^{1}, Zheng Gong

^{1}and Purnima Ratilal

^{1,a)}

### Abstract

An ocean acoustic waveguideremote sensing system can instantaneously image and continuously monitor fish populations distributed over continental shelf-scale regions. Here it is shown theoretically that the areal population density of fish groups can be estimated from their incoherently averaged broadband matched filtered scattered intensities measured using a waveguideremote sensing system with less than 10% error. A numerical Monte-Carlo model is developed to determine the statistical moments of the scattered returns from a fish group. It uses the parabolic equation to simulate acoustic field propagation in a random range-dependent oceanwaveguide. The effects of (1) multiple scattering, (2) attenuation due to scattering, and (3) modal dispersion on fish population density imaging are examined. The model is applied to investigate population density imaging of shoaling Atlantic herring during the 2006 Gulf of Maine Experiment. Multiple scattering,attenuation and dispersion are found to be negligible at the imaging frequencies employed and for the herring densities observed. Coherent multiple scattering effects, such as resonance shifts, which can be significant for small highly dense fish groups on the order of the acoustic wavelength, are found to be negligible for the much larger groups typically imaged with a waveguideremote sensing system.

We thank Nicholas C. Makris for providing the initial motivation for this work. This research was funded by the Office of Naval Research, the National Oceanographic Partnership Program and the Alfred P. Sloan Foundation with administrative support from Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems. This research is a contribution to the Census of Marine Life.

I. INTRODUCTION

II. THEORY

A. Numerical Monte-Carlo model for the statistical moments of the broadband matched filtered fully scattered field from a 3D random distribution of scatterers that includes multiple scattering in a random range-dependent oceanwaveguide

B. Modeling the complex scatter function distribution and 3D spatial distribution for a group of fish

C. Model verification

III. MODELING THE 2006 GULF OF MAINE EXPERIMENT

A. Modeling acoustic propagation in the random range-dependent Gulf of Maine environment

B. Scattering properties of Atlantic herring

IV. ILLUSTRATIVE EXAMPLES

A. Coherent and incoherent broadband matched filtered scattered intensities that include multiple scattering from a fish group compared to environmental reverberation

B. Effects of multiple scattering and dependence on fish density and target strength

C. Attenuation from scattering through a fish group and its dependence on swim-bladder resonance damping

D. Effect of modal dispersion

E. Charting speed in the random range-dependent Gulf of Maine environment

F. Fish areal population density estimation

G. Standard deviation of the broadband matched filtered scattered intensities from the fish group

H. Coherent multiple scattering effects such as resonance shift and sub- and super-resonance local maxima

I. Effect of fish group 3D spatial configuration

V. CONCLUSION

### Key Topics

- Multiple scattering
- 111.0
- Acoustic waveguides
- 38.0
- Oceans
- 29.0
- Oceanic scattering
- 28.0
- Remote sensing
- 26.0

## Figures

(a) Geometry of the bistatic acoustic imaging system in an ocean waveguide. (b) 3D spatial configuration of a large herring group containing 7831 individuals uniformly distributed within a volume similar to an ellipsoid that has axes dimensions of *L _{x} * = 100 m,

*L*

_{y}_{1}= 125 m, and

*L*

_{z}_{1}= 33.33 m, but with cross-range and depth extents cut at ± 50 m and ± 10 m, respectively, to the center of the herring group. The plotted volume has dimensions of

*L*= 100 m,

_{x}*L*= 100 m, and

_{y}*L*= 20 m. (c) Modeled broadband two-way transmission loss

_{z}*TTL*calculated using Eq. (6) from a source array to potential fish locations and from the fish locations back to the receiver array center is shown as a function of fish range and depth in the Gulf of Maine environment for a source waveform of 50 Hz bandwidth centered at

_{W}*f*= 950 Hz. (d) The

_{c}*TTL*obtained by first averaging the broadband two way propagated acoustic intensities over a 20 m thick fish layer centered at

_{W}*z*= 150 m depth and then taking the log transform following Eq. (8). The error bars indicate one standard deviation in the broadband

_{s}*TTL*over the depth layer of the fish.

_{W}(a) Geometry of the bistatic acoustic imaging system in an ocean waveguide. (b) 3D spatial configuration of a large herring group containing 7831 individuals uniformly distributed within a volume similar to an ellipsoid that has axes dimensions of *L _{x} * = 100 m,

*L*

_{y}_{1}= 125 m, and

*L*

_{z}_{1}= 33.33 m, but with cross-range and depth extents cut at ± 50 m and ± 10 m, respectively, to the center of the herring group. The plotted volume has dimensions of

*L*= 100 m,

_{x}*L*= 100 m, and

_{y}*L*= 20 m. (c) Modeled broadband two-way transmission loss

_{z}*TTL*calculated using Eq. (6) from a source array to potential fish locations and from the fish locations back to the receiver array center is shown as a function of fish range and depth in the Gulf of Maine environment for a source waveform of 50 Hz bandwidth centered at

_{W}*f*= 950 Hz. (d) The

_{c}*TTL*obtained by first averaging the broadband two way propagated acoustic intensities over a 20 m thick fish layer centered at

_{W}*z*= 150 m depth and then taking the log transform following Eq. (8). The error bars indicate one standard deviation in the broadband

_{s}*TTL*over the depth layer of the fish.

_{W}Effect of varying inter-fish spacing on the *school* target strength spectra for a small fish group containing 13 individuals. The mean inter-fish spacings are (a) one fish body length, *d* = 40 cm, (b) quarter fish body length, *d* = 10 cm, and (c) 4 times the fish body length, *d = *1.6 m. The coherent, incoherent, total and estimated *school* target strength spectra are calculated via Eqs. (2)–(5), respectively.

Effect of varying inter-fish spacing on the *school* target strength spectra for a small fish group containing 13 individuals. The mean inter-fish spacings are (a) one fish body length, *d* = 40 cm, (b) quarter fish body length, *d* = 10 cm, and (c) 4 times the fish body length, *d = *1.6 m. The coherent, incoherent, total and estimated *school* target strength spectra are calculated via Eqs. (2)–(5), respectively.

(a) The real and imaginary parts of the complex scattering amplitude *S(k)/k* for a herring of mean swimbladder volume 3.68 ml. (b) The target strengths of fish with mean swimbladdder volume (black) and of fish with swimbladder volumes both 1 standard deviation larger (light gray) and smaller (dark gray) than the mean. The ensemble-averaged target strength (dash-dotted black), obtained by averaging over the scattering cross-sections of all fish in the group is also plotted. The normalized histograms (by total number of fish N = 10 000) of (c) the fork length and (d) swimbladder volume for the herring group. The swimbladder volume shown in (d) is inferred from (c) and taking into account the depth of each individual fish in the group given a known neutral buoyancy depth at 85 m using Eq. (5) and (B6), where *p* = 3.35 × 10^{−6} and *q* = 3.35 are empirically determined from length and weight measurements of trawl samples (Ref. 1).

(a) The real and imaginary parts of the complex scattering amplitude *S(k)/k* for a herring of mean swimbladder volume 3.68 ml. (b) The target strengths of fish with mean swimbladdder volume (black) and of fish with swimbladder volumes both 1 standard deviation larger (light gray) and smaller (dark gray) than the mean. The ensemble-averaged target strength (dash-dotted black), obtained by averaging over the scattering cross-sections of all fish in the group is also plotted. The normalized histograms (by total number of fish N = 10 000) of (c) the fork length and (d) swimbladder volume for the herring group. The swimbladder volume shown in (d) is inferred from (c) and taking into account the depth of each individual fish in the group given a known neutral buoyancy depth at 85 m using Eq. (5) and (B6), where *p* = 3.35 × 10^{−6} and *q* = 3.35 are empirically determined from length and weight measurements of trawl samples (Ref. 1).

(a) The areal density plotted as a function of range for a fish group in the Gulf of Maine environment with sand bottom. The vertical lines indicate the range where the fish areal density is half its maximum value. (b) The incoherent Var[Ψ_{s}(*t _{M} *)] and coherent broadband matched filtered fully scattered intensities that include multiple scattering from the fish group imaged using the waveform centered at

*f*

_{c}= 950 Hz with 50 Hz bandwidth and 0 dB re 1 μPa at 1 m source level. The fish scattered intensities are compared to the expected background reverberation estimated from GOME06 data. (c) Identical to (b) but plotted in logarithmic scale. The error bar indicates the standard deviation of the broadband matched filtered fully scattered intensities from the fish group.

(a) The areal density plotted as a function of range for a fish group in the Gulf of Maine environment with sand bottom. The vertical lines indicate the range where the fish areal density is half its maximum value. (b) The incoherent Var[Ψ_{s}(*t _{M} *)] and coherent broadband matched filtered fully scattered intensities that include multiple scattering from the fish group imaged using the waveform centered at

*f*

_{c}= 950 Hz with 50 Hz bandwidth and 0 dB re 1 μPa at 1 m source level. The fish scattered intensities are compared to the expected background reverberation estimated from GOME06 data. (c) Identical to (b) but plotted in logarithmic scale. The error bar indicates the standard deviation of the broadband matched filtered fully scattered intensities from the fish group.

Effect of varying the imaging frequency band on the incoherent matched filtered scattered returns from a fish group. (a) The areal density of the fish group. (b) The incoherent fully scattered intensity Var[Ψ_{s}(*t _{M} *)] that includes multiple scattering and singly scattered intensity Var from the fish group are plotted as a function of the imaging frequency band. The expected background reverberant intensity estimated from GOME06 data is also plotted for comparison. The background reverberation varies by roughly 2 to 3 dB from the lowest to the highest frequency band and the average across the 4 bands is plotted here. (c) Identical to (b) but plotted in logarithmic scale. The error bars show the standard deviation of the broadband matched filtered fully scattered intensities from the fish group at various imaging frequency bands.

Effect of varying the imaging frequency band on the incoherent matched filtered scattered returns from a fish group. (a) The areal density of the fish group. (b) The incoherent fully scattered intensity Var[Ψ_{s}(*t _{M} *)] that includes multiple scattering and singly scattered intensity Var from the fish group are plotted as a function of the imaging frequency band. The expected background reverberant intensity estimated from GOME06 data is also plotted for comparison. The background reverberation varies by roughly 2 to 3 dB from the lowest to the highest frequency band and the average across the 4 bands is plotted here. (c) Identical to (b) but plotted in logarithmic scale. The error bars show the standard deviation of the broadband matched filtered fully scattered intensities from the fish group at various imaging frequency bands.

Effect of varying the fish areal density on the broadband incoherent matched filtered scattered returns from a fish group imaged with frequency band centered at *f _{c} * = 950 Hz. (a) The areal fish densities of three distinct fish groups. (b) The incoherent fully Var[Ψ

_{s}(

*t*)] and singly Var scattered intensities from the fish groups with imaging frequency band centered at 950 Hz are compared to background reverberation. (c) Identical to (b) but plotted in logarithmic scale. The error bars show the standard deviation of the broadband matched filtered fully scattered intensities from fish group with various fish areal densities.

_{M}Effect of varying the fish areal density on the broadband incoherent matched filtered scattered returns from a fish group imaged with frequency band centered at *f _{c} * = 950 Hz. (a) The areal fish densities of three distinct fish groups. (b) The incoherent fully Var[Ψ

_{s}(

*t*)] and singly Var scattered intensities from the fish groups with imaging frequency band centered at 950 Hz are compared to background reverberation. (c) Identical to (b) but plotted in logarithmic scale. The error bars show the standard deviation of the broadband matched filtered fully scattered intensities from fish group with various fish areal densities.

_{M}Effect of varying fish swimbladder damping on the incoherent matched filtered scattered returns with imaging frequency band centered at 950 Hz. (a) The areal fish density of the fish group. (b)–(e) The incoherent fully Var[Ψ_{s}(*t _{M} *)] and singly Var scattered intensities from the fish group are plotted as a function of fish damping coefficient, and compared to background reverberation.

Effect of varying fish swimbladder damping on the incoherent matched filtered scattered returns with imaging frequency band centered at 950 Hz. (a) The areal fish density of the fish group. (b)–(e) The incoherent fully Var[Ψ_{s}(*t _{M} *)] and singly Var scattered intensities from the fish group are plotted as a function of fish damping coefficient, and compared to background reverberation.

Identical to Fig. 7 but for imaging frequency band centered at 1125 Hz.

Identical to Fig. 7 but for imaging frequency band centered at 1125 Hz.

The mean total acoustic intensity *forward* propagated through the fish group in Figs. 7 and 8 are plotted as a function of fish swimbladder damping for imaging frequency bands centered at (a) *f _{c} * = 950 Hz and (b)

*f*= 1125 Hz. The broadband incident and forward propagated (which include multiple scattering from other fish in the group) intensities are averaged over the fish layer depth and over the 100 independent Monte-Carlo realizations.

_{c}The mean total acoustic intensity *forward* propagated through the fish group in Figs. 7 and 8 are plotted as a function of fish swimbladder damping for imaging frequency bands centered at (a) *f _{c} * = 950 Hz and (b)

*f*= 1125 Hz. The broadband incident and forward propagated (which include multiple scattering from other fish in the group) intensities are averaged over the fish layer depth and over the 100 independent Monte-Carlo realizations.

_{c}Identical to Fig. 4 but for the silt waveguide.

Identical to Fig. 4 but for the silt waveguide.

(a) Estimating fish areal population densities from their incoherently averaged broadband matched filtered fully scattered intensities using Eq. (10) illustrated for the fish group in Fig. 4 in the sand waveguide as a function of imaging frequency bandwidth *B* with *f _{c} * = 950 Hz. (b) Identical to (a) but for the fish group in the silt waveguide shown in Fig. 10.

(a) Estimating fish areal population densities from their incoherently averaged broadband matched filtered fully scattered intensities using Eq. (10) illustrated for the fish group in Fig. 4 in the sand waveguide as a function of imaging frequency bandwidth *B* with *f _{c} * = 950 Hz. (b) Identical to (a) but for the fish group in the silt waveguide shown in Fig. 10.

Estimating fish areal population densities from their incoherently averaged broadband matched filtered scattered intensities illustrated for a much larger fish group. (a) True and estimated areal fish densities are plotted as a function of range. (b) The incoherent Var and coherent broadband matched filtered singly scattered intensities from the fish group imaged using the waveform centered at *f _{c} * = 950 Hz with 50 Hz bandwidth and 0 dB re 1

*μ*Pa source level are compared to background reverberation. (c) Identical to (b) but plotted in logarithmic scale. The error bar indicates the standard deviation of the broadband matched filtered singly scattered intensities from the fish group. (d) The true and estimated fish populations integrated as a function of range.

Estimating fish areal population densities from their incoherently averaged broadband matched filtered scattered intensities illustrated for a much larger fish group. (a) True and estimated areal fish densities are plotted as a function of range. (b) The incoherent Var and coherent broadband matched filtered singly scattered intensities from the fish group imaged using the waveform centered at *f _{c} * = 950 Hz with 50 Hz bandwidth and 0 dB re 1

*μ*Pa source level are compared to background reverberation. (c) Identical to (b) but plotted in logarithmic scale. The error bar indicates the standard deviation of the broadband matched filtered singly scattered intensities from the fish group. (d) The true and estimated fish populations integrated as a function of range.

Effect of the 3D spatial configuration on the time harmonic fully scattered field moments, including multiple scattering, as a function of frequency for a monostatic direct-path imaging system examined for (a) a large herring group containing 7831 individuals and (b) a small herring group containing 240 individuals. The coherent, incoherent, total and estimated *school* target strength spectra calculated via Eqs. (2)–(5), respectively, are compared as a function of the fish group configuration.

Effect of the 3D spatial configuration on the time harmonic fully scattered field moments, including multiple scattering, as a function of frequency for a monostatic direct-path imaging system examined for (a) a large herring group containing 7831 individuals and (b) a small herring group containing 240 individuals. The coherent, incoherent, total and estimated *school* target strength spectra calculated via Eqs. (2)–(5), respectively, are compared as a function of the fish group configuration.

Effect of the 3D spatial configuration of a fish group on the statistical moments of its broadband matched filtered fully scattered intensities. (a) The areal fish densities in the two configurations. (b) The incoherent Var[Ψ_{s}(*t _{M} *)] and coherent broadband matched filtered fully scattered intensities that include multiple scattering from the fish groups imaged using the waveform centered at

*f*= 950 Hz with 50 Hz bandwidth and 0 dB re 1

_{c}*μ*Pa at 1 m source level. The fish scattered intensities are compared to the expected background reverberant intensities estimated from GOME06 data. (c) Identical to (b) but plotted in logarithmic scale.

Effect of the 3D spatial configuration of a fish group on the statistical moments of its broadband matched filtered fully scattered intensities. (a) The areal fish densities in the two configurations. (b) The incoherent Var[Ψ_{s}(*t _{M} *)] and coherent broadband matched filtered fully scattered intensities that include multiple scattering from the fish groups imaged using the waveform centered at

*f*= 950 Hz with 50 Hz bandwidth and 0 dB re 1

_{c}*μ*Pa at 1 m source level. The fish scattered intensities are compared to the expected background reverberant intensities estimated from GOME06 data. (c) Identical to (b) but plotted in logarithmic scale.

## Tables

Standard deviations, *σ*, of the broadband matched filtered scattered intensities from fish groups and corresponding estimated charting speeds, *c* _{chart}. The standard deviation of the estimated charting speed is roughly 1 to 2 m s^{−1}.

Standard deviations, *σ*, of the broadband matched filtered scattered intensities from fish groups and corresponding estimated charting speeds, *c* _{chart}. The standard deviation of the estimated charting speed is roughly 1 to 2 m s^{−1}.

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