#### 2.1. Theory: Linear Discrete Array Beamformer

Here, we provide analytic expressions for the beam pattern valid for both the uniformly-spaced and nonuniformly-spaced linear discrete hydrophone array, and use these expressions to define the array gain $AG$, beam width $BW$, and maximum grating lobe height $GLH$. In the next section, these parameters will be quantified for the nonuniformly-spaced subarrays and compared with those for the conventional uniformly-spaced individual subapertures of the ONR-FORA.

Let

$P(\mathbf{r},t)=A(\mathbf{r}){\int}_{-\infty}^{\infty}Q(f)exp\left[j2\pi f({\displaystyle \frac{{\widehat{\mathbf{u}}}_{0}\xb7\mathbf{r}}{c}}-t)\right]\mathrm{d}f$ denote the pressure field of an incident broadband signal at spatial location

$\mathbf{r}$ and time

t in the direction of the unit vector

${\widehat{\mathbf{u}}}_{0}$, where

A is the complex amplitude,

c is the medium sound speed, and

$|Q(f){|}^{2}$ is the signal’s normalized power spectral density (

${\int}_{-\infty}^{\infty}|Q(f){|}^{2}\mathrm{d}f=1$). A nonuniformly-spaced linear array with

N elements is aligned horizontally along the

y-axis in the ocean waveguide with elements located at

${\mathbf{r}}_{\mathbf{l}}=(0,{y}_{l},0)$, where

$l=1,2,...,N$ (

Figure 1). Here, we assume the object that, either generated or scattered, the signal is located in the far field of the receiver array. The pressure field received on the

l-th array element can be considered to be a plane wave given by

where

$-\pi /2\le {\theta}_{0}\le \pi /2$ is the azimuth of the unit vector

${\widehat{\mathbf{u}}}_{0}$ measured from the array broadside (

x-axis).

Let

$T(y)$ be the spatial taper function applied across the array elements. The total pressure field at scan angle

$\theta $ after beamforming is

where

$B(\mathrm{sin}\theta -\mathrm{sin}{\theta}_{0},f)$ is the discrete linear array beam pattern given by

For a uniformly-spaced array with

N elements and total length of

L, applying a rectangular window spatial taper function

$T({y}_{l})=1/N$ for

${y}_{l}$ between

$[-L/2,L/2]$, leads to a beam pattern given by

where

$d=\frac{L}{N-1}$ is the inter-element spacing. For a general nonuniformly-spaced array with

N elements nested into

M subapertures (see

Figure 2 as an example, which is a schematic of

$M=3$ subapertures of the ONR-FORA), its beam pattern can be derived using Equation (

4) by linearly combining the beam patterns of each of the

M uniformly-spaced subapertures and is given by

where

$\lfloor \rfloor $ is the floor operator,

${L}_{k}$ and

${d}_{k}$ are the total length and inter-element spacing of the

k-th subaperture, respectively. A Hamming window

$T({y}_{l})=0.54+0.46cos2\pi {y}_{l}/L$ is often employed instead of a rectangular window as a spatial taper function to minimize side lobes. The resulting beam pattern can be obtained by convolving the Fourier transform of the taper function with the beam pattern from rectangular windowing,

The instantaneous beamformed intensity is

and the time-averaged beamformed intensity

${\sigma}_{B}(\theta )$ is calculated from the integral of the instantaneous beamformed intensity

${I}_{B}(\theta ,t)$ over time

t between

$[-T/2,T/2]$. For sufficiently large integration time

$T\gg 1$, incident plane waves at different frequencies are approximately orthogonal to each other, such that

${\displaystyle \frac{1}{T}}{\int}_{-T/2}^{T/2}{e}^{-j2\pi ({f}_{i}-{f}_{j})t}\mathrm{d}t\approx \delta ({f}_{i}-{f}_{j})$. The time-averaged beamformed intensity simplifies to

For a narrow-band signal, the beamformer output

${P}_{B}(\theta ,t)$ and time-averaged beamformed intensity

${\sigma}_{B}(\theta )$ can be obtained directly from the beam pattern

$B(\mathrm{sin}\theta -\mathrm{sin}{\theta}_{0},f)$ by multiplying with the complex amplitude

A and time dependence

$exp(-j2\pi ft)$. For a broadband signal, these expressions cannot be simplified further unless the normalized power spectral density

$|Q(f){|}^{2}$ is specified. In practice, beamforming is often numerically implemented in the frequency domain by operation on each frequency component following Equation (

3), where the components are obtained via discrete Fourier transform of the broadband signal.

The broadside beam width

$BW$ is estimated as the angle between the half-power (−3 dB) points of the main lobe in the time-averaged beamformed intensity

${\sigma}_{B}(\theta )$ when the array is steered to the broadside direction. The array gain [

19] is calculated as the ratio of the time-averaged beamformed intensity in the direction of the signal

$\theta ={\theta}_{0}$ to that averaged over all directions,

The grating lobe height is calculated as the maximum value of the time-averaged beamformed intensity excluding the main lobe. The maximum possible grating lobe height $GLH$, which occurs when the array is steered in the end-fire direction, is quantified here.

#### 2.2. Experimental Acoustic Data Collection with a Coherent Discrete Hydrophone Array

The Gulf of Maine 2006 Experiment [

1,

2,

7,

8,

11,

12,

13,

24,

50] was conducted from 19 September to 6 October 2006, coinciding with the annual herring spawning period on the northern flank of Georges Bank [

51,

52,

53]. During the experiment, acoustic recordings were acquired using the nonuniformly-spaced 160 hydrophone-element ONR-FORA towed horizontally behind a research vessel along designated tracks north of Georges Bank [

2,

12,

13,

50]. Tukey-windowed linear frequency modulated (LFM) pulses of 1 s duration and 50 Hz bandwidth centered at 415, 735, 950 and 1125 Hz were transmitted by a vertical source array deployed on another research vessel for bistatic measurement of echo returns. The Atlantic herring areal population densities were monitored over instantaneous wide areas using active OAWRS imaging [

1,

2,

8,

12,

13] and calibrated with coincident conventional ultrasonic fisheries echo sounding measurements [

2,

8,

13] with fish species identification and physiological parameters extracted from trawl samples collected over the course of the experiment [

52,

54]. In addition to echo returns of the transmitted source signal, vocalizations from more than eight distinct marine mammal species including fin, humpback, sei, minke, orca, pilot, sperm, and other unidentified baleen and toothed whale species were also passively recorded by the coherent hydrophone array [

1,

4,

5,

12]. The ecosystem-wide spatial distributions of vocal marine mammals from multiple cetacean species were simultaneously mapped by the POAWRS technique [

1,

4,

5,

6]. Detailed information about the Gulf of Maine 2006 Experiment is provided in Refs. [

1,

2,

7,

11,

12,

13,

24,

50,

55], including hydrophone array layout, measurement geometry, temporal and spatial span of acoustic data collection, and oceanographic properties of the environment. Detailed aperture nesting schematic for the ONR-FORA is provided in Ref. [

26] (see Figure 1 and 2 of Ref. [

26]).

Both OAWRS and POAWRS techniques employ beamforming extensively for high-resolution signal bearing estimation. Previously, data acquired by the ONR-FORA was beamformed using conventional uniformly-spaced single subapertures [

1,

2,

7,

8,

11,

12,

13,

24,

50] in both passive and active OAWRS. Here, we demonstrate that the beamformer output can be significantly enhanced by combining the conventional uniformly-spaced subapertures to form nonuniformly-spaced subarrays.

The ONR-FORA contains 160 hydrophone elements nested into four uniformly-spaced subapertures that are the ultra-low-frequency (ULF), low-frequency (LF), middle-frequency (MF), and high-frequency (HF), each of which contains

N = 64 hydrophones with inter-element spacing of 3 m, 1.5 m, 0.75 m, and 0.375 m, respectively, designed to analyze and conventionally beamform acoustic signals with fundamental frequency content below 250 Hz, 500 Hz, 1000 Hz, and 2000 Hz, respectively [

1,

2,

12,

13]. A schematic of three of the four subapertures of the ONR-FORA is provided in

Figure 2, and they are the LF, MF and HF subapertures. These three subapertures are the primary subapertures used in both active and passive OAWRS. By combining multiple conventional uniformly-spaced subapertures, we obtain new nonuniformly-spaced subarrays (see

Table 1).

The pressure-time series measured by hydrophone elements from the newly defined subarrays are beamformed to two-dimensional beam-time series, from which scattering strength images [

2,

3,

13] of the ocean environment are derived following the approaches in active OAWRS imaging [

2,

3,

8,

13,

14]. The beam-time series data is matched filtered with the source transmitted 50 Hz bandwidth LFM waveforms and is then bistatically charted onto geographic space. The beam-time data are linearly converted to beam-range data by multiplying the two-way travel time from the source and receiver locations to the the scattered field location with half the mean measured water-column sound speed

c. The log-transformed scattered intensity images are next corrected for (i) source level; (ii) two-way transmission losses from the source and receiver locations to the scattered intensity locations; and (iii) the range- and azimuth-dependent resolution footprint of the imaging system. The range resolution is determined by the match filter pulse compression, while the cross-range resolution is determined by the array beam width at the output of the beamformer. The two-dimensional beam-time series are also utilized to generate beamformed spectrograms used extensively in POAWRS [

1,

4,

5,

6] for detecting oceanic sound sources. The results generated using the nonuniformly-spaced subarrays are compared to those obtained from the conventional uniformly-spaced single subapertures of the ONR-FORA.