Observed near the seafloor, broadband noise emissions from a vessel passing directly above exhibit frequency bands where potential acoustic energy is greater than kinetic energy while the opposite occurs in neighboring frequency bands. The condition where the dynamic and kinematic energy forms differ in this manner is characteristic to interference involving steep angles or near-normal incidence reflection from the seafloor. Measurements are made at two experimental sites using a research vessel passing above a vector sensor, positioned ∼1.5 m above the seabed, resulting in a vessel horizontal range approaching ∼0. The data are expressed as a ratio of kinetic to potential energy in decibels and yield information on seabed properties. A model for kinetic and potential energy is developed from the method of images using a layered seabed and is used to invert data collected in Puget Sound. A higher-impedance seabed is identified via inversion, which is consistent with the thin Holocene sediments in the region. For data collected on the New England Mud Patch, the model is instead applied directly to nominal seabed parameters originating from prior studies that identify a low-speed mud layer atop a higher-speed transition layer separating the mud substrate from a sediment basement.

## I. INTRODUCTION

When a vessel passes directly above, underwater noise emissions originating from the vessel can dominate the sound field below. In such conditions where sound propagation is nearly vertical, the sound field is reinforced by multiple reflections from the seafloor and sea surface, establishing a standing wave. Potential energy can be an order of magnitude greater than kinetic energy in certain frequency bands with the opposite occurring in neighboring bands. Given that ship noise is a broadband source, the ratio between potential energy, *E _{p}*, and kinetic energy,

*E*, presents a frequency pattern, linked primarily to the position of the sensor above the seafloor and the geoacoustic properties and structure of the marine sediments.

_{k}These conditions are studied in controlled experiments in which two research vessels conduct transits directly above a vector sensor, positioned ∼1.5 m above the seabed, resulting in a horizontal range at closest point of approach (CPA) of ∼0. (In this paper, the term CPA is used with reference to the horizontal range between sensor and vessel; this horizontal range can approach zero when the vessel transits directly above a sensor.)

One case involves R/V *Neil Armstrong* (length 73 m) operating in waters of depth 75–80 m within the New England Mud Patch (NEMP) and the other case involves R/V *Robertson* (length 18 m) operating in waters of depth ∼42 m in Puget Sound. The noise in the 30–1600 Hz band shows a distinct pattern in the energy ratio; differences are due to the sediment type and layering within the seabed at the two experimental sites.

In observations made away from a near-vertical orientation between the vessel sound source and receiver, however, the two energy forms reach parity. For example, in measurements of underwater noise from identifiable cargo ships for which range to CPA was of order ten water depths, underwater noise potential and kinetic energies computed in third-octave bands centered between 25 and 630 Hz were equal within a calibration uncertainty of ±1.5 dB.^{1}

The paper is arranged as follows. Section II outlines a simple model for the ratio of kinetic to potential energy, based on the method of images, that is used to invert the data from Puget Sound to estimate seabed properties and in forward modeling of data from the NEMP. In Sec. III, relevant aspects of measurement instrumentation and frequency domain data processing are described, followed by descriptions of the two controlled experiments and the observations in the form of energy ratios expressed in decibels. In Sec. IV, results of the inversion of the Puget Sound data and forward modeling of the NEMP data are presented. Section V concludes with a summary and discussion.

## II. RATIO OF POTENTIAL TO KINETIC ENERGY, MEASURED DIRECTLY BELOW A PASSING VESSEL

### A. Model formulation

The problem addressed will be through an approximation relating to the energetics observed directly below a passing vessel with the Puget Sound case used to represent the geometry (Fig. 1). The primary approximation is that the kinetic energy is limited to the vertical component, which is justified as a result of the vertical orientation between source and receiver. The model also does not require ship spectral source level information, given that an energy ratio is the essential output to be compared with data rather than energy level. Important specifications, in addition to the seabed properties, are waveguide depth, *H*, and receiver depth, *z _{r}*, giving the height above the seabed of the center-point of the sensor, $H\u2212zr$, and the water sound speed,

*c*, between the sensor and seabed. Although the water sound speed varied with depth in both experiments, a single

_{w}*c*is applied over the entire water column, an approximation consistent with refraction being suppressed at high grazing angles.

_{w}The method of images is used to compute an approximate Green's function, $g(z,z0,f)$, where *f* is frequency, for a harmonic source (time dependence, $e\u2212i2\pi ft$, suppressed) near the air–water interface depth at *z*_{0} and receiver directly below it at depth $z=zr$. This is modified from an exact form^{2} for a rigid or infinite impedance boundary condition at the seabed by setting the receiver range, *r*, equal to zero and replacing the reflection coefficient, *R _{b}* = 1, with finite impedance, frequency dependent form, $Rb(f)$, to be evaluated at normal incidence (grazing angle 90°).

The impedance translation theorem^{3} is used to construct the plane wave reflection coefficient for a fluid seabed. For the Puget Sound case, for which geoacoustic inversion is performed, the seabed is assumed to consist of two layers of thickness, *L _{i}*,

*i*= 1, 2, characterized by density,

*ρ*, and sound speed,

_{i}*c*, and terminated by a half-space characterized by $\rho 3,c3$. For the NEMP, where only forward modeling is performed, an additional layer is used such that

_{i}*i*= 1, 3, with the seabed terminated by a half-space characterized by $\rho 4,c4$.

The sound speeds are made complex by inclusion of an expression for sediment attenuation, *α _{i}*. Depending on the example, attenuation is characterized by a constant, dB/

*λ*expression, where

*λ*is sediment acoustic wavelength, or a frequency-dependent, dB/m expression with both expressions implemented in the model via conversion to an imaginary part of sediment acoustic wavenumber. The particular expression used is made explicit in context.

Note that it might be expected that a spherical wave reflection coefficient form rather than plane wave reflection is applicable for these observations, given the combination of proximity with the seabed and frequency. However, differences between the two forms are greatest near the critical angle, becoming minimal far from this angle and, particularly, at normal incidence.^{4} A shear speed is also not included in the reflection coefficient model to simplify the parameter space, again, with the justification that $Rb(f)$ is limited to normal incidence, where differences between fluid and elastic forms are minimized.^{5}

The image construction starts by forming

to generate distance measures of the four terms to be used in the expression for the total field. Next, put $Rnm=(znm)2$, and $Gnm=eikRnm/Rnm$, where *k* is a wavenumber equal to $2\pi f/cw$ and *m* = 1, 4, identifying the four terms. The contributions from the physical source and first three images are represented by *n* = 0, and successive reflections from the sea surface and seabed yield a series representation for the Green's function, $g(z,z0,f)$, defined as

which converges for *N* greater than about ten, and it is understood that the complex reflection coefficient, $Rb(f)$, is evaluated at normal incidence. This Green's function is used as a surrogate for complex, harmonic pressure, $p\u0302(f)$.

The vertical velocity is derived from the gradient, $\u2202g/\u2202z$. First, identify

then

Analogous to $p\u0302(f)$, the complex, harmonic vertical velocity as a function of frequency, $v\u0302z(f)$, is represented by

where *ρ _{w}* is seawater density.

The model for potential energy spectral density, $Ep(f)$, is $0.25|p\u0302(f)|2/(\rho wcw2)$ and corresponding vertical kinetic energy spectral density, $Ekz(f)$, is $0.25|v\u0302z(f)|2\rho w$. The nondimensional ratio of these two energy quantities as a function of frequency effectively removes the unknown and frequency varying vessel source strength, rendering comparison with field observations convenient.

Although clearly arbitrary, the model result puts vertical kinetic energy in the numerator of this ratio with model output expressed in decibels as $10\u2009\u2009log10Ekz(f)/Ep(f)$. The logarithmic transformation to decibels is important as it tends to generate an approximately zero-mean, stationary series as a function of frequency with informative amplitude variation of this series about the mean value. Hereafter, unless otherwise explicitly stated, any reference to ratio assumes that it is expressed in decibels with the kinetic component in the numerator.

### B. Effect of sensor height above the seabed

The effect of sensor height above the seabed on the ratio of $Ekz(f)$ to $Ep(f)$ can be demonstrated by inspecting the model. For example, for a half-space seabed representation, the model exhibits a sinusoidal oscillation as a function of frequency [Fig. 2(a)]. This repetitive pattern has distinct frequency separation, $\Delta f0$, corresponding to the inverse of the two-way travel time between the sensor and seabed, where

and $\Delta z=H\u2212zr$ [see also Fig. 1(b)]. A change in $\Delta z$ from 1.5 to 1 m off the seabed changes $\Delta f0$ by 250 Hz [compare black and gray lines in Fig. 2(a)], showing that off-bottom distance is a key parameter. Moving the sensor up the water column produces increasing oscillation with frequency (decreasing $\Delta f0$), and this tends to reduce, on average, differences between the two energy quantities.

### C. Effect of sediment properties and layering

The effect of the seabed on the ratio of $Ekz(f)$ to $Ep(f)$ is demonstrated by evaluating the model over different seabed types. For example, increasing the seabed characteristic impedance of the half-space seabed increases the amplitude of the ratio variation with frequency [Fig. 2(a)] with the opposite occurring for a decrease in characteristic impedance.

The effect of sediment layering [Fig. 2(b)] is illustrated using a notional seabed representation for the NEMP. The same $\Delta f0$ scale is maintained for a sensor 1.5 m off of the bottom, but there is now additional modulation. The $\Delta f1$ scale corresponds to the inverse of the two-way travel time to the first dominant, subsurface reflective interface, where

and *L*_{1} and *c*_{1} represent, respectively, the separation between the seabed and this interface or layer thickness and average sound speed of this layer. A second example includes a small layer separating the low-speed mud layer from the terminal half-space. This is parameterized by *L*_{2}, *c*_{2}, and the layer introduces a subtle, yet noticeable, change in the pattern in the frequency range 1300–1400 Hz. Similar effects will be shown in the analysis of the data collected at the NEMP (Sec. III C) and in forward modeling of such data using nominal seabed parameters originating from prior studies (Sec. IV B).

We emphasize that the two frequency scales identified in Eqs. (6) and (7) provide only a guide for an immediate inspection of the data of the kind displayed in Fig. 2 and other figures discussed subsequently (Figs. 3, 5, and 6). Additional frequency scales will exist for more complex layered seabeds with sufficient impedance contrast to yield an observable reflection, with such scales becoming progressively smaller for increasing travel time to the interface.

## III. VECTOR ACOUSTIC OBSERVATIONS OF UNDERWATER NOISE DURING A CONTROLLED VESSEL TRANSIT DIRECTLY ABOVE A SENSOR

### A. Measurement system and frequency domain data processing

The measurements 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 for acoustic acceleration from which acoustic velocity is obtained.^{6} The configuration (IVAR-2) used in these experiments differs from previous studies and is based on a M20-105 sensor (GS, Halifax, Nova Scotia, Canada) from which data are recorded on a multichannel AMAR-G4 recorder (JASCO, Halifax, Nova Scotia, Canada). These components, along with additional instrumentation for oceanographic and video observations, are placed within a custom frame that is shrouded to mitigate current-induced motion of the sensor [see Fig. 1(b)]. Although higher sample rates were initially used in the field, the following analysis related to ship noise is restricted to a sampling frequency (each channel) of 4 kHz as obtained through decimation.

The four-channel data stream is processed in the frequency domain for which there are multiple and nominally equivalent options for windowing and averaging; the course taken here is as follows. A Fourier transform over 1 s of data (uniform window) is applied to the pressure and three acceleration channels with the window advanced to cover the period of interest with 75% overlap. One-sided complex spectra are retained, each normalized to the variance of the corresponding time-domain data. This operation defines $Sp(f;t)$ as the complex spectrum for pressure and $Svj(f;t)$ as the complex spectrum for the *j*th 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 $Svj(f;t)$ originates from the corresponding acceleration channel. The M20-105 sensor calibration curves incorporate the conversion from acceleration to velocity, and a second curve is applied to provide the correct phase. Pressure (dynamic) and the velocity (kinematic) components are expressed in mks units.

The (time-varying) potential energy spectrum, $Ep(f;t)$, is defined as

based on water density, *ρ _{w}*, and water sound speed,

*c*(identified for each case), with angle brackets denoting a 5-s moving time average. Similarly, the kinetic energy spectrum, $Ek(f;t)$, is defined as

_{w}In this analysis, it is useful to distinguish kinetic energy spectral density into its components, $Ekj(f)$, where *j* = 1,2 are the horizontal components and *j* = 3 is the vertical or *z* component explicitly identified as $Ekz(f)$.

### B. Puget Sound

The IVAR-2 system was deployed in Puget Sound at 47.7877° N, 122.4875° W on April 14, 2021. The location is within a small bay on the west side of Puget Sound, containing a Washington State ferry landing at Kingston. The deployment site was approximately 1 km distant from the landing at a depth of 42 m. Important specifications for the interpretation of these data are the nominal height above the seabed of the center-point of the sensor, 1.48 m, and the sound speed over this same interval. Conductivity, temperature, and depth (CTD) measurements made just prior to the deployment indicated that over the lower 30 m of the water column, the sound speed was a nearly constant 1478.3 m/s, along with a similarly constant salinity of 29.8 ppt, density of 1024 kg/m^{3}, and temperature of $\u223c8.5\u2009\xb0$C; a thermocline produced a sound speed increase of about 1.5 m/s from depth of 10 m to the sea surface.

The controlled transit was made directly over the IVAR-2 system by R/V *Robertson* at speed ∼7.5 knots. At the expected vessel position above the IVAR-2 system (or CPA with horizontal range approaching 0 m), the observed vertical kinetic energy density (over the 100–1000 Hz band) exceeded the horizontal counterpart by more than 20 dB, indicating a nearly perfect vertical alignment with the effective radiating source of this 18-m length vessel, assumed to be colocated with the vessel's single propeller. In this study, the relation between vertical and horizontal kinetic energy is used only as a confirmation of positional accuracy.

The energy ratio observations [Fig. 3(a)] demonstrate a hallmark of the direct overpass CPA: a significant exceedance of potential over kinetic energy (and vice versa) at relatively well-defined frequency intervals as suggested by the modeling. The frequency scales, $\Delta f0,\Delta f1$, are evident at approximately 500 and 165 Hz, respectively. With reference to a previous comment that these scales provide only an approximate guide, smaller frequency scales are also evident.

In the absence of a dominant, reflective sub-bottom interface, such as a half-space representation for the seabed, the results would be expected to be modulated in frequency only by $\Delta f0$. The two versions of the ratio, one version based on total kinetic energy spectral density $Ek(f)$ (green line) and the other version based on the vertical kinetic energy spectral density $Ekz(f)$ (black line) are similar, which is expected for this direct overpass geometry.

### C. NEMP

#### 1. Measurements made directly above

A deployment of IVAR-2 from R/V *Neil Armstrong* on May 12, 2022 at location 40.4373° N, 70.5102° W in the southeastern portion of the NEMP was part of the 2022 Seabed Characterization Experiment (SBCEX22). In this case, the center-point of the sensor was 1.45 m above the seabed as a result of a determined 3-cm adjustment from that used in the Puget Sound test. The sound speed, 1486 m/s, is derived from the continuous (12 s interval) CTD measurements made on the IVAR-2 platform (model SBE37, Sea-Bird Scientific, Bellevue, WA) and, thus, represents the speed within the water column between the sensor and seabed. Within this depth interval, the salinity was 33.5 ppt, density was 1026 kg/m^{3}, and temperature was $\u223c9.2\u2009\xb0$C. During SBCEX22, multiple CTD casts show sound speed varying with depth as well as temporal and spatial variability; these data are under study.

A controlled transit was made directly over IVAR-2 at a speed of ∼10 knots, immediately after the deployment at which time (18:30 UTC) the water depth was 78 m. At CPA, the vertical kinetic energy, $Ekz$ (over the 100–1000 Hz band), exceeded the horizontal counterpart by 12 dB or somewhat less than the measurements made in Puget Sound involving a smaller vessel. The primary reason for the difference is the greater length of R/V *Neil Armstrong* relative to the water depth and because the vessel's twin propellers, separated by 7.6 m, complicates such an alignment.

Nevertheless, a similar significant excess of potential over kinetic energy (and vice versa) is observed at relatively well-defined frequency intervals [Fig. 3(b)], although the pattern with frequency differs markedly from that observed in Puget Sound. In this case, the frequency scales, $\Delta f0,\Delta f1$, are approximately 510 and 55 Hz, respectively, and as with the Puget Sound case, smaller frequency scales are also evident.

The small difference in value for $\Delta f0$ between these observations and those from Puget Sound is attributable to the small difference in $\Delta z/cw$ for the two sets of observations. However, the $\Delta f1$ frequency scale is less than that observed in Puget Sound, suggesting that a thicker sediment layer separating a reflective sub-bottom interface from the water–seabed interface dominates these observations. Furthermore, the variation in ratio here is $\u223c\xb1$6 dB compared with $\u223c\xb1$12 dB in Puget Sound, suggesting that seabed reflection at normal incidence is higher for Puget Sound.

#### 2. Measurements at CPA ranges on the order of the water depth

Measurements relating to a ship closing toward an approximate overpassing CPA, then opening, with ship horizontal range to CPA scaled by water depth (Fig. 4), originate from data collected during a second deployment of IVAR-2 from R/V *Neil Armstrong* on May 20, 2022 at a location 40.4416° N, 70.5282° W, or slightly north and west of the prior location on the NEMP. These observations remain under study but are useful in providing a view of the changing relation between kinetic and potential energy as a function of this scaled horizontal range. A time-frequency view of a linear energy ratio, here specifically plotted as a linear ratio of potential, $Ep(f;t)$, to total kinetic, $Ek(f;t)$, spectral energy density, is mapped to vessel approach to and departure from CPA [Fig. 4(a)] with negative and positive horizontal scaled range relative to the white dashed line determined using the vessel's positional information. The phenomenon of acoustic potential energy significantly exceeding kinetic energy for ranges less than a water depth is apparent with this linear view of the energy ratio.

The situation is akin to a standing wave process with the finite impedance of the seabed ultimately placing a bound on the ratio. (Note that the opposite effect of kinetic exceeding potential energy is also occurring, although this is best viewed with the ratio expressed in decibels as in Figs. 2 and 3.) Notably, the corresponding energy spectra at CPA [Fig. 4(b)] do not display strong fades, e.g., as a result of modal interference. However, the spectra each modulate with frequency and follow slightly different patterns, for which a $\Delta f0$ frequency scale in the range of 500 Hz is evident.

The kinetic and potential energy total levels computed within the decidecimal^{7} (third-octave) band centered at 600 Hz, corresponding to the shaded area in Fig. 4(a), are shown varying over a horizontal range to CPA scaled by water depth from approximately ± 8 water depths [Fig. 4(c)]. At approximate CPA or for horizontal range relative to water depth less than ∼1.5, the potential exceeds the kinetic energy by more than a factor of 2 as shown by the ratio (black line) expressed here in decibels as $10\u2009\u2009log10Ek/Ep$ for consistency with Figs. 2 and 3. It is only under such conditions that it is expected that potential will exceed kinetic energy in a significant and sustained manner. For horizontal range relative to water depth greater than two (or less than −2 for approaching CPA), the ratio approximates 0 dB to within calibration uncertainty. The asymmetry about CPA for the third-octave levels, *E _{p}* and

*E*, is interesting and not fully understood but possibly related to the radiation pattern of R/V

_{k}*Neil Armstrong*at this frequency range. More relevant to this study is that the energy ratio is approximately symmetric about CPA.

## IV. GEOACOUSTIC INVERSION AND FORWARD MODELING

### A. Puget Sound data: Geoacoustic inversion

The observations in Puget Sound were made in an area with relatively unknown seabed acoustic properties. Thus, to examine these data with the model of Sec. II, it was first necessary to estimate required model inputs related to seabed acoustic properties through an inversion applied to the data [Fig. 3(a)] as a function of frequency using a Bayesian framework.^{8,9} In this procedure, the model-data mismatch vector, **r**, is the decibel difference between the sequence, **d**, which tends to be approximately centered about ∼0 dB, and a candidate model result, $d(m)$, based on a set of five candidate parameters (length of model parameter vector **m** equals five).

We note that it is important to fix or tie a sediment layer density, *ρ _{i}*,

*a priori*to a candidate sediment layer sound speed,

*c*. The seabed reflection process is governed largely by the characteristic impedance, $\rho ici$, and layer travel times, $Li/ci$. Fixing

_{i}*ρ*as a dependent parameter resolves the uncertainty in parameter estimation caused by coupling of parameters. The reduction in model parameters reduces the estimation uncertainty for layer thickness and depth, which otherwise would be limited to the prior search bound. However, an error in chosen prior sediment density will affect the accuracy of the parameter estimates.

_{i}Some prior information for the Puget Sound observations is available through several studies involving seismic reflection methods to evaluate earthquake fault systems within the region. In one method, the United States Geological Survey (USGS) conducted an extensive sub-bottom profile survey of Puget Sound,^{10} where the majority of the measurement effort was directed toward the deeper, 200–300-m contours characteristic of the Puget Sound basin. Although the site near Kingston was not profiled, the study included a few sites along the western edge of Puget Sound within the same 40–50 m depth contour as Kingston, and these revealed a consistent sub-bottom layer, suggesting an impedance change at sub-bottom depth ∼4 m. This information is used to inform prior search bounds for the depth to the first reflective interface, *L*_{1}.

The Puget Sound observations were also colocated within the vicinity of the Kingston arch, a region of uplifting resulting in a relatively shallow sub-bottom depth to the basement substrate.^{11,12} At the deployment site, sediment maps indicate that the Holocene sediment depth is approximately 5 m.^{13} This information translates to an expectation that the transition layer, *L*_{2}, will not be large before a high-value terminal half-space speed, *c*_{3}, which is more characteristic of basement rock, takes effect. Sediment densities, *ρ _{i}*,

*i*= 1,2, are tied to a polynomial fit of sediment density data from Hamilton,

^{14}where

yielding *ρ _{i}* in kg/m

^{3}for a candidate,

*c*, expressed in km/s (for

_{i}*c*< 2.4 km/s). However, given that much higher candidate speeds for

_{i}*c*

_{3}are used, a density-velocity relation established for shallow bedrock formations within Puget Sound

^{15}is used for corresponding

*ρ*

_{3}, where

and *c*_{3} is expressed in km/s.

Finally, initial testing suggests that the model, when based on informed shallow sediment structure, is not generally sensitive to sediment attenuation provided that some realistic nonzero value for attenuation, *α _{i}*, in dB/

*λ*, of order 0.3 is used; we, thus, set this $\alpha 1,3$ equal to 0.3 dB/

*λ*.

The results of inverting the observations from Puget Sound (Fig. 5) yield plausible geoacoustic parameterization of this seabed. The maximum *a posteriori* (MAP) values for geoacoustic parameters for the Puget Sound data (Table I) are considered to be the optimal estimates. These are found by minimizing data misfit function, $E(m)=(Nf/2)\u2009logerTr$, over the five-dimensional parameter space with *N _{f}* = 524 equal to the number of frequencies used in the evaluation over the frequency range, 30–1600 Hz, taken every 3 Hz. The model based on this MAP parameterization reproduces key features of the data [Fig. 5(a)] such as approximate decibel range and frequency variation.

Parameter . | Unit . | Search bounds . | MAP . | σ
. |
---|---|---|---|---|

c_{1} | m/s | [1500, 3000] | 1750 | 19 |

L_{1} | m | [2,5] | 3.65 | 0.043 |

c_{2} | m/s | [1500, 3000] | 2230 | 73 |

L_{2} | m | [0,4] | 0.45 | 0.044 |

c_{3} | m/s | [1500, 3000] | 2720 | 127 |

Parameter . | Unit . | Search bounds . | MAP . | σ
. |
---|---|---|---|---|

c_{1} | m/s | [1500, 3000] | 1750 | 19 |

L_{1} | m | [2,5] | 3.65 | 0.043 |

c_{2} | m/s | [1500, 3000] | 2230 | 73 |

L_{2} | m | [0,4] | 0.45 | 0.044 |

c_{3} | m/s | [1500, 3000] | 2720 | 127 |

Some frequency ranges exhibit a larger degree of model-data misfit, which suggests a need for more parameters to represent the seabed. Notably, the MAP estimates for sediment layer, *L*_{1}, plus a transition layer, *L*_{2}, combined is 4.1 m, consistent with the thickness map of Holocene sediments. Beneath these layers, the MAP estimate for sound speed, *c*_{3}, suggests a harder bottom substrate.

The one-dimensional (1-D) marginal probability density functions for inverted parameters [Figs. 5(b) and 5(c)] all exhibit mean values approximately colocated with MAP estimates and quantify, to some extent, the ability of the model to resolve features of the data to the extent described by five parameters while using *a priori* densities, *ρ _{i}*. However, as noted previously, there is coupling of these parameters, particularly as they relate to establishing the density within each layer

*a priori*.

This issue is addressed in an approximate manner as follows. The empirical expressions for densities as described by Eqs. (10) and (11) are both assumed to have a relative uncertainty of ± 5%. This is verified by plotting Eq. (11) over the companion set of data (not shown) with ± 5% encompassing the majority of this data. Next, we assume that the inversion using *a priori ρ _{i}* provided optimal or best estimates of $\rho ici$ and travel time, $Li/ci,\u2009\u2009i=1,2$. For example, the characteristic impedance, $\rho 3c3$, based on the MAP estimate of

*c*

_{3}(Table I) and corresponding

*a priori ρ*

_{3}, 2222 kg/m

^{3}, equals 6.044 × 10

^{6}kg m

^{−2}s

^{−1}, and the travel time to the first reflective interface is 2.086 ms based on the MAP estimates of

*L*

_{1}and

*c*

_{1}.

To maintain this same characteristic impedance while using a *ρ*_{3} that varies ± 5% about the *a priori* value puts *c*_{3} between 2590 and 2863 m/s. This analysis is repeated for the MAP estimates of $ci,\u2009\u2009i=1,2$ derived from *a priori* $\rho i,\u2009\u2009i=1,2$, providing equivalent upper and lower range values for these parameters. Finally, the upper and lower range values for $ci,\u2009\u2009i=1,2$, lead to corresponding upper and lower range values for $Li,\u2009\u2009i=1,2$ that maintain travel time.

The upper and lower range values for the five parameters are identified by the horizontal bars centered on the MAP estimates [open symbols in Figs. 5(b) and 5(c)] above each corresponding 1-D marginal probability density function. They represent an estimate of overall uncertainty, which recognizes that *a priori* densities are not without error.

### B. NEMP data: Forward modeling using nominal seabed parameters originating from prior studies

A prominent feature of the NEMP seabed in this location is the low-speed mud layer commencing at the water–seabed interface with thickness of order 10 m.^{16} Other studies provide evidence for increasing sand content in the lower part of the mud layer,^{17–19} constituting a higher-speed transition layer of thickness *O*(1) m between the mud and higher-speed substrate below.

Rather than undertake a geoacoustic inversion, which is not intended to be the thrust of this study, direct forward modeling of the NEMP observations is performed using the model in Sec. II that is based on a nominal seabed description consistent with published studies.

The basic form of the seabed description seeks an approximate representation of the result in Dosso and Bonnel,^{17} as described in depth profiles of marginal probability for geoacoustic properties for sound speed, density, and layer thickness as inverted from SBCEX17 observations on the NEMP at a location approximately 5 km northwest of current observations. The specific results used are from the trans-dimensional inversion, as shown in Figs. 5(a)–5(c) of Dosso and Bonnel^{17} and, thus, profiles of marginal probability do not vary with depth within a layer.

To characterize this depth profile, the plane-wave reflection coefficient is now derived from three layers below the water–sediment interface of thickness, *L _{i}*,

*i*= 1,3 with density,

*ρ*, sound speed,

_{i}*c*, and terminated by a half-space characterized by $\rho 4,c4$. Here, the notional mud layer is represented by the first two layers with

_{i}*L*

_{3}representing a transition layer with increasing sand content.

We identify constant values for sound speed and density for a given layer associated with the highest marginal probability. These layer speeds are adjusted to preserve the index of refraction, which accounts for the change in seawater sound speed between their study (1468 m/s), conducted in March 2017, and our study (1486 m/s), conducted in May 2022. The adjusted layer speeds and identified constant layer densities are listed in Table II.

L_{1} (m) | c_{1} (m/s) | ρ_{1} (kg/m^{3}) |

1.2 | 1493 | 1593 |

L_{2} (m) | c_{2} (m/s) | ρ_{2} (kg/m^{3}) |

9.2 | 1504 | 1649 |

L_{3} (m) | c_{3} (m/s) | ρ_{3} (kg/m^{3}) |

2.6 | 1570 | 1748 |

— | c_{4} (m/s) | ρ_{4} (kg/m^{3}) |

— | 1823 | 1850 |

L_{1} (m) | c_{1} (m/s) | ρ_{1} (kg/m^{3}) |

1.2 | 1493 | 1593 |

L_{2} (m) | c_{2} (m/s) | ρ_{2} (kg/m^{3}) |

9.2 | 1504 | 1649 |

L_{3} (m) | c_{3} (m/s) | ρ_{3} (kg/m^{3}) |

2.6 | 1570 | 1748 |

— | c_{4} (m/s) | ρ_{4} (kg/m^{3}) |

— | 1823 | 1850 |

For layer thickness, *L _{i}*, through experimentation, we find better agreement with our observations by adjusting values that can be inferred from the corresponding peak marginal probability profiles for interface depth [Fig. 5(a) in Dosso and Bonnel

^{17}]. Our determined values for

*L*(Table II) are consistent with the larger body of work on the NEMP, some of which are referenced in this study, but it is emphasized that they are not to be interpreted as a result of a formal inversion.

_{i}Finally, to represent attenuation in the marine mud and transition layers, the sound speeds for *i* = 1, 3 are made complex by inclusion of attenuation with power-law frequency behavior,^{20} $\alpha i=0.14(f/1000)1.72$ in dB/m, with the coefficient adjusted to the slightly lower value of 0.14. For completeness, the half-space speed, *c*_{4}, is made complex using a constant attenuation of 0.3 dB/*λ*, although there is little sensitivity of the model result to this parameter.

The model (Fig. 6) that ran using the Table II parameterization captures well the small frequency scale oscillation described by $\Delta f1$ as well as the large scale $\Delta f0$. As first mentioned in Sec. II, the NEMP data exhibit a subtle effect that is approximately reproducible with the model on inclusion of a transition layer. For example, over the frequency range 1300–1400 Hz, the NEMP observations show that the $\Delta f1$ variation is damped, with this important effect approximately reproduced with the model that embodies a transition layer parameterized by *L*_{3}, *c*_{3}, and *ρ*_{3} (Table II). Note that, absent such a transition layer, the simulation would exhibit the basic $\Delta f0$ variation as shown, on which rides a $\Delta f1$ variation that is effectively unchanging across this frequency band. In short, an effect of the transition layer is manifested in the model and the data.

## V. DISCUSSION AND SUMMARY

The underwater noise field from a ship has been studied where the propagation conditions are governed by a vertical orientation with respect to the ship sound source and receiver, such as a ship CPA horizontal range of order one water depth or less. Two highly controlled overpassing transit experiments were conducted over a vector sensor placed ∼1.5 m above the seabed, where one experiment involved R/V *Neil Armstrong* (length 73 m) operating in waters of depth ∼75–80 m on the NEMP and the other experiment involved R/V *Robertson* (length 18 m) operating in waters of depth ∼42 m in Puget Sound.

These observations, covering the nominal frequency range 30–1600 Hz, reveal a standing wave field directly below the vessel as the vessel transits over the sensor. Such a field exhibits high potential energy, *E _{p}*, relative to kinetic energy,

*E*, within certain frequency bands with the opposite occurring in neighboring bands, depending on sensor height above the seabed and seabed reflective properties.

_{k}This condition, associated with an overpassing CPA, is to be distinguished from the more commonly observed condition where the receiver is sufficiently distant from an overhead CPA, defined here as horizontal range exceeding about two water depths after which waveguide effects, such as mode interference, dominate. In this latter case, an energy imbalance can also be observed for narrow band observations at ranges corresponding to modal interference features, such as dislocations, with typically kinetic exceeding potential energy, although elements of both energy quantities, i.e., pressure and velocity, respond to the interference causing each to reach low level. Note that the energy ratio has also been used as a quality diagnostic to identify nonacoustic effects, such as flow noise,^{21} an effect that is always important to assess but does not influence these observations. To clarify further, conditions associated with a near field, e.g., as delineated by some fraction of an acoustic wavelength from the vessel's primary acoustic emission location such as the propulsion mechanism, still apply but are not relevant to this study.

An analytic model (Sec. II) was developed to describe the potential and kinetic energy observed directly below a passing vessel, with a limitation that the modeled kinetic energy is the vertical component, $Ekz$, only. Model results are presented in the form of energy ratio expressed in decibels, $10\u2009\u2009\u2009log10Ekz/Ep$, as a function of frequency. The transformation to decibels generates an approximately zero-mean, stationary series with informative amplitude variation about the mean value. The model does not require ship spectral source level information given that an energy ratio is used. In the data and model representations, two key frequency scales are evident in the ratio variation over frequency: $\Delta f0$, set by two-way travel time between the sensor and seabed, and $\Delta f1$, set by two-way travel time to the first dominant, subsurface reflective interface.

The modeling has, as its core, the method of images and, thus, shares some similarities with passive fathometry techniques,^{22,23} including an emphasis on noise sources directly over a receiving system. The modeling involves an expression for plane wave reflection coefficient, evaluated at normal incidence with the data similarly restricted to this regime, that is well separated from the critical angle. Although the current model has limitations, future improvements can include sound speed gradients or additional layering options; another avenue of improvement would be to include propagation angles away from normal incidence, generating more sensitivity to sediment attenuation and shear wave effects.

For a sensor located higher in the water column (further off the seabed), the model predicts increasing oscillation with frequency (decreasing $\Delta f0$), thereby reducing, on average, differences between the two energy quantities. For the idealized problem of an isolated source in vertical alignment with a sensor, the relation necessarily changes for frequencies measured by a sensor located within about one-third of an acoustic wavelength of the sea surface.^{1} However, the increasing complexity for the true problem of a sensor in close proximity to a ship's keel and propulsion features, with unknown and variable length scales, is beyond the scope of this study. Of the two cases studied, R/V *Jack Robertson* represents a water-depth to ship-length ratio of ∼2.3, and R/V *Armstrong* represents a water-depth to ship-length ratio of ∼1; we anticipate that for this ratio $\u226a1$, there is added complexity that is not accommodated in the model.

The observations were representative of two different seabed impedances: (1) Puget Sound, colocated within the vicinity of the Kingston arch, a region of uplifting, resulting in relatively shallow Holocene sediments (∼4 m) separating a harder basement substrate, and (2) the NEMP defined by a low-speed muddy sediment layer of thickness ∼10 m. Both sets of observations reveal approximately the same $\Delta f0$ scale (given the similar sensor height above the seabed) with differing $\Delta f1$ scales and with Puget Sound exhibiting a larger energy ratio as a result of the influence of a basement substrate. The observations in each case represented an effective frequency resolution of about 3 Hz.

The results from a second deployment of IVAR-2 in the NEMP were used to demonstrate the ratio of potential to kinetic energy for a ship closing toward CPA then opening with horizontal range scaled by water depth (Fig. 4). For ship position being approximately directly overhead the sensor, the phenomenon of potential energy significantly exceeding kinetic energy is made clear with a linear view of the energy ratio.

For the Puget Sound data, inversion based on the ratio $Ekz(f)/Ep(f)$ expressed in dB yielded a credible description of the seabed, consistent with available geologic information (Puget Sound, Table I), with a model based on this seabed reproducing the primary features of the observations (Fig. 5). With data collected on the NEMP, a forward model informed by published geoacoustic studies of the area reproduces the primary features of the observations (NEMP, Table II and comparisons, Fig. 6).

Finally, the model results suggest that the pattern observed in the energy ratio continues beyond the frequency range studied here with variations, such as $\Delta f1$, more likely damped out owing to sediment attenuation, leaving only the scale $\Delta f0$. We hazard a prediction that the pattern indeed continues toward higher frequencies, up to a frequency range where the vessel emission spectrum is no longer sufficiently above ambient level. However, reliable confirmation of such an effect requires a smaller sensor with characteristic sensor scale,^{24} *L*, such that $L<\lambda /3$, where *λ* is the wavelength of the highest frequency studied.

## ACKNOWLEDGMENTS

This work was supported by the Office of Naval Research (USA).