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  • View in gallery

    Contributions to the sound field at a receiver in the ocean made by an acoustic source in the atmosphere (Urick 1972).

  • View in gallery

    Geometry of the UAV, microphones, hydrophones, and propagation. For , the aircraft noise is reflected from the sea surface. Typical values for = 342 m s−1 and = 1524 m s−1, suggesting .

  • View in gallery

    Geometry for the inverse problem.

  • View in gallery

    Target atmosphere based on LES. The xz coordinate system is such that the positive x axis is in the direction of UAV travel through the sensor array and the z axis is vertical. The origin of the system coincides with the first microphone.

  • View in gallery

    Reconstruction of synthetic atmospheric wind and temperature variations: (left) a representation of the LES data using RBF at the density and resolution used in the tomographic inversion, and (right) the tomographic estimate with errors equivalent to 0.1-ms noise and 0.5% ray pathlength extension.

  • View in gallery

    Target ocean structure based on random coefficients assigned to a uniformly distributed grid of RBF separated by 20 m × 5 m in the x and z directions.

  • View in gallery

    Tomographic reconstruction for an underwater area 200 m long × 50 m deep, generated by randomly assigning coefficients to a uniformly distributed 20 m × 5 m grid of RBF.

  • View in gallery

    Tomographic reconstruction for the same conditions as in Fig. 7, but with two rows of hydrophones separated in depth by 25 m. Errors due to signal processing, flow noise, and ADC jitter set to 1 μs; refraction set to 0.01%, and GPS set to 0.2 cm.

  • View in gallery

    Tomographic reconstruction for the same conditions as in Fig. 7, but with errors due to signal processing, flow noise, and ADC jitter set to 0.1 ms; refraction set to 0.5%; and GPS set to 0.05 m. (top) Image simulates one row of hydrophones at 50 m, and (bottom) image simulates two rows separated by 25 m.

  • View in gallery

    Tomographic reconstruction for the same conditions as in Fig. 9, but the (top) image simulates two rows of hydrophones (separated horizontally by 5 m) at depths of 25 and 50 m, and the (bottom) image simulates four rows (separated horizontally by 5 m) at depths of 12.5, 25, 37.5, and 50 m.

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An Acoustic Tomography Technique for Concurrently Observing the Structure of the Atmosphere and Water Bodies

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  • 1 Defence and Systems Institute, University of South Australia, Mawson Lakes, South Australia, Australia
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Abstract

The opacity of water to radio waves means there are few, if any, techniques for remotely sensing it and the atmosphere concurrently. However, both these media are transparent to low-frequency sound (<300 Hz), which makes it possible to contemplate systems that take advantage of the natural integration along acoustic paths of signals propagating through both media. This paper proposes—and examines with theoretical analysis—a method that exploits the harmonics generated by the natural signature of a propeller-driven aircraft as it overflies an array of surface and underwater sensors. Correspondence of the projected and observed narrowband acoustic signals, which are monitored synchronously on board the aircraft and by both sensor sets, allows the exact travel time of detected rays to be related to a linear model of the constituent terms of sound speed. These observations may then be inverted using tomography to determine the inhomogeneous structures of both regions. As the signature of the aircraft comprises a series of harmonics between 50 Hz and 1 kHz, the horizontal detection limits of such a system may be up to a few hundred meters, depending on the depth of the sensors, roughness of the water surface, errors due to refraction, and magnitude of the sound field generated by the source aircraft. The approach would permit temperature, wind, and current velocity profiles to be observed both above and below the water’s surface.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Professor Anthony Finn, anthony.finn@unisa.edu.au; kevin.rogers@mymail.unisa.edu.au

Abstract

The opacity of water to radio waves means there are few, if any, techniques for remotely sensing it and the atmosphere concurrently. However, both these media are transparent to low-frequency sound (<300 Hz), which makes it possible to contemplate systems that take advantage of the natural integration along acoustic paths of signals propagating through both media. This paper proposes—and examines with theoretical analysis—a method that exploits the harmonics generated by the natural signature of a propeller-driven aircraft as it overflies an array of surface and underwater sensors. Correspondence of the projected and observed narrowband acoustic signals, which are monitored synchronously on board the aircraft and by both sensor sets, allows the exact travel time of detected rays to be related to a linear model of the constituent terms of sound speed. These observations may then be inverted using tomography to determine the inhomogeneous structures of both regions. As the signature of the aircraft comprises a series of harmonics between 50 Hz and 1 kHz, the horizontal detection limits of such a system may be up to a few hundred meters, depending on the depth of the sensors, roughness of the water surface, errors due to refraction, and magnitude of the sound field generated by the source aircraft. The approach would permit temperature, wind, and current velocity profiles to be observed both above and below the water’s surface.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Professor Anthony Finn, anthony.finn@unisa.edu.au; kevin.rogers@mymail.unisa.edu.au

1. Introduction

There are a number of imperatives for observing the structure of the atmosphere and bodies of water: several methods exist. Still, high-resolution inspection of these media is prohibitively expensive if traditional point-source measurements are used. Tomography allows the internal structure of an object to be visualized and its space–time variation to be monitored. It avoids many of the limitations of direct observation and has been applied to all geophysical media. However, we are not aware of any techniques where it has been employed to infer the properties of both the atmosphere and a water body together. If feasible, such a technique may allow for inspection of the variability and interaction of internal waves and other transport, circulation, and mixing processes between these two important environments: interchanges that are of considerable interest to meteorologists, climatologists, and oceanographers—particularly if such information were available in real time.

The research on oceanic tomography derives from work proposed for monitoring meso- and basin-scale structure (Munk and Wunsch 1979). The topic has received considerable attention in the literature since (e.g., Cornuelle 1982; Braun and Hauck 1991; Golan and Dose 2002; Dushaw 2014; Skarsoulis 2004; Munk et al. 1995). Based on interactions between energy emitted by appropriately located transmitters and the observed medium, sound speed values are obtained for intersecting rays. Inversion is then used to infer the structure of the medium by relating the temperature dependence of sound speed in water to the measured travel time of an acoustic signal. Originally, moorings or surface ships were used to carry both sources and receivers, but more recently ambient and seismic noise has been used. Good reviews of the status of research into the field are available in (Dunn 2015; Dushaw et al. 2001).

Atmospheric temperature and wind profiles have also been observed using acoustic tomography. The first reported implementation was based on a series of masts supporting microphones and loud speakers covering an area 200 m × 240 m (Wilson and Thomson 1994). Other arrays have subsequently been built at the University of Leipzig (Ziemann et al. 1999; Arnold et al. 1999) and the Boulder Atmospheric Observatory (Vecherin et al. 2008a), which enables 3D tomography. A variety of other techniques have also been developed (Vecherin et al. 2006, 2007, 2008b,a; Barth and Raabe 2011; Kolouri and Azimi-Sadjadi 2012; Jovanovic et al. 2009; Wilson et al. 2001), including some that use noise sources, such as birds or meteors (Spiesberger and Fristrup 1990) or commercial aircraft (Ostashev et al. 2000; Wilson et al. 2001). A good survey of progress in the field is presented in Ostashev et al. (2008b).

More recently, techniques for performing atmospheric tomography based on the overflight of an unmanned aerial vehicle (UAV) have been developed (Finn and Franklin 2011a; Rogers and Finn 2013c, 2015). As the UAV requires no pilot and has low kinetic energy, it may be flown from altitudes of a few meters to several kilometers, for extended periods (>36 h) and in hazardous or hard-to-reach environments. Subject to the resolution and uncertainty of the derived quantities, which depend upon the sensors, signal properties, signal analysis, UAV flight path, etc.—see Finn and Rogers (2015, section 4)—the technique allows for reconstruction of the surface layer, which typically extends 20–50 m above the ground, and the boundary layer, which extends up to heights of about 2 km.

In this paper a novel remote sounding technique for concurrently inferring the temperature and wind/current movement profiles of both the atmosphere and a body of water is described and examined. The approach imagines use of an array of microphones located on the sea surface and hydrophones suspended directly below. The states of the sensors are envisaged as being monitored using real-time kinematic carrier phase differential GPS and microelectromechanical systems (MEMS) inertial navigation systems (INS). The natural acoustic signature of a manned or unmanned aircraft is monitored synchronously on board the platform and by the sensor sets on and under the water. The signal is assumed to travel only along the direct path (Fig. 1), which is a transitory phenomenon that typically lasts only while the source is almost overhead. Still, as the attenuating effects of water on frequencies below 1 kHz are small, the horizontal detection limits of such a system may be a few hundred meters, depending on the depth of the sensors, the errors due to refraction, and the magnitude of the sound field generated by the source aircraft. The sound speeds of both media are determined from the travel times of signals propagating through the intervening regions, which are augmented by meteorological and hydrological measurements. Finally, tomography is used to infer temperature and velocity structure. The effects of surface roughness and the influence of scattering are assumed to impact background noise rather than the refraction of the direct ray or the imparted Doppler shift. As a result, the technique is unlikely to be extensible to high sea state but may be ideally suited to lakes and other such inland water bodies. Data rates and ranges involved suggest devices such as commercial Wi-Fi devices could be used to communicate the information between nodes and to users in real time (Finn and Franklin 2011b).

Fig. 1.
Fig. 1.

Contributions to the sound field at a receiver in the ocean made by an acoustic source in the atmosphere (Urick 1972).

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

The paper is organized as follows. Section 2 describes the technique and approximations used to derive the coordinates and velocity of the air–sea intercept point. Section 3 then provides an examination of the potential errors that can be tolerated in terms of faithful reconstructions of the atmosphere and water. The final section offers some concluding remarks.

2. Atmospheric–water body tomography

a. UAV signal

Observations of noise produced by an aircraft as it flies over hydrophones located underwater were first reported by Urick (1972). There are also more recent studies (Peng et al. 2010; Kutakov and Maslov 2007; Buckingham et al. 2002b). There are four contributions to the underwater sound field created in this way: direct refraction, bottom reflection, the evanescent wave (also called the lateral wave or inhomogeneous wave), and sound scattered from a rough sea surface (Fig. 1). The relative magnitude of each contribution depends on the horizontal distance of the source from the receiver, the water depth, the depth of the receiver in relation to the wavelength of the noise radiated by the source, and the roughness of the sea surface (Hudimac 1957).

Spectrograms of propeller-driven aircraft show their acoustic signatures comprise strong narrowband tones superimposed onto a broadband random component. These tones correspond closely to its engine firing, rotation, and propeller blade rates (Ferguson and Lo 1999), with each sound source generating its own set of harmonics. The spectrograms also suggest the mean values of each normalized tone (frequency normalized by harmonic number) are approximately the same. This may be exploited to provide accurate estimates of a composite frequency, , where is the frequency of a harmonic, . In practice, is derived from a weighted least squares fit (Finn and Rogers 2015; Rogers and Finn 2013b).

Signal structure is important because, while the lower harmonics suffer least attenuation, in the atmosphere they are typically masked by the effects of wind noise over the microphones. As a result, it is typically the middle-order harmonics that display the highest signal-to-noise ratio (SNR) for measurements in air. Also, while the SNR is generally less for higher harmonics, many still reach the microphones and they have the advantage that their standard error is divided by the harmonic number (Rogers and Finn 2013b), which offers an advantage when trying to accurately estimate a composite frequency. Finally, in the underwater environment, as we are concerned with the direct wave, it is the lowest harmonics that will likely be least attenuated and offer the highest SNR; and, again, exploitation of the linearity between harmonics will likely pay dividends in terms of improving the accuracy with which the frequency received underwater can be computed.

Full characterization of the underwater sound field requires proper treatment of the air–water and water–sediment interfaces and is beyond the scope of this paper. We may note, however, that surface roughness affects the underwater field in four ways: it reduces the coherent sound field; sound is scattered into directions other than the specular direction; breaking waves introduce air bubbles into the water, impacting the structure of underwater noise by reducing SNR; and errors are introduced into the location of the air–water interface point (IP). In this paper, as the signal source comprises continuous narrowband tones—with the propagation delay computed from the Doppler shift of the refracted direct ray—reduction in SNR is dealt with by introducing uncertainty into the estimate of propagation delay and the errors in IP location are similarly modeled (see section 3).

In general, the magnitude of the direct path is 20–30 dB stronger than other contributors to the sound field (Ferguson and Speechley 2009). However, the transmission of aircraft noise across the air–sea interface occurs only when the angle of incidence is less than the critical angle (Fig. 2). For , the aircraft noise is largely reflected from the sea surface. Typical values for the speed of sound in air and seawater— = 342 m s−1 and = 1524 m s−1, respectively—suggest (based on Snell’s law for a motionless, homogeneous medium) .

Fig. 2.
Fig. 2.

Geometry of the UAV, microphones, hydrophones, and propagation. For , the aircraft noise is reflected from the sea surface. Typical values for = 342 m s−1 and = 1524 m s−1, suggesting .

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

Assuming a point source and that the air and sea are isospeed propagation media (and ignoring any scattering and diffraction), Hudimac (1957) showed that the intensity and horizontal range are given by a pair of parametric equations in terms of the angle of incidence. This enabled several researchers (Ferguson and Speechley 2009; Ferguson 1996; Kutakov and Maslov 2007; Peng et al. 2010) to calculate the sound intensity for transiting aircraft as a function of source altitude and receiver depth.

Generally, as only a fraction of the sound generated by a sound source penetrates water—even at a normal incident angle—large aircraft are used as sound sources for underwater detection trials. Indeed, high SNR has been reported at depths of 120 m at distances up to 400 km from the source if appropriate filtering techniques are used (Kutakov and Maslov 2007). However, experiments by Buckingham et al. (Buckingham et al. 2002b; Buckingham and Richardson 2002; Buckingham et al. 2002a) have shown that single-engine light aircraft (Tobago TB10 and Diamond Star DA40) may be tracked, not only through 30 m of water, but also in the sediment of the seafloor to a depth of 1 m. These signals are visible (SNR > 10 dB) 3–4 s either side of the closest point of approach (Buckingham and Giddens 2004). The aircraft used in Buckingham’s work were flown 66 m above the sea surface and at relatively low velocity (60 m s−1), indicating a potential baseline of around 200 m may be feasible.

We are not aware of any underwater sound field measurements conducted using UAVs like the Aerosonde (www.textronsystems.com) as a sound source. However, as the aircraft has a petrol-driven, dual-cylinder, four-stroke piston engine, and a two-blade propeller, a shaft rotation rate of 6000 rpm generates a rich set of useful frequencies ranging from 50 Hz up to about 1 kHz. Furthermore, the acoustic signature of the UAV running at full speed permits detection using time series correlation techniques at ranges approaching 3 km (Lo and Ferguson 2004; Finn and Franklin 2012), with larger UAVs detectable at greater ranges. This is roughly the same range at which the narrowband tones of a Cessna 172SP can be detected using the same techniques (Franklin and Finn 2014; Wyber 2014). Furthermore, removal of the muffler from the UAV’s engine exhaust has been shown to further increase the overall sound field by a further 6–9 dB (M. Gonella 2014, personal communication), indicating detection ranges (and hence baselines) approaching 200 m or more should be possible if the UAV is flown at sufficiently low altitude (<100 m) and the water surface is sufficiently smooth [see section 2b(3)].

b. The forward problem

As we are considering only the direct path, we assume sound waves are emitted by the UAV, travel through the atmosphere to the air–sea interface, whereupon they are refracted through the water body and picked up by the hydrophones. The ray received by a hydrophone does not generally travel the same path as that received by a microphone, except when the sound enters the sea almost vertically, that is, . We therefore solve this problem as the composite of two: one above and one below the sea’s surface.

1) Propagation in air

Estimates of propagation delay through the atmosphere are obtained using existing UAV-based acoustic tomographic techniques (Finn and Rogers 2016b). Microphone positions are measured, as are the aircraft’s location, velocity, and meteorological data sufficient to compute the speed of sound at the UAV and the unit vector normal to the wavefront emitted by it, (Finn and Rogers 2015). In this initial formulation of the forward problem (and for the purposes of simplicity), the microphones are assumed not to move. In section 2b(3), the reader is referred to the techniques for compensating for microphone motion.

The acoustic signature of the UAV is sampled and time-stamped on board the aircraft and at each of the microphones (Finn and Rogers 2016a, 2015; Rogers and Finn 2013a). A discrete Fourier transform (DFT) is then used to compute the instantaneous source frequency and that received at the microphones, , where is the emission epoch, is the travel time of the ray in air, and the superscript m denotes that the parameters are measured as opposed to computed.

Based on an assumption of a stationary microphone and correspondence of with the values computed from Eq. (5.68) in Ostashev and Wilson (2016),
e1
where is the velocity of the medium at the UAV at time t, is the velocity of the UAV, and is the sound speed at the UAV; accurate measurements of atmospheric travel time are possible, as are sound speed, temperature, and wind velocity profiles (Finn and Rogers 2015).

2) Propagation underwater

The total time taken for the signals to arrive at the hydrophones is
e2
where and are the travel times in air and water, respectively; , and are the UAV, IP, and hydrophone locations; and are the speed of sound in air and water, respectively; and is the integration length along the ray path.
Based on the simple 2D geometry of Fig. 2, for a UAV flying at constant altitude such that it passes directly over the row of microphones located on the sea surface (at sea state zero) with hydrophones at depth directly below we have
e3a
e3b
where is the incident angle of the underwater ray with respect to the vertical, and and are the ranges from the hydrophone to the UAV and air–water interface, respectively. Since , from Snell’s law
e4
where is the ratio of the refractive indices. Substituting Eqs. (3a) and (3b) into Eq. (4) gives
e5
Squaring both sides of Eq. (5) we have
e6
Rearranging Eq. (6) we obtain the following quartic equation in :
e7
We may assume a nominal value for or take hydrological measurements at the microphones sufficient to estimate it and calculate . As the parameters , , and are known, and may be estimated from meteorological measurements on board the UAV, we can now solve for each UAV–hydrophone combination by finding the only positive real root of Eq. (7). There are several analytic approaches to solving such equations (Shmakov 2011). Alternatively, may be obtained approximately by assuming small (Rice and Gray 2000),
e8
Estimates of may now be obtained as and are known, and may be computed from the temperature and wind profiles inferred from the atmospheric tomography.
As the hydrophones are assumed stationary, despite being refracted by the air–sea interface, the signals received at the hydrophone, , are also frequency shifted relative to , in accordance with Ostashev and Wilson (2016, their Eq. 5.68); that is,
e9
where . Equation (9) may be expressed as
e10
where , is the nominal propagation delay for the underwater signal, is the nominal distance from IP to the hydrophone, and is the mean sound speed in water. Expanding Eq. (10) as a Taylor series gives
e11
where represents the Taylor terms of the order of and higher (which we ignore). We may rewrite Eq. (11) as
e12
Based on the values of and obtained from observations on board the UAV, we may now compute the direction normal to the wavefront using
e13
where the group velocity of the signal —Eq. (3.43) in Ostashev and Wilson (2016). The direction of the sound ray from the UAV to the IP, characterized by the unit vector , may be estimated from the straight ray path approximation, the magnitude for which is given by
e14
Based on the values observations of , from Eq. (9), and , which we may derive numerically, we are able to derive an estimate of underwater propagation delay, . We may also apply the techniques of Rogers and Finn (2016) to improve these.

3) Errors in the forward problem

There are several sources of error in formulating travel time estimates this way, including the ray path approximation to represent the propagation of sound, the effects of refraction on the propagating rays, a lack of wave coherence due to turbulence, signal and measurement jitter in the UAV tones, statistical errors in the time–frequency analysis of the received signals, wind noise and interference on the microphones, flow noise over the hydrophones, UAV position and velocity errors, microphone location errors, variations in the sampling rate of the analog-to-digital converter (ADC), uncertainty regarding moisture and salinity along the ray path (such as those introduced due to moisture and salinity gradients near the air–water interface), and the structure and nature of any underwater noise.

In any practical system, errors in will be dominated by errors in calculating the atmospheric time delay and the motion of the microphones on the surface of the water. It is important to note, however, that it is uncertainty in microphone location (which is small) rather than displacement from any nominal location (which may be large) that influences errors in time delay estimation. Approaches for reducing errors in are beyond the scope of this paper, but an assessment of the likely magnitude of the many contributions is contained elsewhere (Finn and Rogers 2015; Ostashev et al. 2008a). The latter, typically induced by the roughness of the water surface, will introduce additional Doppler shift in accordance with Eq. (5.43) in Ostashev and Wilson (2016),
e15
where is the unit vector normal to the received wavefront, is the velocity of the medium at the microphone, is the velocity of the microphone, and is the speed of sound at the microphone.

Accurate determination of for a moving receiver requires parameters to be determined at epoch , which is unknown. However, as measurements are obtained at the microphones sufficient to compute these parameters as functions of time, as per Eqs. (11) and (12), they may also be expanded as a Taylor series and solved (Finn and Rogers 2016a), thereby significantly reducing at least one source of major practical error.

In addition to uncertainty in causing errors in , there will also be contributions due to hydrophone motion resulting from currents. Once again, however, it is location uncertainty rather than displacement from any nominal location that influences errors in time delay estimation. Thus, the location of the hydrophones can be accurately monitored using INS.

The interaction of wind perturbations with the water surface—and other background noise—also contributes to the structure and nature of the natural noise underwater, which reduces SNR. It is necessary to obtain accurate estimates of the harmonics in order to obtain an accurate estimate of acoustic propagation time; however, this can be improved by increasing the sampling time or rate. If the frequency is estimated from the fundamental (nominally 50 Hz) alone, then the required estimation accuracy would need to be better than 0.005 Hz, which from Eq. (12) in Rogers and Finn (2013b) implies that an optimal estimator would require for a 1-s observation time, where is the standard deviation of the noise and is the amplitude of the harmonic signal. Moreover, since the harmonics are linearly related, each measurement of a harmonic tone can be used to augment the estimate of the fundamental frequency (Finn and Rogers 2015).

c. The inverse problem

Based on the geometry of Fig. 3, the inverse problems for the air and underwater media are treated sequentially, albeit very similarly, and in accordance with Finn and Rogers (2016b).

Fig. 3.
Fig. 3.

Geometry for the inverse problem.

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

Spatial variations in temperature, wind, and current fields are parameterized about mean values for the atmosphere (, , ) and the water (, , and ) (Jovanovic et al. 2009). For each UAV location i and microphone j, the atmospheric propagation delays may be written as
e16a
and
e16b
where is the distance from the UAV coordinates to the IP coordinates ; is the distance from to the hydrophone coordinates ; and are the mean sound speeds for the atmosphere and water, respectively; and are the mean temperature of each medium, respectively; is the mean wind speed vector; is the mean current speed vector; is the unit vector in the direction of sound ray ij while in air; is the unit vector in the direction of sound ray underwater; and are the temperature deviations at location X; and are the wind and current deviations; and is the integration length along the ray’s path.
Two networks of weighted 2D Gaussian radial basis functions (RBF) of the form are then used to describe , , , and (Wiens and Behrens 2009; Rogers and Finn 2013c), where r is the distance from the spatial center of the RBF and k is a scaling factor. The RBF centers are then distributed evenly on a grid enclosed by (i) the UAV’s flight path and the microphone array (for the atmosphere), and (ii) the microphone and hydrophone arrays (for the subsea environment). The temperature and wind/current fields are given by
e17a
e17b
e17c
e17d
where Nr is the number of RBF; and are the temperature weights for RBF(j) above and below the sea surface, respectively; and and are the wind and current vector weights for RBF(j) in the x and z directions.
Equations (17a)(17d) may be expressed in two sets of matrix notation,
e18
where and are vectors of temperature and wind/current deviations in the x–z plane, respectively; are vectors of coefficients and , respectively; is a matrix of RBF; and the subscripts a and w denote matrices for the air and water environments, respectively.
Equations (17a)(17d) are also constrained using meteorological and hydrological observations taken on board the UAV, at the microphones, and hydrophones (Rogers and Finn 2013c). These measurements are then combined into a matrix relationship for each medium,
e19
where is a vector containing the propagation delays, temperature, and wind or current velocity measurements; and is an (m × n) matrix described in Rogers and Finn (2013c), noting any temporal or spatial resolution is appropriate for the additional observations. Once the atmospheric inversion has been solved, the travel times, , for the underwater rays are computed by subtracting the atmospheric travel times, , from the total travel times for the rays received by each hydrophone.

Equation (19) may be solved using a number of least squares techniques. However, as time delay tomography is inherently an ill-posed problem, regularization (Lingjaerde and Christopherson 1998; Tarantola 2005) is needed to improve the pose of the inversion, and hence the stability and accuracy of the solution (a well-posed problem is one for which a solution exists; there is only one solution, and this solution depends continuously on the observations, whereas a problem is said to be sparse if its solution derives from a small number of nonzero entities; Snieder and Trampert 1999). Varying the “shape” of the UAV’s flight path also improves the pose of the problem and reduces its sparsity. For example, flying a trapezoidal or trigonometric trajectory (low-to-high, high-to-low altitude) produces ray paths that obliquely intersect the lower regions of the RBF grid from both left and right, as well as rays that intersect at steep angles. This improves the condition and rank deficiency of the normal matrices. Unfortunately, additional flight maneuvers such as this also increase the period over which observations are made, and during which the state of the atmosphere and water body may change. It should be noted, therefore, that the tomographic inversion provides a time-averaged representation of the two media, not a crisp “snapshot” of them.

Unfortunately, the IP at the air–sea boundary is by definition always on the surface of the water. Consequently, improvements in pose and sparsity cannot be generated by manipulating UAV flight paths. However, in any practical system microphones will likely be mounted on small floats and hydrophones suspended beneath them. Multiple rows of hydrophones can therefore be deployed below each float, which would potentially deliver similar improvements to pose. It should be noted, however, that the effects of atmospheric and oceanic layers near these platforms could cause some disturbance to the sound field and the environment under observation. Similarly, the physical and thermal design of the UAV and sensor platforms/buoys would likely influence the atmospheric and oceanic temperature and flow conditions in their immediate vicinity. However, for integration paths of around 100 m—and “near field” effects on the order of a few meters—the overall effect along the ray is likely to be minor. A detailed analysis of the effect of these modifications on the sound field and their impact on the integrated path is beyond the scope of the paper.

3. Simulated tomographic reconstructions

A comprehensive simulation with a coupled atmospheric–water body model of temperature, airflow/currents, and sound propagation is required to prove the concept and realistically estimate the uncertainties and limits of the proposed approach. The authors do not have access to such a dataset and its generation is beyond the scope of this paper. We therefore present simulations using numerical models that show results separately for the two media, and it is necessary to understand the simulations (and their intrinsic simplifying assumptions) in these terms, as they may limit the applicability of the results.

a. Atmospheric simulations

The fidelity with which atmospheric profiles may be reconstructed using the UAV-based techniques described above is reported elsewhere (e.g., Finn and Rogers 2016b). Here we simply state and show that—for appropriate error regimes and UAV flight profiles—it is possible to estimate the target structures (Fig. 4) reasonably accurately (Fig. 5).

Fig. 4.
Fig. 4.

Target atmosphere based on LES. The xz coordinate system is such that the positive x axis is in the direction of UAV travel through the sensor array and the z axis is vertical. The origin of the system coincides with the first microphone.

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

Fig. 5.
Fig. 5.

Reconstruction of synthetic atmospheric wind and temperature variations: (left) a representation of the LES data using RBF at the density and resolution used in the tomographic inversion, and (right) the tomographic estimate with errors equivalent to 0.1-ms noise and 0.5% ray pathlength extension.

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

The assessments in Finn and Rogers (2016b) and the target atmosphere above use a realistic set of atmospheric simulations for daytime convective planetary boundary layers generated by Sullivan and Patton (2011). Large-eddy simulation (LES) is used to generate an atmospheric volume of 5120 m × 5120 m × 2048 m (horizontal × vertical) for a uniform grid mesh of 10243 points, that is, 5 m × 5 m × 2 m resolution. “True” time delays for each ray segment are determined by propagating rays through the known LES profile along the straight-line (unrefracted) path between its start and end points. Numerical integration of Eq. (15) is used to determine the total propagation delay for each ray. The left-hand image in Fig. 5 shows a direct fit of RBF to the target dataset of Fig. 4, that is, the best result that can be expected at 20-m resolution of RBFs. The right-hand image in Fig. 5 shows the results of the error-corrupted tomographic inversion.

An approximate approach is used to estimate errors caused by refraction. The effect of sound speed variations along the ray path are ignored in favor of the effects of ray path extension, noting that to a first order both effects combine to deliver a time delay estimate proportional to the integrated ray pathlength. In accordance with the benchmark regimes of an unstable atmospheric boundary layer suggested by Ostashev et al. (2008a)—that is, 0.06%, 0.14%, and 0.67% for low, moderate, and strong wind regimes, respectively (Finn and Rogers 2015)—a percentage of true delay is then added.

Errors due to front-end signal processing, a reduction in SNR due to wind noise, incoherence due to turbulence, UAV engine vibration, and sampling rate jitter were modeled by superimposing zero-mean additive Gaussian white noise (AWGN) onto the refracted ray lengths, as were the position errors used to model uncertainty in the UAV and microphone locations. The upper limits of the signal processing errors (assessed as between 0.1 and 1 ms) were based on field trials equipment (Rogers and Finn 2013b,a). The position error limits were set at levels commensurate with the high-grade, real-time kinematic carrier phase differential global positioning system (GPS) location, that is, 0.05 m.

b. Underwater simulations

To examine the potential performance of the joint atmospheric–water body technique described in this paper, simulation was also used. As we have no equivalent dataset to the LES atmospheres for the underwater environment, foreground spatial variations underwater were modeled as a set of random coefficients assigned to a uniformly distributed grid of RBF. These perturbations were superimposed onto a uniform background. These synthetic profiles were not intended to be representative of real-world underwater environments but were used to demonstrate the effectiveness of the technique. Also, while in practice the region would likely change during the period of a UAV’s overflight, we assumed all profiles were static. Meteorological and hydrological observations, taken on board the UAV and at all sensors, were also included in the inversions. The 1σ simulated accuracy of these observations was 0.1°C and 0.1 m s−1, respectively, with sampling locations corrupted by AGWN at the level of the stated position accuracy.

Figure 6 shows an underwater structure for an area 200 m long × 50 m deep, generated by randomly assigning coefficients to a uniformly distributed 20 m × 5 m grid of RBF, as per Eqs. (17c) and (17d). The foreground variations have peaks/troughs of ±3.7°C and ±8.3 m s−1 and are superimposed onto a uniform background body of water of 15°C and 0 m s−1, respectively. The coordinate system is such that the positive x axis is in the direction of UAV travel through the sensor array and the z axis is vertical. The origin of the system coincides with the first hydrophone.

Fig. 6.
Fig. 6.

Target ocean structure based on random coefficients assigned to a uniformly distributed grid of RBF separated by 20 m × 5 m in the x and z directions.

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

Simulations without errors are unrealistic, so Fig. 7 shows the results of the simulated water structures of Fig. 6 with errors due to signal processing, flow noise, ADC jitter, etc. of 10 μs; residual refraction of 0.01%; and GPS of 0.02 m—that is, the best that could realistically be expected (Finn and Rogers 2016b; Parkinson and Spilker 1996). The root-mean-square errors (RMSE) are 0.6°C and 1.1 m s−1 for temperature and wind speed, respectively.

Fig. 7.
Fig. 7.

Tomographic reconstruction for an underwater area 200 m long × 50 m deep, generated by randomly assigning coefficients to a uniformly distributed 20 m × 5 m grid of RBF.

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

The simulation depicts an array of 20 microphones located along the 200 m baseline, with a hydrophone 50 m below each. The UAV was flown along the line of sensors at a constant speed of 28 m s−1 and at an altitude of 100 m. The RBF grid used in the reconstruction was 20 m × 5 m, that is, aligned with the set used to generate the foreground structure. The sound ray paths propagating between the UAV and the microphones were calculated based on the assumption that the UAV’s engine firing rate was observed using a simulated signal processing regime that uses an ADC sampling rate of 51.2 kHz and a DFT size of 215 with 75% overlapping block, and 4 times the oversampling, that is, samples at a rate of ~5 Hz. The microphones and hydrophones were sampled at the same rate, with all sensor sampling synchronized to the 1 pulse per second timing signal of the GPS.

Figure 8 shows the results of a simulation based on the same conditions as in Fig. 7, except that two rows of hydrophones were used, with each row separated by 25 m. Increasing the number of rows improves results, with the RMSE falling to 0.1°C and 0.15 m s−1. Figure 9 shows the results of a simulation for the same conditions, but with errors increased to 0.1 ms for signal processing, flow noise, ADC jitter, etc.; 0.5% for refraction; and 0.05 m for UAV and sensor location. These errors are close to the upper limit of the inversion technique. The top image of Fig. 9 shows the results for a single set of hydrophones at a depth of 50 m, and the bottom image shows two sets of hydrophones separated in depth by 25 m. The RMSE for these simulations are 0.7°C and 1.9 m s−1 (top) and 0.6°C and 1.4 m s−1 (bottom), respectively. Figure 10 shows results for doubling the horizontal density of sensors (top) and increasing the number of rows of hydrophones to four (bottom).

Fig. 8.
Fig. 8.

Tomographic reconstruction for the same conditions as in Fig. 7, but with two rows of hydrophones separated in depth by 25 m. Errors due to signal processing, flow noise, and ADC jitter set to 1 μs; refraction set to 0.01%, and GPS set to 0.2 cm.

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

Fig. 9.
Fig. 9.

Tomographic reconstruction for the same conditions as in Fig. 7, but with errors due to signal processing, flow noise, and ADC jitter set to 0.1 ms; refraction set to 0.5%; and GPS set to 0.05 m. (top) Image simulates one row of hydrophones at 50 m, and (bottom) image simulates two rows separated by 25 m.

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

Fig. 10.
Fig. 10.

Tomographic reconstruction for the same conditions as in Fig. 9, but the (top) image simulates two rows of hydrophones (separated horizontally by 5 m) at depths of 25 and 50 m, and the (bottom) image simulates four rows (separated horizontally by 5 m) at depths of 12.5, 25, 37.5, and 50 m.

Citation: Journal of Atmospheric and Oceanic Technology 34, 3; 10.1175/JTECH-D-16-0128.1

4. Concluding remarks

A technique for determining temperature and velocity structures of the atmosphere and water bodies based on a single set of sensors is presented and examined through the use of simulation. The approach exploits the mutual transparency of the two media to low-frequency sound waves to obtain a single set of observations from the natural acoustic signature of a UAV as it passes over an array of surface and underwater sensors. The exact travel times of integrated rays are related to a linear model of the constituent terms of sound speed, and these observations then are inverted to estimate the inhomogeneous structure of both regions.

Simulations indicate that for signal processing, refraction, and sensor location, errors of 10 μs, 0.01%, and 0.02 m may add value to hydrological and atmospheric investigations. While these levels of accuracy may be challenging to achieve in practice, they are nevertheless plausible, and the use of multiple rows of underwater sensors improves the accuracy and resolution of the solution. Errors on an order of magnitude less accurate, however, still deliver information about the underwater environment in terms of its broad structure.

As presented, the inverse method is a simple first step. More complex and powerful estimation techniques may yield improvement in accuracy and resolution. Moreover, contributions to the underwater sound field other than those created by the direct path could be exploited to extend its utility. These might include scattering, which would require modeling the sea surface above level zero, and/or detection of the evanescent wave, which would require the sensor depth to be small compared to the wavelength of the acoustic energy. Alternatively, the bottom reflected wave could be used to determine the structure of the water body beneath the hydrophones. Similarly, unmanned underwater vehicles might be used in place of hydrophones, although the motion of both vehicles would need to be accommodated in the Doppler shift equations.

Acknowledgments

The authors are grateful to the Australian Research Council for its support to this project (LP130100695). We are also grateful to our colleagues Maurice Gonella of Aerosonde, Peter May and John Nairn of the Bureau of Meteorology (BoM), and Greg Holland and James Done of the National Center for Atmospheric Research (NCAR), as their ongoing assistance with and support of this project have been most valuable. We are also grateful to Dr. Adrian Jones and Dr. Vladimir Ostashev for their helpful comments on some of the concepts in this paper.

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