domain: phys sde

text:
Benoit Tallon

Philippe Roux
email: philippe.roux@univ-grenoble-alpes.fr

Guillaume Matte

Jean Guillard

Sergey E Skipetrov

Mesoscopic wave physics in fish shoals

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Since the late 1980's, physicists have achieved great progress in the fabrication of strongly disordered materials that would allow for Anderson localization of 'classical' waves (e.g., light, microwaves, sound) in three dimensions [1][2][3]. Anderson localization is a halt of propagation due to disorder [4,5]. Although very few experiments have succeeded [6][7][8], these studies have revealed the 'mesoscopic' interference phenomena that are analogous to that of low-temperature electrical transport [9,[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]: weak localization [START_REF] Van Albada | [END_REF]12], universal conductance fluctuations [13,14], strong fluctuations and long-range correlations of scattered intensity [15,16]. These studies also led to the discovery of new phenomena, such as the slowing down of transport due to scattering resonances [17], random lasing [18], mean path length invariance [19], and transverse localization of transmission eigenchannels [20].Such a set of mesoscopic phenomena have never been observed apart from laboratory experiments, and even less with living matter, as they require carefully designed disordered samples. In acoustics, threedimensional mesoscopic phenomena have been observed exclusively in so-called 'mesoglasses' [6]. Many studies have considered coated particles suspended in a host matrix in both optics and acoustics [21,22] because of their interesting scattering properties. However arduous synthesis and weak stability make those particles rarely employed. Can natural complex media be inspiring for the design of such model systems? Do they scatter waves strongly enough to observe non diffusive wave transport?

Here, we show that shoals of fish trapped in large cages-an example of live, active matter-allow the examination of various mesoscopic interference phenomena in ultrasound scattering for fish densities that are comparable to those encountered in natural fish schools at sea. Fish swim bladder (an organ which allows fish to control their buoyancy) is analogous to an air bubble and thus strongly scatters ultrasonic waves. This strong scattering has been useful for several decades for fish counting with ultrasounds in the single scattering regime [START_REF] Simmonds | Fisheries Acoustics: Theory and Practice[END_REF]. The present study focuses on dense shoals in which single scattering assumption is irrelevant. Comparison with multiple scattering theories reveals the impact of the complex fish structure that can be seen as a coated air bubble. For different fish densities, the scattering strength of fish shoals is demonstrated via measurements of long-range correlations or non-Rayleigh distribution of the intensity speckle, as well as via the dynamic coherent backscattering effect, revealing the lowest energy velocity ever observed in underwater acoustics. Because of their quasi-random movement, fish are also interesting for configurational averaging where spatial ergodicity is usually assumed for laboratory experiments.

To ensure fish shoal control and to prevent avoidance reactions of the fish at sea [START_REF] Draštík | [END_REF], we perform acoustic measurements in large (∼ 5 m ×5 m ×5 m) open-sea fish cages that are typical of fish farms (Fig. 1a,b). A cage typically contains several tens of thousands of fish at a mean density of 10-100 fish per m 3 . The individual fish mass ranges from 10 g for fish larvae up to 1 kg for mature fish. The corresponding fish volume fraction φ ranges from 1% to 10%. Much higher φ, up to 30%, can be reached with fish farms that practice intensive fish farming.

We measure the reflection of short acoustic pulses (∼ 0.1 ms) that are emitted by a cross-shaped array of 2 × 64 acoustic transducers, as shown in Figure 1a, at a central frequency of f = 150 kHz. This is far from the swim-bladder resonance [25]. Successive measurements are repeated several thousands of times at a rate of ∼ 30 shots per second. The natural fish motion at a speed of the order of 5 cm/s is sufficiently slow to ensure that the fish can be considered as immobile during each single shot. At the same time, the fish motion produces independent fish configurations over time, effectively providing us with a huge number of statistically independent measurements that correspond to different configurations of the fish in the cage.

For a single incident pulse, the backscattered acoustic pressure field ψ(r, t) and the intensity I(r, t) = |ψ(r, t)| 2 measured by a transducer at a position r fluctuate widely (see Fig. 1c, dashed line, for typical experimental data), whereas the mean over many shots (and hence many different fish configurations) and over all of the transducers yields a smooth coda, as shown by the symbols in Figure 1c. This is a direct signature of the multiple scattering, and it can be understood by viewing wave transport as a random walk with a velocity v, a step length between scattering events (the scattering length, or mean free path), and an isotropization distance * ≥ (the transport mean free path). For propagation distances greater than * , the ensemble mean intensity |ψ(r, t)| 2 obeys a diffusion equation with diffusivity D = v * /3 [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]. This equation provides an excellent description of |ψ(r, t)| 2 in Figure 1c (solid line). This diffuse behavior of I(r, t) is accompanied by circular Gaussian statistics of ψ(r, t).

Mesoscopic effects manifest as deviations from the of the field and intensity fluctuations, respectively, are shown in Figure 2 for two representative shoals that feature weak (gray asterisks) and strong (red circles) scattering. Weak scattering occurs for the fish fry (i.e., sea bream with mean weight W = 10 g and shoal density η ∼ 6 kg/m 3 ), whereas strong scattering occurs for the dense shoal of adult sea bream (W = 284 g; η ∼ 23 kg/m 3 ). C ψ (∆r) is short-range for both shoals and can be reasonably well fitted according to theory that takes into account the finite size of our acoustic transducers [25] and yields the scattering lengths ∼ λ 1 cm as the best-fit parameters. These small suggest that the Anderson localization of sound expected for 2π /λ 1 (the Ioffe-Regel criterion) [1,4,5] would be reachable in denser fish shoals. In contrast to C ψ (∆r), the intensity correlation function C δI (∆r) features a long-range component C 0 that does not vanish even for ∆r λ, . In our notation, C 0 incorporates contributions from wave interference in the bulk [denoted as C 2 and C 3 in the literature [9,[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]] as well as the near transducers (the genuine C 0 [26]) because our experiments do not . Solid lines are theoretical fits [9] with the dimensionless conductance g as a free-fit parameter.

allow these to be distinguished. C 0 0.4 for the dense fish shoal indicates a breakdown of wave diffusion where C δI (∆r) = |C ψ (∆r)| 2 would be expected. This breakdown is also confirmed by an analysis of the intensity probability density function, as shown in Figure 2a, inset. Fitting of the data to theory [9] allow an effective dimensional conductance g to be attributed to each fish shoal [25]. The variance of the intensity fluctuations is given by δI(r) 2 / I(r) 2 = 1 + 4/3g [9]. Thus, g is a measure of deviations of the scattered field ψ(r) from the Gaussian statistics, which implies δI(r) 2 / I(r) 2 = 1. The large C 0 0.4 and small g 2 for the high-density fish shoal signal substantial deviations from the diffusion picture of wave propagation and confirm the proximity of the Anderson localization regime, for which g < 1 is expected [27].

Coherent backscattering (CBS) represents a mesoscopic effect par excellence that has been measured for light [START_REF] Van Albada | [END_REF]12], ultrasound [28], matter [29] and seismic [30] waves. This is due to constructive interference of waves following time-reversed pairs of paths, and manifests in the static regime as the doubling of the mean scattered intensity in a narrow angular range ∆θ ∼ λ/ * around the back-scattering direction [31]. Examples of our CBS measurements are shown in Figure 3a, for two cages that contain either adult sea bream at a low density (W = 320 g; η ∼ 15 kg/m 3 ) or a dense shoal of croaker fish (W = 886 g; η ∼ 24 kg/m 3 ). The theoretical fits to the data (see Supplementary text) provide the bestfit values of * = 1.7 cm and * = 0.7 cm for the lower and higher fish densities, respectively. These small val-ues of * support our conclusion on the strong multiple scattering of ultrasound in the fish shoals considered.

The diffusivity D can be estimated from the dynamic CBS profile, as shown in Figure 3b. The CBS cone width ∆θ(t) follows the theoretical behavior ∆θ -2 ∝ Dt expected for wave diffusion [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]28] up to t 5 ms (Fig. 3c). The linear growth of ∆θ -2 with time slows at longer times, again indicating strong mesoscopic interference effects and the closeness to the Anderson localization transition [7].

Combining D estimated above with * from the static CBS, we obtain a surprisingly low energy transport velocity v = 3D/ * 35 m/s. This value is much lower than the speed of sound in either water (v 0 1500 m/s) or air (340 m/s)-the two values that might serve as a reference for scattering by an air-filled swim bladder in (possibly) bubbly water. While it has been demonstrated that resonant scattering can slow down diffusive wave transport [17,32,33], narrow-band resonance effects are not clear here since low v values are obtained for any fish size. The only possibility to explain this is to consider the solid multi-layer structure of a fish (see supplementary text) and, in particular, to invoke the slow speed v s ∼ 10 m/s of the shear waves in the fish flesh. By assuming equipartition of the scattered intensity between longitudinal waves in water and shear waves in fish [34], we consider that the wave speed is v s in the fish body and v 0 in between two fish. Then, averaging this along a path that traverses the fish shoal yields in agreement with independent measurements provided by the sea-farm manager. We emphasize that this dramatic slowing down of the ultrasound is not related to the scattering resonances of the fish, and thus cannot be explained by known, resonant mechanisms [17,35].

v = [1 + φ 1/3 /(1 -φ) 1/3 ]/[1/v 0 + φ 1/3 /(1 -φ)
From an applied standpoint for aquaculture, determination of v via dynamic CBS measurements allows the estimation of the shoal density through the fish volume fraction φ. Similarly, measurements of C ψ (∆r) and the stationary CBS yield and * , which are related to the scattering and transport cross-sections σ and σ * of an individual fish, from which a mean fish length can be estimated. The knowledge of both the mean fish length and the fish shoal density opens new perspectives for noninvasive biomass estimation of dense fish shoals.

In conclusion, ultrasound scattering in fish shoals under conditions close to those encountered in nature show such mesoscopic wave phenomena as long-range correlations of scattered wave intensity, CBS, and the slowing down of the diffusion. These phenomena indicate that transition to the Anderson localization might be within reach in experiments with denser fish shoals. The extremely slow energy transport velocity emphasizes the importance of the fish solid structure in the multiple scattering of ultrasound-a phenomenon that has been over-looked up to now. Furthermore, the alliance of mesoscopic wave physics and fisheries acoustics has the potential of being used for monitoring fish biomass, which at present is restricted to the single scattering regime. When transposed to the open sea, CBS measurements might also be applied to study strong fish density variations during day-night schooling transitions [START_REF] Draštík | [END_REF].

The authors thank Prof. J. H. Page for valuable discussions about transport intensity velocity.

N W (g) η (kg/m 3 ) V (m 3 ) C1 (sea bream, adults, Fig. 1c The sea cages are in the Mediterranean Sea (Cannes, France). The distance to the sea bottom is z = 6.5 m for the smallest cage and z = 15 m for the biggest cage, from their lowest points. The parameters relating to the shoals and cages (as provided by the farm manager) are detailed in Table I. To maintain the organic label of the farm and to avoid the need for drug treatments, the fish densities in these cages are lower than in intensive farming facilities (where mass densities η can reach 100 kg/m 3 ).

The fish fry are raised in in-shore tanks and transferred to the open-sea cages when their weight is about 5 g. The feeding procedures are controlled to obtain a calibrated fish size.

Hormones are not used in this farm, so 18 months to 3 years are required to grow the fish to their adult size (which corresponds to the standard commercial weight for a fish fillet).

Species choice is also an organic label constraint as sea bream, croakers and sea bass (not studied here) are native species of the Cannes region.

Ultrasonic antenna

The acoustic device used here is a Mills cross-shaped antenna that consists of two perpendicular linear arrays, each with 64 ultrasonic piezoelectric transducers. Each transducer is narrow band (bandwidth, 10 kHz) with a central frequency of 150 kHz, which corresponds to a wavelength λ of 1 cm in water. The transducers are square in shape and have sides of 0.5 cm λ/2. The antenna (expressly designed by the iXblue company for aquaculture monitoring) is placed just below the water surface, facing the sea bottom.

The Mills cross array allows us to obtain a rapid three-dimensional representation of the ultrasound intensity back-scattered by the fish shoals, by emitting a series of tilted plane waves through one array (with controlled direction α) and recording through the other array. The measurements are projected into the angular β space in the post-processing.

This acquisition sequence lasts less than 1.5 s and provides the time-evolving back-scattered acoustic intensity in the α-β angular space (Fig. S1).

Two types of measurement are used to characterize the shoal: (i) to measure spatial correlations (Fig. 2), only the central element of the antenna emits the acoustic wave, whereas all of the 128 elements are used to record the back-scattered signal; (ii) to measure the intensity profile (Fig. 1c), the probability density (Fig. 2a, inset) and the coherent backscattering (Fig. 3), all of the 128 transducers of the antenna are used as both transmitters and receivers, without applying any emission delay (α = 0). For all of the measurements, the duration of the emitted pulse is about 0.1 ms and the back-scattered waves are recorded over 25 ms immediately after emission. After each acquisition cycle, the system remains inactive for 10 ms to ensure that no residual acoustic signal is detected at the beginning of the next cycle.

Averaging is performed over 2,000 to 20,000 acquisitions (roughly 10-min-long acquisitions), to ensure that many different independent spatial configurations of the fish in the cages are probed. Using an acoustic set-up with phase sensitive detectors allows us to directly access the wave field ψ(r, t) and not only the intensity I(r, t) = |ψ(r, t)| 2 , as in standard optical experiments. The mean amplitude ψ(r, t) of the detected ultrasound is subtracted from each acquisition ψ(r, t) to exclude any residual ballistic contribution (such as ringing artifacts or echos from the cage net).

DATA PROCESSING AND FITTING Diffusion approximation for the average intensity

The mean back-scattered intensity I(t) (Fig. 1c, symbols) is obtained after averaging I(r, t) over the acquisitions and positions r of the transducers of the antenna. The theory used to fit the experimental data is derived from a diffusion equation for I(r, t) [1] that is solved for a disordered medium that occupies the half-space z > 0, with a delta-function source at z = z [2]:

I(t) = I 0 2π +∞ -∞ z 0 exp(-γ 0 z ) D(1 + γ 0 z 0 ) exp(-iΩt)dΩ, ( S1
)
where

γ 2 0 (Ω) = -iΩ D + 1 Dτa , τ a is the characteristic absorption time, z 0 = 2 3 1+R 1-R *

Spatial correlations and probability density

The field correlation function C ψ (∆r) = ψ(r)ψ(r + ∆r) * / |ψ(r)| 2 is calculated in the frequency domain and averaged over the acquisitions and frequencies of the antenna bandwidth. The theoretical expression used to fit the data in Fig. 2a is:

C ψ (∆r) = exp(-∆r/2 ) (k 0 a) 2 {k 0 (a -∆r)Si [k 0 (a -∆r)] + k 0 (a + ∆r)Si [k 0 (a + ∆r)] +2 [cos(k 0 a) -1] cos(k 0 ∆r) -2k 0 ∆rSi(k 0 ∆r)}, ( S2
)
where Si is the sine integral function, k 0 is the wave number in pure water, a is the transducer size, and is the scattering length (mean free path). This expression is obtained by averaging the standard expression for field correlation in a disordered medium [1],

C ψ (∆r) = sin k 0 ∆r k 0 ∆r exp - ∆r 2 , ( S3
)
over transducers of size a for k 0 1, and assuming that transducers are one-dimensional (i.e., each transducer is a line of length a). We checked that relaxing this one-dimensional approximation and considering square transducers of size a × a yields data that are very close to those following from Equation (S2) for the ranges of our parameters, albeit with a slightly different value of a. Using a and as free-fit parameters, we obtain the fits of Figure 2a with = 1 cm and a = 3λ/4.

The fit to the intensity correlation function C δI (∆r) in Figure 2b is obtained using

C δI (∆r) = |C ψ (∆r)| 2 + C 0 , (S4)
where and a are deduced from the fit to C ψ (∆r), so that C 0 is the only free-fit parameter.

The probability density of normalized intensity (Fig. 2a, inset) is also calculated in the frequency domain for the antenna bandwidth and averaged over the acquisitions. The data were binned (over 40 values in the weak scattering case, and 50 values for strong scattering) from I/ I = 0 to I/ I = max{I/ I }. The theoretical function used for the fit, which is valid for an incident plane wave, is [3,4]:

P (I/ I ) = ∞ 0 dv v i∞ -i∞ dx 2πi exp -I/ I v + xv -Ψ(x)) , (S5)
with Ψ(x) = g ln 2 1 + x/g + x/g and g as a free-fit parameter.

Coherent back-scattering

The angular dependence of the back-scattered field is measured by applying the beamforming method [5] to each acquisition. A controlled phase shift is applied to each receiving channel, which allows the projection of the received signals onto angles along the x or y axes. To improve the statistics, we take the means over the two angular intensity profiles, so that the results are shown only as a function of one angle θ.F

The dynamic CBS profiles (Fig. 3b,c) are measured for cages C4 and C5, and integrated over time to estimate the static CBS peak (Fig. 3a). However, our scattering medium is not semi-infinite and the maximum integration time is shorter than the minimum time imposed by the echo from the cage net: t max = 2d/v 0 (with d as the depth of the cage). Nevertheless, our cages are sufficiently large to make the consequent error negligible: a 25% variation in t max induces 10% and 5% variations in the * estimations for C4 and C5, respectively.

The theoretical expression used to fit the CBS profile follows from the same diffusion theory as the expression for the mean intensity (S1):

I(θ, t) = I 0 2π +∞ -∞ z 0 D e -γ 0 z 1 + γ 0 z 0 + e -γz 1 + γz 0 exp(-iΩt)dΩ, (S6)
where γ 2 (θ, Ω) = -iΩ D + k 2 0 sin 2 (θ) + 1 Dτa and γ 0 = γ(θ = 0, Ω). The fits are first performed for the stationary CBS peak I(θ) :

I(θ) = ∞ 0 I(θ, t) dt = I 0 z 0 D e -γ 0 z 1 + γ 0 z 0 + e -γ 1 z 1 + γ 1 z 0 , (S7)
where γ 1 = γ(θ, Ω = 0). * and Dτ a are determined from the fit as best-fit parameters.

These parameters are then fixed to fit the dynamic CBS profile I(θ, t) , with D as a free-fit parameter. These two fits finally yield the energy transport velocity: v = 3D/ * .

Energy velocity of diffusive waves

Here we suggest another approach for the calculation of energy velocity based on theories first developed for multiple scattering of light [8]. In this problem, we consider the scattering of a plane wave (with wavenumber k w = ω/v w ) by the object represented on Fig. A1. As a first approximation, we consider spherical scatterers (the scattering of plane wave by spheroids mainly affects the scattering anisotropy factor and resonant frequencies [6]). The The scattering amplitude f (θ) (with θ the scattering angle) is calculated by solving the Mie problem as detailed in [7]. Thus, the independent scattering approximation (ISA) gives an expression of the effective wavenumber in the fish shoal:

k 2 = k 2 w + 16π 2 3 R 3 1 φf (0) (S8)
The scattering amplitude and effective wavenumber are used for the calculation of energy velocity [8]:

v = v 2 w /v ph 1 + δ , (S9)
with: θ) . Fish size dispersion is taken into account by averaging the v over different particle sizes with a Gaussian distribution [9]. A very low size polydispersity (∼ 1 %) is sufficient to wash out any residual resonant effects that cannot be experimentally observed. The result of energy velocity calculation (with input parameters detailed on Table II) is presented on Fig. S3 for a small frequency range around the working frequency of our sonar probe. The resulting energy velocity (v ∼ 30 m/s) is found to be very close to both our observation and predictions given by Eq. ( 4) in the main text. For comparison, we also In conclusion, the very slow diffusive wave transport has been explained simultaneously both by macroscopic approach (by assuming wave energy equipartition) or microscopic description (by calculating scattering properties of an isolated scatterer). This comparison emphasizes the role of shear wave propagation in the fish body that cannot be considered as a simple air bubble.

δ = 8π 2 3 R 3 1 φ ∂ω ∂k 1 k ∂Re[f (0)] ∂ω π 0 sinθ |f (θ)| 2 ∂ϕ(θ) ∂ω dθ , (S10)
v ph = ω/k and f (θ) = |f (θ)| e jϕ(

FIG. 1 .

 1 FIG. 1. (a, b) Photographs taken underwater (a) and at the surface (b) of an open-sea cage used in these experiments (courtesy of S. Pasta). (c) Intensity (dashed line) and mean intensity (symbols) of the backscattered acoustic signal after emission of a short pulse. Solid line, diffusion theory fit.

  diffusion picture of propagation, due to interference of waves scattered along different paths inside the disordered fish aggregation [9, 10]. We first analyze the statistics of the time-integrated (stationary) wave field ψ(r) and intensity I(r) = |ψ(r)| 2 . Correlation functions C ψ (∆r) = ψ(r)ψ(r + ∆r) * / |ψ(r)| 2 and C δI (∆r) = δI(r)δI(r + ∆r) / I(r) 2 [where δI(r) = I(r) -I(r) ]

  FIG.2. Spatial correlation of field ψ (a) and intensity fluctuations δI (b) for weakly (gray asterisks) and strongly (red circles) scattering fish shoals. The lines show the theoretical fits to the data. While C ψ (∆r) rapidly decays to zero for both shoals, C δI (∆r) is long-lasting and remains appreciable even for ∆r = 5λ, especially for the stronger-scattering shoal. The colored area in (b) represents the noise level. (a) inset: the probability density of the normalized intensity I/ I (symbols) that deviates from the Rayleigh law P (I) = exp(-I/ I )/ I expected for weak scattering (dashed line). Solid lines are theoretical fits[9] with the dimensionless conductance g as a free-fit parameter.

FIG. 3 .

 3 FIG. 3. (a) Stationary coherent backscattering (CBS) profiles for weakly (gray asterisks) and strongly (red circles) scattering fish shoals. The lines are the theoretical fits. (b) Dynamic CBS profile for the strongly scattering shoal. (c) Time evolution of the cone width ∆θ(t) of the dynamic CBS peaks. The linear increase in ∆θ -2 that is expected from diffusion theory is shown by the dashed lines.

  FIG. S1: Three-dimensional acoustic scan of an open sea cage (C2). The cage net is just visible for z = 5 m, as well as the sea bottom for z = 6.5 m.

  is the extrapolation length, with R = 0.99 as the reflection coefficient for the water/air interface, D is the sound diffusion coefficient, and * is the transport mean free path. * and D are deduced from the static and dynamic coherent back-scattering (CBS) measurements, respectively, while the fitting parameters are z = 1.8 mm, τ a = 1.4 ms, and the overall amplitude is I 0 = 5 × 10 6 .

  FIG. S2: Geometry employed to solve the Mie problem and calculate the energy velocity.

  Input parameters used for the energy velocity calculation with the geometry presented on Fig.S3. plotted on Fig.S2calculations for a simple cloud of air bubble in water and for coated bubbles with materials (1) or (2). This results show the importance of the multi-layer geometry: the thin layer of bones and scales (1) helps for the wave polarisation conversion (longitudinal-shear waves) while the thick layer of flesh significantly slows down the wave propagation. The energy equipartition approach employed in the main text assume this polarization conversion and only focus on the flesh shear properties.

  FIG.S3: Geometry employed to solve the Mie problem and calculate the energy velocity.

  1/3 /v s ] 30 m/s for the fish volume fraction φ 10%, which is

TABLE I :

 I Number of fish N and mean fish weight W and shoal density η for the five open sea cages considered in this study. V , cage volume.

	EXPERIMENTAL SET-UP
	Open-sea cages

Supplemental material for "Mesoscopic wave physics in fish shoals" Benoit Tallon, 1 Philippe Roux, 1 * Guillaume Matte, 2 Jean Guillard, 3 and Sergey E. Skipetrov 4 1 Univ. Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France 2 iXblue, Sonar division, 13600 la Ciotat, France 3 Univ. Savoie Mont Blanc, INRA, CARRTEL, 74200 Thonon-les-Bains, France 4 Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France * To whom correspondence should be addressed; E-mail: philippe.roux@univ-grenoble-alpes.fr.

(Dated: October 3, 2021)


Benoit Tallon

Philippe Roux
email: philippe.roux@univ-grenoble-alpes.fr

Guillaume Matte

Jean Guillard

Sergey E Skipetrov

Acoustic density estimation of dense fish shoals

Multiple scattering of acoustic waves offers a noninvasive method for density estimation of a dense shoal of fish where traditional techniques such as echo-counting or echo-integration fail. Through acoustic experiments with a multi-beam sonar system in open sea cages, multiple scattering of sound in a fish shoal, and in particular the coherent backscattering effect, can be observed and interpreted quantitatively. Furthermore, a volumetric scan of the fish shoal allows isolation of a few individual fish from which target strength estimations are possible. The combination of those two methods allows for fish density estimation in the challenging case of dense shoals.

INTRODUCTION

Fish density estimation using acoustic waves has been under investigation for almost 70 years [1,[START_REF] Simmonds | Fisheries Acoustics: Theory and Practices[END_REF]. This interest comes from the strong scattering of acoustic waves by fish, and in particular due to the great acoustic contrast between the fish swim bladder and the surrounding water. Hence, when the fish spacing is large compared to the acoustic wavelength, fish density estimation is relatively straightforward, through the counting of hot spots on echograms [START_REF] Simmonds | Fisheries Acoustics: Theory and Practices[END_REF]. For convenience, the echo-integration method [START_REF] Foote | [END_REF] can be used for large shoals. Furthermore, acoustic scans provide the target strength (TS; dB) [START_REF] Simmonds | Fisheries Acoustics: Theory and Practices[END_REF] of the fish, which depends on their size, species, physiology, and position. However, these traditional acoustic counting methods are only valid under the single scattering assumption: during its propagation, the backscattered signal received on the probe should be scattered at most by one fish. For large or dense shoals (density 10 fish/m 3 ), this assumption does not hold [4], as part of, and indeed most of, the backscattered intensity comes from wave paths that are scattered by several fish between emission and reception. The so-called multiple scattering regime is then reached when the wave propagates over distances greater than the scattering mean free path s , which is defined as the average distance between two scattering events [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]. Therefore, fishery acoustic methods are ineffective, although they remain widely sought after for density estimation in the aquaculture industry due to their nonintrusive aspect. This means that to obtain the main parameters needed (i.e., number of fish, total biomass and/or individual mean size), aquaculture uses manipulation of the fish, with large impact on individuals.

In this Letter, we propose an original method for noninvasive fish-density estimation in open-sea cages. This approach is based on a combination of fishery acoustics and multiple scattering concepts. Multiple scattering of waves in random media is a widely studied phenomenon in optics [START_REF] Wolf | [END_REF], acoustics [7], and geophysics [START_REF] Sato | Seismic Wave Propagation and Scattering in the Heterogeneous Earth[END_REF]. It has applications for medical [START_REF] Derode | [END_REF] and wave control [10] purposes. In particular, it has been shown that wave propagation in random media can result in remarkable "mesoscopic" phenomena [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF], such as the coherent backscattering (CBS) effect [11]. CBS is a wave interference phenomenon that manifests as an enhancement (by a factor of 2) of the average backscattered intensity measured in the direction opposite to the direction of the incident wave. This phenomenon occurs in multiple scattering regimes due to constructive interference of partial waves scattered along reciprocal paths [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]. From the dynamic point of view [7], CBS develops gradually as a wave propagates inside the fish aggregate, and becomes significant for wave propagation distances greater than s . In this way, CBS measurements in fish cages can provide useful information about shoals. In particular, we show below that simultaneous knowledge of the fish TS and the shoal s allows estimation of the fish density even in the challenging cases of dense shoals.

EXPERIMENTS

Experiments were performed with dense salmon shoals that were contained in large open-sea cages on a salmon farm in the North Sea (Eide Fjordbruk, Rosendal, Norway). The cubic cages are 30 m in both width and depth. In this area, the sea depth is about 50 m. The cage for the experiments contained approximately 200,000 Atlantic salmon (Salmo salar ) with an average weight of 6 kg (total length, about 80 cm).

The sonar probe used here was a reversible multibeam antenna (Mills Cross; based on Seapix technological brick [12], iXblue La Ciotat) that can be used for three-dimensional (3D) volumetric scanning. This probe is made of two perpendicular arrays, each of 64 ultrasonic transducers (see Fig. 1a) with a central frequency f = 150 kHz and an inter-element spacing of half a wavelength in water. Each of the 128 transducers can be controlled independently, for precise manipulation of the emission/reception direction of the acoustic waves. A volumetric scan of the whole cage (Fig. 1b) is possible from successive shots in about 1 s, which is sufficiently fast to approximate the fish shoal as 'frozen' between two scans.

Target strength measurement

To determine the fish density inside the cage, an estimation of the individual fish TS is required. To achieve this, we perform a large number of acoustic 3D volumetric scans of the shoal, from which we select a collection of individual targets with propagation distances below s , i.e., in the single-scattering regime. The volumetric scan is constructed as follows: a series of 21 plane waves [START_REF]The targets being small comparing the the propagation distance[END_REF] is sent with array 1 by varying the incidence angle from α = -10 • to α = 10 • (see Fig. 1b). The backscattered acoustic field is recorded with array 2 (perpendicular to array 1) and beamformed after post-processing over angles β = α: for each of the 21 incident angle α, beamforming is applied on the perpendicular array over the 21 angles β. This process was repeated to obtain 550 independent 3D scans of the fish shoal from which 3,800 individual targets were isolated.

From the literature, the TS of an 80-cm salmon is TS th = -26 dB [START_REF] Lilja | [END_REF]. This TS is used to set a detection threshold on the acoustic scan: a spot with TS th -5 dB < TS < TS th + 5 dB is identified as a salmon.

The TS is calculated from the backscattered acoustic intensity I, through the relation:

TS = 10log 10 (I) -SL + 40log 10 (r) + 2ar + NF + ψ, ( 1
)
where SL is the source level (intensity of the incident pulse), a = 0.051 dB/m is the absorption coefficient of sound in sea water, and 40log 10 (r) is a range correction. Furthermore, NF and ψ are the near-field and inter-beam corrections, respectively, which are calculated and measured during the sonar factory calibration.

A (shallow) image of a single 3D scan above the fish shoal is shown in Fig. 1c. This image allows the detection of several individual targets. The collection of individual targets provides the TS distribution (Fig. 2a), which is fitted with a Gaussian law to obtain TS = (-28 ± 1) dB, which spans from -31 dB to -25 dB. Such an enlarged TS distribution is unusual for fish raised under controlled conditions, as it corresponds to 30% fish total length variation [15]. As any TS alterations due to interbeam interference or near-field variations were measured and corrected through laboratory and on-site calibration experiments (Eq. ( 1)), the reason for the distribution width must be the randomness of the fish orientation, which can have a large impact on the TS measurement [START_REF] Lilja | [END_REF]15].

In the literature, the usual definition of TS is [START_REF] Simmonds | Fisheries Acoustics: Theory and Practices[END_REF]:

TS = 10log 10 (σ bs ), (2)
where σ bs is the backscattering cross-section; i.e., the normalized scattered intensity in the backward direction. In the present case where the salmon size is much larger than the wavelength, the measured σ bs corresponds to the acoustic intensity scattered mainly by the swimbladder (the most reflective organ in the fish body).

As an additional tool, if the scanning process is fast enough (the 3D image acquisition takes 1.02 s here), the fish movement can be observed for two or more successive scans. A histogram of fish velocities can be constructed by measuring the distance traveled by each fish between these two images [16]. Figure 2b shows the velocity histogram for the salmon cage that follows a Rayleigh law with mean v = 0.19 m/s. This means that during the duration of a 3D scan, each fish might have moved over a distance greater than the wavelength, but much smaller than the individual fish size. Furthermore, the Rayleigh velocity distribution confirms the visual observation that the dynamics individual fish are random inside the shoal. On the time scale of this experiment (∼10 min), no variation in the mean velocity was observed. However, the mean velocity estimation can be used over a longer time scale to monitor the fish activity for feeding optimization, for example.

Scattering mean free path measurements

Coherent backscattering is a wave interference phenomenon that is manifested as a pronounced angular dependence of the average backscattered acoustic intensity in the multiple scattering regime. More precisely, the intensity in the exact backscattering direction (θ = 0 • ) is twice that for large scattering angles θ [11]. The backscattered intensity shows a cone that narrows with time t (or depth z = v 0 t/2 with v 0 =1500 m/s, the speed of sound in sea water) [7]. Figure 3a shows the measurement of CBS in the salmon cage by the beamforming method [17] with the Seapix probe [18]: the incident plane wave is generated using all of the 128 transducers and spatial Fourier transform is performed over the array after reception in order to probe the angular dependence of backscattered acoustic intensity. The CBS is measured with a depth resolution dz = 0.1 m but for the sake of clarity, it is plotted in Figure 3a only for times corresponding to three different depths z. When the acoustic wave propagates deeper into the fish shoal, it undergoes more scattering events and gets closer to the multiple scattering regime. The peak in the intensity at θ = 0 • increases gradually with depth.

The rise of the CBS peak can be characterized by the intensity enhancement factor EF(z) = I(θ = 0, z)/I(θ max , z), where θ max is the angle for which the intensity profile becomes flat. In this case, the maximum angle of observation θ max = 6 • appears to be sufficient since the intensity I(θ max , z) seems to be independent of the depth z. In the single scattering regime, the intensity profile shows no fine structure and EF(z) = 1. Once the multiple scattering regime is reached, the intensity is halved for large angles, and EF(z) tends to 2. Finally, single and multiple scattering contributions are equivalent for EF(z) ≈ 4/3, which corresponds to a propagation distance equal to the scattering mean free path s [START_REF] Derode | [END_REF]. Measurement of the enhancement factor is shown in Figure 3b. From Figure 3b, it is clear that the multiple scattering regime is not fully reached for depths z < 10 m, as the enhancement factor grows with z. A linear fit EF(z) = Az + 1 to the 'transitional regime' together with the condition EF( s ) = 4/3, yields an accurate estimation of the scattering mean free path s = (4/3 -1)/A = (4 ± 0.3) m.

RESULTS AND DISCUSSION

During these experiments, there were no currents in the fjord, and therefore no fish polarisation [19] was observed, as can be seen for other at-sea cages under strong currents from tidal effects. Thus, we can reasonably assume that the fish are randomly oriented in the azimuthal plane, and we do not expect complex effects, such as the anisotropic light diffusion that occurs in liquid crystals [20]. Furthermore, the reasonable fish density (∼ 10 fish/m 3 ) allows us to neglect correlations between scatterers [21] and to use the relation [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF]:

η = 1 σ s , ( 3
)
where η is the fish density and σ is the total scattering cross-section σ = σ bs /φ(γ = π). The phase function φ(γ) reflects the anisotropy of sound scattering by a fish [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF]. For isotropic scattering by an infinite cylinder, φ(γ) = 1/2π. In the present case, considering the length L of the fish, we approximate its swimbladder as an immersed air cylinder with radius [23] R = 0.0245L. By numerically solving the scattering problem [START_REF] Van De Hulst | Light Scattering by Small Particles[END_REF] for such a scatterer, this gives φ(γ = π) δγ = 9 × 10 -2 , where φ(γ = π) δγ is the phase function averaged over a small angular range δγ = 10 • around the backscattering direction γ = π, to take into account the angular spectrum of emission of our ultrasonic probe. Thus, the simultaneous knowledge of the backscattering cross-section and the mean free path gives a straightforward estimation of the fish number density η = (14 ± 3) fish/m 3 . However, this estimation corresponds to the fish density in the shoal and not in the cage. Indeed, because of its spherical shape, the shoal does not occupy the whole volume of the cubic cage (see Fig. 1). Thus, the measured fish density has to be corrected by the volume ratio between the cubic cage and its inscribed sphere: π/6. The effective fish density in the cage is then η × π/6 = 7.4 fish/m 3 , which agrees with the farmer estimations (∼7 fish/m 3 ). Note that during a feeding sequence, the shape of the shoal can change rapidly and approaches a torus. Therefore feeding sequences were excluded from the data analysis.

CONCLUSION

The combination of fishery acoustics and mesoscopic physics provides new opportunities for fish density estimation, by taking advantage of the multiple scattering of sound. Experiments were performed in salmon cages, although the method is a priori not limited to any particular fish size or species. By taking into account the avoidance phenomena [START_REF] Brehmer | [END_REF], this CBS density estimation approach can also be applied to fish shoals in their natural environment. For example, CBS can be used for density estimation of dense herring shoals (η ∼ 60 fish/m 3 ), which is at present a key challenge [START_REF] Simmonds | Fisheries Acoustics: Theory and Practices[END_REF] for fishing resources monitoring. However, for such high densities, one has to be careful about strong mesoscopic interference effects that can impact the CBS temporal evolution [18]. Such effects appear when the scattering mean free path is so low that k s ∼ 1 (where k is the wave number). Thus, high shoal density can be probed with CBS provided that fish average T S is low enough to fulfill the condition k s 1. The CBS density estimation method presented here has some limitations. Indeed, some species, such as sea bream, live in very dense shoals and thus the acoustic waves are immediately multiply scattered when they penetrate inside the fish shoal [18]. It can then difficult to identify and isolate enough individual targets to obtain a satisfactory TS estimation. In this case, TS measurements have to be performed by other means, such as acoustic characterization on a limited number of fish or on isolated fish. Additionally, the spherical shape of the shoal is an approximation, and this could be improved by accurately measuring the effective volume occupied by the fish shoal in the cage.

The authors wish to thank Mikkel Straume from Aquabio, and Eide Fjorbruk for allowing access to their salmon cages.

FIG. 2 .

 2 FIG. 1. (a) Scheme of the Seapix sonar probe positionned at the surface of the open sea cage. (b) Snapshot of a volumetric scan of a cage (backscattered acoustic intensity I). (c) Isosurface representation of the shallow scan (z < 2 m). Hot spots represent the closed volumes for which TS >-31 dB.

FIG. 3 .

 3 FIG. 3. (a) Angular dependence of the intensity for three different depths z (b) Depth dependance of the enhancement factor EF (z). The dashed line represents the linear fit used to measure the scattering mean free path s.