Mode coherence at megameter ranges in the North Pacific
Ocean
Kathleen E. Wage
Department of Electrical and Computer Engineering, George Mason University, Fairfax, Virginia 22030
Matthew A. Dzieciuch and Peter F. Worcester
Scripps Institution of Oceanography, La Jolla, California 92093
Bruce M. Howe and James A. Mercer
Applied Physics Laboratory, University of Washington, Seattle, Washington 98195
共Received 27 April 2004; revised 22 July 2004; accepted 23 July 2004兲
This article analyzes the coherence of low-mode signals at ranges of 3515 and 5171 km using data
from the Acoustic Thermometry of Ocean Climate 共ATOC兲 and Alternate Source Test 共AST兲
experiments. Vertical line arrays at Hawaii and Kiritimati received M-sequences transmitted from
two sources: the 75-Hz bottom-mounted ATOC source on Pioneer Seamount and the near-axial
dual-frequency 共28/84 Hz兲 AST source deployed nearby. This study demonstrates that the
characteristics of the mode signals at 5171-km range are quite similar to those at 3515-km range. At
75 Hz the mode time spreads are on the order of 1.5 s, implying a coherence bandwidth of 0.67 Hz.
The time spread of the 28-Hz signals is somewhat lower, but these signals show significantly less
frequency-selective fading than the 75-Hz signals, suggesting that at the lower frequency the
multipaths are temporally resolvable. Coherence times for mode 1 at 75 Hz are on the order of 8 min
for the 3515-km range and 6 min for 5171-km range. At 28 Hz mode 1 is much more stable, with
a magnitude-squared coherence of greater than 0.6 for the 20-min transmission period. © 2005
Acoustical Society of America. 关DOI: 10.1121/1.1854551兴
PACS numbers: 43.30.Qd, 43.30.Re, 43.30.Bp 关AIT兴
I. INTRODUCTION
Broadband receptions at megameter ranges have a complicated structure, which consists of a series of multipath
arrivals. Tomographers typically separate these arrivals into
two categories for processing: the early steep-angle arrivals,
which are analyzed using ray models, and the energetic finale
arrivals, which are analyzed using normal-mode models. One
of the difficulties in using any of these arrivals for tomography is the fluctuations caused by internal waves. The internal
wave effects on the rays have been studied extensively using
the path-integral theoretical framework, summarized in the
monograph by Flatté et al.1 By comparison, the energetic
late arrivals associated with the low-order modes have received much less attention. Some theoretical aspects of mode
propagation in a fluctuating ocean have been developed in
the literature. Dozier and Tappert’s seminal work2,3 describes
a theoretical analysis of mode intensity statistics. They used
numerical simulations to verify their predictions. In several
papers4,5 Sazontov and others studied the impact of internal
waves on mode coherence. Their theoretical work focuses on
how mode coherence affects signal processing techniques
such as matched filtering6 and beamforming.7–9 In terms of
experimental results, Colosi et al.10 analyzed the reception
finale at 1000-km range using the SLICE89 data set. They
showed that the broadening of the transmission finale is due
to internal wave effects. Later, Colosi and Flatté11 studied
mode coupling for megameter range propagation using parabolic equation 共PE兲 simulations.
The Acoustic Thermometry of Ocean Climate 共ATOC兲
experiment provided a unique opportunity to study longJ. Acoust. Soc. Am. 117 (3), Pt. 2, March 2005
Pages: 1565–1581
range propagation in the North Pacific. In a series of
papers12–14 ATOC researchers used the resolved ray arrivals
at ranges of greater than 3 Mm for tomographic inversions.
Others considered the problem of source localization at
megameter ranges using ATOC transmissions.15 Several investigators have studied different aspects of the reception
finale. Dzieciuch et al.16 developed the turning-point filter to
analyze the transition between the ray-like and mode-like
arrivals in basin-scale transmissions. Colosi et al.17,18 used
data from the ATOC Engineering Test 共AET兲 to analyze intensity fluctuations in the late arrivals. In this issue, Colosi19
considers the statistics of the acoustic field in the finale region of the ATOC receptions. Wage et al.20 analyzed the
mode arrivals in the ATOC vertical array receptions at
3515-km range using short-time Fourier transform 共STFT兲
processing. The STFT analysis showed that the lowest ten
modes have complicated multipath arrival patterns characterized by frequency-selective fading and a high degree of temporal variability. Based on the ATOC measurements at 3515
km, Wage et al.20 concluded that the lowest ten modes have
roughly equal power and are incoherent, which agrees with
Dozier and Tappert’s theoretical predictions. Coherence
times of the peak mode arrivals were estimated to be on the
order of 5.5 min.
This paper extends the previous mode analysis by making use of the extensive data set compiled by the ATOC
project. In addition to the 3515-km range receptions discussed above, the ATOC experiment also recorded receptions
at a range of 5171 km using a second vertical line array
共VLA兲. The two VLAs recorded 9 months’ worth of recep-
0001-4966/2005/117(3)/1565/17/$22.50
© 2005 Acoustical Society of America
1565
tions from the ATOC 75-Hz source located on Pioneer Seamount off the California coast. In conjunction with ATOC,
another experiment called the Alternate Source Test 共AST兲
transmitted signals at two carrier frequencies simultaneously
over a period of 9 days.21 The ATOC VLAs recorded these
28-Hz and 84-Hz AST transmissions, facilitating a study of
mode fluctuations at different carrier frequencies. Since the
AST source was deployed close to the ATOC source, the
transmission ranges are comparable. One key difference between the two sources is that the ATOC source was bottommounted, while the AST source was deployed in the water
column in deeper water. The two sources excited the loworder modes differently.
The ATOC and AST experiments provide a wealth of
data with which to study the low-order mode arrivals for
different ranges, carrier frequencies, and source excitations.
Using these data sets, this paper characterizes the statistics of
the low-mode arrivals. Specifically, the analysis has three
objectives: 共1兲 to investigate how well the conclusions20
from the 3515-km range receptions apply to the receptions at
5171 km; 共2兲 to examine the dependence of the mode statistics on source frequency by comparing the ATOC and AST
transmissions; 共3兲 to assess whether there are measurable differences in the low-mode arrival patterns due to using
bottom-mounted vs suspended sources. Along with offering
some new insights into the nature of mode fluctuations at
megameter ranges, this paper presents a set of experimentally determined statistics that may be useful in designing
better signal processing for the finale and for evaluating stochastic models of mode scattering.
The outline of this article is as follows. Section II describes the relevant details of the ATOC and AST experiments and presents examples of receptions at each range and
carrier frequency. This section also discusses the observed
differences between receptions from the bottom-mounted
ATOC source and the suspended AST source. Simulations
indicate that these differences are due to bathymetric interactions near the seamount source. Section III describes the
mode-filtering techniques used for the ATOC and AST data
sets and illustrates them with examples of the mode 1 estimates for a set of sample receptions. The final part of Sec. III
discusses the source and receiver motion corrections that
must be applied to the data prior to mode coherence calculations. Section IV presents the mode statistics estimated
from the data and compares them to simulation results. Section V concludes the paper with a summary of results and a
discussion of how they might be applied in future research.
II. EXPERIMENT DESCRIPTION
In 1995–1996 low-frequency sources located off the
California coast transmitted broadband pulses that were recorded by a network of receivers in the North Pacific, including two VLAs and a number of U.S. Navy Sound Surveillance System 共SOSUS兲 arrays.22 This paper focuses on the
VLA data sets since those are the most appropriate for analyzing the axial mode arrivals. Figure 1 shows the geodesic
paths to the two 40-element VLAs, which were located near
Hawaii and Kiritimati. The following sections review the
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FIG. 1. Geodesic paths from the Pioneer seamount source to the ATOC
vertical line arrays at Hawaii and Kiritimati.
relevant details of the ATOC and AST transmissions and describe the receptions used in the mode analysis.
A. Transmissions
The ATOC source was located on Pioneer Seamount 共indicated in Fig. 1兲 at a depth of approximately 939 m. Ranges
from the ATOC source to the Hawaii and Kiritimati VLAs
were 3515 and 5171 km, respectively. The Pioneer source
transmitted phase-encoded pseudorandom sequences at a
center frequency of 75 Hz, with a 3-dB bandwidth of 37.5
Hz. With 2 cycles/digit, the 1023-digit M-sequence provided
a time resolution of 26.67 ms. Each transmission consisted of
44 repetitions of the 27.28-s M-sequence. The ATOC Marine
Mammal Research Program determined the transmission
schedule. During specified periods, the source transmitted for
20 min every 4 h.
Transmissions from Pioneer Seamount were temporarily
discontinued during the AST, which took place in late June
and early July of 1996. A source was deployed from the M/V
INDEPENDENCE to a depth of 652 m at a position 7 nm southwest of the ATOC Pioneer source. Ranges from the AST
source to the Hawaii and Kiritimati VLAs were 3502 and
5157 km, respectively. This suspended source transmitted
pseudorandom sequences at two carrier frequencies simultaneously: 28.0425 Hz 共fundamental兲 and 84.1276 Hz 共third
harmonic兲. Each had a 3-dB bandwidth of approximately 10
Hz. With 3 and 9 cycles/digit, respectively, these signals provided a travel-time resolution of 106.98 ms. Similar to the
ATOC Pioneer transmissions, each M-sequence was 27.28 s
long. Most transmissions were for 20 min, occurring at intervals of 4 h, but there were five longer 共40-min兲 transmissions over the course of the experiment. See Worcester
et al.21 for additional details about the AST transmissions.
B. Receptions
The VLAs at Hawaii and Kiritimati each consisted of 40
hydrophones with 35-m spacing. To conserve disk space over
the long deployment, the hardware computed four-period avWage et al.: Mode coherence at megameter ranges
FIG. 2. Receptions at the Hawaii array during the ATOC and AST experiments. Each plot represents one four-period average recorded by the receivers. The top plot shows a reception of a 75-Hz ATOC source transmission.
The bottom two plots are examples of AST receptions at the 84- and 28-Hz
center frequencies. Although the plots have different starting times, the timewidth of all plots is 6 s. Note that the bottom half of the Hawaii array failed
prior to the AST experiment; thus, the AST signals were received by only
the top half of the VLA. In addition, the tenth hydrophone from the surface
on the Hawaii VLA failed prior to the AST experiment.
erages of the M-sequences broadcast by the sources. The
VLAs recorded ten of these four-period averages 共spanning
18.2 min兲 for each regular ATOC/AST transmission and recorded 20 four-period averages for the long transmissions
from the AST source. The position of the VLAs was tracked
using a long-baseline acoustic navigation system. Navigation
data were recorded immediately before and after each transmission, and the mooring motion for each sensor was derived from these measurements. For additional information
on the ATOC receiver hardware, see the paper by Howe
et al.22
After the arrays were recovered, the time series for each
of the four-period averages was complex demodulated and
matched filtered. Preprocessing for all the receptions discussed in this paper consists of demodulation and pulse compression, followed by application of the mooring corrections
共assuming a nominal wave number兲. Figure 2 shows representative examples of receptions at the Hawaii array during
the ATOC and AST experiments. The top plot shows the
received pressure field as a function of time and depth for
one of the four-period averages of an ATOC 75-Hz source
transmission. The bottom two plots show examples of fourperiod averages for the 84- and 28-Hz signals from the AST
J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
source at the same array. Since the bottom half of the Hawaii
VLA failed after approximately 5 months, the AST transmissions were recorded only on the top half of this array. In
addition, hydrophone 10 at the Hawaii VLA appears to have
failed prior to AST. Note that the starting time for the ATOC
reception plot is different from that of the AST plots, due to
the fact that the path to Hawaii from the ATOC source is
slightly longer than for the AST source. All three of the plots
in Fig. 2 display a 6-s time interval.
The time series in Fig. 2 are characteristic of receptions
at megameter ranges. The familiar ‘‘accordion’’ pattern visible in the early part 共2371.5 to 2373 s兲 of the 75-Hz reception shown in Fig. 2 is due to the constructive interference of
the higher-order modes, which travel faster than the lower
modes in deep water. The diffuse finale region 共2373.5 to
2375.5 s兲 consists of the lower-order mode arrivals. The AST
receptions at 84 and 28 Hz exhibit similar features, with a
few notable differences. First, the difference in the time resolution 共bandwidth兲 of the ATOC and AST sources, 26.67 vs
106.98 ms, is evident in the plots. Second, comparing the
three plots in Fig. 2 indicates that the AST receptions have a
sharper final cutoff, which appears to be tapered in depth,
i.e., the shallower hydrophones have earlier cutoffs than the
deeper hydrophones. The following section postulates an explanation for this feature. Third, the 28-Hz AST reception is
dominated by a strong arrival with spatial characteristics
similar to mode 1, whereas the finale regions of the 75- and
84-Hz receptions are characterized by much more complicated interference patterns.
Figure 3 shows representative examples of receptions at
the Kiritimati array during the ATOC and AST experiments.
For this VLA the tenth hydrophone from the surface appears
to have calibration problems, at least for the 75- and 84-Hz
receptions. Unlike at the Hawaii array, the 75-Hz reception at
Kiritimati is very similar to the 84-Hz reception. Both the
75-Hz data and the 84-Hz data show a sudden cutoff, with
little tapering in depth, i.e., the shallow and deep sensors cut
off almost simultaneously. Similar to Hawaii, the 28-Hz reception shown in Fig. 3 has a strong arrival with spatial
characteristics similar to mode 1.
The ATOC and AST data sets are extensive, including
receptions over a 9-month period from late December 1995
until the VLAs were recovered in September 1996. Table I
summarizes important information about the receptions in
the data sets for each of the experiments. The receptions
from the Pioneer source at the Hawaii and Kiritimati arrays
are denoted by HVLA75 and KVLA75, respectively. For this
analysis the HVLA and KVLA data sets were divided into
groups of receptions, clustered over 2- to 4-day periods. During these periods the Pioneer source nominally transmitted
for approximately 20 min every 4 h. The Kiritimati data set
includes more groups than Hawaii because this study ignores
HVLA75 receptions after yearday 509, which is when the
lower half of the Hawaii VLA failed, making mode filtering
much more difficult.
The AST experiment took place over 9 days from 30
June 1996 to 8 July 1996. During this time period 34 good
receptions were recorded on each of the arrays. Of those, five
were for the longer 40-min transmissions; the remaining
Wage et al.: Mode coherence at megameter ranges
1567
FIG. 3. Receptions at the Kiritimati array during the ATOC and AST experiments. Each plot represents one four-period average recorded by the
receivers. The top plot shows a reception of a 75-Hz ATOC source transmission. The bottom two plots are examples of AST receptions at Kiritimati
for the 84- and 28-Hz center frequencies. Note that the top plot has a different starting time than the bottom two plots; the timewidth of all three
plots is 6 s.
were recordings of standard 20-min transmissions. The
28-Hz carrier frequency receptions at each of the arrays are
denoted by HVLA28 and KVLA28, and the 84-Hz transmissions by HVLA84 and KVLA84, respectively.
Figure 5 shows the actual bathymetry along the path
from Pioneer to the Hawaii VLA. At its steepest, the slope of
the Pioneer-Hawaii path is approximately 20 deg. To examine the effects of the source bathymetry on long-range propagation, consider two broadband simulations implemented using the RAM PE code.24 The first simulation places the
source on the seamount, using the sloping bathymetry shown
in Fig. 5. The second simulation keeps the source at the same
depth, but removes the seamount, i.e., the bathymetry is as
shown by the dashed line in Fig. 5. In both simulations the
background sound speed is taken from temperature and salinity profiles for winter, derived from the World Ocean
Atlas.25,26 The background environment in both cases is perturbed by internal wave fluctuations at 21 Garrett-Munk
strength, generated using the method of Colosi and Brown.27
The center frequency for these simulations is 75 Hz.
The upper set of plots in Fig. 6 shows the results for the
simulation using the measured Pioneer seamount bathymetry.
The top plot in this set is the synthesized pressure time series
at Hawaii. Note that the finale region is not clearly defined
by a sharp cutoff, similar to the ATOC reception discussed
above. Below the pressure time series is the modal time series obtained by projecting the field 共finely sampled in depth兲
onto the mode functions at the receiver. In the seamount
simulation, modes 1–5 have strong arrivals around 2375 s,
followed by almost 1 s of weaker arrivals. The bottom set of
plots in Fig. 6 shows the results of the simulation with the
seamount removed. In contrast with the seamount simulation, the time series for the suspended source exhibits a very
sharp cutoff. Modes 1–5 again have strong arrivals around
2375 s, but they are not followed by any weaker arrivals.
Note that the finale region of the suspended source simulation resembles the finale of the AST reception shown in Fig.
2. While more accurate modeling of the near-source bathymetric interaction could be done using a propagation code
that incorporates shear effects 共such as the study by Sperry
et al.28兲, these simulations provide strong evidence that the
more diffuse finale region seen in the ATOC Hawaii receptions is due to bottom interaction near the Pioneer seamount
source.
C. Comparison of receptions from bottom-mounted
and suspended sources
As noted above, the AST receptions at Hawaii exhibit a
much sharper cutoff as compared to the ATOC 75-Hz receptions. This effect is even more pronounced when the signal is
averaged over the entire recording period 共⬇18.2 min or 40
M-sequences兲. Figure 4 shows the 40-period average for an
ATOC reception and an AST reception. The ATOC reception
from the Pioneer source clearly shows between 0.5 to 1.0 s
of weaker arrivals following the most energetic axial arrivals, whereas the AST reception has a very sharp cutoff. Since
the source frequencies and ranges of these two receptions are
quite similar, the primary difference is the bathymetry at the
source. Heaney23 attributes the weaker arrivals in the ATOC
finale to bathymetric interaction near the seamount source.
The following simulations support this theory and predict the
effects that should be evident in the low order mode estimates for the ATOC and AST HVLA receptions.
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J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
III. MODE PROCESSING
This section describes the mode-filtering methods used
to analyze the ATOC and AST data sets. Mode estimation is
a well-known problem that numerous authors have
addressed.29–32 The mode processor designed for the ATOC/
AST receptions uses a standard pseudoinverse solution for
the spatial filter, coupled with a Fourier decomposition to
facilitate broadband processing. Section III A summarizes the
parameters for each of the mode filters 共designed for the six
data sets兲 and discusses other implementation details. Section III B presents the results of mode processing for each of
the sample receptions given in Sec. II. These examples illustrate some of the essential characteristics of the mode arrivals at the three source frequencies and two ranges. Finally,
Sec. III C describes how the source motion and mooring moWage et al.: Mode coherence at megameter ranges
TABLE I. Summary of the ATOC and AST data sets. Yeardays are referenced to 1995.
No. receptions
Data set
Group
Yeardays
Month
20 min
HVLA75
1
2
3
4
5
6
7
8
9
10
11
12
13
362–366
370–372
376 –377
394 –396
407– 409
427– 428
432– 436
465– 467
470– 473
477– 481
485– 489
499–502
506 –509
Dec.
Jan.
Jan.
Jan.
Feb.
Mar.
Mar.
Apr.
Apr.
Apr.
Apr.–May
May
May
20
10
10
16
9
11
20
14
14
24
14
13
13
40
40
40
40
40
40
40
40
40
40
40
40
40
KVLA75
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
362–366
370–372
376 –377
394 –396
407– 409
427– 428
432– 436
465– 467
470– 473
477– 481
485– 489
499–502
506 –509
534 –537
540–542
566 –568
581–584
590–593
597–598
604 – 606
612– 612
Dec.
Jan.
Jan.
Jan.
Feb.
Mar.
Mar.
Apr.
Apr.
Apr.
Apr.–May
May
May
Jun.
Jun.
Jul.
Aug.
Aug.
Aug.
Aug.
Sep.
20
11
6
9
11
11
0
2
14
19
15
18
16
13
8
11
19
23
11
12
5
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
HVLA28
547–556
Jun.–Jul.
29
5
19
KVLA28
547–556
Jun.–Jul.
29
5
39
HVLA84
547–556
Jun.–Jul.
29
5
19
KVLA84
547–556
Jun.–Jul.
29
5
39
tion corrections were applied. These corrections are crucial
for the estimation of mode coherence statistics, as discussed
in Sec. IV.
Mode processing assumes the pressure measured by an
array consists of a weighted sum of modes plus noise, i.e.,
for a vertical array
共1兲
where p is the pressure vector at frequency ⍀ and range r, ⌽
is the matrix of sampled mode shapes, a is the vector of
narrow-band mode amplitudes, and n is the vector of observation noise. For a broadband source, the measured vector
time series ⌿(r, ␶ ) can be written as the Fourier synthesis of
the narrow-band components defined in Eq. 共1兲
J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
⌿共 r, ␶ 兲 ⫽
冕
⍀
No. hydrophones
共 ⌽共 r,⍀ 兲 a共 r,⍀ 兲 ⫹n共 ⍀ 兲兲 e j⍀ ␶ d⍀.
共2兲
The goal of mode filtering is to determine estimates of the
Fourier mode amplitudes, a(r,⍀), or the corresponding
mode time series ␣共␶兲
A. Broadband mode processors for ATOC and AST
signals
p共 r,⍀ 兲 ⫽⌽共 r,⍀ 兲 a共 r,⍀ 兲 ⫹n共 ⍀ 兲 ,
40 min
␣共 r, ␶ 兲 ⫽
冕
⍀
a共 r,⍀ 兲 e j⍀ ␶ d⍀.
共3兲
In general the mode shapes vary as a function of frequency, meaning that mode filtering requires a Fourier decomposition prior to the spatial processing to avoid problems
with mode shape mismatch.33 The simplest solution34 –36 to
the broadband mode filtering problem is to take a fast Fourier transform 共FFT兲 of the entire reception, do narrow-band
spatial processing in each FFT bin, and inverse transform to
obtain the estimated modal time series. An alternative
approach20 is to use short-time Fourier transform 共STFT兲
Wage et al.: Mode coherence at megameter ranges
1569
FIG. 4. Forty-period averages for receptions at the Hawaii VLA. The top
plot shows an ATOC 75-Hz reception and the bottom plot shows an AST
84-Hz reception.
processing, where the signals are bandpass filtered and mode
filtering is done for each band. The result is time-varying
spectra for each of the modes. Each approach has its advantages. The FFT processor produces a single broadband time
series for each mode that is useful for analyzing the multipath arrival patterns. Each individual FFT bin is comparable
to the output of a narrow-band PE run, which facilitates comparisons between experiment and predication without requiring computationally intensive broadband simulations. The
main advantage of the short-time Fourier approach is that it
permits a study of the characteristics of individual arrivals
within a multipath pattern 共rather than combining all the
paths together兲, provided they are temporally separable by
the STFT processor. The ATOC and AST analysis relies on
both techniques.
Once the signal is separated into bins, mode estimation
reduces to a classical linear inverse problem. The pseudoinverse 共PI兲 filter31 is a standard solution where the estimated
mode amplitudes are defined to be the product of pseudoinverse of the sampled modeshape matrix and the measured
pressure vector. A more general form37 allows for diagonal
loading of the pseudoinverse to accommodate ill-
FIG. 5. Bathymetry for the PE simulations. The solid line is the measured
bathymetry along the track from Pioneer seamount to the Hawaii VLA. The
dashed line is the bathymetry used in the suspended source simulations 共the
seamount is removed兲. A large dot indicates the source location.
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J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
1
FIG. 6. Broadband PE simulations for winter environment perturbed by 2
Garrett–Munk-strength internal waves. The seamount simulation uses the
measured bathymetry for the Pioneer seamount ATOC source. The midwater column simulation uses a source at the same depth as Pioneer seamount, but the seamount is not included in the bathymetry. Top plot for each
simulation is the pressure time series: 20 log10兩 p(t) 兩 ; the bottom plot is the
associated time series for the first 20 modes.
conditioning due to inadequate spatial sampling of the
modes, i.e.,
â⫽ 共 ⌽T ⌽⫹ ␥ I兲 ⫺1 ⌽T p,
共4兲
where â is the vector of estimated mode amplitudes and I is
the identity matrix. The parameter ␥ controls the amount of
diagonal loading.
The spatial resolution of the PI mode filter is determined
by how well the array samples the modes. Figure 7 shows
plots of the mode shapes38 at the Hawaii and Kiritimati arrays, respectively. Each plot shows the shapes at two frequencies: 75 Hz 共ATOC center frequency兲 and 28 Hz 共lower
AST center frequency兲. The plots also indicate the hydrophone depths for each array. Note that the arrays span more
of the first 10 modes at 75 Hz than they do at 28 Hz.
Although the continuous modeshape functions are orthogonal, the sampled modeshapes may not be. The sampled
mode shape correlation matrix, defined by ⌽T ⌽, indicates
how orthogonal the sampled modes are and predicts how
well the array can separate them. Figure 8 shows the sampled
mode shape correlation matrices at 28- and 75 Hz for the two
array locations. Each of the 40-element VLAs was designed
to estimate the first 10 modes at 75 Hz. The mode shape
correlation matrix for HVLA75 confirms that the sampled
mode shapes up to mode 10 are effectively orthogonal;
above mode 10, neighboring modes are correlated. This is in
sharp contrast to the 28-Hz results for the Hawaii array,
Wage et al.: Mode coherence at megameter ranges
FIG. 8. Sampled mode shape correlation matrices for the Hawaii and Kiritimati VLAs. Color scale is in dB. The Hawaii VLA 75-Hz results are for
the full 40-element array, whereas the 28-Hz results are for the 19 elements
that were operational during the AST experiment. The Kiritimati VLA results are for the 39-element VLA 共i.e., a 40-element VLA with one bad
phone兲.
FIG. 7. Mode shapes at 75- and 28 Hz for the Hawaii and Kiritimati VLAs.
The hydrophone depths are shown at the far right. The ‘‘⫹’’ symbols indicate phones that remained operational throughout the experiment, and the
‘‘*’’ symbols indicate phones that failed prior to the AST part of the experiment.
which indicate a high degree of correlation between neighboring modes. The correlation is caused by inadequate sampling, due to the fact that only 19 hydrophones were operational during the AST experiment. For the Kiritimati array,
the 75-Hz results are similar to those at Hawaii, with modes
up to 10 being largely uncorrelated. The small amount of
crosstalk between modes 3, 4, and 5 is due to the nonuniform
sampling caused by the failure of hydrophone 10. At 28 Hz
the figure shows that modes up to 4 are approximately orthogonal, as sampled by the 39-element Kiritimati VLA.
As Fig. 8 indicates, modes 1–10 of the ATOC 75-Hz
signal are effectively orthogonal as sampled by both of the
arrays; thus, they can be estimated with a standard PI filter
with no diagonal loading. Some diagonal loading is required
to process the Kiritimati 28-Hz data set since the VLA does
not sample these lower-frequency modes as well. In addition,
diagonal loading is required to stabilize the mode filters designed for the degraded Hawaii VLA in order to compensate
for the failure of 21 of the 40 hydrophones. Table II summarizes the parameters for the PI mode filters designed for the
ATOC and AST data sets. The rationale for including modes
higher than 10 共which the VLAs were designed for兲 in the
filters for the KVLA28 and HVLA84 data sets is that this
forces more of the crosstalk from modes not included in the
estimate into the higher-order modes. See the article by Wage
et al.20 for a more thorough discussion of mode filter design
J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
in the context of the ATOC experiment. The fourth column of
Table II lists which modes are expected to be reliably estimated by these filters, i.e., which modes have low crosstalk
from other modes. The estimates of the number of ‘‘good’’
modes are somewhat conservative. For instance, the table
indicates that only mode 1 can be estimated for the Hawaii
AST receptions 共HVLA28 and HVLA84兲. This is due to the
large amount of nearest-neighbor crosstalk for the degraded
共19-phone兲 Hawaii VLA. Modes higher than 1 could be estimated in groups, e.g., mode 2 and 3 together, but this study
focuses only on the mode 1 signals, which are better separated.
B. Mode 1 processing examples
As an example of the processing techniques described
above, consider the mode 1 estimates derived from the six
sample receptions discussed in Sec. II. Figures 9 and 10
show the estimates for the Hawaii and Kiritimati receptions
at the three carrier frequencies. In each plot the short-time
Fourier mode spectrum is shown above the time series derived from the FFT processor. Short-time Fourier mode estimates consist of a time series for each of the frequency bins
contained in the processor. The STFT processor uses the design parameters selected for the previous analysis20 of the
ATOC Hawaii data. Specifically, the processor uses a 0.4-s
TABLE II. Mode filter parameters for ATOC and AST analyses.
Data set
Modes in PI
Diagonal loading
‘‘Good’’ modes
HVLA75
KVLA75
HVLA28
KVLA28
HVLA84
KVLA84
1–10
1–10
1–5
1–20
1–20
1–10
None
None
5%
10%
10%
None
1–10
1–10
1
1– 4
1
1–10
Wage et al.: Mode coherence at megameter ranges
1571
FIG. 9. Mode 1 estimates for the Hawaii receptions shown in Fig. 2. The
color images are the short-time Fourier estimates, and the line plots are the
time-series output of the FFT processor. The 0-dB level in the color plots
corresponds to the predicted noise floor.
FIG. 10. Mode 1 estimates for the Kiritimati receptions shown in Fig. 3. The
color images are the short-time Fourier estimates, and the line plots are the
time-series output of the FFT processor. The 0-dB level in the color plots
corresponds to the predicted noise floor.
Hanning window, which provides a frequency resolution of
approximately ⫾2.5 Hz. With this window, the processor can
begin to temporally resolve arrivals that are 0.2 s apart. Note
that for the plots in Figs. 9 and 10, the bin spacing is 1.25
Hz; thus, neighboring frequency bins are highly correlated.
Oversampling by a factor of 2 merely improves the appearance of the plots, but does not change the underlying frequency resolution of the STFT processor. The color scale for
the STFT plots is set so that the 0-dB level corresponds to
the predicted noise floor.
Figure 9 illustrates several general characteristics of the
mode estimates for the ATOC and AST data sets. As noted in
a previous article,20 the HVLA75 results indicate that there is
no dominant arrival in mode 1 at 3515-km range. Instead
there is a complicated multipath arrival pattern, attributed to
internal wave-induced scattering. The short-time Fourier estimates show significant frequency-selective fading, which
occurs when two signals with different phase characteristics
arrive at the same time 共within the 0.2-s resolution of the
processor兲. The constructive and destructive interference of
multipath arrivals can produce large peaks and deep fades in
the arrivals. As expected from the discussion of receptions
for the bottom-mounted source, the mode 1 estimate does not
have a sharp cutoff; rather, it trails off with approximately
0.5 s of weaker arrivals.
Figure 10 demonstrates that the mode estimates at
5171-km range are comparable to those at 3515-km range.
The KVLA75 mode 1 estimate exhibits frequency-selective
fading that is quite similar to the HVLA75 estimate. The
arrivals are spread over approximately 1.5 s, which is almost
identical to the HVLA75 result, assuming that the 0.5 s of
weak arrivals at the end of the HVLA75 estimate are ignored.
Figures 9 and 10 show that the 84-Hz estimates are similar to the 75-Hz estimates, though they have sharper cutoffs,
as expected for the suspended AST source. It is important to
note that the STFT bins for the ATOC arrivals span a 20-Hz
bandwidth from 65 to 85 Hz, whereas the STFT bins for the
AST arrivals span half that bandwidth 共79 to 89 Hz兲. The
frequency-selective fading in the 84-Hz results is comparable
to that observed at 75 Hz, i.e., arrivals consistent over bandwidths of approximately 5 Hz.
The 28-Hz mode 1 estimates shown in Figs. 9 and 10
contrast sharply with the higher-frequency results. At 28 Hz,
mode 1 is dominated by a single large arrival. The HVLA28
plot in Fig. 9 shows a strong unfaded final arrival, i.e., an
arrival that is consistent across the entire source band. The
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J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
Wage et al.: Mode coherence at megameter ranges
FIG. 11. Plot of short-time Fourier mode estimates in the 28-Hz bin for ten
successive 4-period averages in a single reception at the Kiritimati VLA.
The x axis represents arrival time within a single four-period average, and
the y axis represents lag time over the approximately 20-min source transmission. The black ⫻’s mark the peak arrival times in the first four-period
average.
lack of frequency-selective fading indicates that this arrival
is temporally separable from the earlier multipaths 共within
the resolution of the STFT processor兲. The KVLA28 plot in
Fig. 10 also shows a large final mode 1 arrival, but it fades a
bit toward the upper edge of the band.
C. Source and VLA motion compensation
The results in Figs. 9 and 10 are for single receptions,
i.e., one 4-period average of the received M-sequence. In
order to study the temporal variations across multiple receptions, the mode processing algorithm must account for any
source and receiver motion. This section describes how motion compensation was implemented for the ATOC and AST
data sets.
Since the ATOC source was bottom-mounted, those receptions require no source correction. The AST source was
deployed from the M/V INDEPENDENCE; thus, the AST receptions must be corrected for fluctuations due to source motion.
During the experiment the ship was dynamically positioned
using P-code GPS and maintained its station to within about
20 m.21 Since the acoustic tracking of the suspended source
failed, the only available information about source motion is
the ship GPS record. To see how well the observed fluctuations match the predicted fluctuations due to source motion,
consider an example for the Kiritimati VLA. Figure 11
shows the short-time Fourier estimates in the 28-Hz bin for
successive 4-period averages within a single 20-min source
transmission. The black ⫻’s on the plot mark the location of
peaks in the first 4-period average that are at least 12 dB
above the estimated noise floor. The black lines track those
peak arrival times through the estimates at successive lag
times. The peak arrivals appear fairly consistent for this reception, though the amplitudes do vary across the 20 min.
Figure 12 shows how the phases of the six peak arrivals vary
as a function of lag time. Phases are relative to the phase of
the first 4-period average for each peak 共which has been
subtracted out兲. In this reception, all of the peaks show a
sharp negative phase trend, with an overall phase shift of
approximately ␲/2 radians within the first 5 min.
Given a change in range, it is possible to predict the
phase changes in mode 1 as a function of the lag time t, i.e.,
⌬ ␪ 共 t 兲 ⫽k 1 ⌬r 共 t 兲 ,
J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
共5兲
FIG. 12. Relative phase as a function of lag time for the six peak arrivals
shown in Fig. 11. Each peak is offset by ␲ radians along the x axis, e.g., the
curve starting at 2␲ corresponds to the second peak from the left in Fig. 11.
Solid lines correspond to the measured peak phases. The dashed lines
marked with circles correspond to the phase changes predicted by the ship
GPS record.
where k 1 is the range-averaged wave number for mode 1,
and ⌬r(t) is the change in range along the geodesic path.
The source GPS data provide an estimate of the range
changes as a function of time. To account for the 4-period
averaging that is inherent in the ATOC processing, the GPS
prediction of the phase changes is defined as
冉兺
4
⌬ ␪ GPS共 t 兲 ⫽arg
n⫽1
冊
e jk 1 ⌬r n 共 t 兲 ,
共6兲
where ⌬r n (t) corresponds to the range associated with the
nth M-sequence in the 4-period average. Figure 12 shows the
GPS phase prediction for the KVLA28 reception described
above. The plot shows six identical copies of the GPS prediction as dashed lines with circle markers alongside the
phase data 共solid lines兲 for each of the six peaks. Over the
first 5 min where the steepest phase transition occurs, the
GPS prediction and the observed data are in close agreement.
After that, the fourth peak appears to track the GPS prediction, but the other peaks show an additional phase trend that
is not in the prediction.
The above example is typical of results for other receptions. Since the GPS data appear to predict a significant component of the observed phase fluctuations, they are used to
apply a source motion correction to the AST receptions during mode processing. There will be a residual error due to the
fact that the true source motion is not exactly represented by
the GPS for the source ship. Some of this error may be
removed by the procedure used to correct for the VLA mooring motion, as discussed below.
As indicated in Sec. II, long-baseline navigation systems
tracked each VLA during the experiment. Navigation measurements taken immediately before and after each reception
yield good estimates of hydrophone depths and their positions relative to the anchor reference point at the beginning
of the reception. The velocity of the mooring is expected to
be low, on the order of 1 cm/s maximum.39 In previous
work,20 Wage et al. used an approximate mooring velocity
estimated from the navigation data to correct for motion of
the VLA during the 20-min transmission. While the results
appeared reasonable, it is not clear that all of the effects of
mooring motion were removed by that correction procedure.
Motivated by Colosi’s use of linear phase corrections to reWage et al.: Mode coherence at megameter ranges
1573
FIG. 14. Histogram of mooring velocity estimates for the 188 good ATOC
receptions at the Hawaii VLA.
FIG. 13. Relative phase as a function of lag time for the six peak arrivals in
Fig. 11 after application of the source motion correction. Each peak is offset
by ␲ radians along the x axis, e.g., the curve starting at 2␲ corresponds to
the second peak from the left in Fig. 11. Solid lines correspond to the
measured peak phases. Dashed lines marked by circles correspond to a
linear fit to the estimated phase trend for the reception.
move residual mooring motion in his analysis19 of the pressure field, this paper considers a similar approach for the
mode analysis.
To understand how the VLA motion was estimated, consider Fig. 13, which shows the relative phases of the peaks in
KVLA28 example reception 共discussed above兲 after the
source correction has been applied. Five out of the six peaks
have phases that decrease approximately linearly as a function of lag time. The dashed line marked with circles plotted
next to each peak represents a linear least-squares fit to the
phase associated with the third peak. The third peak was
chosen because it has the most consistent amplitude across
all the lag times. The linear approximation appears to fit all
the peaks reasonably well, with the exception of peak 4.
Note that peak 4 shows the most fading in amplitude across
the 20-min transmission 共see Fig. 11兲. Assuming that the
linear trend in phase corresponds to mooring motion, the
slope is directly related to the mooring velocity. For example, the velocity inferred from Fig. 13 is ⫺1 cm/s, which
is reasonable given prior expectations about the ATOC moorings.
This example suggests a simple procedure for implementing mooring corrections. First, estimate the linear phase
trend of the peak mode 1 arrival at the center frequency that
has the most consistent amplitude across all lags. Use this to
infer the velocity of the mooring. Then, given the mooring
velocity and the initial VLA location, determine the position
of the array elements at the lag times corresponding to each
4-period average. Finally, use the estimated array positions
共hydrophone depths and ranges兲 to implement the mode processing for each 4-period average. Applying this procedure
to all of the 28-Hz AST data yields good results. During the
AST experiment, the estimated mooring velocities for the
Hawaii VLA are between ⫺2 and ⫹2 cm/s, while the velocities of the Kiritimati VLA are between ⫺1.5 and ⫹1.5 cm/s.
Since the 28- and 84-Hz transmissions were simultaneous,
the mooring velocities obtained from the 28-Hz data can be
used to compensate the 84-Hz receptions.
A similar approach can be applied to determine the
mooring corrections for the ATOC receptions, though the
amount of fading at 75 Hz makes it more difficult to find
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J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
arrivals that are consistent across the extent of a 20-min
transmission. The algorithm searches for peak arrivals that
fade by less than 3 dB in amplitude over 20 min to use for
the velocity estimation. The search begins with mode 1 in the
75-Hz bin. If no peaks in mode 1 meet the criteria, the program looks for peaks in modes 2 and 3. If no suitable peaks
are found at 75 Hz, the search expands to the 72.5-, 70-, and
77.5-Hz bins. This procedure produced reasonable VLA motion estimates for both the Hawaii and Kiritimati arrays. Figure 14 shows a histogram of the mooring velocity estimates
for the 188 good receptions in the HVLA75 data set. The
estimates are concentrated between ⫹1 and ⫺1 cm/s.
While this approach to the VLA motion corrections is ad
hoc and does not satisfy any optimality criteria, it does produce good results. In particular, as will be shown in the next
section, this motion compensation algorithm leads to longer
coherence times for the modes than were observed in previous work40 with the same data sets.
IV. MODE STATISTICS
As the examples in the previous section illustrate, the
low-mode signals are quite complicated at megameter ranges
away from a broadband source. The frequency-selective fading observed in the mode arrivals is typical for propagation
through time-varying multipath channels. In the communications literature, fading multipath channels are characterized
by their statistics. The texts by Van Trees41 and Proakis42
provide useful introductions to statistical models for such
channels. The purpose of this section is to compute the statistics of the ATOC and AST mode arrivals that are relevant
to the future development of a channel model. Specifically,
the following sections present estimates of the multipath
time spread, coherence bandwidth, cross-mode coherence,
and temporal coherence. Experimental results are compared
to PE simulations.
A. Time spread
At megameter ranges, the low-mode signals consist of
multiple arrivals spread over time. To quantify the multipath
time spread, consider the average power as a function of
arrival time, i.e.,
K m 共 ␶ 兲 ⫽E 关 ␣ m 共 ␶ 兲 * ␣ m 共 ␶ 兲兴 ,
共7兲
where E is the expectation operator, ␣ m ( ␶ ) is the time series
for the mth mode, and * indicates complex conjugation.
Proakis42 calls this quantity the multipath intensity profile of
Wage et al.: Mode coherence at megameter ranges
FIG. 15. Multipath intensity profiles for modes 1–10 at Hawaii computed
using data from the first group of ATOC receptions.
FIG. 16. Multipath intensity profiles for modes 1–10 at Kiritimati computed
using data from the first group of ATOC receptions.
the channel. Assuming that the channel is relatively stationary over a period of days, K m ( ␶ ) can be estimated by averaging over the FFT-processed mode estimates for a reception
group 共as defined in Table I兲. Figure 15 shows the multipath
intensity profiles for the first 10 modes of reception group 1
in the HVLA75 data set. This plot highlights several characteristics of the low-mode arrivals at 3515-km range. First, the
arrivals in modes 1–10 are spread over several seconds, with
the higher modes having slightly larger time spreads than the
lower modes within that group of 10. Second, the average
arrival time is a function of mode number, e.g., the mean
arrival time for mode 10 is smaller than that for mode 1.
Third, the plot shows that the main arrivals in each mode are
followed by the approximately 0.5 s of weaker arrivals that
Sec. II attributes to bathymetric interaction near the Pioneer
source.
To compare the 3515-km results with those at 5171 km,
Fig. 16 shows the multipath intensity profile for the first 10
modes in reception group 1 of the KVLA75 data set. The
plot shows that, similar to the Hawaii results, the arrivals in
modes 1 through 10 are spread over several seconds. At Kiritimati the first 10 modes are more tightly grouped in arrival
time than at Hawaii, i.e., there is less dispersion between the
modes. Worcester et al.21 note that the Pioneer-Kiritimati
path is more oceanographically complex, particularly near
the equator. Additional mode-to-mode scattering would tend
to erase the mode-dependent trends in arrival time. Note that
in this reception group, there appear to be a set of weaker
arrivals trailing the main arrivals, particularly in modes 1 and
2. Interestingly, this is not a consistent feature in the
KVLA75 data set. Many of the other reception groups show
a much sharper cutoff. It is not clear whether these late arrivals can be attributed to bathymetric interaction near the
source.
Figures 17 and 18 show how the multipath intensity profiles for mode 1 vary over the course of the ATOC experiment. A particularly strong shift in arrival time is evident
between the first and second reception groups in each plot.
These shifts are likely due to mesoscale activity since they
are not in agreement with the predicted seasonal trends.12,20
Figure 19 shows the multipath intensity profiles for
mode 1 measured at Hawaii and Kiritimati during the AST
experiment. The plots contrast the results for the two AST
source frequencies. In the Hawaii data there is a significant
difference between the profiles for the 28-Hz source and the
84-Hz source. At 28 Hz, one arrival appears to dominate,
whereas at 84 Hz the arrivals are much more spread out. At
Kiritimati, the overall spread is roughly the same for the 28and 84-Hz profiles, but the 28-Hz result indicates some distinct individual arrivals rather than the blur of arrivals seen
in the 84-Hz profile.
The leading and falling edges for each mode can be
defined as the minimum and maximum times, respectively,
that the multipath intensity profiles exceed a threshold. To
see how to pick the threshold, consider the example shown in
Fig. 20. The plot shows the profile for mode 1 in group 13 of
J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
Wage et al.: Mode coherence at megameter ranges
1575
FIG. 17. Multipath intensity profiles for mode 1 for reception groups in the
HVLA75 data set.
the HVLA75 data set. Dashed lines indicate the estimated
noise floor, and the 3- and 8-dB thresholds above the noise
floor. Note that the 8-dB threshold is high enough to avoid
detecting the weaker arrivals that follow the primary arrivals
in mode 1. A review of all reception groups indicates that an
8-dB threshold works reasonably well to detect the main
arrivals and ignore the weaker trailing arrivals.
Figure 21 shows the mode time spreads, which are defined as the difference between the leading and falling edges
determined by the 8-dB threshold. The HVLA75 and
KVLA75 results shown in the plot represent the mean over
all reception groups. For modes 1–10 in the HVLA75 data,
the standard deviations of mean time spread lie between 0.13
and 0.3 s. The highest variance is associated with mode 1,
which is due to the fact that the 8-dB threshold does not
always suppress all of the weaker trailing arrivals. For the
KVLA75 data, the standard deviations are somewhat smaller,
with a minimum of 0.07 s and a maximum of 0.19 s.
Figure 21 shows that the average time spreads observed
in ATOC and AST are on the order of 1.3 to 1.8 s. The
spreads at the two ATOC receiver ranges 共3515 and 5171
km兲 are quite comparable. Spread is a function of mode
number, with the lowest modes showing somewhat less
spread than the higher modes. Mode 1 in the HVLA28 data
set shows the least amount of time spread, which is not surprising given that at 28 Hz the Hawaii mode 1 receptions are
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J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
FIG. 18. Multipath intensity profiles for mode 1 for reception groups in the
KVLA75 data set. Note that there are no good receptions at the Kiritimati
VLA for reception group 7 共see Table I兲.
FIG. 19. Multipath intensity profiles for mode 1 at Hawaii and Kiritimati
during the AST experiment. Both plots show a 4-s interval, but have different starting points.
Wage et al.: Mode coherence at megameter ranges
FIG. 22. Average power in the center frequency bin for modes 1–10 for the
HVLA75, KVLA75, and KVLA84 data sets. The HVLA75 and KVLA75
results are for reception group 1.
FIG. 20. Closeup of the multipath intensity profile for mode 1 in reception
group 13 of the HVLA75 data set. Dashed lines mark the level of the noise
floor, and the 3- and 8-dB thresholds above the noise floor.
dominated by one arrival. Mode 1 for the HVLA84 data set
shows the most spread, but it is possible that this an artifact
due to crosstalk in the processor designed for the degraded
Hawaii VLA.
The multipath spread is a measure of the coherence
bandwidth41,42 of the channel, i.e., a time spread of T m s
implies a coherence bandwidth of 1/T m Hz. With time
spreads on the order of 1.5 s, the coherence bandwidth for
the low modes is on the order of 0.67 Hz. This relatively
short coherence bandwidth is consistent with the frequencyselective fading observed in the STFT mode estimates. The
exception is the 28-Hz mode estimates, which show significantly less frequency-selective fading, even though their
time spreads aren’t substantially different. This indicates that
at 28 Hz mode 1 is dominated by a few multipath arrivals
that are temporally resolvable by the STFT processor 共which
has a resolution of 0.2 s兲.
B. Cross-mode coherence
In two seminal papers,2,3 Dozier and Tappert predicted
that scattering due to internal waves decorrelates the modes
and results in an equipartitioning of energy at long ranges.
Using the HVLA75 data, Wage et al.20 showed that these
FIG. 21. Mode time spread for the ATOC and AST experiments. The
HVLA75 and KVLA75 data represent the means over all reception groups
in the experiment.
J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
predictions are valid for the lowest 10 modes at 3515 km. As
discussed below, data from the Kiritimati array confirm that
this result holds for a second, longer path.
A useful coherence metric is the magnitude-squared coherence 共MSC兲, defined as:
MSC共 ␻ 兲 ⫽
兩 S mn 共 ␻ 兲 兩 2
,
S mm 共 ␻ 兲 S nn 共 ␻ 兲
共8兲
where S mn is the cross-power spectral density and S mm and
S nn are the auto power spectral densities. By definition the
MSC lies between 0 and 1. Carter43 provides a thorough
discussion of the statistical properties of the MSC. For the
modes the cross-power spectrum can be estimated by averaging over L measurements in each STFT bin, e.g., an estimate of the cross spectrum for modes m and n is
L
Ŝ mn 共 ␻ 兲 ⫽
1
â 共 ␻ ,l 兲 â n* 共 ␻ ,l 兲 ,
L l⫽1 m
兺
共9兲
where â m and â n are the amplitude estimates for modes m
and n. Setting m⫽n in Eq. 共9兲 produces auto spectrum estimates. For the ATOC and AST data sets, the series of measurements needed to estimate the auto- and cross spectra are
obtained by subsampling the output of the STFT processor.
Specifically, samples are taken every 0.15 s, which corresponds to a 62.5% overlap of the 0.4-s processing window
used in the STFT. The sampling interval is determined by the
leading and falling edges defined by the 8-dB threshold, as
discussed in Sec. IV A.
Using the HVLA75 data, Wage et al.20 showed that the
average power in modes 1–10 is approximately constant.
Figure 22 demonstrates that this conclusion is also true for
the Kiritimati propagation path. The plot shows the average
power estimates (10 log10兩 Ŝ mm 兩 ) for the center frequency bin
in the HVLA75 共group 1兲, KVLA75 共group 1兲, and KVLA84
data sets. The average powers for the first 10 modes are
equal within about 1 dB. Other bins and other reception
groups show similar behavior. Dozier and Tappert’s prediction of equipartioning of energy appears to be approximately
true for the lowest 10 modes at megameter ranges.
Dozier and Tappert also predict that internal wave scattering will cause the modes to decorrelate. Again, the ATOC
and AST data sets support this conclusion. For the HVLA75,
KVLA75, and KVLA84 data sets, the maximum MSC between the low-order modes is 0.06. These results were obtained by averaging across all reception groups in each data
Wage et al.: Mode coherence at megameter ranges
1577
set. Note that the HVLA28, HVLA84, and KVLA28 data are
excluded from this analysis because the diagonal loading required for mode processing introduces mode crosstalk, making mode cross-coherence estimation difficult. A set of PE
simulations, which will be discussed more in the following
section, also confirms that mode signals at 3.5 Mm or greater
ranges are incoherent. The maximum cross-mode coherence
for modes 1–10 estimated from narrow-band simulations at
both 28- and 75 Hz is less than 0.1.
C. Temporal coherence
In previous work Wage et al.20,40 demonstrated that the
mode signals fluctuate significantly over time. Quantifying
the temporal variability of these signals is an important prerequisite to developing a stochastic model for the mode arrivals. This section computes the coherence times of the low
modes in the ATOC and AST data sets and compares them to
estimates obtained from narrow-band PE simulations. Recall
that each ATOC and AST transmission consisted of 40
M-sequences. The receiver recorded 10 four-period averages
for each transmission. One way to quantify the temporal
variability is to compute the MSC between the first fourperiod average in each transmission and successive fourperiod averages. In this study the MSC was estimated two
ways: 共1兲 using one bin of an FFT of the entire mode signal
共between the leading and the falling edges兲, and 共2兲 using
samples from the desired bin of the STFT mode estimates
共similar to how the cross-mode coherence estimation was
implemented兲. Since the two approaches yield very similar
results, only the STFT estimates are presented below. Note
that in the following discussion the coherence time is defined
as the lag time at which the MSC reaches 0.5. This threshold
is somewhat arbitrary and may not be appropriate for all
applications. Figures 23–26, which display the MSC as a
function of lag time, can be used to determine the coherence
time associated with any desired threshold.
Figure 23 shows the temporal coherence as a function of
lag time for the 75-, 84-, and 28-Hz receptions at the two
VLAs. The HVLA75 and KVLA75 results were obtained by
averaging over all the reception groups listed in Table I. At
75- and 84 Hz, mode 1 decorrelates rather rapidly. The
HVLA75 result shows the longest coherence time, reaching
an MSC of 0.5 at approximately 8 min. Note that this is a
slightly longer coherence time than previously reported; the
previous result20,40 was an MSC of 0.5 at 4.5 min. The difference is due to the revised mooring motion corrections discussed in Sec. III C. The new approach should do a better job
of compensating for mooring-related phase changes. In addition, it is possible that this technique is removing a linear
phase component associated with the internal wave fluctuations. The simulations discussed below shed more light on
this issue.
Figure 23 also indicates that at 75 Hz the mode 1 signal
at Kiritimati 共5171-km range兲 is less coherent than at Hawaii
共3515-km range兲. The MSC decays to 0.5 in approximately 6
min instead of 8 min. Both of the AST 84-Hz data sets show
somewhat less coherence in mode 1 than the corresponding
ATOC data sets. While this may be due to the increase in
center frequency, it seems highly likely that residual source
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J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
FIG. 23. Temporal coherence estimates for mode 1 in the ATOC and AST
data sets. The ATOC results represent averages over all reception groups.
motion 共not removed by the GPS-based correction兲 is a contributing factor. It is also important to remember that the
mooring motion corrections for the 84-Hz receptions were
derived from the peaks in the simultaneous 28-Hz receptions
共since the lower frequency peaks are easier to track兲. Assuming that the temporal fluctuations at 28 Hz are independent of
those at 84 Hz, the phase corrections derived from the 28-Hz
data would not remove any of the internal-wave-induced
variability at 84 Hz.
By comparison, the mode 1 signals at 28 Hz are much
more coherent than at 75 Hz. Figure 23 also shows the MSC
FIG. 24. Comparison of temporal coherence estimates for modes 1, 5, and
10 in the HVLA75 data set. Results are averages over all reception groups.
Wage et al.: Mode coherence at megameter ranges
FIG. 25. Comparison of mode 1 coherence times for narrow-band PE simulations at 75- and 28 Hz with three GM levels: 1, 0.5, and 0.25 GM. The
dashed line in each plot shows the Hawaii experimental results prior to the
application of mooring corrections.
FIG. 26. Mode 1 coherence time for PE simulations at three GM levels after
application of a linear phase correction. Subplots show results for 75- and 28
Hz. The dashed line in each plot shows the Hawaii experimental results after
the application of mooring corrections.
as a function of lag time for the HVLA28 and KVLA28 data
sets. As with the 75-Hz data, mode 1 appears to be slightly
more coherent at the shorter 共3515-km兲 range than at the
longer range. At the maximum lag, the coherence is 0.77 for
Hawaii and 0.64 for Kiritimati.
Figure 24 illustrates the general trend in coherence time
as a function of mode number using the results for the
HVLA75 data set. The plot shows the MSC for modes 1, 5,
and 10. Mode 1 is the most coherent. The two higher-order
modes decay to an MSC of 0.5 a few minutes earlier. This
may be another consequence of how the mooring motion
corrections were computed. The phase corrections were usually derived from mode 1 共assuming it had a peak that was
trackable across all the four-period averages兲. Thus, it is possible that some of the internal wave variability associated
with mode 1 is being removed, while this is not true for the
other modes.
It is instructive to compare the experimental results to
PE simulations of propagation through time-varying internal
waves. The background environment for these simulations is
identical to the one used in Sec. II C, and the suspended
source is used. For this analysis, 30 realizations of soundspeed perturbations were generated for nine different lag
times over a 20-min interval using the method of Colosi and
Brown.27 Using the perturbed environments, the RAM PE
code24 generated realizations of the narrow-band pressure
field at 28- and 75 Hz for a range of 3515 km. Projecting the
pressure field onto the mode functions yields the Fourier
coefficient for each mode. These were averaged to obtain the
auto- and cross spectra and finally the temporal coherence
estimates.
Figure 25 shows the time coherence at 75 Hz for mode
1, derived from simulations using three different energy levels for the Garrett–Munk 共GM兲 spectrum. As expected, the
higher the GM strength, the lower the temporal coherence.
Comparing Fig. 25 to the 75-Hz experimental results shown
in Fig. 23 indicates that the measured mode 1 signals are
more coherent than the simulations. Interestingly, the simulation results are comparable to the measured coherence for
mode 1 obtained prior to applying the revised mooring corrections 共shown as a dashed line in the figure兲. This suggests
that these corrections, which are estimated by tracking arrival peaks, are removing more than just mooring motion.
Figure 25 also shows the simulation results for 28 Hz.
As for the 75-Hz case, the temporal coherence for 28-Hz
signals propagating through internal waves at 1 GM strength
is lower than for those propagating through 21 GM or 41 GM
internal waves. Also similar to 75 Hz, the 28-Hz simulations
in Fig. 25 appear to be less coherent than the real data in Fig.
23. The dashed line in Fig. 25 indicates that if the mooring
correction is not applied to the 28-Hz experimental data, the
J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
Wage et al.: Mode coherence at megameter ranges
1579
estimated temporal coherence lies between the 41 and 12 GM
simulation results.
The discrepancy between experimental and simulation
results raises the possibility that the revised mooring corrections could be removing more than just mooring motion. To
make a fairer comparison between experiment and simulation, a similar phase correction should be applied to the
simulation data prior to coherence estimation. Since the
simulations are narrow band, it is not possible to incorporate
exactly the same type of phase correction as for the real data.
Instead of using a linear fit to the phases of the detectable
peaks in the STFT mode estimates, the phase correction for
the simulated data is derived from a linear fit to the phase
trend 共across time兲 in the single frequency bin generated by
the PE code. This removal of the linear trend is similar to
what Colosi does for the finale coherence calculations.19 Figure 26 shows the MSC estimates at 75- and 28 Hz, respectively, after the linear phase correction has been incorporated
into the coherence calculation. For reference, the plots also
include the results for the mooring-corrected experimental
data.
At 28 Hz, there is good agreement between the phasecorrected simulations and the mooring-corrected data. The
experimental data lies between the 0.5- and 1-GM results. At
75 Hz, the experimental measurement falls between the 0.25and 0.5-GM results for most lag times; however, the trends
in the simulated and experimental data don’t really match.
With the incorporation of the phase correction, the MSC is
not monotonically decreasing. This is due to the fact that a
linear fit to a nonlinear phase characteristic is being used for
the correction. For many of the PE realizations, the leastsquares solution for the phase trend results in a better fit near
the endpoints than in the middle. Thus, the coherence decreases initially and then increases. The real data do not exhibit this behavior, but this is not surprising given that the
phase corrections were implemented differently.
V. CONCLUSION
This article presented a detailed study of the mode arrivals observed in the ATOC and AST experiments. The
analysis confirmed that the receptions at 5171 km share
many of the same characteristics as the receptions at 3515
km analyzed previously. This study noted the difference in
the cutoff of the ATOC and AST receptions. The simulations
presented here suggest that bathymetric interaction near the
Pioneer Seamount source is responsible for the generation of
the weak, late arrivals observed in the ATOC receptions.
Mode processing was implemented using both FFT and
STFT techniques. The new approach to correcting for VLA
motion worked well, resulting in longer mode coherence
times than estimated in previous work. Simulations indicated
that the simple phase correction for mooring motion likely
removed some of the internal wave variability. Section IV
provided a detailed set of mode statistics for the ATOC and
AST data sets, including multipath intensity profiles, crossmode coherence, and temporal coherence. At 75 Hz the
mode time spreads are on the order of 1.5 s, giving a coherence bandwidth of 0.67 Hz. This explains the high degree of
frequency-selective fading observed in short-time Fourier
1580
J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 2, March 2005
mode estimates. The time spread of the 28-Hz signals is
somewhat lower, but these signals show significantly less
frequency-selective fading. This suggests that at the lower
frequency, the multipaths are temporally resolvable. At these
ranges, the experimental analysis indicates that modes 1–10
have equal powers and that the coherence between these
modes is effectively zero. Coherence times for mode 1 at 75
Hz are on the order of 8 min for the 3515-km range and 6
min for 5171-km range. At 28 Hz mode 1 is much more
stable, with a magnitude-squared coherence of greater than
0.6 for the 20-min ATOC/AST transmission period.
This investigation raises several interesting issues for
future study. First, Colosi et al.19 computed similar coherence statistics for the total acoustic field 共rather than splitting
the signal into modes兲. A key question is, how do the mode
statistics derived in this paper relate to the statistics of the
overall field? The temporal coherence trends are quite
similar,44 but further work is needed to understand the relationship between these results. Second, it is intriguing that a
simple correction designed to remove mooring motion also
removed a nontrivial portion of the ocean variability, particularly in the 28-Hz case. This raises the question of whether
more sophisticated signal processing can be designed to
equalize the mode signals and produce observables for tomography.
ACKNOWLEDGMENTS
Kathleen Wage gratefully acknowledges the support of
an Office of Naval Research Ocean Acoustics Entry-Level
Faculty Award. Additionally, this research was supported by
the Office of Naval Research 共Grant N00014-95-1-0800 to
APL-UW and Grant N00014-95-1-0589 to SIO兲 and by the
Strategic Environmental Research and Development Program through Defense Advanced Research Project Agency
Grant MDA972-93-1-0003. The two HLF-6A low-frequency
acoustic sources used in the AST experiment were modified
for operation at depths of up to 1300 m and a specialized
handling system was developed for them as part of the Joint
Environmental Test Initiative sponsored by SPAWAR PMW182. The mode filters were designed using environmental
profiles from the ECCO database, which is a contribution of
the Consortium for Estimating the Circulation and Climate
of the Ocean funded by the National Oceanographic Partnership Program. The authors acknowledge the significant contributions of Arthur Baggeroer, Theodore Birdsall, Kurt
Metzger, Walter Munk, and Robert Spindel to the design and
implementation of the ATOC and AST experiments. In addition they thank the anonymous reviewers for their suggestions.
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Sec. IV.
Wage et al.: Mode coherence at megameter ranges
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