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Bearing calibration of the Cape Leeuwin hydroacoustic station

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Calibration of bearing accuracy was conducted for the hydroacoustic station (HA01) deployed in the Indian Ocean off Cape Leeuwin, Western Australia, as part of the International Monitoring System of the Comprehensive Nuclear-Test-Ban Treaty. Both the random and systematic components of the bearing error were investigated using the azimuth measurement of various underwater events detected at the Cape Leeuwin station. The RMS value of the random component of azimuth estimation was examined using long-lasting low-frequency underwater events, such as harmonic tremor signals from drifting iceberg and seismic events including Sumatra earthquakes (main shock and aftershock). The random bearing errors were associated with horizontal deviation of hydrophones' moorings from the position based on a model of mooring motion. The systematic component was estimated through inversion of the signal travel time difference to the HA01 hydrophones from a number of underwater explosions made in the Indian Ocean at known locations. It is shown that the standard deviation of bearing estimates due to the random component is around 0.5 degree. The systematic error, which is about 0.8 degree clockwise, can be compensated by small correction of moorings' coordinates. Potential effects on azimuth estimation of horizontal refraction along cross-ocean acoustic propagation paths are also considered through numerical modelling.
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Acoustics 2008 1
Acoustics 2008
Geelong, Victoria, Australia 24 to 26 November 2008
Acoustics and Sustainability:
How should acoustics adapt to meet future
demands?
Bearing calibration of the Cape Leeuwin hydroacoustic
station
Binghui Li (1), Alexander Gavrilov (1) and Alec Duncan (1)
(1) Centre for Marine Science & Technology, Curtin University of Technology, Perth WA 6845, Australia
ABSTRACT
Calibration of bearing accuracy was conducted for the hydroacoustic station (HA01) deployed in the Indian Ocean off
Cape Leeuwin, Western Australia, as part of the International Monitoring System of the Comprehensive Nuclear-
Test-Ban Treaty. Both the random and systematic components of the bearing error were investigated using the azi-
muth measurement of various underwater events detected at the Cape Leeuwin station. The RMS value of the random
component of azimuth estimation was examined using long-lasting low-frequency underwater events, such as har-
monic tremor signals from drifting iceberg and seismic events including Sumatra earthquakes (main shock and after-
shock). The random bearing errors were associated with horizontal deviation of hydrophones’ moorings from the po-
sition based on a model of mooring motion. The systematic component was estimated through inversion of the signal
travel time difference to the HA01 hydrophones from a number of underwater explosions made in the Indian Ocean at
known locations. It is shown that the standard deviation of bearing estimates due to the random component is around
0.5 degree. The systematic error, which is about 0.8 degree clockwise, can be compensated by small correction of
moorings’ coordinates. Potential effects on azimuth estimation of horizontal refraction along cross-ocean acoustic
propagation paths are also considered through numerical modelling.
INTRODUCTION
As part of International Monitoring System (IMS) of Com-
prehensive Nuclear-Test-Ban Treaty (CTBT), the Cape
Leeuwin hydroacoustic station (HA01) was deployed about
150 km north-west of
Cape Leeuwin, Western Australia. The
station consists of three hydrophones (triplet) with horizontal
spacing of around 2 km from one another. The hydrophones
are submerged near the SOFAR acoustic channel axis at a
depth of about 1100m. Because of its location and design, the
station has the capability of long-range acoustic reception and
bearing estimation. Since its deployment in 2001, the HA01
station has played an important role in monitoring various
hydroacoustic events in the Indian and Southern Oceans,
which include earthquakes and other tectonic activity, for
example Indian Ocean ridge seismicity (Jeffrey and Roger,
2005&2006) and the Great Sumatra-Andaman Earthquake
(Jeffrey and Roger, 2005; Tolstoy and DelWayne, 2006), as
well as ice-related noise, such as noise produced by drifting
icebergs (Emily et al, 2005; Jacques et al, 2006) and tran-
sient signals from ice breaking or rifting activities (Li and
Gavrilov, 2006&2008; Gavrilov and Li, 2007&2008). In
order to enhance the bearing accuracy of hydroacoustic moni-
toring, especially for the analysis of the localization and
analysis of the spatial distribution of distant hydroacoustic
events, it is essential to carry out the bearing calibration of
the HA01 station.
Assuming both variable and permanent horizontal deviations
of hydrophones’ position from their nominal locations, the
bearing error of Cape Leeuwin station was considered to
consist of random and systematic components. Based on this
assumption and using various underwater events detected at
Cape Leeuwin station, both components of the bearing error
were investigated. Potential effects of horizontal refraction
along the acoustic propagation paths on the bearing accuracy
were also examined through numerical modelling.
BEARING ERROR ANALYSIS
For the long-range hydroacoustic monitoring, the back-
azimuths of observed hydroacoustic events at HA01 station
can be estimated by the application of Plane Wave Fitting
(PWF) method (Del Pezzo and Giudicepietro, 2002). The
covariance matrix of the slowness vector p with two compo-
nents (p
x
,
p
y
) can also be derived as follows [Menke, 1984]:
cov(p) = [(x
T
x)
-1
x]cov(t)[ (x
T
x)
-1
x]
T
.
(1)
Where t is the vector of travel time differences between each
pair of hydrophones t
i,j
, and x is the vector of relative geo-
metric positions of the hydrophones. Based on this equation,
the standard deviation of the back-azimuth estimation can be
obtained (Li and Gavrilov, 2006). From Eq. (1) we can see
that the back-azimuth estimate via PWF is constrained by the
errors of both the different travel time estimates t and relative
position of hydrophones x. The differential travel time es-
timates t
i,j
measured through cross-correlation of the signals
at two receivers i and j depends on the quantization, the Sig-
nal-to-Noise Ratio (SNR), the signal bandwidth and its dura-
Proceedings of ACOUSTICS 2008 24-26 November 2008, Geelong, Australia
2 Acoustics 2008
tion. In the case of remote hydroacoustic observation, the
received signals for analysis are low-frequency intense
broadband signals. Therefore, based on our previous study
(Li and Gavrilov, 2006), the contributions of cross-
correlation uncertainties to the travel time difference estimate,
and consequently to the back-azimuth estimate, are negligible.
Apart from the contribution from the travel time difference
estimates, the bearing errors are also caused by the deviation
of the relative geometric positions of the hydrophones from
the relative touchdown mooring position x. The deviation
may be due to the temporal horizontal motion of the hydro-
phones mounted on long vertical moorings, which has been
thoroughly investigated in our previous study (Li and
Gavrilov, 2006). We refer to the bearing error caused by this
deviation component as random bearing error. Possible per-
manent deviation of the receivers’ positions relative to their
nominal locations may induce a time-independent or system-
atic bearing error. Although the systematic bearing error is
superimposed on the random bearing errors, it can be esti-
mated statistically given the exact bearings of enough known
sample events. Based on the observation of a number of un-
derwater explosions with know positions, the systematic
bearing error of HA01 will be explored by the inversion of
relative deviations of receivers using modelled and measured
travel time differences between pairs of receivers.
RANDOM BEARING ERROR ESTIMATION
Hydroacoustic recordings over six years, from December
2001 to January 2008 have so far been collected from the
HA01 station. These continuous sea noise recordings were
divided into 20-second fragments and only the fragments
with high coherence at three receivers were selected for fur-
ther analysis. The cross-correlation between noise signals on
any pair of hydrophones must exceed a threshold of 0.5 in at
least one of four different frequency bands: 3-15 Hz, 15-30
Hz, 30-60 Hz, and 60-100 Hz. Based on characteristics of the
waveform and spectrogram of these selected signals, the
coherent events were divided into different classes. Among
them is the group of harmonic tremor signals which is char-
acterized by a fundamental frequency below 10 Hz with sev-
eral harmonics at higher frequencies. Tremor events have
various durations from tens of minutes to several hours or
even longer. These events are believed to be related to drift-
ing icebergs. Signals from seismic events, such as earth-
quakes, display extremely high energy concentration at very
low frequencies below 5 Hz, and the duration of the signals
can be from days up to months. Based on the duration and the
high SNR in the low frequency band, a number of tremor and
seismic events were selected for the estimation of HA01
random bearing error, as shown in Table 1. In the selection
process, only stationary sections of those events, when the
mean value of measured back-azimuth did not change, were
considered.
Table 1. The azimuthal mean values and the standard devia-
tions (SD) of six long-lasting Antarctic tremor events and
four seismic events. TR: tremor event; SE: seismic event; MS
and AS: the main shock and the aftershock of the Great Su-
matra-Andaman Earthquake; Mean and SD represent the
mean value and standard deviation of azimuth respectively.
Events
Lasting time
[year/day]
Mean
[
0
]
SD
[
0
]
TR 1 02/151.54 ~ 02/152.36 163.927 0.09
TR 2 03/217.38 ~ 03/217.46 162.740 0.16
TR 3 04/170.14 ~ 04/170.88 195.571 0.18
TR 4 05/061.83 ~ 05/075.11 180.242 0.19
TR 5 06/003.87 ~ 06/004.65 181.952 0.09
TR 6 07/214.75 ~ 07/224.07 191.770 0.11
SE 1 03/233.54 ~ 03/367.50 121.254 0.40
SE 2 07/273.25~ 08/009.30 128.086 0.31
SE 3 (MS) 04/361 ~ 05/010 331.560 0.65
SE 4 (AS) 05/087 ~ 05/101 333.274 0.24
For the six selected tremor signals with azimuths ranging
from ~160 to ~200 degrees, the standard deviation of azimuth
estimates is mostly below 0.2 degrees. If compared with the
modelled result shown in Fig. 3 in the reference [Li and
Gavrilov, 2006], such small values of the standard deviation
correspond to only a few metres deviation of the HA01
hydrophones from their nominal position. The standard de-
viation values of azimuth estimates of the four earthquake
events, including two events from the Southern Ocean and
another two from the main shock and aftershock of the Great
Sumatra-Andaman Earthquake, are larger than those values
from the tremor events. This is expected considering that the
dimension of the region of those seismic events is larger than
that of ice-related events. If we attribute the azimuth variation
due to the dimension of seismic events to that from the
hydrophones' deviation, even in the extreme scenario as in
the mainshock of great Sumatra-Andaman Earthquake, the
SD of azimuth measurement is only as much as 0.65 degree,
which correlates with slightly over 10 m SD of horizontal
motion for each HA01 hydrophone. Therefore based on the
analysis of SD value of measured azimuth from both tremor
and earthquake events shown in Table 1, it can be concluded
that the random bearing error of HA01 is less than half a
degree, which is consistent with our previous preliminary
result [Gavrilov and Li, 2007].
SYSTEMATIC BEARING ERROR ESTIMATION
Based on a simple geometric model described in Appendix 1,
the horizontal deviation of the hydrophones relative to their
nominal position in Cartesian coordinate system can be ex-
pressed by Eq. (A8), providing a number of calibration
events with their exact coordinate.
Blackman proposed some experiments aimed at calibrating
Table 2.
The sources of underwater acoustic explosions and their shot times, coordinates, shot depths, original and measured azi-
muths from HA01, inverted group velocities and the azimuth residual values. The azimuth residual is the value of measured azimuth
minus predicted azimuth using HA01 receivers’ mooring coordinates.
Sources of
Explosion
Shot time
[year/day/hour]
Coordinates
[latitude longitude]
Shot
Depth
(m)
Original
Azimuth
[
o
]
Group
Velocity
(km/s)
Measured
Azimuth
[
o
]
Azimuth
Residual
[
o
]
A6 SUS3 03/149/04.3394 [-22.0848 72.7422] 915 278.18 1.472 278.62 0.44
A7 SUS3 03/151/11.8699 [-18.4374 80.9182] 915 290.38 1.466 291.69 1.31
A8 SUS2A 03/152/09.2333 [-17.1759 83.6751] 610 295.13 1.467 296.11 0.98
A10 SUS3 03/158/03.9227 [-12.2133 96.7966] 915 320.85 1.464 321.69 0.84
A11 SUS 03/160/00.6261 [-13.1980 104.6944] 915 336.05 1.466 337.52 1.47
Bengal Bay 1 04/126/15.4678 [10.14 89.07] - 327.33 1.466 327.94 0.61
Bengal Bay 2 04/126/16.2794 [10.01 89.50] - 327.75 1.467 328.41 0.67
Proceedings of ACOUSTICS 2008 24-26 November 2008, Geelong, Australia
Acoustics 2008 3
the CTBT hydroacoustic stations in the Indian Ocean and
some of them have been implemented in the past few years
[Blackman et al, 2003, 2004, 2005 and 2007]. Due to high
transmission loss and some data missing in the HA01 re-
cordings, only signals from 5 SUS explosions, made in 2003
during the cruise of R/V Melville across the Indian Ocean
from Cape Town, South Africa to the Cocos Islands [Black-
man et al, 2003], were detected at HA01 with sufficient SNR
to be used for bearing calibration. The recordings of two
strong explosions made in the Bay of Bengal on May 5, 2004
at known coordinates [Roger et al, 2005] were also used for
calibration. The sources of the seven calibration events, and
their shot times, coordinates, shot depths and azimuths from
HA01 are shown in the first five columns of Table 2. All of
these explosions were made in deep water regions and the
signals from these explosions underwent multi-path propaga-
tion in the SOFAR acoustic channel. To account for multi-
path propagation effects, a Progressive Multi-Channel Corre-
lation (PMCC) method [Cansi, 1995] was used for measuring
the signal travel time difference to the HA01 hydrophones
needed for slowness and back-azimuth estimates by the plane
wave fitting algorithm. In the PMCC method, the correlation
of signals at three receivers i, j, k, is calculated within sliding
windows and in a series of frequency bands to obtain the
consistency of the following criterion:
0
=
+
+
kijkijijk
tttr
(3)
Where
ij
t
is the time delay between the arrivals of a signal
at receivers i and j. Due to the background noise and the
finite sampling rate, the consistency condition in Eq. (3)
might have slight deviation from zero for fully coherent sig-
nals. Therefore we set a threshold of the consistency of 0.02 s.
Signals filtered in different frequency bands were considered
to be suitable for azimuth estimation, if the consistency crite-
rion did not exceed this threshold. The waveform of the cali-
bration event arrivals has a relatively short and sharp peak
and, therefore, a single time window of about one second
long was selected for the correlation analysis, rather than a
series of sliding windows. The passband of 20 Hz was se-
lected for the sliding frequency window, and was applied in
the frequency range from 10 Hz to 70 Hz, thus excluding the
frequency bands where the coherence of background noise
was high. As additional criteria for signal acceptance, the
cross correlation coefficient was tested to be at least 0.5 and
the group velocity estimates to be within 1.40 - 1.50 km/s.
The inverted group velocities, azimuths and the difference of
the measured and actual azimuths are shown in the last three
columns of Table 2. Note that the measured back-azimuths to
all seven calibration events have a small clock-wise deviation
from their actual values.
Fig. 1 shows the HA01 triplet patterns before and after the
relative coordinate calibration in the Cartesian coordinate
system. The size of the calibrated triplet pattern is inversely
proportional to the sound speed in the geometric model as
demonstrated in Appendix 1. To draw the corrected triplet
pattern shown in Fig. 1, the sound speed was set to be the
mean value of the inverted group velocities given in the 6th
column of Table 2. After calibration, the shape of the HA01
triplet appeared to be anti-clockwise rotated relative to the
original pattern. Using the original and calibrated HA01 trip-
let coordinates, the systematic bearing error was calculated as
a function of azimuth, which is shown in Fig. 2. The system-
atic error is slightly azimuth dependent and the average
clockwise deviation is around 0.8 degree.
-2 -1.5 -1 -0.5 0
-1
-0.5
0
0.5
1
1.5
x, km
y, km
Figure 1. The patterns of the HA01 triplet in the Cartesian
coordinate system before and after correction. The blue line
represents the pattern based on the relative positions of the
moorings determined during deployment; the red line is the
result obtained after the relative coordinate calibration. The
coordinates of hydrophone one were set as the reference posi-
tion.
Figure 2. Systematic bearing error as the function of azimuth
calculated based on the original and calibrated relative coor-
dinates of HA01 triplet.
AN ANTARCTIC ICEBERGE COLLISION
OBSERVATION
A series of harmonic tremor signals from late Julian day 260
till middle of Julian day 262 in 2003 was observed at HA01.
The back-azimuth to these events was calculated before and
after correction of the HA01 relative triplet coordinates,
which is shown in Fig. 3. One can see the calibration offset
of around 1.2
0
from the original bearing estimates. Over the
1.5-day observation period the mean value of calibrated azi-
muths to the observed tremor events varied gradually from
163
0
to 163.7
0
with the standard deviation of 0.2
0
.
0 50 100 150 200 250 300 350
0
0.2
0.4
0.6
0.8
1
Azimuth, degree
Syst ematic error, degree
1
2
3
Proceedings of ACOUSTICS 2008 24-26 November 2008, Geelong, Australia
4 Acoustics 2008
An investigation of Antarctic iceberg activity for this time
period and the area on the Antarctic continental shelf that
corresponded to the measured azimuth was conducted using
the Antarctic Iceberg Tracking Database
(http://www.scp.byu.edu/data/iceberg/database1.html). The
observed tremor signals are believed to be generated by colli-
sions of iceberg C008 with the ice shelf off Victor Bay. This
iceberg has been tracked from the middle of 1999 to early
2008 using satellite images, during which time it has drifted
along almost half of the Antarctic coastline from the Com-
monwealth Bay to the Weddell Sea. Fig. 4 shows the loca-
tions of iceberg C008 at different times and the bearing lines
(bars) from HA01 drawn for the original and corrected azi-
muth estimates as shown in Fig. 3. Remarkably, the back-
azimuth to these events estimated after correction of the
HA01 triplet position, indicates exactly the part of the ice
shelf edge that the iceberg C008 drifted by and most likely
collided with on Julian day 261, whereas the back-azimuth
derived from the original position of HA01 indicates at the
region where C008 was observed on Julian day 268. Accord-
ing to the overall variation of the azimuth and the duration of
this series of tremor signals, the total range that iceberg C008
had drifted, scraping the ice shelf, was about 35.6 km with an
average drifting speed of 0.93 km per hour.
This observation is further evidence for the systematic bear-
ing error.
EFFECT OF HORIZONTAL REFRACTION ON
BEARING ESTIMATION
To examine all possible errors in locating remote underwater
acoustic events by the CTBT stations, it is necessary to inves-
tigate the effect on the bearing estimation of horizontal re-
fraction of sound propagation in the ocean. Both large-scale
spatial variations of oceanographic characteristics and
changes in the bottom topography can induce horizontal re-
fraction [Jensen et al, 2000; Doolittle et al, 1988]. In this
study, we followed the computational procedure proposed for
the analysis of the Perth-Bermuda propagation experiment
results [Heaney et al, 1991]. It involves the combination of an
adiabatic mode theory in the vertical dimension and a ray
theory in the horizontal dimension and, therefore, takes into
account horizontal refraction of individual modes due to both
transverse sound speed gradients and bottom interaction over
the continental slopes and sea mounts. The ray model was
constructed on the surface of the Earth represented by an
ellipsoid of rotation and expressed in terms of the parame-
ters
φ
,
λ
, and
, where
φ
and
λ
are the latitude and lon-
gitude respectively, and
the azimuth angle measured
clockwise from the north. The ray equations on an ellipsoid
are:
)(/cos
φµαφ
=
& (4a)
Figure 3. The back-azimuth to a series of harmonic tremor
signals received at HA01 as the function of signal arrival
times. Blue and red dots represent the azimuth measured
using original and corrected relative coordinates of the
HA01
triplet respectively.
Figure 4. A satellite image showing the location iceberg C008 drifted along the ice shelf off Victor Bay on Julian days 261 and 268
in 2003. The blue and red bars are the regions seen along the back-azimuths to the tremor signals measured before and after correc-
tion of the HA01 triplet position respectively.
Proceedings of ACOUSTICS 2008 24-26 November 2008, Geelong, Australia
Acoustics 2008 5
φφαλ
cos)(/sin v=
&
(4b)
n
vv
κ
λφφ
α
φφµ
α
φ
φ
α
α
log)
cos)(
cos
)(
sin
(tan
)(
sin
=
&
(4c)
where
n
κ
are the horizontal wavenumbers of modes and the
variables µ and ν are:
2/3222
)sin1/()1()(
φεεφµ
=
eq
r
2/122
)sin1()(
φεφ
=
eq
rv
(5)
and
eq
r
and
ε
are the equatorial radius and eccentricity of
the Earth respectively. The last term in Eq. (4c) accounts for
distortion of the ray paths due to gradients of the horizontal
wavenumber
n
κ
based on the Snell's law. If this term is ne-
glected, the solutions of Eq. (4) are geodesics on the ellipsoid
[Bomford, 1980, P649].
The modal horizontal wavenumbers were calculated using
the KRAKEN program [Porter and Reiss, 1984] on a hori-
zontal grid with the grid size of half degree. The sound speed
profiles were derived from climatology salinity and tempera-
ture data gridded to 1-degree resolution in the World Ocean
Atlas 2005 [Locarnini et al and Antonov et al, 2006] and then
interpolated into a half-degree grid. The bathymetry data
were taken from the ETOPO2 Global 2-Minute Gridded Ele-
vation Data
(http://www.ngdc.noaa.gov/mgg/fliers/01mgg04.html). The
system of ordinary differential equations Eq. (4) can be
solved using a 4-th or 5-th order Runge-Kutta method [Wil-
liams et al, 2007]. During the integration process, the grid
size of the modal wavenumbers was set to be equal to the
maximum integration step to reduce numerical integration
errors.
Figure 5. The map projection of the mode 1 horizontal
wavenumber matrix in the region of Indian and Southern
Oceans. The frequency is at 20 Hz and climatological data
are taken for the winter season.
Fig. 5 shows the map projection of the matrix of mode 1
wavenumbers at 20 Hz in the winter season. As can be seen,
in deep water regions, the wavenumber has strong depend-
ence on the sound speed profile rather than bathymetry. In
the Indian Ocean north of the Antarctic Convergence Zone
(ACZ), the wavenumber is almost uniform except for the
region around the equator. The strongest gradient of the
wavenumber is in the ACZ, across which the sound speed
profile evolves from the polar upward-refracting shape in
Southern Ocean to the temperate ocean shape with a deep
SOFAR channel in the temperate ocean. The water depth is
much shallower over the Antarctic continental shelf, and
hence the modal wavenumber undergoes noticeable depend-
ence on depth.
Fig. 6 gives the one-degree resolution map of bearing devia-
tion at HA01 due to horizontal refraction in the Indian Ocean,
across the ACZ and in the Southern Ocean region over the
Antarctic continental shelf, numerically modelled using the
horizontal wavenumber matrix shown in Fig. 5. The bearing
deviation from the geodetic azimuth at each grid point was
calculated as the residual
21
θ
θ
θ
=
, where
1
θ
is the true
azimuth to the grid point as seen from HA01 and
2
θ
is the
back azimuth to the end point of the refracted ray derived
from Eq.(4) for the same launch angle. The ray has the same
length as the geodesic. The bearing deviation induced by
horizontal refraction for the most part of Indian Ocean region
north of ACZ does not exceed 0.2
0
because of small gradients
of the wavenumber. Strong wavenumber gradients across the
ACZ introduce considerable azimuth dependent deviation of
the acoustic propagation path to the locations within the ACZ
and south of it. The azimuth deviation in the Southern Ocean
south of the ACZ has both negative values in the western
region and positive values in the eastern region with a transi-
tion zone around the azimuth of about 203
0
from HA01,
along which the propagation path is nearly perpendicular to
the ACZ.
Figure 6. The map projection of bearing deviation at the
HA01 receiving station for noise sources located in the In-
dian and Southern Oceans. Deviation is due to horizontal
refraction calculated for the horizontal wavenumber matrix
shown in Fig. 5.
The dependence of the wavenumber gradient on mode num-
ber and frequency was investigated for the Southern Ocean
region. It is found that 1) for a certain mode, the gradient of
the horizontal wavenumber
n
κ
across the ACZ increases
with frequency and 2) for a fixed frequency, the gradient
decreases with mode number. Such dependence takes place
because higher order modes at lower frequencies penetrate
deeper in the water column and hence they are less sensitive
to rapid change in the sound speed in the upper water layer
across the ACZ.
CONCLUSIONS
In this paper, bearing errors of the HA01 station for long-
range low-frequency hydroacoustic monitoring are consid-
ered to contain a systematic component in addition to the
Proceedings of ACOUSTICS 2008 24-26 November 2008, Geelong, Australia
6 Acoustics 2008
random one due to motion of the HA01 hydrophones. The
systematic component results from the limited accuracy of
positioning of the HA01 moorings that was performed after
deployment. An analysis of a number of long-lasting, low-
frequency underwater harmonic tremor signals and seismic
events, as well as some signals from explosions made in the
Indian Ocean at known locations was conducted for bearing
calibration of the HA01 station. It is demonstrated that the
random component of HA01 bearing errors due to motions of
the receivers is below half a degree, which is in agreement
with the estimates made before, while the systematic compo-
nent is around 0.8 degrees. The horizontal refraction effect
also contributes considerably to HA01 bearing errors for the
Ocean region within and beyond the ACZ. The effect is azi-
muth dependent. All these bearing errors must be taking into
account when locating ice events using CTBT hydroacoustic
stations.
APPENDIX A
A simple geometric model for HA01 bearing calibra-
tion
This analysis is made in the Cartesian coordinate system by
projecting coordinates from the Geodetic Earth Model. Under
the condition that the underwater explosions are far enough
from the hydroacoustic station, the propagation paths can be
represented in the horizontal coordinates x and y ignoring the
depth difference. Let the coordinates of mooring 1 of the
HA01 triplet, (
0
1
x
,
0
1
y
), be a reference position for both
original and corrected systems, and the relative coordinates
of moorings two and three are (
0
2
x
,
0
2
y
) and (
0
3
x
,
0
3
y
) re-
spectively. Let also the coordinates of the n-th underwater
explosive event be (
n
r
x
,
n
r
y
). The corrected relative coordi-
nates of moorings 2 and 3, (
2
x
,
2
y
) and (
3
x
,
3
y
) respec-
tively, have deviations (
2
x
δ
,
2
y
δ
) and (
3
x
δ
,
3
y
δ
) from the
original positions. The travel time differences
n
ij
T from the
nth explosion to a pair of hydrophones i and j, can be ex-
pressed as a function of the vector of deviations
),,,(
~
3322
yxyx
δδδδδ
=
T
as:
)
~
(
δ
n
ij
n
ij
fT =
i , j = 1,2,3 & i
j , (A1)
where the subscript T denotes the matrix transpose operation.
If the deviations of hydrophones from the original positions
are small compared with the dimension of the triplet, the
travel time differences
n
ij
T
can be expanded in a power series
of deviations from the original coordinates and only the first
two low-order terms of the expansion can be kept:
3
~
3
3
~
3
2
~
2
2
~
2
0
00
00
)()(
)()(
)
~
(
y
y
f
x
x
f
y
y
f
x
x
f
fT
n
ij
n
ij
n
ij
n
ij
n
ij
n
ij
δ
δ
δ
δ
δ
δ
δ
δ
δ
δδ
δδ
+
+
+
+=
(A2)
where the original deviations
),,,(
~
0
3
0
3
0
2
0
2
0
yxyx
δδδδδ
are
set to be a zero vector.
The formula for the residuals of the travel time difference can
be obtained from (A2):
3
~
3
3
~
3
2
~
2
2
~
2
00
00
00
)()(
)()(
)
~
(
y
y
f
x
x
f
y
y
f
x
x
f
fT
n
ij
n
ij
n
ij
n
ij
ij
n
ij
δ
δ
δ
δ
δ
δ
δ
δ
δ
δδ
δδ
+
+
+
=
(A3)
In the matrix notation, one can express (A3) as:
δ
~
=
nn
AY
(A4)
where the vector of the residual of the travel time difference
is
Y
n
= [
0
~
2121
δ
nn
fT
0
~
3131
δ
nn
fT
0
~
2323
δ
nn
fT
]
T
(A5)
and
n
A
is a 3×4 matrix of derivatives of the travel time dif-
ference. Therefore, for total N explosive events, the following
equation can be obtained:
δ
~
= AY
(A6)
where Y is a one-column vector with 3N elements, and A is a
3N×4 matrix:
Y = [Y
1
Y
2
Y
n
]
T
; A = [A
1
A
2
A
n
]
T
(A7)
Then the least square solution of
δ
~
can be obtained:
YAAA
TT 1
][
~
=
δ
(A8)
Assuming the sound speed along the propagation path to be
constant v km/s, the travel time differences between the
HA01 hydrophones from nth explosive events can be calcu-
lated as:
))()(
(
1
)
~
(
2
22
2
22
22
2121
yyyxxx
yx
v
fT
n
r
n
r
n
r
n
r
nn
δδ
δ
+
+==
(A9)
))()(
(
1
)
~
(
2
33
2
33
22
3131
yyyxxx
yx
v
fT
n
r
n
r
n
r
n
r
nn
δδ
δ
+
+==
(A10)
))()(
)()((
1
)
~
(
2
33
2
33
22
2
22232 3
yyyxxx
yyyxxx
v
fT
n
r
n
r
n
r
n
r
nn
δδ
δδδ
+
+==
(A11)
ACKNOWLEDGEMENTS
The authors thank Dr. David Jepsen of Geoscience Australia
for providing us with the HA01 acoustic data and Dr. Donna
K. Blackman of IGPP, Scripps Institution of Oceanography
for providing NBP0701 Hydroacoustics Project Cruise Re-
port.
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