The depth-dependence of rain noise in the Philippine Sea
David R. Barclay and Michael J. Buckinghama)
Marine Physical Laboratory, Scripps Institution of Oceanography, University of California, San Diego,
9500 Gilman Drive, La Jolla, California 92093-0238
(Received 14 July 2011; revised 13 March 2013; accepted 20 March 2013)
During the Philippine Sea experiment in May 2009, Deep Sound, a free-falling instrument platform,
descended to a depth of 5.1 km and then returned to the surface. Two vertically aligned hydrophones
monitored the ambient noise continuously throughout the descent and ascent. A heavy rainstorm
passed over the area during the deployment, the noise from which was recorded over a frequency
band from 5 Hz to 40 kHz. Eight kilometers from the deployment site, a rain gauge on board the R/V
Kilo Moana provided estimates of the rainfall rate. The power spectral density of the rain noise shows
two peaks around 5 and 30 kHz, elevated by as much as 20 dB above the background level, even at
depths as great as 5 km. Periods of high noise intensity in the acoustic data correlate well with the
rainfall rates recovered from the rain gauge. The vertical coherence function of the rain noise has
well-defined zeros between 1 and 20 kHz, which are characteristic of a localized source on the sea
surface. A curve-fitting procedure yields the vertical directional density function of the noise, which
is sharply peaked, accurately tracking the storm as it passed over the sensor station.
C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4799341]
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Pages: 2576–2585
I. INTRODUCTION
Little is known about high frequency (above 2 kHz) ambient acoustic noise in the deep ocean, especially at and
below the critical depth, where the sound speed is the same
as that at the sea surface. Indeed, measurements of deep
ocean noise at any frequency are scarce, not least because of
the challenge of deploying conventional cabled acoustic
arrays to great depths. The few arrays that have been
deployed generally operated below 500 Hz at a fixed depth,
located around either the sound channel axis or the critical
depth (Morris, 1978; Marshall, 2005; Gaul et al., 2007).
Operating at even lower frequencies, ocean-bottom seismometers (OBSs) have been placed directly on the seafloor
(Dorman, 1993; McCreery et al., 1993) to return noise data
from the bottom of the water column. Since the arrays and
OBSs were all in fixed positions, they provided a sampling
of only a small proportion of the water column.
To profile the ambient noise over the full depth of the
ocean, a different approach is needed. This is provided by
the acoustic recording platform Deep Sound (Barclay et al.,
2009), which is a free-falling (untethered) instrument platform, designed to descend under gravity and, at a preassigned depth, release a drop weight, allowing it to return to
the surface under buoyancy. Two vertically aligned, broadband (30 kHz) hydrophones on board Deep Sound record the
ambient noise continuously throughout the descent and
ascent. From these time series, the second-order statistics of
the noise, including the power-spectral and cross-spectral
densities, along with the coherence function, may be computed as continuous functions of depth over the full water
a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
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J. Acoust. Soc. Am. 133 (5), May 2013
column. Such profiles are acquired over a period of several
hours, depending on the maximum depth of the deployment.
The first deep-sea trials of Deep Sound took place as
part of the North Pacific Acoustic Laboratory (NPAL)
Philippine Sea experiment (Worcester and Spindel, 2005),
April to May 2009, known as PhilSea09. Working from the
R/V Kilo Moana, three deployments were made on Julian
Days 106, 110, and 119 to depths of 5.1, 5.5, and 6.0 km. On
Julian Day 106, a rainstorm happened to pass over the
experiment site, allowing continuous recordings of undersea
rain noise to be made from a depth of a few hundred meters
down to 5 km. To help identify rain noise effects in the
acoustic data, direct measurements of rainfall rates were
made on board the R/V Kilo Moana, drifting some 8 km
away from the deployment site.
Noise due to splashes and bubble formation by small
and medium size raindrops are observed in the data over the
entire water column. From the time series acquired by the
two hydrophones, the vertical coherence of the rain noise
was computed, from which the vertical directionality was
estimated using an inversion technique based on the wellestablished integral relationship that exists between spatial
coherence and directionality (Cox, 1973). To facilitate the
inversion, the vertical directionality of the noise, which is
quite peaky due to the localized nature of the rainstorm on
the sea surface, is represented as a piecewise, multiparameter function, with the unknown parameters being
determined from a coherence-data fitting procedure. The resultant vertical directional density function shows a maximum at an elevation angle corresponding to the position of
the storm on the sea surface.
Before describing the rain-noise data and the associated
inversion technique for estimating the angular position of the
storm, the features of Deep Sound that are relevant to the
deployment on Julian Day 106 are briefly discussed. Further
0001-4966/2013/133(5)/2576/10/$30.00
C 2013 Acoustical Society of America
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Author's complimentary copy
PACS number(s): 43.30.Nb [JAC]
II. DEEP SOUND
Deep Sound, shown in Fig. 1, is an autonomous, untethered, free-falling instrument platform (Barclay et al., 2009),
designed to descend from the sea surface to a preprogrammed depth, which may be as great as 9 km, whereupon a disposable weight is dropped using a burn-wire
release, and the platform returns to the surface under buoyancy. The rates of descent and ascent are similar at, nominally, 0.6 m/s. A pressure housing in the form of a Vitrovex
glass sphere contains data acquisition, data storage, and system control electronics. External to the sphere, two vertically
aligned High-Tech, Inc. HTI 94 SSQ hydrophones with a
spacing of 0.5 m record the ambient noise fluctuations during
the descent and ascent. Each of the phones is mounted on a
horizontal arm (Fig. 1) that places it away from the turbulent
flow around the body of the platform. Even so, the trailing
hydrophone returns enhanced noise levels since it lies in the
turbulent wake of the leading hydrophone. The two channels
of acoustic data are simultaneously sampled, each at a frequency of 204.8 kHz, with a dynamic range of 24 bits over
an analogue input range of þ/5 V.
According to the manufacturer, the nominal sensitivity
of the phones is 165 dB referenced to 1 V/lPa over the
range 5 Hz to 40 kHz. Our own independent calibration, performed in a high-pressure chamber over the 2 Hz to 2 kHz
FIG. 1. (Color online) Deep Sound prior to attaching the drop-weight (vertical cylinder at left), in preparation for deployment in the Philippine Sea.
The two hydrophones, aligned vertically, are mounted on arms attached to
the foreground corner of the instrument. The glass sphere containing the
electronics is within the high-density polyethylene (HDPE) “hard hat,” on
top of which are mounted three recovery antennas.
J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013
frequency band, with pressures up to 60 MPa (equivalent to
6 km ocean depth), returned a variation in sensitivity of less
than 2 dB overall. Although the manufacture specifies the
maximum operating depth of the 94 SSQ as 6096 m, our sensitivity tests showed that these sensors function satisfactorily
to equivalent depths as great as 12 km, with less than a 3 dB
change in the response under such extreme pressures. Figure 2
shows our calibration of the frequency response, at a depth
of 1 m, of the two hydrophones used on Deep Sound. In fact,
these curves also represent the frequency response of the
complete data acquisition system, since the cut-off frequencies of the high-pass and anti-aliasing filters on the analogto-digital converters lie well outside the frequency band
covered by the hydrophones.
Electronically, the noisiest components of the data acquisition system are the pre-amplifiers on the hydrophones.
According to the manufacturer, the pre-amp noise lies below
Sea State zero ambient noise at 100 Hz and between Sea
States zero and 0.5 at 1 kHz (38 dB re 1 lPa2/Hz), where the
ambient noise levels are taken to be the minimum values
reported by Wenz (1962).
In addition to the hydrophones, several system and environmental sensors are on board Deep Sound. A solid-state
flux-gate compass, recorded at a rate of 1 Hz, returns pitch,
roll, and yaw, each accurate to 1 , as the instrument platform
descends and ascends through the water column. Hydrostatic
pressure and seawater temperature are also recorded at a rate
of 1 Hz, corresponding to a sampling of the water column
every 0.6 m. The pressure sensor, which is accurate to within
1.1 m, or 0.01% of the instrument’s full ocean depth rating,
provides not only depth but also the rate of descent or ascent
throughout the deployment. The accuracy of the temperature
sensor is 1 mK .
Since the instrumentation on board the original version
of Deep Sound, as used in PhilSea09, does not include a conductivity sensor [unlike later versions which include a full
conductivity-temperature-depth (CTD) probe], the salinity
profile is derived from Temperature-Salinity relations established for the region for May 2009 by the Global
FIG. 2. Frequency response of HTI 94 SSQ hydrophones, measured from
2 Hz to 40 kHz.
D. R. Barclay and M. J. Buckingham: Depth-dependence of rain noise in the ocean
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information on the engineering and operational details of
Deep Sound can be found in Barclay et al. (2009).
Temperature and Salinity Profile Programme (Sun, 2010).
With temperature, pressure, and salinity known, the depth
and sound speed are estimated for each deployment from the
Fofonov and Millard algorithms (Fofonoff et al., 1983).
III. THE PHILIPPINE SEA RAIN NOISE DEPLOYMENT
PhilSea09 was conducted in the northern Philippine
Basin, located southeast of Taiwan, where the average water
depth is 5.5 km (Smith, 1997). The three deployments of
Deep Sound were made at different sites, as shown in Fig. 3,
around each of which the local bathymetry is relatively flat.
On Julian Day 106, the occasion of the first deployment,
Deep Sound descended to 5.1 km and returned to the surface
after 4 h 20 min. The deployment and recovery positions
were 21 13.0550 N, 125 57.3250 E and 21 13.5360 N, 125
57.6840 E, respectively, corresponding to a lateral drift during the descent and ascent of approximately 1 km. From the
bridge of the R/V Kilo Moana, drifting 8 km away from the
deployment site, local shipping was monitored by radar, radio transmissions, and visual sightings, but no vessels were
detected during the deployment.
A rainstorm passed over the area, beginning 20 min into
Deep Sound’s descent and lasting for 2 h 45 min. Since rain
had not been anticipated, a basic but effective rain gauge
was hastily assembled, consisting of a bucket of known
cross-sectional area (214 cm2), a cylinder graduated in milliliters, and a stopwatch. The accumulated volume of water in
the bucket was measured every 15 min, long enough to
smooth the spatial variability of the rainstorm and allowing
significant volumes (from a few to several hundred milliliters) of rainwater to accumulate, even during periods of light
drizzle. The rainfall measurements were performed on the
fantail of the R/V Kilo Moana, away from overhead
structures and areas of potential sea spray. Qualitative observations of raindrop size and rain type (drizzle, light, and
heavy rain) were recorded in a notebook and photographs
were taken of the sea state.
Drizzle fell for an hour at a rate of 0.2 mm/h, and 90 min
into the deployment a heavier fall rate was measured at
2 mm/h, climbing to 12 mm/h over the following 60 min.
This was followed by a 15-min pause, then by light rain with
occasional squalls; and rainfall rates then varied between 0.8
and 3.2 mm/h for the remainder of the deployment. Figure 4
shows the measured rainfall rate superimposed on the instrument’s depth versus time profile.
The wind speed was monitored and recorded continuously by a ship-mounted anemometer. During the entire
deployment, winds were measured at 9 to 11 knots with the
occasional squall of 12 to 15 knots. Some white cap coverage was observed, corresponding to Sea State 4.
The sound speed profiles on the descent and ascent,
computed from the temperature and pressure data from Deep
Sound in conjunction with a salinity profile derived from the
Temperature-Salinity relations established for the region for
May 2009 (Sun, 2010), are shown overlaid in Fig. 5. The
sound channel axis was at 856 m on the descent and 805 m
on ascent, although in both cases the minimum is quite
broad, making precise identification difficult. The surface
conjugate depth, where the sound speed at depth is equal to
that at the surface, was at 3950 m. Below the sound channel
axis, where the hydrostatic pressure governs the sound
speed, the two profiles are indistinguishable, indicative of
the stability of the deep ocean environment, at least on the
time scale of the deployment.
On both the descent and ascent, a constant yaw slowly
rotated the instrument about the vertical axis with a period
of 2 min. The mean pitch and roll during the descent were
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D. R. Barclay and M. J. Buckingham: Depth-dependence of rain noise in the ocean
Author's complimentary copy
FIG. 3. (Color online) Bathymetry of the
Philippine Sea, with the Deep Sound rainfall experiment site depicted by the white
circle at 21 13.050 N, 125 57.320 W. The
locations of the other two deployments
of Deep Sound during PhilSea09 are
identified by the white squares.
0.6 and 7 , respectively, with standard deviations of 0.9
and 0.8 . During the ascent, the instrument rebalanced due to
the release of the drop weight and the mean pitch and roll
both became 3 with standard deviations of 3 .
IV. RAIN NOISE POWER SPECTRA
To compute the power spectra of the ambient noise fluctuations at the two hydrophones, the time series were split
into contiguous segments approximately 20 s long, each of
which was further divided into 40 sub-segments of 0.64 s duration, with no overlap. A 2^17 point fast Fourier transform
(FFT) was applied to each sub-segment, from which a power
spectrum was formed with a frequency cell width of 1.56 Hz.
These sub-segment power spectra were then ensembleaveraged to obtain the spectrum of the original 20 s segment
of each time series.
From these power spectra, a spectrogram was constructed showing the frequency-depth dependence of the
noise spectral level over the entire deployment, as illustrated
FIG. 5. The sound speed profiles measured by Deep Sound during the
descent and ascent, showing the sound channel axis at 800 m and the critical
depth at 3950 m.
J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013
FIG. 6. (Color online) Spectrogram showing the evolution of the noise field
versus time (bottom axis) and depth (top axis). Labels E1–E5 depict elevated background noise due to increased rainfall rates.
D. R. Barclay and M. J. Buckingham: Depth-dependence of rain noise in the ocean
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FIG. 4. The depth of Deep Sound versus time (left axis, solid line) and the
rainfall rate from the rain gauge on the R/V Kilo Moana (right axis, dashed
line) during the deployment.
in Fig. 6. For both phases of the deployment, descent and
ascent, the data in Fig. 6 are from the leading hydrophone,
which was not contaminated by a turbulent wake, unlike its
trailing counterpart. The spatial (depth) resolution of the
spectrogram in Fig. 6 is the speed of travel through the water
column times the total FFT time, which is approximately
0.6 20 ¼ 12 m.
Each vertical slice in the spectrogram in Fig. 6 shows
the ambient noise field as it evolves in both time (lower horizontal axis) and depth (upper horizontal axis) during the
descent and ascent. The onset of the rainstorm can be seen in
the spectrogram around 20 min after deployment, when the
instrument was at a depth of 800 m. The level of high frequency noise, between 3 and 12 kHz, rises abruptly, indicating the addition of rain noise to the ambient noise field. This
initial rain noise level, corresponding to the observed rainfall
rate of 0.2 mm/h, persists until 70 min into the deployment,
at which point the first of three periods of increased rainfall
appears in the spectrogram, labeled as Event 1 (E1) in Fig. 6.
This event lasted approximately 10 min. The second and
third periods of increased rainfall (labeled E2 and E3) are
both 15 min in duration and centered, respectively, at the
100 and 125 min marks of the spectrogram. Elevated noise
levels in the 5 to 20 kHz band occur during these periods in
the spectrogram. Event E3, the third and loudest of these
intense rainfall periods, corresponds with the 12 mm/h rainfall rate measurement made on board the ship, and coincides
with Deep Sound’s arrival at 5.1 km, where the drop weight
was released and the ascent began.
The ascent phase of the deployment begins at the 130min mark in the spectrogram in Fig. 6. The marked decrease
in low-frequency noise after the turnaround is most likely
due either to a decrease in mechanical noise associated with
vibrations of the main body of Deep Sound or to a reduction
in the self-induced flow noise around the leading hydrophone. Both mechanisms could be linked (Urick, 1967) to
FIG. 7. Ambient noise spectrum (black line) with 95% confidence levels
(gray lines), recorded at 4.9 km depth during an 11 mm/h rainfall. The uniform slope of the Knudsen spectrum is shown for comparison.
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The spectral evidence in Fig. 7 for the presence of medium-to-large raindrops as well as small raindrops is consistent with the rain-gauge rainfall rate of 12 mm/h and with the
visual observations of drop size made from the deck of the
R/V Kilo Moana. Moreover, the observed spectral level of
68 dB re 1 lPa2/Hz at 5 kHz at a depth of 5130 m for the
rainfall rate of 12 mm/h is comparable with previous measurements from shallow sensors in lakes (66 dB re 1 lPa2/Hz)
(Bom, 1969), and in shallow seawater (63 dB re 1 lPa2/Hz)
(Nystuen et al., 1993). It is, however, a little higher than the
empirical prediction for a shallow hydrophone in deep water
(59 dB re 1 lPa2/Hz), as obtained from the equation (Ma and
Nystuen, 2005; Anagnostou et al., 2008)
I5 kHz ¼ aRb ;
(1)
where R is the rainfall rate in mm/h, 10 log10(I5 kHz) is the
sound pressure level in dB re 1 lPa2/Hz at 5 kHz,
a ¼ 104.25 ¼ 1.78 104, and b ¼ 1.54.
Figure 8 shows the depth dependence of the noise, averaged over octave frequency bands, as Deep Sound rose from
its maximum depth to the surface. The storm weakened during the ascent from 5 to 4 km, with the rain gauge indicating
a reduction in rainfall rate from 12 to 0.6 mm/h. A corresponding reduction occurs in the octave bands spanning
400 Hz to 13 kHz, while the bands below 200 Hz are hardly
affected. Such behavior is consistent with the spectrum in
Fig. 7, where most of the rain noise can be seen to occur at
frequencies above 1 kHz.
During the ascent from 3 to 1.5 km, the rainfall rate, as
observed on board the R/V Kilo Moana with the rain gauge,
remained relatively constant at 0.8 mm/h, despite passing
squalls. For a center frequency of 5 kHz and a bandwidth of
100 Hz, the noise spectral level averaged over depths from 3
to 1.5 km is approximately 61 dB re 1 lPa2/Hz, which is
about 20 dB higher than the level predicted by the empirical
relationship in Eq. (1). It is, however, within the limits of
variability of previously reported rain noise measurements
(Ma and Nystuen, 2005).
FIG. 8. Octave-band-averaged ambient noise spectral levels during Deep
Sound’s ascent.
D. R. Barclay and M. J. Buckingham: Depth-dependence of rain noise in the ocean
Author's complimentary copy
the difference between the speed of travel on the descent
(0.63 m/s) and the ascent (0.60 m/s). Be that as it may, the
higher-frequency rain noise can be readily distinguished during the remainder of the deployment. By the 150-min mark,
the heavy rainfall has diminished, evident not only in the
spectral levels but also in the mechanically measured rainfall
rate. For the remainder of the deployment, the rain gauge
rainfall rate was between 0.8 and 3.2 mm/h, which is consistent with the acoustic levels in Fig. 6, with the exception of
the 5-min squalls appearing as vertical stripes in the spectrogram, labeled as E4 and E5.
A single spectrum from the upper hydrophone, shown in
Fig. 7 and computed as Deep Sound ascended from 5134 to
5122 m, reveals further evidence that the enhanced ambient
noise levels are due to noise from rain falling on the surface
of the ocean. The spectrum exhibits broad peaks over the 1.5
to 10 kHz and 15 to 25 kHz bands, consistent with previous
observations of rain noise (Nystuen, 2005). As illustrated in
Fig. 7, these peaks show a distinct departure from the
expected wind-driven ambient noise spectral shape of f 5/3
(Wenz, 1962). Incidentally, the spike in the spectrum at
93.3 Hz is an artifact created by the acoustic noise from a
hard drive, located within the glass sphere, which, while
acquiring data, spins at a rate of 5600 rpm. In later versions
of Deep Sound, the hard drive has been replaced by solidstate memory.
The noise of rain on the ocean surface is comprised of
two parts: The impact noise of the drops on the surface, and
the ringing of bubbles entrained by the drops (Franz, 1959;
Medwin et al., 1992; Nystuen and Medwin, 1995). In Fig. 7,
the enhanced spectral energy over the broad frequency band
from 1.5 to 10 kHz is associated with the impact splash of
medium (1.2 to 2.0 mm) or large (2.0 to 3.5 mm) raindrops,
as well as the resulting turbulent entrainment of bubbles of
irregular size. The spectral peak between 15 and 25 kHz corresponds to bubble entrainment by small (0.8 to 1.2 mm)
drops (Pumphrey et al., 1989).
V. VERTICAL COHERENCE AND DIRECTIONALITY
The coherence function relating two time series, x1(t)
and x2(t), is defined as the cross-spectral density function
normalized to the geometric mean of their power spectra
X1 ðxÞX2 ðxÞ
ffi;
C12 ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jX1 ðxÞj2 jX2 ðxÞj2
(2)
where x is angular frequency, Xk(x) is the Fourier transform
of xk(t), k ¼ 1 or 2, and the overbar and asterisk denote,
respectively, an ensemble average and complex conjugation.
Cox (1973) has shown that, for hydrophones configured vertically, as in Deep Sound, the coherence function of a plane
wave noise field is uniquely related to the vertical directionality of the noise through a finite Fourier transform over the
vertical angle:
ð
1 p
Fðx; hÞeix cos h sin hdh;
(3)
C12 ðxÞ ¼
2 0
pffiffiffiffiffiffiffi
where i ¼ 1 and F(h) is the directional density function,
a dimensionless measure of the noise power per unit angle
incident on the receivers from the polar (declination) angle
h, as measured from the zenith. The normalized angular fre is defined as
quency, x,
¼
x
xd
;
c
(4)
where d is the separation of the sensors in a medium of sound
speed c. Since the noise is fully coherent when the two sensors
are coincident, it follows from Eq. (3), with d ¼ 0, that
ð
1 p
Fðx; hÞsin hdh ¼ 1;
(5)
2 0
which is a normalization condition on F(h).
In general, the coherence function in Eq. (3) is complex
but for the special case in which the directional density function is symmetrical (even) about the horizontal, the imaginary part of C12 (x) is zero and the coherence function is
real. If the directional density function is independent of x,
rather than x
then the coherence function depends only on x
and d individually, in which case plots of C12 (x) versus x
will all collapse onto the same curve, regardless of the separation of the sensors.
With the two sensors aligned vertically, as in Deep
Sound, the axial symmetry prevents any azimuthal variations
J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013
in the noise field from being detected. In effect, the noise
incident at polar angle h is an integration of the contributions
from the azimuthal interval [0 2p].
VI. TURBULENCE
It is implicit in Eq. (3) that the noise field is spatially homogeneous, consisting entirely of independent, acoustic plane
waves. The noise fields observed with Deep Sound, however,
do not wholly satisfy this condition because the trailing hydrophone lies in the wake of the leading hydrophone and is therefore subject to turbulent-flow pressure fluctuations in addition
to the plane wave acoustic noise of interest.
To establish the effect of turbulence on the coherence
function, let the total noise recorded at each hydrophone be
written as the sum of an acoustic component, x(t), and a turbulent fluctuation, n(t), as follows:
sk ðtÞ ¼ xk ðtÞ þ nk ðtÞ;
(6)
where the subscript k ¼ 1 or 2 identifies the hydrophone in
question. From the definition in Eq. (2), the coherence
between the two signals is
½X1 ðxÞ þ N1 ðxÞ½X2 ðxÞ þ N2 ðxÞ
ðsÞ
;
C12 ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jX1 ðxÞ þ N1 ðxÞj2 jX2 ðxÞ þ N2 ðxÞj2
(7)
where the upper case letters are the Fourier transforms, with
respect to time, of their lower case counterparts. The superscript (s) in Eq. (7) identifies the coherence as being that due to
the combined effect of the acoustic and turbulence fluctuations.
Since Deep Sound descends and ascends at a steady
rate, the level of turbulence noise at each of the hydrophones
should remain constant over time, and the acoustic noise is
taken to be statistically stationary, at least over the averaging
time used to compute the coherence. Under these conditions,
the turbulence spectrum may be expressed as a fixed fraction
of the acoustic noise spectrum through a frequency dependent signal-to-noise coefficient ek(x),
jNk ðxÞj2 ¼ ek ðxÞjXk ðxÞj2 :
(8)
It follows that Eq. (7) reduces to the form
1
X1 ðxÞX2 ðxÞ
ðsÞ
ffi;
C12 ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½1 þ e1 ðxÞ½1 þ e2 ðxÞ jX ðxÞj2 jX ðxÞj2
1
2
(9)
where the last term is just the coherence of the acoustic
noise, C12 (x). Thus, the observed coherence is a scaled version of the ambient noise coherence
ðsÞ
C12 ðxÞ ¼ G12 ðxÞC12 ðxÞ;
(10)
where
1
G12 ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
½1 þ e1 ðxÞ½1 þ e2 ðxÞ
D. R. Barclay and M. J. Buckingham: Depth-dependence of rain noise in the ocean
(11)
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The disparity between the Deep Sound measurement
and Eq. (1) could be associated with the fact that the latter
was established for relatively shallow depths, no more than
98 m. At the significantly greater depths of the measurement,
acoustic sources covering a much larger area of the sea surface contribute to the overall noise level. This allows for the
possibility that the measurement is an average over squalls
or patches of heavy rain that may not have been registered
by the rain gauge, which would lead to an underestimate of
the noise level from Eq. (1).
is the frequency dependent scaling factor.
If either of the ek(x) is much greater than unity, implying that turbulence noise is dominant at one or both hydrophones, the function G12(x) approaches zero and the
ðsÞ
observed coherence, C12 (x), is negligibly small. However,
for lesser values of the ek(x), representing lower levels of
turbulence noise, the observed coherence will be a reduced
version of the coherence due to the acoustic noise alone. In
this case, the critically important properties of C12 (x), notably the zero crossings, should be recoverable.
VII. RAIN NOISE COHERENCE
VIII. RECOVERING F(h) FROM THE COHERENCE
FUNCTION
J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013
assume that the directional density function is independent
of frequency, at least over the principal rain noise band from
1 to 10 kHz. To determine the vertical directionality of the
rain noise, the directional density function is represented as a
piecewise continuous function in the form of a linear combination of angular segments
FðhÞ ¼
N
X
An ½uðan dÞ uðan þ dÞ;
(12)
n¼1
where An and an are, respectively, the intensity and arrival
direction of the nth segment, d is the angular half-width of
each segment, N ¼ p/(2d) and u(.) is the Heaviside unit
step function. The next step is to determine the unknown
intensities, An, from the rain noise data, at which point the
directional density function in Eq. (12) will be fully
characterized.
From Eq. (3), the vertical coherence function corresponding to the directional density function in Eq. (12) is
¼
CðxÞ
Since the multi-frequency storm sources are all located
in the same region of the sea surface, it is reasonable to
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FIG. 9. (a) The real and (b) the imaginary parts of the vertical coherence of
ambient noise at 4.9 km depth during an 11 mm/h rainfall.
N
X
n¼1
An
cos an cos dsin½x
sin an sin d
exp½ix
:
x
(13)
D. R. Barclay and M. J. Buckingham: Depth-dependence of rain noise in the ocean
Author's complimentary copy
By using an ensemble averaging procedure similar to
that described above in connection with the power spectra,
the coherence function was computed from the noise
acquired by the two vertically aligned hydrophones on Deep
Sound. For the coherence computations, the total averaging
time remained the same at 20 s but the length of the individual FFTs was reduced to 2^16, half that used for the power
spectra. This doubles the frequency cell width to 3.125 Hz,
but the depth resolution is unchanged at 12 m. In essence,
the reduced FFT length doubles the number of terms in each
ensemble average, which leads to smoother coherence
curves. The coherence is significant, with 95% confidence,
for values greater than 0.27 (Thompson, 1979).
Figure 9 shows the real and imaginary parts of the coherence function, plotted on semi-logarithmic axes over the
frequency band from 5 Hz to 20 kHz. The data, which are
from a depth of 5 km on the ascent of Deep Sound, were
recorded during an intense period of rainfall, estimated from
the rain gauge at 11 mm/h. As in the power spectrum in
Fig. 7, the spike at 93.3 Hz is an artifact due to the spinning
hard drive on board Deep Sound.
Over the 1 to 10 kHz frequency band in Fig. 9, the coherence data are dominated by rain noise, implying that at
these frequencies the turbulence parameters e1(x) and e2(x)
are negligible. Above 10 kHz and below 300 Hz the coherence is not significantly different from zero, implying large
values for e1(x) and e2(x). In the transition region between
300 Hz and 1 kHz, clearly visible in the semi-logarithmic
plot, the coherence is a blend of incoherent turbulence noise,
which increases with decreasing frequency (McGrath et al.,
1977; Finger et al., 1979), and surface generated rain noise.
The reasonably strong signature of the rain noise in
Fig. 9 suggests the possibility of using the directional density
function of the noise to locate the storm (at least within an
annulus coaxial with the hydrophones) as it moves over the
sea surface. If the noise were white, with an infinitely broad,
flat power spectrum, the directional density function would
be obtainable directly as a mapping of the cross-correlation
function (Buckingham, 2011). Since the rain noise is band
limited with a spectrum that is far from white, an alternative
technique is required.
A further constraint on the intensities An derives from the
normalization condition in Eq. (5), which requires that
n¼1
An sin an ¼
1
:
sin d
(14)
To derive the unknown intensities, an iterative procedure
is used in which Eq. (13) is evaluated for a given set of An and
compared with the rain noise coherence data. The values of
the An are then adjusted to minimize the difference between
the modeled and measured coherence over the frequency
band from 1 to 10 kHz. The fit between theory and data is
achieved using a nonlinear constrained minimization algorithm in which the normalization condition in Eq. (14) serves
as the constraint. The search algorithm is centered on sequential quadratic programming (Nocedal and Wright, 1999), a
constrained version of the Newtown-Raphson method. In
order to accommodate the uncertainty in the data, a reasonable best-fit tolerance for the nonlinear constrained minimization algorithm was determined by first applying a brute-force
search to a single inversion, from which the minimum rootmean-square difference between the measured and modeled
coherence was calculated. This systematic search over the
entire model domain showed that the optimized solution was
unique, within the resolution of the search.
The angular resolution of the discrete directional density
function used in the model is 18 (d ¼ 9 ), which corresponds to N ¼ 10 in Eq. (13). The value of N determines the
number of equations in the constrained linear inverse problem stated in Eqs. (13) and (14) and therefore must be chosen with care. When N is too small, the root-mean-square
difference between the measured and modeled coherence is
large and the uncertainty on the inversion is large. When N
is too large the system becomes over-determined and the
search algorithm responds by setting the weighting coefficients [the An in Eq. (13)] to zero. A suitable value of N was
found by running the inversion iteratively, increasing N with
each step, which decreased the root-mean-square error
between the modeled and measured coherence, but stopping
before any weighting coefficients were found to be zero.
Figure 10 shows the best-fit coherence curves, as
derived from the constrained minimization algorithm, superimposed on the real and imaginary parts of the rain noise coherence data from a depth on the ascent of 4.9 km, about
1 km below the critical depth. It is clear that the model
matches the data remarkably well over most of the frequency
band. Note, in particular, that the zeros between 1 and
10 kHz are extremely well represented by the model. It is
also worthy of comment that these rain noise zeros differ significantly from the zeros of the coherence function predicted
by the Cron and Sherman theory (Cron and Sherman, 1962,
1965; Buckingham, 2011), which is based on a uniform distribution of noise sources located immediately beneath the
surface of an isovelocity ocean.
FIG. 10. (Color online) (a) The real and (b) the imaginary parts of the measured (jagged curve) and best-fit (smooth line) vertical coherence at 4.9 km
depth.
between model and data in Fig. 10, is shown in Fig. 11. Any
bias from the pitch and roll of the Deep Sound platform has
been removed from the estimated directionality. Most of the
noise is downward traveling and concentrated in a lobe about
30 wide, centered in the region of 50 below the vertical.
Since the depth was 4.9 km, this places the storm center at a
IX. RAIN NOISE DIRECTIONALITY
The discrete directional density function, computed
from the set of intensities, An, as derived from the match
J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013
FIG. 11. Vertical directional density function derived from the best-fit to the
noise coherence data recorded at 4.9 km depth during an 11 mm/h rainfall.
D. R. Barclay and M. J. Buckingham: Depth-dependence of rain noise in the ocean
2583
Author's complimentary copy
N
X
horizontal distance from Deep Sound of approximately
6 km.
The same fitting procedure may be used to obtain the
intensities, An, for all depths, returning the directional density function versus depth, as shown in Fig. 12 for the 120min ascent from 5100 to 800 m, close to the sound channel
axis. In computing this figure, the FFT averaging procedure
was the same as that used for the power spectra, providing a
depth resolution of 12 m. At all depths, the arrival angles of
the noise lie within a 70 cone about the vertical, with essentially no noise arriving from below the horizontal.
Several moments of intense rainfall, as well as the angular positions of the storm, can be seen in Fig. 12. In several
cases, the rain conditions on the surface can be correlated
with the rain gauge observations made on board the R/V
Kilo Moana. For instance, the peak between 4.6 and 5 km is
associated with the storm event marked as E4 in Fig. 6, for
which the measured rainfall rate was 11 mm/h. Twelve
minutes later, with Deep Sound at a depth of 4.5 km, the
peak at h ¼ 25 indicates that the storm had moved toward
the zenith. By the time Deep Sound had ascended to 4.2 km,
the intensity was relatively weak in all directions, corresponding to a 15-min pause in the rainfall rate observed on
the Kilo Moana. At a depth of 3.8 km, the storm had intensified again, with most of the acoustic energy arriving at Deep
Sound from more or less directly overhead. This corresponds
to the brief 5-min squall designated E5 in Fig. 6. Throughout
the remainder of the ascent, the measured rainfall rates
stayed below 3 mm/h but a few intense squalls were
observed on the surface, with corresponding features appearing in the directional density function in Fig. 12, at depths
from 3 km upwards.
X. CONCLUDING REMARKS
During PhilSea09 in May 2009, the instrument platform
Deep Sound, with two vertically aligned hydrophones on
board, made three descents, to depths of 5.1, 5.5, and 6 km.
On the first of these deployments, the noise from a local rainstorm was recorded during the descent and ascent. The rainfall rates varied between 0.2 and 12 mm/h, as measured on a
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J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013
ACKNOWLEDGMENTS
The authors wish to acknowledge Fernando Simonet
and Kevin Hardy for their assistance with the development
of Deep Sound. The efforts of the captain and crew of the R/
V Kilo Moana to ensure the safe deployment and recovery
of Deep Sound during the Philippine Sea experiment are
greatly appreciated, as is the assistance of the NPAL scientists during the research cruise. The research reported in this
paper was supported by the Office of Naval Research, Ocean
Acoustics Code 322OA, under Grant Nos. N00014-10-10092 and N00014-10-1-0286, sponsor Dr. Robert Headrick.
D. R. Barclay and M. J. Buckingham: Depth-dependence of rain noise in the ocean
Author's complimentary copy
FIG. 12. (Color online) Vertical directional density function as it evolves
over depth (left axis) and time (right axis) during the ascent of Deep Sound.
rain gauge on board the R/V Kilo Moana, some 8 km away
from the drop site.
The spatial and temporal variability of the rainstorm are
reflected in the ambient noise spectra returned by Deep
Sound. Two spectral peaks appear, centered at 5 and 30 kHz,
which are associated with the impact of medium to large
drops and bubble entrainment, respectively. Even at a depth
of 5 km, the effect of the heavier rain was to raise the noise
spectral level by as much as 20 dB.
From the noise fluctuations recorded at the two hydrophones, the vertical coherence function was computed over a
bandwidth of 20 kHz. During periods of intense rain, the
coherence function is very well defined, showing zeros that
are characteristic of a localized surface source. These zeros
differ significantly from those expected from a uniform distribution of noise sources over the whole sea surface, suggesting that the directional density function of the storm
noise is sharply peaked in the direction of the storm.
To determine the directional density function, a
curve-fitting procedure is introduced in which a model of the
coherence function is fitted to the coherence data. A set of
coefficients is returned which, in effect, specify the directional
density function. It turns out that the rain noise is, indeed,
strongly directional, consisting of downward traveling waves
propagating at angles lying within a cone which may extend
out to 70 from the vertical, depending on the relative position
of the storm on the sea surface.
During periods of heavy rainfall, the directional density
function shows a strong lobe centered at an angle that
depends on depth as well as the horizontal distance between
the storm and Deep Sound. At a depth of 5 km, this rainnoise lobe is about 50 from the vertical, from which the
storm is estimated to have been at a distance of 6 km from
the instrument. Fortuitously, the storm passed more or less
directly over Deep Sound and, as it approached, the peak in
the noise directional density function showed a corresponding movement toward the zenith.
The acoustic measurements from Deep Sound during
PhilSea09 provide a means of tracking the storm continuously, at least in horizontal range, that is to within an
annulus centered on the sensor station. In more recently
developed versions of Deep Sound, the axial symmetry of
the vertically aligned sensors is broken by the addition of
several horizontally displaced hydrophones, opening up
the possibility of tracking a storm in both range and
azimuth.
J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013
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