Feasibility of Ocean Acoustic Waveguide Remote Sensing (OAWRS) of Atlantic Cod with Seafloor Scattering Limitations

Abstract

Recently reported declines in the population of Atlantic cod have led to calls for additional survey methods for stock assessments. In combination with conventional line-transect methods that may have ambiguities in sampling fish populations, Ocean Acoustic Waveguide Remote Sensing (OAWRS) has been shown to have a potential for providing accurate stock assessments (Makris N.C., et al. Science 2009, 323, 1,734–1,737; 54th Northeast Regional Stock Assessment Workshop (54th SAW) US Department of Commerce, Northeast Fisheries Science Center, 2012). The use of OAWRS technology enables instantaneous wide-area sensing of fish aggregations over thousands of square kilometers. The ratio of the intensity of scattered returns from fish versus the seafloor in any resolution cell typically determines the maximum fish detection range of OAWRS, which then is a function of fish population density, scattering amplitude and depth distribution, as well as the level of seafloor scattering. With the knowledge of oceanographic parameters, such as bathymetry, sound speed structure and attenuation, we find that a Rayleigh–Born volume scattering approach can be used to efficiently and accurately estimate seafloor scattering over wide areas. From hundreds of OAWRS measurements of seafloor scattering, we determine the Rayleigh–Born scattering amplitude of the seafloor, which we find has a f^2.4 frequency dependence below roughly 2 kHz in typical continental shelf environments along the US northeast coast. We then find that it is possible to robustly detect cod aggregations across frequencies at and near swim bladder resonance for observed spawning configurations along the U.S. northeast coast, roughly the two octave range 150–600 Hz for water depths up to roughly 100 m. This frequency range is also optimal for long-range ocean acoustic waveguide propagation, because it enables multimodal acoustic waveguide propagation with minimal acoustic absorption and forward scattering losses. As the sensing frequency moves away from the resonance peak, OAWRS detection of cod becomes increasingly less optimal, due to a rapid decrease in cod scattering amplitude. In other environments where cod depth may be greater, the optimal frequencies for cod detection are expected to increase with swim bladder resonance frequency.

1. Introduction

Reported declines [1–5] in the population of Atlantic cod have led to calls for additional survey methods for stock assessments, specifically the use of Ocean Acoustic Waveguide Remote Sensing (OAWRS) in the Gulf of Maine [6–8]. Recently, OAWRS techniques have been shown to be capable of instantaneous wide-area sensing of marine life over thousands of square kilometers [9,10]. Dwindling cod populations have a potential to affect long-term ecological balance [11–13] and the sustainability of the cod fishery along the US northeast coast [14]. Cod assessments typically incorporate data collected from conventional acoustic and trawl line transect surveys [15–19] that may lead to ambiguities in population estimates [17,20,21]. The combination of conventional methods and OAWRS techniques, however, has a potential for accurately estimating fish stocks over ecosystem-scale areas [9,22]. Here, we assess the feasibility of OAWRS detection and enumeration of cod in typical continental shelf waveguide environments along the US northeast coast. We do so by combining ocean-acoustic waveguide propagation modeling [9,10] that has been calibrated in a variety of continental shelf environments for OAWRS applications with a model for cod scattering that matches well with measured data [23] for cod scattering.

Both OAWRS detection range and minimum detectable fish population density are limited by scattered returns from the seafloor in the same resolution cell as the fish. Efficient and reliable estimates of seafloor scattering are then necessary to determine OAWRS fish detection limitations. Here, we use a Rayleigh–Born volume scattering approach, where multimodal propagation effects, which appear only in the waveguide Green function, can be mathematically separated from seafloor scattering effects. Our approach differs from traditional investigations of seafloor scattering using free-space plane-wave approaches [18,24], which often do not efficiently or effectively translate to multimodal propagation and scattering in an acoustic waveguide [25–27]. We estimate seafloor scattered returns by applying the Rayleigh–Born approximation to Green’s theorem [28]. From hundreds of measurements of seafloor scattering in continental shelf waveguide environments along the US northeast coast, we determine key statistics of the Rayleigh–Born seafloor scattering amplitude for various source frequencies. We show that OAWRS techniques are capable of robustly detecting Atlantic cod above seafloor scattering at and near their swim bladder resonance frequencies. These frequencies also happen to be optimal for long-range propagation in an ocean waveguide. Away from the swim bladder resonance frequencies, OAWRS detection of cod can become less optimal, due to rapid decreases in the cod scattering amplitude.

2. Data Collection

2.1. Seafloor Scattering Data Collection during OAWRS Experiments

We used OAWRS seafloor scattering data acquired in 2003 on the New Jersey continental shelf [10,29] and in 2006 on the northern flank of Georges Bank in the Gulf of Maine [9,30] (Figure A.1). The areas investigated for study here had relatively constant bathymetry with sandy bottoms of mean density and sound speed, 1.9 g/cm³ and 1700 m/s, respectively [27,30,31]. Hundreds of sound speed profile samples of the water column were taken during the experiments to allow accurate characterization of both the mean and randomly fluctuating components of the continental shelf waveguide. Here, by “scattered returns” we mean the OAWRS source signal that is scattered from a target, such as fish or seafloor, and received at the OAWRS horizontal receiver array, typically measured in decibel units (dB re 1 μPa).

2.2. Measured Distribution of Atlantic Cod in Massachusetts State Waters

We used cod body length distributions and aggregation densities measured during recent spring surveys conducted from New Hampshire during May–June, 2011, described in [32,33]. The surveys were conducted just south of the Isle of the Shoals and entered the Gulf of Maine Cod Spawning Protection Area just a few nautical miles east of the northern Massachusetts coastline [34]. A scientific echo sounder with a 120-kHz split beam transducer produced range-depth images of cod aggregations [32,33], as shown in Figure 1. Spawning aggregations imaged along the survey tracks were found to have areal population densities of roughly 0.01 cod/m² [32,33]. Concurrent trawls found a mean cod length of roughly 66 cm (Figure 2) [32,33]. The most dense cod aggregations were found within roughly 10 m of the seafloor, where the water depth varied between roughly 50 to 90 m (Figure 1) [32,33]. These recent measurements of cod during spawning are used to provide examples of cod distributions, from which the feasibility of using OAWRS to detect cod aggregations is assessed. Since no cod were found in trawls during the Northeast Fisheries Science Center Georges Bank Atlantic Herring survey [35] that was conducted in conjunction with an OAWRS survey during the peak fall herring spawning in 2006, no ground truth experimental comparisons between OAWRS detections and known cod aggregations are available.

3. Methods

3.1. Estimation of Rayleigh–Born Seafloor Scattering Amplitude Statistics from Measured Seafloor Scattering

The expected magnitude squared of the Rayleigh–Born seafloor scattering amplitude, 〈|A_S|²〉, is estimated by minimizing the mean squared error between the measured and modeled scattered field (Equation (11)) over range with respect to ${|\widehat{A_S}|}^2$. For the m^th radial beam of the j^th transmission, an estimate, ${\widehat{{|A_S|}^2}}_m^j$, is obtained by minimizing the sum of the mean squared error over L OAWRS range resolution cells, so that:

$$[{{|\widehat{{\text{A}}_S\hspace{0.17em}(f_c)}|}^2}_m^j]=\underset{{|{\text{A}}_S|}^2}{\text{min}}\frac{1}{L}\sum _{i=1}^L{\left|10\hspace{0.17em}\text{log}(\frac{{|{\mathrm{\Phi }}_{m^{\mathit{Data}}}^j({\mathbf{r}}_{S_i}|\mathbf{r},{\mathbf{r}}_0,f_c)|}^2}{{\mathbb{P}}_{\mathit{ref}}^2})-10\hspace{0.17em}\text{log}(\frac{\text{Var}({\mathrm{\Phi }}_m({\mathbf{r}}_{S_i}|\mathbf{r},{\mathbf{r}}_0,f_c))}{{\mathbb{P}}_{\mathit{ref}}^2})\right|}^2$$

(1)

where ${|{\mathrm{\Phi }}_{m^{\mathit{Data}}}^i({\mathbf{r}}_{S_i}|\mathbf{r},{\mathbf{r}}_0,f_c)|}^2$ is the measured scattered field and Var(Φ_m(r_Si |r, r₀, f_c)) is the variance of the modeled scattered field; a good approximation to the second moment of the scattered field (Equations (A.17) and (A.18)), at the i^th OAWRS range resolution cell centered on r_Si and ℙ_ref, is 1 μPa/Hz in water. The estimate of 〈|A_S(f_c)|²〉 for each source frequency, f_c, is then the mean of ${{|\widehat{{\text{A}}_S\hspace{0.17em}(f_c)}|}^2}_m^j$ over M radial beams and J transmissions:

$$〈|\widehat{{\text{A}}_S(f_c)}{|}^2〉=\frac{{\sum }_{j=1}^J{\sum }_{m=1}^M{{|\widehat{{\text{A}}_S\hspace{0.17em}(f_c)}|}^2}_m^j}{\mathit{JM}}$$

(2)

We restrict our analysis to beams that are not contaminated by bio-clutter or ship noise and to regions that are within approximately 30 km of the receiver array (Figure A.1). We have excluded the beams where the scattered returns from the seafloor fall below the local ambient noise in the same beam. Seafloor scattered returns can be distinguished from fish scattered returns in OAWRS images, because (i) they have different frequency dependence from fish scattered returns [9,10,30] and (ii) they are statistically stationary in time as opposed to non-stationary returns from fish aggregations [9,10,27,30,37].

Comparisons between the measured and modeled scattered field level in the New Jersey continental shelf and Georges Bank for various source frequencies are shown in Figures A.2 and A.3, respectively. We find that the mean modeled seafloor scattered level matches well (within one standard deviation) with the respective mean measured scattered level over a given resolution cell along the entire sensing range for most frequencies that have been used for OAWRS sensing in the past. The 1,325-Hz New Jersey continental shelf scattered field data (Figure A.2C) may be contaminated by diffusely distributed scatterers in the water column that affect the range-dependence of the scattered field.

3.2. Fish-to-Seafloor Scattering Ratio

For a given fish aggregation density of N fish in the water column per unit area, the second moment of total scattered field from all N fish [10,38] is:

$$\langle {|{\mathrm{\Phi }}_F(N)|}^2\rangle =N\langle {|{\mathrm{\Phi }}_F(1)|}^2\rangle$$

(3)

where Φ_F (1) is the scattered field from one fish per unit area. For the fish aggregation to be detectable, the expected magnitude squared of the scattered field from N fish, 〈|Φ_F(N)|²〉, should be greater than that from the seafloor, 〈|Φ_S|²〉, over the same area:

$$N\langle {|{\mathrm{\Phi }}_F(1)|}^2\rangle >\langle {|{\mathrm{\Phi }}_S|}^2\rangle$$

(4)

or:

$$\text{FSR}=\frac{N\langle {|{\mathrm{\Phi }}_F(1)|}^2\rangle }{\langle {|{\mathrm{\Phi }}_S|}^2\rangle }>1$$

(5)

where FSR is the fish-to-seafloor scattering ratio. Then, the minimum required number of fish per unit area, N_min, above which fish scattering is greater than seafloor scattering is:

$$N_{\text{min}}=\frac{\langle {|{\mathrm{\Phi }}_S|}^2\rangle }{\langle {|{\mathrm{\Phi }}_F(1)|}^2\rangle }$$

(6)

As a direct consequence of Equation (5), FSR decreases with decreasing fish aggregation density, and so, it becomes harder to detect less dense fish aggregations above seafloor scattering within a given OAWRS resolution footprint.

4. Results

4.1. Experimental Estimation of Rayleigh-Born Seafloor Scattering Amplitude

Using OAWRS measurements of long-range seafloor scattering in typical sandy continental shelf waveguide environments along the US northeast coast, we estimate the magnitude squared of seafloor scattering amplitude per coherence volume (Figure 3) over the range of frequencies that are conducive for long-range waveguide propagation and where the scattering amplitude of fish aggregations is high due to swim bladder resonance. We find that seafloor scattering amplitude squared follows a roughly f^2.4 frequency dependence, varying as f^2.7 in the New Jersey continental shelf and as f^2.1 in Georges Bank (Figure 3). At all source frequencies, the magnitudes of seafloor scattering amplitudes in the two continental shelf environments match well with each other (Figure 3), consistent with the similarity in sandy sediments found in these two regions [9,10]. Seafloor scattering is modeled using a Rayleigh–Born approach in terms of the incoming Green function from an acoustic source to a seafloor scattering volume element, the Rayleigh–Born seafloor scattering amplitude and an outgoing Green function from the seafloor scattering volume element to a receiver, integrated over the entire scattering volume in the seafloor (Section 4.3). Multimodal propagation effects appearing in waveguide Green functions [28,39] can be mathematically separated from seafloor scattering effects. In order to account for random fluctuations in the ocean waveguide, the moments of Green functions and their derivatives are calculated by Monte Carlo realizations over range- and depth-dependent sound speed structures [29,30,37].

4.2. OAWRS Detection of Atlantic Cod Aggregations and Individuals

We find that at swim bladder resonance frequencies, robust detection and imaging of cod spawning aggregations with OAWRS is possible (Figures 1 and 2) in continental shelf waveguide environments along the US northeast coast. Our analysis shows that scattered returns from bottom-dwelling spawning cod aggregations, which were observed by echosounders during recent spring surveys in Ipswich Bay [32,33], with population densities of 0.01 fish/m², can be detected roughly 20–27 dB above seafloor scattered returns at swim bladder resonance, roughly 300 Hz in this case, when cod are neutrally buoyant close to the seafloor (Figure 4A,B). This spawning cod aggregation density is roughly 2–3 orders of magnitude larger than the minimum density necessary for OAWRS detection above seafloor scattering for ranges up to and beyond 30 km, as shown in Figure 4B. Similarly robust OAWRS detection is possible for cod spawning aggregations during mid-water column feeding [40,41] up to 30 km and beyond, when probed at a swim bladder resonance frequency of roughly 200 Hz, for cod neutrally buoyant in the mid-water column (Figure 4C,D). Individually dispersed cod located in the mid-water column can also be robustly detected within a few kilometers using OAWRS when probed at the swim bladder resonance frequency (Figure 5). In the extreme case of cod being neutrally buoyant at the sea surface and, so, at their negative buoyancy limit anywhere in the water column [42–45], the swim bladder resonance frequency will increase and the cod scattering amplitude will decrease (Figure 2C), so that the fish-to-seafloor scattering ratio (FSR) (Equation (5)) will decrease by roughly 8 dB for the bottom dwelling case of Figure 4A,B and by roughly 5 dB for the mid-water column case of Figure 4C,D.

The spawning aggregation density then would be roughly 1–2 orders of magnitude higher than the minimum density required for OAWRS detection above seafloor scattering, again enabling robust OAWRS detection. At spawning sites with denser cod aggregations [46], the FSR is expected to be higher, and so, the cod will be robustly detectable over a wider range of frequencies near swim bladder resonance. In the case of cod found in deeper Gulf of Maine waters located and neutrally buoyant close to the seafloor [47], the swim bladder resonance frequency increases and cod scattering amplitude decreases (Figure 2D), so that the fish-to-seafloor scattering ratio (FSR) (Equation (5)) decreases. For example, this decrease is roughly 5–6 dB from that shown in Figure 4A for cod at 300 m of depth with a population density of 0.01 fish/m². This aggregation density then would be roughly 1–2 orders of magnitude higher than the minimum density required for OAWRS detection above seafloor scattering, again enabling robust OAWRS detection. The scattering amplitude of Atlantic cod is modeled as a function of frequency using the Love model [36] for a given body length distribution and an occupancy depth in the water column. Our theoretical predictions of cod swim bladder resonance from the Love model [36] match well with past swim bladder resonance measurements of cod [23] made at similar depths. The scattered field from cod are modeled using this scattering amplitude and the waveguide Green function determined by parabolic equation calculations given known water column and sediment sound speed structures and bathymetry and using the methods described in Appendix C of Jagannathan et al. [37]. Cod densities are assumed to be uniform within the resolution footprint of the OAWRS system. This is easily attained for OAWRS range resolution Δr = c/(2B), which is typically on the order of 15 m and set by the signal bandwidth, B [10]. It is also typically attained in cross-range ΔS = R(λ/L) out to ranges of at least 30 km, where ΔS typically is less than the roughly 1 km extent of spawning cod aggregations (Figure 1) [32,48] for typical 128-element receiver arrays sampled at half-wavelength spacing [9,10]. The seafloor scattering amplitude in the Ipswich Bay environment considered here is expected to be similar to that estimated in the New Jersey continental shelf and Georges Bank environments (Section 4.1, Figure 3), due to similar [49] sediment types.

Spawning cod aggregations in the coastal waters under investigation here have recently been shown to produce sounds in the 50–500 Hz frequency range [50,51]. The fact that these cod sounds are in the same frequency range as our theoretical predictions of swim bladder resonance for spawning cod aggregations is expected, because the cod sound production mechanism involves stimulation of the swim bladder [52,53], which produces the largest response at resonance. These cod sounds would not affect active OAWRS cod sensing, because their levels are well below active OAWRS scattered returns and are also incoherent with respect to OAWRS scattered returns, which reduces their effect as noise even further by the coherent matched filtering used in OAWRS detection and imaging [9,10].

4.3. Theoretical Derivation of Rayleigh–Born Seafloor Scattering Amplitude

The seafloor scattered field at center frequency, f_c, can be written from Equation (A.5) as:

$$\begin{array}{ll}{\mathrm{\Phi }}_S({\mathbf{r}}_S|\mathbf{r},{\mathbf{r}}_0,f_c)=\hfill & (4\pi ){\iiint }_{V_S}[k^2{\mathrm{\Gamma }}_{\kappa }({\mathbf{r}}_t)G({\mathbf{r}}_t|{\mathbf{r}}_0,f_c)G(\mathbf{r}|{\mathbf{r}}_t,f_c)\hfill \\ \hfill & +{\mathrm{\Gamma }}_d({\mathbf{r}}_t)\nabla G({\mathbf{r}}_t|{\mathbf{r}}_0,f_c)\cdot \nabla G(\mathbf{r}|{\mathbf{r}}_t,f_c)]{\mathit{dV}}_t\hfill \end{array}$$

(7)

where Φ_S(r_S|r, r₀, f_c) is the seafloor scattered field for an OAWRS source located at r₀, a scattering seafloor region centered on r_S and an OAWRS receiver located at r; Γ_k is the fractional change in seafloor compressibility; Γ_d is the fractional change in seafloor density; V_S is the volume of the scattering seafloor patch determined by the resolution footprint of the sensing system; k = 2πf_c/c is the acoustic wavenumber; c is the sound speed; G(r_t|r₀, f_c) is the Green function from the source to the elemental patch centered on r_t within V_S; and G(r|r_t, f_c) is the Green function from the patch to the receiver.

Since the mean scattered field from diffuse inhomogeneities in a fluctuating waveguide vanishes [54–57], the second moment of the seafloor scattered field is well approximated by its variance (Equations (A.17) and (A.18)):

$$\begin{array}{l}\text{Var}({\mathrm{\Phi }}_S({\mathbf{r}}_S|\mathbf{r},{\mathbf{r}}_0,f_c))={(4\pi )}^2{\iiint }_{V_S}{V}_c({\mathbf{r}}_S,z_t)k^4\text{Var}({\mathrm{\Gamma }}_{\kappa })\langle {|G({\mathbf{r}}_t|{\mathbf{r}}_0,f_c)|}^2{|G(\mathbf{r}|{\mathbf{r}}_t,f_c)|}^2\rangle {\mathit{dV}}_t,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{(4\pi )}^2{\iiint }_{V_S}{V}_c({\mathbf{r}}_S,z_t)\text{Var}({\mathrm{\Gamma }}_d)\langle {|\nabla G({\mathbf{r}}_t|{\mathbf{r}}_0,f_c)\cdot \nabla G(\mathbf{r}|{\mathbf{r}}_t,f_c)|}^2\rangle {\mathit{dV}}_t,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{(4\pi )}^2{\iiint }_{V_S}{V}_c({\mathbf{r}}_S,z_t)k^2\text{Covar}({\mathrm{\Gamma }}_{\kappa },{\mathrm{\Gamma }}_d)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\langle 2\Re \{G({\mathbf{r}}_t|{\mathbf{r}}_0,f)G(\mathbf{r}|{\mathbf{r}}_t,f_c)\nabla G^{*}({\mathbf{r}}_t|{\mathbf{r}}_0,f_c)\cdot \nabla G^{*}(\mathbf{r}|{\mathbf{r}}_t,f_c)\}\rangle ]\hspace{0.17em}{\mathit{dV}}_t\end{array}$$

(8)

where V_c is the coherence volume of inhomogeneities and ℜ{.} represents the real part. Here: (i) the integral term with waveguide Green functions sums monopole scattering contributions; (ii) the integral term with the gradients of waveguide Green functions sums dipole scattering contributions; and (iii) the integral term with Green functions and their gradients contains cross terms. Since the integrals (ii) and (iii) are found to be proportional to the integral (i) in Section A.2, we define proportionality constants F_d and F_c, such that:

$$\begin{array}{l}\text{Var}({\mathrm{\Phi }}_S({\mathbf{r}}_S|\mathbf{r},{\mathbf{r}}_0,f_c))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\iiint }_{V_S}{k}^4{V}_c[\text{Var}({\mathrm{\Gamma }}_k)+F_d\text{Var}({\mathrm{\Gamma }}_d)+F_c\text{Cov}({\mathrm{\Gamma }}_k,{\mathrm{\Gamma }}_d)]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{(4\pi )}^2\langle {|G({\mathbf{r}}_t|{\mathbf{r}}_0,f_c)|}^2{|G(\mathbf{r}|{\mathbf{r}}_t,f_c)|}^2\rangle {\mathit{dV}}_t\end{array}$$

(9)

The variance of the seafloor scattered field can also be written as:

$$\begin{array}{l}\text{Var}({\mathrm{\Phi }}_S({\mathbf{r}}_S|\mathbf{r},{\mathbf{r}}_0,f_c))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\iiint }_{V_S}\frac{{(4\pi )}^2}{V_c}\langle {\left|\frac{S(f_c,{\mathbf{r}}_t)}{k}\right|}^2\rangle \langle {|G({\mathbf{r}}_t|{\mathbf{r}}_0,f)|}^2{|G(\mathbf{r}|{\mathbf{r}}_t,f)|}^2\rangle {\mathit{dV}}_t\end{array}$$

(10)

where S(f, r_t) is the scatter function of seafloor inhomogeneities contained within a single coherence volume centered on r_t (Section A.3). Then, for relatively constant moments of fractional changes in seafloor density and compressibility across OAWRS spatial scales, we can write Equation (9) as:

$$\begin{array}{l}\text{Var}({\mathrm{\Phi }}_S({\mathbf{r}}_S|\mathbf{r},{\mathbf{r}}_0,f_c))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle {|{\text{A}}_S(f_c,{\mathrm{\Gamma }}_k,{\mathrm{\Gamma }}_d,V_c)|}^2\rangle {\iiint }_{V_S}{(4\pi )}^2\langle {|G({\mathbf{r}}_t|{\mathbf{r}}_0,f_c)|}^2{|G(\mathbf{r}|{\mathbf{r}}_t,f_c)|}^2\rangle {\mathit{dV}}_t\end{array}$$

(11)

where the expected magnitude squared of the seafloor scattering amplitude per coherence volume is defined as:

$$\begin{array}{ll}\langle {|{\text{A}}_S(f_c,{\mathrm{\Gamma }}_k,{\mathrm{\Gamma }}_d,V_c)|}^2\rangle \hfill & =\frac{1}{V_c}\langle {\left|\frac{S(f_c)}{k}\right|}^2\rangle \hfill \\ \hfill & =k^4{V}_c[\text{Var}({\mathrm{\Gamma }}_k)+F_d\text{Var}({\mathrm{\Gamma }}_d)+F_c\text{Cov}({\mathrm{\Gamma }}_k,{\mathrm{\Gamma }}_d)]\hfill \end{array}$$

(12)

which is a function of insonification frequency and seafloor properties, such as changes in seafloor density, compressibility and coherence length scale.

Detailed derivations of the broadband seafloor scattered field and its moments and approximations that enable efficient wide-area estimation of seafloor scattering using the Rayleigh–Born approach are provided in Appendices A.1–A.4.

5. Discussion

A lower frequency OAWRS system [9,10] spanning the cod swim bladder resonance frequencies may be used to continuously detect, monitor and enumerate cod populations and to quantitatively describe their temporal and spatial behavior. Widely separated population centers (Figure 6A) can be identified by the continuous spatial coverage of OAWRS imaging, but may be missed by conventional acoustic and trawl line transect approaches. A typical scenario for wide-area sensing using OAWRS is given in Figure 6, where almost the entire Massachusetts and New Hampshire coastal region, including the Gulf of Maine Cod Spawning Protection Area [34], can be surveyed in 75 s within the corresponding 100-km diameter OAWRS imaging window. The OAWRS coverage area in Figure 6B includes some of the prime spawning locations of Atlantic cod in the Gulf of Maine, where they are known to form dense aggregations [47,58] that should be robustly detectable using OAWRS techniques. The OAWRS approach typically employs a towed receiver array that moves along tracks and, so, produces hundreds of instantaneous overlapping imaging windows [9,10]; some of these are shown in Figure 6B, where the centers of the circles correspond to the location of the tow ship when the image was formed. Since detection ranges in OAWRS typically span many tens to hundreds of water column depths, ocean depth is very shallow compared to the OAWRS sensing range in continental shelf environments [9,10] and enables hundreds to thousands of propagating acoustic modes to fill the entire water column at OAWRS frequencies [24,59–61]. Only at potential nulls within an eighth of the acoustic wavelength from potential pressure release surfaces at the water column’s upper and lower boundaries may fish detection become difficult to effectively impossible, as in any acoustic sensing system. Consistent application of National Oceanic and Atmospheric Administration (NOAA) guidelines on sound in the ocean [62] suggests that many ships with continuous engine noise, such as cruise ships, cargo vessels and whale-watching boats [63–65], should monitor for marine mammals within a few to tens of kilometers of the ship, but that OAWRS, which uses short, infrequent pulses of low duty cycle (roughly 0.01) and whose received sound pressure levels never exceed those of natural marine mammal vocalizations [66–68], should monitor for marine mammals within tens to hundreds of meters from the OAWRS source.

It may be possible to remotely classify cod by OAWRS spectral analysis, since they have such a strong response at their swim bladder resonance, which is not found in seafloor scattering. Ground truth from capture, as well as behavioral analysis to set the context would still likely be necessary, since a number of other species of large fish, such as haddock, hake and pollock [70], may have resonant responses in similar frequency ranges (Figure 7). It should, however, be possible to remotely distinguish cod from smaller groundfish species that have resonance responses at frequencies hundreds of Hertz greater than those of cod, as shown in Figure 7 for plaice and redfish. Cod surveys are typically conducted during the spawning season, when cod aggregations are largest and densest [34,58], making concurrent ground truth verification possible by capture trawl. During off-spawning seasons, however, capture trawl may not be practical for the typical widely dispersed cod populations, so increasing the need for classification by remote techniques, such as OAWRS spectral analysis.

6. Conclusions

We find that Ocean Acoustic Waveguide Remote Sensing (OAWRS) is capable of instantaneously and robustly detecting spawning cod aggregations over wide areas spanning thousands of square kilometers for observed spawning cod densities and configurations in US northeast continental shelf waters at cod swim bladder resonance frequencies. At swim bladder resonance, OAWRS can also be used to detect individual cod within a few kilometers of the OAWRS system. Such robust detections of cod are possible, because the cod scattering amplitude has a strong low frequency resonance peak spanning the roughly two-octave range of 150–600 Hz within water depths of roughly 100 m, in contrast to the relatively uniform seafloor scattering frequency dependence of roughly f^2.4, which we find below roughly 2 kHz in US northeast continental shelf environments. This cod resonance frequency range is also optimal for long-range ocean acoustic waveguide propagation, because it enables multimodal acoustic waveguide propagation with minimal acoustic absorption and forward scattering losses. As the sensing frequency moves away from the resonance peak, OAWRS detection of cod becomes increasingly less optimal, due to a rapid decrease in cod scattering amplitude. In other environments where cod depth may be greater, the optimal frequencies for cod detection are expected to increase with swim bladder resonance frequency.