On the equivalence of scalar-pressure and vector-based acoustic dosage measures as derived from time-limited signal waveforms^a)

Vector acoustic observations in fish behavioral and ecological studies are pregnant with interesting research possibilities.^1–3 Vector acoustic measurements also play an important role in probing underwater sound propagation physics and studies of ambient noise,^4–7 including geoacoustic inversion studies to obtain seabed properties.^8–11 The focus of this study is on vector acoustic measurements as they relate to dosage metrics for biological impact assessment. We examine kinematic (acoustic acceleration and velocity) and dynamic (acoustic pressure) properties of airgun signals and other time-limited signals such as from explosive sources and transient sonar signals. The study relates directly to the recent programmatic thrust¹² towards an evaluation of the potential effects of sound-producing activity such as airgun sources on aquatic life, sponsored by the Joint Industry Programme (JIP), including benchmark simulations of propagation from a seismic airgun source.

The objective of this paper is to demonstrate, both numerically and experimentally, that for time-limited or transient signals and under the most realistic scenarios, the dose metric obtained from a vector sensor positioned within the water column can be evaluated equivalently using acoustic pressure measurements. The study is motivated by the significant differences in both instrument and analysis costs for vector acoustic versus hydrophone measurements.

Time-limited signals here refer to those of limited time extent¹³ and thus of finite duration T as determined by the waveform start t₁ and end t₂ times such that $T = t 2 − t 1$⁠; the term transient signal also applies. Of primary importance to our study is that the signal embodies a non-zero frequency bandwidth, i.e., to be distinguished from a continuous, pure tone waveform. The work is thus relevant to many anthropogenic activities producing transient signals such as airguns, pile driving, and underwater explosive sources, as well as more generic broadband sources such as sonars.

This study commences with acoustic simulations as prescribed by one of the modeling verification scenarios¹⁴ for acoustic propagation from a seismic airgun source, referred to therein as the B1 scenario. The B1 scenario involves the propagation of acoustic pressure and acceleration fields in a Pekeris waveguide (water depth 50 m) based on an airgun source signature, for which the source and receiver depths are 5 and 15 m, respectively, and two ranges are studied. Here, the B1 scenario is further expanded to study a parameter space defined by four receiver ranges between 75 and 3000 m and receiver depths that span the water column. A mode-based formulation for the acoustic field is used to compute time-domain results for acoustic pressure and acceleration that can be compared with other results for the scenario; our study will also include an evaluation of acoustic velocity.

The airgun source numerical study is followed by an analysis of at-sea observations using vector acoustic measurements obtained from a vector sensor positioned 1.45 m off the seabed in 75 m of water. In this second case two, time-limited signals are studied, one originating from underwater explosive sources (SUS charges) at various ranges, and the other from a towed sonar source.

For both the airgun numerical study and at-sea observations, we evaluate and compare measures of acoustic dosage based on squared first-order acoustic variables such as acoustic pressure, velocity, and acceleration. The first involves an acoustic acceleration-based dosage that is compared with a surrogate based on acoustic pressure. The time-integrated squared pressure and decibel equivalent sound exposure level (SEL) and the time-integrated squared acceleration and decibel equivalent acceleration exposure level (AEL) are analyzed. Both are defined formally and supported by references from the literature. The SEL metric has widespread use as a measure of dosage or explanatory variable in studies involving the potential effects of impulsive signals such as impact pile driving¹⁵ and underwater explosives^16,17 on fish and marine mammals.¹⁸ A recent study involving the effect of seismic airgun signals on caged fish,¹⁹ for which vector acoustic recordings were also made, involved both SEL and AEL metrics.

The first comparison involves acoustic pressure and acceleration. Measures of SEL and AEL are explored together with an alternative measure of AEL constructed from the pressure field data via a spectral form of SEL. It is shown that over nearly the entirety of the water column, an estimate of AEL is derived accurately from the SEL spectrum. The second comparison involves acoustic pressure and velocity. In this case, potential (based on acoustic pressure) and kinetic (based on acoustic velocity) energy densities are computed with analogous equivalence found, which is consistent with a recent analytical derivation relating frequency-integrated kinetic and potential energies.²⁰ Although the underwater noise from ships and machinery per se is not addressed in this study, such signals have relevance to this study when typical averages are taken, e.g., a third-octave frequency band over some finite observation time yields equivalences in kinetic and potential energy spectra.²¹ 

Overall, for the air gun simulations absolute differences between AEL and the surrogate estimate based on acoustic pressure were typically less than 1 dB (absolute error) at a range of 75 m and differences between kinetic and potential energy densities were less than 0.2 dB (absolute error). Comparable results (given subsequently) were obtained with field measurements from explosive sources and transient sonar signals.

The remainder of this paper is organized as follows. In Sec. II, a (single frequency) mode-based formulation of the acoustic first-order quantities, acoustic pressure, velocity, and acceleration properties are outlined. Based on this formulation, time series of these quantities based on multiple frequencies are generated. Section III defines acoustic dose metrics for biological impact assessment based on acoustic pressure, acceleration, and velocity. The numerical simulation and field results are presented in Secs. IV and V, respectively, and a summary and discussion are provided in Sec. VI.

The objective of this study is to compare pressure-based and vector acoustic-based measures of acoustic dose, specifically the related kinematic measures of acoustic dose. Such kinematic measures of dosage, where the term particle motion is often used to describe the responsible first order acoustic variable, e.g., a_z, in some instances may not have the traditional currency shared by pressure-based, or dynamic measures of dosage, such as SEL. To the extent possible, we provide applicable correspondence with International Organization for Standardization (ISO) definitions.²⁵ 

The equivalent estimate of AEL is $E ̃ a , T$ expressed in dB re μm²/s⁴ s. The basis for Eq. (15) is the equivalence of potential and kinetic acoustic energies that necessarily emerges upon the inclusion of some degree of temporal or frequency averaging, such as in third-octave observations of ship noise²¹ or general broadband signals.²⁰ The equivalence is demonstrated in this study with both numerical (the B1 scenario) and experimental results.

An exception to the equivalence of potential and kinetic energies can happen in the near field (see Sec. IV), and also when an isolated acoustic source (as distinct from broadly distributed sea surface noise) is located directly above a vector sensor positioned near the seabed. For example, broadband noise emissions from a vessel passing directly above a sensor will manifest frequency bands where potential acoustic energy is greater than kinetic energy while the opposite occurs in neighboring frequency bands.²⁷ The condition where the dynamic and kinematic energy forms differ in this manner is characteristic of interference involving steep angles or near-normal incidence reflection from the seafloor.

For the simulated data, we use a sliding averaging window²⁸ of $T =$ 50 ms to compute the mean square, with the maximum windowed value taken. This approach more optimally accommodates simulated pulses of short duration, particularly at close range. This second definition of T, as distinguished from the one corresponding to 90% of the pulse energy, is not fundamental to the study given comparison is only between simulated estimates of w_p and w_k based on the same approach.

In the following, the B1 benchmark scenario¹⁴ for propagation from a seismic airgun source in a Pekeris waveguide is evaluated numerically following the approach in Secs. II and III. Results demonstrate an approximate equivalence between the acceleration exposure E_a and the surrogate based on acoustic pressure $E ̃ a$ (henceforth T is removed from the notation), as well as between the kinetic (w_k) and potential (w_p) energy densities. For reference, the parameters of the waveguide are: water depth 50 m, water column sound speed c_w 1500 m/s, seabed sound speed and density 1700 m/s and 2000 kg/m³, respectively, and seabed attenuation 0.5 dB/λ.

Simulations corresponding to source depth z_s = 5 m, receiver depth 15 m, and range r = 3 km are summarized in Fig. 1 and show the source signal s(t), as well as the time-domain received pressure, acceleration, and velocity. Propagation effects are clearly visible, with multimodal time dispersion apparent in the received signals; vertical components of both acceleration and velocity also tend to be less energetic than horizontal counterparts.

Central to this study are metrics for impact assessment and the related sound exposure $| P ( f ) | 2$ and acceleration exposure $| A ( f ) | 2$⁠; spectral densities (Fig. 2), corresponding to receiver ranges 75 m and 3 km (again based on z_s = 5 m, receiver depth 15 m), are demonstrative. The peak frequency of the Dublin waveform source spectrum is ∼10 Hz, which defines a peak wavelength, λ_p equal to 150 m. At a range of 75 m, corresponding to $λ p / 2$⁠, the peak frequency is still evident in $| P ( f ) | 2$ [Fig. 2(a)] by range 3000 m [Fig. 2(b)], however, the low-frequency content has vanished.

The pressure surrogates for acceleration exposure spectral density $| A ̃ ( f ) | 2$ [Figs. 2(c) and 2(d)] suggest that vector quantities and their respective pressure surrogates are similar, with each manifesting a shift towards higher frequency content. Some significant differences occur at frequencies corresponding to destructive interference, though at such frequencies both acceleration and pressure quantities are low (with pressure slightly lower).

Time averaging over the $T =$ 50-ms sliding window of the pressure and velocities time series corresponding to these ranges yields potential energy density w_p via Eqs. (7) and (21), and kinetic energy density w_k via Eqs. (19) and (20). Similarly, frequency integration of $| A ( f ) | 2$ and its pressure surrogate $| A ̃ ( f ) | 2$ over 2.8 Hz to 2.8 kHz yields E_a and $E ̃ a$⁠, respectively. Small differences in these spectra have a negligible impact on the frequency-integrated quantities. Decibel equivalents (Table I) of these measures suggest that the acceleration and velocity-based values differ little from the pressure-based counterparts.

The computations from Sec. IV A are extended to multiple receiver positions spanning the water column and ranges 75, 300, 600, and 3000 m [Fig. 3]. The concurrences are again evident between potential w_p and kinetic w_k energy densities [Fig. 3(a)] and between acceleration exposure E_a and its pressure surrogate $E ̃ a$ [Fig. 3(b)].

The greatest difference occurs between potential w_p and kinetic w_k energy densities at shallow receiving depths, with ∼1 dB differences commencing at depths less than ∼3 m. Although the 10 Hz peak in the source spectrum identifies a wavelength scale λ_p, the 50-m water depth imposes additional constraints on propagation and 10 Hz is below the cut-off frequency of this waveguide. The spectral content of the pressure and velocity fields beyond the range ∼75 m is better characterized by a mean frequency. A measure of the mean frequency is determined using the sound exposure spectral density $| P ( f ) | 2$ for a given receiver depth and range, and converting to a probability density function (PDF) by normalizing the frequency integral to unity, from which frequency moments are computed. Estimated in this manner the mean frequency is ∼100 Hz, with a corresponding wavelength scale of ∼15 m. Disparities between w_k and w_p tend to be limited to depths less than ∼5 m or one-third the wavelength and consistent with an earlier study on broadband ship noise.²¹ 

To compare the acceleration exposure E_a and its pressure surrogate $E ̃ a$ [Fig. 3(b)], a measure of the mean frequency is appropriately determined using the acceleration exposure spectral density $| A ( f ) | 2$⁠, or its pressure surrogate that is proportional to $f 2 | P ( f ) | 2$⁠, each appropriately normalized to a PDF. The mean frequency necessarily increases to ∼300 Hz with a corresponding wavelength ∼5 m, and disparities between E_a and its pressure surrogate $E ̃ a$ tend to be limited to one-third of this new wavelength, which is considerably less.

The differences in kinetic w_k and potential w_p energy, and acceleration exposure E_a and its pressure surrogate $E ̃ a$⁠, manifested near the sea surface (Fig. 3), ought to be viewed in the context of the overall reduction in the acoustic field within this region. For example, the kinetic density, or alternatively mean square acoustic velocity, is not necessarily high-valued close to the sea surface, being typically less than values within the water column. These results (Fig. 3) are fully consistent with a recent analytical derivation.²⁰ 

As noted, the range 75 m (Figs. 2 and 3) equates to a range of $λ p / 2$⁠, based on the peak frequency of the Dublin waveform source spectrum. The effects of the near field, where kinetic w_k will exceed potential w_p energy, are expected to commence for ranges less than a waveguide depth, or 50 m in the B1 scenario, and determining more precisely the range where this happens is of interest.

Not all waveguide geometries nor source signatures can be addressed, but using the Dublin waveform source spectrum (10 Hz peak) and the 50-m water depth Pekeris waveguide from the B1 scenario yields $w k ≈ w p$ (within 0.5 dB) for ranges exceeding 30 m. This result (again using the Dublin waveform) is duplicated upon doubling the water depth to 100 m and it can be assumed that the result applies to water depths greater than about 50 m. With this restriction in mind, we take 30 m, or range-from-source greater than $∼ λ p / 5$⁠, as a reasonable estimate for delineating the near field.

This section extends the previous numerical results to at-sea data collected during the Seabed Characterization Experiment 2017 (SBCEX).²⁹ The experiment was conducted in spring 2017 in waters known as the New England Mud Patch, located about 95 km south of Cape Cod. Several types of sound sources were used to test remote sensing methods toward further understanding sound propagation in fine-grained, muddy sediments. Signals originating from two such sources are assessed here to compare acoustic dosage based on pressure and vector measurements.

The first, representing an impulsive signal, originates from Mk-64 explosive SUS charges,³⁰ detonated at an approximate depth of 18 m and deployed from R/V Armstrong. The second, representing a controlled sonar-like pulse, originates from a J-15 Acoustic Source towed from R/V Endeavor at an approximate depth of 40 m. In this case, a frequency modulated (FM) pulse of duration 1 s, bandwidth 150–250 Hz, was transmitted. These data were recorded over an approximately 15-h period commencing at 0000 UTC, 18 March 2017, during which sound speed measurements confirmed a well-mixed water column characterized by c_w equal to 1468.3 m/s.

Both signals were recorded on a vector sensing system referred to as the Intensity Vector Autonomous Receiver (IVAR),³¹ positioned 1.45 m off the seabed at location 49.48655°N, 70.63831°W, with mean water depth 74.4 m ± 0.4 m owing to tidal variation. The IVAR system used in this study consists of low- and high-gain receive networks, differing by 30 dB, with explosive SUS charges recorded on the low-gain and the J-15 sonar pulses recorded on the high-gain network. Each network recorded four coherent channels of acoustic data continuously (25 000-Hz sample rate), one for acoustic pressure and three for acoustic acceleration in three orthogonal directions. The IVAR accelerometer channels have a sensitivity of −11.3 dB re 1 V/(m/s²) over the operational frequency band of 15–3000 Hz. The pressure channel has flat sensitivity over this band, and though capable of receiving higher pressure levels, the upper bound of pressure-velocity fidelity on the low-gain network is represented by peak pressure of approximately 178 dB re μPa. Note that the peak frequency range of the explosive source was 30–100 Hz, and for the sonar-like pulse was ∼150 Hz, which, notably, puts the vector sensor position (1.45 m) within fractions of the signal dominant acoustic wavelength (of order 10 m) from the seafloor.

Impulsive signals originating from SUS charges were recorded at ranges between about 4 and 13 km. The average sound exposure spectral density $⟨ | P ( f ) | 2 ⟩$ received at a range 5.5 km is shown in Fig. 4(a). This is based on the average of five, notionally identical, SUS charges, i.e., with $E p , T$ as used in Eq. (7) based on this average. (Detailed views in the time domain of acoustic pressure and acoustic velocity in the x, y, and z directions for one of the five signals are presented in Fig. 5 of Ref. 31). The corresponding five-signal average SEL for these data upon frequency integrating $⟨ | P ( f ) | 2 ⟩$⁠, or using $E p , T$ directly, is 155 dB re μPa² s.

Figure 4(b) shows the corresponding average acceleration exposure spectral density $⟨ | A ( f ) | 2 ⟩$ (black line) with $E a , T$ in Eq. (14) based on the same five-signal average. A surrogate version, $| A ̃ ( f ) | 2$⁠, is derived from acoustic pressure data $⟨ | P ( f ) | 2 ⟩$ (orange line) through Eq. (15), where, in this case, $Δ F$ spans the frequency range from 10 to 2000 Hz. The decibel equivalent of E_a yields an AEL of 93.1 dB re μm²/s⁴ s, and that of $E ̃ a$ is 92.7 dB μm²/s⁴ s. Obtaining closer agreement is not realistic in view of the necessary precision required for the independent calibrations associated with the three acoustic accelerometer channels and the acoustic pressure channel. The complete set of SUS observations in the form of decibel equivalents of E_a and $E ̃ a$ [Fig. 5(a)] and kinetic w_k and potential w_p energy density [Fig. 5(b)] show similar agreement. On average E_a and $E ̃ a , T$ are within 0.3 dB absolute error, and w_k and w_p are within 0.5 dB absolute error (one outlier at range 3 km).

Vector acoustic measurements of transient signals originating from the towed source were recorded at a much closer range to the IVAR sensor, starting at ∼100 m, made possible owing to the lower amplitude of this source signal (source level of 160 dB re 1 μPa at 1 m) relative to the SUS charge,³⁰ with the transient signal measurements ending at range ∼8500 m. Compared to the source waveform s(t) measured by a source monitoring hydrophone [Fig. 6(a)], the received pressure p(t) at range 500 m [Fig. 6(b)] shows time modulation indicative of waveguide interference, with the SEL for this signal equal to 112 dB re μPa² s.

The towing transect was approximately aligned with the x-axis channel of the IVAR sensor so a_x and v_x are representative of the acoustic horizontal acceleration and velocity signals [Figs. 6(c) and 6(d), respectively], with vertical acceleration and velocity shown by Figs. 6(e) and 6(f), respectively.

Analogous to Fig. 5(a) for the SUS signal, Fig. 7(a) shows AEL measured with a vector sensor as a function of range with each range and equivalent AEL measured with a pressure sensor. In this case, the bandwidth $Δ F$ is limited to that defined by the FM pulse, or between 150 and 250 Hz. Similarly, the kinetic energy density measured with a vector sensor is compared with pressure-based potential energy [Fig. 7(b)], which is analogous to Fig. 5(b). On average E_a and $E ̃ a$ are within 0.8 dB (absolute error), while w_k and w_p are typically within 0.1 dB (absolute error). Note that range sampling for the FM pulse is much finer, thus revealing a small increase in acceleration and energy levels at a range ∼1.5 km that is consistent with modal waveguide propagation in these waters for a center frequency of ∼200 Hz.

The acoustic pressure, velocity, and acceleration fields generated from sources that produce a time-limited or transient signal of finite bandwidth have been studied with an emphasis on comparing time-integrated squared measures of these quantities. The study is restricted to the water column where a neutrally-buoyant vector sensing system might plausibly be located for purposes of environmental assessment of anthropogenic sound exposure to fishes and aquatic life. This is to be distinguished from sensing the kinematic features of the acoustic field in or on the seabed, where ocean bottom seismometers are well suited for such purposes. The quantities, SEL for acoustic pressure and AEL for acoustic acceleration, have been used as dose metrics for biological impact assessment of seismic airgun activity.¹⁹ For acoustic velocity, it is kinetic energy density w_k that is compared to pressure in the form of potential energy density w_p. We emphasize that any dose metric involving kinematic, vector acoustic quantities, such as acoustic velocity or acceleration, must as a practical matter be in magnitude form, as is the case for AEL, which is based on acoustic acceleration magnitude.

It was shown through numerical simulation of sound propagation from a seismic airgun source that an estimate of AEL is derived accurately from the spectral counterpart of a pressure-based estimate of SEL, the pressure surrogate, with the derivation applicable over the majority of the water column. Similarly, kinetic w_k and potential w_p energy densities were nearly equal, and thus mean square acoustic velocity is related linearly to mean square pressure. The numerical study was followed by an analysis of vector acoustic observations of an impulsive signal generated from an explosive source and a transient sonar signal generated by a towed sonar source. In both cases, the vector acoustic measurement system was positioned 1.45 m off the seabed. Collectively, with both numerical simulations and the field observations, differences between vector or kinematic-based and pressure or dynamic-based measures tended to be less than 1 dB.

The study identifies two regions within the water column requiring closer attention. First, a near-field zone is delineated where kinetic exceeds potential energy as defined by ranges less than $∼ λ p$/5, where λ_p is a characteristic acoustic wavelength established by the peak frequency (10 Hz) of the source based on the Dublin waveform. It is not possible to address all waveguide geometries and source signatures, nonetheless, conducting biological impact assessments within this relatively small zone (using a hydrophone or vector sensor) is largely impractical owing to instrumentation requirements concerning dynamic range.

Second, kinetic can exceed potential energy for depths less than about one-third of an acoustic wavelength from the air–water interface, although it must also be understood that both energy components are reduced relative to values away from the interface. In this case, wavelength is established by the mean frequency of the sound exposure spectral density $| P ( f ) | 2$ for a given receiver depth and range, rather than by source spectrum. Finally, since the acceleration exposure spectrum $| A ( f ) | 2$ amplifies higher frequency content, the zone for this near-surface effect is further reduced when comparing AEL and the corresponding pressure surrogate.

The current study's emphasis on transient signals, such as those originating from seismic and explosive sources, pile driving, and sonar, is complemented in a previous theoretical and numerical study²⁰ based on a more generic acoustic field of high bandwidth. Moreover, a previous study on underwater noise from shipping activity,²¹ reports that potential and kinetic energies computed in decidecimal bands centered between 25 and 630 Hz are equal, representing further evidence towards the inference of kinematic properties from pressure-only measurements for broadband signals outside the near-field zone. The measures AEL and kinetic energy (yielding a mean square acoustic velocity) do not represent all possible dose metrics to be assessed in this manner, e.g., a formal relation between peak pressure and corresponding peaks in velocity and acceleration magnitude may be of interest. For peak velocity, and referring to time-limited signals as studied here, we anticipate equivalence, for example, as shown previously with an explosive SUS arrival.³¹ For peak acceleration, the relation will be more subtle, and signal bandwidth and frequency play a role that requires further study.

This study, as well as the previously-mentioned companion works,^20,21 all include some degree of non-zero bandwidth, e.g., continuous wave (CW) tone signals are not addressed. A recent numerical study³² provides an interesting example of such a narrow-band treatment, as if the signal is CW propagating at a precise, single frequency. Their results illustrate an acoustic pressure–velocity relation via scaled impedance; the relation tends toward greater equality upon depth averaging over the water column. We note that a precise equivalence emerges upon depth averaging over the entire computational space including the seabed.³³ However, in this and the companion studies, the strategy is different; the small amount of frequency averaging, i.e., in realistic signals of non-zero bandwidth, equivalences between pressure-based and vector acoustic-based dosage measures emerge. A closer look at vector acoustic field measurements and modeling of a truly narrow band of signal⁷ shows fine details in the acoustic field that provide information on the seabed.

It is also important to understand the limitations of using vector sensors to evaluate the potential effects of sound-producing activity on aquatic life. The accelerometer-based, commercially available vector sensors used in this study have an upper frequency limit set by the sensor dimensions, limiting the analysis to frequencies less than a few kilohertz. A practical lower limit of typically ∼10 Hz (if not somewhat higher) exists owing to the rising noise floor of the acoustic acceleration field with decreasing frequency that encumbers recovery of reliably useful data. However, we note that vector acoustic measurements at frequencies as low as 2 Hz have been made using three, orthogonally oriented seismic accelerometers, mounted within a neutrally buoyant sphere.³⁴ The measurements were made in close vicinity to an off-shore wind turbine foundation, and spectral data (2–200 Hz) showed similar characteristics to spectra derived from measured mechanical vibrations of the turbine structure. For frequencies less than 2 Hz, the study also observed a correlation with sea surface activity, suggesting motion induced by sea surface gravity waves can influence this frequency range.

Finally, close attention should be paid to the limited dynamic range of acceleration based vector sensors, relative to hydrophones, given the kinematic channels are typically limited to acoustic fields characterized by maximum acoustic pressures of less than 185 dB re μPa. Thus, placement of the sensor in close proximity to, for example, an airgun source to document the near field, will be highly problematic. Furthermore, calibration and analysis costs associated with vector acoustic measurements are necessarily also greater relative to well-established methods based on acoustic pressure.

Though entirely expected, the results of this study comparing dynamic and kinematic measures of transient signals as they might compose an acoustic dosage, have, to our knowledge, not been shown, and they point to a practical field calibration check on vector acoustic systems. However, our results also demonstrate a viable path to convert pressure-based measurements of transient, or broadband pressure signals in the spectral domain to obtain an equivalent estimate of the AEL or exploit equivalence between kinetic and potential energies to assess dosages based on acoustic velocity. In summary, the results presented here at a minimum should lead to a questioning of the benefit and efficacy of vector acoustic measurements made within the water column for purposes of obtaining a dosage or otherwise explanatory variable, compared with more conventional hydrophone measurements, to evaluate potential effects of sound-producing activity on aquatic life.

This work was supported by the Office of Naval Research (USA). The authors appreciate the helpful reviews provided by the anonymous reviewers and those of the Guest Editor for this Special Issue.

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.