The Effect of Attenuation from Fish Shoals on Long-Range, Wide-Area Acoustic Sensing in the Ocean

Abstract

Acoustics is the primary means of long-range and wide-area sensing in the ocean due to the severe attenuation of electromagnetic waves in seawater. While it is known that densely packed fish groups can attenuate acoustic signals during long-range propagation in an ocean waveguide, previous experimental demonstrations have been restricted to single line transect measurements of either transmission or backscatter and have not directly investigated wide-area sensing and communication issues. Here we experimentally show with wide-area sensing over 360° in the horizontal and ranges spanning many tens of kilometers that a single large fish shoal can significantly occlude acoustic sensing over entire sectors spanning more than 30° with corresponding decreases in detection ranges by roughly an order of magnitude. Such blockages can comprise significant impediments to underwater acoustic remote sensing and surveillance of underwater vehicles, marine life and geophysical phenomena as well as underwater communication. This makes it important to understand the relevant mechanisms and accurately predict attenuation from fish in long-range underwater acoustic sensing and communication. To do so, we apply an analytical theory derived from first principles for acoustic propagation and scattering through inhomogeneities in an ocean waveguide to model propagation through fish shoals. In previous experiments, either the attenuation from fish in the shoal or the scattering cross sections of fish in the shoal were measured but not both, making it impossible to directly confirm a theoretical prediction on attenuation through the shoal. Here, both measurements have been made and they experimentally confirm the waveguide theory presented. We find experimentally and theoretically that attenuation can be significant when the sensing frequency is near the resonance frequency of the shoaling fish. Negligible attenuation was observed in previous low-frequency ocean acoustic waveguide remote sensing (OAWRS) experiments because the sensing frequency was sufficiently far from the swimbladder resonance peak of the shoaling fish or the packing densities of the fish shoals were not sufficiently high. We show that common heuristic approaches that employ free space scattering assumptions for attenuation from fish groups can lead to significant errors for applications involving long-range waveguide propagation and scattering.

1. Introduction

Acoustics is the primary means of long-range and wide-area sensing in the ocean due to the severe attenuation of electromagnetic waves in seawater. While it is known that densely packed fish groups can attenuate acoustic signals during long-range propagation in an ocean waveguide, previous experimental demonstrations have been restricted to single line transect measurements of either transmission or backscatter and have not directly investigated wide-area sensing and communication issues. Here we analyze the effects of attenuation from multiple forward scattering on instantaneous imaging of fish population densities over thousands of square kilometers with Ocean Acoustic Waveguide Remote Sensing (OAWRS) [1,2,3]. We experimentally show with wide-area sensing over 360° in the horizontal and ranges spanning many tens of kilometers that a single large fish shoal can significantly occlude acoustic sensing over entire sectors spanning more than 30° with corresponding decreases in detection ranges by roughly an order of magnitude. Such blockages can comprise significant impediments to underwater acoustic remote sensing and surveillance of underwater vehicles and geophysical phenomena as well as underwater communication. Understanding the effects of attenuation from fish is also important for conservation efforts. Climate change and overfishing have led to massive declines in marine populations, with an estimated 33 percent of marine fish stocks being harvested at unsustainable levels [4]. In order to effectively regulate fishing quotas and stabilize marine populations, it is increasingly important to develop methods for accurately surveying and monitoring fish populations over ecosystem scale areas. Low frequency (<3 kHz) acoustic sensing systems currently provide the only means of instantaneously sensing fish populations over kilometer-scale areas [2,3], however attenuation within fish shoals can reduce the strength of acoustic signals and bias population density estimates.

The potential limitations in detection range imposed by attenuation from fish makes it important to understand the relevant mechanisms and accurately predict attenuation from fish. To do so, we apply an analytical theory derived from first principles for acoustic propagation and scattering through inhomogeneities in an ocean waveguide [5] to model propagation through fish shoals. Our formulation treats multiple scatter in the forward direction. Multiple scattering in non-forward directions was found to be negligible given the packing densities and target strengths of the fish shoals we encountered based on the work of Andrews et al. [6].

The formulation used here for modeling attenuation combines waveguide scattering theory [7,8] and a differential slab marching approach introduced by Rayleigh for the mean field to derive the first two statistical moments of the acoustic field in a waveguide with inhomogeneities [5]. This formulation has been previously shown to be consistent with experimental measurements of attenuation and temporal coherence loss in the presence of internal waves for both mid-frequency signals (415 Hz) in a continental shelf environment and low-frequency signals (10-70 Hz) in a deep ocean waveguide [9,10,11].

In previous experiments, either the attenuation from fish in the shoal or the scattering cross sections of fish in the shoal were measured but not both, making it impossible to directly confirm a theoretical prediction on attenuation through the shoal [12,13]. Here, both measurements have been made and they experimentally confirm the waveguide theory presented. We study the effects of attenuation due to multiple forward scattering on sensing of herring, cod and capelin shoals in an ocean waveguide. A review of developments over past decades in wide-area underwater acoustic sensing of marine life appears in Jagannathan et al. [1]. No attenuation was measured in previously published OAWRS imagery where the acoustic sensing frequencies were off the resonance peak of the shoaling fish, including Mid-Atlantic Bight herring [3], Gulf of Maine herring [2] and Lofoten cod [14], as shown in Appendix A. Attenuation resulting in significant reductions in sound pressure level (≥10 dB) is observed in OAWRS imagery presented here of dense herring shoals collected in February 2014 near Ålesund, Norway where swimbladder resonance was near the sensing frequency.

We examine how the shadowing effects of dense fish shoals along the propagation path can be accounted for in OAWRS data to produce accurate estimates of fish areal population density using the waveguide attenuation model of Ratilal and Makris [5]. We also explore methods to predict attenuation from fish in various continental shelf environments and examine how sensing frequency can be adjusted to yield the largest detection ranges.

Attenuation from fish has also been observed in more conventional fisheries acoustics sensing methods that rely on ultrasonic sensors such as downward directed echosounders and sonars [15]. Since ultrasonic fisheries acoustic sensing employs direct paths between sensors and scatterers, modal waveguide propagation effects are not present and simple theoretical formulations derived earlier for multiple scattering in free space are valid [16,17]. These have traditionally been employed in the analysis of fisheries acoustic data where attenuation due to dense fish groups has been studied [18,19,20,21,22,23]. While free space formulations for attenuation from fish are valid for downward directed echosounders, we find that common heuristic approaches that employ free space scattering assumptions for attenuation from fish groups in long-range waveguide propagation can lead to significant errors [24,25,26].

2. Materials and Methods

Evidence of attenuation is observable in OAWRS scattering strength images of herring shoals near Ålesund, Norway. These scattering strength images are generated by beamforming, matched filtering and charting scattered returns and then correcting for source level, transmission loss and areal resolution footprint (Appendix B). Since scattering losses are not incorporated into the transmission loss, attenuation effects are visible when the scattered returns from a distant shoal diminish after an occluding shoal blocks the propagation path (Figure 1). In addition, attenuation within a single shoal is observable when the scattered returns from the shoal are strongest at the edge closest to the sensing system, with received sound pressure level decreasing with range. Ambient seafloor scattering in Figure 1 is not significantly attenuated behind the occluding shoal. This is because there is a right/left ambiguity about the horizontal line-array’s axis, so if seafloor scattering or ambient noise on one side of the ambiguity is significantly attenuated, the received signal will tend to be dominated by seafloor scattering or ambient noise from the ambiguous side. Ambiguity about the location of fish shoals in OAWRS images was resolved mainly by varying receiver ship heading [27].

Here, we apply the formulation of Ratilal and Makris [5] to attenuation from fish shoals and demonstrate that models that ignore waveguide physics can lead to significant error (>5 dB for a given large shoal) in predicting attenuation in acoustic propagation in an ocean waveguide.

In a waveguide without scatterers, the pressure field at receiver $\mathit{r}$ from source at ${\mathit{r}}_{\mathbf{0}}$ can be expressed as a sum of modes:

$${\mathrm{\Psi }}_i\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)=\sum _n{\mathrm{\Psi }}_i^{\left(n\right)}\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)$$

(1)

where ${\mathrm{\Psi }}_i^{\left(n\right)}\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)$ is the contribution to the field by mode n. ${\mathrm{\Psi }}_i^{\left(n\right)}\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)$ can be expressed as

$${\mathrm{\Psi }}_i^{\left(n\right)}\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)=4\pi \frac{i}{d\left(z_0\right)\sqrt{8\pi }}{e}^{-i\pi /4}{u}_n\left(z\right)u_n\left(z_0\right)\frac{e^{i{\xi }_n\rho }}{\sqrt{{\xi }_n\rho }}$$

(2)

where $\rho$ is the horizontal range, z is water depth, $d\left(z_0\right)$ is the medium density at source depth $z_0$ and $u_n\left(z\right)$ is the amplitude of acoustic mode n in the waveguide, which depends on the sound speed profile. The effects of continuous seafloor and sea surface scattering on each mode is incorporated in its complex horizontal wavenumber ${\xi }_n$.

Each $u_n\left(z\right)$ is normalized such that

$${\int }_0^{\infty }\frac{1}{d\left(z\right)}{u}_n\left(z\right)u_m\left(z\right)dz={\delta }_{nm}$$

(3)

In the formulation developed by Ratilal and Makris [5], the total mean forward field in an acoustic waveguide with scatterers can be expressed as

$$〈{\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)〉=\sum _n{\mathrm{\Psi }}_i^{\left(n\right)}\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)e^{i{\int }_0^{\rho }{\nu }_n\left({\rho }_s\right)d{\rho }_s}$$

(4)

where each dispersion and attenuation coefficient ${\nu }_n$ describes the change in the horizontal wavenumber of mode n as it propagates through the scatterers. Dispersion and attenuation coefficients depend on the horizontal wavenumber ${\xi }_n$, the shape of the corresponding mode $u_n$, the volume density of the scatterers $n_V$ and the scatter function of an individual scatterer S. A general formulation for ${\nu }_n$ can be found in Equation (60a) of Reference [5]. Since long-range ocean sensing systems typically operate at low frequencies where the acoustic wavelength is larger than the dimensions of a fish, individual fish will be compact scatterers and the dispersion and attenuation coefficients for fish shoals can be obtained from Equations (19) and (60a) of Reference [5]:

$${\nu }_n\left(\rho \right)={\int }_0^{\infty }\frac{2\pi }{k}\frac{1}{{\xi }_n}\frac{1}{d\left(z_t\right)}{\left(u_n\left(z_t\right)\right)}^2〈s(\rho ,z_t)〉d{z}_t$$

(5)

where $〈s(\rho ,z)〉$ is the expected scatter function density of the scatterers. Since the spacing between individual fish is larger than the acoustic wavelength, the fish are incoherent scatterers and the expected scatter function density of a group of fish can be expressed as $〈s(\rho ,z)〉=n_V(\rho ,z)〈S(\rho ,z)〉$. Equations for calculating the scatter function S of an individual fish are shown in Appendix C.

The variance of the forward field can be expressed as

$$\mathrm{Var}\left({\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)\right)=\sum _n\frac{2\pi }{d^2\left(z_0\right)}\frac{1}{\left|{\xi }_n\right|\rho }{\left|u_n\left(z_0\right)\right|}^2{\left|u_n\left(z\right)\right|}^2{e}^{-2\Im \left[{\xi }_n\rho +{\int }_0^{\rho }{\nu }_n\left({\rho }_s\right)d{\rho }_s\right]}\left(e^{{\int }_0^{\rho }{\mu }_n\left({\rho }_s\right)d{\rho }_s}-1\right)$$

(6)

where ${\mu }_n\left(\rho \right)$ is the exponential coefficient of modal field variance. A general formulation for ${\mu }_n$ can be found in Equation (94a) of Reference [5]. Since the scatter function of an individual fish is omnidirectional in long-range ocean sensing applications, the exponential coefficient of modal field variance for fish shoals can be obtained from Equations (19), (72) and (94a) of Reference [5]:

$${\mu }_n\left({\rho }_s\right)=\sum _m\sqrt{\frac{\rho }{2\pi {\xi }_m{\rho }_s(\rho -{\rho }_s)}}\frac{1}{\left|{\xi }_m\right|}{\int }_0^{\infty }\frac{4{\pi }^2}{k^2{d}^2\left(z_t\right)}{\left|u_n\left(z_t\right)\right|}^2{\left|u_m\left(z_t\right)\right|}^2{V}_c\left(z_t\right)\mathrm{Var}\left(s(\rho ,z_t)\right)d{z}_t$$

(7)

where the scatter function coherence volume $V_c$ quantifies the spatial scale over which the scatter functions of two fish are correlated. The variance of the scatter function density for incoherent scatterers can be obtained from Equations (A23) of Reference [5]:

$$\mathrm{Var}\left(s\right)=\frac{1}{V_c}{n}_V\mathrm{Var}\left(S\right)$$

(8)

Note that ${\mu }_n$ (Equation (7)) is independent of the coherence volume $V_c$ since the scatterers are incoherent.

The mean intensity or the second moment of the forward field is the sum of the coherent intensity and incoherent intensity:

$$〈{\left|{\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)\right|}^2〉={\left|〈{\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)〉\right|}^2+\mathrm{Var}\left({\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)\right)$$

(9)

Finally, the decrease in sound pressure level due to attenuation in the forward field from scattering can then be expressed as

$$\Delta SPL=10{log}_{10}{\left|{\mathrm{\Psi }}_i\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)\right|}^2-10{log}_{10}\left(〈{\left|{\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)\right|}^2〉\right)$$

(10)

As derived in Appendix D, the decrease in sound pressure level due to attenuation during two-way propagation to-and-from a uniform distribution of scatterers with mean depth $z_0$ and height H positioned at ${\mathit{r}}_{\mathit{t}}=({\mathbf{\rho }}_{\mathit{t}},z_t)$ within resolution footprint $A_R\left({\rho }_C\right)$ can be similarly defined as

$$\begin{array}{cc}\hfill \Delta SP{L}_{2way}& =10{log}_{10}\left({\int }_{A_R\left({\rho }_C\right)}{\int }_{z=z_0-H/2}^{z=z_0+H/2}{\left|{\mathrm{\Psi }}_i\left({\mathit{r}}_{\mathit{t}}|{\mathit{r}}_{\mathbf{0}}\right)\right|}^2{\left|{\mathrm{\Psi }}_i\left(\mathit{r}|{\mathit{r}}_{\mathit{t}}\right)\right|}^{2}d{z}_{t}d{\mathbf{\rho }}_{\mathit{t}}^{\mathbf{2}}\right)\hfill \\ \hfill & -10{log}_{10}\left({\int }_{A_R\left({\rho }_C\right)}{\int }_{z=z_0-H/2}^{z=z_0+H/2}〈{\left|{\mathrm{\Psi }}_T\left({\mathit{r}}_{\mathit{t}}|{\mathit{r}}_{\mathbf{0}}\right)\right|}^2〉〈{\left|{\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathit{t}}\right)\right|}^2〉d{z}_{t}d{\mathbf{\rho }}_{\mathit{t}}^{\mathbf{2}}\right)\hfill \end{array}$$

(11)

where ${\mathrm{\Psi }}_i\left({\mathit{r}}_{\mathit{t}}|{\mathit{r}}_{\mathbf{0}}\right)$ is the incident field from source ${\mathit{r}}_{\mathbf{0}}$ to target ${\mathit{r}}_{\mathit{t}}$ and ${\mathrm{\Psi }}_i\left(\mathit{r}|{\mathit{r}}_{\mathit{t}}\right)$ is the field scattered from target ${\mathit{r}}_{\mathit{t}}$ to receiver $\mathit{r}$ (both defined by Equation (1)), while $〈{\left|{\mathrm{\Psi }}_T\left({\mathit{r}}_{\mathit{t}}|{\mathit{r}}_{\mathbf{0}}\right)\right|}^2〉$ is the total mean intensity from source ${\mathit{r}}_{\mathbf{0}}$ to target ${\mathit{r}}_{\mathit{t}}$ and $〈{\left|{\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathit{t}}\right)\right|}^2〉$ is the total mean intensity from target ${\mathit{r}}_{\mathit{t}}$ to receiver $\mathit{r}$ (both defined by Equation (9)).

In the unique case where a fish shoal is uniformly distributed through the water column, we find that attenuation in a waveguide has a form approximately like that found in free space. Specifically, in this case $〈s(\rho ,z)〉=n_V\left(\rho \right)〈S\left(\rho \right)〉$ is independent of z, so that, with Equation (3), Equation (5) can be simplified to

$${\nu }_n\left(\rho \right)=\frac{2\pi }{k}\frac{1}{{\xi }_n}{n}_V\left(\rho \right)〈S\left(\rho \right)〉{\int }_0^{\infty }\frac{1}{d\left(z_t\right)}{\left(u_n\left(z_t\right)\right)}^{2}d{z}_t=\frac{2\pi }{k}\frac{1}{{\xi }_n}{n}_V\left(\rho \right)〈S\left(\rho \right)〉$$

(12)

resulting in a total mean field equal to

$$〈{\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)〉=\sum _n{\mathrm{\Psi }}_i^{\left(n\right)}\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)e^{i\frac{2\pi }{k}\frac{1}{{\xi }_n}{\int }_0^{\rho }{n}_V\left({\rho }_s\right)〈S\left({\rho }_s\right)〉d{\rho }_s}$$

(13)

If the horizontal wavenumbers of the propagating modes do not significantly deviate from the wavenumber (${\xi }_n\approx k$), we can further reduce Equation (13) using Equation (1) to yield

$$〈{\mathrm{\Psi }}_T\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)〉={\mathrm{\Psi }}_i\left(\mathit{r}|{\mathit{r}}_{\mathbf{0}}\right)e^{i\frac{2\pi }{k^2}{\int }_0^{\rho }{n}_V\left({\rho }_s\right)〈S\left({\rho }_s\right)〉d{\rho }_s}$$

(14)

which has the same form as the mean field of a free space plane wave propagating through a medium with scatterers, following Rayleigh [28]. If the mean intensity is dominated by the coherent field, attenuation in a waveguide will then follow the free space form of Equation (14). We model attenuation in Ålesund waters in a case where a herring shoal uniformly covers the water column. As shown in Figure 2, the full waveguide model and the free-space-like factored approximation effectively agree, with a negligible difference between the total attenuation predicted by the two models (less than 0.5 dB).

While attenuation from fish in a waveguide approximates free space attenuation when the fish uniformly cover the water column, it can differ significantly when the fish shoal is concentrated at a specific depth. We model attenuation in Ålesund waters in cases where herring are concentrated at the upper, middle and lower water column and we compare the results with a free-space-like factored approximation for attenuation of Equation (14) (Figure 3). In order to apply a free-space-like factored formulation to concentrated fish shoals, we modify the expected wavenumber change to ${\nu }_{freespace}=(2\pi /k^2)(n_A/D)〈S〉$ where $n_A$ is the areal density of the shoal and D is the water depth. This replacement effectively "spreads" a concentrated fish shoal across the water column while retaining the total number of fish. Previous studies have modeled attenuation from fish in a waveguide using this formulation [12,29].

We find that attenuation of each mode depends on the amplitude of the mode at depths where the fish are concentrated and the resulting attenuation will be strongly depth-dependent. In the upward-refracting environment modeled here, lower-order acoustic modes concentrated near the sea surface will pass through fish shoals near the seafloor with negligible attenuation and a receiver in the upper water column will measure significantly less attenuation than a receiver near the seafloor. Free-space-like factored formulations that ignore these depth-dependent modal attenuation effects can result in significant error (>5 dB for a given large shoal).

3. Results and Discussion

3.1. Consistency between Backscattered Returns and Attenuation

Here we demonstrate consistency between concurrently measured backscattering and attenuation levels from a single fish shoal using an example of attenuation from herring in Ålesund waters. In Figure 1B, there is no occluding shoal between the sensing system and a distant shoal and no attenuation is observed. In Figure 1C, the monostatic sensing system is positioned so that an occluding shoal in the propagation path attenuates the scattered returns of the distant shoal. We expect theoretical consistency between the scattering strength of the occluding shoal and the attenuation caused by the occluding shoal, where attenuation is measured as the decrease in scattered returns from the distant shoal after the occluding shoal moves into the propagation path.

The scattering strength and attenuation levels from the occluding shoal are measured using OAWRS data shown in Figure 1. The distant shoal and the occluding shoal are segmented by smoothing the scattering strength image using a circular averaging filter of radius 90 m and selecting regions with scattering strength values greater than 5 dB above the background seafloor scattering. Total two-way attenuation from the occluding shoal ($\Delta SP{L}_{2way,data}$) is measured as the difference between the scattering strength of the distant shoal before attenuation (Figure 1B) and after attenuation (Figure 1C). Note that this measurement of $\Delta SP{L}_{2way,data}$ assumes the areal density and vertical position of the occluding shoal and the distant shoal remain unchanged in the 17 min between the observations shown in Figure 1B,C.

When measuring scattering strength of shoals in cases where attenuation is present, care must be taken to exclude the effects of scattering loss within the shoal. We calculate the scattering strength of the occluding shoal ($S{S}_{data}$) by selecting pixels in the shoal with scattering strength values above the 20th percentile, effectively isolating pixels at the edge of the shoal closest to the sensing system, where scattering strength is strongest and attenuation effects are negligible. This analysis is performed for all six operating frequencies used in the 2014 OAWRS experiment.

The modeled scattering strength $S{S}_{model}$ of a shoal with mean shoal depth $z_0$, shoal thickness H and neutral buoyancy depth $z_{nb}$ at frequency f is calculated [30,31] from

$$S{S}_{model}(z_0,H,z_{nb},n_A,f)=10{log}_{10}\left(\frac{1}{H}{\int }_{z_0-H/2}^{z_0+H/2}{\int }_l{\left|\frac{S(z,z_{nb},l,f)}{k}\right|}^{2}g\left(l\right)dldz·n_A\right)$$

(15)

where $S(z,z_{nb},l,f)$ is the far-field scatter function of a single fish, k is the wavenumber in free space, l is the fork length of an individual fish, $g\left(l\right)$ is the Gaussian probability density function of the fork length and z is the fish depth. Equations for modeling $S(z,z_{nb},l,f)$ are shown in Appendix C. Total two-way attenuation $\Delta SP{L}_{2way,model}(z_0,H,z_{nb},n_A,r_h,f)$ is modeled from Equation (11) using the same five parameters used to calculate $S{S}_{model}$ and additionally assumes the horizontal extent of the occluding shoal along the propagation path, $r_h$. Note that $r_h$ cannot be directly determined from the scattering strength image in Figure 1 because attenuation within the occluding shoal masks the scattered returns at the far edge.

The standard deviation of the water depth along the propagation path is less than 5 m in the region where our measurements are made, which is consistent with the range-independent seafloor assumed by the current model. We verify this by using a parabolic equation model [32] to compare transmission between a flat waveguide and a varying-bathymetry waveguide using bathymetric data from the region where the experiment took place and we find the difference in transmission to be less than 1 dB.

Modeled scattering strength and attenuation values are simultaneously matched with the data using the maximum likelihood estimate of fish shoal parameters $z_0$, H, $z_{nb}$, $n_A$ and $r_h$. Since the acoustic field can be described as a circular complex Gaussian random variable (CCGR) and the time-bandwidth product of the acoustic measurements $\mu =\left(1\mathrm{second}\right)\left(50\mathrm{Hz}\right)\gg 1$, intensity measurements in the logarithmic domain such as $SS$ and $\Delta SP{L}_{2way}$ can be well-approximated as Gaussian random variables with variance independent of the mean [33,34]. Since scattering strength and attenuation are dependent on many of the same parameters ($z_0$, H, $z_{nb}$ and $n_A$), we treat $SS$ and $\Delta SP{L}_{2way}$ as correlated random variables. We then estimate fish shoal parameters $\widehat{\mathbf{\theta }}=[\widehat{z_0},\widehat{H},{\widehat{z}}_{nb},{\widehat{n}}_A,{\widehat{r}}_h]$ by maximizing the log-likelihood function $\ell \left(\mathbf{\theta }\right)$ assuming $SS$ and $\Delta SP{L}_{2way}$ are correlated Gaussian random variables with variance independent of the mean, as derived in Appendix E:

$$\begin{array}{cc}\hfill & \ell \left(\mathbf{\theta }\right)=\sum _{i=1}^{N_f}(-\frac{{\left(S{S}_{model}(\mathbf{\theta },f_i)-S{S}_{data}\left(f_i\right)\right)}^2}{{\sigma }_{SS}{\left(f_i\right)}^2}-\frac{{\left(\Delta SP{L}_{2way,model}(\mathbf{\theta },f_i)-\Delta SP{L}_{2way,data}\left(f_i\right)\right)}^2}{{\sigma }_{\Delta SPL}{\left(f_i\right)}^2}\hfill \\ \hfill & +2{\sigma }_{SS,\Delta SPL}\left[\frac{(S{S}_{model}(\mathbf{\theta },f_i)-S{S}_{data}\left(f_i\right))}{{\sigma }_{SS}{\left(f_i\right)}^2}\times \frac{(\Delta SP{L}_{2way,model}(\mathbf{\theta },f_i)-\Delta SP{L}_{2way,data}\left(f_i\right))}{{\sigma }_{\Delta SPL}{\left(f_i\right)}^2}\right])\hfill \end{array}$$

(16)

where ${\sigma }_{SS}{\left(f_i\right)}^2$ is the variance of the measured scattering strength in dB at frequency $f_i$, ${\sigma }_{\Delta SPL}{\left(f_i\right)}^2$ is the variance of the measured attenuation in dB at frequency $f_i$, ${\rho }_{SS,\Delta SPL}\left(f_i\right)$ is the sample covariance between measured scattering strength and attenuation at frequency $f_i$ and $N_f$ is the number of frequencies where $SS$ and $\Delta SP{L}_{2way}$ are measured. The maximum of the log-likelihood function is determined through an exhaustive grid search across the five unknown parameters ($z_0,H,z_{nb},n_A,r_h$). The ranges of these parameters are determined such that the herring shoal physically stays within the water column ($H<D$ and $H/2<z_0<D-H/2$, where D is the water depth) and the areal number density is physically realistic ($n_A<10$ fish/m², as determined by echosounder observations).

When fish shoal parameters are inverted by maximizing the log-likelihood function, the modeled scattering strength and attenuation values match measured values within a standard deviation, as shown in Figure 4. The inverted fish shoal depth, vertical thickness and areal density are consistent with synoptic echosounder measurements collected during the experiment (Figure A3). This consistency between scattering strength and attenuation demonstrates that the waveguide attenuation model can accurately predict attenuation levels in ocean sensing applications.

3.2. Attenuation Prediction and Frequency Selection

Using the waveguide attenuation model, we demonstrate how sensing frequency can be adjusted to yield the largest detection range for acoustic sensing of fish shoals in various continental shelf environments. Scattering strength and attenuation are modeled for each fish species and environment and we search for the sensing frequency that maximizes scattering strength uncorrected for two-way attenuation from fish ($SS-\Delta SP{L}_{2way}$), which in turn maximizes the detection range of the sensing system. Here $SS$ is modeled using Equation (15) and $\Delta SP{L}_{2way}$ is modeled using Equation (11).

We model the scattering strength uncorrected for attenuation of fish shoals in four environments: herring in the Gulf of Maine, herring in Ålesund waters, cod in Lofoten waters and capelin in Finnmark waters (Figure 5) using environmental and species-specific parameters determined from synoptic echogram data and trawl samples (

Table 1. Environmental parameters used for attenuation calculations in Figure 5.

Environment/Species	Areal Density (fish/m2)	Water Depth (m)	Shoal Depth (m)	Shoal Vertical Thickness (m)	Neutral Buoyancy Depth (m)	Mean Fish Length (cm)
Gulf of Maine herring	2 a	200 a	150 a	30 a	82 a	24 a
Ålesund herring	2.5 b	120 b	60 b	40 b	0 c	34 d
Lofoten cod	0.05 e	100 e	75 e	50 e	75 e	83 e
Finnmark capelin	10 b	300 b	30 b	40 b	10 f	17 d

^a Gong et al. [27], ^b Measured from echogram data collected during the experiment, ^c Figure 4, ^d Measured from trawl samples collected during the experiment, ^e Makris et al. [14], ^f Jagannathan et al. [1].

). We observe a tradeoff between (1) frequencies near swimbladder resonance, where backscattering is highest but attenuation can weaken the signal and (2) frequencies off-resonance, where attenuation is negligible but backscattering is weaker. When the horizontal extent of fish along the propagation path is less than 1 km, attenuation is less of a concern and frequencies near fish swimbladder resonance are optimal. As the horizontal extent of high-density shoals in the propagation path increases, attenuation will severely limit sensing near resonance and off-resonance frequencies will become more favorable.

The predictions of the waveguide attenuation model shown in Figure 5 are in agreement with attenuation levels observed in previous OAWRS experiments. Significant attenuation from Ålesund herring ($\Delta SPL$ > 10 dB) is predicted at the sensing frequencies used in the 2014 experiment, which is consistent with observations from that experiment. The absence of attenuation from Gulf of Maine herring in 2006 and Lofoten cod in 2014 can be explained by off-resonance sensing frequencies. Attenuation from Finnmark capelin was less than 2.5 dB even at resonance because the horizontal extent of observed capelin shoals was not large enough to produce significant attenuation (<0.25 km).

This analysis shows the waveguide attenuation model can be used to determine the conditions for significant attenuation and select sensing frequencies that maximize detection range. For example, attenuation from Ålesund herring can be avoided by choosing a sensing frequency above or below the swimbladder resonance of the fish shoals. The results of our model are specific to lower-frequency acoustic sensing, where the dimensions of a fish swimbladder are smaller than the acoustic wavelength.

3.3. Estimating Scattering Strength with Attenuation

We developed a method for accounting for attenuation in scattering strength measurements using the waveguide attenuation model. A scattering strength image uncorrected for scattering losses is generated from OAWRS data using the method described in Appendix B. In regions where scattering is dominated by fish, each scattering strength pixel can be corrected according to the following equation:

$$SS(r_0,{\theta }_0)=\widetilde{SS}(r_0,{\theta }_0)+\Delta SP{L}_{2way}(r_0,{\theta }_0)$$

(17)

where $(r_0,{\theta }_0)$ are polar coordinates defined with respect to the monostatic sensing system, $SS$ is the true scattering strength, $\widetilde{SS}$ is the scattering strength uncorrected for scattering losses and $\Delta SP{L}_{2way}$ is two-way attenuation due to fish calculated from Equation (11). As shown in Equation (5), $\Delta SP{L}_{2way}(r_0,{\theta }_0)$ will depend on the volume density of the fish between the sensing system and the target, $n_V(r\in [0,r_0],{\theta }_0)$, which can be calculated at each pixel as a function of $SS$ according [27] to

$$n_V(r,\theta )=\frac{n_A(r,\theta )}{H}=\frac{1}{H}{10}^{\left(SS(r,\theta )-TS\right)/10}$$

(18)

where $n_A$ is the areal density of the fish, H is the shoal thickness and $TS$ is the target strength of an individual fish (modeled in Appendix C). As a result, each calculation of scattering strength, $SS(r_0,{\theta }_0)$, depends on the scattering strength at pixels between the sensing system and the target, $SS(r\in [0,r_0],{\theta }_0)$, so Equation (17) must be iteratively applied to pixels closest to the sensing system before correcting pixels further out.

Using this method, we modify the OAWRS scattering strength image shown in Figure 6A to include scattering losses. We designate regions where scattering is dominated by fish by smoothing the scattering strength image using a circular averaging filter of radius 90 m and selecting regions with scattering strength values greater than 5 dB above the background seafloor scattering. Corrections are made to pixels within these regions assuming the fish are uniformly distributed between depths 40 m and 80 m with neutral buoyancy depth 0 m, as determined by the parameter inversion in Equation (16).

After modifying the scattering strength image to include scattering losses, shadowing effects caused by attenuation are no longer visible (Figure 6B). The scattering strength of the fish groups are comparatively uniform in space and are not biased by the horizontal extent of fish in the propagation path. It is important to note that scattering loss corrections cannot be made in cases where the attenuated signal is masked by ambient noise or background seafloor scattering. As a result, detection range limitations caused by attenuation from fish can only be improved by selecting a proper sensing frequency, as discussed in Section 3.2.

4. Discussion

We experimentally and theoretically demonstrate how propagation through vast and dense shoals of swimbladder-bearing fish can attenuate acoustic signals and reduce detection range for long-range sensing in the ocean as a consequence of scattering and absorption by the fish. We modeled attenuation from fish shoals in continental-shelf environments using a normal-mode based formulation derived from first principles for acoustic propagation through inhomogeneities in a waveguide. In previous experiments, either the attenuation from fish in the shoal or the scattering cross sections of fish in the shoal were measured but not both, making it impossible to directly confirm a theoretical prediction on attenuation through the shoal. Here, both measurements have been made and they experimentally confirm the waveguide theory presented. We show that common heuristic approaches that employ free space scattering assumptions for attenuation from fish groups can lead to significant error (>5 dB for a given large shoal) for applications involving long-range waveguide propagation and scattering. This same basic theory applies to fish without swimbladders, except there will be no resonance effects and attenuation is expected to be increasing with frequency as in Rayleigh-Born scattering.

5. Conclusions

We experimentally show with wide-area sensing over 360° in the horizontal and ranges spanning many tens of kilometers that a single large fish shoal can significantly occlude acoustic sensing over entire sectors spanning more than 30° with corresponding decreases in detection ranges by roughly an order of magnitude. Such blockages can comprise significant impediments to underwater acoustic remote sensing and surveillance of underwater vehicles, marine life and geophysical phenomena as well as underwater communication. We apply a rigorous formulation for modeling multiple forward scattering through inhomogeneities in an ocean waveguide to model attenuation through fish shoals in a continental shelf environment. The approach is found to accurately predict attenuation observed in Ocean Acoustic Waveguide Remote Sensing measurements over a variety of species and environments, including herring in the Georges Bank and the Nordic Seas, as well as cod and capelin in the Nordic Seas. The approach can be applied to both active and passive waveguide propagation and sensing through fish. We show that previous attenuation models that ignore waveguide physics can lead to sound pressure level predictions that are in error by at least an order of magnitude in many practical scenarios. We show that for sensing through fish shoals at frequencies near the peak swimbladder resonance of the fish, attenuation can adversely affect sensing within and beyond the shoal. In these cases, we find that the detection range of sensing systems can be maximized by choosing sensing frequencies off swimbladder resonance where attenuation is negligible. We also show how waveguide attenuation modeling can be used to more accurately estimate scattering amplitudes and fish population densities in regions where shadowing from attenuation is present.