Measuring phase difference to sense small-scale ocean sound-speed structure Open Access

We explore the effects of ocean fine structure and internal waves on mid-frequency sound propagation using a series of broadband receptions at a 1.8 km range. We analyze a 30-min segment of acoustic transmissions sampled every 63.5 ms between a source at 200 m depth and a receiver at 130 m depth from a deep water acoustic experiment.¹ The two downward refracted acoustic paths between source and receiver at this range are nearly horizontal and travel between 100 and 200 m depth. The second arrival is a shadow-zone arrival but is treated like a ray arrival. The travel time of the two arrivals is separated by roughly 2 ms, which varies by $±0.25$ ms due to sound-speed variability. The phase difference between the two arrival phases is sensitive to the meter-scale displacement of the sound-speed profile. The displacement of the sound-speed profile by internal waves is measured within 0.5 km of the propagation paths' midpoint. We compare the measured phase difference to the predicted phase difference using the in situ measurements of the sound-speed profile.

Although broadband transmissions with small travel time separation or overlapping arrivals are frequently observed in mid-frequency experiments, the temporal behavior and spatial behavior of these types of arrivals are not well understood.^2,3 While the acoustic observations in our experiment are similar to those of previous experiments, here we have oceanographic measurements at sub-meter vertical resolution located in the transmission path to model the arrival structure in detail. By deterministically comparing the observed phase difference to the forward model near the first turning point, we can explore the response of triplicated ray arrivals to the vertical advection of small-scale structures in the ocean at the time scale of minutes to seconds. Even though the ray paths of these arrivals traverse similar parts of the water column, the travel time for each path is not fully correlated, which causes a changing phase difference between arrivals over the 30-min transmission.

The temporal evolution of the phase difference is relevant for sensing ocean variability at small scales and understanding the statistical properties of mid-frequency interference effects. While not comprehensive, this study demonstrates that meter-scale internal wave variability can be sensed acoustically and that the phase difference between triplicated ray arrivals varies with internal wave motion. Although the arrivals in our study are temporally resolved, in general, triplicated rays have travel time separations that can be smaller than the temporal resolution of transmitted mid-frequency pulses, causing interference effects.³ Since the effects of the multipath interference can be inferred from the phase difference between ray arrivals and respective amplitudes,⁴ studying the time behavior of the phase difference between resolved triplicated ray paths can help us understand the temporal statistics of micro-multipath interference and how it is related to the structure of the sound-speed profile.

On 7 May 2021, maximum-length phase-modulated sequences were transmitted from the R/V Roger Revelle for 30 min. The drifting receiver array recorded those sequences at a nominal range of 1.8 km. Sequences are repeated every 63.5 ms with 2 cycles per digit and a 4 kHz carrier frequency. The effective bandwidth of these sequences is 4 kHz. During the transmission period, a drifting thermistor chain measured the temperature and pressure within 0.5 km of the acoustic propagation's center point. The Global Positioning System (GPS) positions of the source, receiver, and thermistor chain are shown in Fig. 1(a). During the transmission period, the receiver drifted toward the ship, shortening the source-receiver range by approximately 100 m. The transmission range divided by the average sound speed is plotted in Fig. 1(c). The average source depth was 200 m, and the average receiver depth was 130 m. The source and receiver depths vary by about $±4$ m and $±1.5$ m, respectively, due to surface-wave heave on the ship and receiver buoy.

The recorded sequences are demodulated and matched-filtered⁵ using a replica sequence to produce amplitude, travel time, phase, and Doppler frequency shifts. The matched-filtered output $c(τ)$ for each time delay, $τ$⁠, is complex-valued. The amplitude of each arrival is evaluated as the magnitude, $C=|c(τ)|$⁠, and the instantaneous phase, $ϕ=tan−1[Im{c(τ)}/Re{c(τ)}]$⁠, is the angle between the real and imaginary parts. Figure 1(b) shows the real part, amplitude, and phase of the matched-filtered output for a single reception. The matched-filtered amplitude is maximal at time delays (arrival times) of the channel impulse response. The peak amplitude of these arrivals is calculated using a peak-finding algorithm that evaluates the derivative of the matched-filtered output above a 15 dB threshold and looks for zero points of the derivative. After the maximum point is found, a parabola is fit to the four points surrounding the maximum to interpolate to the maximum matched-filtered amplitude between sampled points. The peak travel time, phase, and amplitude are evaluated at this interpolation point.

There are 28 320 matched-filtered sequences in our 30-min-long transmission. Four peaks appear in the matched-filtered output [see Fig. 1(b)]. The first two arrival peaks are from downward refracted paths, while the third and fourth arrival peaks are from surface reflected paths.¹ The travel times of the first and second arrivals are shown in Fig. 1(c). The drifting motion of the receiver dominates the travel time signal of the first two arrivals. The source-receiver range carries some uncertainty because the position of the source and receiver is not the same as the GPS position of the buoy or ship. However, the changes in travel time of the arrivals are similar, suggesting they may be similarly sensitive to the change in transmission range. Thus, the difference between the two travel times constrains some of the range-related variability.

We found that the difference in instantaneous phase between arrivals is a better measurement than the difference in travel time for this experiment.⁶ The phase provides information about the travel time, $τ$⁠, because they are linearly related, $ϕ=2πfcτ$⁠, by the carrier frequency, $fc=4$ kHz. The phase has a $2π$ cycle ambiguity so the travel time cannot be estimated directly from the phase, but the time change in travel time can be estimated with the unwrapped phase. The unwrapped phase difference [see Fig. 1(c)], amplifies the signal related to the change in sound speed while dampening the signal related to the change in source-receiver range. The observed phase difference includes minute-scale variability and smaller amplitude second-scale variability. The two arrivals differ in phase by $±2π$ over 30 min—a 0.5 ms excursion in travel time difference.

The sound-speed profiles from 0 to 200 m were estimated from the thermistor-chain temperature and pressure measurements during the transmission. Sound-speed profiles [see example in Fig. 2(a)], are sampled every minute and are estimated from a collection of thermistor-chain temperature and pressure measurements sampling at 2 Hz.¹ The thermistors are vertically spaced roughly every 2 m between 0 and 200 m depth. The vertical resolution of the thermistor chain is increased to a few centimeters by using the thermistor chain's vertical motion as a synthetic aperture.¹ The salinity is estimated from the thermistor temperature measurements using the relationship between temperature and salinity derived from shipboard conductivity–temperature–depth (CTD) casts.

Range-independent ray paths between the source and receiver are calculated to predict the sensitivity of the arrival phases to the change in sound speed. The ray model uses measurements from the acoustic experiment: source/receiver depth, range, and sound speed. The change in sound-speed-vertical derivative between 100 and 150 m depth allows for two ray paths [see Figs. 2(a) and 2(b)], with launch angles of 3.3° and 5.5° that connect the source and receiver. The two ray paths have a maximal vertical separation of 40 m. The second ray path does not intersect the receiver; however, this ray path is connected to the energy that ensonifies the receiver.

Since the second arrival in the observations is in the shadow zone of the ray simulation, we use a range-independent parabolic equation (PE) model⁷ using the same environmental parameters as the ray trace. Each frequency in the source's transmitted frequency band, 2–6 kHz, is simulated at 1 Hz frequency intervals. The spectrum of the source transmission emulates the spectrum of the matched-filtered maximum-length sequence (MLS). This removes the need to apply the matched filter to the simulated transmission. The simulated spectrum at the range and depth of the receiver is composed of the simulated amplitude and phase of each frequency. The simulated spectrum is inverse-Fourier transformed to produce the time domain impulse at the receiver.

The range-independent ray and PE simulation of the acoustic reception at 1.8 km range in the vicinity of the receiver [see Fig. 2(c)] demonstrates two interesting effects in the acoustic propagation. There is a triplication along the ray arrivals at 1.8 km range between 105 and 115 m depth, where the derivative of the ray depth with respect to launch angle reverses once at 105 m and then again at 115 m. The second arrival recorded on the receiver is in the geometric shadow zone of the ray arrivals; however, the PE model demonstrates that energy from the ray triplication extends into the shadow zone.

The extension of sound energy into the shadow zone near triplications and caustics is well-known theoretically⁸ and has been observed in experiments.¹ The PE model replicates the phase distribution of the first and second arrivals observed over the array of hydrophones with fairly consistent agreement [see Figs. 2(d) and 2(e)]. The phase of the second arrival changes at a constant rate over the array, suggesting that it acts like a plane wave. The PE arrival branch and the ray triplication branch [yellow dashed-dotted line in Fig. 2(c)] have the same approach angle. Thus, we assume that the ray nearest to the bottom caustic, marked with a star, can be used to calculate the phase perturbations of the second observed arrival. The observed travel time of path 2 [see Fig. 2(d)] is about 1.3 ms greater than the ray estimate because of the distance between the ray arrival (star) and the hydrophone at 130 m. This travel time offset is roughly constant during the transmission and so does not affect the comparison between the modeled and observed change in phase difference.

Over the 30-min transmission, the sound speed near the ray paths decreases due to the upward heaving of the water column temperature structure by internal waves [see Fig. 3(a)]. Crucially, the decrease in sound speed over each ray path is not uniform, even though both rays have small vertical separation and the observed internal wave heave is nearly the same. Due to the non-same sound-speed changes in depth, the upper ray path experiences a larger sound-speed decrease than the lower path. The heave of the internal waves, illustrated with isotachs [see black lines in Fig. 3(a)], is vertically correlated. In contrast, the vertical standard deviation in sound speed is not uniform in depth due to the change in sound-speed-vertical derivative between 100 and 130 m. The change in sound-speed-vertical derivative over depth decorrelates the travel time/phase of these two paths [see Fig. 3(c)]. The sound-speed observations indicate that the decorrelation between ray paths can happen on a much smaller vertical scale than the vertical decorrelation scale of internal wave displacement. In other words, the scale of the changing sound-speed-vertical derivative determines the time scale for the change in the acoustic phase, rather than the modal structure of internal wave displacement.

In our case, the ray paths are nearly horizontal, with $θsrc$⁠, $θrcv$ close to zero (see the Ray angle columns in Table 1), thus reducing the effect of the phase difference due to range perturbations. The scale of the range, depth, and internal wave changes during the experiment are shown in Table 1 to give a sense of how sensitive the phase and the phase difference are to each parameter. The largest phase variability, $δϕ=262.5$ cycles, for paths 1 and 2 comes from the range variability, $δRrcv=98$ m, due to the receiver drift. However, when the phases are subtracted between paths 1 and 2, the phase variability is reduced to 1.31 cycles (see the $δRrcv$ column in Table 1), which is one reason the phase difference is used in this experiment.

The change in phase difference from the changing sound speed, range, and depth are summed to estimate the total phase difference. The estimated phase difference is compared to the observed phase difference in Fig. 4(a). Since we are interested in the time evolution of the phase difference, the time mean of the phase difference is subtracted from the observed and estimated differences. The observed change in phase difference roughly matches the predicted change in phase difference over the 30-min interval. The summed components that make up the estimated phase difference are shown individually in Figs. 4(b) and 4(c).

Despite their similar magnitudes, the distinct time scales of the depth, range, and sound-speed perturbations help to identify the different signals contributing to the total phase difference (see Fig. 4). The $±5$ [rad] oscillation in the phase difference over 30 min is the most salient feature in the observations [see the black line in Fig. 4(a)] and is driven by the internal wave displacement of the sound-speed profile. The majority of the variability in the phase difference observations are correlated (⁠ $R2=88%$⁠), with phase change from sound speed [see the red line in Fig. 4(b)]. The internal wave accounts for the change in the phase difference between minutes 5 and 17 of around 1 cycle over 12 min. When the decrease in transmission range [see the gray line in Fig. 4(b)] is accounted for, the $R2$ value increases to $90%$⁠. Finally, accounting for the phase difference signal due to depth changes in the source and receiver [see the orange and blue lines in Fig. 4(c)], the $R2$ value increases to $94%$⁠.

The surface wave signature of the source depth vertical motion on the phase difference is highlighted in Fig. 4(d). The coherence between the source depth and phase difference gives us confidence that our plane wave approximation is appropriate for the perturbations related to the changing location of the source and receiver. The correlation between observations and the model suggests that our extrapolation of the phase from a ray arrival at 115 m down to the receiver depth, 130 m, is sufficient to track $94%$ of the variance in the observed phase difference.

The residual between the predicted and observed phase difference is shown in Fig. 4(e). The second-scale variability in the residual signal is likely from deviations in the ray angle, which is fixed in our simulation. Deviations in the ray angle change the magnitude of the phase perturbations from the motion of the source and receiver [see Eq. (2)]. The minute-scale deviations in the residual are possibly from the uncertainty of the range measurement or the uncertainty of the sound-speed profile measurement, specifically the range dependence of the sound-speed profile. Higher precision measurements of these parameters would increase the accuracy of our prediction.

At short ranges, two ray paths stemming from fine structure in the sound-speed profile allow for a phase difference measurement that is sensitive to internal wave advection. The phase difference is sensitive to the $±5$ m internal wave displacement of the sound-speed profile over 30 min. The small vertical scales of the sound speed structure allow us to gain information about the internal wave displacement because the vertical changes in sound speed decorrelate the phase of the two observed arrivals. The phase difference, a measurement of the decorrelation, tracks the displacement of the profile. These results demonstrate that meter-scale internal wave motion at the minute-to-second time scale can be tracked deterministically using mid-frequency broadband observations. Furthermore, the agreement between observation and model implies that the phase of arrivals in the geometric shadow zone can be extrapolated from a ray-based model.

The phase and amplitude fluctuations of mid-frequency acoustic receptions are often modeled statistically due to the complications of the medium. These finely sampled sequential observations of instantaneous phases between two mid-frequency broadband receptions show that the propagation of triplicated ray paths can be modeled deterministically at short ranges with a well-resolved estimate of the medium. Even though this is not practical for all cases, these observations suggest that modeling the temporal variability of interfering broadband arrivals requires an accurate model of the varying shape and position of the fine structure in the sound-speed environment. Conversely, these types of arrivals provide information that resolves vertical differences in the sound-speed profile, a necessary step for inverting the fine structure.

This research was supported by the Office of Naval Research under Grant Nos. N00014-19-1-2635 and N00014-23-1-2557. Special thanks goes to Matthew Dzieciuch, who provided acoustic propagation modeling code for both RAM and ZRAY, and to David Ensberg, who processed the recorded MLS data from the experiment.

The authors have no conflicts to disclose.

The acoustic data are available from W. Hodgkiss, subject to sponsor restrictions.