Vector acoustic and polarization properties of underwater ship noise

Salient properties of the vector acoustic field associated with underwater noise emissions from two cargo ships are described. The data originate from the 2017 Seabed Characterization Experiment (SBCEX17) conducted from March–April 2017 over an approximately 10 × 30 km seabed area known as the New England mud patch, located about 95 km south of Cape Cod.¹ The primary goal of the experiment is the understanding of sound propagation in fine-grained, muddy sediments.^2–5 

Observations discussed here are in the form of acoustic pressure and acoustic velocity in horizontal and vertical directions, forming, respectively, potential and kinetic energy densities. Pairs of the velocity signal components are taken to constitute a bivariate signal, from which the non-dimensional indicators, known as circularity and degree of polarization, are formed. A primary contribution of this study is the reconciliation of these indicators with the Stokes framework as applied to ocean acoustics,⁶ demonstrating its usefulness to describe experimental ship noise data.

The observations were made with the Intensity Vector Autonomous Recorder (IVAR), a system that records four coherent channels of acoustic data continuously: one channel for acoustic pressure and three channels associated with a neutrally buoyant tri-axial accelerometer from which acoustic velocity is obtained. The system is described by Dahl and Dall'Osto⁷ in relation to measurements from broadband SUS (Signal, Underwater Sound) explosive charges made during SBCEX17.

Two ships were observed (on different days) over the course of a closest point of approach (CPA) with the IVAR system, and Automatic Identification System (AIS) data are available for both merchant vessels. These direct, identifiable observations of underwater ship noise are of higher energy than from distant shipping, or traffic noise, for which the exact ship source is unidentified, with traffic noise representing a widespread anthropogenic component of underwater ambient noise.^8–11 Further, in our observations, the ratio of ship range to water depth at CPA is of order 10. Therefore, we do not address sound propagation conditions in more vertical orientations (e.g., near field).

The paper is arranged as follows: in Sec. II, the data, measurement period, location, and conditions are described. Section III presents an overview of the observations in time and frequency; first in terms of kinetic and potential energy, followed by circularity showing interference patterns. Section IV provides a closer look at circularity and the degree of polarization, and how these quantities relate to Stokes parameters, including a detailed examination of the degree of polarization as it relates to ship bearing estimation. A summary is provided in Sec. V.

IVAR was deployed from the R/V Endeavor during SBCEX17 for two periods: 7 March 16:45 to 9 March 14:15 UTC at 40.4716° N, 70.6061° W, and 17 March 23:30 to 18 March 20:30 UTC at 49.48655° N, $70.63831°$ W. Underwater noise originating from the transit of M/V Alice Oldendorf [International Maritime Organization (IMO) number 9183776, length 190 m], was recorded on 8 March. The analysis is restricted to data ±10 min about CPA (Fig. 1) to exploit the higher signal level as the vessel was transiting approximately east to west at speed $≃$ 8 knots. The CPA range, 2470 m, and speed are based on the ship's AIS record. At CPA time, 21:23 UTC, the water depth at the IVAR sensor location was 75.5 m, as measured by IVAR. These data are the subject of the geoacoustic inversion study presented by Dahl and Dall'Osto.³ Similarly, data are restricted to ±10 min about the CPA range, 420 m, for the transit of M/V Oregon Highway (IMO number 9381655, length 200 m), recorded by IVAR on 17 March moving approximately west to east at speed ∼14 knots; at CPA time, 04:50 UTC, the water depth at the IVAR sensor location was 74.6 m.

Note that a complete accounting of the identified large merchant ship traffic in the SBCEX17 experimental area, over a much longer period that includes the IVAR deployments, has recently been reported by Escobar–Amado et al.,¹² where they apply such data towards seabed classification based on deep-learning algorithms.

The IVAR system measures acoustic pressure p and related acoustic acceleration $a→=−∇p/ρ$ with sampling frequency (in 2017) f_s = 25 kHz at a single point 1.25 m above the seabed, where $a→$ can be converted to acoustic velocity $v→$ in either the frequency or time domain. The two horizontal axes of IVAR are arbitrary, determined upon final instrument landing on the seabed. During the first deployment and observations of M/V Alice Oldendorf, the $x‐$axis was aligned with true bearing 198.5° and $y‐$axis with 288.5°, and during the second deployment and observations of M/V Oregon Highway, the $x‐$axis was aligned with true bearing 216.3° and $y‐$axis with 306.3°. For subsequent manipulation of vector acoustic data, we assume a water density $ρ=1027.6$ kg/m³, and a depth-average water sound speed c_w, given that the water column was well mixed during SBCEX17. Sound speed measurements made during the experiment put this average at c_w = 1470 m/s and 1468.3 m/s for the 8 and 17 March observations, respectively.

The IVAR four-channel data stream is processed in the frequency domain as outlined by Dahl and Dall'Osto.³ The data are first decimated by a factor of 5, setting sampling frequency to 5 kHz, after which a Fourier transform of 1 s (Hanning window) is applied to pressure and three acceleration channels. The 1 s window is advanced to cover the 20 min period of interest with 50% overlap. One-sided complex spectra are retained, each normalized to the variance of the corresponding time-domain data. This operation defines $Gp(f;t)$ as the complex spectrum for pressure and $Gvj(f;t)$ as the complex spectrum for the jth component of acoustic velocity, where t is time (i.e., to index each data segment), f is frequency, and j represents one of the x, y, z components of velocity. Note that $Gvj(f;t)$ originates from the corresponding acceleration channel and includes division by $−i2πf$⁠. For subsequent time-averaging of these quantities, a 10 s sliding window average is taken.

For purposes of comparison with other results, a time domain approach is used to yield potential and kinetic energies integrated over decidecade (third octave) frequency bands. A preliminary step involves obtaining time series of real-valued acoustic pressure p(t) and 3-axis acoustic velocity $vx,y,z(t)$ within a given decidecade¹³ frequency band, where the velocities are obtained through time integration of the corresponding filtered acceleration time series. The band integrated potential energy density E_p in J/m³ is equal to $0.5⟨p2(t)⟩/(ρc2)$⁠, with kinetic energy E_k counterpart $0.5ρ∑j=13⟨vj2(t)⟩$⁠, where $⟨ · ⟩$ is the time averaging operator.

An assumption in the estimate of kinetic energy is that there is no background fluid motion, or mean flow, U₀, which can impart an error proportional to the Mach number, $|U0|/cw$⁠, of the mean flow¹⁴ (this relation is modified slightly to apply to energy as distinct from intensity). Although bottom currents were not measured directly at the IVAR locations, video images of marine snow taken from the IVAR platform bound $|U0|$ to $≲$ 1 m/s. This limits the Mach number to $O(10−3)$ which translates to the same relative error on an estimate of E_k, so effects of mean flow can be ignored.

An informative summary of the energetics of underwater ship noise is made upon taking the kinetic and potential energy spectral densities, averaged over the period ±60 s about each CPA (T = 120 s), and further summed within decidecimal bands (Fig. 2). Each estimate of E_p expressed in dB re J/m³ can be mapped to sound pressure level (SPL) in dB re μ Pa² upon adding 216.5 dB to account for the MKS (meter, kilogram, second) units used in our study and conversion based on averaged sound speed and density. The estimated SPL generated by the two ship sources during SBCEX17 can be compared to SPL values over the same frequency range for distant shipping based on long-term time averages reported by Ainslie et al.¹⁰ (see their Fig. 5). The basic spectral shapes are similar, but because the SBCEX17 data were obtained at closer range and involve shorter time averages, the SPL estimates exceed the long-term average counterparts by approximately 20 dB.

Of interest is the approximate concurrence displayed in the time-averaged kinetic and potential energy levels, for which the frequency span of Fig. 2 encompasses greater than 95% of the total energy. Statistical uncertainty for potential energy within each band is approximately¹⁵ $20 log10(1+1/TB)$⁠, where T and B are, respectively, the integration period and bandwidth. For example, uncertainty for the first band studied, centered at 25 Hz, is $≃$ 0.3 dB, given the 120 s integration period and bandwidth $∼6$ Hz; this decreases with increasing frequency and by center frequency 200 Hz, uncertainty is reduced to $≃$ 0.1 dB. Additionally, the three-component variance contribution to kinetic energy is expected to reduce this uncertainty somewhat relative to the single-component variance contribution to potential energy,¹⁶ though we have not further quantified this reduction as it applies to the case of a progressive underwater sound field. Regardless, any closer agreement is, as a practical matter, untenable given the calibration requirements for the four channels involved (one pressure and three for acceleration), with nominal calibration uncertainty of ±1.5 dB applicable to each energy measure.

The potential and kinetic energy densities (Fig. 2) provide respective overviews of the dynamic and kinematic properties of underwater ship noise measured at relatively close range during SBCEX17. Greater detail is available through study of the magnitude and phase relationship among separate components of the acoustic velocity field $v→$⁠. Circularity $Θ→$ is a non-dimensional vector quantity that characterizes these properties; when constructed from velocities directly, it is defined by the vector product

where $0≤|Θ→|≤1$⁠. In Eq. (1)$v→$ represents the complex counterpart to $v→$ in the time domain, e.g., via Hilbert transform, with $v→⋆$ representing the conjugate. The term circularity is used here because it has shared currency and interpretative power in underwater acoustics^17–19 and seismology;²⁰ Eq. (1) is also referred to as the polarization vector in acoustics.^21,22

Consider two components of $v→$ forming a bivariate signal, say v_x and v_y. If they were in phase, the motion (trajectory) in the plane of the component axes is linear along a line and the corresponding component of circularity, Θ_z, equals 0. Similarly, if v_x, v_y were offset by a constant phase, the trajectory is elliptical with $0<|Θz|≤1$⁠, with the ratio²² of minor to major axis e given by $|Θz|=2e/(e2+1)$⁠. In the case of an exact $π/2$ phase offset, together with $|vx|=|vy|$⁠, the trajectory is purely circular and $|Θz|=1$⁠. Finally, the sign of circularity identifies the circulation direction. In an oceanic waveguide, reflections from boundaries are a common cause of elliptical motion. For example, a change in sign of circularity created by a reflection from the air–water interface is visualized in Fig. 1 of Dall'Osto and Dahl.¹⁹ The change in sign of circularity is revisited in Sec. IV.

In the following, circularity is presented under the assumption of cylindrical symmetry in the field about a vertical z axis, as is commonly applicable in underwater acoustics, where the acoustic field is described in a range (r) and depth (z) plane. Circularity is thus a scalar, and we present observations of $Θ(f;t)$⁠. Furthermore, it can be shown^17,21,23 that $Θ→$ is equivalent to a normalized measure of the curl of active intensity, which provides a convenient pathway to estimate $Θ(f;t)$⁠.

Estimates of $Θ(f;t)$ are computed by finding active I_j and reactive Q_j intensity components

and

Next, put

and

The two horizontal components (⁠$j=$ 1,2) are combined to estimate radial (or r-component) active and reactive intensity magnitude as follows: $Ir(f;t)=I12+I22$⁠, and $Qr(f;t)=Q12+Q22$⁠, and vertical components I_z, Q_z are associated with j = 3.

The estimate of circularity $Θ(f;t)$ is thus given by

Equation (6) is a bounded indicator of the vector acoustic field (between ±1), responsive to the interference structure of the acoustic field.^3,17,23 The scalar form generated using I_r and Q_r allows convenient comparison with models for an underwater waveguide based on cylindrical symmetry that support only horizontal and vertical components of acoustic velocity.³ There is a small approximation in effect, insofar as I_r and Q_r can only be positive. However, instances where they can be negative occur in certain waveguide interference features that are spatially rare and have an extremely low signal level.²⁴ Such events are likely only observed in controlled settings in conjunction with controlled sources with more general observability obscured by ambient noise, and thus, this minor drawback is compensated by computational convenience with little effect on results.

A general feature of $Θ(f;t)$ is that the magnitude increases and approaches 1 corresponding to regions of destructive interference of the acoustic field, responding to the phase difference between horizontal and vertical acoustic velocity components approaching $±π/2$⁠. The magnitude of $Θ(f;t)$ is less, typically <0.25, corresponding to regions of constructive interference of the acoustic field. This is only a basic framework, within which the precise pattern of $Θ(f;t)$ corresponding to multi-modal waveguide propagation takes many forms.

This pattern, however, does share some properties with a spectrogram of pressure from a moving underwater broadband source (Figs. 3 and 4). For example, both consist of a set of hyperbolas with vertices occurring at CPA, but with $Θ(f;t)$⁠, the normalization effectively eliminates changes in received level as the vessel closes and opens about CPA, which benefits forward modeling of this quantity. Furthermore, $Θ(f;t)$ displays the structure of modal interference more clearly, which, we believe, provides more information (e.g., for environmental inversion operations³) than the pressure-based spectrogram.

The patterns are different for the slower moving M/V Alice Oldendorf [Fig. 3(a)] that closes and opens in range from IVAR from $≃$3.5 km to the CPA of $≃$2.5 km in the span of $≃$20 min, versus the faster moving M/V Oregon Highway [Fig.4(a)] that closes and opens in range from $≃$ 5 km to CPA of $≃$0.5 km also in 20 min. This large difference in range variation over the same observation period is the primary reason, while the underwater waveguide characteristics are similar.

Furthermore, experimental activities during SBCEX17 were less on 8 March, so the observations (Fig. 3) are relatively free of other broadband ship and active narrowband experimental sources, lending themselves to interpretative modeling and inversion.³ On 17 March (Fig. 4), however, the R/V Neil Armstrong continues to close in range with the IVAR location [best seen in the $Θ(f;t)$ data], reaching a comparable or shorter range from IVAR than the M/V Oregon Highway (although different bearing) over the final $∼200$ s of the observations. Another vessel, R/V Endeavor, is initially closer to IVAR, then at comparable range over the first 100 s and its characteristic emission in the 260–300 Hz band is evident.

A general framework to study the polarization of bivariate signals relies on the Stokes parameters,²⁵ a set of four real scalars that fully describe the polarization of the signal, which are widely used in optics.^26–28 Our focus is limited to bivariate velocity signals, one set being the two horizontal components, v_x and v_y, and another set upon resolving these into a single range component v_r paired with the vertical component v_z, for the bivariate signal in the range–depth plane. The $n−$ variate case or three-component case in particular, is also defined in the literature.^29–31 The Stokes formalism has been applied by Bonnel et al.⁶ to describe the polarization of the velocity pair $[vr(t),vz(t)]$⁠; it can be extended to any velocity pair {e.g., $[vx(t),vy(t)]$} and thus has a direct connection to this study.

Consider any two real-valued velocity components v_i and v_j, with $i≠j$⁠. The Stokes parameters are defined in the time–frequency domain using the previously defined spectral quantities as^6,32

Normalized Stokes parameters, with values between –1 and 1, can further be obtained as $s1=S1/S0, s2=S2/S0$ and $s3=S3/S0$⁠.

Here, we also need to estimate the Stokes parameters using time averaged values of narrowband, or bandpass, real-valued velocity components, for which a complex (analytic) signal is made upon forming a Hilbert transform pair. For example, take the complex counterparts of real-valued velocity components $vi,j$ as equal to $vi,j$⁠, then, for example, Eq. (7) is implemented as

and similar equations hold for S1, S2, and S3.

Alternative descriptors of the bivariate signal are generated via the combination of Stokes parameters⁶ and are listed for completeness here

Of primary interest in this study are: (1) the relationship between the Stokes parameters and circularity, and (2) the degree of polarization $Φ$ representing a statistical measure of the dispersion of the generally elliptical bivariate velocity trajectories over some observation period. Regarding circularity, the scalar definition is made clear if the considered complex bivariate signal is $[vr(t),vz(t)]$ and applied to Eq. (1). In this case, as previously demonstrated by Bonnel et al.,⁶ the frequency-domain version of Eq. (6) leads directly to the fact that the normalized Stokes parameter s₃ equals the scalar $Θ(f;t)$⁠. This does require the projection of the two horizontal velocity components into one source–receiver direction,⁷ whereas computing $Θ(f;t)$ using I_r and Q_r [Eq. (6)] yields the desired result without this projection.

The degree of polarization $Φ$ is a bounded indicator taking values in $[0,1]$⁠. In seismic studies, it is used, for example, to extract more stable signals independent of their amplitude³³ or in signal enhancement through noise suppression,³⁴ where signal stability might be defined using a threshold, e.g., $Φ>0.8$⁠.³³ Here, we hypothesize that $Φ$⁠, when applied to the study of a horizontal velocity bivariate relation between $vx(t)$ and $vy(t)$⁠, can be used to assess the quality of the bearing estimate of the ship source, with a low $Φ$ implying a high uncertainty. The hypothesis is evaluated subsequently in Sec. IV C, but as a general interpretation, the stability or consistency of the trajectory of the bivariate signal projected onto the plane of the two corresponding axes due to noise from other interfering sound sources and/or environmental fluctuations will be manifested in $Φ$⁠.

The parameters $χ,θ$⁠, and κ describe the shape of the polarization ellipse, for example, as shown in Fig. 2(a) in Bonnel et al.⁶ Note that a useful alternative interpretation of these parameters emerges by way of a 2 × 2, symmetric covariance matrix of real-valued components, with elements $⟨vivj⟩$⁠, i = 1, 2. In this case, $tan |χ|$ equals $λ2/λ1$ where λ₁ is the larger of the two eigenvalues, and $κ cos χ$ equals $2λ1$⁠. (The $2$ is required owing to the real-valued matrix, given κ is derived from their complex counterparts.) Finally, the angle of the largest eigenvector of this (again, real-valued) matrix with respect to horizontal, equals θ.

Two frequency cuts of estimates of $Θ(f;t)$ corresponding to M/V Alice Oldendorf observations [Fig. 3(a)], centered at f = 63 and f = 64 Hz, are compared with corresponding model solutions for $Θ(f;t)$ (Fig. 5). Details of the model are described in Dahl and Dall'Osto.³ Briefly, the model produces a replica of the data based on the sum of normal modes yielding complex pressure and velocity fields as functions of range and frequency. Necessary seabed parameters were estimated through inversion of the observations of $Θ(f;t)$⁠, i.e., as shown in Fig. 3(a), but with inversion limited to frequencies less than 140 Hz. The modeled complex fields are then used to construct the intensity variables in Eq. (6). Each observation of $Θ(f;t)$ at time t is mapped to a specific range, an estimate for which was determined through inversion but also verified with AIS-determined range. The model replica for $Θ(f;t)$ is reproduced over the frequency range 20–140 Hz, and the two model solutions in Fig. 5 are extracted from that replica.

It is apparent that at these frequencies, $Θ(f;t)$ passes through about eight local extrema reaching close to 1 in absolute value during which the vessel closes and opens in range to IVAR, from $≃$ 3.5 km to the CPA of $≃$ 2.4 km. The times of these extrema correspond to destructive modal interference that are observable, though not as clearly, with a time-frequency display of kinetic or potential energy density within a narrow frequency band (⁠$≃1$ Hz) corresponding to either 63 Hz or 64 Hz.^3,35 Although beyond the scope of this study, the interesting shift between 63 Hz and 64 Hz in both model solutions and observations of $Θ(f;t)$ is related to the slope of the striations in Fig. 5, and thus, the waveguide invariant, a quantity that has been used to describe the spectrogram of broadband ship noise.³⁶ 

Observations defined as $Θ(f;t)≃0, Θ(f;t)≃1$⁠, and $Θ(f;t)≃−1$ correspond to the time of CPA, with range to IVAR $≃$ 2470 m, and two more times post-CPA, both within about 150 m of CPA range, respectively (Fig. 5, square symbols). We anticipate that the points with $|Θ(f;t)|≃1$ correspond to elliptical trajectories for a bivariate signal composed of v_r, v_z, with changing sign predicting opposite rotation. Similarly, the $Θ(f;t)≃0$ case would predict a linear trajectory or approximately so. The pressure field (potential energy) is expected to reach a local minima at the two ranges corresponding to $|Θ(f;t)|≃1$⁠, between which there has been a sign change in $Θ(f;t)$⁠.¹⁷ It will be shown that kinetic energy follows a similar pattern, also reaching local minima when $|Θ(f;t)|≃1$⁠.

These properties in $Θ(f;t)$ and energy level are manifest in 3D representations of the 5 s snapshots of the bivariate signals $[vr(t),vz(t)]$⁠, as obtained after narrowband filtering (bandwidth $≃1$ Hz) centered at 63 Hz (see Fig. 6). Note that to obtain these, it is first necessary to project the two horizontal components $vx,y(t)$ into the single radial component $vr(t)$⁠, based on the known IVAR $x‐$axis and $y‐$axis fixed alignment combined with the slowly varying bearing of M/V Alice Oldendorf. Although the latter can be estimated directly from the horizontal active intensity vector field,³⁵ here the AIS bearing data are used for convenience.

A sequence of data plots corresponding to the specific times where $Θ(f;t)$ is represented by $∼0$⁠, $∼1$⁠, and $∼−1$⁠, gives an unusually detailed look at the kinematic properties of the underwater noise field originating from a ship source of opportunity within a narrow frequency band (Fig. 6). The corresponding estimates of Θ, $Φ$⁠, and the kinetic (KE) and potential (PE) energy density expressed in decibels re 1 J/m³ over the 5 s duration are listed above each plot. Note that Θ and $Φ$ are estimated via the Stokes parameters using the complex representation for the two velocities (Sec. IV A). These estimates of Θ necessarily differ in final precision from the corresponding estimates shown in Fig. (5), simply owing to different averaging and processing methods.

The horizontal (red) and vertical (blue) acoustic velocities, as well as the corresponding bivariate signal (black) signal plots are limited to a 0.1 s time window (rather than the full 5 s), which shows the expected $≃$ 6 cycles (Fig. 6). Also represented is a complete record of the trajectory for the full (5 s) bivariate signal, projected onto the r – z plane. This record is in the form of a 2D histogram. It shows the occurrence of the bivariate signal location in the r – z plane, with color from blue (low occurrence) to yellow (high occurrence).

The range of each 2D histogram identifies the overall “footprint” of the trajectory record (Fig. 6). For the two cases with $|Θ(f;t)|≃1$ [Figs. 6(b) and 6(c)], the larger magnitude of circularity is predictive of elliptical trajectories with the ensemble of trajectories filling an elliptical area. For the case $Θ(f;t)≃0$ [Fig. 6(a)], the smaller magnitude of circularity is predictive of the quasi-rectilinear trajectories as confirmed by the footprint of the ensemble of trajectories approximating a line. Plotted over each footprint is a single trajectory (white line) representing that of a deterministic, stationary bivariate time series with the same θ, χ, and κ as that from the 5 s of data. Such a bivariate signal is given by^6,25

with, in the specific case under study, f = 63 Hz. The estimates of $Φ$ and the single trajectory defined by Eqs. (16) and (17) are related as follows. A bivariate signal with $Φ≃1$ should have a complete trajectory histogram very narrowly defined around the ellipse [Eqs. (16) and (17)], while a signal with lower $Φ$ will show a more dispersed trajectory histogram.

For the two cases of $Θ(f;t)≃1$ and $≃−1$ [Figs. 6(b) and 6(c)], respectively, a sense of the direction of the elliptical motion including its change is inferred by the “corkscrew” property describing the time-evolution of the rotating 2D vector (black line). The energy levels for these cases are also reduced by approximately 6 dB relative to those shown in Fig. 6(a), although there has been only a $≃$ 6% change in slant range between the vessel and IVAR location over the $≃$ 200 s period encompassing these observations. The lower energy levels, combined with $|Θ(f;t)|$ is $≃1$⁠, indicate the observations are from an interference region, with attendent drift in an ensemble of trajectories consistent with depressed values of $Φ$⁠. Predicting this energy reduction is beyond the scope of the current study and of less importance in view of the narrowband property of these data. Notably, however, estimates of $Θ(f;t)$ (Fig. 5) based on time-averaged estimates of the requisite second-order quantities tend to be much more stable and suggestive of higher information content. Linking such data to the companion estimate $Φ$⁠, however, introduces stochastic features not embodied by circularity.

Observations of $Θ(f;t)$ for frequencies greater than about 250 Hz [Fig. 4(a)] appear somewhat less related to the time evolution of striations associated with the M/V Oregon Highway transit. The approximately constant feature at frequency $≃$ 263 Hz has some relation to tonal emissions from M/V Oregon Highway, but is within the characteristic band of R/V Endeavor. The influence of R/V Neil Armstrong also enters during the last $≃$ 250 s of the observations, primarily for frequencies above 125 Hz. In contrast, a decidecimal band centered at 63 Hz (nominally 56–71 Hz) looks to be relatively free of other sources over the 20 min observation period. With these observations in mind, it is of interest to examine horizontal velocity components $vx,y$ and the influence of multiple sources on $Φ$ within two decidecimal bands: one centered at 63 Hz and one centered at 250 Hz (nominally 223–281 Hz), with the upper end of the 250 Hz band entering into the noise emission band of R/V Endeavor.

For this analysis, our interest is in $Φ$ tied with an estimate of ship bearing. The real-valued time series for the x and y components of acoustic velocity and pressure from the M/V Oregon Highway observations within the filtered 63 Hz and 250 Hz decidecimal bands are parsed into increments of 5 s, spanning the 20 min observation period. The estimate of bearing is based on the active intensity⁷ computed from the 5 s time-average $Ix,y=⟨p(t)vx,y(t)⟩$ (Umov vector) with bearing constructed from the (four quadrant) inverse tangent relation between active intensity components $Ix,y$⁠.²³ The estimate of $Φ$ is derived upon converting $vx,y$ via Hilbert transform to complex counterparts, and then upon 5 s averaging Eq. (15) is applied. Note that the bearing span for this study (⁠$<160°$⁠) fits within the $180°$ range afforded by the parameter θ as defined by Eq. (13); thus, based on complex values of $vx,y$ alone without pressure data a continuous bearing estimate results, which is the same as that using $Ix,y$⁠. This illustrates the difference and tradeoff for bearing estimation using active intensity (⁠$Ix,y$⁠) compared with the Stokes framework (parameter θ): the first requires acoustic pressure and velocity to provide an absolute bearing, while the second relies exclusively on velocity but has an intrinsic $180°$ ambiguity.

Examining first the 63 Hz center frequency case, estimates of $Φ$ made over the 5 s averaging duration remain $≳0.8$ over the entire observation period suggesting a steady, single arrival angle (Fig. 7) that follows the expected sigmoidal pattern and is in agreement with the AIS data. With the 250 Hz case, however, estimates of $Φ$ reveal the influence of multiple sources. During the initial $≃$ 250 s of the observations (⁠$t<−400$ s), the range of M/V Oregon Highway from IVAR exceeds that of R/V Endeavor by a few hundred meters. Interestingly, $Φ$ has a value of $≃0.6$⁠, suggestive of greater stability in trajectory, and the corresponding bearing appears a steady $≃$ 210°. This is exactly the bearing of R/V Endeavor with respect to IVAR as known from AIS data. After this period, i.e., $−400<t<−200$ s, the M/V Oregon Highway closes in on the IVAR location whereas R/V Endeavor does not. We hypothesize that the two competing sources at different bearings cause low estimates of $Φ<0.4$⁠. As anticipated, the bearing estimates are also degraded during this period. However, once CPA is approached (⁠$−100<t<100$ s), energy from the M/V Oregon Highway dominates; $Φ$ reaches $≃$ 0.95, and bearing estimates improve (the bearing estimated around 250 Hz is similar to the one estimated around 63 Hz). Post CPA (t > 300 s) other sources, including R/V Armstrong suppress (episodically) estimates of $Φ$ and degrade bearing estimates.

An additional interpretation is supported by plotting the real-valued horizontal velocities $vx(t)$ vs $vy(t)$ for the 250 Hz band, corresponding to some representative periods over the course of the 20 min observation period (Fig. 8). The example $≃$ 500 s before CPA has a moderately high $Φ$⁠, and attendant quasi-stable estimate of bearing (Fig. 7), though of the R/V Endeavor. A later example, $≃$ 375 s before CPA, the low $Φ$ predicts the lack of a consistent, rectilinear relation between $vx(t)$ and $vy(t)$ sufficient to yield a reasonable estimate of bearing. The remaining examples, ±50 s about CPA and at CPA [which for this vessel $vx(t)$ dominates owing to the orientation of the IVAR x, y axes] are all characterized by $Φ>0.9$⁠, with an expectation of stable estimates of bearing.

Finally, we note that Thode et al.³⁷ provide an intriguing alternative to $Φ$ as it applies to the influence of multiple sources and degradation of an estimate of bearing. In that work, the normalized indicator U_c is used to assess Arctic Oean noise directionality, where U_c given in terms of variables used in this study equals $cw−1Ix2+Iy2/(Ek+Ep)$⁠. The energies E_k and E_p are computed as in Sec. III, with E_k limited to the horizontal components $vx(t)$ and $vy(t)$⁠. The non-normalized version $cwUc$ is known as the speed of acoustic energy density transport³⁸ in the horizontal direction. We find U_c clearly correlates with $Φ$⁠, for example, reaching ≃ 1 in Fig. 7 for both the 63 and 250 Hz frequency bands near CPA. However for the 250 Hz case, and over the period $−400<t<−200$ s where bearing is less certain, $Uc>Φ$⁠, and for the 63 Hz case over the period $t<−400$ s where bearing remains more certain, $Uc<Φ$⁠. Thus, on this basis, it might be argued that $Φ$ is the better measure of bearing dispersion. The main difference between U_c and $Φ$ is that U_c can be modeled⁷ and thus possesses deterministic features related to the waveguide, while $Φ$ characterizes the stochastic behavior of the bivariate signal. It is therefore possible that a combination of both indicators might provide a better measure.

Salient properties of the acoustic pressure and velocity fields associated with underwater noise radiation from ships have been presented. The noise is from two, identifiable merchant vessels observed on different days during the course of SBCEX17. Measurements were made with the IVAR that records acoustic pressure and acceleration $a→$ at a single point 1.25 m above the seabed, with $a→$ converted to three Cartesian components of acoustic velocity $vx,y,z$ in either the frequency or time domain. The CPA of these two vessels with the IVAR location was $≃$ 2400 m (M/V Alice Oldendorf) and $≃$ 450 m (M/V Oregon Highway).

The relation between the kinematic (velocity) and dynamic (pressure) fields is first described in terms of kinetic and potential energy spectra, and for both vessels, there was approximate concurrence in the time-averaged energy levels within decidecimal bands, over a frequency span that encompasses greater than 95% of the total energy. We anticipate this result to apply for ship CPA ranges exceeding the water depth. For the observations in this study, the receiver was near the seabed. A larger ratio of kinetic to potential energy would be expected for a near-surface receiver (e.g., located within about one-third of an acoustic wavelength). The kinetic energy in this near-surface zone would also be reduced relative to that deeper in the water column. With these limits in mind, a practical implication is that reasonable estimates of the root-mean-square acoustic velocity within a decidecimal (third-octave) band are available from the pressure-only measurement.

In terms of velocity, the focus is on bivariate acoustic velocity signals—one set composed of the two horizontal components, v_x and v_y, and another upon resolving these into a single range component v_r paired with the vertical component v_z, yielding the bivariate signal in the range–depth plane. These data are described through their relation to Stokes parameters, a framework to describe a bivariate signal with a set of four real scalars $S0,1,2,3$ that fully describe the polarization of such a signal; two relations obtained from these parameters are circularity Θ and the degree of polarization $Φ$⁠.

An overview of the time-varying observations from each vessel over a 20 min period was provided in terms of circularity, for which a scalar version of this otherwise vector quantity as a function frequency f and time t, $Θ(f;t)$⁠, is formed. The scalar version describes a bivariate relation in a range–depth (r, z) plane and is amenable to modeling based on an approach that assumes cylindrical symmetry in the field about a vertical z axis. Striation features produced by frequency-dependent modal interference as each vessel closes and opens in range illustrate the high information content of $Θ(f;t)$ for purposes of geoacoustic inversion, including source range and, to a lesser extent, depth information.

Greater detail is seen by presenting two frequency cuts of the estimates of $Θ(f;t)$ corresponding to the M/V Alice Oldendorf observations during which it closes and opens in range to IVAR from $≃$ 3.5 km to the CPA of $≃$ 2.4 km. These data centered at f = 63 Hz and f = 64 Hz and corresponding model solutions for $Θ(f;t)$ (from another study³) based on the geoacoustic inversion of these data are in good agreement. Three time periods from the 63 Hz observations are selected to provide an explicit demonstration of a bivariate velocity signal in the r – z plane. For this, the acoustic velocity components $vx,y,z$ are narrowband filtered (bandwidth $≃1$ Hz), with the two horizontal components $vx,y(t)$ projected into a single radial component $vr(t)$ based on the known IVAR $x‐$axis and $y‐$axis fixed alignment combined with the known (via AIS) and slowly varying bearing of M/V Alice Oldendorf.

A detailed and nuanced picture of such a bivariate signal emerges with three cases defined by: $Θ(f;t)≃0,Θ(f;t)≃1$⁠, and $Θ(f;t)≃−1$⁠. The two, $|Θ(f;t)|≃$ 1 periods show bivariate trajectories that are more elliptical in character and are coincident with reduced kinetic and potential energy, possibly local minima. In the period between the two $≃1$ examples, the sign Θ changes, the manifestation of which is visualized by a change in direction of rotation of the bivariate vector. The period $Θ(f;t)≃0$ shows an approximately rectilinear trajectory in the r – z plane and higher kinetic and potential energy.

The full trajectory history over the 5 s averaging period is also shown for each case. For the rectilinear, $Θ(f;t)≃0$ case the trajectories are relatively confined, which is consistent with the corresponding estimate of $Φ=0.99$⁠. In the limit of Φ = 1, this trajectory will be confined to a single line. For the two elliptical $|Θ(f,t)|≃$ 1 cases, the envelope is more distributed within a range of trajectories filling an elliptical area, consistent with corresponding estimates of $Φ≃0.65$⁠, and in the limit of Φ = 1, the trajectory is confined to a single ellipse.

Finally, we demonstrate the horizontal bivariate relation composed of $vx(t)$ and $vy(t)$ observations originating from the M/V Oregon Highway transit and observations of the bearing. In this case, decidecimal bands, centered at 63 Hz and 250 Hz, are studied, from which estimates of $Φ$ are derived based on complex counterparts. The estimate of bearing is based on the Umov vector equal to $⟨p(t)vx,y(t)⟩$⁠. With bearing span confined to $<160°$⁠, the parameter θ as defined by Eq. (13) yields the same bearing estimate upon resolving the $180°$ ambiguity. The choice of center frequencies is motivated by the lower band remaining free of interference over the 20 min observation period while the higher band receiving contributions from other ship sources. Estimates of $Φ$ over time at 250 Hz are significantly depressed relative to those at 63 Hz, owing to interference from other ship sources, their locations known with AIS. The results suggest that when $Φ≲0.4$ bearing estimates are degraded and when $Φ≳0.6$ estimates are stable.

This work was supported by the Office of Naval Research (USA) and the Direction Generale de l'Armement (France). The authors thank J. Flamant, N. Le Bihan, and D. R. Dall'Osto for helpful discussions. The incisive comments by the reviewers of this study were very much appreciated.