Estimation of peak dosage of kinetic acoustic measures from pressure measurements as derived from time-limited signal waveforms

In the study by Dahl et al.,¹ the practical equivalence of some scalar pressure and vector-based acoustic dosage measures as potentially used to assess the impact of anthropogenic underwater sound on marine life is discussed. An assumption behind the equivalence is the prevalence of time-limited signal waveforms of finite bandwidth composing this sound, such as signals originating from pile driving, airguns, or explosive sources. Dosage measures derived from acoustic pressure, velocity, and acceleration fields were studied with an emphasis on comparing time-integrated, squared measures of these quantities.

The context and motivation for this study remain the same as in the previous work. However, in this study, peak values associated with the kinematic acoustic field, acoustic acceleration, velocity, and displacement are examined and compared with the peak value associated with the dynamic acoustic field (i.e., acoustic pressure), and a means to directly relate the two is presented. The key relevance of this study thus applies to the use of peak values as dosage measures to assess the impact of anthropogenic underwater sound on marine life^2,3 and how such measures, either vector-based or via scalar pressure, are equivalent. As in the previous study, the current study is also restricted to the water column, where a neutrally buoyant vector-sensing system might plausibly be located for environmental assessment of anthropogenic sound exposure to fishes and aquatic life.

Section 2 presents a derivation for the estimation approach on how pressure can be used to approximate certain aspects of the kinematic time series. In Sec. 3, the equivalence is demonstrated using explosive sources as measured during the Seabed Characterisation Experiment 2022 (SBCEX2022)^4–6 and short continuous wave pulses measured during the Targets and Reverberation Experiment 2013 (TREX2013).^7,8 Section 4 concludes with a discussion.

Results from two field experiments are used to demonstrate the accuracy of deriving peak estimates of acoustic displacement, velocity, and acceleration via measurement of peak acoustic pressure, using broadband signals in one case and more narrow band signals in another case.

During the 2022 Seabed Characterisation Experiment, which took place approximately 95 km south of Cape Cod, multiple MK64 Sound Underwater Source (SUS) explosive sources were deployed for purposes of studying sound propagation in fine-grained, muddy sediments. The related previous work¹ also compared time-averaged acoustic dosages based on pressure and vector measurements of signals received from SUS explosive sources deployed at this same site in 2017. In 2022, however, a different vector-sensing system referred to as the Intensity Vector Autonomous Recorder (IVAR2) sensor was used,^4,5 with a Geospectrum model M105 vector sensor positioned approximately 1.5 m off the seabed in waters 75 m deep.

The measurements involve the deployment of 164 SUS at a nominal depth of 18 m from R/V Armstrong with signals recorded from IVAR2 at ranges from 2 to 22 km. All signal channels are high-pass filtered at 30 Hz to remove any DC offset and then low-pass filtered at 3200 Hz. Figure 1 shows a time domain representation of MK64 SUS explosion measured by IVAR2 at a range of 12 km. The pressure time series in linear units [Fig. 1(a)] provides a sense of the signal level and duration (time axis is relative). In the following, the measured quantities (red) represent the absolute value as a function of time for acoustic displacement [Fig. 1(b)], velocity [Fig. 1(c)], and acceleration [Fig. 1(d)] plotted in dB units. These curves are compared with corresponding inferred versions (blue) generated from the pressure time series, following Eqs. (1)–(3). These estimates show that, when the transient pressure signal is present (0.8–1.8 s), the measured kinematic values and the corresponding inferred estimate match to ±1.5 dB. Due to the differing noise floors of the vector sensor compared to the hydrophone, the estimate is not valid outside this time window as this is primarily low-magnitude ambient noise.

An interesting but subtle feature is the tendency of magnitude displacement [Fig. 1(b), red curve] and velocity [Fig. 1(c), red curve] to not fall to the same low level as the corresponding integrated [Fig. 1(b), blue curve] and differentiated [Fig. 1(c), blue curve] dynamic signal. It can be shown that in these low-level (or interference-like) regions, the vertical component alone defines the lower bound of the kinematic signal.^1,5 Jacobsen and Molares¹² largely confirm this result by noting that as the kinematic signal has more degrees of freedom compared to the dynamic signal (3 vs 1), the variance must be lower and the minima higher.

The peak displacement, velocity, and acceleration are compared with the corresponding pressure-based estimates for all 164 recorded SUS (Fig. 2) representing ranges from 2 to 20 km. The peak velocity estimates [Fig. 2(b)] fall most closely upon the dashed line representing exact correspondence, with variability about this line equal to ±1 dB. Acoustic displacement [Fig. 2(a)] and acceleration [Fig. 2(c)] follow similar patterns in terms of variability, although peak displacement is somewhat underestimated via Eq. (2) by about 1 dB. In comparison, peak acceleration is somewhat overestimated via Eq. (3) by about 1 dB.

During the 2013 TREX, which took place approximately 4 km offshore of Panama City, FL, quasi-continuous wave sonar pulses were transmitted from a source lowered from a research vessel at a depth of 12 m in waters 19 m deep. An ITC-2010X source transmitted a pulse of duration 100 ms consisting of four simultaneously transmitted tones centered at 1025, 2050, 3075, and 3950 Hz. The signals were coherently recorded by an intensity vector receiver composed of four channels: a three-axis accelerometer-based vector sensor (Ocean Applied Acoustics, China, model VHS-100) combined with a single pressure hydrophone (ITC model 1042, International Transducer Corp.). This system was positioned at a depth 2 m above the seafloor, as was the case for a variant of this study involving the same equipment and pulse suite conducted a few days earlier.⁸ Transmissions from the research vessel were made every 5 s as the vessel moved out in range starting from 190 to 1000 m from the intensity vector receiver. An example of the broadband TREX signal is Fig. 3 where, like Fig. 1, the magnitude of the kinematic signals (red) is compared against the inferred values from pressure signal (blue). A total of 300 pulses are bandpass filtered with their measured and inferred peak values shown in Fig. 4.

For acoustic displacement and acceleration, peak values are inferred from peak pressure in two ways. First, time domain integration and differentiation are used in a manner analogous to the SUS pulses to yield estimates of acoustic displacement [Fig. 4(a)] and acceleration [Fig. 4(c)]. Results are again plotted against a dashed line representing the exact correspondence between true measured (via the vector sensor) and inferred from pressure. In the second [Figs. 4(d) and 4(e)], a single frequency scaling is used, which produces identical values to the ones produced above [Figs. 4(a) and 4(c)]; a result anticipated given the signal is relatively narrow-band with an effective bandwidth of 15 Hz. Finally, we may also assess the entire, unfiltered TREX pulse consisting of the four narrow band tones [Figs. 4(a)–4(c), white circles]. For peak displacement and acceleration, the complete time integration and differentiation are required to infer a reasonable estimate from peak pressure. The inferred peaks of the three kinematic signals follow the dashed line closely with a variation of ±4 dB for the single frequency estimates and ±2 dB for the full bandwidth estimate.

An interesting feature in the multi-frequency TREX data, and not evident with the SUS data, is the approximate “reversal” of peak levels for displacement and acceleration. Specifically, in the frequency domain, differentiation and integration produce an increased or decreased expression in high-frequency components, respectively. Comparing the peak levels of the integrated [Fig. 4(a)] and differentiated [Fig. 4(c)] levels, the highest frequency component (blue circles) trends away from the total displacement and toward the total acceleration (white circles), respectively. Correspondingly, the lowest frequency component (red circles) trends toward the total displacement and away from the total acceleration as expected.

A simple approach to estimating the magnitude time series of kinematic acoustic variables is demonstrated on measured vector acoustic data. This allows for peak levels of the kinematic sound field to be estimated from peak pressure in time-limited signals. The deviation between pressure-based and vector-based peak measurements is less than 3 dB, which is within the combined calibration uncertainty of the hydrophone and the vector sensor.

Estimates for displacement and acceleration depend on performing numeric integration and differentiation, respectively, to the pressure data. Two field datasets were used in the analysis. The first (SBCEX data), involving impulsive SUS data, required full numeric integration or differentiation owing to its large bandwidth. The second (TREX data), involving more narrow band (but still time-limited) signals, required only the dividing by 2πƒ as a replacement for integration and multiplying by 2πƒ as a replacement for differentiation. The TREX results show a larger variance in the individual narrow-band frequency estimates than the full bandwidth result. This increased variance can be accounted for by Flamant and Bonnel,¹⁰ where the ratio between potential and kinetic energy has increased variance with decreased bandwidth. As in the previous work, restrictions at the near field and air-water proximity also apply to the peak estimates.¹ Similar to Dahl et al.,¹ however, the receiver depth for the impulsive SUS signals put the receiver close to the reflecting seabed, i.e., within a fraction (∼0.1) of the characteristic wavelength of the signal (based on the peak frequency of the explosive SUS signal¹).

Regardless of the filter bandwidth, the estimation of the peak kinematic level for the TREX signal has a higher variance than the SUS signals. This increased variation can potentially be explained by differences in self-noise and calibration of the TREX and SBCEX sensors and noise in the acoustic environment. However, we posit that the bandwidth of the respective signals is the main driver of the variance. Being much more impulsive in duration, the SUS signal, therefore, has a much larger bandwidth than the time-limited but narrow-band TREX data. We note, however, that the concept of peak level as a dosage is best applied to impulsive sources with more tone-like signals better characterized by an averaged level, such as root mean square.^2,3,13

These results extend the equivalency between pressure-based and vector-based dosages to include the peak value. The correlation between the kinematic and dynamic signals shows that, from the outset, it can be anticipated that peak kinematic values offer no additional advantage as explanatory variables over peak pressure to assess the impact of anthropogenic underwater sound on marine life.

This work was supported by the Office of Naval Research. The authors appreciate the helpful reviews provided by the anonymous reviewers.

The authors have no conflicts of interest to disclose.

The observed and subsequently processed data that support the findings in this study, and in the form displayed here, are available from the corresponding author upon reasonable request.