In this paper, a model for the bistatic reverberation associated with seafloor scattering of sound from a moving, narrowband source in an ocean waveguide is developed. Studies of the Doppler effect for moving sources in waveguides have typically focused on the forward propagating field where the Doppler shift leads to a splitting or broadening of the received spectrum. In contrast, the contributions to the scattered field come from all directions and as a consequence the spectrum of the received energy is spread across the entire range of Doppler-shifted frequencies possible for the speed of the source. The model developed here uses rays for the incident field, ray-mode analogies for the scattering, and normal modes to propagate the scattered field to the receiver. Results from this model are compared with data collected using a towed source during the Target and Reverberation Experiment 2013. The possible applications of this Doppler reverberation for seafloor characterization are also considered.

## I. INTRODUCTION

The detection of targets using active sonar in the shallow water environment is often reverberation limited. One technique used to isolate the target from the reverberation is to exploit the Doppler shift of the target return. This assumes that the target is moving relative to the scatterers in the environment. This assumption can break down in the presence of a rough sea surface where spectral spreading of the reverberation from the surface can be induced by both the wave motion itself (Brown and Frisk, 1974; Lynch and D'Spain, 2012; Wild and Joyce, 1995) and by the motion of entrained bubbles (McCammon and McDaniel, 1990). This can increase the relative importance of sea surface scattering in sonar performance prediction in shallow water environments where seafloor scattering is typically greater than scattering from the sea surface (Thorsos *et al.*, 2011). When the sonar itself is also moving, the Doppler of the transmitted signal will be present in the seafloor scattering contribution to the reverberation. If the source has a narrow beampattern, the Doppler shifts across the beam have been observed to lead to beating in the time series response of the sonar (Hurdle *et al.*, 1964). For a source with a broad beampattern, the Doppler shifts of the outgoing sound will broaden the frequency response of the seafloor reverberation. This spread in the Doppler spectrum was observed in a set of bistatic reverberation measurements collected in the Gulf of Mexico in 2013. This paper discusses those measurements and develops a model of the Doppler spectrum of the seafloor reverberation.

The measurements were made as part of the Target and Reverberation Experiment conducted in the Spring of 2013 (TREX13). The primary goal of the experiment was to simultaneously measure mid-frequency (2–10 kHz) reverberation and transmission loss with extensive environmental characterization (Yang *et al.*, 2018). As part of the effort to measure transmission loss at the site, two vertical line arrays of receivers were deployed and a source transmitting 15-s long tonals was towed at mid-water depth past each array. Details of these measurements are given in Sec. II. While the main peak in the received energy occurred at a Doppler-shifted frequency consistent with the velocity of the source relative to the array, *f*_{0} + Δ*f*, where *f*_{0} is the transmitted frequency and Δ*f* is the Doppler shift, there is an appreciable amount of energy distributed in the band between *f*_{0} – Δ*f* and *f*_{0} + Δ*f*.

This spectral response is due to sound leaving the source in directions away from the direct source-to-receiver path and being scattered back to the array. While the direct path propagation will have a Doppler-shifted frequency corresponding to the relative velocity between the source and the array, the reverberation will occur at Doppler-shifted frequencies corresponding to the velocity of the source relative to the scattering location in the environment. The sound scattered from this location to the array will then have the same Doppler-shifted frequency as the incident sound regardless of the direction of the array from the scatterer. For the nearly omnidirectional source used in the experiment, the reverberation will therefore cover the band of Doppler-shifted frequencies defined by *f*_{0} ± Δ*f*. A similar effect is observed in airborne, pulse Doppler radar where the side lobe returns can be spread across the Doppler space (Skolnik, 2001). This issue has obviously received more attention in the radar community since aircraft speeds are more than an order of magnitude faster than ship speeds and the spread in Doppler is much more significant as a result.

The modeling approach used here is based on the reverberation models developed by Bucker and Morris (1968) and Ellis (1995) and is illustrated in Fig. 1. The incident field is modeled using a ray approximation and takes into account the Doppler shift of the field. The scattering is modeled using ray-mode analogies and the propagation of the scattered field is described using normal modes. The model is developed in Sec. III and comparisons of the model to data are given in Sec. IV. Section V discusses the implications of the Doppler broadening of the seafloor reverberation for possible applications of the effect for environmental characterization and sonar performance prediction.

## II. MEASUREMENTS AT TREX13

The Target and Reverberation Experiment was conducted in the Spring of 2013 (TREX13), 2 km off the coast of Shell Island near Panama City Beach, Florida. The primary goal of the experiment was to simultaneously measure mid-frequency (2–10 kHz) reverberation and transmission loss with extensive environmental characterization (Yang *et al.*, 2018). Reverberation was primarily measured using a bottom-mounted source and horizontal line array which were deployed from the moored R/V Hugh R. Sharp at the center of the experiment site. In order to limit the area within which environmental characterization was needed, the reverberation analysis focused on an area along a bearing of 129° which was designated as the “main reverberation track.” The location of the R/V Sharp and the bathymetry measured along the main reverberation track are shown in Fig. 2.

To measured transmission loss, two vertical line arrays (VLAs) were deployed along the main reverberation track at ranges of 2.4 and 4.2 km from the reverberation source (see Fig. 2). Each array consisted of 32 elements equally spaced at increments of 0.2 m for a total aperture of 6.2 m. The lowest element was located 5 m above the seafloor. The VLAs were autonomous, self-recording systems with a sampling rate of 25 kHz for each element. In order to measure acoustic propagation as a function of range along the main reverberation track, a mid-frequency source was deployed from the R/V Walton Smith and suspended in the middle of the water column. The source was towed along the path shown in Fig. 2. The source was an ITC-2015 which transmitted a tonal comb consisting of 15 s-long continuous wave (CW) signals from 1503 to 4003 Hz in 500 Hz increments. The analysis and modeling presented below will focus on the *f*_{0} = 3003 Hz tone which was transmitted with a source level of 173.2 dB re *μ*Pa @ 1 m. During the tow, the signal was transmitted with a 50% duty cycle, i.e., 30 s between transmissions.

The data recorded on each element during the tow was partitioned into 30 s long segments which captured the received tones from each transmission. This 30 s long time series of pressure was Fourier transformed and the absolute value squared was taken. The mean was taken across the 32 elements of the array and divided by the transmission time of 15 s to obtain the power spectral density (PSD). The PSD received on the VLAs during the out-going tow (away from the R/V Sharp) as a function of distance from the closet point of approach (CPA) for each array is shown in Fig. 3.

The average speed of the R/V Smith, $\u27e8|v|\u27e9$, during the source tow was 2.0 m/s which produces a direct path, Doppler-shifted frequency of $f0(1+\u27e8|v|\u27e9/cw)=3007\u2009Hz$ when the source is moving towards the receiver and $f0(1\u2212\u27e8|v|\u27e9/cw)=2999\u2009Hz$ when the source is moving away, where *c _{w}* is the water sound speed. The frequency of the peak in the power spectral density in Fig. 3 is consistent with these extremes when the source is far from the receiver and transitions through the tonal frequency as the source passes through the CPA. While the direct path from a moving source in a waveguide should broaden due to the Doppler shifts of the individual propagating modes, the broadening of the direct arrival peaks in Figs. 3 and 4 is wider than expected given the frequency and speed of the source. This additional broadening could be due to small accelerations of the source during the tow. Variations in the speed of the source on the order of 0.2 m/s is enough to account for the broadening of the direct arrival observed in Fig. 4. Another source of the broadening could be additional Doppler shifts imposed by the motion of the sea surface during the tow. The root mean square (RMS) height of the sea surface varied over 0.10–0.15 m during the towed and stationary transmissions.

In addition to this direct path energy, throughout the tow there is also energy arriving at all frequencies within the band defined by these two Doppler frequencies (2999 and 3007 Hz). This is seen more clearly in Fig. 4 which shows the PSD measured on VLA #1 when the source is 2.1 km from the CPA and moving with a velocity of −2.1 m/s. At this range and velocity the direct arrival has a Doppler-shifted frequency of 2998 Hz and spectral level of 112 dB. The energy spread across the 2998−3008 Hz band is clearly seen and while it is roughly 30 dB below the direct path level, it is still well above the noise floor which is at 67 dB. As explained in the Sec. I and in more detail in Sec. III, this broad spectral response across the entire Doppler band is due to sound leaving the source in directions away from the direct source-to-receiver path and being scattered back to the array.

## III. DOPPLER REVERBERATION MODELING

In this section, we will develop a model for the bottom reverberation produced by a CW source moving through an ocean waveguide. Prior to developing this model, it is useful to conceptually distinguish the approach used to model the reverberation in the frequency domain from the approach typically used to model reverberation in the time domain. Figure 5(a) shows an idealized representation of the bistatic reverberation measured as a function of time for a broadband pulse transmitted from a stationary source. The time series of the intensity peaks with the direct arrival which is then followed by a decay in intensity as a function of time (Ellis, 1993). At any given time, *t*_{0}, the field measured at the receiver is due to scattering from an elliptical patch of the seafloor for which the propagation path from the source to the seafloor, *r*_{1}, and from the seafloor to the receiver, *r*_{2}, satisfies *t*_{0} = (*r*_{1} + *r*_{2})/*c _{w}*. An example of this elliptical scattering patch is shown in Fig. 5(b).

The reverberation measured as a function of frequency for a moving, narrowband source is shown in Fig. 5(c). In this example the source is moving away from the receiver and the large spectral level corresponds to the direct arrival. Sound leaving the source in directions other than those corresponding to the direct path will experience different Doppler-shifts and scatter from the seafloor. This scattered energy can then propagate back to the receiver and contribute to the spectral levels across the Doppler band. The Doppler-shifted frequency, *f*, is associated with sound leaving the source at some angle *θ _{x}*, the angle between the out-going acoustic wave vector and the velocity vector of the source, which will be assumed to be aligned with the

*x*axis, and expressed as

where *v _{x}* is the velocity in the direction of motion of the source. The surface of constant frequency forms a cone about the velocity vector and the intersection of this cone with the seafloor defines the scattering patch which is a hyperbola on the seafloor. In the case of a source and receiver above a half-space, a single hyperbolic scattering patch would exist for each frequency. In a waveguide, the sound will reflect multiple times from the seafloor and sea surface and as a result multiple scattering patches will exist for a given Doppler-shift as shown in Fig. 5(d).

Since the source will initially be assumed to be omnidirectional, the incident grazing angles on the seafloor will range from 0° to 90°. In order to account for the sound at very steep grazing angles, the incident field will be modeled using a ray approximation. Also, since we are interested only in the intensity of the reverberation, spherical wave effects at the bottom reflection will be neglected and a plane wave approximation will be used. For the scattered field, we will use a single scattering approximation; the sound can scatter into a range of angles which will then propagate without further scattering down the waveguide. To account for the angular spread in scattered energy, the propagating field will be modeled using normal modes. The scattering into the modes will be determined using ray-mode analogies and the scattered field is assumed to be dominated by sound traveling at grazing angles below the critical angle. Hence only the trapped modes will be used.

The source is assumed to be omnidirectional, moving at height, *H*, above the seafloor with constant velocity in the x-direction, and transmitting a continuous tone of frequency, *f*_{0}. The ocean waveguide is treated as isovelocity with sound speed, *c _{w}*, and is range-independent with water depth,

*D*. The seafloor sediment is characterized by the sediment/water density ratio,

*ρ*, the sediment sound speed,

*c*, and the ratio of the imaginary to real wave number in the sediment,

_{s}*δ*. The sea surface is assumed to be flat, while the seafloor has small-scale roughness at its interface and heterogeneities in its volume, both of which can contribute to the bistatic scattering cross-section,

where *σ _{r}* and

*σ*are the roughness and volume scattering cross sections per unit area, respectively. The incident grazing angle,

_{v}*θ*, the scattered grazing angle,

_{i}*θ*, and the bistatic angle,

_{s}*ϕ*, follow the convention used in Jackson and Richardson (2006, Fig. 2.4).

_{s}As discussed above, the cones of constant frequency defined in Eq. (1) intersect the seafloor to form hyperbolas described by

as shown in Fig. 6. While these hyperbolas are defined for the direct path from the source to the seafloor, sound which reflects from the sea surface prior to reaching the seafloor can be modeled using an image source at height, *H*_{1} = (*D* − *H*) + *D*, above the seafloor. After multiple reflections from the seafloor, the incident sound can be modeled using an image source at height,

where *n* = 0 for sound that reflects first with the seafloor after leaving the source, *n* = 1 for sound that reflects first from the sea surface, and *m* = 0, 1, 2,… is the number of reflections from the seafloor. For these image sources, the associated hyperbolas for the sound incident on the seafloor become

Assuming that the source is transmitting at intensity, *I*_{0}, and using a plane-wave approximation, the intensity of the sound at the seafloor is

where *V _{nm}* is the plane-wave reflection coefficient,

the incident grazing angle is given by

and the distance from the real or image source to a point along the hyperbola on the seafloor is

To use the cross-section given by Eq. (2) to determine the scattering, a narrow band of frequencies, Δ*f*, at the Doppler-shifted frequency will be used to determine the size of the scattering patch on the seafloor. For this band of frequencies, the change in Doppler can be written as

Assuming that the bandwidth is small and Δ*θ*$\u226a$*θ _{x}*, the range of angles subtended by this band can be approximated as

The width of the seafloor scattering patch corresponding to this range of angles can be found from Eq. (5) and can be written using the approximation in Eq. (12) as

The scattering strength at the patch can be calculated for a given scattering angle using the expression for the incident grazing angle given by Eq. (9) and the bistatic angle given by

The scattered field will be modeled using normal modes where the field due to a unit point source can be approximated as

where Φ_{j}(*z*) are the mode functions, *k _{j}* are the mode wavenumber,

*z*

_{0}is the depth of the source, and

*r*is the horizontal range from the source. To calculate the scattered field, we will follow the approach used by Ellis who used a ray-mode analogy to determine the amplitudes of the scattered modes (Ellis, 1995). In that approach the modes were written as a sum of upward and downward propagating waves,

where *A _{j}*(

*z*) is the amplitude of the mode function and

*ψ*(

_{j}*z*) is the vertical phase function. For the general case of a horizontal stratified waveguide, the mode amplitude is depth dependent and the phase function can be expressed using the Wentzel−Kramers−Brillouin (WKB) approximation (Jensen

*et al.*, 2011). For the isovelocity case considered here, the mode amplitude is depth independent and the vertical phase function becomes

*ψ*(

_{j}*z*) =

*k*where $kjz=k2\u2212kj2$. In this case the mode amplitudes can be found from Eq. (16) and its derivative at the seafloor,

_{jz}zThe scattered intensity from a point, *r*_{0}, on the seafloor propagating in mode *j* can now be approximated as

where *α _{j}* is the imaginary part of

*k*and $r\u2032=|r\u2212r0|$.

_{j}Using the incident field given by Eq. (6), the scattering cross-section, and the incoherent sum of the scattered modes, the frequency-dependent reverberation can be expressed as

where the integration is along the hyperbolas given by Eq. (5) and the width of the scattering patch is given by Eq. (13). Note that Δ*x _{nm}*,

*I*, and

_{nm}*σ*are functions of the Doppler frequency and as a result, the angle

_{nmj}*θ*. The modes are also a function of the Doppler frequency, but for the source speeds used in TREX13, a good approximation is to use the wavenumbers and mode functions at the source frequency,

_{x}*f*

_{0}, for all frequencies across the Doppler spectrum.

While the model assumes an omnidirectional source, it is straightforward to include a source with an arbitrary beam pattern by modifying the expression for the incident intensity given by Eq. (6). Assuming the directivity of the intensity, *B*(*θ*, *ϕ*), is known and the grazing angle, *θ*, is positive for rays incident on the seafloor, the intensity can be rewritten as

where the azimuthal angle is given by Eq. (14).

## IV. DATA/MODEL COMPARISONS

To apply the model developed in Sec. III, we will use the TREX13 environmental parameters shown in Table I. A majority of these parameters were derived from measurements conducted at the TREX13 site. The mean water depth was found from the bathymetry survey conducted by DeMoustier and Kraft (De Moustier, 2014) and which is shown in Fig. 2. The water sound speed and density were determined from CTD casts along the main reverberation track. The sediment sound speed was measured by both time-of-flight (Yang and Tang, 2017) and by reflection measurements (Dall'Osto *et al.*, 2016; Holland *et al.*, 2017) and the mean value across these measurements has been used. Attenuation was also measured using these methods but, as is often the case for mid-frequency seafloor environmental characterization, the measured values have a significant amount of uncertainty. The measurements of attenuation at the site vary from 0.2 to 1.7 dB/*λ* and the value used in this paper, 0.5 dB/*λ*, falls within this range and is typical for sand sediments. The sediment density value is also typical of sand sediments and falls within the range of measurements made on diver cores collected at the site.

Parameter . | Value . |
---|---|

Mean water depth (H) | 19.6 m |

Water sound speed (c) _{w} | 1530 m/s |

Water density (ρ) _{w} | 1000 kg/m^{3} |

Sediment sound speed (c) _{s} | 1670 m/s |

Sediment density (ρ) _{s} | 2000 kg/m^{3} |

Sediment attenuation (β) | 0.5 dB/λ |

Roughness spectral strength (w_{2}) | $0.0015\u2009m4\u2212\gamma 2$ |

Roughness spectral exponent (γ_{2}) | 4.23 |

Roughness characteristic length (L) | 0.2 m |

Lambert scattering strength (μ) | −31 dB |

Parameter . | Value . |
---|---|

Mean water depth (H) | 19.6 m |

Water sound speed (c) _{w} | 1530 m/s |

Water density (ρ) _{w} | 1000 kg/m^{3} |

Sediment sound speed (c) _{s} | 1670 m/s |

Sediment density (ρ) _{s} | 2000 kg/m^{3} |

Sediment attenuation (β) | 0.5 dB/λ |

Roughness spectral strength (w_{2}) | $0.0015\u2009m4\u2212\gamma 2$ |

Roughness spectral exponent (γ_{2}) | 4.23 |

Roughness characteristic length (L) | 0.2 m |

Lambert scattering strength (μ) | −31 dB |

For the scattering cross-section, we will use only the roughness scattering cross-section. Measurements of the volume scattering indicate that there may be a homogenous sand layer over a sub-bottom layer within which a majority of the volume heterogeneities are confined. As a consequence the scattering from the volume is weak. For the roughness scattering cross sections, two scattering models will be considered here. This first is described by the well-known Lambert's law,

where *μ* is the Lambert scattering strength. Note that this expression is typically used to model backscatter and has no dependence on the bistatic angle. Refined versions of Lambert's law have been developed in the past to improve the performance for bistatic scattering, particularly in the specular direction (Ellis and Crowe, 1991). However, since the contribution of the field scattered into the specular direction is obscured by the propagating field, we will use the original version given in Eq. (21). This bottom scattering model has been applied to the stationary, monostatic reverberation data collected at TREX13, and a value for the scattering strength of –31 dB was used to match the data (Ellis *et al.*, 2017). This value will also be used here.

The second roughness scattering model used here is first-order small-roughness perturbation theory. In addition to the sediment parameters, this model requires information about the roughness power spectrum of the seafloor. Direct measurements of the rough surface found that the spectrum could be described using a von Karman spectrum,

where the spectral strength, *w*_{2}, and spectral exponent, *γ*_{2}, were found from a fit to the mean spectrum determined from measurements at multiple locations around the TREX13 site (Hefner, 2017). The characteristic length, *L*, was not directly measured and was arbitrarily chosen to prevent the large scale roughness from becoming too large. The impact of this parameter on the model results will be discussed later.

The ITC-2015 used in the measurements described in Sec. II is a free flooded ring transducer which has a toriodal beam pattern. While the beam pattern for the particular transducer used in TREX13 was not measured at 3003 Hz, a different ITC-2015 was characterized prior to the 2006 Shallow Water experiment (Yang *et al.*, 2012). The beam pattern was axisymmetric about the *z*-axis, with a maximum value (0 dB) around the circumference of the transducer in the *x*-*y* plane and dropping at the *z*-axis to –9 dB at the top and –18 dB at the bottom. For the following data/model comparisons, we assume that the beam patterns of the two transducers are similar and use the measured beam pattern in Eq. (20).

The output of the model for both scattering cross-sections is compared to the measured PSD in Fig. 7. For this example, the integral in Eq. (19) was evaluated as a summation over *y* from –1000 to 1000 m with Δ*y* = 1 m and used a maximum value of *m* = 10 for the number of reflections of each ray from the seafloor. Since the model treats the scattered field as incoherent, the model output is independent of the receiver position within the array. In order to reduce computation time, the model was evaluated at a single depth, *z* = 10 m. Note also that the model does not account for the direct propagation from the source to the receiver which produces the large peak in the PSD at 2999 Hz.

The two scattering strength models give very similar results, both of which compare well to the measurement. The small-roughness perturbation theory result gives a slightly higher value than Lambert's law over most of the frequency range with the greatest difference occurring at those frequencies which correspond to the direct path peak. The scattering in this region is near specular and hence the perturbation result is a function of the low-wavenumber part of the roughness spectrum. This part of the spectrum is controlled by the choice of characteristic length. The reverberation level below *f*_{0} can be increased by increasing the characteristic length and vice versa. While the choice of characteristic length used here was somewhat arbitrary, decreasing it beyond the value of 0.2 m will begin to affect the spectral fit in the measured band of wavenumbers (14.41–33.63 rad/m). Increasing it above this value will begin to significantly overestimate the spectral strength measured by Holland *et al.* at these low wavenumbers (Holland *et al.*, 2017). They used the measured bathymetry to estimate the spectral levels at the low wavenumbers and found the spectral levels are much lower than would be predicted if the characteristic length were allowed to become large.

At high frequencies, both model results predict a peak in the reverberation spectrum as the maximum Doppler frequency is approached. To understand the origin of this peak, consider the sound incident on the seafloor along the hyberbola corresponding to zero Doppler shift (*f* = *f*_{0} or $cos\u2009\theta x=0$). Along this hyperbola, the incident sound is above the critical angle, and hence suffering significant bottom loss until *θ _{nm}* =

*θ*, where

_{crit}*θ*is the critical angle. As the amount of Doppler shift increases, the portion of the hyperbola for which the incident sound is above the critical angle decreases. When the Doppler shift reaches the point where

_{crit}*θ*=

_{x}*θ*, all the incident sound is below the critical angle and the bottom loss is greatly reduced. Just after this point, the reverberation peaks and then decreases as the scattering strength falls off as $\u2009sin2\theta $. This is seen clearly in Fig. 8 where the contributions of each of the multiply reflected rays are shown relative to the frequencies at which all rays become subcritical.

_{crit}Figure 9 shows a comparison of the model output about Δ*f* = 0 Hz to the spectral levels measured on both VLAs along each leg of the tow. Again both scattering strength expressions give similar results and compare well to the measurements when the source is greater than 500 m from the CPA. Near the CPA, the PSD at Δ*f* = 0 Hz is dominated by the direct propagation which the model does not take into account. These range dependent model results use the local velocity of the ship which can vary along the legs causing slight differences in the reverberation levels. There are, however, no significant variations in either the model or the measurements and the range dependence is very similar with no dependence on the direction of the tow or distance from the VLA. This is consistent with the narrowband, monostatic reverberation measurements made using a fixed source and horizontal line array at the TREX13 site. In those measurements, while it was possible to resolve range-dependent fluctuations in the broadband reverberation, the narrowband pulses ensonified larger portions of the seafloor and the reverberation level was a function of the mean scattering strength within those scattering patches. For the Doppler reverberation, the scattering patch size is determined by the frequency resolution, Δ*f*, and the length and number of the ensonified hyperbolas [see Fig. 5(d)]. The smallest ensonified scattering patch therefore corresponds to Δ*f* = 0 Hz for which the scattering patch is essentially a strip of seafloor directly beneath the source and extending out perpendicular to the direction of travel. Even in this case, however, evaluation of the integral in Eq. (19) as a function of the limits of the integral finds that the length of the patch needs to be at least 1.6 km to provide an accurate result. As a result, this scattering patch averages out the fluctuations in the scattering strength detected in the narrowband reverberation yielding the smooth range dependence seen in Fig. 9.

## V. DISCUSSION

While the TREX13 measurements discussed above were unable to detect any variation in the seafloor scattering, it may be possible to design a sonar that could exploit the Doppler reverberation for seabed characterization. Suppose that instead of an omnidirectional or toroidal source, one used a long thin source similar to side-scan and multibeam sources. Those sources are designed to have a very broad beam pattern in one direction with a narrow beam pattern in the perpendicular direction. In multibeam and side scan applications, the wide part of the beam is oriented such that it ensonifies a large swath of the seafloor underneath the tow body or vessel that is perpendicular to the direction of travel. If instead the source is rotated such that the wide portion of the beam is parallel to the direction of travel, only the center of the hyperbolas in Fig. 5(d) will be illuminated, greatly narrowing the portion of the seafloor from which the scattering is being measured. This is analogous to the Mill's cross configuration used in many multibeam sonars where the fan beams of a source and receiver are aligned perpendicular so that pencil beams can be formed. Here the frequency resolution provided by the Doppler reverberation replaces the beam pattern of the receiver.

In TREX13, the Doppler reverberation utilized a towed source and was measured on two, self-recording arrays of receivers deployed along the track. Compared to the co-located source and receiver required to measure direct-path scattering or monostatic, long-range reverberation, the sonar system considered above requires more equipment and presents additional logistical challenges to make a similar measurement. However, a direct path scattering measurement is often limited by the timing of multipath arrivals and only a portion of the bistatic scattering cross section can be measured (Holland *et al.*, 2000). A monostatic reverberation measurement avoids this issue by measuring the reverberation over a longer time and incorporating the multipath into the model fit. The difficulty with this measurement is that the model fit requires that the transmission loss is also modeled well since uncertainties in the transmission loss can produce uncertainties in the scattering cross-section estimate. This method also only samples the subcritical grazing angles in the scattering cross section since the sound propagating at higher angles is lost more rapidly to the seafloor.

The model fit to the Doppler reverberation measurement also requires modeling the transmission loss, however the bistatic source and receiver configuration provides the opportunity to make a simultaneous, independent measurement of the transmission loss. The towed source additionally makes this loss measurement a function of range. The Doppler reverberation measurement also provides a broader range of angles for the scattering strength, sampling angles both above and below the critical angle. It may also be possible to co-locate the source and receiver on the same moving platform. This would introduce an additional Doppler shift due to the moving receiver and also eliminate the transmission loss measurement, but it would reduce the equipment needed and allow for wide-area assessments of the seafloor scattering.

The model developed here considered only the Doppler spectrum but to fully consider this reverberation in the context of sonar performance prediction, the time component needs to be considered as well. This will require a combination of both scattering diagrams shown in Figs. 5(b) and 5(d) and the model developed in Sec. III can serve as a starting point for a time-frequency model. This would also make it possible to explore broadband transmissions from moving platforms and how the Doppler reverberation impacts sonar performance for different waveforms. The current model should also be expanded to consider additional environmental effects such as arbitrary sound speed profiles, sea surface roughness, and bathymetric variations.

## VI. SUMMARY

A model for the frequency dependence of a narrowband signal transmitted from a moving source has been developed using a combination of rays, normal modes, and ray-mode analogies. Using environmental parameters consistent with values from environmental measurements conducted during TREX13, the model results were compared to towed source data also collected during that experiment. The model results compared favorably with the data both in frequency and range along the measurement track. The model used both Lambert's law and small roughness perturbation theory and although both provided similar results, perturbation theory used direct measurements of the seafloor properties while Lambert's law used an input parameter determined from a fit to previously analyzed data. The Doppler reverberation could potentially form the basis of a seafloor characterization technique if a source with limited directionality were used. The Doppler reverberation due to seafloor scattering may have an impact on sonar performance prediction particularly with regards to the use of Doppler to discriminate targets from reverberation and clutter.

## ACKNOWLEDGMENTS

This research was supported by the U.S. Office of Naval Research under Grant Nos. N00014-16-1-2999 and N00014-12-1-0513.