the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Acoustic wave propagation in rivers: an experimental study

### Ludovic Michel

### Sébastien Zanker

### James Robert Rigby

This research has been conducted to develop the use of passive acoustic monitoring (PAM) in rivers, a surrogate method for bedload monitoring. PAM consists in measuring the underwater noise naturally generated by bedload particles when impacting the river bed. Monitored bedload acoustic signals depend on bedload characteristics (e.g., grain size distribution, fluxes) but are also affected by the environment in which the acoustic waves are propagated. This study focuses on the determination of propagation effects in rivers. An experimental approach has been conducted in several streams to estimate acoustic propagation laws in field conditions. It is found that acoustic waves are differently propagated according to their frequency. As reported in other studies, acoustic waves are affected by the existence of a cutoff frequency in the kilohertz region. This cutoff frequency is inversely proportional to the water depth: larger water depth enables a better propagation of the acoustic waves at low frequency. Above the cutoff frequency, attenuation coefficients are found to increase linearly with frequency. The power of bedload sounds is more attenuated at higher frequencies than at low frequencies, which means that, above the cutoff frequency, sounds of big particles are better propagated than sounds of small particles. Finally, it is observed that attenuation coefficients are variable within 2 orders of magnitude from one river to another. Attenuation coefficients are compared to several characteristics of the river (e.g., bed slope, surface grain size). It is found that acoustic waves are better propagated in rivers characterized by smaller bed slopes. Bed roughness and the presence of air bubbles in the water column are suspected to constrain the attenuation of acoustic wave in rivers.

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## 1.1 Context of this study

Bedload transport monitoring is a challenging issue for river management. Geomorphological changes may be driven by anthropogenic uses of rivers (e.g., hydroelectricity, sediment dredging, embankment, mining, land use changes) or by changes in available sediment loads related to extreme events or climate changes. Bedload transport is a dominant factor governing fluvial morphology but monitoring bedload transport is a difficult task. Direct sampling of bedload flux requires intensive field work and is difficult to accomplish during flood conditions. Long-term, automatic and continuous measurements of bedload materials have already been achieved with direct sampling (e.g., Turowski and Rickenmann, 2009) but such a monitoring is typically expensive and technically challenging. This is why the development of surrogate (or indirect) methods has been studied in recent decades. The report of Gray et al. (2010) gives an overview of available techniques. One of these methods concerns the use of bedload self-generated noise (SGN). When bedload particles impact the river bed, an acoustic noise is created that propagates in the water column. Bedload SGN can be measured using hydrophones that are deployed in the river. Theoretical and experimental studies have shown that the acoustic power monitored by hydrophones can be related to bedload fluxes using power laws (Barton et al., 2010; Geay et al., 2017a; Johnson and Muir, 1969; Jonys, 1976; Marineau et al., 2016; Rigby et al., 2016a; Thorne, 2014). Some relations are also observed between bedload granulometry and spectral characteristics of SGN signals (Geay et al., 2017a; Thorne, 2014).

However, monitored signals are not only dependent on bedload SGN but also on propagation effects (Geay et al., 2017b; Rigby et al., 2016a). When propagating in rivers, bedload SGN suffers from geometrical spreading losses (Medwin, 2005), multiple diffractions on rough boundaries (Wren et al., 2015) or from other attenuation processes, for example, related to the occurrence of suspended load (Richards et al., 1996). Therefore, acoustic waves are modified by the environment along their propagation paths, from noise sources to hydrophone measurements. It has been shown that the river could be modeled as an acoustic wave guide where acoustic waves are partially trapped between the water surface and the river bed (Geay et al., 2017b). The occurrence of a cutoff frequency (related to the Pekeris wave guide) has been observed in field experiments (Geay et al., 2017b; Lugli and Fine, 2007) and reported in a theoretical review (Rigby et al., 2016b). A laboratory study focused on the role of river bed roughness as a source of attenuation process (Wren et al., 2015): an increase of 4 dB with an increasing bed roughness of 20 mm has been observed. There is comparatively little literature in the range of frequencies of interest (i.e., 0.1 to 100 kHz) and none of these studies have done specific experiments to define acoustic propagation laws in field experiments. For this reason, we designed a new protocol enabling the determination of propagation laws in rivers. These experiments result in experimental laws that are useful for building direct or inverse models, which is necessary to analyze bedload SGN signals. For example, it could be used to better understand the measurement range of a hydrophone in a river, a question which remains unknown.

The next section of the paper relates a simple theoretical framework that is used to analyze field data. The second part of this paper describes the protocol which is based on emitting a known signal with an active source (i.e., an underwater speaker) and on measuring this same signal at several distances from the source. The third part is related to the application of this protocol in a set of rivers that have different morphology (e.g., water depth, slope, flow velocities, bed roughness). The variation of propagation properties is observed from one river to another and related to river characteristics.

## 1.2 Theoretical framework

Acoustic measurements are in part determined by the ability of the
environment to propagate sounds. In this section, an acoustic theory is
proposed to model the loss of acoustic power with the distance of
propagation. At a first stage, without attenuation processes, the monitored
power (*P* – µPa^{2}) of a sound source decreases
with distance from the point source as the energy spreads in space:

where *r* (m) is the distance from the source to the sensor,
*P*_{@1 m} (µPa^{2} @1 m) is the initial
power of the sound source monitored at 1 m in a free field, and
*G* is a function depicting geometrical spreading. The
geometry of the river is simplified as a rectangular channel with a uniform
water depth, denoted *h*. For underwater acoustic waves
propagating in a river, the medium is bounded by the water surface and the
river bed. The effect of river banks is not explicitly considered in this
study. It is assumed that banks act as efficient sound absorbers. At the
upper and lower interfaces, reflection coefficients are variables, depending
on the geo-acoustic parameters of the river bed (Geay
et al., 2017b) and on the roughness of the interfaces
(Wren et al., 2015). Two extreme cases can be
assumed. First, when the interfaces are perfectly reverberant, acoustic
waves are totally trapped into the water column and acoustic waves propagate
in a cylindrical way. For large distance of propagation (i.e.,
*r*>*h*),

Secondly, when the interfaces are highly absorbing (as in an anechoic chamber), acoustic waves propagate in a spherical mode as in a free space:

In the following, both propagation laws (spherical or cylindrical) will be tested to fit field data.

Acoustic waves not only suffer from geometrical spreading but also from losses from other processes that attenuate sounds like absorption or scattering effects. As stated in ocean studies, it is not really possible to distinguish both effects in field experiments (Jensen et al., 2011). In this study, we propose to quantify these effects in a single exponential term as written in the following equation:

where *α* is a coefficient of attenuation
(nepers m^{−1}), *α*>0.

The attenuation of acoustic waves is a process which is frequency dependent.
That is why it is common to express the coefficient of attenuation as a
function of wavelength (Jensen et al., 2011),
denoted here *α*_{λ} (nepers):

where *λ* is the wavelength (m), *c* is the velocity of sound in water
(m s^{−1}), and *f* is the frequency (Hz).

The goal of this study is to experimentally determine the values of the
attenuation coefficients for acoustic waves in rivers, for frequencies
between 1 and 100 kHz. This range of frequency corresponds to the expected
range of frequencies generated by bedload self-generated noise of particles
size between 10^{−1} and 10^{−3} m (Thorne, 2014).

An experimental setup was designed to measure the loss of acoustic power with distance of propagation in natural streams. A controlled sound source emits a known signal at a fixed position on the river bed and this same signal is monitored with a hydrophone at several distances from the point of emission. The equipment and the protocol are described hereafter.

## 2.1 Sound source

The sound source is generated by an underwater loudspeaker (Lubell Labs, LL 916H) controlled by an electronic device designed by the RTSys Company. The loudspeaker has a frequency response of ±10 dB between 0.5 and 21 kHz, enabling the generation of sounds in this spectrum. The generated sound is determined by a theoretical signal (i.e., a wave file) and reproduced with a bias linked to the transfer function of the loudspeaker. The theoretical signal, chosen for this study, is a logarithmic chirp varying from 0.2 to 50 kHz in 1 s, a bit larger than the theoretical frequency response of the loudspeaker. This signal is continuously emitted by the loudspeaker in an endless loop. In a preliminary study, several tests have been conducted in Lac du Bourget (France) to characterize the response of the system.

To measure the generated sound at different angles from the speaker, four
hydrophones (HTI-96) were placed at a fixed distance of 0.7 m from the sound
source (Fig. 1a). HTI-96 hydrophones have a flat
frequency response between 2 Hz and 30 kHz (±2 dB), enabling absolute
measurement of the acoustic power in this frequency range. The entire system
was deployed in a lake with an aluminum structure to ensure the relative
position of the sensors (Fig. 1b). To minimize
the effect of this structure, all the sensors were attached to the structure
with free ropes of 10 cm length. Several measurements of the emitted sounds
were made by varying the depth of the system from 0.5 to 3.5 m and by
changing the orientation of the loudspeaker (horizontal or vertical). The
power spectral density (PSD) of each emitted chirp monitored by the four
hydrophones has been computed and plotted all together
(Fig. 2). It can be observed that the generated
sounds have a spectral power between 10^{12} and 10^{14} µPa^{2} Hz^{−1} but do not have a flat frequency response due to the transfer
function of the system. Overall, we observed that the monitored PSD was
variable between the different tests that were conducted. The monitored
power varied between ±3 dB between the 25th and 75th quartiles, and between ±10 dB between the minimum and the maximum. The most important
parameter influencing the emitted sounds was the directivity of the
loudspeaker (horizontal or vertical positions). The emitted signals also did
not vary when repeating the same signal in a fixed configuration of
emission.

This preliminary study indicated that we would not be able to precisely
predict the power emitted by the sound source during our experiments. The
loudspeaker is deployed with a weighted rope from a bridge so that its
orientation is uncertain when deployed on the riverbed. We therefore have
an uncertainty concerning the initial power of the sound source
(*P*_{@1 m}) defined in the Eq. (1). This
parameter will therefore be estimated for each experiment.

## 2.2 Hydrophone measurements at varying distances

Acoustic measurements were performed with HTI-96 hydrophones plugged to a EA-SDA14 recorder (RTSys company). Acoustic signals were stored in wav files at a sampling frequency of 156 kHz. The acoustic recorder and the hydrophone are shared by a Carlson river board, drifting during the measurements (Fig. 3). Lagrangian measurements were preferred to fixed-position measurements to optimize the signal-to-noise ratio. By measuring when drifting, the noises generated by the resistance of the river board against the flow are drastically reduced. The hydrophone was located under the river board at a constant depth from the water surface. The underwater loudspeaker is deployed at a fixed position on the river bed and emits a logarithmic chirp with an infinite loop of 1 s. During this time, several drift trajectories were made with the river board along the cross-section. As a first step, acoustic measurements were positioned using a synchronized GPS. This GPS equipment was damaged during the first field experiments requiring another way to position the hydrophone during the drifts. The cross-sectional distance of the hydrophone was monitored at start positions and considered as constant during the drift (i.e., drifts are considered parallel to the river banks). Secondly, longitudinal positions of the hydrophone during the drift were computed knowing the start position and by assuming a constant velocity of the river board:

where *x* and *y* are respectively the
cross-sectional and longitudinal positions of the hydrophone (m);
*x*_{0} and *y*_{0}
are the initial positions of the hydrophone monitored at the beginning of
the drift; *v*_{drift} is the mean velocity of
the river board during the drift (m s^{−1}), computed as the traveled distance
divided by the duration of the drift. The assumption of parallel drifts at
a constant velocity was supported by the fact that our field sites are
straight reaches.

Finally, the position of the hydrophone is known over time. The next section describes how the hydrophone signals were processed.

## 2.3 Signal processing of the monitored acoustic waves

The use of a matched filter was chosen to detect the chirps in the
hydrophone signals. When a chirp is detected, the position of the
measurement is computed by matching the time of detection with the position
of the hydrophone. Finally, knowing the position of the loudspeaker, the
distance *r*, between the sound source and the measurement,
is computed.

For each located chirp, a short-term spectrogram is computed using Hamming
windows of 2^{12} points with 50 % overlapping
(Fig. 4). Based on this spectrogram, several
PSDs are computed. First, the PSD of the studied chirp (noted
PSD_{r}) is computed by using the signal
contained inside the black lines. The black lines correspond to the upper
and lower limits of the octave band centered around the instantaneous
frequency of the chirp. Secondly, the 95th percentile of the monitored power
is computed (Merchant et al., 2013) in each
frequency band. This PSD is used to represent the power of the ambient noise
(PSD_{95}). In this example
(Fig. 4), one can particularly observe the
harmonics generated by the loudspeaker when reproducing the theoretical
logarithmic chirp. The ambient noise depends on the sounds that are
naturally generated in the river (e.g., bedload impacts). To ensure that the
chirp is not affected by ambient noise, we decided to keep only the chirps
that are at least twice as powerful as the ambient noise (i.e.,
PSD_{r}>2 PSD_{95}).

At this point, we can propose a protocol to monitor the PSD of an emitted chirp at varying distances from its point of emission.

## 2.4 Fitting propagation laws

The acoustic power of each chirp measured at a distance *r*
was computed by integrating PSD_{r} in
third-octave bands. For the *j*th third-octave band, *P*_{i,j} is the
acoustic power of the *i*th measurement made at a distance *r*_{i} from
the loudspeaker. Using the theoretical model (Eq. 4) and assuming one model
of geometric spreading loss (cylindric or spherical), the estimated acoustic
power $\stackrel{\mathrm{\u0303}}{{P}_{i,j}}$ as a function of *r*_{i} is

where *P*_{@1 m,j} and *α*_{j} are parameters to fit for each third-octave band *j*. These parameters were estimated with a non-linear least-square algorithm on the log values of power. It means that *P*_{@1 m,j} and *α*_{j} were estimated by minimizing the following term:
$\sum _{i=\mathrm{1}}^{N}{\left[\mathrm{log}\left(\stackrel{\mathrm{\u0303}}{{P}_{i,j}}\right)-\mathrm{log}\left({P}_{i,j}\right)\right]}^{\mathrm{2}}$, where
*N* is the total number of observed chirps.

For each frequency band, the fit was characterized by a coefficient of
correlation between the log values of the estimated power
($\stackrel{\mathrm{\u0303}}{{P}_{i,j}}$) and the log values of the measured power (*P*_{i,j}).
Finally, the residuals (dB) of the fits were computed using the following
relationship: $\frac{\mathrm{1}}{N}\sum _{i=\mathrm{1}}^{N}\left|\mathrm{10}\mathrm{log}\left(\frac{\stackrel{\mathrm{\u0303}}{{P}_{i,j}}}{{P}_{i,j}}\right)\right|$. The residuals
are the average variation of the data set around the fitted law, it
represents the dispersion of the data set.

In summary, the source power (*P*_{@1 m}) and the attenuation coefficient
(*α*) were estimated by fitting a propagation law (Eq. 4) to
power measurements made at several distances from the loudspeaker.
Estimations were made by considering third-octave bands, therefore enabling
the estimation of *P*_{@1 m} and *α* in several frequency bands. Note
that these estimations were done by considering either a cylindrical (Eq. 2)
or spherical model (Eq. 3).

## 2.5 Field sites

The protocol presented in the previous section was applied in seven field sites
located in the French Alps. Their characteristics are presented in
Table 1. The mean bed slope of the studied reaches
varies from 0.05 % to 1 %, and the width of the cross-section from 8 to
60 m. The roughness (or the surface particle size distribution) of the river
bed is a difficult parameter to measure, particularly in rivers that are not
wadable. This aspect of bed roughness was approached by doing Wolman
measurements on the closest emerged bars. The surface *D*_{84} of emerged
bars varies from 20 to 150 mm. Hydraulic parameters (discharge, surface
velocity and mean water depth) were obtained by using several methods
(acoustic Doppler current profiler, surface velocity radar (SVR) gun or existing gauging
station) depending on the field sites. Finally, the measurement of suspended
sediment load was achieved with a turbidimeter (Visoturb, WTW).

## 3.1 The Leysse River

Data from the Leysse River are presented in
Fig. 5 (see also the Supplement). It
represents the acoustic power received by the hydrophone at different
distances from the underwater loudspeaker. As an example, the results
obtained with the third-octave band centered on 1 kHz are shown in Fig. 5;
this data set has been obtained with 27 drifts of the river board. The
effect of source location has been tested by varying the source location in
the river cross-section. It has been found that the result was insensitive
to source location in this river. Spherical and cylindrical models of
propagation losses have been fitted with a least-square procedure on the
logarithmic values of the acoustic power. Two parameters are obtained: the
initial power of the sound source (*P*_{@1 m}) and
an attenuation coefficient (*α*). This procedure was
repeated on each third-octave band to obtain the variation of these
parameters with frequency (see the Supplement).

Results of the fits are shown in Fig. 6 for
the Leysse River experiment. Logically, attenuation coefficients that are
estimated with cylindrical spreading loss exhibit higher values than
coefficients estimated with spherical spreading loss. However, they behave
similarly with frequency variations. At low frequency, approximatively below
10^{3} Hz, the attenuation coefficient is higher. This result was expected
because of the existence of a cutoff frequency (Geay et
al., 2017b). The cutoff frequency is dependent on the water depth (mean
water depth of 0.95 m), the sound speed in water (assumed to be equal to
1500 m s^{−1}) and the sound speed in the sediment layer. Typical values of sound
speed in sea floor materials (from silt to gravel) were observed to vary
between 1550 and 2000 m s^{−1} (Jensen et al., 2011),
depending on many factors such as the type of materials, grain sizes or
porosity (Hamilton and Bachman, 1982). Using sound speed of 1550
and 2000 m s^{−1} in the sediment leads to cutoff frequencies of 1500 and
600 Hz, respectively, which is consistent with our observation.

Above 10^{3} Hz, attenuation coefficient increases with frequency:
acoustic waves are more attenuated at higher frequencies. Considering the
estimation of the sound source power, it is observed that the cylindrical
model best reproduces the power monitored in the experiment made in Lac du
Bourget (the median value is represented in
Fig. 6b). Using a spherical model, we
overestimate the power of the sound source by approximatively 1 order of
magnitude. However, as we will see for other experiments, the best
estimation of the sound source power is sometimes obtained with spherical
spreading loss model.

In Fig. 6c, the residuals of the regression represent the dispersion of the data around the fit. It has been computed as the mean square difference between data and fits. In the Leysse River, we observed that the power of the reception fluctuates between 2 and 3 dB around the fits.

Finally, considering the correlation coefficients of the fitted laws (Fig. 6d), we cannot make a distinction between spherical or cylindrical spreading loss models.

## 3.2 Propagation laws in several rivers

Propagation properties of several rivers were investigated. For some of the
rivers, experiments were done at different hydrodynamic conditions
(Table 1). For the discharge investigated,
hydrodynamic conditions were not variable enough to observe major
differences in the results. We therefore decided to gather data to propose a
unique result for each river. The data set is presented in the Supplement, for all rivers and for different frequency bands. A first result
concerns the estimated power of the sound source
(*P*_{@1 m}) emitted during the experiments
(Fig. 7). Compared with the measurements made in
Lac du Bourget, it can be observed that the estimation of the sound source
power is overestimated when using a spherical model and underestimated when
using a cylindrical model of the geometric spreading loss. Considering the
correlation coefficients of the data to the fits, we did not observe a
significant difference between the models. Based on these observations, we
are not able to argue that geometric spreading is cylindrical or spherical
in these rivers. In the following, all the results are presented by assuming
a cylindrical, spreading loss model.

The attenuation coefficients obtained for each river are presented as a function of frequency (Fig. 8). From the Isère to the Arve river, we can observe that the attenuation coefficient varies by more than 1 order of magnitude (Fig. 8b). Looking at the linear representation (Fig. 8a), we see that the variation of the attenuation coefficient with frequency is different from case to case. It increases faster for rivers having the largest attenuation coefficients. Note that minimal and maximum frequencies of the observations are variable from one river to another. At low frequency, observations are limited by the cutoff frequency which is inversely proportional to the water depth (Geay et al., 2017b). At high frequencies, measurements are limited by too-strong attenuation of the emitted acoustic waves.

Table 2 contains, for each river, a summary of the results obtained by fitting a cylindrical propagation model to the data. All the parameters indicated in this table are an average of the values obtained between 1 and 10 kHz. It can be observed that the correlation coefficients vary from 0.4 to 0.8. We observed that the lowest correlation coefficients were obtained for the largest rivers (Isère and Romanche rivers with section widths of 60 and 33 m, respectively) and may be representative of cross-sectional variations that have not been considered in this study. The residuals vary from 2 to 6 dB. Rivers having largest attenuation coefficients seem to have larger residuals: the dispersion of the monitored acoustic power is larger when the attenuation is larger. Finally, the maximum distance of the monitored chirps represents the maximum distance from the hydrophone to the underwater speaker where we were able to record the chirps with a sufficient signal-to-noise ratio. The smaller the attenuation coefficient, the larger the maximum distance of the observation. Note that the maximum distance is also dependent on operational issues.

## 4.1 Attenuation processes in rivers

During our field campaign, it has been found that attenuation coefficients
were variable from one river to another. The attenuation due to freshwater
varies from 10^{−9} to 10^{−3} nepers m^{−1} from 1 to 100 kHz
(Fisher and Simmons, 1977). The attenuation due to water
only does not explain the coefficients of attenuation that were found in this
study. In this section, we wonder how propagation properties are related to
typical characteristics of the rivers (e.g., slope, water depth). As shown in
Fig. 8, the dependency of the attenuation
coefficient to frequency does not follow a simple law.

At low frequency, around 1 kHz, acoustic wave propagation should be affected
by wave guide properties. The river could be considered as an acoustic wave
guide where sounds are partly trapped between the water surface and the
river bed (Geay et al., 2017b): this problem is known
as the Pekeris wave guide. Theoretically, in a perfect medium without
attenuation, it can be shown that acoustic waves having frequencies lower
than the cutoff frequency are exponentially decaying with horizontal
distance (Jensen et al., 2011). The cutoff
frequency *f*_{cutoff} (Hz) is dependent on the
wave guide characteristics, water depth and sediment layer acoustic
properties, as shown in the following equation:

where *h* is the water depth (m),
*c*_{s} and *c*_{w}
are sound celerity (m s^{−1}) in the sediment layer and in water, respectively.
Cutoff frequencies have been estimated in each river, by assuming a fixed
sound speed of 1600 m s^{−1} in the sediment layer and using the mean water depth
monitored (Fig. 8b). Estimated cutoff frequencies
are approximatively located around the minimum of the observed attenuation
coefficient. Our ability to precisely determine a cutoff frequency is
limited. First, the acoustical properties of river beds are unknown,
depending on lithology, grain sizes, porosity and heterogeneity of the
materials constituting the river bed. Secondly, the water depth is not
constant over the investigated sections but varies from the banks to the
middle of the river. For these reasons, cutoff frequencies are rough
estimates and do not perfectly correspond to the observed local minimum of
attenuation coefficient. Note also that different hydrodynamic conditions
were investigated for some rivers. Varying water depth results in different
cutoff frequencies and relative positions of the hydrophone between water
surface and streambed. These two parameters have been observed to modify the
response of the hydrophone (Geay et al., 2017b) in the
lower frequency range, around the cutoff frequency. The range of
hydrodynamic conditions that was investigated in this study did not enable
the observation of such effects.

The variation of attenuation coefficients at higher frequencies is
discussed here. As attenuation properties are frequency dependent, it is common
to characterize the attenuation in mediums by giving a value of the
attenuation coefficient per wavelength (Eq. 5). Attenuation coefficients per
wavelength (nepers) are presented in Fig. 9
for frequencies higher than the local minimum of *α*
(nepers m^{−1}). Except for the Isère river, we can observe that
*α*_{λ} is almost constant with
frequency, which in turns means that *α* (nepers m^{−1})
varies almost linearly with frequency. Note that the maximum frequency
analyzed in Fig. 9 varies from one river to another and shows some
inverse correlation with the average *α*_{λ}. In rivers with high attenuation (high
*α*_{λ}), the recording of the
emitted chirps was not possible in the highest frequency range, due to a too-fast decrease of chirp power with distance. This observation reveals that
the measurement of bedload sounds generated by the smallest particles (i.e.,
highest frequencies generated) should be local or even impossible in rivers
having too-high attenuation coefficients. In the following, each river is
characterized by an average value of *α*_{λ} (Table 3) and is compared to river characteristics
(Table 1; Fig. 10).
Looking at the relationship between *α*_{λ} and the slope measured at the
local reach (i.e., 100 m downstream and upstream from the bridge where
experiments were undertaken), we can observe that there is good correlation:
higher attenuation coefficients were obtained for steeper rivers. As for
slope, surficial granulometry of the emerged bars
(*D*_{84}) is also well correlated with
*α*_{λ}: larger roughness (i.e.,
larger *D*_{84}) induces larger attenuation of
the acoustic waves. Surface velocity or water depth seems to be a less robust
explanatory variable of *α*_{λ}.
The possible influence of typical nondimensional numbers has also been
tested. The ratio of the water depth over
*D*_{84} and the Froude number were used by
Tonolla et al. (2009,
2010). They found that they were the main hydrogeomorphological variables
explaining the differences in passive acoustic signals in field experiments.
Small ratio of the relative submergence (i.e., small
*h*∕*D*_{84}) induces breaking waves or water
plunging directly in the water column, entraining bubbles in the water
column. These hydraulic mechanisms are sources of noise generated by
oscillating air bubbles in the water column, as it is observed for breaking
waves in a marine environment (Deane,
1997; Norton and Novarini, 2001). In our study, entrained air bubbles could
explain the increase of attenuation coefficient in rivers having rough beds.
It is indeed known that the presence of air bubbles increases the
attenuation of acoustic waves (Deane,
1997; Norton and Novarini, 2001) because of the heterogeneity of the medium
constitution of water and air which have very different acoustic impedances.
Also, as observed in a flume experiment (Wren et
al., 2015), the bed roughness itself is a source of attenuation, with larger
roughness involving higher attenuation. Finally, both processes (rough
boundaries and entrained air bubbles) could explain our observations by
causing concomitantly higher attenuation of the acoustic wave. The river bed
roughness should be the best characteristic enabling the prediction of
acoustic wave propagation properties in rivers. However, this parameter is
not easy to measure. It is sometimes difficult to access the riverbed, and
surface grain size distributions are known to be variable in space. The
local slope of the reaches is easier to measure and, even if less
meaningful, should be a more robust parameter to infer propagation
properties of a river.

## 4.2 Recommendation for monitoring bedload with hydrophones

This study was done to improve our ability to better use the measurements of
bedload self-generated noise in rivers. This section aims at giving an
example on the use of attenuation coefficients in a simple case. Let us
consider an infinite river bed with a homogeneous repartition of sound
sources over the river bed. Bedload impacts generate a constant spectral
power per surface unit noted PSD_{s} (µPa^{2} Hz^{−1} m^{−2}). If sound sources are random and independent noise sources (Thorne, 2014), the acoustic power measured by a
hydrophone can be written as a sum of the power of all sound sources:

where PSD_{h} is the spectral power monitored
by a hydrophone in a fixed position (µPa^{2} Hz^{−1}), *h* is the water
depth (m), *d* is the distance of the hydrophone above the river bed (m), and *r* is the
horizontal distance from the hydrophone (m). From Eq. (9), it follows that

Considering that $\mathrm{0}<\mathit{\alpha}\ll \mathrm{1}$, it follows that
PSD_{h} is inversely proportional to the
attenuation coefficient as the exponential term tends to 1. This has several
implications for the use of bedload monitoring using passive acoustics.
First, as the attenuation coefficient could be variable from one reach to
another, the acoustic power of bedload SGN could be variable from one reach
to another even if bedload fluxes are similar. Secondly, as observed in
Fig. 8, attenuation coefficients are variable
with frequency. It means that the frequency content of bedload SGN spectra
is modified by propagation effects, which in turns means that the shape of
monitored spectra is not only related to grain size distributions
(Petrut et al., 2018; Thorne, 2014) but also
to propagation properties. Therefore, in order to estimate grain size
distribution, measured spectra should be corrected for propagation effects
before any inversion procedure. From Eq. (10) (*α*≪1), a better estimate of the sound generated by bedload transport
could be done by multiplying the monitored sound pressure levels by the
attenuation coefficient (*α*>0):

The power generated by bedload sounds is proportional to the power of
measured sounds multiplied by the attenuation coefficient. This simple
operation enables us to get an unbiased measurement of the sound generated
by bedload impacts and therefore a more robust proxy for bedload transport
monitoring in rivers. To achieve the estimation of sounds that are generated
by bedload transport (PSD_{s}), both measurements of propagation properties
(*α*) and ambient sounds (PSD_{h}) are needed. Note that Eq. (11)
was obtained by assuming sound sources (i.e., bedload fluxes) that are
homogeneously distributed. As this hypothesis will rarely be valid, more
realistic inverse methods should be invented to estimate the real sounds
(PSD_{s}) generated by bedload transport and its spatial distribution.

A simple model for acoustic wave propagation in rivers has been investigated
in this study. It considers that the power of acoustic waves decreases with
distance by spreading effects (cylindrical or spherical models) and with an
additional exponential term including other propagation effects (e.g., volume
attenuation, scatter by rough boundaries). The model was used to interpret
the attenuation properties of a controlled sound source in several rivers
having different hydrogeomorphic characteristics. Our tests were not able to
distinguish whether spherical or cylindrical models should be used, both
models being valid. The exponential attenuation coefficient
(*α* in nepers m^{−1}) has been found to vary with frequency and
with the type of river considered. Two types of attenuation regimes have
been observed. Below the cutoff frequency, which is inversely proportional
to the average water depth, *α* decreases with
increasing frequency until a local minimum is reached. Reaches with large
water depth should therefore be selected for doing passive acoustic
measurements. The cutoff frequency should be sufficiently low to listen to
the coarsest grains of bedload transport. Above the local minimum (i.e., the
cutoff frequency), attenuation coefficients increase almost linearly with
frequency. The higher frequency regime has been characterized by a constant
attenuation coefficient per wavelength (*α*_{λ} in nepers). It has been found that
*α*_{λ} was well correlated with the
slope of the riverbed reaches (and to the surface *D*_{84} of the emerged
bars as well), where *α*_{λ} is
higher for higher bed slopes of the river. Assuming that riverbed slope and
surface *D*_{84} of bars are good proxies for the riverbed texture, it can
be concluded that attenuation properties are dominated by processes related
to the riverbed roughness at high frequencies, including the entrainment of
air bubbles in the water column and scattering effects on rough boundaries.
As shown in the discussion, the acoustic power monitored by a hydrophone, in
a fixed position, is almost inversely proportional to the attenuation
coefficient at a given frequency. As consequences, the spectra of bedload
SGN that are measured in rivers are modified by the variations of
attenuation coefficients with frequency. As attenuation is higher at high
frequencies, acoustic signals that are monitored by a hydrophone are shifted
to lower frequencies compared to the sound really generated by bedload
impacts. As shown for an idealized case with an infinite riverbed and
homogeneous bedload sound sources, the real sounds generated by bedload can
be estimated by correcting the hydrophone signal by the propagation laws of
acoustic waves in rivers.

Data supporting the content of this paper can be obtained by contacting Electricité de France (sebastien.zanker@edf.fr).

The supplement related to this article is available online at: https://doi.org/10.5194/esurf-7-537-2019-supplement.

TG developed the signal processing tools and prepared the manuscript with contributions from all co-authors. TG, LM and SZ designed and carried out he experiments. JRR helped in designing the experiments thanks to previous experience and was integral in the analysis of the results.

The authors declare that they have no conflict of interest.

The authors are grateful to all the contributors to this work; particularly abundant during field experiments was Guy Cellier (EDF), who designed the equipment to record sounds with a river board. This work has also been carried out with the help of engineering consultants and numerous students.

This project was essentially founded by Eléctricité de France with additional subsidy from Agence de l'eau Rhone-Alpes, the Rhone-Alpes region (ARC3-Environment) and Grenoble University (OSUG).

This paper was edited by Jens Turowski and reviewed by two anonymous referees.

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