a medium with no damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 119
4.30
Dispersion curves for three-dimensional shear
motion in a medium with no damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 120
4.31
Dispersion curves for three-dimensional longitudinal
motion normalized with respect to wavelength.
Small spacing between receivers ................................. 125
4.32
Dispersion curves for three-dimensional longitudinal
motion normalized with respect to wavelength.
Large spacing between receivers ................................. 126
4.33
Dispersion curves for three-dimensional shear
motion normalized with respect to wavelength.
Small spacing between receivers ................................. 127
4.34
Dispersion curves for three-dimensional shear
motion normalized with respect to wavelength.
Large spacing between receivers ................................. 128
4.35
Dispersion curve obtained from ths phase of the transfer
function for two-dimensional longitudinal motion in a medium
with no damping and Poisson's ratio = 0.25 ...................... 131
4.36
Dispersion curve obtained from the phase of the transfer
function for two-dimensional in-plane shear motion in a
medium with no damping and Poisson's ratio = 0.25 ............... 132
4.37
Dispersion curve obtained from the phase of the transfer
function for two-dimensional antiplane shear motion in a
medium with no damping and Poisson's ratio = 0.25 ............... 133
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4.38
Dispersion curve obtained from the phase of the transfer
function for three-dimensional longitudinal motion in a
medium with no damping and Poisson's ratio = 0.25 ............... 134
4.39
Dispersion curve obtained from the phase of the transfer
function for three-dimensional shear motion in a medium
with no damping and Poisson's ratio = 0.25 ...................... 135
5.1
Apparent damping ratio for in-plane longitudinal motion
in a medium with no damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 150
5.2
Apparent damping ratio for in-plane longitudinal motion
in a medium with no damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 151
5.3
Apparent damping ratio for in-plane shear motion in a medium
with no damping and Poisson's ratio = 0.25. Small spacing
between receivers ............................................... 152
5.4
Apparent damping ratio for in-plane shear motion in a medium
with no damping and Poisson's ratio = 0.25. Large spacing
between receivers ............................................... 153
5.5
Apparent damping ratio for antiplane shear motion in a medium
with no damping and Poisson's ratio = 0.25. Small spacing
between receivers ............................................... 154
5.6
Apparent damping ratio for antiplane shear motion in a medium
with no damping and Poisson's ratio = 0.25. Large spacing
between receivers ............................................... 155
5.7
Apparent damping ratio for three-dimensional longitudinal
motion in a medium with no damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 156
5.8
Apparent damping ratio for three-dimensional longitudinal
motion in a medium with no damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 157
Y
5.9
Apparent damping ratio for three-dimensional shear motion
in a medium with no damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 158
5.10
Apparent damping ratio for three-dimensional shear motion
in a medium with no damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 159
5.11
Apparent damping ratio for in-plane longitudinal motion in a
medium with five percent damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 161
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5.12
Apparent damping ratio for in-plane longitudinal motion in a
medium with five percent damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 162
5.13
Apparent damping ratio for in-plane shear motion in a medium
with five percent damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 163
5.14
Apparent damping ratio for in-plane shear motion in a medium
with five percent damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 164
5.15
Apparent damping ratio for antiplane shear motion in a medium
with five percent damping and Poisson's ratio = 0.25.
Small spacing between receivers ................................. 165
5.16
Apparent damping ratio for antiplane shear motion in a medium
with five percent damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 166
5.17
Apparent damping ratio for three-dimensional longitudinal
motion in a medium with five percent damping and Poisson's
ratio = 0.25. Small spacing between receivers .................. 167
5.18
Apparent damping ratio for three-dimensional longitudinal
motion in a medium with five percent damping and Poisson's
ratio = 0.25. Large spacing between receivers .................. 168
5.19
Apparent damping ratio for three-dimensional shear motion
in a medium with five percent damping and Poisson's ratio =
0.25. Small spacing between receivers .......................... 169
5.20
Apparent damping ratio for three-dimensional shear motion
in a medium with five percent damping and Poisson's ratio
0.25. Large spacing between receivers .......................... 170
A.1
Amplitude decay with distance (short distances) for
two-dimensional longitudinal motion in a medium with
no damping and Poisson's ratio = 0.25 ........................... 180
A.2
Amplitude decay with distance (long distances) for
two-dimensional longitudinal motion in a medium with
no damping and Poisson's ratio = 0.25 ........................... 181
A.3
Amplitude decay with distance (short distances) for
in-plane shear motion in a medium with no damping and
Poisson's ratio = 0.25 .......................................... 182
A.4
Amplitude decay with distance (long distances) for
in-plane shear motion in a medium with no damping and
Poisson's ratio
0.25 .......................................... 183
'A
•
,1~
.J.
xiv
Figure
Page
A.5
Amplitude decay with distance (short distances) for
antiplane shear motion in a medium with no damping and
Poisson's ratio = 0.25 .......................................... 184
A.6
Amplitude decay with distance (long distances) for
antiplane shear motion in a medium with no damping and
Poisson's ratio = 0.25 .......................................... 185
A.7
Amplitude decay with distance (short distances) for
three-dimensional longitudinal motion in a medium with
no damping and Poisson's ratio = 0.25 ........................... 186
A.8
Amplitude decay with distance (long distances) for
three-dimensional longitudinal motion in a medium with
no damping and Poisson's ratio = 0.25 ........................... 187
A.9
Amplitude decay with distance (short distances) for
three-dimensional shear motion in a medium with no
damping and Poisson's ratio = 0.25 .............................. 188
A.10
Amplitude decay with distance (long distances) for
three-dimensional shear motion in a medium with no
damping and Poisson's ratio = 0.25 .............................. 189
A.11
Amplitude decay with distance (short distances) for
two-dimensional longitudinal motion in a medium with
five percent damping and Poisson's ratio = 0.25 ................. 190
A.12
Amplitude decay with distance (long distances) for
two-dimensional longitudinal motion in a medium with
five percent damping and Poisson's ratio = 0.25 ................. 191
A.13
Amplitude decay with distance (short distances) for
in-plane shear motion in a medium with five percent
damping and Poisson's ratio = 0.25 .............................. 192
A.14
Amplitude decay with distance (long distances) for
in-plane shear motion in a medium with five percent
damping and Poisson's ratio = 0.25 .............................. 193
A.15
Amplitude decay with distance (short distances) for
antiplane shear motion in a medium with five percent
damping and Poisson's ratio = 0.25 .............................. 194
A.16
Amplitude decay with distance (long distances) for
antiplane shear motion in a medium with five percent
damping and Poisson's ratio = 0.25 .............................. 195
A.17
Amplitude decay with distance (short distances) for
three-dimensional longitudinal motion in a medium with
five percent damping and Poisson's ratio
0.25 ................. 196
xv
Figure
Page
A.18
Amplitude decay with distance (long distances) for
three-dimensional longitudinal motion in a medium with
five percent damping and Poisson's ratio = 0.25 ................. 197
A.19
Amplitude decay with distance (short distances) for
three-dimensional shear motion in a medium with
five percent damping and Poisson's ratio = 0.25 ................. 198
A.20
Amplitude decay with distance (long distances) for
three-dimensional shear motion in a medium with
five percent damping and Poisson's ratio = 0.25 ................. 199
A.21
Amplitude decay with distance (short distances) for
two-dimensional longitudinal motion in a medium with
no damping and Poisson's ratio = 0.4 ............................ 200
A.22
Amplitude decay with distance (long distances) for
two-dimensional longitudinal motion in a medium with
no damping and Poisson's ratio = 0.4 ............................ 201
A.23
Amplitude decay with distance (short distances) for
in-plane shear motion in a medium with no damping and
Poisson's ratio = 0.4 ........................................... 202
A.24
Amplitude decay with distance (long distances) for
in-plane shear motion in a medium with no damping and
Poisson's ratio
0 ........................................... 203
A.25
Amplitude decay with distance (short distances) for
three-dimensional longitudinal motion in a medium with
no damping and Poisson's ratio = 0.4 ............................ 204
A.26
Amplitude decay with distance (long distances) for
three-dimensional longitudinal motion in a medium with
no damping and Poisson's ratio = 0.4 ............................ 205
A.27
Amplitude decay with distance (short distances) for
three-dimensional shear motion in a medium with no
damping and Poisson's ratio = 0.4 ............................... 206
A.28
Amplitude decay with distance (long distances) for
three-dimensional shear motion in a medium with no
damping and Poisson's ratio = 0.4 ............................... 207
C.1
Cross-correlation function of two-dimensional P-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 5%
......................................... 216
C.2
Cross-correlation function of two-dimensional SV-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 5%................................................ 217
.4
%I.*/
'"q,
xvi
Figure
Page
C.3
Cross-correlation function of two-dimensional SH-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 5% ................................................ 218
C.4
Cross-correlation function of three-dimensional P-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 5% ................................................ 219
C.5
Cross-correlation function of three-dimensional S-motion
records obtained in the near field. Poisson's ratio = 0.25
and damping = 5%................................................ 220
C.6
Cross-correlation function of two-dimensional P-motion
records obtained in the near field. Poisson's ratio = 0.4,
Damping = 0%.................................................... 221
C.7
Cross-correlation function of two-dimensional SV-motion
records obtained in the near field. Poisson's ratio = 0.4,
Damping = 0%.................................................... 222
C.8
Cross-correlation function of two-dimensional SH-motion
records obtained in the near field. Poisson's ratio = 0.4,
Damping = 0%.................................................... 223
C.9
Cross-correlation function of three-dimensional P-motion
records obtained in the near field. Poisson's ratio = 0.4,
Damping = 0%.................................................... 224
C.10
Cross-correlation function of three-dimensional S-motion
records obtained in the near field. Poisson's ratio = 0.4,
Damping = 0%.................................................... 225
C.11
Cross-correlation function of two-dimensional P-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 5%................................................ 226
C.12
Cross-correlation function of two-dimensional SV-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 5%.
.........................................
227
C.13
Cross-correlation function of two-dimensional SH-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 5%................................................ 228
C.14
Cross-correlation function of three-dimensional P-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping = 5%................................................ 229
C.15
Cross-correlation function of three-dimensional S-motion
records obtained in the far field. Poisson's ratio = 0.25
and damping
5%.
.........................................
230
%
xvii
Figure
Page
C.16
Cross-correlation function of two-dimensional P-motion
records obtained in the far field. Poisson's ratio = 0.4
and damping = 0% ................................................ 231
C.17
Cross-correlation function of two-dimensional SV-motion
records obtained in the far field. Poisson's ratio = 0.4
and damping = 0% ................................................ 232
C.18
Cross-correlation function of two-dimensional SH-motion
records obtained in the far field. Poisson's ratio = 0.4
and damping = 0% ................................................ 233
C.19
Cross-correlation function of three-dimensional P-motion
records obtained in the far field. Poisson's ratio = 0.4
and damping = 0% ................................................ 234
C.20
Cross-correlation function of three-dimensional S-motion
records obtained in the far field. Poisson's ratio = 0.4
and damping = 0% ................................................ 235
D.1
Dispersion curves for two-dimensional in-plane longitudinal
motion in a medium with five percent damping and Poisson's
ratio = 0.25. Small spacing between receivers .................. 238
D.2
Dispersion curves for two-dimensional in-plane longitudinal
motion in a medium with five percent damping and Poisson's
ratio = 0.25. Large spacing between receivers .................. 239
D.3
Dispersion curves for two-dimensional in-plane shear motion
in a medium with five percent damping and Poisson's ratio
0.25. Small spacing between receivers .......................... 240
D.4
Dispersion curves for two-dimensional in-plane shear motion
in a medium with five percent damping and Poisson's ratio
0.25. Large spacing between receivers .......................... 241
D.5
Dispersion curves for two-dimensional antiplane shear motion
in a medium with five percent damping and Poisson's ratio
0.25. Small spacing between receivers .......................... 242
D.6
Dispersion curves for two-dimensional antiplane shear motion
in a medium with five percent damping and Poisson's ratio =
0.25. Large spacing between receivers .......................... 243
0.7
Dispersion curves for three-dimensional longitudinal motion
in a medium with five percent damping and Poisson's ratio
0.25. Small spacing between receivers .......................... 244
D.8
Dispersion curves for three-dimensional longitudinal motion
in a medium with five percent damping and Poisson's ratio =
0.25. Large spacing between receivers .......................... 245
B'
v
rWTW. WCrv~r.~v
-..
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xvii i
Figure
Page
D.9
Dispersion curves for three-dimensional shear motion in a
medium with five percent damping and Poisson's ratio = 0.25.
Small spacing between receivers.
...........................
246
D.10
Dispersion curves for three-dimensional shear motion in a
medium with five percent damping and Poisson's ratio = 0.25.
Large spacing between receivers ................................. 247
D.11
Dispersion curves for two-dimensional in-plane longitudinal
motion in a medium with no damping and Poisson's ratio = 0.4.
Small spacing between receivers ................................. 248
D.12
Dispersion curves for two-dimensional in-plane longitudinal
motion in a medium with no damping and Poisson's ratio = 0.4.
Large spacing between receivers ................................. 249
D.13
Dispersion curves for two-dimensional in-plane shear motion
in a medium with no damping and Poisson's ratio = 0.4.
Small spacing between receivers ................................. 250
D.14
Dispersion curves for two-dimensional in-plane shear motion
in a medium with no damping and Poisson's ratio = 0.4.
Large spacing between receivers ................................. 251
D.15
Dispersion curves for two-dimensional antiplane shear motion
in a medium with no damping and Poisson's ratio = 0.4.
Small spacing between receivers ................................. 252
D.16
Dispersion curves for two-dimensional antiplane shear motion
in a medium with no damping and Poisson's ratio = 0.4.
Large spacing between receivers ................................. 253
D.17
Dispersion curves for three-dimensional longitudinal motion
in a medium with no damping and Poisson's ratio = 0.4.
Small spacing between receivers ................................. 254
D.18
Dispersion curves for three-dimensional longitudinal motion
in a medium with no damping and Poisson's ratio = 0.4.
Large spacing between receivers ................................. 255
D.19
Dispersion curves for three-dimensional shear motion in a
medium with no damping and Poisson's ratio = 0.4.
Small spacing between receivers ................................. 256
D.20
Dispersion curves for three-dimensional shear motion in a
medium with no damping and Poisson's ratio = 0.4.
Large spacing between receivers ................................. 257
D.21
Dispersion curves obtained from the phase of the transfer
function for two-dimensional longitudinal motion in a medium
with five percent damping and Poisson's ratio
0.25 ............ 258
S%
-%
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%
.
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xix
Figure
Page
D.22
Dispersion curves obtained from the phase of the transfer
function for two-dimensional inplane shear motion in a medium
with five percent damping and Poisson's ratio = 0.25 ............ 259
D.23
Dispersion curves obtained from the phase of the transfer
function for two-dimensional antiplane shear motion in a medium
with five percent damping and Poisson's ratio = 0.25 ............ 260
D.24
Dispersion curves obtained from the phase of the transfer
function for Three-dimensional longitudinal motion in a medium
with five percent damping and Poisson's ratio = 0.25 ............ 261
D.25
Dispersion curves obtained from the phase of the transfer
function for three-dimensional shear motion in a medium
with five percent damping and Poisson's ratio = 0.25 ............ 262
D.26
Dispersion curves obtained from the phase of the transfer
function for two-dimensional longitudinal motion in a medium
with no damping and Poisson's ratio = 0.4 ....................... 263
D.27
Dispersion curves obtained from the phase of the transfer
function for two-dimensional inplane shear motion in a medium
with no damping and Poisson's ratio = 0.4 ....................... 264
D.28
Dispersion curves obtained from the phase of the transfer
function for two-dimensional antiplane shear motion in a medium
with no damping and Poisson's ratio = 0.4 ....................... 265
D.29
Dispersion curves obtained from the phase of the transfer
function for Three-dimensional longitudinal motion in a medium
with no damping and Poisson's ratio = 0.4 ....................... 266
D.30
Dispersion curves obtained from the phase of the transfer
function for three-dimensional shear motion in a medium
with no damping and Poisson's ratio
0.4 ....................... 267
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LIST OF SYMBOLS
= auto spectrum of displacement
A
wr
Cs
0
Cp
C
c
= phase velocity
C
= compressional wave velocity
CpD
= damped compressional wave velocity
CPI
= D
cp
*
cp, imaginary part of complex compressional wave velocity
cp + iCip
cs
= shear wave velocity
CsD
= damped shear wave velocity
s
cS
=
c, imaginary part of complex shear wave velocity
= cs + iCsl
CR(T) = cross correlation function
CS(f) = cross spectrum function
d
= distance
dI
= distance from source to first receiver
d2
D
= distance from source to second receiver
= hysteretic damping ratio
Dp
= damping ratio in compression-extension
Ds
= damping ratio in shear
e
= base of natural logarithmics
E
= Young's modulus of elasticity
f
= frequency in Hz
fmax
= maximum frequency of the FFT
F
= amplitude of loading function
xxi
S.
%
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. 4-,
.S
%
%
xxii
FFT
= fast Fourier transform
G
= shear modulus of elasticity
GI
= 2GDs, imaginary part of complex shear modulus
G*
= G(1+2iDs), complex shear modulus
H(w)
= Green's function or transfer function
H0 (2) = zero order Hankel function of the second kind
i
=Ve-T
k
= wave number
K°' K,
2 = modified Bessel functions of the second kind and orders zero,
one, and two, respectively
*'%
L
=wavelength
LpP
= primary wave in a record of longitudinal-motion (far-field wave)
Ls
= secondary wave in a record of longitudinal-motion (near-field wave)
LSV
=
M
= constrained modulus of elasticity
MI
= 2MDp, imaginary part of constrained modulus
M*
= M(1+2iDp), complex constrained modulus
N
= number of points of FFT
r
= distance from source to receiver
t
= time variable
T
= 1/f, time period
T
= total duration of function to transform with the FFT
Tp
= primary wave in a record of transverse-motion (near-field wave)
Ts
= period of loading function
Ts
= secondary wave in a record of transverse-motion (far-field wave)
TSH
= secondary wave in a record of transverse antiplane motion
TSV
= secondary wave in a record of transverse in-plane motion (far-field
wave)
*_€_?_
/.%'%
" I, ,,
,
secondary wave in a record of in-plane longitudinal-motion (nearfield wave)
\v*
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xxiii
u
= displacement
u
= displacement in x-direction
U
= linear spectrum of displacement
v
= velocity
v
= displacement in y-direction
V
= apparent velocity
w
= displacement in z-direction
W
= elastic strain energy stored when the strain is a maximum
S=
attenuation coefficient
6
= logarithmic decrement
At
= sampling rate for the FFT
AW
= energy dissipated per stress cycle
A
= Lame's constant of elasticity
I
= wavelength of shear wave
v
= Poisson's ratio
ir
= 3.14159...
p
= mass density
S
= inclination of geophone with respect to a vertical line
T
= time delay
0
= phase of cross spectrum or phase of transfer function
= damping capacity factor
= frequency in rad/sec
.
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CHAPTER
ONE
INTRODUCTION
1.1 SEISMIC METHODS IN CIVIL ENGINEERING
Steel, concrete, soil, rock, pavement and other engineering
materials behave like elastic bodies at small strains.
For soil,
rock and pavement materials, small strains can be considered any
strains on the order of o.ool percent or less.
The small-strain
range in steel and concrete is considerably higher.
Stresses
produced by a source releasing energy into the medium which creates
small strains will propagate away from the source as elastic waves.
The velocities at which these waves propagate depend mainly on the
elastic moduli of the materials that compose the medium.
Therefore, if one generates small-strain stress waves and measures
propagation velocities, elastic material properties can be
calculated.
Such measurements are typically categorized as seismic
measurements.
It is the purpose of this study to understand more completely
seismic measurements as used for engineering applications.
Most
applications of seismic methods, to date, have been by exploration
geophycisists, seismologists and petroleum engineers. The main
efforts of these groups have been directed toward determining
abrupt and broad geological changes in the interior of the earth,
generally at great depths within rock formations.
Estimation of
the structure of the earth or location of the position of natural
resources was typically the goal of those measurements. This
I%I %
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.
.
.
.
.
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2
information was generally obtained by studying the arrivals of
waves (usually compression waves) that were reflected or refracted
at the interfaces of layers with very different elastic properties.
Engineers have also been interested in locating abrupt
structural changes in earth materials. In addition to
interpretation of the structural bedding surfaces, engineers have
been attempting to determine the physical properties and
stratigraphy of the uppermost zone of the earth. In certain cases
near the surface of the earth, conventional sampling techniques can
be used. In other cases, at greater depths or under circumstances
where conventional sampling techniques and laboratory testing are
not feasible or do not provide an adequate evaluation of the
physical properties of the materials being tested, seismic methods
offer the engineer a viable tool for evaluating engineering
materials.
The emphasis in this research is placed on developing
analytical formulations with which synthetic seismic records can be
generated. These synthetic records are then processed following
standard field procedures used in engineering applications. The
applicability and limitations of the uses and processing of seismic
records are studied. The thrust of this work is placed on wave
velocity measurements (hence elastic stiffnesses), but attenuation
measurements (material damping) are also addressed.
I
."
1.1.1 Uses and benefits of seismic methods
As analytical techniques to analyze physical problems are
becoming more and more sophisticated, so are the laboratory testing
methods used to determine the properties of the materials involved
Z'
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in these problems.
There are, however, several difficulties
inherent with laboratory testing of sampled materials that do not
exist with in situ methods.
One of these difficulties is trying to
minimize the disturbance caused by sampling methods in testing
materials (especially in geotechnical materials).
This
complication is almost completely avoided by most seismic field
methods since there is no need to obtain any physical samples.
Another shortcoming of laboratory techniques is caused by the
inability to simulate properly in situ stress conditions.
With
field testing techniques, in situ stress conditions do not need to
be simulated since they are, in most cases, the stresses acting in
the materials at the time the tests are performed.
A further
obstacle in laboratory testing occurs when one tries to obtain
representative samples of the zone of interest.
Many times the
specimens tested are only characteristic of a small area within a
larger zone having some irregularities (like in most jointed rock
and in soils with primary and secondary structure).
With most of
the seismic methods, however, the areas sampled by the propagating
waves are large enough so that the properties obtained are
representative of the zone of concern.
When compared with other in situ testing methods, such as the
standard penetration test and the cone penetration test (in soils),
seismic methods have the advantages that: 1) they can be tailored
to sample small or large zones of materials; 2) they have a strong
theoretical basis (the theory of elasticity) and are not dependent
on empirical correlations; and 3) measured wave velocities are
independent of equipment used and do not need correction factors
like many of the other in situ testing methods.
The most important properties usually obtained from seismic
tests are the elastic shear modulus (G), which is obtained from the
1.-
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-
.
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.
.
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4
propagation velocity of shear waves, the constrained modulus (M),
which is obtained from the propagation velocity of longitudinal
waves, and Poisson's ratio (v) which can be calculated if shear and
longitudinal wave velocities are known.
Other properties like mass
density (p), material damping (D), strain amplitudes and strain
rates can be inferred or estimated from the analysis of wave
propagation records.
One of the main objectives of this study is
to understand better what and how variables affect wave velocities
calculated from various propagation measurements as outlined in
Section 1.3.
Elastic properties obtained from seismic methods are those
corresponding to very low-amplitude strains (strains less than
0.001 percent for most geotechnical materials).
Elastic moduli at
these small strains are usually called initial tangent moduli.
Initial tangent moduli are used directly as the stiffness of
engineering materials in vibration problems where the amplitudes of
the vibrations are kept very small (as is the case of most problems
concerning vibrations of soils and foundations).
For larger-
amplitude straining, initial tangent moduli represent key
parameters needed to fit nonlinear material models.
For anisotropic materials like soils, it is possible to
determine the small-strain elastic properties in different
i
directions by generating waves whose motions are polarized in
different planes.
In fact, one of the possible future uses of
seismic methods in soils is to evaluate the state of stress from a
three-dimensional picture of the seismic wave velocities (Lee and
I
Stokoe, 1986).
0
In the same way that geophysicists are interested in
determining structural changes and discontinuities in subsurface
Pi
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materials, so are engineers interested in locating anomalies of
engineering materials.
In this respect seismic methods can be used
to determine layering and properties of pavement systems; to detect
cracks and other anomalies like voids in structural members, rock
formations or pavement systems; to locate tunnels; in the integrity
testing of piles and piers and other structures where construction
specifications are difficult to check; and for many other purposes.
In Geotechnical Engineering, one of the major areas for the
use of seismic methods and the area of focus in this study, shear
and constrained moduli and their variation with strain are
essential in characterizing soil behavior under dynamic loadings
such as those created by earthquakes and blasting, in determining
dynamic stiffnesses of foundations for soil-structure interaction
analyses, and in evaluating the liquefaction susceptibility of
cohesionless soils by the strain approach.
Seismic methods often
used to determine the variation of initial shear and constrained
moduli with depth include the crosshole, downhole, surface
refraction, surface reflection, steady-state Rayleigh-wave method
and the spectral-analysis-of-surface-waves (SASW) method. Of these
techniques the crosshole and the downhole methods (body wave
methods) are the most widely used methods today for engineering
applications and the ones that provide the most reliable results.
The SASW method, a variation of the steady-state Rayleigh-wave
method, is a promising new technique that has all of the advantages
of the steady-state method without having its disadvantages.
The
surface refraction and surface reflection techniques are not
suitable for most engineering applications since one is not able to
obtain detailed moduli profiles with these methods.
-.
-
-.......
.-...
...
-.....-
%
6
1.2 SEISMIC METHODS BASED ON BODY WAVES
Body wave methods are based on the fact that the velocity at
which seismic waves travel through the interior of materials
depends on the elastic properties of the materials.
Measurements
are made by generating a seismic disturbance at one point in the
medium and measuring the time required for the disturbance to
travel to one or more seismic receivers.
If the distances
travelled by the measured waves are known, body wave velocities can
be calculated by dividing the distances by the corresponding travel
times.
From the theory of plane wave propagation in a homogeneous
isotropic elastic body, it can be shown that two types of body
waves propagate.
The first type, usually called the
compressional-, primary- or P-wave, propagates at a velocity cp (or
Vp; both notations are used in this study) which is given by
Cp2
M/p
(1.la)
or
C
=
( + 2G)/p
(1.lb)
2
cp
P
-
p(1 + E(1
v)(1- v)
- 2v)
(1.1c)
or
i
where,
e.
4..
%.5%
of
U~k.
U
~
-f
q4
Vs,
.4
-- I
7
M is constrained modulus of elasticity,
p is mass density,
x is Lame's constant of elasticity,
G is shear modulus of elasticity,
E is Young's modulus of elasticity, and
v is Poisson's ratio.
The second kind of body wave is called the shear-,
transverse-, secondary- or S-wave and propagates at a velocity cs
(or vs) given by
Cs2 = G/p
(1.2)
For plane wave propagation, P-waves are characterized by
particle motion along the line of the wave advance, while S-waves
are distinguished by having particle motion perpendicular to the
direction of wave propagation.
A look at Eqs. 1.1 and 1.2 shows
that, in the same material, P-waves propagate at a much faster
speed than S-waves, with the difference in velocities depending on
the value of Poisson's ratio.
The most common methods to determine wave propagation
velocities in the field for engineering applications are the
crosshole and downhole (or uphole) methods.
In the crosshole
method, the time required for body waves to travel horizontally
between several points located at the same depth is measured.
By
repeating the process at different depths, a profile of shear and
compressional wave velocities with depth can be obtained.
In the
downhole method, a seismic disturbance is generated at a point on
the surface, and the waves are monitored at several points in the
interior of the soil mass.
By measuring the travel times, wave
K.
V
velocities can be calculated after travel distances have been
determined.
Elements required in downhole and crosshole seismic surveys
are boreholes, sources, receivers and recording and triggering
systems. Typical field procedures are shown schematically in
Figs. 1.1 and 1.2. A detailed analysis of the field processes and
data analysis can be obtained in Woods (1978), Stokoe and Hoar
(1978), Corps of Engineers EM 1110-1-1802 (1979), Patel (1981) and
Hoar (1982).1
1.3 ORGANIZATION AND OBJECTIVES OF THIS REPORT
If increasingly broader and more accurate conclusions and
information are to be drawn from seismic records, it is essential
to understand better the wave propagation signatures and the
methods to analyze these signatures. Geophysical applications
usually involve analysis of seismic waves that are recorded at far
distances from the source. At these long distances, body waves are
usually considered plane waves (plane fronts). The fact that these
waves were generated by a point source and therefore are
propagating with a spherical wave front is usually neglected
without incurring serious errors. For engineering applications,
however, seismic waves are recorded at short aistances from the
source and should not be considered plane if a precise evaluation
of elastic velocities is necessary. An analytical formulation to
generate wave propagation records produced by a point source in a
full space (and therefore spreading in a spherical pattern) is
explained in Chapter Two. It is not intended in this chapter to
give a general overview of the theory of wave propagation in a--11
linearly elastic, homogeneous and isotropic body. However, by
e
77,;
0
V.
%
.o
%
%
I%
N
V%
"
-
f-
II
.P,
10
-ru
------,r
HaImmer Blow
Csing
Generation of
Body Waves
Trigeloit
I
I
Trnsdger
Wed~eclOn
Hamme
ncli
.ed
'.%
3-D Velocity Transducer
Wedged in Place
Fig. 1.2
-
Downhole seismic method (after Stokoe and Hoar,
1978).
'
.
.
. .
.
...
..
...
...
.
.
. .
.
11
S
1%
making use of the Green's functions formulation of displacements
produced by a point load in the interior of a full-space (for which
case closed-form solutions exist), a set of synthetic wave
propagation records is generated.
Typical features of these
synthetic records are studied in Chapter Three.
It is also necessary to develop new methods of analysis of
the field data that make better use of all the information
contained in the wave propagation records of body waves.
In this
respect, Chapter Four is devoted to the analysis of existing and
new techniques to evaluate propagation velocities from which the
elastic properties of the propagating medium can be determined.
The subjective nature of visual identifications of times of
arrivals is demonstrated, especially for wave propagation in a
medium with material damping.
The cross-correlation function is
shown to be an excellent method for determining elastic constants.
d
Finally spectral analysis techniques based in cross spectrum and
transfer functions are shown to be excellent tools to determine
elastic wave propagation velocities if properly applied.
Care must
be taken not to use in the near field.
A technique to estimate material damping is presented in
Chapter Five.
The technique is based in spectral analysis methods
and although is at a preliminary phase of development seems to hold
promise in the field determination of material damping.
A summary of this work, conclusions and recommendations are
then presented in Chapter Six.
-
..-
'.
CHAPTER
ANALYTICAL
TWO
FORMULATION
WAVES
IN
A
FOR
FULL
BODY
SPACE
2.1 INRODUCTON
In the crosshole and downhole seismic methods, body waves
(longitudinal and shear waves) are generated by a source at one
point and monitored at one or more other points as the waves pass
these points.
Direct as well as reflected and refracted waves are
recorded by the receivers.
It is assumed, however, in most of the
analyses performed of the wave propagation records that only direct
waves arrive at the receivers or that the effect of the reflected
and refracted waves is negligible (compared to the effect of the
direct waves).
It can be assumed under these conditions that the
waves behave as propagating in a full space.
Although the validity
of this assumption is questionable, it is intended (as a first step
.
in the process of understanding the behavior of the waves generated
by seismic sources) to analyze wave propagation records produced by
point and line loads in the interior of a three-dimensional space.
The mathematical formulation that leads to the analytical
generation of wave propagation records in a full space is explained
-
in this chapter.
13
N.......
.
...
...
.
.
..
a
.....
-
""a,1
14
2.2 THEORETICAL BACKGROUND
One of several methods to study wave propagation phenomena in
a linearly elastic medium is by superposition of the response to
steady-state (or harmonic) excitations.
The method, known as
Fourier superposition, Fourier synthesis or frequency domain
analysis, provides an easy way to study complicated transient
events when the solution to the steady-state problem is known.
Assume, for instance, that the solution to a harmonically vibrating
point load is known for all frequencies of vibration.
Then, the
response of the medium to any transient point load excitation can
be calculated by expressing the load in terms of its harmonic
components, evaluating the response of the system to each
component, and superposing the harmonic solutions to obtain the
final results.
If the solution to the point load is known, the
solution to loads over any area can also be obtained by integrating
the point load solutions over the area.
Superposition techniques are limited to linear systems.
Internal dissipation of energy in a truly nonlinear system is
simulated in a linear system by assuming a complex stiffness, G*,
of the form G* = G(1 + 2iD) in which G is the elastic shear modulus
of the material, i = V-1, and D is the hysteretic damping ratio.
In the following, the asterisk will be dropped from G* for
simplicity, and G will be used as a complex number when the
material exhibits a hysteretic type of damping.
The notation will
be clarified at each point in the text when confusion might arise.
A hysteretic type of damping is considered, in a frequency
domain analysis, to be frequency and strain independent.
..
%
While the
15
first assumption is usually true for most geotechnical materials in
the frequency ranges of interest herein (1 to looo Hz), the second
assumption is not.
It has been experimentally observed that the
energy dissipated per vibration cycle can generally be considered
independent of frequency but depends on the amplitude of the
vibration.
For low-strain amplitudes such as those associated with
seismic waves, however, damping can be considered to be independent
'S.
of strain (Johnston et al, 1979; Toksoz et al, 1979). The Fourier
superposition technique is, thus, an approximate method to study
waves propagating through a dissipative medium.
Even for a
material with true strain and frequency independent hysteretic
damping, the solution would be approximate due to some mathematical
problems created by the fact that hysteretic damping does not
satisfy the principle of causality.
The method offers, however, a
very good approximation, particularly for materials with low
damping.
Assume that the response of a medium to an excitation of the
form p(t) is desired.
As a first step in the Fourier superposition
method, the function p(t) is decomposed in its different frequency
components by means of a Fourier transform, P(w), as
i4,*
P(w) =
fP(t)e lwtdt
(2.1)
6
where
--
j
Pt)
P(w)
etdw
(2.2)
and
t is the time variable in seconds,
.
I
'p.
,V
'2
'
S.
i.*
.
•
.
,
4
-.
. . ." -....
,
,'
.- .,-...• .
.'-,
',o -...',.
'..
."
.'.
,. -
-. ,
.
4.
.-
.. .'% '.-....',. ..
•,
.* •.-
.:
.
- .
.- .
.
.
.
-. .
. -,
.. - .. - ?
. .* .
*:
,,
.
S
4
4
4
.
4
16
w is frequency in rad/sec,
e is the base of the natural logarithmics,
i = V-1, and
= 3.14159...
The two integrals in Eqs. 2.1 and 2.2 are known as a Fourier
transform pair. P(w) represents the harmonic components of the
loading function p(t).
The response of the medium to an excitation of the form •iWt
is defined by the Green's function or fundamental solution. The
fundamental solution gives the displacement at one point of the
medium due to a unit harmonic force applied at any other point of
the medium. If the Green's function of the problem under
consideration, H(w), is known and the superposition principle is
applied, the response of the system, u(t), to the loading function,
p(t), defined in Eq. 2.2 is then given by
u(t)
L
-
.P(w)
H(w)
•
dw
(2.3)
This means that the response of the system to any excitation
is given by the inverse Fourier transform of the product of the
fundamental solution by the Fourier transform of the excitation.
In other words, the Fourier transform of the response, U(w), is the
product of the Fourier transform of the excitation and the Green's
function and can be written as
U(w) =P(w)
H(w)
(2.4)
Forward and inverse Fourier transforms can be efficiently
calculated in a digital computer by means of a Fast Fourier
%
%
%
%0
%p.
17
Transform (FFT) algorithm.
Details on Fourier Transform theory and
FFT algorithms are explained in references such as Bracewell, R.N.,
1965; Brigham, E.O., 1974; and Newland, D.E., 1975.
Two- and three-dimensional wave propagation in a homogeneous,
isotropic, linearly elastic medium with or without linear
hysteretic damping is considered in this and the following two
chapters.
Fundamental solutions for two-dimensional plane strain
(in-plane) and out-of-plane (antiplane) elastodynamic motion as
well as three-dimensional elastodynamic motion are presented in the
following sections.
2.2.1 Two-dimensional antiplane motion
Antiplane shear motion is illustrated in Fig. 2.1a.
This
type of transverse motion (SH-motion) is characterized by particle
movement perpendicular to the plane of propagation of the wave.
The displacement caused by a unit concentrated harmonic force at a
distance r is (Achenbach, 1973)
i
w(W)
(2)
r-"
H
(2.5)
where
w is the value of the displacement,
w is the circular frequency of the vibration,
G is the complex shear modulus,
r is the distance from the source to the target point,
cs =G/p, is the shear wave velocity of the material
(complex value),
'kV
' ,'K
~ ,'.',.
-.
-i-'--i-"
.i,. ,"., °"-,
f%--"
-
-,'.'-'-.','
'.,
',.I'
' '
,,
A
18
e~
iwt
a. Antiplane motion
iI
w
u*e w
b. In-plane motion
Fig. 2.1
motion for antiplane and inpiane
-Particle
loading.
d,
%I
.1
e
V
~fl- ~~
~ ~ ~ ~ ~ ~ rJwgv.~.w
F1~~W
-.
wtrw
WV...-
J
PV
r'rw~f
Vr . I
WT
lp ry
-
p
y MP
2.~,~p
-
-w
m~ps-a
n
19
p is the density of the material, and
Ho( 2 ) is the zero order Hankel function of the second kind.
It should be noticed that the displacement, w, is a complex
value.
Since this displacement is the response of the medium to a
steady-state excitation, the magnitude of the complex number
represents the amplitude of the steady-state displacement, and the
phase gives the phase difference between the excitation and the
response.
2.2.2 Two-dimensional in-plane motions
In-plane excitation leads to two kinds of motions, a
longitudinal motion and a transverse or shear motion.
The
longitudinal motion (P-motion) is characterized by particle
displacement in the direction of propagation of the wave.
In the
shear motion (SV-motion), particle movement is perpendicular to the
direction of wave propagation (Fig. 2.1b).
These motions cause a
compression (or extension) and a shear distortion, respectively.
The displacement in the direction of the load at a point
along the line of excitation (P-motion) caused by a unit
concentrated harmonic force at a distance r is (Cruse and Rizzo,
1968)
u(w) = [1/(21PCs)] [*-x)
(2.6)
where,
u is the value of the displacement,
p is the density of the material, and
and x are given by the formulas:
%%
. ' %,
, ,V;,W
,
.. ,
.
,...
•
#,.
,....-
,,,
,
.%,,.,
./
/.
,
-
*'"
,
..
.
.,
.
,
.>,.",,
.
,
,
,
"
,,..
'
,,,
-
.1.'
,,
'
20
=
Ko(iao) + (1/ia o ) [Kl(ia o ) - (cs/cp)Kl(ibo)]
(2.7)
K2 (iao) - (cs/cp)2 K2 (ibo)
(2.8)
x
where,
ao = wr/cS,
b0 = wr/cp,
= [(A+2G)/p] 1/2 , is the compression wave velocity of the
material (complex value),
A is the complex Lame's constant, and
K0 , K, and K2 are the modified Bessel functions of the second
kind and orders zero, one and two, respectively.
The displacement in the direction of the load at a point on a
line perpendicular to the direction of excitation and passing by
the point of impact (SV-motion), caused by a unit concentrated
harmonic force at a distance r is (Cruse and Rizzo, 1968)
v(w) = [1/(2rpcs))
(2.9)
where,
v is the value of the displacement, and
is defined in Eq. 2.7.
2.2.3 Three-dimensional notions
The three-dimensional notation is illustrated in Fig. 2.2.
The fundamental solution for longitudinal motion (P-motion) is
defined by
u(w)
2
[1I(4ffpcs ][r-E]
%
s
(2.10)
%
.1
N
%
Or
%N
%
%%"%
21
lidt
ve.iwt
rp
Fig. 2.2
-Particle
motion in three-dimensional case.
It,
22
and the fundamental solution for shear motion (S-motion) is
[1/(4rpcs )) • r
v(w)
where r and
(2.11)
are given by
r = {[l+(1/iao) - (1/ao) 2/r} e- iao
-i
- (cs/cp) 2 {[(1/ib o ) - (1/bo) 2]/r} eibo
=
[i + 3(1/iao) - 3(1/ao) 2]/r} e
-
(Cs/Cp)2 {[
+ 3(1/ib o
(2.12)
iao
o) ]/r
e b0
(2.13)
2.3 TIME AND FREQUENCY DOMIN PARAMETERS
The loading function considered in this study is one cycle of
a sine wave of amplitude F and period Ts as illustrated in Fig.
2.3. The amplitude F has a value of one unit and the duration of
the impulse, Ts, is one second. The material considered as the
medium of propagation has a shear wave velocity of 100 units and a
mass density of 3.1 units. Note that no specific units are needed
(except for the time). If a specific set of units was to be used,
then all units would have to be compatible. For instance if forces
are expressed in lbs, and the wave velocities in ft/sec, then the
mass density should be in lb-sec 2 /ft4 and so on. If the density
was set in kg/m 3 , then velocities should be in m/sec and forces in
Newtons. The results are presented in dimensionless form, in most
of the cases.
Therefore, the actual magnitude of the force used,
%1
%
'
lop,
23
I-
CL'
°00
0-
U.
•t
I
U.
"
o
-5
..Z%
,4,'
5--
%5
%
%
%-%..
r.
24
the value of the shear wave velocity or the duration of the impulse
are irrelevant when presenting the results.
The fast Fourier transform parameters used in this study are
a sampling rate, At = 0.01 sec, and a total number of sampling
points, N = 1024.
(A few studies were done with, at = 0.05 sec and
N = 512, when the target point was far from the source).
In
choosing these parameters, the following conditions were
considered:
a) The sampling time, At, should be small enough: 1. to
reproduce accurately the loading and response functions, and 2. to
allow a sufficiently high maximum frequency, fmax to be obtained.
b) The total duration of the function to transform, Tp,
should be long enough:
1. to minimize the effects of the
periodicity inherent in the FFT algorithm, 2. to cover the time
range of interest, and 3. to produce a small frequency domain
.
sampling rate, af.
A singularity occurs in the fundamental solutions at zero
frequency (Eqs. 2.5 through 2.13).
In the present study, this
singularity was avoided using the fundamental solution at a
frequency equal to Af/lO as the solution for the static case.
The studies presented in this chapter and in the following
chapters have been conducted from records of particle displacement
with time.
Similar studies could have been performed from time
histories of particle velocity or acceleration, but trends are
i
easier to identify from waveforms of particle displacement than
from records of particle velocity or acceleration.
As an
illustration, displacement, velocity and acceleration records for
the five cases of motion considered in this report (two-dimensional
P-, SV- and SH-motions and three-dimensional P- and S-motions) are
%'N
U
25
presented in Figs. 2.4 through 2.8.
The notation used in these
figures is the following:
u is particle displacement,
v is particle velocity,
a is particle acceleration,
G is shear modulus of the material (real value),
F is amplitude of loading force,
cs is shear wave velocity (real value),
t is time in seconds,
d is distance from the source to the target point,
=s . T s ("wavelength of loading function", not to be
confused with Lame's constant),
D is damping ratio in percent, and
v is Poisson's ratio.
Velocity or acceleration records can be obtained from
Velocity or acceleration fundamental solutions, or by
differentiating the displacement records.
All velocity and
acceleration records in this study were obtained by using
fundamental solutions of velocity or acceleration.
It can be observed in Figs. 2.4 through 2.8 that the velocity
and acceleration waveforms are more complex than the displacement
records.
Even the simple case of the two-dimensional (2-D)
SH-motion in Fig. 2.6 seems easier to understand in terms of
displacement than in terms of velocity or acceleration.
A further
study of the displacement waveforms is done in the following
chapters.
I7.
,,...,-
-.
26
zp
a
j
U
W
U~%5%
a-t/
-
I
*
0
05
I
1.01.5
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27
1Ch
1
0
2.6
S.
us
U 40
_j
U
'-a
1.v
0.
*
Fig2.
__
_ ___ _ _
Dipaeet
fo
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__
vlct
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an
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ace
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rio
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re
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_
rd
tw-iesinli-paesea-oin
r
J
.1
e
Wk~
Z:.<
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a'
28
_
W
_
_
_
_
_
_
_
_
_
_
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_
_
_
_
Y
.
IA
W_3
o
CC
C
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WI
____._________________4__
00-
1.71
c,.N/I
29
.~
0.
U 4
0.
2
1.
we
D
5%
t4t/
4.L
P
z7 8
6
% %
4.
30
UA
I-
C
a-
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I.
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. "'p
CS.
.j
cc
Nb
U
a-t/
1.0
'Vp
P*'% _ %
%
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1.5
31
2.4 SUMARY
Analytical formulations to compute wave propagation records
of displacement, velocity or acceleration generated by a point load
in the interior of two- or three-dimensional, isotropic,
homogeneous, elastic full-spaces has been presented in this
chapter. The formulations are based on Fourier synthesis of the
Green's functions. The loading conditions and fast Fourier
transform parameters used in the computations for the examples
shown in the next two chapters are given in Section 2.3.
61
%
%'
0
'-4
;
Oi
A
110-1110Mx_
Jun
CHAPTER
A:-
THREE
CHARACTERISTICS
OF
BODY
WAVE
SIGNATURES
3.1 INTRODUCTION
Evaluation of in situ soil properties and site
characterization from wave propagation records is gaining
acceptance in engineering practice.
However, the level of use is
very superficial because the understanding of the wave record is
basically limited to "first-arrival" determinations.
In some cases
estimation of the "apparent" wave arrival may even lead to
erroneous values of elastic moduli. A better understanding of the
waveforms, the effect of different parameters on the time of
arrival of the waves and a better use of all the information
F
provided by the wave record is necessary for a more efficient use
of seismic methods. A series of analytical studies on the time
histories of body wave signatures is presented in the following
sections.
3.2 ANALYSIS OF TIME RECORDS
Waveforms obtained in the time domain at a dimensionless
distance, d/X = 2
are presented in Figs. 3.1 through 3.10 for
different kinds of body wave motion (where d is the distance from
the point source to the receiver and A = cs • Ts is the
"wavelength of the loading function"). All results are presented
33
the p
s
c
t
er..
k
J
-
V UW
ZVU
.
-
-
-
V
.
-W
-
U' i
34
in dimensionless form.
The wave amplitude is presented as a
dimensionless displacement uG/F for the two-dimensional cases
(since the force F is per unit length), and as uXG/F for the threedimensional cases (point forces).
dimensionless form as cst/d.
The time is presented in
In this way, the theoretical time of
arrival of the shear wave controlled by the elastic stiffness,
ts = d/cs, always corresponds to a value of one, and the
theoretical time of arrival of the longitudinal wave, tp
d/cp,
corresponds to
cs • tp/d = cs/cp = [(1-2v)/(2-2v)] 1/ 2
(3.1)
which depends on the value of Poisson's ratio, v. For most figures
presented in this chapter, a value of 0.25 was used for Poisson's
ratio. This ratio results in a theoretical arrival time of the Pwave of 0.577.
Also, in these expressions and in all figures
presented in this and other chapters, G, cs and cp represent the
real values of the shear modulus, shear wave velocity and
compressional wave velocity, respectively.
3.2.1 Medium with no material damping
The waveforms presented in Fig. 3.1 correspond to twodimensional in-plane longitudinal motion.
Consider the upper part
of the figure which corresponds to a wave travelling in a medium
with no material damping. The trace shown is composed of three
basic parts. The first part on the left of the trace represents a
quiet zone.
This results from the fact that the impulse has been
applied at the excitation point, but no energy has yet arrived at
the target point.
The second part, starting at the point indicated
I
I
U
I
S
mv
Wn v~rnNwm
uffKE
FEN7%mrum
w
rnx
Ar~w
% r~waw~
FWX
LW wI~r~xIM
35
LP
0%
0
2. 0
o d/)X
.
2.0
1.5
1.0
0.5
'0.0
Co td
UU
0
5%
1. 52.
OL70 051.0
Fig. 3.1
in-plane longitudinal motion.
Effect of different damping ratios.
-Two-dimensional
%a
%
36
by the solid arrow, includes a big excursion of the wave and lasts
until a second excursion arrives at a dimensionless time of one,
where the third part of the trace begins.
The point where the wave
arrives corresponds to a dimensionless time cst/d = 0.577.
This
time coincides precisely with the time of arrival of a wave
traveling at the compressional wave velocity in a medium with
Poisson's ratio equal to 0.25 (as used in this case).
The second
part of the wave (third part of the trace), starting at the point
denoted by the dashed arrow, arrives at a dimensionless time equal
to one, meaning that it is traveling at the shear wave velocity.
Hence, the first part of the waveform is called the primary wave
(P-wave), because it arrives first, and is denoted by the symbol Lp
for reasons that will become obvious soon.
The second part of the
waveform is called the secondary wave, because it arrives second,
and is denoted by LSV.
It should be noticed that even though the
second part of the wave is traveling at the shear wave velocity, it
does not represent a shear motion.
This second part of the wave is
actually a longitudinal motion (compression and extension) since
the particle motion is in the direction of propagation of the wave.
This second part of the waveform is called the near-field wave (or
additional near-field wave) of the P-motion and is discussed in
later sections.
A second set of records, similar to those presented in
Fig. 3.1 for two-dimensional P-motion, is presented in Fig. 3.2 for
two-dimensional SV-motion.
Consider again the wave propagating in
a medium with no material damping (upper trace).
After a quiet
period, a perturbation arrives at the target point.
This arrival,
indicated in the figure by the dotted arrow, comes at a time
corresponding to the compressional wave velocity (cst/d = 0.577).
The second and main part of the wave arrives at a dimensionless
time equal to one (corresponding to the shear wave velocity).
%
%
%
"%
I
37
*T
U.
T
I?
O
N -
'0.0
2. 0
0.5
1.0
1.5
2.0
C~t/d
LL..
57..
aa/
%
'0.
% %%
0.O10
.
%/d
.%
38
Again, the first wave arrival (between the two arrows) is called
the P-wave, because it arrives first (primary), but is denoted by
Tp, because it represents a transverse or shear motion. The second
part (starting at the point indicated by the solid arrow) is called
the S-wave (or SV-wave) and is denoted by TSV. The P-wave
(additional near-field in this case) represents that part of the
energy transmitted at the compressional wave velocity, and the
S-wave represents that part of the energy transmitted at the shear
wave velocity (although in both cases the motion is in the same
direction). Note that, in general, when P-motion (longitudinal
motion) is being measured, as in the case in Fig. 3.1, most of the
energy is carried by the P-wave, Lp, (indicated by the large
amplitude excursion of this wave). On the other hand, when shear
motion is being monitored, the bulk of the energy is transmitted by
the shear wave, TSV (Fig. 3.2). This is true at distances that are
far from the source (which d/x = 2 corresponds to). When the
receiver is located near the source, the additional near-field wave
may carry as much energy as the other wave. Records of waveforms
monitored at other distances are presented in Figs. A.1 through
A.4. Notice that, in these figures, the wavelength is constant
from one graph to the next. Because the horizontal axis is
normalized with respect to the distance from the source to the
receiver and this distance varies from one graph to the other, the
horizontal scale is changing giving the impression that the
wavelength is varying.
Two-dimensional SH-motion at a point located at a distance
d/A = 2 from the source is shown in Fig. 3.3. This kind of motion
is characterized by a pure shear wave. All energy is transmitted
at the shear wave velocity as can be observed in the upper trace of
Fig. 3.3. SH-motion will help clarify certain topics in the
following sections. But for the moment, notice that there is no
w-
%%
'-P.
.
I
S
%--
TS H
I
.p
0
-
/
d/N. .2. 0
C
10.0
0.5
1.0
1.5
2.0
C t/d
0
TS H
a.
Cat/.
Fig. 3.3
antiplane shear motion.
of different damping ratios.
-Two-dimensional
Effect
40
P-wave energy coupled with the SH-motion.
Additional antiplane
motion (SH-motion) records of wave propagation in a medium with no
material damping are included in Figs. A.5 and A.6.
Three-dimensional motion is presented in Figs. 3.4 and 3.5.
The same conclusions drawn for the two-dimensional motions apply to
the three-dimensional case.
The secondary wave arriving at a
dimensionless time of one in Fig. 3.4 does not present a clear
arrival.
This is due to the coupling of the primary and secondary
waves which distorts the shape of the waves.
Actually the sharp
angle in Fig. 3.4 is caused by the P-wave energy terminating at
that point and not to the arrival of the S-wave energy.
It should
also be noticed that, in the three-dimensional case, there is no
pure S-wave.
This occurs because of the inherent coupling between
P- and S-waves generated by the point source.
Records of waveforms
monitored at other distances are presented in Appendix A (Figs. A.7
through A.1O).
A look at these records further clarifies the
topic.
A first conclusion can be reached at this point.
If
compression wave velocity is to be determined from wave propagation
records, it is easier and more accurate to measure the time of
arrival from records of P-motion than from records of S-motion.
i
Similar conclusions can be made with respect to shear wave
velocity.
In terms of practical application, this conclusion means
that three-dimensional sensors oriented along the directions of
particle motions must be used in seismic testing to obtain the most
precision.
S
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Fig. 3.4
"Three-dimensional longitudinal motion.
of different damping ratios.
%%
Effect%
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42
0
0
d/-
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0.5
'0.0
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ct/d
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Effect of
shear motion.
-Three-dimensional
.
.
,,
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.
43
3.2.2 Medium with material damping
i
Records of wave propagation in a medium with no material
damping have been considered so far.
In the lower traces of
Figs. 3.1 through 3.5 and in Figs. A.11 through A.20, synthetic
records of waves propagating in a medium with five percent material
damping are presented.
It can be observed in these figures that
the general shape of the waveforms remains the same as those for
waves propagating in a medium with no damping.
Both primary and
secondary waves are encountered in two-dimensional P- and
SV-motions as well as in three-dimensional motions; and a pure
shear wave is encountered in only two-dimensional SH-motion.
It is
further observed that the amplitude of the waves has decreased
compared to the amplitude of the waves in a material without
damping; but the most important fact is that the time of arrival is
no longer easy to identify.
The arrival of the wave is not marked
by a sudden change of slope, as in the case with no damping, but is
represented by a smooth change.
It has been noted that for real materials, the time of
arrival chosen depends many times on the amplification applied to
the time history record (Ricker, 1953; Hoar and Stokoe, 1978).
The
higher the amplification, the sooner the waves seem to arrive.
The
I
synthetic records shown in Figs. 3.1 through 3.5 and A.11 through
A.20 clearly exhibit this point.
It is also observed in the
synthetic records that the waves arrive earlier than the arrival
,11
times calculated from the theoretical velocities of propagation, cs
or cp (cs and cp being the real values). The theoretical
propagation velocities (csD and cpD) caused by the use of complex
shear and compressional wave velocities (cs* = cs + icsl and
cp*
.',..
cp + icpl) are
ii
44.
Cp D :[Cp 2 + (Cpl)2]/Cp
(3.1)
[Cs 2 + (Csl) 2 ]/cs
(3.2)
and
cs
=
in which
Cpl and CsI are the imaginary parts of the complex
velocities, and
CpD and csD are the damped propagation velocities.
The damped velocities, cp0 and csD, are practically identical
to cs and cp, respectively (with differences smaller than 0.25
percent for values of the damping ratio smaller than five percent).
A discussion and derivation of the relationships between elastic
and damped body wave velocities and moduli is presented in Appendix
B.
The theoretical times of arrival (t = d/cs, or t = d/cp)
marked with arrows in Figs. 3.1 through 3.5.
are
-
Times obtained from
the first arrival of the wave will lead to estimates of the shear
or compression wave velocities of the medium which are slightly too
high.
This "sooner" arrival of the waves is a problem encountered
in the field and should always be kept in mind when analyzing real
time records.
Other approaches that eliminate the uncertainty in
the arrival-time estimation are presented in the next chapter.
3.2.3 Effect of Poisson's ratio
Finally the effect of Poisson's ratio, v, is studied in
Figs. 3.6 through 3.10.
In the upper part of these figures are the
records of waves propagating in a medium with a low Poisson's ratio
%
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-
Two-dimensional in-plane shear motion.
of Poisson's ratio.
Effec
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antiplane shear motion.
of Poisson's ratio.
-Two-dimensional
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50
(v = POI = 0.25) and in the lower part are corresponding waves in a
medium with a higher Poisson's ratio (v = POI = 0.4). All the
results are for a material with no damping.
The first observation that can be made is that Poisson's
ratio does not affect the secondary wave characteristics.
Secondly, the P-wave velocity increases as v increases. According
to Eq. 3.1 the dimensionless time of arrival for a Poisson's ratio
of 0.4 should be 0.408 as is the case. Finally it can be observed
that, due to the higher stiffness, the amplitude of the P-waves
V
decrease with increasing Poisson's ratio. A complete set of
records at other distances from the source is given in Appendix A.
3.3 NEAR-FIELD EFFECTS
It was indicated in the previous section that, in all the
motion records except for those of two-dimensional SH-motion, two
types of waves appeared in the waveform, one travelling at the
longitudinal wave velocity (P-wave) and the other travelling at the
shear wave velocity (S-wave). If one inspects the time history
records of longitudinal motion at different distances from the
source (presented in Appendix A), it can be observed that the
amplitude of the secondary wave (Lsv) in these P-motion records is_
attenuating at a much faster rate than the main event (the P-wave).
At a certain distance from the source, the amplitude of the S-wave
(Lsv-wave) becomes insignificant compared with the amplitude of the
P-wave (Lp-wave). Therefore, the S-wave (Lsv-wave) in a pure
longitudinal-motion (P-motion) record (no shear motion) only
"exists" at distances that are close to the source and is commonly
referred to as the additional near-field wave or near-field wave
(Aki and Richards, 1980; White, 1983).
-% IlN
51
Similarly, in a shear motion (transverse motion) record,
there are two events present: one travelling at the compressional
wave velocity (a P-wave, Tp) and a second travelling at the shear
wave velocity (Tsv). The first event (Tp) is, in this case, the
one that decays at a faster rate and represents the additional
near-field effect.
To illustrate this point, consider the case of threedimensional shear-motion in Eqs. 2.11 and 2.12.
The first
exponential in Eq. 2.12 represents a wave that is propagating at
i t = eiw[(r/cs)-t)
while
the shear wave velocity, cs , (e-iao
the second exponential is a wave propagating at the compressional
wave velocity, cp.
The amplitude of the wave travelling at the
2
shear wave velocity is composed of terms that vary with 1/r, 1/r
and 1/r3 , whereas the amplitude of the wave propagating at the
.
compressional wave velocity has terms varying at 1/r2 and 1/r3 .
Therefore, at large distances from the source (large r's), the
controlling term is that attenuating at a rate 1/r.
This term is
usually called the far-field term, since it is the only one
existing at far distances from the point of excitation.
The other
terms, those propagating at the shear and compression wave
velocities are called the additional near-field terms (or nearfield terms). Notice that the wave travelling at the shear wave
velocity also has near-field terms.
However, since these
components are travelling with the far-field component, they are
undistinguishable in the wave record, and the term near-field is
usually applied only to the components of the P-wave. An analogous
reasoning can be made for the three-dimensional longitudinal-motion
and for the two-dimensional motions.
'.A
Further illustrations are
given in Section 3.5.
.......
°'
52
In summary, when monitoring a pure longitudinal-motion (Pmotion), the P-wave represents the main event called the far-field
effect, and the S-wave represents the near-field effect. When
exciting a pure shear motion (S-motion), the S-wave is the main
event (far-field effect) and the P-wave is composed only of nearfield terms.
3.4 POLARITY REVERSALS UPON REVERSING THE IMPULSE
It should be noted that when the direction of the impulse is
reversed, both near- and far-field terms change polarity. However,
when measuring transverse motion in crosshole and downhole tests,
it is frequently found that S-waves change polarity but P-waves
keep the same polarity when the direction of the impulse is
reversed. To explain this behavior, several conditions should be
considered. First, to generate a shear motion (transverse motion)
with a crosshole source, a vertical impulse is applied to the
source. Since the source is of finite length and is usually wedged
in a borehole when the vertical impulse is applied, a longitudinal
motion is generated along with the shear motion. Second, when
reversing the direction of the vertical impulse, the direction of
the transverse motton is reversed, but the direction of the
longitudinal motion remains the same.
For the conditions cited above, a receiver will only monitor
the transverse motion and will not record any of the longitudinal
motion if the receiver used to record the transverse motion is
placed in a perfect vertical position (perpendicular to the
direction of propagation) . Under these conditions, the P-wave
recorded at the vertical receiver will be exclusively produced by
near-field terms of the transverse motion. Therefore, when the
-
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p~WZY~N~~W".%W
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9-
53P
direction of the impulse is reversed, both P- and S-waves should
reverse polarity.
However, when placing "vertical" receivers in
the field, the receivers are very seldom in a perfect vertical
position, and they are most commonly imperfect receivers.
exhibit some cross-sensitivity.)
(They
The small inclination of the
receiver and/or the small cross-sensitivity present in receivers
will allow the recording of a small component of the longitudinal
motion.
The waveform record will then be composed of both S- and
P-motions with their corresponding near- and far-field terms.
When
the direction of the impulse is reversed, the transverse motion
reverses but not the longitudinal motion.
Therefore, unless the
near-field wave associated with the transverse motion recorded is
larger than the far-field component of the part of the longitudinal
motion recorded by the receiver, the polarity of the P-wave will
not change.
To illustrate this point, longitudinal- and transverse-motion
records produced by horizontal and vertical point loads applied at
a distance of two wavelengths from the source are presented in
Fig. 3.11 as recorded by a perfect vertical receiver (zero crosssensitivity and perfect vertical orientation, 0 = 0 degrees).
Time
histories of forward and reversed vertical impulses as monitored by
the perfect vertical receiver are shown in Fig. 3.12.
It can be
observed that the P-wave is exclusively composed of the near-field
effect in the transverse-motion record and that both traces are
opposite (reversed) but otherwise identical.
The case of a
receiver which is inclined 30 degrees (0 = 30 degrees, roe
illustration purposes in which part of the inclination could
correspond to cross-sensitivity effects) with the vertical is
considered in Fig. 3.13.
The lower trace in this figure is
composed of seven-tenths of the shear motion produced by the direct
impulse and by half of the longitudinal motion.
In the upper
54
LC)L
0.
I..,o
cco
CD
cLL
0
0
E
u.,,,.
cL.7
EI
jU-No dS (
Uu
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6
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.. . .
\.
.. . . . . .,.....
.. .
. .
* *...
. . . . .:,
55
LO
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57
trace, that corresponds to the reverse impulse, the shear motion
has reversed but the longitudinal motion has remained with the same
sign.
Since one half of the P-motion carried more P-wave energy
than seven-tenths of the S-motion record, the combined waveform
resulted in initial P-waves that did not reverse polarity when the
direction of the impulse was reversed.
A similar line of reasoning can be used when the source and
receiver are not at the same depth for a crosshole test, or for
other cases in a downhole test.
3.5 WAVE AMPLITUDE DECAY
The amplitudes of seismic waves decrease as the waves
propagate through a medium. This decay of wave amplitude is caused
by two mechanisms: 1) spreading of wave energy from a source,
generally called geometrical or radiational damping, and 2)
dissipation of elastic energy due to primarily frictional losses in
the material itself, commonly known as attenuation, material or
internal damping.
Records of displacement with time at different distances from
the source are presented in dimensionless form in Appendix A. The
records are for two- and three-dimensional wave propagation in
media with material damping of 0 or 5 percent and Poisson's ratio
of 0.25 or 0.4. The input to the medium is one cycle of a sine
wave, as described in Section 2.3, and the parameters used in these
figures are the same as those described in Section 3.2.
Two-dimensional motion in a medium with no damping and
Poisson's ratio of 0.25 is shown in Figs. A.1 through A.6. It can
WV.,U\
~rUr ;
,;
.
~C-n
;i.
uq WW L
1,
.'W
,V 21
W
F._ I 2. -,
IV
-
58
be observed that for motions in which the primary and secondary
waves are coupled, the coupling occurs mainly at short distances
from the source (d/A < 2).
At larger distances, the P-wave has
separated from the S-wave, and the two waves can be clearly
distinguished from each other.
The minor wave (S-wave in the case
of P-motion and P-wave in the case of SV-motion) attenuates at a
faster rate than the major wave.
These minor waves, the additional
near-field waves, only exist at short distances from the source.
It should be noticed, however, that at short distances the energy
carried by the near-field terms can be a substantial amount of the
total energy in the waveform.
Wave amplitude decay in these
figures is due exclusively to radiation damping, since material
damping, D, is zero.
A similar situation arises for motions in a three-dimensional
full-space (Figs. A.7 through A.10).
However, the amplitude of the
V.
waves decreases much faster than in the two-dimensional cases.
This faster decay is due to the spreading of the energy from a
point source in a spherical pattern while in the two-dimensional
case the energy spreads in a cylindrical pattern from a line
source.
.9
Motion in a medium with five percent material damping is
presented in Figs. A.11 through A.20. The behavior is similar to
that in the medium with no material damping but amplitudes are
decreasing with distance at a faster rate due to the energy lost in
each cycle because of internal friction.
Finally a set of records corresponding to propagation in a
medium with no damping and a Poisson's ratio of 0.4 is presented in
Figs. A.21 through A.28.
The behavior is similar to that described
when Poisson's ratio was 0.25.
The amplitude of the P-wave
%.
"
59
component of the motion is, however, smaller when Poisson's ratio
is 0.4.
No records are included for SH-motion since they are the
same as those when Poisson's ratio is 0.25.
To study the attenuation behavior of body wave amplitude with
distance due to radiation damping, the variation of wave amplitude
with distance for steady-state motion has been plotted in
Figs. 3.14 through 3.18.
In these figures the displacement
amplitude is normalized as uG/F for two-dimensional motions and as
uAG/F for three-dimensional motions (as in the figures in Appendix
A), and the distance from the steady-state source to the receiver
(d) is normalized by dividing by the wavelength of the shear wave
(A = cs/f). The material considered as the propagating medium has
a Poisson's ratio of 0.25 and no material damping.
the parameters are explained in Section 2.3).
(The rest of
A frequency of
100 Hz was used for the computations.
The amplitude decay for a two-dimensional event is shown in
Figs. 3.14, 3.15 and 3.16 for P-, SV- and SH-motions, respectively.
It is observed that the general trend indicates that the amplitudes
decay with the square root of the distance.
V
This is particularly
clear in Fig. 3.16 for SH-motion amplitude decay.
For very small
distances or very low frequencies (d/A < 0.2), all wave amplitudes
decay at a rate that is slightly smaller than with the square root
of the distance.
For P- and SV-motion the amplitude decrease is in
proportion with the square root of the distance only at long
distances.
The value of this "long distance" seems to be of the
order of two or three shear wavelengths (A) for the shear motion
and ten shear wavelengths for the longitudinal motion.
(This
distance translates to approximately five compression wavelengths).
In the range of distances where the coupling of the P- and S-wave
~V
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65
is important, the decay in amplitude is not proportional to the
square root of the distance.
Three-dimensional wave-amplitude decay due to radiation is
shown in Figs. 3.17 and 3.18.
It can be observed that the general
trend indicates that the amplitudes decay proportionally to the
distance. The relation is valid, again, for long distances where
the coupling of the P- and S-waves is almost insignificant. A
further look at these figures indicates that at very short
distances from the source, the decay is also proportional to the
distance and that it is only in certain ranges of d/x where the
amplitudes of the waves do not decrease in proportion to the
distance.
A close inspection of Eqs. 2.10 through 2.13 indicates
that the amplitude of the three-dimensional waves is formed by
terms that decrease in proportion with the distance, the square of
the distance and the cube of the distance.
In the far field, the
only significant terms are the far-field terms that decrease in
proportion to the distance.
In a mid-range of distances (say,
1 < d/A < 10), the decay is a combination of terms that decay in
proportion to the distance, the square of the distance and the cube
of the distance.
Thus, the oscillating behavior of the decay
curves in this mid-range of distances.
Finally, in the very near
field (d/X on the order of tenths or less), the terms that decay in
proportion with the square of the distance and the cube of distance
essentially cancel, and amplitude decay is again nearly
proportional to the distance.
Wave amplitude decay graphs for waves propagating in mediums
with two and five percent material damping are presented in Figs.
3.19 through 3.23 and Figs. 3.24 through 3.28, respectively.
It is
noticeable in these figures that, at large distances (d/x greater
than about two), the amplitudes of the waves are decreasing at a
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much faster rate than when the waves were propagating in a medium
with no material damping.
At short distances, the waves have gone
through very few cycles of motion (if any), and the effect of
material damping has had a very minor influence.
For this reason
the amplitudes in the short range of distances are practically the
I,.
same as when no damping was considered.
3.6 SUMARY
Waveforms of particle displacement of body waves produced by
line loads in a three-dimensional space (in-plane and antiplane
point loads in two-dimensional notation) and by point loads also in
a three-dimensional space have been analyzed in this chapter. From
examination of these syntheti.c records, the following conclusions
V
k.
-V.
can be drawn.
1.
The most precise evaluation of P- or S-wave velocities
is obtained with receivers aligned parallel to the direction of
particle motion.
If the receivers are inclined, wave arrivals are
more difficult to identify because of the additional coupling
between waves. This conclusion is especially true for the S-wave,
the one of most concern in geotechnical engineering applications.
2.
When the material in which the body waves propagate
presents a dissipative behavior (as represented by a hysteretic
type of damping), it is more difficult to estimate the time of
arrival of the waves than when the material is not dissipative.
In
those cases, the apparent time of arrival of the wave is very much
influenced by the amplification applied to the wave record.
The
higher the amplification, the sooner the wave seems to arrive.
%I
77
In all waveforms except in those for two-dimensional
SH-motions, the waveforms are composed of two types of waves,
3.
S
P-waves travelling at the compressional wave velocity and S-waves
travelling at the shear wave velocity.
In a two-dimensional
in-plane transverse-motion record, the P-wave (Tp wave) only exists
at short distances from the source (distances smaller than
approximately two shear wavelengths), but the secondary wave (Tsv
wave) prevails at longer distances. It is for this reason that the
TSV wave is called the far-field wave while the Tp wave is called
the additional near-field wave (or near-field wave).
".
In records of
two-dimensional in-plane longitudinal-motion, the far-field wave
corresponds to the primary wave (Lp), and the near-field wave is
the SV-wave (Lsv).
In two-dimensional antiplane motion (SHmotion),
there is only one wave (TsH) which propagates at the shear wave
velocity.
4.
For waves generated by a point source in a three'5
dimensional space, the wave fronts spread in a spherical manner
from the point source.
There are, again, near- and far-field waves
both in longitudinal- and transverse-motion records.
The far-field
wave in a longitudinal motion record (Lp) travels at the
compressional wave velocity while the near-field wave (LS) travels
at the shear-wave velocity.
In a transverse-motion record, the
far-field wave (TS) travels at the shear wave velocity while the
near-field wave (Tp) propagates at the compressional wave velocity.
The near- and far-field waves have components with amplitudes
3
varying in proportion to 1/r2 and 1/r.
The far-field wave has,
in addition, terms that diminish in proportion to the distance.
Those terms with amplitudes decreasing with the square and the cube
of the distance are called the near-field components (or near-field
terms), while the terms decreasing with the distance are known as
the far-field components.
LIN.
The expression "near-field term" or
N.
78
"near-field component" is sometimes applied, however, to the nearfield wave, while the expression "far-field term" or "far-field
component" is sometimes applied to the far-field wave.
5.
Near-field effects in longitudinal-motion (P-motion)
records are more important than near-field effects in transversemotion records. In addition, near-field effects in longitudinalmotion records last longer (to distances of about ten times the
shear wavelength) than near-field effects in transverse-motion
records (to distances of about two shear wavelengths).
6.
Wave polarity reversals, which are frequently
encountered in the field upon reversing the direction of the
impulse, were explained in Section 3.4 by considering the effect of
near-field waves.
7.
For a medium in which energy dissipation occurs
exclusively because of radiation damping (geometrical spreading of
the energy), it was observed that wave amplitude decreases, in
general, in proportion to the distance for point sources and in
proportion to the square root of the distance for line sources.
These general rules apply more precisely to the far-field range of
distances which is defined as those distances (or frequencies) for
which the ratio d/x is greater than two for transverse motions and
ten for longitudinal motions. For waveforms generated from a point
source, this rule of energy dissipation (radiation damping) also
applies correctly for values of d/X that are smaller than about
1/2. When internal dissipation of energy is considered by the use
of a hysteretic, frequency- and strain- independent, material
damping, the amplitudes in the far-field decrease at a much faster
rate.
9....
:;Z
.1
%N
CHAPTER
TECHNIQUES
FOUR
OF
WAVE
EVALUATING
BODY
VELOCITIES
.-
4.1 INTRODUCTION
Many of the variables that can affect in situ seismic tests
such as sources, receivers, recording equipment and triggering
devices have been studied extensively in the literature (Stokoe and
Woods, 1972; Stokoe and Hoar, 1978; Hoar, 1982; Patel, 1981) and do
not require further extensive studies.
However, techniques of
determining wave velocities and attenuation have received little
attention. With the advent of digital waveform processing in the
field, several new techniques are available for velocity and
attenuation measurements.
In this chapter, new techniques for
velocity measurements based on sophisticated waveform processing
and data analysis are presented along with an extensive study of
the traditional time-domain technique.
Attenuation measurements
are presented in Chapter Five.
4.2 WAVE VELOCITIES FROM DIRECT TIMES OF ARRIVAL
The cheapest and most commonly used method to calculate
seismic wave velocities is by measuring the travel time of a wave
from a source point to a target point.
the direct travel time.
This time is usually called
By dividing the travel time into the
distance between the two points, the propagation velocity of the
.
79
.1
%
~~~~L
L&*..*w%
/
800
wave is obtained.
Wave velocity calculations by measuring direct
travel times have been analyzed in Chapter Three (Section 3.2) and
will not be repeated herein. Suffice it to say that, even under
ideal field conditions, some judgment is required when "picking"
the wave arrival time.
4.3 WAVE VELOCITIES FROM INTERVAL TIMES OF ARRIVAL
It is common practice in the field to determine wave
velocities by monitoring the waves passing by two or more different
receivers. In such cases, the time for body waves to travel
between several points within the soil mass is measured. Use of
interval travel times between two or more receivers has the
advantage that it eliminates possible triggering errors of the
recording device and also minimizes other field errors that might
occur when measuring direct travel times from the source to the
first receiver (Hoar, 1982).
Records of wave trains monitored at dimensionless distances
d/ = 2 and d/A = 8 from the source are presented in Figs. 4.1
through 4.10 (x being the predominant shear wavelength,
A= cs
Ts). The dimensionless time is defined in these figures
as cst/X, to avoid the distortion of the waves produced by the
distance (d) in the factor cst/d. All the records presented are
Iq
for waves propagating in a medium with a Poisson's ratio of 0.25.
A first approach to calculating wave velocities is by
identifying the interval time between first arrivals of the wave
from the first to the second receivers. Similar approaches can be
used when interval velocities are calculated in which the time it
takes the wave to travel from the first to the second receiver is
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91
determined by measuring the time difference between other reference
points such as first troughs, first peaks or zero crossings.
An
example of these procedures is illustrated in Fig. 4.3, for the
case of two-dimensional SH-motion in a material with no damping.
It can be observed that, since the wave does not change its shape
with distance in this case, all the approaches give the same
interval times.
Similar results would be obtained from the other
records of waves propagating in a medium with no damping (Figs.
4.1, 4.2, 4.4 and 4.5).
This conclusion is particularly true when
the receivers are placed in the far field.
If the receivers are
located in the near field, then the coupling of near- and far-field
waves can obscured the arrival of the waves.
When the medium in which propagation occurs is composed of a
material with hysteretic damping, several problems arise.
First,
times of arrival of the waves are no longer easy to identify (as
was previously noted in Section 3.2), and evaluation of interval
times from first arrivals may lead to slightly erroneous results.
In this regard, interval times from first peaks or zero crossings
might be easier to estimate.
The use of first troughs can be very
troublesome for conditions in which the near-field and far-field
waves are coupled together.
The first trough attributed to the
far-field wave might correspond or be obscured by the near-field
wave, as could happen in cases similar to those in Figs. 4.6 and
4.9.
A second problem arises from the fact that the shape of the
waves vary-with distance as shown in Figs. 4.6 through 4.10.
This
may lead to interval velocities that depend on the relative
distance between the recording points. The change in shape is
principally caused by the fact that the two components of the
waveform (the P-wave and the S- wave) are traveling at different
speeds.
Therefore, short distances from the source (near field),
the P- and S-waves are still coupled, and the combined wave can
,,o .....
o
. .
92
have a very irregular shape. In the far field, the P-wave, which
is traveling faster than the S-wave, arrives much earlier, and the
displacements caused by both waves do not interfere with each
other. In addition, in the far field, the near-field components
have disappeared.
In Fig. 4.8, possible points to use for interval time
measurements are indicated with arrows. It is clear that choosing
the points is a subjective matter (especially the ones
corresponding to times of the initial arrival). Even for a simple
case like Fig. 4.8 (SH-motion), the choice is not clear. The
dimensionless interval times obtained from first arrivals, first
peaks and first zero crossings are 5.7, 5.9 and 6.1, respectively,
compared to the theoretical value of 6.0. In his field research
a
with the crosshole method, Hoar (1982) has experimentally shown
similar relationship between travel times.
It should be noted that the conclusions drawn here for the
first peak of a wave whose first excursion goes up, will correspond
to the first trough of a wave whose first excursion goes down.
4.4 WAVE VELOCITIES FROM CROSS-CORRELATION RECORDS
,
.
4.4.1 Cross-correlation function
A different technique to calculate wave velocities from
seismic records is based on the cross-correlation function of waves
traveling by two receivers. Cross-correlation, CR(t), of two
functions, g(t) and h(t), is given by the integral:
I el
%
%-
93
CR(T)
f_:g(t) h(t+t) dt
(4.1)
where g(t) and h(t) represent the time records of the waves passing
by the first and second receivers, T is the time delay and t is the
variable of integration.
To interpret this integral, assume the second wave is shifted
by a time T. The sum of the products at each point of the shiftedsecond wave by the first wave will give one value of the crosscorrelation function at time shift t. If the process is repeated
for other time shifts, the cross-correlation function is defined.
Assume now that the two functions to be correlated are the same (in
terms of the shape of the amplitude versus time function) but that
one lags the other by a time t*. Then the cross-correlation
function will exhibit a maximum at a value of T equal to t*. If
these two shifted but otherwise identical functions represent the
waveforms recorded at two receivers of a seismic array, the time t*
where the peak of the cross-correlation occurs represents the time
it takes the wave to travel between the two receivers.
Unfortunately this idealization does not occur exactly in reality
because body waves change shape as they travel in the medium.
Nevertheless, when cross-correlating waves from two seismic records
that contain energy from predominately one type of body wave, a
maximum in the correlation function will occur at a time that
corresponds approximately with the time it takes the predominate
body wave to travel between the two receivers. If the distance
between the two receivers is divided by this time, a shear or
compression wave velocity is obtained.
-
'
.
I
- •
94
4.4.2 Analysis of cross-correlation records
Cross-correlation functions of some of the records presented
in Chapter Three are shown in Figs. 4.11 through 4.20 and in the
figures in Appendix C. Two geophone spacings were considered.
In
the first case, the receivers were located in the near field
(d/X2), Figs. 4.11 through 4.15 and Figs. C.1 through C.10.
In
the second case, the receivers were located in the far field, Figs.
4.16 through 4.20 and Figs. C.11 through C.20.
The ratio of the
distance from the source to the first receiver to the distance from
the source to the second receiver, d2 /d1 , was equal to 2, which is
common practice for crosshole seismic measurements (ASTM, 1985).
Values of Poisson's ratio (v = POI.R. in the figures) of 0.25 and
0.4 were considered, and damping ratios (D) of 0 and 5 percent were
used.
The cross-correlation function is presented in dimensionless
form by means of a normalized cross-correlation factor.
The
normalized cross-correlation is defined as the ratio of the value
of the cross-correlation at any time delay to the maximum absolute
cross-correlation value.
The time delay is presented in
dimensionless form as cst/(d 2 -dl), where t represents the time
delay.
In this way the shear wave velocity of the material would
correspond to a value of the abscissa equal to 1, while the
compression wave velocity would correspond to values of 0.577 and
0.408 for Poisson's ratios of 0.25 and 0.4, respectively. The
cross-correlation records are presented for two-dimensional, P-,
SV- and SH-motions as well as three-dimensional, P- and S-motions.
From inspection of the cross-correlation records, the
following conclusions can be made.
1.
The cross-correlation function provides an easy and
accurate method to calculate wave velocities from these synthetic
seismic records. As such, the procedure represents a powerful tool
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105
However, a key
because it has the potential for full automation.
aspect to application of this method is generation and recording of
seismic waves of predominately one type (P or S).
2.
Velocities calculated with the cross-correlation
function are, in general, as accurate as (or more accurate than)
those determined using times of arrival or interval times from time
histories.
Travel times obtained from the maximum of the
cross-correlation function were in almost every case within one or
two percent of the exact value.
Interval travel times obtained in
this manner have the following important characteristics: they are
unique values; they are not subject to interpretation as are the
times of arrival of the waves; and they are not affected or are
only slightly affected by the value of the damping ratio.
3.
The cross-correlation method can be used with
confidence when the receivers are located in the far field.
.
In the
near field, the compression wave velocities obtained from
longitudinal-motion records on materials with high values of
Poisson's ratio require careful interpretation (as in the case in
Fig. C.9 where the peak of the cross-correlation function
corresponds to the value of the shear wave velocity instead of the
value of the compression wave velocity as would be expected from a
longitudinal-motion record).
-
These conclusions apply to the "clean" time histories shown
in Appendix A. For actual seismic records obtained in the field,
the cross-correlation curves may be not be as simple as those
presented here.
In those complicated cases, however, the time
histories will be complex as well and the estimation of the times
of arrival will also present some difficulties.
A practical
application of the cross-correlation technique to actual crosshole
'
"
field tests can be found in Stokoe et al (1985).
..
106
4.5 WAVE VELOCITIES USING SPECTRAL ANALYSIS TECHNIQUES
4.5.1 Theoretical background
In recent years great improvements have been achieved in
electronic equipment to monitor the response of dynamic systems.
The development of microprocessors and computational algorithms has
significantly increased the capability of the instrumentation for
recording, measuring and analyzing systems in both time and
frequency domains.
Recording and analysis of signals can be easily
e
and rapidly done in the field with modern spectral analyzers.
Frequency domain analyses have several advantages.
First,
trends that might be difficult to identify in the time domain can
be easily studied in the frequency domain.
Second, operations with
signals in the time domain (such as integration or differentiation)
can be greatly simplified if done in the frequency domain.
frequency domain techniques allow full automation of data
reduction.
Third,
Finally, averaging of signals, which requires extreme
care in the time domain, is easily done in the frequency domain.
The spectrum of a signal is a representation of that signal
in terms of its frequency components.
The tool used to transform a
signal from its time history to its frequency components is the
Fourier transform.
Forward and inverse Fourier transform formulas
are given in Section 2.2, along with a brief introduction to the
Fast Fourier Transform (FFT) algorithm to compute Fourier
transforms.
The FFT forms the basis for an additional method to
calculate wave velocities (compared with those presented in the
previous sections) based on the cross spectral density function (or
cross spectrum).
%
%a
I-0 e-%.
107
The cross spectrum, CS(f), of two functions, u1 (t) and u2 (t),
is defined as:
heeCS(f) = U1(f) • _Mf)
(4.2)
where,
4
U1 (f) is the Fourier transform of ul(t),
U2 (f) is the Fourier transform of u2 (t),
t is the time variable,
f is the frequency in Hz, and
indicates complex conjugate.
The cross spectrum is, therefore, a complex function of frequency.
Assume that the functions ul(t) and u2 (t) are the time
records at two sampling points. For each frequency the phase of
the cross spectrum of ul(t) and u2 (t) will give the phase
difference in radians (or degrees) of the corresponding harmonic.
Since the period (T) of that harmonic is known (T = 1/f), a travel
time between target points can be obtained for each frequency, f,
by:
t(f) = [T • s(f)]/2r =
(f)/2lf
(4.3)
€
where,
t(f) is travel time for a given frequency, and
O(f) is the phase in radians of the cross spectrum at each
frequency.
The distance between *ne two receivers is a known parameter.
Therefore the apparent wave velocity at a given frequency is simply
calculated by:
der.%
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*. .. . -.
e
.
el
•
.... • ,
-
108
V(f)
(4.4)
d/t(f)
where,
d = d2 - d1 , is the distance between target points, and
d, and d2 are the distances from the sour:e to the first and
second receiver respectively.
The term "apparent velocity" is used since it does not
represent the phase velocity of a pure plane wave because of the
interactions between near- and far-field components.
Using these formulas a curve of apparent wave velocity versus
frequency can be calculated.
Such a curve is called a velocity
dispersion curve (or simply dispersion curve) in this study.
Dispersion curves are also defined by other authors as relations of
velocities with wavelengths, wave numbers or wave periods.
Once
one form of the dispersion curve is known, the others can be easily
obtained since frequency, wavelength, wave number and period are
.N'
related by:
(4.5)
%.*JW
= 2rf
(4.6)
%
V(f) = Lf
(4.7)
wT = kL
27
where,
is the frequency of the harmonic in rad/sec,
'
T Is the period in sec,
k is the wave number, and
*
L is the wavelength of the harmonic wave.
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109
Variables that affect apparent wave velocity are: frequency
(w or f), distance from the source to the first receiver (dl), and
distance from the source to the second receiver (d2 ), in addition
to the elastic variables. The number of variables can be reduced
if they are expressed in dimensionless form.
In this study the
distance from the source to the first receiver and the frequency
are considered in dimensionless form as dl/X, where X is the
wavelength of the shear wave (X = cs/f). The distance between the
d
two receivers is considered as d2 /dI . The apparent velocity is
presented in non-dimensional form as V/cp for compressional
(longitudinal) motions and as V/cs for shear (transverse) motions.
The other variables studied are Poisson's ratio and damping ratio.
Any impulse can be considered as the excitation force.
However, since the only information that is needed to calculate the
dispersion curves (for this theoretical study) is the phase of the
cross spectrum, this relative phase can be calculated from the
phases of the Green's functions at the two receivers. This
calculation is very convenient because it can be done without
performing any of the Fourier transforms. To illustrate this
point, assume that the load applied at the source point is any
arbitrary function p(t) and that the Green's functions (or transfer
functions) at the two receivers (points 1 and 2) are H1 (f) and
H2 (f). At each frequency the response at points 1 and 2 will be
(see Eq. 2.4)
Ul(f)
P(f)
HI(f)
U2 (f)
P(f) • H2 (f)
(4.8)
- •
and
(4.9)
%0%
7 "'
110
respectively.
P(f) is the Fourier transform of p(t).
The phase of
the cross spectrum of ul(t) and u2 (t) is then the phase of Ul(f)
minus the phase of U 2 (f), which in turn is the same as the phase of
Hl(f) minus the phase of H2 (f).
4.5.2 Dispersion curves from cross spectrum function
A first set of dispersion curves for two- and threedimensional motions are presented in Figs. 4.21 through 4.30 for a
material with a Poisson's ratio of 0.25 and no material damping.
It is first observed in Figs. 4.21 through 4.30 that the dispersion
curves depend on the relative position of the source and receivers.
For a constant ratio d2 /dl, the apparent velocity fluctuates as
dl/A varies (A is the wavelength of the shear wave).
This can be
interpreted as the apparent velocity varying with frequency (or
wavelength) for a given distance dj, or as the apparent velocity
varying with distance d1 , for a given frequency (or wavelength).
The fluctuations are large in the near field (small distances, dl,
or low frequencies, f) and decrease as the distance d, increases
(or the wavelength decreases).
The dispersive effect also decreases as the spacing between
the receivers increases (or as d2 /dI increases).
This can be
observed in Figs. 4.21 and 4.22 for two-dimensional P-motion,
Figs. 4.23 and 4.24 for two-dimensional SV-motion, Figs. 4.27 and
4.28 for three-dimensional P-motion and Figs. 4.29 and 4.30 for
three-dimensional S-motion.
The dispersive effect seems to be more
important in longitudinal motions (P-motions) than in transverse
motions (S-motions).
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The nature of this dispersive behavior in the two-dimensional
P- and SV-motions and in the three-dimensional, P- and S-motions,"0
is due to the coupling between the additional near-field and the
far-field waves.
As was mentioned in Chapter Three, the coupling
effect is more important in the near-field than in the far-field,
and so is the dispersive behavior.
This conclusion is further
reinforced by looking at Figs. 4.25 and 4.26 in which dispersion
curves for two-dimensional SH-motions are presented.
The shear
wave generated in 2-D, SH-motion is not coupled with any other type
of wave; it is a "pure shear wave."
It can be observed in these
last two figures that the apparent shear wave velocity coincides
exactly with the shear wave velocity of the medium except for very
small values of dl/A.
The reason why the apparent velocity at low
values of dj/X (low frequencies) is not equal to the shear wave
velocity of the medium is not entirely known.
It should be
noticed, however, that the dynamic solution for the Green's
-
functions does not app'v for the static case and that this solution
is probably not accurate for small frequencies.
A second set of dispersion curves is shown in Appendix D
(Figs. D.1 through D.1O).
This group of dispersion curves
corresponds to waves propagating in a medium with 5 percent
material damping.
It is observed that the amplitudes of the
fluctuations corresponding to P-motions have decreased slightly
when compared to the respective curves for a medium with no
material damping.
This behavior is encountered in two-dimensional
as well as three-dimensional motions.
For the two-dimensional
SV-motion and the three-dimensional S-motion, the fluctuations for
the material with 5 percent damping are slightly larger than for.
the material with no damping.
However these differences a e so
small that the corresponding dispersion curves can be considered
almost identical.
I
Ila*
I
122
The two-dimensional antiplane shear motion presents no
dispersion whether the waves propagate in a medium with no damping
(Figs. 4.25 and 4.26) or in a medium with material damping
(Pigs. D.5 and D.6). It becomes clear from Figs. D.5 and D.6 that
strain and frequency independent material damping of the hysteretic
type does not produce any dispersive effect on body waves
propagating in a linearly elastic full space.
The effect of Poisson's ratio on dispersion curves is
investigated in Figs. D.11 through D.20. In these figures are
shown the dispersion curves of waves propagating in a medium with a
Poisson's ratio of 0.4 and zero material damping. From comparison
of these figures (Figs. D.11 through D.20) with the figures
corresponding to waves propagating in a material with no damping
and Poisson's ratio value of 0.25 (Figs. 4.21 through 4.30), the
following conclusions can be drawn. First, for longitudinal motion
(two- or three-dimensional) the dispersive effect is stronger for
the waves propagating in the medium with higher Poisson's ratio.
Second, for two-dimensional, SV-motion and three-dimensional,
S-motion, the dispersive effect is more important for the waves
propagating in the medium with Poisson's ratio of 0.4 than for
those propagating in the medium with Poisson's ratio of 0.25. The
difference is almost insignificant at high frequencies (or large
values of dj/X), but at low frequencies, where the influence of the
P-wave (near-field effect) is more important, the dispersive effect
is greater when Poisson's ratio is 0.4. Third, Poisson's ratio
does not affect the two-dimensional SH-motion dispersion curves.
This result is expected since Poisson's ratio does not affect the
two-dimensional SH-motion fundamental solution (Eq. 2.5). Finally
it should be noticed that in some of the P-motion dispersion curves
the apparent velocities fell out of the limits of the graphs and
were not included in the plots. These high velocities were
.
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123
obtained when the phase of the cross spectrum approached a value of
zero.
In some cases it was found that the phase of the cross
spectrum became negative for a small range of frequencies and
resulted in negative apparent velocities.
Two conclusions can be made at this point.
1. If elastic
properties are to be back-calculated from dispersion curves it is
better to space the receivers so as to obtain a large value of
d2 /d1 . Of the configurations studied so far, that corresponding to
a value of d2 /dI = 4 gives dispersion curves with apparent
velocities closer to the velocities cs and cp from which the
elastic properties can be calculated.
2. The agreement between
apparent and phase velocities (cs and cp) is best for high values
of dl/A.
It is important to notice that in the previous figures the
horizontal axis, distance (or frequency), has been normalized with
respect to the wavelength of the shear wave, since it was a known
parameter for these theoretical studies.
However for practical
applications, it would probably be more convenient to normalize
with respect to the wavelength of the propagating wave, L
V/f.
In the field the parameters that are known are the distances (d1
and d2 ), the apparent velocities (V) and the frequency (f).
It is
interesting then to know at what value of dl/L apparent velocities
and the elastic velocities of plane waves (cs and cp) become equal
so as to filter out any frequencies smaller than that corresponding
to the critical value of dl/L.
Dispersion curves that would be obtained if this new
normalization were done would be, for the case of transverse
motion, very similar to those previously presented (since in this
case V and vs are quite similar).
V...
*.
However, curves obtained for
,
V
124
longitudinal motions would be shifted to the left by a factor of
about two.
(This factor depends on the value of Poisson's ratio
and is about 1.7 for v = 0.25, about 2 for v = 0.33 and about 2.4
For v = 0.4.)
To give an indication of how the new curves look,
dispersion curves obtained for three-dimensional motion in a medium
with a value of Poisson's ratio of 0.25 and no material damping
have been plotted in Figs. 4.31 and 4.32 for longitudinal motion
and Figs. 4.33 and 4.34 for shear motion.
As a practical application it can be concluded that for the
measurement of body wave velocities in which the ratio of distances
from the source to the second and to the first receivers (d2/dl) is
larger or equal to 2, the apparent compression (or shear) wave
velocity can be assumed equal to the compression (or shear) wave
velocity of the elastic medium at values of dj/L greater than 2. A
value of dl/L of one could be used without incurring any
significant errors (errors greater than about five percent).
However, when considering longitudinal-motion, dispersion curves
-
produced by a point load in a medium with a high value of Poison's
ratio (Fig. 0.18) present large fluctuations and a higher value of
dl/L should be considered (of at least two).
A similar problem
occurred when analyzing cross-correlation functions for these
material properties (Section 4.4.2 and Fig. C.9).
The ratio of dl/L > 1 means that since wavelength can be
expressed as
L
V/f
(4.10)
then
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129
dl/L = dl*f/V > 1
(4.11)
Since velocity can be expressed as
V = (d2 - dl).3600 f/
(4.12)
(with 0 in degrees), then
d1 .o/3600 (d2 - d1 ) > 1
(4.13)
o/3600 > ( 2 - d)/dI
(4.14)
or
For the case of d2/d= 2, The rule means that
(4.15)
* > 3600
or the number of cycles has to be greater than one before the phase
should be used to calculate apparent wave velocities. For cases
where d2 /d1 is greater than two, the limiting value of dl/L would
be smaller than one.
4.5.3 Dispersion curves from transfer functions
It is not always necessary to know the displacements
(velocities or accelerations) at two different points in order to
obtain a phase difference from which velocities can be calculated.
If one knows the transfer function of the response at one point due
..
,. , , , , ,
, .. ., . ., . . , . ..
.
.. ..... , . ..
~.
.. .. .S .. .
.. . . . .,
130
to an input force at other point, it is possible to calculate the
phase difference between the source and receiver from the phase of
the transfer function.
If displacements are being used to monitor
the response (as is the case in this theoretical study), then force
and displacements are in phase at the source and one could use
directly the phase of the transfer function of displacement due to
a force to obtain the phase difference in between source and
receiver.
(For a material with damping there will be a small phase
difference in between force and displacement).
In the field it is
common practice to use instrumented hammers that by means of an
accelerometer (or a load cell)
record the input to the system.
attached to the head of the hammer
In these cases it would be
convenient then to use accelerometers to monitor the response at
the target point so as to avoid any extraneous phase shifts
produced by using different types of sensors.
If velocity
transducers are used at the target point and an accelerometer is
attached to the hammer then a correction of 90 degrees would have
to be made to account for the use of a transfer function of a
W.
velocity output due to an acceleration input.
Dispersion curves obtained from the phase of the transfer
function of displacement due to a unit force (Green's function) are
presented in Figs. 4.35 through 4.39 and in Appendix 0, Figs. D.21
through D.30.
Dispersion curves based on two-dimensional wave
motions for a medium with no damping and Poisson's ratio of 0.25
are presented in Figs. 4.31. 4.32 and 4.33.
When compared with the
dispersion curves obtained from the phase difference at two
re
ivers (Figs. 4.21 through 4.26).
it can be observed tnat tre
S
apparent velocities in Figs. 4.31. 4.32 and 4.33 do not give as
good an estimate of the elastic velocities. c. or c s. as tme
dispersion curves in Figs. 4.21 through 4.26 did (even at hgn
values of dl/,).
%-
The apparent velocities always inde-est'mate *me
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131
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2-D IME NS.
POI. R.
P-MOT ION
0 . 25
D -0O
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Fig. 4.35
.
.
.
..
,
-1"."
,,.--
6".,',",*","*1-" "
, -. ' "."-" ,' '-"
Dis persion curve obtained from the phase of the
'"l
transfer function for two-dimensional
-"
longitudinal motion in a medium wit~h no
damping and Poisson's ratio =0,25."'
132
CI4I
114
00
0
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0.
dpingRan Pis0.25
0.5
Dai
133
oM
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2-D IME NS.
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Fig. 4.37
-
0.25
SH-MOT ION
D =- .
curve obtained from the phase of the
,i~.4
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___:_____00_______20_________0________0
transfer function for two-dimensional
,*
-Dispersion
.*|~~!
antiplane
damping
4
~ 80*-7 ~~~
shear motion in a medium with no
and Poisson's ratio
0.25.
134
Go
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0
3-DIMENS.
POI. R. - 0.25
*I
0 0
I
P-MOTION
D - OX
I.'.
4.0
2.0
6.0
8.0
d1
Fig. 4.38 - Dispersion curve obtained from the phase of the
transfer function for Three-dimensional
longitudinal motion in a medium with no
0.25.
damping and Poisson's ratio
,Z.
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136
elastic velocities by a few percent (about five percent), but at
low values of dl/A (less than two), the velocities are
significantly underestimated.
Dispersion curves based on three-dimensional wave motions
(Figs. 4.38 and 4.39), however, give better estimates. At values
of dl/A > 2, the velocities obtained are accurate predictions of
the elastic velocities. At short values of dj/x, say one, the
transfer function gives apparent velocities that are closer to the
elastic velocities than the velocities obtained from using the
$
cross spectrum information at two receivers, Figs. 4.27 through
4.30, (especially for low values of d2/dl).
,/.
In Appendix D are included the dispersion curves derived from
the transfer function approach for materials with five percent
damping and 0.25 for Poisson's ratio and zero damping and a
Poisson's ratio of 0.4, Figs. D.21 through D.25 and Figs. D.26
through D.30, respectively. The first set of curves is almost
identical to the ones with no damping (Figs. 4.35 through 4.39) and
the same comments apply. The curves corresponding to a value of
0.4 for Poisson's ratio (Figs. D.26 through 0.30) did not present
any special advantages when compared to the curves obtained from
the cross spectrum information. The poorest comparisons between
apoarent wave velocities and elastic velocities occurred for threedimensional P-motion in mediums with high values of Poisson's
ratio. In this case it seems better to use the phase of the cross
spectrum at two receivers with a high d2 /dI ratio than to use the
phase of the transfer function at one receiver.
".
'S
,
• , , , , ,. , -,'
- \or
......- , , -
. -
-
.. - -..-.
,
,...-.........-.......•.-..-..,...
137
4.6 SUMMARY
.JN
Several methods to calculate body wave velocities from
seismic records have been reviewed in this chapter.
The methods
include those based on the visual estimation of the time of arrival
at one receiver (direct times), those based on the visual
estimation of interval times between two receivers, those based on
the cross-correlation function, and those based on dispersion
curves obtained from the phase of the cross spectrum or transfer
function.
Visual determinations of various times of arrival of waves
are the most common methods to calculate wave velocities.
Velocities obtained from direct travel times are slightly higher
than those corresponding to the elastic constants (cs and cp).
Many times the differences depend on the individual reducing the
field data and on the amplification applied to the
time signals.
These small errors can partially be reduced if interval times from
first arrivals are used, but the choice of an arrival time is still
a personal matter.
This subjective evaluation of interval times
can be avoided if interval times are calculated from first peaks,
first troughs or zero crossings.
The use of interval times also
eliminates possible triggering errors.
The use of the cross-correlation function to calculate wave
velocities seems to be a very valuable method.
Velocities obtained
by this method were accurate representations of the elastic wave
velocities for most cases analyzed.
The only erroneous results
were those corresponding to P-wave velocities obtained from
longitudinal-motion records at receivers in the near-field
(d1/X < 2, with dl being the distance from the source to the first
receiver and x the predominant shear wavelength of the time record)
,,.. . ..
. .. .o, ,
9.
138
of a medium with a high value of Poisson's ratio (greater than
about 0.4, Fig. C.9) . A very important advantage of the use of
the cross-correlation function is that calculations of wave
velocities can be easily automated with the use of a computer.
However, for the successful use of this technique " , the field, one
-.
still has to generate and record only one predominate wave type.
The use of dispersion curves of apparent wave velocity also
turned out to be an excellent tool for evaluation of elastic wave
velocities. Apparent velocities at high values of dl/L (values in
general greater than 2) were equal to elastic wave velocities of
the material (with L being the wavelength corresponding to the
apparent wave velocity). When dispersion curves were calculated
from the cross spectrum of the signals at two receivers, it is
better to have a source-receiver configuration with high values of
d2 /dI (for a fixed value of d, and with d2 being the distance from
the source to the second receiver). For a typical crosshole setup
(d2 /dI = 2), the frequencies should be filtered so as to eliminate
U
any frequencies that give values of dl/L < 1. For cases involving
materials with high values of Poisson's ratio (say 0.4) and when
measuring P-wave velocities from longitudinal-motion records, the
-
filtering criteria should be more strict, and values of dl/L > 2
should be used.
Wave velocities can also be evaluated from the transfer
function as discussed in Section 4.5.3. However, dispersion curves
obtained from the phase of the transfer function do not present any
particular benefit when compared to dispersion curves obtained from
the phase information of the cross spectrum for two-dimensional
cases (line sources). For point sources (three-dimensional
motions), apparent wave velocities obtained from the transfer
'-jI
-
I
I,_
function were good estimates of elastic wave velocities when values
*.. -
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" ,-."%'-"-,
139
of dj/L were greater than one, for all the cases but for threedimensional longitudinal motion in a medium with a high value of
Poisson's ratio. The principal advantage of using the transfer
,1
function is that only one receiver is needed.
iII
-5,
%.
.5
-%
%*
T
• --.
CHAPTER
EVALUATION
FIVE
OF
U.
BODY
WAVE
"
ATTENUATION
5.1 INTRODUCTION
Most uses of seismic methods in engineering applications have
been to determine wave velocities.
Very little effort has been
applied to the study of techniques to obtain damping values.
main emphasis of this report is on wave velocities.
The
However, a
preliminary analytical investigation on damping has been carried
out.
A brief theoretical background of wave amplitude attenuation
along with the description of a technique to evaluate material
damping is presented in this chapter.
5.2 THEORETICAL BACKGROUND
The amplitudes of seismic waves decrease as the waves
propagate through a medium.
This attenuation of wave amplitude is
caused by two mechanisms: 1) spreading of wave energy from a
source, generally called geometrical or radiational damping, and 2)
dissipation of energy due to mainly frictional losses in the
material itself, commonly known as attenuation, material or
internal damping.
There are several measures of energy losses in a dissipative
medium.
For a plane harmonic wave, assuming that all energy losses
141
-Z
--. u
-
142
are due to frictional forces (case of a plane wave front), the
decay in wave amplitude can be defined as
=
e-a(d2- d )
(5.1)
where,
u I is the amplitude of the displacement at a distance d, from
the source,
u2 is the amplitude of the displacement at a distance d2 from
the source,
a is the attenuation coefficient, and
e is the base of natural logarithms.
Eq. 5.1 can be derived from the solution to the plane wave
equation by assuming that there is a phase shift between stresses
and strains (White, 1983; Toksdz and Johnston, 1981; Gordon and
Davis, 1968). The attenuation coefficient is not necessarily the
same for different types of waves. Shear waves generally present
different attenuation characteristics than longitudinal waves.
The natural logarithm of the ratio of the amplitudes at two
successive points spaced apart by a distance equal to one
wavelength of the harmonic wave (L) is defined as the logarithmic
decrement, 6, and is used as another measure of frictional losses.
Logarithmic decrement can be expressed as
6 = ln[ul(ddl)/u 2 (d=dl+L)]
(5.2)
The relationship between the logarithmic decrement and the
coefficient of attenuation is easily found to be
eV
0'%
°.
%
° , °.
% •-0"",...................................................................................."...
'- .%• '-
.% •
143
6
oL
(5.3)
Another parameter that measures energy dissipation is the
"damping capacity" factor (or specific damping capacity).
Damping
capacity, T, is defined as the energy dissipated in one stress
cycle divided by the elastic strain energy stored when the strain
is a maximum and is expressed as
: AW/W
(5.4)
where,
AW is the energy dissipated per stress cycle, and
W is the elastic strain energy stored when the strain is a
maximum.
Equation 5.4 is based on the assumption of a hysteretic type
of behavior.
The damping capacity has been found to be frequency
independent but strain dependent.
At very low strains, however,
(like those associated with most wave propagation phenomena in
engineering applications, strains below 10-6) the damping capacity
appears to be independent of strain amplitude.
A similar measure of internal friction is obtained with the
material damping coefficient (or hysteretic damping ratio), D,
which is defined as
D
T/47
AW/(4rW)
(5.5)
In a material with small damping, the hysteretic damping
ratio can be defined as half of the ratio of the imaginary to the
real part of a complex elastic modulus, as mentioned in Section 2.2
.-.
-
:*.
%.
%
144
If the complex shear modau's.
dnd used through out this study.
'or
instance, is defined as
G* : G + iG
(5.6)
then the damping ratio in shear is
D
(5.7
GI/2G
where G is the real shear modulus and G I is the imaginary Dart of
the complex shear modulus.
5.3 MATERIAL DAMPING CALCULATED FROM SEISMIC RECORDS
Special interest and effort nave been devoted
the past to
determination of attenuation paramters from $ie'd meas. romtmts
The field procedure itself presents severa' prob ems. suc" as
precise calibration of the receivers 's necessary, 2, "-te-e-e-ce
of reflected and refracted waves adverse'y afects t'e
measurements; 3) electrical or mecnanca' -o'se car aacerseo
affect the results; 4) perfect coup,'g oetweer
the receivers is necessary
-e-tat'o
5) the tee-dme-s'a
the receivers must be accurate'y :ortrc'e.
as
between near- and far-fie'd compoments
Several procedures tc
-e Doeec es a-c
ta-"
parameters have been croocse c
al, 1982) and an estems',e
ns by 'ovsoz ar- C.cr.-'been 1compiled
.amc-q
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calculate damping ratio, based on spectral analysis tecrnniques 's,.
suggested in the following paragraphs.
For a medium with a linear stress-strain relation, wave
amplitude, u, is proportional
W1/
2
Hence, Eq.
to the square root of the energy,
5.5 can be expressed as
0 : u/(27vu)
where .u
(5.8)
is the amplitude decay per cycle.
Therefore, damping
ratio can be expressed as
D = -[(L/2ru)(du/dr)]
(5.9)
where r is the distance in the direction of propagation.
9
Upon
integrating Eq. 5.9 one finds that
..".
n (u/uO
: "2rD/L (r-rO )
)
(5.10 )
".0
where uo is the amplitude at a reference point ro .
The relation between damping ratio, 0, and the attenuation
coefficient, a, can be obtained directly from Eq. 5.10 as
.9
9%
= 2 D/L
(5.11)
%
The phase velocity of a plane wave, v, is related to the
vave~ength. L, by: v
L/2r.
define from Eq. 5.10 as
.'
d 4
/Vo
•
,
. •
'
_A% -.
:,*
.
•
.
Therefore damping ratio can be
..
'
.,, U
,..
9
- .
%*.
•
.9
.
%
5
9
*
'*
.
*
o
i
p
"
,.5
,.nere.
's
Eq. 5.12
frequency in raa/sec.
'he damping
at'o ootaned Dy
s constant with frequency
A convenient way to ca'.culate phase
apparent velocities. V. 's descriea
apparent velocities
"
eoc-t'es. ,.
Section 4.5.
nstead of phase 4elocity
'n
Ea
4
"om
:f one
.ises
5 :12 osu'ts
n
5
LL, -ol
-[('uu),~
*53
where D(-.) represents an apparent damping ratio.
In Eq. 5.13, dissipation of energy occurs only from "nterna'
friction in the material and no dissipation of energy arises fromV
geometrical spreading (plane waves have been considered)
if the
seismic waves originate from a point source (as usual'y haPDOenS.
correction must be applied to Eq. 5 13 to account for energy
due to geometrical spreading.
a
osses
It is shown in Chapter Three that
wave amplitude decay due to radiational damping is proportional to
the square root of distance for two-dimensional motion (cylimdrica
fronts) and is proportional to the distance for three-dime,*siona.
motion (spherical fronts).
;
The relations are valid for the far
field, but they can be considered good approximations for a v'de
range of frequencies (or distances).
ON
If one assumes that the
relations are valid for any frequency range, the exoression for
apparent damping ratio becomes
%I
.,5 °
40,
"S.,
"S"
147
\I)(w).ln [(u/u 0 )(r/r0)1 /2 ]
0(w
0)
(5.14)
-w(r-r
for two-dimensional motion, and
D~w)
0(w)
-
V(w)1ln [(u/u 0 )(r/r0)]
w(r-r 0 )
(.5
(.5
for three-dimensional motion.
The expressions derived so far apply to steady-state harmonic
motion. Any other motion has to be decomposed into its different
harmonics by applying a Fourier transform. Then, by using the
notation introduced in Section 4.5.2 and Eqs. 4.3 and 4.4, the
equation for the apparent damping ratio becomes
U2 (f)
In[
0(f)
0)
(f)T,(5.16)
1
-
.%V
d2
(
-7-.
for twa-dimensional motion, and
U2 (f)
In[ U
0()
d2
dy
(5.17)
(f)
for three-dimensional motion.
In the development of Eqs. 5.16 and 5.17. it has been
assumed that the response of a medium to an impulse has been
%%-
%'~
%~
N"
148
decomposed into its harmonics by means of a Fourier transform.
these equations
in
D(f) is apparent damping ratio,
f is frequency in Hz,
*(f) is phase of cross spectrum in radians (or difference
between the linear spectrum phases),
U1 (f) is amplitude of the linear spectrum at the first
receiver,
U2 (f) is amplitude of the linear spectrum at the second
receiver, and
d, and d2 are distances from the source to the first and
second receivers, respectively.
Equations 5.16 and 5.17 can be written in terms of the auto
spectrum, A, instead of the linear spectrum as
A2 (f)
ln[A 2
D(f)
1 (
-
*( f
d2 ]
2
1)(5
18)
and
A2 (f)
In{[ A
0(f) :
d2
(5
)9
*(f)
for two- and three-dimensiona, motion%, respecte
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149
5.4 CURVES OF APPARENT DAMPING RATIO
Apparent damping ratios for two and three-dimensional motions
In the first case. the
have been calculated for two materials.
apparent damping ratio is determined for a medium with no material
In the second case, a medium
damping and Poisson's ratio of 0.25.
with 5 percent material damping and Poisson's ratio of 0.25 is
considered.
Figures 5.1 through 5.10 show the curves of apcarent damping
ratio versus dl/, for a medium with D
=
0% (. is wavelength of the
These curves were obtained using the same spacings
shear wavF).
between the receivers as those used to study the velocity
dispersion curves.
and d2/0.
The behavior (fluctuations with d1-
of these curves is similar to that found for the velocity
dispersion curves.
For the two-dimensional SN-motion (FPgs
5
and 5.6), the apparent damping ratio coincides precisely with the
material damping ratio.
low di/,
(The disagreement at low freQuencies or
is similar to that obtained for the
ooocity jispers', r
curves and the reasons for this Dehavior are not tota 'Y Known
For the cases with coupled motions 1two-dimensiona' P- and TVapparent jamp ng
lamo'ng -ato c' Ie
motions and three-dimensional P- and S-motionsu.
ratio presents fluctuations around the actue
These fluctuations are 'arge for
material
ow values of d2/d 1 . out ,- the faf
for
a'e )f
and/or high freQuencies .he
m,"at
rapidly approaches the actua
interesting
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For the cases where the material damping ratio was 5 percent
rather than zero (Figs. 5.11 through 5.20), the tendency is the
same as when D = 0 percent.
For typical field measurements where d2 /dI has a value of the
order of 2 (such as in crosshole measurements), apparent damping
ratio can be assumed to equal the actual material damping ratio at
a value of dl/x = 2. The apparent damping ratio could be used,
therefore, to calculate material damping in the field.
Although only one unique value of damping has been used for
both shear and longitudinal motions, different values could have
been used as well.
In a field experiment, the value of damping
obtained from longitudinal motion records would be that associated
with the constrained modulus while the value obtained from
transverse motion records would be that associated with the shear
modulus.
5.5 SumMRY
A brief theoretical background of wave amplitude attenuation
is presented in this chapter.
-
A method to calculate damping ratios
from seismic records is also presented
.
The method is based on
spectral analysis techniques and requires calculation of the auto
spectrum at two different receivers.
For configurations such that
d2 /dj is equal to two (typical of crosshole seismic testing), the
method seems to give values of apparent damping ratio that are good
approximations of the actual damping ratio for values of dl/L
greater than one.
As in the case of apparent wave velocities, for
a fixed value of d1 , the best values of damping ratio are obtained
for higher ratios of d2 /d1 . Although good from a theoretical point
A
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171
of view, vaues of d2 /d, >> 4 will probably present complications
,n tne field since there will be very little energy arriving at the
second receiver. In this respect it is recommended that some
experimental studies be carried out to determine the best sourcereceiver configuration.
CHAPTER
SUMMARY,
SIX
CONCLUSIONS
AND
RECOMMENDATIONS
6.1 IIRODUCTION
Seismic methods are used in engineering to determine the
elastic properties and damping characteristics of materials at low
strains (strains less than o.ool percent for most geotechnical
materials).
Seismic methods employ both body and surface waves.
Body waves are compression and shear waves. These waves are
typically used to sample within the interior of solid media (such
as geotechnical sites).
In this study, body waves are investigated
analytically to improve the understanding and uses of them.
The
use of seismic methods in engineering applications presents many
advantages.
For instance, seismic methods: 1. reduce sampling
disturbance; 2. model properly in situ stress conditions; 3. test
S
large or small representative zones of interest; and 4. are based
on sound theoretical principles.
In most engineering applications of seismic methods using
body waves, the receivers typically have to be placed very close to
the points of excitation.
This configuration creates several
complications that do not occur in most geophysical applications.
In addition, the use of information provided by the seismic records
has been routinely limited to the first arrival of the waves.
Therefore, an analytical investigation was undertaken to better
understand and use seismic methods in engineering. Results,
173
,1
,.
,
174
conclusions and recommendations from this study are summarized in
the following paragraphs.
-
6.2 BODY WAVE CHARACTERISTICS
An analytical formulation to generate synthetic wave forms of
body waves created by point or line sources and propagating in a
full space was explained. The fact that body waves spread from
point source in a spherical fashion and from line sources in a
cylindrical manner can be an important factor in using these
seismic waves in engineering applications. At long distances from
the sources, spherical wave fronts can be considered plane, and the
classical theory of plane wave propagation is valid. This case is
the most common for geophysical applications where distances from
the source to the closest receivers are of the order of hundreds of
meters. Distances that are involved in most engineering
applications, however, are a few meters. The characteristics of
body waves at these short distances was studied, and an
understanding of the limitations of using the theory of plane wave
propagation was developed.
The behavior of spherical or cylindrical body waves at close
distances to the source was found to be in some cases quite
different from plane wave behavior. An analysis of the
characteristics of generated wave records from point or line
sources was performed. From these waveforms, it was found that, in
wave propagation records of longitudinal motion, there is a first
wave (denoted by Lp) propagating at the compressional wave velocity
and a second wave (denoted by LS) that exhibits longitudinal motion
but propagates at the shear wave velocity. The amplitude of the
second wave (LS) decreases at a much faster rate than that of the
'kS
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.
175
primary wave (Lp).
Even though the contribution of the LS wave at
distances near the source is quite important, at distances far from
the source the LS wave is almost insignificant compared with the
primary wave (Lp).
For these reasons, the primary wave in a
longitudinal-motion record is called the far-field wave (because it
lasts even in the far field), and it is the wave typically referred
to in the literature as the P-wave.
The LS wave is called the
near-field wave, the near-field effect, the additional near-field
wave or the additional near-field effect because it only "exists"
in the near field.
Similar conclusions were found for transverse motion records.
In this case, however, the near-field wave (denoted as Tp)
propagates at the compressional wave velocity, and the far-field
wave (denoted as TS) propagates at the shear wave velocity.
All of these observations apply to three-dimensional (point
sources) and two-dimensional in-plane motions (line sources).
For
two-dimensional antiplane motion (sometimes called SH-motion), only
one wave propagating at the shear wave velocity appears, and there
are no near-field terms.
For time domain records, near-field and far-field distances
have been defined as those distances from the source, d, such that
the value d/A is smaller or greater than two, respectively, for
transverse motions (with A being the shear wavelength).
Near- and
far-field field distances are defined as those distances such that
d/A is smaller or greater than about ten for longitudinal motions
(A is again the shear wavelength).
Some of the polarity reversals in the wave forms encountered
in the field when the direction of the source is reversed were
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176
found to be explained by the behavior of the near-field waves. In
addition, receiver cross-sensitivity and imperfect alignment
contribute to the lack of polarity reversal in the P-wave train.
6.3 BODY WAVE VELOCITIES
I.
Techniques to evaluate body wave velocities were studied in
Chapter Four. The effects that frequency and strain independent
material damping of the hysteretic type have on wave records and
resulting wave velocities were also investigated. Waves
propagating in this type of material arrive at an earlier time than
that predicted by the compressional and shear speeds, cp =
[(X + 2G)/p] and cs = (G/p)i, respectively. To obtain a good
estimate of the elastic properties, it is convenient to pick
arrival times of the wave from the source to the receiver a little
after the actual arrival of the wave. It is concluded that the
best results are obtained when interval times from first peaks or
first troughs are used. The results are always better when the
receivers are located in the far field.
The cross-correlation technique to determine interval times
is found to be an appropriate method to calculate wave velocities.
This technique has the advantage with respect to the other timedomain methods that it can be easily automated. The results were
found to be accurate and reliable for all cases analyzed, except
for the case when compressional wave velocities were determined
from longitudinal-motion records in media with high values of
Poisson's ratio, values on the order of 0.4 or greater. (This case
of high Poisson's ratios also presented problems for some of the
other techniques.)
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177
Spectral analysis techniques to calculate wave velocities
from the phase of the cross spectrum of two time records, or from
the phase of the transfer function of one record, are effective
tools to calculate elastic moduli.
It is shown that, when two
receivers are used and the cross spectrum phase is employed, the
best values of elastic wave velocities are obtained when:
1.
for a fixed dj, the ratio of the distances from the
source to the second and first receivers, d2 /dl, is of
%.1
the order of four or higher.
2.
For a fixed value of d2 /dl, the best comparisons are
obtained at high values of dl/x.
For a typical crosshole setup, where d2 /dI is equal to two,
apparent velocities are equal to the plane wave phase velocities at
values of dj/L greater than one. Both procedures, using the cross
spectrum and using the transfer function present the advantage that
they can be easily automated.
The only significant advantage that
the use of the transfer function seems to have with respect to the
use of the cross spectrum is of an economical nature since the
number of receivers needed is only one.
6.4 WAVE ATTENUATION AND MATERIAL DAMPING
Wave records at several distances from the source and plots
of wave amplitude attenuation with distance are presented in
Chapter Three.
It was found that, for point sources, the amplitude
decay due to only geometrical damping is inversely proportional to
the distance from the source at distances that are far from the
source (d/A > 2). In the very near-field range (d/A < 0.5), the
decay is also inversely proportional to the distance. Only in a
small range of distances (0.5 < d/A < 2 for transverse motions and
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178
0.5 < d/A < 10 for longitudinal motions) is this relation not
valid. For line sources, the amplitude reduction due to
geometrical damping is in proportion to the square root of the
distance only in the far-field. In the very near-field
(d/A < 0.5), the amplitudes decay at a smaller rate. When material
damping is included, wave amplitudes decay at much faster rates
than those corresponding to only geometrical damping for distances
greater than L (with L being the wavelength of the propagating
wave). At distances smaller than L, the decay is practically equal
to that corresponding to geometrical damping.
A technique to estimate damping ratios from seismic records
that is based on spectral analysis techniques is suggested in
Chapter Five. Apparent damping ratios calculated by this method,
fluctuated around the correct value of material damping ratio, but
at large values of dI/L, the fluctuations were minimal. The
analytical results indicate that this method can give a good
prediction of damping ratio if the two receivers are spaced at a
.
ratio d2 /d1 > 2 and the apparent damping ratios used to calculate
the actual damping are taken for frequencies such that dl/L is
greater than one. Like the other spectral analysis techniques to
calculate velocities, the method to compute damping can be easily
automated.
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APPENDIX
RELATIONSHIP
DAMPED
B
BETWEEN
BODY
WAVE
MODU
ELASTIC
VELOCITIES
AND
AND
LI
Material damping ratio is presented in Chapter Five as a
convenient way to account for energy losses due to frictional
forces.
The definition of a frequency and strain independent
damping ratio is given in Eq. 5.5.
A convenient way to express
material damping ratio, for small values of damping, is by the use
of complex moduli.
The damping ratio is then defined as one half
of the ratio of the imaginary part of the complex modulus by its
real part.
For a material in which energy losses in shear are
different than in compression-extension types of motion, it is
preferable to define two damping ratios, one for shear and another
for compression-extension.
Therefore, the damping ratio in shear,
Ds, is defined as
DS = GI/2G
(B.1)
and the damping ratio in compression, Dp, as
Dp
M 1 /2M
(B.2)
where
G and G, are the real and imaginary parts of the shear
modulus, and
20
209
!i
'p.
210
M and M I are the real and imaginary parts of the constrained
modulus.
In this way the complex shear modulus, G*, and complex constrained
modulus, M*, are given by
G=
G + iGI = G(1 + 2iD
S)
(B.3)
and
M
=
(B.4)
M + iMI = M(I + 2iDp)
If one uses shear and compressional wave velocities as the
parameters to define the stiffness of the material, then the energy
loses associated with material damping are more easily accounted
for by using complex wave velocities.
The counterparts to Eqs. 8.3
and B.4, in terms of complex shear and compression wave velocities,
(cs* and Cp,
respectively) are
cs
cs + icsl = cs(l + iDS)
(B.5)
C
Cp + icpl
cp(l + iDp)
(B.6)
and
where
cs and csl are the real and imaginary parts of the shear wave
velocity, and
C
and cpl are the real and imaginary parts of the
compressional wave velocity, respectively.
*
..-
V ..
C.
%.
C..•.-
-.
-
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.%
.
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.
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-
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go211""
Equations B.5 and B.6 are good approximations of Eqs. B.3 and
BA for small values of damping (as usually happens for the small
strain amplitudes encountered in most seismic engineering
applications).
_
In other words one can define the complex shear
modulus as
G*
pCs* 2
pcs 2 (1 + 2iD
s -
DS2 )
pCs 2 (1 + 2iDs) =
G(I +
2 iOs)
(B.7)
.
which is a good approximation of Eq. B.3 for small values of the
damping ratio, D S .
A parallel reasoning can be used for
constrained modulus and compressional wave velocities.
%
,
It should be noted that these complex velocities do not
represent propagation velocities but are approximate mathematical
tools to account for energy dissipation.
In this respect, consider
a plane harmonic wave of the form
exp[iw(t - x/c)
(B.8)
This wave propagates parallel to the x-axis of a non-dissipative
medium with a propagation velocity represented by the variable c
(the phase velocity).
If the phase velocity is considered complex
to account for energy losses, then the wave will be of the form
exp[iw(t
-
x/c*)]
.°1
,a
212
exp{iw[t-
x/(c + icl)]}
exp[-wclx/(c 2 + c12) ] • exp{iw[t
-
cx/(c 2 + c,2 )]}
(B.9)
Equation B.A represents a plane wave that propagates along
the x-axis with a velocity, cD, the damped wave propagation
velocity, which has a value of
cD = (c2 + ci2 )/c
(B.10)
and whose amplitude attenuates with distance with a coefficient of
attenuation, a of
a = Wcl/(c 2 + c12 )
(B.11)
By expressing Eqs. B.1O and B.11 in terms of the damping
.
of propagation is given by
ratio, the damped velocity
,
J*
.
,.
CD
c.(1 + D2 )
(B.12)
and the coefficient of attenuation is given by
*,°*
a= w-.D/[c(1 + D2 )]
W.D/c = 21D/L
(B.13)
where L is the wavelength, (L
'-
2vc/w).
It can be observed from Eq. B.12 that a damped wave
propagates at a velocity which is practically the same as that of
% ,
'
U
U
*****
-UoU.
.
....
.
.°%U
V V-1
F_77
I.-M.-
VLIWL
213
the undamped wave.
For instance, if D = 5 % the damped wave
velocity is only 0.25 percent higher than the undamped elastic
velocity, and if D = 20 % (very high value) the damped velocity is
still only 4 percent higher than the undamped elastic velocity.
• .,
%~%
-0
W~ U..
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1 Jx~ 7~ ~
APPENDIX
M
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RECORDS
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APPENDIX
BODY
WAVE
D
DISPERSION
CURVES
2337.
233
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