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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

α

, where

φ

and

λ

are the latitude and lon-

gitude respectively, and

α

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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