Incipient Heterogeneity of First Year Arctic Ice
by
James C. Parinella
B.S.M.E., Case Western Reserve University, (1987)
Submitted to the Department of Ocean Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Ocean Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 1995
@ Massachusetts Institute of Technology 1995. All rights reserved.
i
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Author .......
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Department of Ocean Engineering
July 20, 1995
Certified by.................
........
........ *........
Robert Fricke
Assistant Professor
hsesis Supervisor
"J.
Accepted by......
.....
.. - .•..-b
..
.
. ..........
g a Carmichael
Chairman, Departmental Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAY 2 6 1998
LIBRARIES
Incipient Heterogeneity of First Year Arctic Ice
by
James C. Parinella
Submitted to the Department of Ocean Engineering
on July 20, 1995, in partial fulfillment of the
requirements for the degree of
Master of Science in Ocean Engineering
Abstract
The propagation of low-frequency seismo-acoustic waves in the Arctic ice pack is
examined through the use of geophone data collected during a 1992 experiment on
uniform first-year ice in Allen Bay within the Canadian archipelago.
Extensive analysis of vertical particle velocities of the flexural wave revealed that the
inhomogeneous and anisotropic nature of the ice plate can not be neglected. Geophones as close as six meters to each other displayed drastically different responses to
hammer blows. The relative responses were also highly dependent on the direction of
propagation. It was found that there was a preferred direction of propagation. Longitudinal and flexural wavespeeds were higher in that direction, the spectral densities of
the flexural wave were higher, and the properties were reversed for the perpendicular
direction. The coherence of geophones was also calculated. The coherence between a
pair of geophones was higher when the incoming wave was propagating perpendicular
to the axis between the two of them than when it was propagating parallel to that
axis. For adjacent geophones, coherence was high below approximately 35 Hz. This
corresponds to a quarter wavelength correlation length of 3 m.
Additionally, this thesis provides a technique to remove reflections from the shallow
sea bottom. The bottom has a negligible effect on the characteristics of the waves
in the ice plate, but it contributes a significant amount of energy that would otherwise cause the spectral density to be overestimated. The demonstrated technique is
repeatable and removes the effect of bottom bounces.
Thesis Supervisor: J. Robert Fricke
Title: Assistant Professor
Acknowledgments
I would first like to thank my advisor, Professor Rob Fricke, for his help and patience
through my time here. At times the tasks here seemed monumental, but his experience
at MIT on both ends of the advisor/advisee relationship helped make life at the
firehose more manageable.
I would like to thank Professor Henrik Schmidt, who provided financial and technical
support near the end.
I hereby acknowledge my debt to Peter Stein, who brought me into the world of
acoustics several years ago, and has been a great source of information about the
wonderful world of sound.
I'd like to thank the unwitting inspirations, the Bruce Miller's and Kevin Lepage's,
whose theses I used as references-for content, for style, for quality. It makes it go
so much more smoothly when you have something to aim for.
To my officemates in the basement, Brian, Rama, Qing, Hua, Henry, J.T., and everyone else, thank you for all your help with school, MATLAB, little tricks to get around
the computer systems, and everything else. I enjoyed the worldly feel of our office.
Without people like you we are all doomed to a gloomier existence. I leave you my
frisbee as a reminder that not everything is related to the ocean.
I owe my sanity (or perhaps my insanity) to my friends on the frisbee scene. I've
learned so much about life and responsibility and accomplishment through the pursuit
of trying to be the best. Let's add a few more titles before we call it quits. And I can't
help but acknowledge my fellow members of the Tea Party, my constant companions
over the past five years. We've argued quantum mechanics, genetics, the role of H.L.
Mencken in society, frisbee philosophy, you name it. My life without you guys would
be, well, let's just say different.
To my parents, if you are still reading this, thank you for all your help and love over
the years. Just try to remember that I have taken the road less traveled, and that
has made all the difference.
And to my dearest Imelda, words can not express what you have meant to me and
what you have done for me over these last four years. I am far better for having
known you and loved you.
Contents
1 Introduction
1.1
Motivation . . . . . . . . . . . . . .
1.2 Thesis Objectives . . . . . . . . . . .
1.3 Thesis Content . . . . . . . . . . . .
2 Theory
2.1 Introduction . . . . . . . . . . . . . .
2.2
Thin elastic plate theory . . . . . . .
2.3
Hammer blow (vertical impulse) . . .
3 Experiment
3.1
Background ..............
3.2 Array Layout .............
3.3
Data . . . . . . . .. .. . . . . . . .
3.4
Geophone Data ..............................
39
4 Processing
44
4.1
Introduction ..
4.2
Source Spectrum . . . . . . . . . . . . .
4.3
Spectral Analysis . . . . . . . . . . . . .
4.4
Bottom bounces................
.. ... .... .. ... .... .... .
4.4.1
Analysis ..............
4.4.2
Removal of bottom bounces . . .
44
5 Results
5.1
5.2
Demonstration of Inhomogeneity
. . . .
5.1.1
Center geophones, broadside . . .
5.1.2
Center geophones, endfire
5.1.3
Center geophones, ambient noise
5.1.4
Discussion . . . . . . . . . . . . .
. . . . . .
Demonstration of Anisotropy
5.2.1
Total power ............
5.2.2
Spectral Density
.
.
. . ..
.
. .... .. ..... ..
90
5.3
5.2.3
Dispersion . . . . . . . . . . . . .
95
5.2.4
Longitudinal and SH Wave Speed
115
Coherence .................
116
6 Conclusions
6.1
6.2
Summary
138
.........
. . . . . . . . . . . . . . . . . . . . . . 138
. . . . . . .
6.1.1
Bottom reflections
6.1.2
Inhomogeneity and Anisotropy
6.1.3
Coherence ............
Recommendations for future work . . .
139
139
. . . . . . . . .
140
140
List of Figures
2-1 Air-ice-water system with up- and down-going waves . . . . . . . . .
2-2 Symmetric and antisymmetric mode for ice plate . . . . . . . . . . .
2-3 Wavenumber diagram. Eigenmodes in wavenumber domain for low
frequencies for lossless materials . . . . . . . . . . . . . . . . . . . .
2-4 System and mass-dashpot model for hammer blow impact . . . . . .
2-5 Velocity of the hammer blow during impact and the resulting force
applied to the plate . ..........................
2-6 Energy spectrum of hammer blow for model . . . . . . . . . . . . . .
3-1 Geophone layout in array. ........................
3-2 Response of all axes of all geophones to HB 5-1 . . . . . . . . . . . .
3-3
Response of all axes of all geophones to HB 7-3 . . . . . . . . . . . .
3-4 Response of all axes of all geophones to HB 10-1 . . . . . . . . . . .
3-5 Response of all axes of all geophones to HB11e2 . . . . . . . . . . . .
3-6 Response of z axis geophones to HB 11-3 . . . . . . . . . . . . . . . .
3-7 Source time series for HB 11-3 .....................
3-8
Hodograph of initial reaction at G11 to HB 11-3. The response immediately after impact is in the y direction . . . . . . . . . . . . . . . .
3-9
Response of z axis geophones to HB 11-3 .. . . . . . . . . . . . . . ...
4-1 Block diagram of measurement of hammer blows . . . . . . . . . . .
4-2 Comparison of MLM and conventional FFT-based PSD estimate. . .
4-3 Bottom reflections for shallow water . . . . . . . . . . . . . . . . . .
4-4 Inversion for bottom depth and acoustic wave speed, using response at
G 1 to H B 5-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-5 Multiple paths for n=2 bottom bounces. Ice thickness greatly exaggerated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....
4-6 Magnitude of reflection and transmission coefficient for air-ice-waterbottom interfaces ..............................
4-7 Phase angle of reflection and transmission coefficient for air-ice-waterbottom interfaces ..............................
4-8 Magnitude of overall reflection coefficient for two bottom bounce multiple paths. ......
.. .. .........
.............
4-9 Phase angle of overall reflection coefficient for two bottom bounce multiple paths. ................................
4-10 HB 5-1, response at G10. Amplitude of second bottom bounce is
largest, and subsequent reflections are of the same order as the first..
4-11 HB 11-3, response at G1. First bottom bounce has significantly greater
amplitude than others. By fifth bounce, no distortion is noted .....
4-12 "Surgical removal" of bottom bounces from time series of G1 due to
HB 11-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-13 Close up of first bottom bounce from Figure 4-12 . . . . . . . . . . .
4-14 Comparison of vertical velocity for HB 11-2 of original signal and after
bottom bounces have been removed . . . . . . . . . . . . . . . . . . 64
4-15 Sample waveform with simulated bottom bounces . . . . . . . . . . .
4-16 Interference from bottom bounce . . . . . . . . . . . . . . . . . . . .
4-17 Comparison of PSD's of waveforms . . . . . . . . . . . . . . . . . . .
4-18 Effect of bottom bounce on ESD at G1 due to HB 5-1 . . . . . . . .
4-19 Effect of bottom bounce on ESD at G1 due to HB 7-3 . . . . . . . .
4-20 Effect of bottom bounce on ESD at G1 due to HB 10-1 . . . . . . . .
4-21 Effect of bottom bounce on ESD at G1 due to HB 11-3 . . . . . . . .
4-22 ESD at G2 due to hammer blows at G10. This demonstrates the repeatability of the hammer blows as well as the bottom removal technique.
5-1
Geometry of center geophones, in relation to a hammer blow at Gl.
5-2 Response of center geophones to HB 7-6. Note the change in phase
between G1 and G9 between 0.6 and 1.2 s. Note also the response of
G 2 at 0.9 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
Response of center geophones to HB 11-3. Note that the amplitudes
still differ, but all the geophones are in phase . . . . . . . . . . . . .
5-4 Energy spectral densities of center geophones due to hammer blows at
G 7.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-5
Energy spectral densities of center geophones due to hammer blows at
G il ......................................
5-6 Transfer function comparing ESD's of other center geophones to ESD
of G8 for hammer blows at G7. .....................
5-7 Transfer function comparing ESD's of other center geophones to ESD
of G8 for hammer blows at G7 .....................
5-8 Response of center geophones to HB 5-1 (endfire) . . . . . . . . . . .
5-9 Response of center geophones to HB 10-1 (endfire) . . . . . . . . . .
5-10 ESD of center geophones at endfire due to HB 5-1 . . . . . . . . . ..
5-11 ESD of center geophones at endfire due to HB 10-1 . . . . . . . . . .
5-12 Ambient noise at center geophones
. . . . . . . . . . . . . . . . . . .
5-13 Ratio of ambient noise at each center geophone to ambient noise at G8.
In contrast to Figures 5-6 and 5-7, the ratios here are nearly constant
across frequency. .............................
5-14 Normalized velocity squared for each geophone, with lines added for
mean ± standard deviation .......................
5-15 Normalized velocity squared for each hammer blow. Lines added for
mean ± standard deviation .......................
5-16 Coherence of G1 in response to HB 11-2 and HB 11-3 . . . . . . . .
5-17 Average ESD at G1 to hammer blows at all locations . . . . . . . . .
5-18 Average ESD at G2 to hammer blows at all locations . . . . . . . . .
5-19 Average ESD at G3 to hammer blows at all locations . . . . . . . . .
84
5-20 Average ESD at G4 to hammer blows at all locations . . . . . . . . ..
99
5-21 Average ESD at G5 to hammer blows at all locations . . . . . . . . ..
100
5-22 Average ESD at G6 to hammer blows at all locations . . . . . . . . ..
101
5-23 Average ESD at G7 to hammer blows at all locations . . . . . . . . ..
102
5-24 Average ESD at G8 to hammer blows at all locations . . . . . . . . ..
103
5-25 Average ESD at G9 to hammer blows at all locations . . . . . . . . ..
104
5-26 Average ESD at G10 to hammer blows at all locations . . . . . . . ..
105
5-27 Average ESD at Gil to hammer blows at all locations . . . . . . . ..
106
5-28 Average ESD at G12 to hammer blows at all locations . . . . . . . ..
107
5-29 Direction of best propagation for each geophone . . . . . . . . . . . ..
108
5-30 Direction of worst propagation for each geophone . . . . . . . . . . .. 109
5-31 Peaks used in calculating the group velocity for HB 10-1, response at
G i .... .... .... .. .... .... ... ... ... .... ... 111
5-32 Dispersion curve at Gl using second order polynomial curve fit ...
112
5-33 Dispersion curve at G6 using second order polynomial curve fit ...
113
5-34 Dispersion curves of center geophones in response to HB 10-1 . . . . . 114
5-35 Directions of fastest and slowest wave speeds for longitudinal and SH
waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....
.. . 117
5-36 Coherence of center geophones with G1 for HB 11-2 (broadside). G1
is at (0,0) and 350 m from the source ...................
119
5-37 Coherence of center geophones with G1 for HB 10-1 (endfire). G1 is
120
at (0,0) and 376 m from the source ....................
5-38 Coherence with G1 of geophones with large spatial separation in response to HB 11-2. G1 is at (0,0) and 350 m from the source.
. .
>.
5-39 Coherence with G1 of geophones with large spatial separation in response to HB 10-1. G1 is at (0,0) and 376 m from the source . ..
121
122
5-40 Coherence with G4 of geophones with large spatial separation but similar range in response to HB 10-1. G4 is at (155,0) and 531 m from
the source .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5-41 Coherence pair locations. Each pair is plotted so that the midpoint
between the two geophones is at the origin. Arrow indicates G4 paired
up with center geophones. ........................
125
5-42 Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bot129
tom ) for 10-15 Hz. ............................
5-43 Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bottom ) for 15-20 Hz. ............................
130
5-44 Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bot131
tom) for 20-25 Hz. ............................
5-45 Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bottom ) for 30-35 Hz. ............................
132
5-46 Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bot133
tom) for 40-45 Hz. ............................
5-47 Difference in Spatial Mean Square Coherence for HB 10-1 and HB 11-2
for 10-15 Hz. Negative values corresponds to higher coherence for HB
11-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5-48 Difference in Spatial Mean Square Coherence for HB 10-1 and HB 11-2
for 20-25 Hz. Negative values corresponds to higher coherence for HB
11-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5-49 Difference in Spatial Mean Square Coherence for HB 10-1 and HB 11-2
for 30-35 Hz. Negative values corresponds to higher coherence for HB
11-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5-50 Difference in Spatial Mean Square Coherence for HB 10-1 and HB 11-2
for 40-45 Hz. Negative values corresponds to higher coherence for HB
11-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
List of Tables
3.1
Geophysical data from experiment . . . . . . . . . . . . . . . . . . ..
29
3.2
Location of geophones within array . . . . . . . . . . . . . . . . . . ..
32
5.1
Mean and standard deviation of normalized power, sorted by geophone
location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
5.2 Mean and standard deviation of normalized power, sorted by hammer
blow location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
5.3 Best and worst propagation paths . . . . . . . . . . . . . . . . . . ..
95
5.4 Longitudinal and Shear (SH) Wavespeed . . . . . . . . . . . . . . . . 116
5.5 Spatial separation for geophone pairs . . . . . . . . . . . . . . . . . . 126
Chapter 1
Introduction
1.1
Motivation
The Arctic is a vast, largely unexplored area that can help us better understand our
world. Historically, the Cold War provided an impetus to learn more about communication across and underneath the Arctic canopy, since the two great superpowers
could conduct surveillance under the ice and also move around in relative secrecy.
However, changing global conditions have brought about an environmental crisis.
Global warming is a great concern, but has been difficult to prove or even to measure. The Arctic has a great mass of ice that would be especially susceptible to small
changes in global temperature. Study of the elastic properties of the Arctic ice pack
will enable scientists to come up with more accurate models of the state of the Arctic.
The flexural wave in Arctic ice has been the subject of intensive study over the past ten
years. Yang and Yates[22] and Yang and Giellis[21] have recently published papers
experimentally characterizing the nature of the flexural wave. To the best of the
author's knowledge, these were the first published works that obtained attenuation
coefficients for the flexural wave. Originally, this thesis attempted to corroborate
their results. However, the variations in spectral estimates caused by the latent
inhomogeneity of the ice prevented the calculation of attenuation.
1.2
Thesis Objectives
The objective of this work is to display the heterogeneous nature of Arctic pack ice
and attempt to quantify that. To do this, data sets collected in 1992 during a seismoacoustic experiment at an ice camp near Resolute, Canada, are analyzed. An array of
three-axis geophones measured particle velocities due to a series of hammer blows at
various locations within the array. Initially, signal processing in the frequency domain
was intended to produce coefficients for the attenuation of the flexural wave. However,
variations within the ice overwhelmed any of the measurable effects of propagation.
Therefore, the focus turned toward finding out more about the incipient heterogeneity
of the ice. This report documents this nature in several ways, in both the time domain
and the frequency domain.
1.3
Thesis Content
Chapter 2 follows the derivation of fluid loaded thin elastic plate theory to establish
the existence and the properties of the flexural wave. The theory has been well
developed, so this thesis only highlights the major equations. The interested reader
can review more in-depth references[6][5][17] for the full development. Also, in this
chapter, the spectrum for the impact of a hammer onto an ice plate is reviewed.
Chapter 3 details the experiment performed near Resolute, Canada, in 1992. Twelve
three-axis geophones recorded the response to hammer blows on the ice. A PC-based
data acquisition system recorded the particle velocities at the surface of the ice. This
chapter reviews that data and identifies the various waves.
Chapter 4 reviews the signal processing necessary to estimate spectral densities and
attenuation. The most significant contribution in this chapter is the identification of
bottom reflections in the signal and the subsequent manual removal from the time
traces. Although the presence of the bottom does not influence the flexural wave at all,
reflections of the compressional wave from the bottom create a significant interference
with the time signal, leading to incorrect estimates of the spectral density.
Chapter 5 demonstrates the heterogeneous nature of apparently uniform ice. The signals are analyzed in both the time and the frequency domain to show the anisotropic
nature of the ice plate. First, responses of neighboring geophones are compared to
show that each small section of ice is not perfectly coupled with the adjacent patch
of ice. Then, individual geophones responses were examined using sources from all
directions to show that the ice plate did not respond equally in all directions. The
propagation characteristics were best in the WNW-ESE direction, and worst in the
N-S direction. The wavespeeds of the flexural and longitudinal waves were higher in
the preferred direction, and the spectral density was higher in that direction as well.
Finally, the coherence between geophones was examined. The geophone responses
were incoherent above approximately 35 Hz for neighboring geophones. Also, any
given pair of geophones had a markedly higher coherence when the incoming wave
was propagating perpendicular to the axis connecting them than they did for a wave
coming in along the axis. In this section, the energy spectral density, the dispersion
relationship for the flexural wave, the phase speeds of the longitudinal and shear
waves, and the mean square coherence were calculated.
Chapter 6 summarizes the contributions of this thesis and makes recommendations
for future work.
Chapter 2
Theory
2.1
Introduction
In this chapter the plane wave propagation in a floating ice sheet will be derived
using thin elastic plate theory. In a situation involving parallel boundaries such
as an ice plate, plane waves will reflect off the boundaries and constructively and
destructively interfere with each other. The boundaries guide the waves into two
dimensions instead of three. The theory has been well developed, dating back to
Ewing, Crary and Thorne's[6] and Ewing and Crary's work in 1934[5]. In general,
the derivation of Stein[17] will be followed, but with the notation and wavenumber
representation of Schmidt[16].
AIR (VACUUM)
2H
alj31'pl
ICE (ELASTIC)
---
B-+
Al-
A2+
x
BI-
a2, p2
WATER (FLUID)
y
z
A2-
Figure 2-1: Air-ice-water system with up- and down-going waves.
2.2
Thin elastic plate theory
Consider an infinite, isotropic, homogeneous elastic ice plate of thickness 2H floating
on an infinite half-space of water and bounded on top by an infinite half-space of air.
An elastic solid will support compressional (P) waves and shear waves (horizontally
polarized (SH) waves and vertically polarized (SV) waves) propagating in three dimensions, while the fluid half-space will only propagate compressional waves. If air
can be treated as a vacuum, no sound waves will propagate into the air.
The compressional waves have wave speed a,, and are designated by A' for upgoing
waves and Al for downgoing ones. The shear waves travel at speed /1 and are
designated B + and BT for upgoing and downgoing waves, respectively. The acoustic
waves in the water travel at speed a 2 and are labelled A+ and A-. For convenience,
the origin is set in the middle of the ice plate (see Figure 2-1).
The velocity can be expressed in terms of the scalar 4 (compressional) and vector T
(shear) displacement potentials. These are defined by
u=V
+Vx X.
(2.1)
Then the potentials will satisfy the Helmholtz equations,
I 1 = 0,
(2.2)
1
= 0,
(2.3)
k 2 (12
0,
(2.4)
V 2( 1 + k
V 2'I 1 + k2,
V2 ()2 +
=
2O.
(2.5)
0.
For plane waves, propagation is independent of y, so o/cy = 0. The displacements
become
=
d
d'Y
(2.6)
dzx
dxz
dx
(2.7)
u dx
V--"
dz
dz'
d,,
dz
W= T+
The relevant boundary conditions are, for the ice-air interface (z=-H),
* no normal stress (azz = 0),
* no shear stress (azz = 0),
and at the ice-water interface (z = H),
* vertical particle velocity continuous,
* normal stress continuous, and
(2.8)
* no shear stress(azz = 0),
With harmonic time dependence, the potentials can be expressed as
k Xe
i4 1 = (ADe'*1 + Ae-atz)eika
F1 = (B+ef'z + B-e-8'z)eik•iXe
'2= (A+ea2z + Ae-a•2z)e
ika2X e-'
,
(2.9)
'' ,
(2.10)
.
(2.11)
In order to satisfy the radiation condition at z = oo (finite amplitude),
A + = 0.
(2.12)
This leaves five equations (the boundary conditions) and five unknowns (the amplitudes). Setting the determinant of the resulting matrix to zero leads to the characteristic equation. Solutions to this equation include the symmetric and anti-symmetric
eigenmodes, as shown in Figure 2.2.
The first symmetric mode is known as the longitudinal mode, which has a wave speed
slightly lower than the compressional wave. The particle displacement in this mode
is symmetric with respect to the centerline of the ice sheet. Since the wavenumber
is smaller then the water wavenumber (see Figure 2-3), the longitudinal wave will
propagate into the water column. This is known as a leaky mode, since energy leaks
into the fluid.
The first antisymmetric mode is known as the flexural or bending wave at low frequencies. It is a subsonic (compared to water) wave, so it suffers an exponential decay
in the water and will not leak energy into the water column. Therefore, the flexural
wave can propagate through the ice sheet. The bulk of the remainder of this work will
focus on the flexural wave. Higher order symmetric and antisymmetric modes such as
Plate displacement
u
---
---
-
FU
SYMMETRIC MODE
(D symmetric
A- = A+
T anti-symmetric
B- = -B+
Plate displacement
----------------------F----~r---------u
w
w
ANTISYMMETRIC MODE
anti-symmetric
Ssymmetric
A-= -A+
B-= B+
Figure 2-2: Symmetric and antisymmetric mode for ice plate.
the Stoneley wave and Rayleigh wave also exist at higher frequencies. They will not
be treated here since the frequency range of interest is below the cutoff frequencies
for these modes.
If the wavelength is long compared to ice thickness, the plate bending wave equation
can be used. Satisfying the boundary conditions leads to the characteristic equation
for the flexural wave is
4
k -
AW2
B2
- B(
Ctanh(h()
-
g)
= 0,
(2.13)
where
A
-
1 2 pi(1
- v 2)
(2H) 2E
(2.14)
X
-Poles of characteristic equation
Propagating in
water column
\/
\/ I
/\
/\
kat
kL
-
Evanescent in
water column
-
-
-
k
Sx
kf
Figure 2-3: Wavenumber diagram. Eigenmodes in wavenumber domain for low frequencies for lossless materials.
12p2(l -3 v2)
(2H) E
(2.15)
and C is the vertical attenuation coefficient in the water
(=
/k 2 - k 2 .
(2.16)
This equation must be solved numerically. For this Arctic experiment, the effects of
gravity and bottom depth can be ignored, leaving
k4 - Aw
2
- B-
2w
= 0.(
(2.17)
The phase velocity is given by
(2.18)
ccp=p W
and the group velocity is
4k 3 + Bw2 k
c
dw
c9 =• dk
2.3
2w(A +
)
=
=(2.19)
Hammer blow (vertical impulse)
The hammer blow may be considered a vertical impulse on a thin elastic plate. Lyon
[11] has studied impact as a source of vibration. The system can be modeled as a
mass-dashpot as in Figure 2-4. The hammer of mass m strikes the plate, which acts
as a pure dashpot with no stiffness or mass reactance component. The loss in the
dashpot represents the energy propagated away by vibration. The plate has resistance
R=force
R
-
vel
-orce
8prcj;
(2.20)
where
pS = mass / unit area= p(2H),
a = radius of gyration= 2H/x/ii,
cl = longitudinal wave speed.
The velocity and therefore the force will exponentially decay during the impact (see
Figure 2-5). The Fourier transform of the force is then
F(w) =f
Rvoe-Rt/me-'wtdt =
Rvo
S fo
Rim + iw
(2.21)
mL R8
I, R= 8oxc
-------------I
I
s
I
"
K
F
(t
Figure 2-4: System and mass-dashpot model for hammer blow impact.
and the energy spectrum is given by
E(w)=-
27r
IF(w) 2 =
1
(Rvo) 2
27r w2 + (R/m)2
(2.22)
and is shown in Figure 2-6. At low frequencies, the energy is a constant proportional
to the momentum squared of the hammer before impact.
Vo
v (t)
Vo e
-Rt/m
F (t)
Figure 2-5: Velocity of the hammer blow during impact and the resulting force applied
to the plate.
10
CO= Rim
Figure 2-6: Energy spectrum of hammer blow for model.
Chapter 3
Experiment
3.1
Background
In March 1992, a sea ice mechanics experiment was performed on first year ice in
shallow water near Resolute, Canada. The data were taken in Allen Bay (74.67 0 N,
95.17 °) southwest of Cornwallis Island. In addition to acoustics experiments, there
were also meteorological observations and ice and snow cover measurements. Relevant
data are summarized in Table 3.1. For a fuller summary, see Lewis et al[10].
Table 3.1:
Air temperature
Wind speed
Snow cover thickness
Ice core thickness
Ice density
Geophysical data from experiment.
-20 to -30 degrees C
0 to 13 m/s
4 to 32 cm (mean 15 cm, std dev 7 cm)
1.41 to 1.71 m (mean 1.57 m, std dev 0.11 m)
.91 kg/m 3
Lewis et al. [10] reported the ice to be a relatively smooth, undeformed first-year ice
floe, apparently landfast. Occasional small, isolated upthrust blocks were observed
at various locations within the site. The relief of such features was less than 30 cm.
No continuous pressure ridges were observed within the immediate area.
3.2
Array Layout
Twelve three-axis geophones were frozen into the ice to form an array. Later, the
accuracy of each was checked using a 30 Hz generator and established to be accurate
within 1 dB. The geophones measured particle velocity, with a conversion factor of .28
V/cm/s, with a maximum reading of 2.5 V. The 36 channels were high pass filtered
to 200 Hz and digitized directly to an optical disk at 500 Hz using a PC based digital
data acquisition system.
The geophones were arranged in a cross with roughly logarithmic spacing (see Figure
3-1), with a maximum separation of 827 m. Table 3.2 lists the location of all the
geophones. They were carefully aligned to ensure that the orientation of all geophones
was the same. For each one, the y axis was aligned with the N-S leg of the array,
the x axis with the E-W leg, and the z axis pointed down. The z, y, and x axes, in
that order, were recorded for each geophone before moving on to the next. So, for
example, channels 10, 11, and 12 are the z, y, and x axes, respectively, for geophone
4.
At least three types of events were studied:
1. Fractures of the ice floe due to thermal and wind stress,
2. Horizontal and vertical blows with a sledgehammer onto an I-beam frozen into
x G6
E
0
C
as
z
-400
-300
-200
-100
0
100
E-W distance (m)
200
300
Figure 3-1: Geophone layout in array.
400
500
Table 3.2: Location of geophones within array.
Geophone number
E-W location (m)
N-S location (m)
1
2
3
4
5
6
7
8
9
10
11
12
0
5.9
24.8
154.8
450.4
0
0
-6.1
-24.8
-376.3
0
0
0
0
0
0
0
95.8
305.4
0
0
0
-350.2
10.2
the ice, and
3. Vertical blows with the sledgehammer directly onto the ice.
The third type, hereafter referred to as hammer blows, will be studied in this work.
They approximate a vertical impulse.
On the morning of 31 March 1992, at each of the four outlying geophones (5, 7, 10,
and 11), a set of five to seven hammer blows was recorded, with fewer than 10 seconds
in between each one, with 10 to 25 minutes between the start of each set. For each
hammer blow, 4.1 seconds (2100 samples) were later isolated for analysis, as well as
2.0 seconds of ambient data before each set. Additionally, there were ten hammer
blows done in the southwestern quadrant on a 100 meter radius circle around the
origin, but these have not been analyzed.
3.3
Data
Figures 3-2 to 3-5 show one of the hammer blows at each location. Each set of three
traces are the three axes for a particular geophone. In the plots, each of the geophones
is scaled equally, with the exceptions of the geophones next to the hammer blow, which
have been scaled down to fit on the plot. Therefore, the relative amplitudes between
geophones or between the three axes of a single geophone can be compared quite
easily. For comparison's sake, a typical maximum vertical velocity for G1 (channel 1
on each of the time traces) is about .08 cm/s.
For a completely vertical hammer blow onto a smooth, uniform ice plate, there would
only be vertical and radial plate velocity components. However, a real hammer blow
has some horizontal component to it, and a real ice plate has irregularities which
cause coupling into other waves (for example, the horizontally polarized shear (SH)
waves) as well as scattering into all directions.
The flexural wave has particle displacement in both the vertical and transverse directions, but the signal to noise ratio is greater in the vertical direction, so analysis
will focus primarily on only the z axis of each geophone. Figure 3-6 shows only the
z axis velocity in response to the third hammer blow at geophone 11. Twenty-five
such hammer blows were analyzed. For simplicity, hammer blows will be referred to
by the location of the nearest geophone and which hammer blow in the set that it
was, so the example above would be HB 11-3. Also, geophones will be abbreviated
Gx, where x is the geophone number, such as Gl.
3.n
25
-A----
S20
E
c
0
15
I~
10
OfT"-
I
0
0.5
c---~-
iII
I
I
1
1.5
|
I
I
I
I
I
I
2.5
3
3.5
2
Time
·
I
4
Figure 3-2: Response of all axes of all geophones to HB 5-1.
34
..0
E
,--i
C:
cc
O0
iI
0.5
1
I
I
I
I
I
I
1.5
2
2.5
3
3.5
4
Time
Figure 3-3: Response of all axes of all geophones to HB 7-3.
:35
30
25
. 20
E
.15
15
10
P-
0
0
I
I
I
0.5
1
1.5
II
2
Time
2.5
3
3.5
4
Figure 3-4: Response of all axes of all geophones to HB 10-1.
I.
CD
E
Z
.i
JC
0
0.5
1
1.5
2
Time
2.5
3
3.5
4
Figure 3-5: Response of all axes of all geophones to HB11e2.
-----.IIAAA ,III........
...
...........
10
I
.....
0.5
I
I
1
1.5
I
2
Time
I
iI
¢I
II
2.5
3
3.5
4
Figure 3-6: Response of z axis geophones to HB 11-3.
3.4
Geophone Data
Figure 3-7 shows the response to HB 11-3 of G11, which was about 2 m from the
source. The peaks of the vertical velocity trace are clipped at a maximum of 2.5 V
(8.9 cm/s), so it is unknown what the true maximum amplitude or waveform was.
There is also a significant response on the other axes. For this hammer blow, the
y axis has a large negative response at first, much larger than for the x axis, so it
appears that the hammer struck the ice just north of the geophone. Figure 3-8 shows
a hodograph comparing the x and y velocities at each instant of time. The initial
response is shown with the solid line and can be seen to have an immediate response
in the y direction, and shortly thereafter moves in both directions. The same patch of
ice within a few inches was struck for each of the hammer blows at a given location.
All three geophone axes have some ringing, quick oscillations that continue for up to
one second after impact.
Figure 3-9 shows the response of G1 to HB 11-3, at a range of 350 m. The hammer
blow occurred at t = 0.112 s. The first arrival is the longitudinal wave on the radial
axis at about t = 0.22 s. The SH wave arrival can be seen on the transverse axis at
about t = 0.32 s. The dispersive flexural wave appears on both the vertical and radial
axes beginning around t = 0.4 s and continuing for the remainder of the trace. The
higher signal-to-noise ratio for the vertical axis makes this one much more favorable
for study.
For most of the time traces, the x and y velocities translate almost directly into
radial and transverse velocities. In the example above, the y axis was aligned with
the direction of propagation, so it was the radial component, and the x axis was
aligned perpendicular to that, so it corresponds to the transverse component. For
other experiments or for ambient noise traces, this will not be the case. However, in
this experiment, all of the hammer blows being studied were struck on one of the axes
33
32.5
.Q
E
c
32
C
31.5
31
30.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time
Figure 3-7: Source time series for HB 11-3.
1
0
0)
-2.5
-2
-1.5
-0.5
0
-1
X axis velocity (cm/s)
0.5
1
Figure 3-8: Hodograph of initial reaction at Gil to HB 11-3. The response immediately after impact is in the y direction.
CL
0
0.2
0.4
0.6
0.8
1
Time
1.2
1.4
1.6
1.8
Figure 3-9: Response of z axis geophones to HB 11-3.
42
2
of the array far away from the center, so except for the furthest outlying geophones
on the opposite leg of the array, the propagation path was almost directly N-S or
E-W, matching the orientation of the geophones.
Also noticeable are the echoes from the bottom. Sometimes as many as six bottom
bounces create a significant distortion in the vertical time series.
These bottom
bounces maintain the pulse shape of the source. Their effect on the spectral analysis
will be treated in the next chapter.
Chapter 4
Processing
4.1
Introduction
Many things affect the spectral analysis of the geophone data. The hammer strikes
the ice in a non-linear collision. The ice plate acts as a dashpot and dissipates
the hammer blow in the form of vibrational energy. The (mostly) parallel boundaries
reflect the waves, creating traveling modes in the ice. Each heterogeneity scatters and
diffracts the waves. Energy is radiated into the surrounding air and water. When the
wave finally passes through its destination, it leaves only a history of its motion, and
analysis of this motion is needed to reveal its spectral composition. These actions
and more all go into the final result, as Figure 4-1 shows.
Each block in the diagram has its own transfer function relating input and output.
Although some such as geometric spreading are well known, others are difficult or
--------- I
I
MEDIUM
ABSORPTION
GEOMETRIC
SPREADING
SCATTERIN[G FROM
INHOMOGE:NE1TIES
WAVEGUIDE
MODES
I
I...
OUTPUT
Figure 4-1: Block diagram of measurement of hammer blows.
impossible to predict and can only be examined on a statistical level. In Figure 4-1,
the blocks within the dashed line can be collectively considered as the ice response.
This response is highly dependent on the local characteristics of each patch of ice,
the direction of propagation, and even shifting ambient conditions. As modelers and
experimenters, scientists and engineers try to eliminate as many of these transfer
functions as possible to reduce the task to a simple yet accurate representation. This
report attempts to capture the transfer function of the ice response by showing that
the other transfer functions in Figure 4-1 can be neglected.
4.2
Source Spectrum
In Chapter 2, the hammer blow was modeled as a vertical impulse on a thin elastic
plate. During the experiment, data were taken only for a point roughly 2 m from the
impact, and there is a significant velocity component on all three geophone axes. It
is difficult to estimate how much of this is related to the horizontal component of the
impact, how much is due to the coupling into other waves, and how much is just the
energy being carried away in the propagating waves.
Unfortunately, as mentioned in Chapter 3, the vertical axis was saturated by the
impact, and the next closest geophone was at least 200 m away, so no source spectra
are available. Using the mass-dashpot model from Chapter 2, the source level is
approximately constant up to the rolloff frequency fT
fr
-~
-
262Hz,
which is above the Nyquist rate for this experiment, for
(4.1)
m = 10 kg,
3
p = .91kg/m ,
ci = 3180 m/s, and
2H = 1.57 m.
4.3
Spectral Analysis
Since the source is nearly a uniform spectrum, and since the ice transfer function
should not have any rapid changes, the output should be a slowly varying broadband
spectrum. In Kay and Marple's paper on spectrum analysis[9], the maximum likelihood method (MLM) of Capon[2] appears to provide the best representation of a
broadband spectrum.
The MLM was originally developed for seismic array frequency-wave number analysis.
In the MLM, a set of narrow-band filters estimates the power at each frequency
independently of the other frequencies. The filters at each frequency differ, in general,
whereas in a conventional Fast Fourier Transform (FFT)-based method, they are the
same. This provides for better wavenumber (or frequency) resolution. Poor frequency
resolution is a common complaint with traditional methods.
The MLM uses finite impulse response (FIR) filters with p weights
A = [ao a ... ap-1]T.
(4.2)
The weights for each frequency fo are chosen so that the variance of the output process
is minimized, that is,
in = AHRxA,
(4.3)
where R,, is the covariance matrix of the input, and so that an input sinusoid at
frequency fo is passed without distortion, that is,
EHA = 1,
(4.4)
E = [1 exp(i2rfoAt)...exp(i27r(p- 1)foAt)]T,
(4.5)
where E is the vector
and H denotes the complex conjugate transpose.
This gives an optimal solution for the filter weights
R-1E
R-JE
AoPt = EHRp1 E'
(4.6)
1
EHR- E"
(4.7)
and the variance is then
2
rmin
Then, the Power Spectral Density (PSD) for the MLM is given as
Sf)
At
(4.8)
where At is the time increment between samples.
Figure 4-2 compares the MLM to a conventional FFT-based PSD for the response at
geophone 1 to hammer blow 11-2. As noted, the MLM spectral estimate is smooth,
whereas the conventional PSD has rapid fluctuations.
j ^-2
MLM vs Conventional PSD, HB 11-2, response at 1
U/)
0
C.
a,-
Frequency
Figure 4-2: Comparison of MLM and conventional FFT-based PSD estimate.
4.4
4.4.1
Bottom bounces
Analysis
In the analysis of the flexural wave in Chapter 2, it was shown that in practice there
was no waveguide effect of the water being bounded from below. For the flexural
mode, the water appears infinitely deep. However, the bottom reflection will provide
another path for sound waves to propagate.
Figure 4-3 shows the first three bottom reflections. The hammer strikes the ice, a
compressional wave is transmitted to the water, it strikes the bottom at angle T, and
it reflects back to the surface. Because the minimum grazing angle in this experiment
was Omin = 200, refraction due to sound speed gradients is negligible. Therefore, all
paths are straight lines. For a uniform bottom, the nth bottom bounce will arrive at
t
2n
• ( r)2 + d2,
c
2n
(4.9)
where
c = average acoustic wave speed,
r = range to receiver, and
d := bottom depth.
The average wave speed c = 1432 m/s and bottom depth d = 153 m were determined
by inverting the first three bottom bounces received at geophone 1 from HB 5-1,
as shown in Figure 4-4 However, the typical Arctic profile has a steep sound speed
gradient near the surface, and the acoustic wave speed just below the surface is
important for analyzing the characteristic modes, so the usefulness of an average
r.
___
2H
___
"'~~'
Figure 4-3: Bottom reflections for shallow water.
sound speed in water without any information on the gradient is limited.
Figure 4-5 shows some of the possible paths for two bottom bounces. Examination
reveals two paths with slightly different lengths. Path 1 is the most direct. Path 2
is almost the same, but the wave travels through the water-ice interface before being
reflected off the ice-air interface, resulting in an extra path length of roughly twice the
ice thickness. Each interface will have a reflection or transmission coefficient, which
in general has both magnitude and phase, and for a fluid-fluid interface is given by
m sin 0 - (n2 - cos 2 0) 1/2
S msin 0 + (n2 - cos 2 0)1/2
(4.10)
and
2m sin 0
Tij
m sin 0T+ (n 2 - -cos 2 0)1/2=(4.11)
where
m=
Pi
,
(4.12)
IOU
-7I
159
158
...
...
...
..
i....
...
..........
i................. i....
...
.........
157 -. 156...
. ...
. ...i.................i......
156 .
E
15
-
.155
. . . . . . . . .. . . . . . ...
15
15.........
15
.. .. . . . .
.. . . .. . . .
.
..........................
............
.. ... .
-.
.................. ............
-
First bottom bounce
-
15
1425
--......
-
.
I
1430
I
-
Third bottom bounce
I
I
1445
1435
1440
Acoustic wave speed (m/s)
I
1450
1455
Figure 4-4: Inversion for bottom depth and acoustic wave speed, using response at
GI to HB 5-1.
52
'ER
R 34
4- BOTTOM
Figure 4-5: Multiple paths for n=2 bottom bounces. Ice thickness greatly exaggerated.
n --
ca
.
(4.13)
This analysis is intended merely an order of magnitude calculation, not a thorough
analysis of bottom characteristics. For an elastic solid, there will also be transmission
and reflection into shear waves, which would tend to reduce the coefficient for the
compressional wave, which is what is being calculated here. However, the general
conclusions here will still be valid.
Figures 4-6 and 4-7 show the magnitude and phase, respectively, of the reflection
and transmission coefficients as functions of grazing angle for typical values of Arctic
conditions. Additionally, there will be some downward directivity associated with the
hammer blow. Fiet[7] suggests that the hammer blow has the directivity of a dipole,
which varies as sin 0. So Path 1 would have an overall reflection coefficient of
R 1 = T23 R 34R 32R 34 T32 sin 9,
(4.14)
and Path 2 would have
R2
T23 R 34 T 32 R21T23 R 34 T32 sin 0.
(4.15)
Figures 4-8 and 4-9 show the overall reflection magnitude and phase, respectively,
for specular reflection for two bottom bounces. Over the range of interest for this
experiment (9 = 360 - 700), the two reflections are of the same order.
Additionally, there will be an infinite number of paths resulting from forward scattering from near-specular and near-specular reflections, such as Path 3 in Figure 4-5.
This will cause dispersion of the reflection arrival. However, Urick[19] indicates that
for all but very rough bottoms, non-specular scattering is negligible compared to specular and near-specular. In general, though, multiple bottom bounces in the measured
data tended to be more spread out in time than did single bounces.
The shape of Figure 4-8 also indicates that for certain geophone separations, the
magnitude of the second bottom bounce could be larger than for the first, which
initially seems counterintuitive. To illustrate the point, consider the largest separation
of 827 m between G5 and G10. For n = 1, 9 = 200, and for n = 2, 0 = 360. From
Figure 4-8, the overall reflection coefficient for the direct path for n=2 is about 0.5.
For n = 1,
Rn== T23 R 34 T32 sin a1 = 0.07.
(4.16)
C)
Grazing angle (degrees)
Figure 4-6: Magnitude of reflection and transmission coefficient for air-ice-waterbottom interfaces.
mevs• a eeee, eeZ,ee e
/
a~~.
I1 •...
e
l
l
I
I
I
I
I
I
.eea
e easo. so0
/"
/"
/
. .
-
A
-0
-
I
I
I
10
20
30
HR21
R32
-. R34
.....
T32
x T23
S--
/::
I
I
I
I
I
70
80
E
40
50
60
Grazing angle (degrees)
90
Figure 4-7: Phase angle of reflection and transmission coefficient for air-ice-waterbottom inte rfaces.
·
·
·
·
-
0.9
-
0.8
-
0.7
-
0.6
-
O.S
0.5
-
O..4
-
0.3
-
0.2
-
0.1
-
0
10
20
30
Grazing
40
angle
sO
(degrees)
0
60
70
P-ath
-
1
Path 2
80
90
Figure 4-8: Magnitude of overall reflection coefficient for two bottom bounce multiple
paths.
For smaller separations, the effect disappears. For example, for HB 11-3, the separation between the hammer blow and geophone 1 is 351 m. For n = 1, 0 = 410, and for
n = 2, 0 = 600. Again from Figure 4-8,
(4.17)
Rn=2 = 0.07,
and
Rn= = T2334T
.19.
(4.18)
These calculations are confirmed by examining the time series for HB 5-1 and HB
I
-~
1
8
I,
Figure 4-9: Phase angle of overall reflection coefficient for two bottom bounce multiple
paths.
11-3, reproduced for the given geophones in Figures 4-10 and 4-11. For HB 5-1, the
second arrival is the largest amplitude, and the third, fourth, and fifth are all about
as large as the first. In HB 11-3, though, the first arrival dwarfs the others, and by
the time of the fifth reflection, no distortion is observed.
4.4.2
Removal of bottom bounces
The acoustic wave speed and bottom depth can be used to calculate the approximate
arrival times for all bottom bounces. Depending on the range, anywhere from two
to five bottom bottom bounces can create a significant distortion to the signal. Frequently one or more of these reflections would arrive in the middle of the flexural
wave. Even with the multiple paths for each reflections, the pulse largely keeps its
Figure 4-10: HB 5-1, response at G10. Amplitude of second bottom bounce is largest,
and subsequent reflections are of the same order as the first.
shape upon arrival, and as noted earlier in this chapter, would have a broadband
spectrum. Without knowing the source spectrum and bottom characteristics, it is
impossible to bypass filter the data to remove the bottom bounces.
Instead, the bottom bounces were "surgically removed" by hand. First, MATLAB
routine predicted the arrival times for the first five bottoms bounces. Then, if there
were a significant disturbance in the signal, the signal was corrected so that it matched
the surrounding points. Usually, the bottom bounce would interfere with the signal
for much less than one period, so the spectrum in that window would be extremely
narrowband. In effect, the signal was time-gated, and narrow-bandpass filtered manually around the instantaneous frequency. An automated routine is possible, but would
require a priori knowledge of the dispersion characteristics and acoustic parameters.
At the higher frequencies (over 60Hz), the interference would typically last for one or
two of the periods, but the amplitude of the signal was small and relatively constant
from peak to peak. (Incidentally, the word "period" should be loosely interpreted to
mean the time between consecutive peaks or troughs, since the frequency content of
the dispersive flexural wave is constantly changing.) Two of the highest SNR hammer
blows at each location were chosen and the bottom bounce surgically removed from
each at the z axis time series. Typically, only two or three removals were necessary at
the shorter ranges and four or five at the further ranges. Figures 4-12 and 4-13 show
the original and corrected signal at geophone G1 due to HB 11-2. Figure 4-14 shows
the z axis for all twelve geophones, with the uncorrected signal below the corrected
one for each geophone. In Figures 4-15 and 4-16, a 5000 point sample wave form was
created to demonstrate the effect of removing the bottom bounce. Then, three points
were changed to simulate the reflection from the bottom. In Figure 4-17, it is evident
that the bottom bounce contributes to the whole frequency spectrum.
The effect on the spectral estimate often proved to be significant. Figures 4-18 to
4-21 show the changes in the MLM estimate of the PSD for G1 for one of the hammer
blows at each location. In some cases, such as HB 5-3 (Figure 4-18), the first bottom
bounce arrives before the flexural wave, so it is the second and third bottom bounces
which affect the PSD. In other cases such as HB 11-3 (Figure 4-21), only the first
reflection has a large amplitude, so just removing the first bounce will give the correct
spectrum.
Little error is introduced by this manual act of correction, especially below 50 Hz.
Figure 4-22 compares the spectral estimates of two hammer blows at G10. Comparing the spectra of the original signals shows the repeatability of the hammer blows,
with little deviation below 50 Hz. The changes above that frequency may have resulted from statistical differences in scattering because of the subtle shift in hammer
blow location. Importantly, the spectrum shift caused by removal of the bounces is
repeatable, again with little difference in the spectra below 50 Hz.
Figure 4-11: HB 11-3, response at G1. First bottom bounce has significantly greater
amplitude than others. By fifth bounce, no distortion is noted.
61
Time
(s)
Figure 4-12: "Surgical removal" of bottom bounces from time series of G1 due to HB
11-2.
0.015
-
-
Original signp I
sig
Corrected
al
-|
0.01
-i
0.005
-
0
-t
-!
-0.005
-0.01
00150.3
0.32
0.34
0.36
0.4
Time (s)
0.38
0.42
0.44
0.46
0.48
Figure 4-13: Close up of first bottom bounce from Figure 4-12.
63
0.5
12
10
4I -I-8
E
S
=1
C
C1
Z:
CL
0
6a
•
•
A/•/•
......
/•
•
••.•
•
-
.......
4
..................
2
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
4
Figure 4-14: Comparison of vertical velocity for HB 11-2 of original signal and after
bottom bounces have been removed.
64
Figure 4-15: Sample waveform with simulated bottom bounces.
Figure 4-16: Interference from bottom bounce.
__
-0
bounce=
3
VWith bottor
-20
. . . . . . . . . .i . . . . . . . . . . :. . . . . . . . . .
. . . . . . . . . .: . . . . . . . . . . .i.
. . . . . . . . . . :. . . . . . . . . . .•
. . . . . . . . . . .!.
.. . . . . . . ..;. . . . . . . . .
-40
-80
. . ..
.. . . . . .
... .. ... ...
. . . . . . . . . . :. . . . . . . . . . .
. .- . . . . .
. . . .. .
.
-100
.. . . . :. . . . . . . . . .
. . . . . . ...
.. . . . . . . . . ..
: .. .. .. ... .. ... .. .
. .....
-120
i
-140
0
0.1
. ... . ... ..
0.2
0.3
0.4
0.5
Frequency
0.6
0.7
0.8
Figure 4-17: Comparison of PSD's of waveforms.
66
0.9
00
0
0o
C
0.
w
100
101
Frequency (Hz)
102
Figure 4-18: Effect of bottom bounce on ESD at G1 due to HB 5-1.
-65
-70
-75
A
-
-80
S-85
a1)
i
LU
I.,
-95
-Uncorrected
-
signal
First bounce removed
....- First two bounces removed
-100
.First
.4nf
100
three bounces removed
101
102
101
102
Frequency (Hz)
Figure 4-19: Effect of bottom bounce on ESD at G1 due to HB 7-3.
68
--U
ID
U)
Cu
'a
"0
U)
LI=
aJ
100
10'
Frequency (Hz)
10z
Figure 4-20: Effect of bottom bounce on ESD at G1 due to HB 10-1.
-70
-75
IV
-
-80
E
Go
S-85
CL.-1
_
-.
C
E. -90
a)
-95
-
.-100
- S-
Uncorrected signal
First bounce removed
- First two bounces removed
...... First three bounces removed
IU0
0
100
I
I
I
10
101
Frequency (Hz)
Figure 4-21: Effect of bottom bounce on ESD at G1 due to HB 11-3.
70
t-
C1
0)
V-
a)
,2
Frequency (Hz)
Figure 4-22: ESD at G2 due to hammer blows at G10. This demonstrates the repeatability of the hammer blows as well as the bottom removal technique.
Chapter 5
Results
The original intent of this thesis was to calculate the attenuation of the flexural waves
as a function of frequency. This was to have been done by estimating the spectra
for each hammer blow and comparing the amplitude as a function of range from the
source. Unfortunately, preliminary analysis unveiled strange things. What began as
an attempt to measure the attenuation of flexural waves soon turned into a quest
for reasons why the accurate measurement of attenuation was not possible. The ice
plate was remarkably uniform and the data appeared reasonable, indeed even good clean, repeatable, few extraneous phenomena. However, drastically different spectral
levels were being observed for nearby geophones, discrepancies that should not have
been as large as they were.
5.1
Demonstration of Inhomogeneity
Lewis et al[10] also experimentally characterized the physical properties of the ice
during this experiment.
They were able to to quantify the vertical variations in
salinity, temperature, density, and air and brine volumes of the ice, by examining
ice cores. Additionally, they measured ice and snow thickness and reported on the
variations, which are summarized in Table 3.1. From this information they developed
a thermal stress model to predict stresses and fracture count caused by ambient
conditions. However, no previously published work has attempted to capture the
spectral variations in flexural waves.
The inhomogeneity of the ice was first observed and can most easily be seen by
examining the five geophones in the center of the array (in order from west to east,
9, 8, 1, 2, and 3). The inner three (8, 9, and 2) are less than a wavelength apart for
all frequencies of interest for the flexural wave.
5.1.1
Center geophones, broadside
For a wave originating from either the north or south, the five center geophones can
be considered an array viewed broadside. Figure 5-1 shows the range and bearing for
each of the center geophones for a hammer blow from the south at G11. The ranges
are within a meter of each other, much less than a wavelength, and the bearings are
within 8.20. Therefore, the range and propagation paths are nearly identical for the
five center geophones, especially the inner three. However, the geophones responded
noticeably different. Figure 5-2 depicts the time series for the center geophones due
to HB 7-6. Several points should be noted. First, G8 has a much smaller amplitude
than the others. Second, G2 and G3 responded similarly until t = 0.8 s, then differed
(-25,0)
(-6,0)
G9
G8
(0,0) (6,0)
G1
(25,0)
G2
G3
10
o
RANGE, BEARING (0 = N)
\
/
4.1"
G9
G8
G1
G2
(0,-350)
X
351.1 m -4.1 O
350.25 m -1.0 0
350.2 m
0.00
350.25 m 1.00
G3 351.1
4.10
G11
NOTE: FIGURE NOT TO SCALE
Figure 5-1: Geometry of center geophones, in relation to a hammer blow at Gl.
appreciably. Third, G1 and G2 were similar for the earlier arrivals. Fourth, at t =
0.6 s, G1 and G9 are nearly in phase, but over the next 0.6 s became approximately
900 out of phase..
Figure 5-3 shows the response to a hammer blow coming from the opposite direction,
still broadside. The responses to HB 11-3 are much more uniform. The geophones
all remain nearly in phase for the whole time trace. However, the amplitudes differed
significantly, and in a different ratio from HB 7-6.
a
3I
. ..
_
0
__
0.2
__
0.4
__
__
0.6
0.8
. . .
. . .
__
1
Time
1.2
1_
1.4
__
1.6
__
1.
2
Figure 5-2: Response of center geophones to HB 7-6. Note the change in phase
between G1 and G9 between 0.6 and 1.2 s. Note also the response of G2 at 0.9 s.
6%
3
..
.........
.....
........
......
....
....
....
2
....
......
..................
.............
8
j
9
...
.........
...
...
.. ......
..
..
V V-%0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
2
Figure 5-3: Response of center geophones to HB 11-3. Note that the amplitudes still
differ, but all the geophones are in phase.
The differences were even more noticeable in the energy spectral densities. Figure
5-4 shows the average energy spectral density for the hammer blows at G7. There is
an average spread of 10 dB between G8 and G9. In Figure 5-5, the maximum spread
for the hammer blows at G11 averages about 5 dB, which agrees with the earlier
observation about the increased uniformity. In both figures, G8 has the smallest
response, which is confirmed by the comments on the time series amplitudes.
The geophones can not be related by a simple function of frequency. Figures 5-6 and
5-7 demonstrate that there is a non-constant transfer function between geophones.
The two plots consider G8 to be the input, since it has the smallest response, and
each of the other geophone ranges to be the output. As can be seen, each geophone
ranges over at least a factor of 2 compared to other geophones. Again, the differences
can not be accounted for by differences in range propagation path.
5.1.2
Center geophones, endfire
Next consider a signal originating from either the east at G5 or the west at G10.
Now, the propagation path is exactly the same, but the geophones are all at different
ranges, up to a maximum spread of 50 m. Therefore, the arrival times for each frequency component will be different. At such range differences, though, the frequency
responses should be nearly identical.
The time series for HB 5-1 is presented in Figure 5-8. Although the direction of
propagation could still be discerned, the traces are quite dissimilar, as if they came
from different sources. The response to HB 10-1 shown in Figure 5-9 is much more
uniform, after the range dependence is taken into consideration. The energy spectral
densities in Figures 5-10 and 5-11 confirm this.
CO
E
CL
C.
16-)
LU
2
Frequency (Hz)
Figure 5-4: Energy spectral densities of center geophones due to hammer blows at
G7.
78
U)
C0
CI
w
2
Frequency (Hz)
Figure 5-5: Energy spectral densities of center geophones due to hammer blows at
Gil.
C)
C)
0
C.
eD
a)
0C1
CM
.C"
0
as
Ca
.o)
0
75
cc)
100
101
Frequency (Hz)
Figure 5-6: Transfer function comparing ESD's of other center geophones to ESD of
G8 for hammer blows at G7.
106
101
Frequency (Hz)
10
Figure 5-7: Transfer function comparing ESD's of other center geophones to ESD of
G8 for hammer blows at G7.
81
DirectI
I
prpgto
I
Direction: of propagation
3
.-
AA
A
1
Time (s)
1.2
.
8
9
0.2
0.4
0.6
0.8
1.4
1.6
1.8
2
Figure 5-8: Response of center geophones to HB 5-1 (endfire).
82
0
I
I
I
I
III
..
....
.
.
. . .
.
2
......_
8
J
Iirection of propagation
0
0.2
0.4
0.6
0.8
1
Time (s)
1.2
1.4
1.6
1.8
2
Figure 5-9: Response of center geophones to HB 10-1 (endfire).
83
CO
CL
CoL
C"L
CD
a),
w;
1%01
0
S"10
10
1
2
10
Frequency (Hz)
Figure 5-10: ESD of center geophones at endfire due to HB 5-1.
84
N
U)
"0
V
o
C)
0.
U)
0o
w
10 0
101
102
Frequency (Hz)
Figure 5-11: ESD of center geophones at endfire due to HB 10-1.
5.1.3
Center geophones, ambient noise
For the ambient noise traces, there are no obvious paths of propagation, so the received
waves are assumed to be coming in from all directions. Therefore, any comparisons
based on broadside or endfire are meaningless. However, the center geophones are still
spatially close, so a homogeneous plate would have a uniform response. Figure 5-12
proves that this is not the case. Figure 5-13 plots the ratios between the geophones as
in Figures 5-6 and 5-7, again using G8 as the benchmark. In this instance, however,
the ratios are nearly constant across the frequency range.
5.1.4
Discussion
At first glance, the ambient spectra suggest that each geophone was functioning
properly but had a unique gain associated with it. However, the same geophones
were used in a later experiment, and they were shown to be accurate to within 1 dB
at 30 Hz, which is in the middle of the frequency range of interest. Therefore, the
problem was not in the geophones. The geophones were frozen quite solidly into the
ice by experienced experimenters. Notes from the experiment as well as conversations
with the people who ran the experiment eliminated the possibility of the coupling
between the geophones and the ice varying from geophone to geophone. This points
to inhomogeneity of the ice as the reason for the differing frequency responses.
This suggest that a floating ice plate should not be viewed as a homogeneous layer,
but instead as a collection of small, interconnected platelets, some rigidly fastened to
their neighbors, others nearly uncoupled. These platelets shall next be examined for
anisotropy.
m
Co
0)
"-o
m
CO
C-
CL)
L)
0•.
C,0,
w
2
Frequency (Hz)
Figure 5-12: Ambient noise at center geophones
3.5
-
3
(D
.C*
O
CC=
• 2.5
-
G1 (0,0)
-
(D
G2 (6,0)
G3 (25,0)
...... G9 (-25,0)
- -
0
0.
O
(D
O
0r)
-
0
-
cz 2
ID
(I)
0
41
CD
r=
13)
1B
]51.5
IE
,--
Cr
------------------,
0.5
F-
,,
rI
,
,
,
,
,
,
L
100
101
Frequency (Hz)
Figure 5-13: Ratio of ambient noise at each center geophone to ambient noise at
(G8. In contrast to Figures 5-6 and 5-7, the ratios here are nearly constant across
frequency.
88
5.2
Demonstration of Anisotropy
Since each geophone appeared to be on its own ice plate, an alternate tack was chosen
to try to determine the attenuation of the flexural waves. If the ice behaved as an
isotropic material, or at least as a transversely isotropic material, then each platelet
would have its own transfer function relating its output to the "true" output. The
response of each plate could be viewed as a perturbation to a mean field, and that
response would not vary with direction. The intended technique was to make the
analysis independent of the perturbed field by considering the response of an individual geophone to hammer blows at different locations and ranges. Then, since energy
spectral density as a function of range would be known, it would then be a simple
matter to back out the attenuation. Unfortunately, the ice behaved anisotropically.
5.2.1
Total power
Using Parseval's theorem, the total power in a signal is related to the average of the
velocity squared. The flexural wave in a plate is confined to two dimensions and does
not radiate into the water, so it experiences cylindrical spreading and loses power in
inverse proportion to the range. Total power can then be expressed in the form of a
normalized velocity as
rms
1
2.
(5.1)
Figure 5-14 shows the normalized velocity for all geophones in response to hammer
blows at all locations, as well as the mean plus and minus the standard deviation. No
consistent pattern emerges. Figure 5-15 depicts the values by hammer blow location.
Table 5.1: Mean and standard deviation of normalized power, sorted by geophone
location.
Geophone Mean Std dev
0.0054 0.0013
1
2
0.0082 0.0021
0.0100 0.0024
3
0.0112 0.0046
4
0.0049 0.0035
5
6
0.0083 0.0051
0.0058 0.0038
7
0.0030 0.0014
8
9
0.0083 0.0025
10
0.0035 0.0007
11
0.0030 0.0008
12
0.0084 0.0032
Table 5.2: Mean and standard deviation
location.
Geophone
5
7
10
11
of normalized power, sorted by hammer blow
Mean
0.0077
0.0055
0.0080
0.0063
Std dev
0.0034
0.0027
0.0037
0.0048
Table 5.1 summarizes the average value and standard deviation for each geophone,
and Table 5.2 shows the values sorted according to hammer blow location. Both show
a statistically significant variation.
5.2.2
Spectral Density
There are differences in the response of an individual geophone due to hammer blows
at different locations. If the ice were isotropic, then the response at a particular
geophone would depend on only the range to the hammer blow. Each hammer blow
AAln
0
2
4
6
8
10
12
Figure 5-14: Normalized velocity squared for each geophone, with lines added for
mean ± standard deviation.
I
SI
I
I
I
G10
0.018
0.0116
0.014
0.012
0.01
I
G11
-
-
-
-
-
0.008
-
0.006
-
0.004
H
0.002
H
0
0.5
1
1.5
·
·
·
·
·
2
2.5
3
3.5
4
4.5
5
Figure 5-15: Normalized velocity squared for each hammer blow. Lines added for
mean ± standard deviation.
within a set displayed a large amount of consistency to the next, so it is assumed that
each set is consistent to the other sets. Figure 5-16 shows the coherence between the
flexural response at G1 to two hammer blows at G11, HB 11-2 and HB 11-3. The
two data sets are highly coherent up until about 80 Hz, which encompasses the whole
region of interest. This is only a measure of the repeatability of the hammer blows,
not a measure of how the field varies across the ice plate.
The energy spectral densities for all 12 geophones in response to all hammer blows
are shown in Figures 5-17 to 5-28. Since the transfer function of the individual plate
is removed from the process now, inhomogeneity is irrelevant. Similarly, geophone
coupling can be eliminated as a problem. Geometric spreading is known, the attenuation should be a monotonic function of range, and all other transfer functions in
the system model of Figure 4-1 have been removed. Yet the energy spectral densities
do not correlate with range. For example, in Figure 5-17, the largest response at G1
was due to a hammer blow at G10, which was further from G1 than both G7 and
G11. Furthermore, the relative amplitudes change over the frequency range. There
were no large scale variations in the ice that may have scattered the energy.
No hammer blow consistently provided the maximum response, as mentioned in the
previous section. In Figure 5-21, the response of G5 to the hammer blows at G11 was
about 15 dB lower than the response to G7 between 15 and 100 Hz, and the two were
approximately the same distance away. However, Figure 5-26 shows that G10 had
about a 5 dB greater response to G11 than it did to G7. In these two examples, the
WNW-ESE path had a larger degree of propagation of the flexural wave than did the
N-S path, suggesting that there is a favorable alignment for flexural wave propagation
in that direction. This anisotropic nature is also evident at other geophones. Table
5.3 summarizes the best and worst directions for propagation for each geophone. The
data are presented in Figures 5-29 and 5-30. The best and worst propagation paths
are more evident here. All of the best paths were either E-W or NW-SE, and all of
the worst paths except one were either N-S or NE-SW. This may be related to the
0
E
E
LU
w
C
o
t3
U0
U
I.)
o
0
50
100
Frequency
150
200
250
Figure 5-16: Coherence of G1 in response to HB 11-2 and HB 11-3.
Table 5.3: Best and worst propagation paths.
Geophone Location Best propagation Worst propagation
1
(0,0)
10
7
2
(6,0)
10
7
3
(25,0)
10
7
4
(155,0) no preferred path no preferred path
5
(450,0)
7
11
6
(0,96)
5
11
7
(0,305)
5
10/11
8
(-6,0)
10
7
9
(-25,0)
10
11
10
(-376,0)
11
7
11
(-350,0)
10
5
12
(0,10)
10
7
strong sea-axis alignment of the ice.
5.2.3
Dispersion
At the geophone, the flexural wave appears as a non-stationary signal. The frequency,
amplitude, and phase of the sinusoidal components change over time. Therefore, the
results from any spectral analysis will be time dependent. Speech signals and Doppler
radar signals also fall into this category.
For a plate in air, the phase speed is proportional to the square root of the frequency,
so for a given range, the frequency is inversely proportional to the time squared.
In radar systems, this type of signal is known as a chirp signal. The dispersion
relationship is more complicated for a fluid-loaded plate and hence does not yield a
second power relationship, but it can be approximated as one.
In this work, the anisotropic nature of the dispersion will be emphasized rather than
(0
aJ
,)
,,
c:
43
_0
ci)
QuJ
01
U3
100
101
102
Frequency (Hz)
Figure 5-17: Average ESD at G1 to hammer blows at all locations.
CM
<W
V
73
0
U
CL
_,
0a,1
0.
w
t-
I8
12
Frequency (Hz)
Figure 5-18: Average ESD at G2 to hammer blows at all locations.
0-
(A
LU
U,
"D
C
U,
0,
U,·
'I
10
102
10'
Frequency (Hz)
Figure 5-19: Average ESD at G3 to hammer blows at all locations.
98
C14
T
cin
V
"a
t.
ci,
0)
w)
CL
LUI
3
100
10'
10
Frequency (Hz)
Figure 5-20: Average ESD at G4 to hammer blows at all locations.
-M
..
i
,..
--
G7 (0,305)
-- G10 (-376,0)
...... G11 (0,-350)
-65
·- '"-
H
~c'·.
'·.
-70
H
.75
.4
\
H
\'
-80
'
E
'
-85
I'·
-90
I I
E
I I
*1\
'* I
I
-95
F
I
I
C.
-100
-1Ai•
I
'
0)
10
I
I
1
I
I
I
I
I
I
I
I
I
I
ill
101
Frequency (Hz)
Figure 5-21: Average ESD at G5 to hammer blows at all locations.
100
h
in
"1o
v,
Co
,)
Co
C
VC.)
Uj
0.U
O)
100
101
10
Frequency (Hz)
Figure 5-22: Average ESD at G6 to hammer blows at all locations.
101
-70
-75
• -80
E
2
-85
o
e
-90
a)
-95
-.100
1 rh5
100
102
101
Frequency (Hz)
Figure 5-23: Average ESD at G7 to hammer blows at all locations.
102
04
h
ci)
In
U)
0)
0)
CL
C
w
u,
,2
Frequency (Hz)
Figure 5-24: Average ESD at G8 to hammer blows at all locations.
103
_1
-60
-65
-.70
-75
,,
CD
-80
Oi
÷- -85
C1
CD
C"
U -90
-95
-100
4n
100
101
102
Frequency (Hz)
Figure 5-25: Average ESD at G9 to hammer blows at all locations.
104
h
a)
"o
CU
v
0.
t-U)
a)
w
100
10
101
Frequency (Hz)
Figure 5-26: Average ESD at G10 to hammer blows at all locations.
105
(D
CL.
C,;
at-
z-I.
r:
IlD
4'
CLL
]2
Frequency (Hz)
Figure 5-27: Average ESD at G11 to hammer blows at all locations.
106
m
a)
V
a,
0.
a,
C
w
10i
101
Frequency (Hz)
10O
Figure 5-28: Average ESD at G12 to hammer blows at all locations.
107
G7
G12
r\
10,,
G9
G8
/
C
i
\/
\1
Y
!I
1l G2 G3
m-
Gil1
Figure 5-29: Direction of best propagation for each geophone.
108
9
G7
N
G6
G12
X
G10
G9
G8
l1 G2 G3
X
X
G4
Gl1
Figure 5-30: Direction of worst propagation for each geophone.
109
G5
the technique used to derive the curves or the actual curves themselves. Wang's[20]
research on this data uses more refined techniques to derive the phase and group
velocity versus frequency. Here, a first order examination is sufficient.
The instantaneous period for a non-stationary signal such as the flexural wave can
be approximated to be the time between two adjacent peaks or troughs. Figure 5-31
reproduces the flexural wave at G1 due to HB 10-1. At two adjacent peaks tl and t2 ,
the frequency and group velocity are given by
f(t =
ti
+t2
2 )
1
tl
(5.2)
2
and
cg(t= t~
r- t
2
(5.3)
where
r = range to hammer blow, and
to = time of hammer blow.
Figures 5-32 and 5-33 show the resulting dispersion curves at G1 and G6, respectively, for hammer blows at all locations, along with the calculated dispersion curve
from Equation 2.19. In both cases, over the range of 20-50 Hz, there is about a 50
m/s difference between the curves for hammer blows at G10 compared to the other
locations. In comparison, the dispersion curves for the center geophones shown in
Figure 5-34 in response to HB 10-1 are nearly identical.
110
Figure 5-31: Peaks used in calculating the group velocity for HB 10-1, response at
G1.
111
Co
E
C.)
(D
0e
0:
ou
(_
0
1
20
40
30
50
60
Frequency (Hz)
Figure 5-32: Dispersion curve at G1 using second order polynomial curve fit.
112
100
90
80
70
60
0
0
= 50
O
40
30
20
10
0
10
20
3w
Frequency (Hz)
40
5u
Uo
Figure 5-33: Dispersion curve at G6 using second order polynomial curve fit.
113
(I)
2,
CL
E
0
1
20
40
30
50
60
Frequency (Hz)
Figure 5-34:: Dispersion curves of center geophones in response to HB 10-1
114
The window size is also important. If too long of a window is chosen, ambient noise
gets included. If too short of a window is used, then some part of the flexural wave
response gets chopped off. In order to capture the essence of the flexural wave, just
the right amount must be used.
One way around this is to do a short time Fourier transform (STFT)[13], or its
modulus squared, called the spectogram. In an STFT, the characteristics of the signal
are assumed constant over the length of the (short) window, and as the window slides
along the length of the signal, a picture develops tracking the frequency history. A
longer window will provide higher frequency resolution, with a corresponding decrease
in time resolution. A shorter wideband window will have poor resolution in fr quency
but good resolution in time.
Several variations attempt to improve on the resolution by varying the window length
according to the instantaneous frequency. The Wigner distribution[4], the exponential
distribution[3], the reduced interference distribution[8], and the wavelet transform[15]
all attempt to provide high resolution in both time and frequency. Again, however,
the objective here is to display the anisotropic nature rather than demonstrate the
effectiveness of new techniques.
5.2.4
Longitudinal and SH Wave Speed
An earlier unpublished study[14] examined the longitudinal and shear wave speed of
this data set. Arrival times for the radial axis for the longitudinal wave and for the
transverse axis for the horizontally polarized (SH) wave were used to calculate phase
velocities. The values are summarized in Table 5.4. In general, these were lower than
previous in situ measurements[1]. More importantly for this work, the wave speeds
displayed anisotropic behavior.
115
Table 5.4: Longitudinal and Shear (SH) Wavespeed
Hammer blow location
5
7
10
11
Receiver
1
7
10
11
1
5
10
11
1
5
7
11
1
5
7
Longitudinal (m/s) Shear (m/s)
2771
1560
2875
1649
2840
1552
2926
1638
2773
1568
2861
1638
2782
1652
2788
1582
2824
1569
2805
1560
2715
1653
2747
1562
2753
1587
2726
1639
2753
1574
10
2764
1554
Figure 5-35 summarizes the directions of fastest and slowest wave speeds for both
types of waves. The longitudinal wave had the same relationship to direction that
the flexural wave did, namely fastest in the WNW-ESE direction and slowest in the
N-S direction, and the horizontal shear wave had the opposite directional relationship. A further look at the particle velocities of each wave type reveals consistency.
The longitudinal wave has only radial particle displacement, the flexural wave has
both radial and vertical, and the SH wave has transverse displacement. The particle
velocity for the fastest propagation in all three cases is WNW-ESE.
5.3
Coherence
The coherence measures the extent to which two signals are similar. In scattering
theory, a field can be expressed as the sum of a coherent mean field and the incoherent
116
G7
G10
G7
G10
G5
G11
SLOWEST SHEAR
G11
FASTEST SHEAR
G7
G10
/
G5
G7
I
G10
G1
"ýý
~----·-------------
G5
/
G11
G11
SLOWEST LONGITUDINAL
FASTEST LONGITUDINAL
Figure 5-35: Directions of fastest and slowest wave speeds for longitudinal and SH
waves.
117
scattered field. In this case, the coherence function will be used to examine the flexural
waves.
The mean square coherence (MSC) is defined as
G(x, y, f) =
Gx(f)12
Gxx(f)G* (f)
(5.4)
where Gy is the cross spectral density and Gx and GO are the auto spectral densities
of the two signals being compared. For a homogeneous, isotropic plate, the coherence
would depend only on range.
Figures 5-36 through 5-39 use G1 as a benchmark to demonstrate the coherence. For
Figure 5-36, the signal is coming in perpendicular or broadside from HB 11-2 to the
E-W leg of the array that the center geophones are on, and the geophones are all
350-351 m away from the source, so range is not a factor. Below about 25 Hz, the
geophones displayed high coherence. The same geophones are compared in Figure
5-37 for HB 10-1 coming in direct line or endfire with the geophones. Up to about 20
Hz, the two plots are similar, but above that, the geophones are much more coherent
for the endfire case, even though the ranges vary by as much as 50 m. In Figures 5-38
and 5-39, the MSC for HB 11-2 and HB 10-1, respectively, is shown for geophones
with large spatial separation.
Figure 5-40 compares the coherence with G4 of the three geophones with the most
similar ranges for HB 10-1. The geophones display a high coherence up to approximately 25 Hz.
Each geophone can be paired up with every other geophone, giving a total of 55
unique pairs for each hammer blow location. Table 5.4 lists the spatial separations
118
04
CU
W
E
LL
0
C
ou
r'I)
0
0)
-c
0
o
{D
0
10
20
30
Frequency (Hz)
40
50
60
Figure 5-36: Coherence of center geophones with Gl for HB 11-2 (broadside). G1 is
at (0,0) and 350 m from the source.
119
CJ
IJ•
X
()
Ca
Uj
0
C'
(3
UCD
cD
C-)
Frequency (Hz)
Figure 5-37: Coherence of center geophones with G1 for HB 10-1 (endfire). GI is at
(0,0) and 376 m from the source.
120
0
a)
E
0)
C
0
LL
0)
Q)
0
0
0
Frequency (Hz)
Figure 5-38: Coherence with G1 of geophones with large spatial separation in response
to HB 11-2. G1 is at (0,0) and 350 m from the source.
121
CM
<_.
W
a)
E:
w
.4,
Iii
LLI
a)
t-:
0
=0
C;
oL
C)
U..
C:
a;
0
10
20
30
Frequency (Hz)
40
50
--
60
Figure 5-39: Coherence with G1 of geophones with large spatial separation in response
to HB 10-1. G1 is at (0,0) and 376 m from the source.
122
I
I
I
I
·
,I\
G3 (25,0) 401 m
- - G7 (0,305) 485 m
.- - G11 (0,-350) 514 m
I'
0.9
Xii
A
~i~
I \1*
-
I
I'I
*
ji
I
II
I
I
*I
~ ~'
0.8
-
I
Ii
I
I
i
II
II
I.
0.7
I
II
.
IIII
t,,
I
*I I
ii
II
I
I
-ITi
0.6
II
I
ijI i
f'II
S ii
E
I
j
I I
Ii
,I
-1
-
II
I.
o 0.5
\ii
II
II
11
I'
i
I
T 0.4
l
-
II II
IiII
'I
o
LL
0)
II
Ii
I
I~I
II
ii
II II
'ii
I
I iII
'-
0.3
I
*
'
0.2
'V
?;n:
I
1Y
II
II
n
,
I~i
* I* *.I
'ii
0.1
I
•~~
Ii .1' Il
.
If
:I!
*'r
I
I(
''I
WI
I
Frequency (Hz)
Figure 5-40: Coherence with G4 of geophones with large spatial separation but similar
range in response to HB 10-1. G4 is at (155,0) and 531 m from the source.
123
for all pairs. In each case, the lower number geophone is listed first. Of course, each
pair has two parts, and the signs are reversed when the geophone order is reversed. In
Figure 5-41, each pair is plotted so that the midpoint between the two geophones is
at the origin, so the plot is symmetric about the origin. Because of the approximately
logarithmic spacing of the geophones, there are more pairs clustered near the origin.
Several patterns should be pointed out that will help in interpretation.
* A column corresponds to a particular geophone matched up with each of the
geophones on the N-S leg of the array, and a row is for the E-W array.
* In many places there are five points next to each other plus one either just above
or below. These are the center geophones and G12. For example, the cluster
near (-80,0) is for G4 and the center geophones, while the ones immediately
above and below correspond to the other geophones on the N-S leg (from top
to bottom, G7, G6, and G10).
* Each pair on the x axis (N-S separation = 0) corresponds to two geophones that
are both on the E-W leg of the array, and each pair on the y axis represents a
pair on the N-S leg. The points away from either axis are for one geophone on
the E-W leg and the other on the N-S leg.
* The only closely spaced geophones are in the center of the array.
* There are few points away from either axis, so the coherence matrix will be
rather sparse. Therefore, when plotting the spatial coherence, little information
will be lost by plotting just slightly more than one quadrant (as shown by the
dotted line in Figure 5-41), with the benefit of increased resolution in the plots.
Figures 5-42 to 5-46 show the spatial MSC in 5 Hz-wide frequency bins ranging from
10-15 Hz to 40-45 Hz. In each case, the coherence for HB 10-1 is shown on top and
124
300
-7
200
100
-|
0
6
12
- - --12
-100
7
11
-200
5
10
4
-300
I
-400
I
-300
I
-200
I
-100
I
I
0
100
_ _I
200
300
400
Figure 5-41: Coherence pair locations. Each pair is plotted so that the midpoint
between the two geophones is at the origin. Arrow indicates G4 paired up with
center geophones.
125
Table 5.5: Spatial separation for geophone pairs
Geophone pair
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
2
3
2
4
2
5
2
6
2
7
2
8
2
9
2
10
2
11
2
12
3
4
3
5
3
6
3
7
3
8
3
9
3
10
3
11
3
12
4
5
4
6
4
7
4
8
4
9
4
10
4
11
4
12
5
6
5
7
5
8
5
9
5
10
5
11
5
12
6
7
6
8
6
9
6
10
6
11
6
12
7
8
7
9
7
10
7
11
7
12
8
9
8
10
8
11
8
12
9
10
9
11
9
12
10
11
10
12
11
12
E-W separation (m)
-6
-25
-155
-450
0
0
6
25
376
0
0
-19
-149
-445
6
6
12
31
382
6
6
-130
-426
25
25
31
50
401
25
25
-296
155
155
161
180
531
155
155
450
450
457
475
827
450
450
0
6
25
376
0
0
6
25
376
0
0
19
370
-6
-6
352
-25
-25
-376
-376
0
126
N-S separation (m)
0
0
0
0
-96
-305
0
0
0
350
-10
0
0
0
-96
-305
0
0
0
350
-10
0
0
-96
-305
0
0
0
350
-10
0
-96
-305
0
0
0
350
-10
-96
-305
0
0
0
350
-10
-210
96
96
96
446
86
305
305
305
656
295
0
0
350
-10
0
350
-10
350
-10
-360
for HB 11-2 on bottom. In order to improve readability, each point is represented by
a cross.
For the hammer blow at G10, the pairs on the x axis all have the same propagation
path, so the coherence of these pairs is solely a measure of the range and inhomogeneity. The pairs on the y axis but still near the origin represent pairs that are
approximately the same distance from the source but along different propagation
paths. The coherence of these is a measure of the inhomogeneity and the anisotropy.
The pairs further away from the origin and the ones away from the axes are better
analyzed through the use of conventional coherence plots such as Figure 5-40. For
the hammer blow at G11, the situation is similar except the roles of the x and y axes
are reversed.
Several observations can be made by considering all of the plots:
1. The center geophones had a high coherence for a greater frequency range for
the hammer blow at G10 than for at G11. This agrees with the observations
made earlier of best and worst propagation paths.
2. Even adjacent geophones displayed low coherence above about 35 Hz.
3. Geophones spatially distant but at similar ranges to the source displayed low
coherence above 25 Hz.
4. For almost all geophone pairs, the coherence was higher when the wave was
coming in broadside rather than at endfire. Figure 5-47 to 5-50 show the difference in coherence between HB 10-1 and HB 11-2. The dark colored points
had a higher coherence for HB 11-2, and the lighter colored ones had a higher
coherence for HB 10-1. Almost all the points on the x axis are dark, almost
all the points on the y axis are light. As mentioned earlier, the points on the
x axis are at endfire for HB 10-1 and broadside for HB 11-2, and the reverse is
true for the y axis.
127
The upper limit of frequency can be used to calculate a characteristic scattering length
for the ice. In scattering theory, waves are coherent if they are nearly in phase. If
several sinusoids at a single frequency are no more than 7r/2 radians out of phase,
their peaks will nearly match at a given location. If they are out of phase more
than this cutoff amount, some of the sinusoids will be at a peak, some will be in a
trough, and others will be in between, so there will be some cancellation. For a given
frequency, a wavelength is specified, and the phase requirement becomes a wavelength
requirement. The quarter wavelength correlation length then becomes the measure
of the scattering coherence. At the upper limit of 35 Hz for coherence, the phase
velocity is approximately 400 m/s. The correlation length becomes
S-
Acoh
c=_p
400
4 4f (4)(35)
2.9 m.
(5.5)
The closest separation between geophones is 6 m. Therefore, for this experiment, all
geophone pairs must be considered as incoherent above 35 Hz. At lower frequencies,
because of the dispersion characteristics of the flexural wave, the correlation length
becomes longer. At 20 Hz, the phase velocity is about 300 m/s, so the correlation
length becomes approximately 3.8 m. All of the Arctic experiments researched in
preparation of this thesis had minimum spacing between geophones of at least 10 m.
This suggests that for high frequency data, coherent signal processing may not be
valid.
128
200
I
I
I
4.
150
100
50
,16. 4.
4,
-
-
-
.~44
+4U
÷ .,4,. ÷
+
~~
I
-300
0
-300
-200
-250
4.4.4•
·
-150
-50
-100
IAI ean•frtirn (m\
0.1
0.2
0.3
-200
-250
0.5
0.4
0.6
-100
-150
0.7
0.9
0.8
1
0
-50
EAI epnor~tinn (m
F
-
0.2
0.3
0.4
W
0.5
vmman
0.6
m
0.7
0.8
0.9
1
Figure 5-42: Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bottom)
for 10-15 Hz.
129
·
200
150
100
1
·
·
-
44.
-
4
4i
flf
4
50
-
-300
-250
-200
-150
AIAI
-50
-100
u I
\
OMUMLO +1~~,iji
Lau
II1
LUll
it
0
0.1
-300
0.2
-250
0.3
-200
-
0
0.1
0.2
0.3
0.4
0.5
-150
I• IAInr6 ,
CM
,H~
r-yL
n~Llnlnllllll
0.4
0.5
0.6
-100I
· \
IIIII
0.6
0.7
0.8
-50
0.7
0.9
1
0.9
1
0
0.8
Figure 5-43: Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bottom)
for 15-20 Hz.
130
r
IN
0.1
0.2
0.3
-200
-250
-300
-100
m
\AA ~rnr~*inn
~ZYU
~lmltinn (mlII
-150
-200
-250
-300
:::
0.4
0.5
-100
-150
F-W
0.6
Yninfinn
0.7
0.1
0.2
0.3
0.4
0.5
0.9
0.8
1
o
u
-50
(m)
::i::~
0
50
0
-50
0.6
0.7
0.8
A A
0.9
J
1
Figure 5-44: Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bottom)
for 20-25 Hz.
131
20(0
++4 +
+
150
E
ca 100
-
a
(n
;z 50
0
I
--300
0
I
-250
0.1
-200
0.2
0.3
I
-150
0.4
I
-100
0.5
-50
0.6
0.7
0
0.8
50
0.9
1
20 0
150
4 +4. 4
0
100 t',
50.
I 50
÷7
U
I
II
-.
300
0
-250
0.1
-200
0.2
0.3
-150
0.4
-100
0.5
0.6
-50
0.7
0
0.8
50
0.9
1
Figure 5-45: Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bottom)
for 30-35 Hz.
132
2
I'
I
I
-Q
150
i
100
50
4 .*
-
u
-150
-200
-250
-300
4I
IF-~LY
I
1.
-100
-50
-50
4.
I
s~rli~rilll~~llIIIII
:::"::
i:iii-ili
"
0.2
0.3
0.7
0.6
0.5
0.4
0.9
0.8
1
euu·
ZUU
150
100
-
-
~p·
-300
II
-200
-250
44*
4'
4.+
50
~4-
~~fi·
I-50
-150
tAE
-100I\
•,l:~
T"
F-w
senamilan
-50
fm)
'
0.1
0.2
0.3
0.4
'4.
0.5
0.6
0.7
0.8
0.9
1
Figure 5-46: Spatial Mean Square Coherence for HB 10-1 (top) and HB 11-2 (bottom)
for 40-45 Hz.
133
__1
2
C
cc
CL
z
.ZU
-Ou
-U.0
-u.4
-U.i
-1bu
-100
E-W seDaration (m)
-0.2
-0.1
0
0.1
-50
0.2
0
0.3
0.4
I
0.5
Figure 5-47: Difference in Spatial Mean Square Coherence for HB 10-1 and HB 11-2
for 10-15 Hz. Negative values corresponds to higher coherence for HB 11-2.
134
E
C
0c
0.
CO
zz
-250
-200
-150
-100
-50
0
E-W separation (m)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
I
0.5
Figure 5-48: Difference in Spatial Mean Square Coherence for HB 10-1 and HB 11-2
for 20-25 Hz. Negative values corresponds to higher coherence for HB 11-2.
135
-250
-0.5
-200
-0.4
-0.3
-150
-100
E-W seParation (m)
-0.2
-0.1
0
0.1
-50
0.2
0
0.3
0.4
I
0.5
Figure 5-49: Difference in Spatial Mean Square Coherence for HB 10-1 and HB 11-2
for 30-35 Hz. Negative values corresponds to higher coherence for HB 11-2.
136
20C
E 15C
CO
50
C
C
-250
-200
-100
-150
-50
0
E-W separation (m)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
I
0.5
Figure 5-50: Difference in Spatial Mean Square Coherence for HB 10-1 and HB 11-2
for 40-45 Hz. Negative values corresponds to higher coherence for HB 11-2.
137
Chapter 6
Conclusions
The final chapter summarizes the significant results of Chapters 4 and 5 and makes
some recommendations for future work.
6.1
Summary
This experiment has already been the subject of much research. It has spawned a
study on the geophysical properties of and thermally induced stresses in the ice[10],
been subjected to a preliminary analysis of wavespeeds and attenuation[18], and been
a proving ground for the use of the wavelet transform in elastic wave analysis[20].
There are certainly enough good data left to provide several more papers. It is the
hope of the author that this thesis be a reference for any future research on this
excellent source of information.
138
6.1.1
Bottom reflections
In the Arctic environment, the flexural wave mode is independent of the water column depth. However, single and multiple reflections of a shallow bottom may interfere
with the measurement of the particle velocity associated with the flexural wave, often
overwhelming the signal. Unchecked, these bottom bounces would lead to significant
errors in the energy spectral density estimates. In this paper, an effective and repeatable way to "surgically remove" the bottom bounces was demonstrated, leading
to truer estimates of the energy contained in the flexural wave. Additionally, the
average sound speed and bottom depth were calculated to be 1432 m/s and 153 m,
respectively.
6.1.2
Inhomogeneity and Anisotropy
Probably the most significant contribution of this paper was to demonstrate the heterogeneous nature of seemingly uniform ice. Originally, the most remarkable thing
about the system seemed to be its homogeneity and the clean data generated. Soon,
however, preliminary spectral analysis revealed large discrepancies in the frequency
responses of neighboring geophones. Eventually, geophone calibration, geophone coupling, experimental repeatability, and analytic error were all ruled out, leaving the
ice itself as the only possible explanation.
Geophones as close as 6 m responded as if they were each on their own individual
plate, with each abutting the next, sometimes strongly coupled, other times not.
The degree of coupling also seemed to depend on the direction of the incoming signal.
Miller[12] had earlier shown that some discrepancies between calculated and measured
values in the Arctic ice canopy could be explained as a result of two abutting semiinfinite plates. Wang[20] is currently researching the effect of uniformly placed cracks
139
in an otherwise uniform plate. However, to the best of the author's knowledge, no
previously published work has looked this much in detail at the subtle differences in
propagation in a situation like this.
Flexural waves propagated best in a WNW-ESE direction, and worst in the N-S
direction. This is quite likely related to the direction of crystallization, which would
depend on the underlying current in Allen Bay. An earlier unpublished study[14]
examined the anisotropic nature of the longitudinal and shear wave speed. The
directions of fastest and slowest propagations agreed with the information found here.
6.1.3
Coherence
The spatial mean square coherence (MSC) measured the coherence between geophones. Above approximately 35 Hz, even nearby geophones displayed very low
coherence, although this could be tied to the low signal to noise ratio in that frequency range. Above 25 Hz, geophones far away from each other but equidistant to
the source displayed low coherence. For a particular geophone pair, the coherence
was higher when the wave was coming in perpendicular to the axis connecting the
two than when the wave was parallel to the axis. Using the upper limit of 35 Hz, the
correlation length was calculated to be approximately 3 m, less than the minimum
spacing between geophones.
6.2
Recommendations for future work
The great deal of variation in the elastic parameters studied suggest that much more
work should be done with this data. Future work should focus on the as yet unana140
lyzed set of hammer blows done in a quarter circle arc 100 m from the center of the
array. This data should be able to provide better estimates of the source spectrum,
which would eliminate one of the difficulties encountered in this work estimating the
attenuation. Also, there will be more geophone pairs that are nearly equidistant from
the source but have significantly different propagation paths, which should lead to a
more robust study of the anisotropic nature.
The bottom reflections could be studied in more detail to get a better estimate of the
bottom properties. This could lead to an automated bottom bounce removal system.
Future experiments examining the heterogeneous nature of ice should attempf to do a
controlled study for anisotropy and inhomogeneity. One suggestion would be to place
several geophones in a line on both sides of a known crack in the ice. Then hammer
blows should be done perpendicular to, parallel to, and at an angle to the fault line.
Several of the geophones should be placed within the obtained correlation length of
6 m to establish more firmly the length and whether the length is dependent on the
direction of propagation relative to the crack. Also, even though the hammer blows
seemed repeatable, perhaps a guillotine-like machine could be used to strike the ice,
guaranteeing a purely vertical impact. The machine could also include the capability
of striking a beam frozen into the ice horizontally using a pendulum motion.
141
References
[1] G.H. Brook and J.M. Ozard. Underwater Acoustic Data Processing,pages 113-
118. Kluwer Academic Press, 1989. In situ measurement of elastic properties of
sea ice.
[2] Jack Capon. High-resolution frequency-wavenumber spectrum analysis. Proceedings of the IEEE, 1969.
[3] Hyung-Ill Choi and William Williams. Improved time-frequency representation
of multicomponent signals using exponential kernels. IEEE Transactions on
Acoustics, Speech and Signal Processing, 1989.
[4] Leon Cohen. Time-frequency distributions, a review. Proceedings of the IEEE,
1989.
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