Coherent reflection from surface gravity water waves during reciprocal acoustic transmissions

Reflection and scattering of sound from the sea surface have long been studied. Significant experimental and modeling efforts have been focused on statistical representations of acoustic field fluctuations resulting from surface interactions.^1–3 Ocean surface waves often consist of dominant frequencies and directions, which influence the mechanisms of acoustic scattering and reflections. Arrival time and intensity structure of the acoustic field over time are two properties affected.⁴ Recent attention has been focused on deterministic descriptions of the coherent surface returns.^5,6 In some cases, the passage of the wave crests can result in multiple focused micro-paths when the gravity water wave crests propagate overhead. These focused micro-paths may have strong intensity, sometimes even stronger than the direct return. Under some conditions, acoustic focusing and scattering from a dynamic surface can lead to caustic ray behaviors.^5,6 The acoustic focusing events also depend on source receiver geometries, as reported in a very shallow, well-mixed water column.⁵ The resulting time-varying surface arrival structures have implications in applications such as high frequency acoustic communication.⁵ 

The field data presented here demonstrated acoustic focusing resulting from moving surface wave crests during the Kauai acoustic communication experiment during the summer of 2011, referred to as the KAM11 experiment. Two sound stations, each with transmission and reception capabilities, were deployed on the seafloor for reciprocal acoustic transmissions at a center frequency of 10 kHz. Monitoring hydrophones were tethered in the upper water column as additional receivers.

The present study is unique from previous work^5,6 in several aspects. First, the observations took place in a deeper waveguide with larger source-receiver ranges. The experimental site had a depth of 100 m with a thermocline at around 60 m. In the downward refracting environment, therefore, the surface returns were further modulated by the stratified water column. The source-receiver ranges were 500 and 1000 m, corresponding to five and ten times the surface wave lengths. By contrast, the surf zone experiment⁵ was conducted in 6 m deep water. Its source-receiver range was 40 m. Second, the experiment had reciprocal transmissions and receptions from two sound stations deployed on the seafloor as well as monitoring hydrophones in the upper water column. Such an experimental setting allowed examination of (1) directionality of the surface effects and (2) combined effects from an evolving surface and the stratified water column. Third, we present full wave modeling results through the use of a two-dimensional (2D) split-step parabolic equation model, which can address scattering and reflection from an evolving surface.

The KAM11 experiment was conducted during the summer of 2011 in a 100 m shallow water region, west of the coast of the Kauai Island, Hawaii.⁷ During the experiment, a pair of acoustic stations was deployed twice for reciprocal transmissions. The two acoustic stations were configured identically, each having an eight-element receiving array and a source mounted on a fixed tripod structure. Acoustic transmissions at two center frequencies, 25 kHz on July 6–7 and 10 kHz on July 10–11, were conducted. In this letter, acoustic signals at 10 kHz around 01:52 on July 11 in coordinated universal time (UTC) were reported and analyzed.

As shown in Fig. 1(a), the two acoustic stations were separated by about 1 km. At each station, the transducer atop the tripod structure was positioned about 5 m above the seafloor. The receiving array had an aperture of 3.5 m, with an element spacing of 0.5 m and its bottom element 1 m above the seafloor. To distinguish between the two acoustic stations, they are referred as to their deployed locations, Sta05 for the left acoustic station and Sta07 for the right one as in Fig. 1(a). A Waverider buoy was deployed at Sta06 midway between the two acoustic stations to collect directional surface wave spectra. A thermistor string was deployed at Sta08, which was about 0.5 km away from Sta07. The research vessel Kilo Moana was dynamically positioned close to the Waverider and stayed in a fixed station during the transmissions. Two monitoring hydrophones, one at 15 m depth and the other at 25 m depth, were tethered from the research vessel as additional receivers. The monitoring hydrophones were positioned slightly east of the Sta05-Sta07 line. Their distance to Sta05 and Sta07 was about 570 m and 460 m, respectively. The top view of the experiment is shown in Fig. 1(b).

At the specific time of interest (around 01:52 UTC on July 11), the water column was well-mixed down to 60 m indicated by the sound speed profile [Fig. 1(c)], obtained from the thermistor string. There existed a sharp sound speed gradient below 60 m, creating a downward-refracting acoustic environment. The sound speed profile had small range dependence, shown from measurements at multiple thermistor string moorings at the site (plots not provided in this letter). The measured surface spectrum is shown in Fig. 1(d). The dominant surface frequencies were concentrated in the 0.1–0.25 Hz range, which is typically associated with gravity wind waves. The dominant wave period was about 8–10 s and the significant wave height was about 1.4 m. As shown in the top view of the experiment [Fig. 1(b)], the dominant wave direction was from the north-east (42°). The angles between the dominant wave direction and the three acoustic tracks were about 27°, 12°, and 42°, for Sta05-Sta07, Sta05-Ship, and Sta07-Ship, respectively. The averaged wind speed was about 6 m/s at 01:52 UTC.

Linear frequency modulated (LFM) signals from the experiment were analyzed. The single LFM chirp swept from 8 to 13 kHz in 48 ms. Multiple LFM chirps were repeatedly transmitted every 144 ms for 30 s. The two sources alternated transmission every two minutes. For two minutes, the first acoustic station transmitted and the second station was in the reception mode. Two minutes later, the second station started transmission and the first station switched to the reception mode. A matched filter was applied to extract impulse responses of the acoustic channels.

Figure 2 shows the impulse responses measured at two acoustic stations deployed on the seafloor (2 m above the seafloor) and at the monitoring hydrophone in the upper water column (25 m below the sea surface). These subplots are arranged so the top panels show the acoustic signals at the upper water column and the bottom panels show the acoustic signals at the lower water column. The source-receiver range for the top panels was about 0.5 km while the range for the bottom panels was about 1 km [see Figs. 1(a) and 1(b)]. The left panels show the acoustic signals emitting from the source at Sta05 and the right panels show the acoustic signals from the source at Sta07. The time difference between the signals emitting from Sta05 and those from Sta07 was two minutes. To distinguish from arrival time, the time at which LFM signals were transmitted is referred to as geotime in Fig. 2.

The measured impulse response exhibited interesting coherent focused surface structures that are related to the surface wave. As shown in Fig. 2(a), the impulse responses between Sta05 and the monitoring hydrophone had strong direct and bottom paths marked as “1” and “2,” respectively. Except for slight arrival time fluctuations, these two paths were rather stable in intensity. They were not dispersive over the arrival time axis. The surface and bottom-surface paths, marked as “3” and “4,” respectively, showed contrasting characteristics. First, these paths were highly fluctuating over the geotime axis and remarkably dispersive over arrival time. The single surface interacting paths started around arrival time 7 ms. There were still observable surface returns around arrival time 20 ms. Further, there existed intense late coherent surface arrivals shown in Fig. 2(a), where some of these returns are marked in white lines. These returns had increasing delay over time.

These coherent returns were the result of surface wave crest focusing. Due to the curvature of the surface wave crest, multiple local reflections from the wave crest reached the receiver. With similar arrival time and phases, these reflections created acoustic focusing at the receiver, thus leading to enhanced intensity. As the wave crest moved away from the specular point, the intense acoustic focusing had increasing delay. The coherent structure disappeared when the receiver was not in the focusing area of the moving wave crest. During the first several seconds in the geotime axis, multiple coherent returns appeared simultaneously and created a striation pattern of enhanced intensity. This may be the result of acoustic focusing from different wave crests.

Details of acoustic focusing are determined by the exact profile of the air-water interface.⁴ The only geometrical difference between Figs. 2(a) and 2(b) was the source-receiver directions, which were opposite. Due to the shape of the wave crests, the acoustic focusing showed clear directionality. In Fig. 2(a), the late coherent surface returns had increasing delay over time. In Fig. 2(b), they had decreasing delay over time.

The received signals close to the seafloor displays a slightly different arrival structure, as shown in Figs. 2(c) and 2(d). The direct and bottom paths overlapped, marked as “1 + 2,” since the receivers were positioned close to the seafloor. It is noted that at a longer range and deeper depth, some coherent surface returns are still observable in Figs. 2(c) and 2(d), though not as clear as the ones shown in Figs. 2(a) and 2(b). Since the surface returns at the bottom mounted receivers penetrated through the thermocline, their intensities were much weaker.

We used a version of the 2D Monterey–Miami parabolic equation (MMPE) model⁸ to simulate the impulse responses during the KAM11 experiment. The MMPE model employs a split-step Fourier marching algorithm for efficient implementation. Rough surface scattering was incorporated through the use of the field transformation technique of Tappert and Nghiem-Phu and was implemented using a wide-angle approximation, described by Smith.⁹ 

To reproduce the surface effects observed here, an evolving linear sea surface model was combined with the MMPE model.¹⁰ An initial surface was constructed from the measured directional surface spectrum.^11,12 The directional surface spectrum was first transformed into a 2D amplitude spectrum. Combining the 2D amplitude spectrum with random phases, the initial 2D water surface was created through a 2D Fourier transform. A fourth-order Runge–Kutta integrator was then used to make the water surface evolve over time. When the surface progressed over time, the snapshots of evolving surface roughness were fed to the MMPE model. The MMPE model calculated the acoustic field for each surface snapshot by treating the ocean and its surface as frozen. As a result of multiple successive runs, the model generated a time-varying acoustic field in the frequency domain. A broadband calculation, or calculations at multiple frequency bins, produced time-varying impulse responses in the time domain. The modeling results are shown in Fig. 3, where the subplots correspond to those of Fig. 2.

Through comparison between Figs. 2 and 3, it is noted that the acoustic measurements and the model results show good agreement in several aspects. First of all, the arrival times between data and modeling results match well. This is true for all four cases. Also, the MMPE model generates similar fluctuating surface features. As shown in Fig. 3(a), the model surface returns fluctuate over time and disperse over arrival time, similar to Fig. 2(a). The striation patterns appear in the model result in Fig. 3(a). In both Figs. 2(a) and 3(a), the focused surface returns can be observed as late as at arrival time 20 ms. Directionality of the striation pattern is reproduced for Figs. 3(a) and 3(b), consistent with the measured data.

For the impulse responses shown at the bottom receivers [Figs. 3(c) and 3(d)], the modeled surface returns are weaker than the modeled direct and bottom returns. This is consistent with the measurements shown in Figs. 2(c) and 2(d). It is important to note that there exist some differences between the measured data and the model results at the bottom mounted receivers. For example, the surface returns in Figs. 2(c) and 3(c) both show the striation feature. The model results in Fig. 3(c) show striations with interfering up and down slopes. Similar differences exist between Figs. 2(d) and 3(d), which would require additional analysis beyond the scope of this paper.

In this letter, data from acoustic reciprocal transmissions during the KAM11 experiment were presented. The passage of moving surface wave crests was shown to generate focused and intense acoustic surface returns at the center frequency of 10 kHz in 100 m depth shallow water. The resulting late coherent surface returns had varying delay over time, which were observed in both the upper and lower water columns. These coherent structures demonstrated strong directionality, depending on the angle between the acoustic track and the direction of surface wave crests. It was also shown that a rough surface 2D parabolic equation model with an evolving sea surface reproduced similar surface features, suggesting that such features contain deterministic information about the evolving rough surface.

This research was supported by the Office of Naval Research (ONR) Code 322OA through Grants Nos. N00014-10-1-0396 and No. N00014-10-1-0345 and a Multidisciplinary University Research Initiative (MURI) (Grant No. N00014-07-1-0739). The authors wish to thank all participants of the KAM11 experiment. Special thanks are due to Dr. William S. Hodgkiss for leading the MURI project and providing monitoring hydrophone data. We also thank Jing Luo, Joseph Senne, and Entin Karjadi for their various help.