Proceedings of the Institute of Acoustics

 

Is acoustic noise and variability a nuisance or a potential tool?

 

C. H. Harrison Emeritus, Centre for Marine Research and Experimentation, Viale S. Bartolomeo 400, 19126 La Spezia, Italy

 

 

1 INTRODUCTION

 

 

Fluctuations, variability, uncertainty, incoherence, ambient noise and reverberation are some of the things that acousticians have to put up with when trying to use sound to detect, localize and classify objects under water or in air. Each of these topics leads to interesting domains of physics and engineering research in its own right, and an integral part of this research is the signal processing or data processing that is used to extract whatever is deemed to be the "signal".

 

 

Recent developments into robotics have introduced the possibility of multiple interactions and communications between autonomous underwater vehicles (AUVs) and gliders with real time on board modelling and processing. Analysis of such systems involves game theory, pursuit theory, and much more complicated operational analysis. In addition to obvious defence applications, on board sensors and processors can be used in oceanographic and geophysical applications where a glider, for instance, can continue zig-zagging between surface and seabed, recharging at the surface, almost indefinitely.

 

 

Many man-made sounds from shipping, machinery or sonars may also be loud enough or persistent enough to be irritating to humans and marine mammals, and so any studies of these sources and propagation from them usually tend towards reducing or getting rid of them. Although ambient wind and wave noise may be a hindrance for a detection system it contains a huge amount of encoded information about the ocean boundaries which may be extracted in various ways. Even though propagation from distant sources may encounter the surface and seabed many times the bottom reflection loss may be determined from the directionality of the noise as received, for example, by a vertical array^1,2,3. A completely different technique originated in ultrasonics^4,5,6, and has been developed in seismology ("seismic interferometry")^7,8 and in the ocean acoustic horizontal plane with ships^9-12 and the ocean vertical plane ("passive fathometry")^13,14. In all cases there are two "signals", either from spatially separated receivers or one receiver and its image in a layered seabed or a target, whose cross-correlation is related (as will be shown in Sect. 2) to their Green's functions or the impulse response from one to the other.

 

 

A summary of various techniques was given in Ref.15; here we concentrate on passive fathometry, where a drifting 32-element vertical array with no man-made source can produce a sub-bottom profile with two-way bottom penetration of 50m or so^13,14, that is indistinguishable from that produced by a boomer or low frequency echo sounder. A variant of this method, also discussed, has already been shown to be capable of detecting a point target¹⁶. An interesting question now is, can these same techniques be applied with more compact arrays from quiet platforms such as gliders that move (intentionally or unintentionally) in the vertical as well as the horizontal. This paper addresses some of the problems involved.

 

 

2 CROSS CORRELATION OF NOISE IN GENERAL

 

 

The cross-correlation of the time series received on a pair of hydrophones in a noise field is closely related to the impulse response that would be received on replacing one of the pair by a sound source. More formally it can be stated that the Green’s function between two points can be recovered from the cross-correlation of the noise time series measured at those points. Various conjectures, assertions and proofs exist in slightly different contexts. Here we follow Ref. 10 in asserting that the Green’s function is proportional to the time derivative of the cross-correlation function. The credibility of this assertion can be reinforced by a simple example (taken from Ref. 10). The normalised cross-spectral density between a pair of points in a uniform noise field is

 

 

By taking the Fourier transform this becomes the cross-correlation function (– a function of time)

 

 

The time derivative is then

 

 

which is the sum of the time domain Green’s functions between the hydrophones, taken either way round.

 

 

To understand this generally, imagine the paths between a single noise source (on the sea surface, say) and two separated receivers. The travel time difference between these paths has a maximum or stationary point when the noise source is on the projection of the straight line through the two receivers. Thus there is a sudden truncation in correlation offset time which corresponds exactly to the separation of the two hydrophones. The derivative with respect to time of this step function is a delta function, so the result is the time domain Green’s function between the hydrophones.

 

 

In the case of sub-bottom profiling (passive fathometry) the real, 'single' receiver is an array of hydrophones which is simultaneously steered to endfire up and endfire down. The 'up' beam sees the noise source directly and the 'down' beam sees its reflection in the seabed (or in a target). In effect there is the real array and an image array supplying two signals that are then cross- correlated. Given sufficient integration time the result is identical to the impulse response from the centre of the array.

 

 

3 SUB-BOTTOM PROFILING WITH NOISE

 

 

Siderius, et al.¹³ applied this approach to the domain of sub-bottom profiling by cross-correlating all possible pairs of hydrophones from a drifting vertical array and then aligning their peaks in time according to position on the array. This clearly demonstrates the link between correlating hydrophone pairs and correlating beam-formed time series. Harrison and Siderius¹⁴ derived a formula for the absolute value of the normalised cross-correlation of the upward- and downward- steered beam time series and also the decorrelated background.

 

 

The derivation assumed there to be uncorrelated time sequences emitted from random points on the sea surface, received by a M-element vertical array a distance z above the reflecting seabed of reflection coefficient R. The absolute peak value of the forward difference (indicated by ) of the normalised correlation of the time series (sampled at f_s) is then

 

 

where c is the speed of sound, L is the array length, y is the ratio of sample frequency to design frequency for the array: y = f_s/f_o. The term β is a factor, of order unity, that allows for the variation of noise spatial coherence across the (wide) processing band. In the intermediate term of Eq. (1) one can recognise the impulse response multiplied by c / f_s . In contrast the normalised background variance is the reciprocal of the number of independent samples in the time series. Given white noise this number corresponds to the total number of samples during the integration time. Otherwise with filtering or a finite band the number will be smaller and the background will be higher.

 

 

This formula for peak values and background levels was reconciled with a number of simulations using straight ray acoustics and a large number of random number sequences emanating from spatially randomised locations above a 32-element vertical array [Fig. 2(a,b)]. It was shown¹⁴ that one can indeed recover a signed impulse response from multiple layers, and that a correlation peak is still obtained when the seabed is tilted although naturally one must use upward and downward beams (both) steered at the same angle as the tilt. Another processing option is to take the Hilbert transform of the cross-correlation (or its time derivative). The price for the added robustness is loss of the impulse response’s sign [Fig. 2(c)].

 

                                                                                                                           

 

                                                                           

 

Figure 2: Simulation of the cross-correlation resulting from (a) a single 11s file showing a peak at two-way travel time corresponding to 60m, (b) 81 coherently added 11s files, (c) the Hilbert transform corresponding to 81 files.

 

 

The technique has produced sub-bottom profiles with a drifting VLA in four series of experiments to date BOUNDARY2002, BOUNDARY2003, BOUNDARY2004 and CLUTTER2007 in the vicinity of the Malta Plateau and Ragusa Ridge south of Sicily. Snapshots of layering can also be derived from various moored VLA experiments. These profiles have been verified against seismic boomer records along the same drift track¹³ and layering has been seen as deep as 55m inside the sediment. In addition the correlation amplitudes were shown by Harrison and Siderius¹⁴ to agree well with the predictions of Eq. (1). An example from 2003 using a 32 element array with a band between 2 and 4kHz is shown in Fig. 3. The correlation during each 10 second file was incoherently averaged over about 10 files, i.e. a couple of minutes in total, which provides adequate spatial resolution at this drift speed. The Hilbert transformed cross-correlation, i.e. the impulse response, for time 23:00:00 is shown in Fig. 4(a). Figure 4(b) shows a blow-up of the main signed peak near 115m path length with its Hilbert envelope. One can see that there are phase differences between the various layer arrivals.

 

                                                                                                 

 

Figure 3: Sub-bottom profile from drifting MFA on the Malta Plateau (2003) showing deep echoes.

 

                                                                                                  

 

Figure 4(a):  Hilbert transformed impulse response (average of 100 files) during the 2003 drift (time 23:00:00), (b) blow-up of the main peak showing correlation amplitude with Hilbert envelope.

 

 

4 TARGET LOCATION WITH NOISE

 

 

It was noted in the 2003 and 2004 experiments¹⁴ that there was a persistent echo at respectively 21m and 24m two-way path length. This was suspiciously close to the path length expected for the array’s ballast weight. During the CLUTTER2007 experiment an effort was made to measure the positions of all potential scatterers, floats, weights, etc. hanging from the array assembly, in order to make a more systematic check of travel times. In addition, experiments were done 16 with and without three glass spheres attached below the array. Figure 5(a) shows that reflections were indeed seen from the ballast weight at 19.5m below centre (39m round trip), the stainless steel end cap of the array at 6.5m (13m round trip), and two of the three glass spheres when they were attached [Fig. 5(b)]. Whether one thinks of the source as a small patch of sea surface noise sources above the array or an effective point impulse source in the middle of the array one might expect the lower spheres and the ballast weight to be slightly obscured by the top sphere. This is believed to be the reason for the unchanged end cap response but weakening lower sphere and ballast responses in Fig. 5(b) when compared with Fig. 5(a)).

 

                                                                                      

 

Figure 5: Correlation amplitude, to be interpreted as an impulse response, showing labelled returns (a) excluding the glass spheres, (b) including the glass spheres. Black triangles indicate measured two-way path lengths to the ballast weight, the array end cap, and the three spheres.

 

 

The strength of the echoes from the end cap, the unobscured ballast weight, and the upper glass sphere could be predicted by a slightly modified version of Eq. (1). Using the normalisation implicit in Eq. (1) the bottom echo is independent of reflection coefficient magnitude since, with up and down beam time series denoted respectively by U, D, the cross-correlation numerator is U * D and the denominator is the product of the standard deviations {< U² >< D² >}^1/2. Thus the magnitude of the reflection coefficient, which is contained only in D, cancels. One could just as easily have used a normalization where the denominator was <U²> instead. Thus the magnitudes of the noise sources still cancel but the result does depend on the magnitude of the reflection coefficient. In the context of demonstrating a target detection it is desirable to avoid the first normalization because, although the peak height in the numerator would depend only on the target (through D), the standard deviation of D in the denominator would also depend on reflections from the seabed. With the second normalization the resulting peak height depends only on the target.

 

Thus for a point target of target strength TS = 20 log₁₀ ( s ) at a depth z beneath the array the peak strength is

 

 

Dependence of this amplitude on distance goes as z² exactly as one would expect for a point scatterer a distance z from a monostatic active sonar.

 

 

5 TARGET DETECTION GEOMETRIES IN GENERAL

 

 

In the above experiments the targets were immediately below the array, but it can be seen (by analogy with the tilted seabed simulation^14,16) that targets can also be detected and ranged when away from the vertical line. The fact that in laboratory ultrasonic experiments (on centimetre scales) targets have been detected by autocorrelation of a single transducer adds credibility to this assertion⁴. Indeed this is not the first underwater demonstration of target detection with noise, but the cross-correlation approach is novel in resolving the target range – it operates like a passive radar. In this respect it is quite distinct from acoustic daylight^17,18,19 which, being an analogue of daylight vision, resolves a two-dimensional angle.

 

 

Generally one can determine three-dimensional geometries for which noise-cross-correlation will produce a "target echo" by considering a surface of distant sources that surrounds a target and a receiver. The essential ingredient is a stationary value of travel time difference between the direct path from a source to the receiver and the indirect path from the same source via the target. Now construct a straight line through the target and receiver out as far as the sources in both directions. Sources on that line and behind the receiver must result in a well-correlated direct path and a target backscattered path with path difference being twice the range from receiver to the target. Similarly sources behind the target will result in a well-correlated direct path and a (weaker) forward scattered path with a path difference of zero. Whether this latter path could be distinguished in practice depends on details of the processing since one already expects artefacts near zero delay. However the first type of "target echo" is always available given sound sources behind the receiver.

 

 

In the horizontal plane the backscattered paths will provide a detection range given sufficient integration time. A horizontal or planar array would both reduce the integration requirement and provide horizontal directionality. The same reasoning applies to the vertical plane and the experiments described in Sect. 4 except that, typically, the noise sources only exist above the target / array pair and not below. However, because the sources extend to the horizons the target can, in principle, be detected anywhere as long as its depth is greater than the array depth.

 

 

6 SOME OTHER IMPORTANT CONSIDERATIONS

 

 

6.1 Noise directionality and coherence

 

 

All the above methods rely on ambient wind noise directionality and its resulting coherence, a formula for which was given by Harrison²⁰. Making the substitutions in terms of elevation angle θ, (complex) reflection coefficient R , and water depth H

 

 

this can be rewritten for a pair of vertically separated hydrophones at depths z_n , z_m as

 

 

This makes the symmetry of the coherence clear (in the limiting case of infinite integration time). The first two terms depend on hydrophone separation whereas the second two depend on average depth. If the separation is uniform, when the coherence is written as a cross-spectral density matrix (CSDM) the first pair form a Toeplitz matrix Tn,m with bands parallel to the main diagonal; the second pair form a Hankel matrix Hn,m with bands in the anti-diagonal direction. Thuss

 

 

This separation is important because it is clear from Eqs.(4,5) that the Toeplitz part contains no information whatsoever about bottom or layer depths (contained in the complex argument of D). Similarly the Hankel part is always real and cannot discriminate between up and down components as is necessary for bottom reflection loss estimation¹.

 

In practice the CSDM is estimated by Fourier transforming blocks of hydrophone data and time- averaging, the aim being to average over all angles the product of arrivals at z_n, z_m. Thus at each frequency.

 

 

However, since the hydrophones are omni-directional, each element of the vector x is already averaged over angle. Effectively the product in Eq.(7) contains angular cross-terms that need to be eliminated by averaging. It can be shown that, surprisingly, the Toeplitz and Hankel matrices can be found separately and directly from the vector of inputs x at each frequency. Taking the spatio- angular Fourier transform of x produces an estimate of the far-field complex source amplitude aˆ(θ) for all angles –π/2 < θ < π/2. Then

 

 

 

and

 

 

where z_T is the offset from the diagonal (z_T = z_n − z_m) and z H is the offset from the anti-diagonal (z_T = z_n + z_m).

 

 

In all of the above techniques, whether comparing up-to-down intensity or cross-correlation, one needs to extract a response term (at each frequency) of the form v^†Cw where v and w are potentially different steering vectors: for passive fathometry v would be an up beam and w would be a down beam; for reflection loss measurement they would be both the same for each steered angle. This mean response < v^†Cw > can be computed in three ways: 1) by averaging C then using matrix algebra, as is, 2) by noting that v^†<C>w = <v^†xx^†w> =  <v^†x><w^†x>^† and computing the latter two terms separately without matrix algebra, or 3) by computing T and H as above then summing them to form C . There are clear differences in computation time (method 2 is the fastest), but the main interest is in the potential for reduction in integration time by Fourier transforming from x to aˆ. Unfortunately this is an illusion since the results are identical.

 

 

6.2 Array robustness

 

 

The BOUNDARY experiments used a fairly robust array of 32 elements occupying about 6m of tubing several cm in diameter with a steel strengthening cable and 150kg ballast weight. One further aim, before the advent of gliders and AUVs, was to construct a demonstrator for an air- dropped vertical array with processor and radio transmitter, etc. It is straightforward enough to build a light weight, possibly expendible, array and on-board processor, but the experimental version cannot be expendible and it needs some kind of radio buoy. The problem is that even a specially built small radio buoy with a mast and batteries is still big enough to drag the 'vertical array' to near horizontal and also the inevitable swaying of the mast with wave motion tends to jerk the array and make unwanted local noises. This annoying 'Catch-22' meant that although the light weight array could be used to investigate bottom reflection loss through incoherent noise directionality measurements as in Ref.1, it could not be used for cross-correlation measurements.

 

 

6.3 Adaptive Beam Forming (ABF)

 

 

Siderius²¹ demonstrated that interference from local shipping could be dramatically reduced by using adaptive beam forming (ABF) to form the up and down beam (see Ref.22 for exact formulation). In addition it is possible to use a wider frequency band. At very low frequencies all hydrophones tend to be highly correlated and the steering vectors consist entirely of '1's so for conventional beam forming (CBF) low frequencies need to be filtered out. However ABF seems to be more tolerant of these effects and can increase the band from the design frequency down to quite close to zero.

 

A disadvantage of ABF is that it requires inversion of the CSDM, and for this one needs either a full- rank matrix, which takes more averaging time, or to apply palliatives, such as diagonal loading²³. Another point is that the up beam is calculated independently of the down. So although, by definition, the amplitude of the optimal beams in the up and down direction is the same (i.e. unity), their beam width may not be the same so the response to locally uniformly distributed noise may be different. This could be a problem for reflection loss estimation, however it can be shown by experiment, simulation, and theory that, surprisingly, the resulting ABF impulse response is identical to the CBF but with reversed sign²² 

 

 

6.4 Rank and invertibility of the coherence matrix

 

 

The rank of a matrix is the number of significant eigenvalues it has. Without averaging (or changes to the noise field) the term xx^† in Eq.(6) has rank equal to one, because each row is demonstrably a simple multiple of the first row, whereas summing N realisations of a N x N matrix reaches full rank, N.

 

 

One can estimate the rank of the CSDM by constructing a circulant matrix that is close to it. A circulant matrix is a special case of a Toeplitz matrix where each row is a repeat of the first row but cyclically shifted. This has the useful property that its eigenvalues are exactly the discrete Fourier transform (DFT) of its first row. With a noise directionality D (sin θ) and design frequency f_o the νth eigenvalue λ_ν at frequency f can be shown to be

 

 

 

 

So the distribution of eigenvalues mirrors the noise directionality function D (sin θ) but scaled by f_o /( Nf ). If the directionality has width sin  o then the rank of the CSDM is

 

 

 

 

Thus for typical acoustic applications, regardless of averaging, this is always less than N so the CSDM is always rank-deficient and therefore not invertible without employing palliatives such as "diagonal loading" (see Ref. 23). There are a couple of alternatives before reverting to CBF: one is to calculate the hydrophone weightings in a separate run since the interference field changes only slowly; another is to use a noise model (such as Eq.(4)) to calculate fixed ABF weightings.

 

 

6.5 Array vertical motion and Doppler

 

 

The cross-correlation technique was less successful in the 2004 experiments¹⁴ for two reasons: firstly the location was deliberately more ambitious being over a rocky seabed, and secondly the sea state was high. As a rule wave noise and 'white horses' are beneficial, but a swell leads to vertical oscillation of the array which spoils the time correlation. Although the array was well away from the surface it was connected to its radio buoy by a curved line of slightly buoyant floats, and the ballast mass and the buoyancy force predicted a roughly 10 sec period with unknown amplitude. For most of the 2004 data integration times much longer than 10 sec were required, which results in blurred depth resolution, however one section was found²⁴ where there were indeed clear oscillations of about 1m vertically with period about 7s.

 

 

There are two problems. One can be seen as a simple time domain problem where the correlation peak simply shifts during the integration and causes blurring. Another problem is the influence of Doppler on the correlation itself. When the array moves downwards, although its velocity may be extremely low, the direct path from the surface is down-shifted in frequency, i.e. stretched in time, while the reflected path is up-shifted in frequency and shrunk in time. This shrink and stretch over the integration time leads to a slight mismatch. One can see that this is not the same phenomenon as the simple blurring by imagining the array to remain stationary at all the up / down positions but jumping from one to the next. This results in blurring but with no Doppler.

 

 

Simulations have shown that provided one knows the motion one can counteract it and increase the integration time beyond the oscillation period. However, other than exhaustive trial and error there appears to be no 'hands-off' solution where one can process experimental data without knowing the motion.

 

 

7 CONCLUSIONS

 

 

It is possible to extract information about reflection coefficients and sub-bottom layering from ambient wind noise with a drifting directional array. By steering an 'up' and 'down' beam then cross-correlating their respective time series one can create a sub-bottom profile equivalent to that of an active system. Generally the time differential of the cross-correlation is equivalent to the impulse response (or Green's function) when there is a stationary value of a travel time difference. Experimental examples have been seen with two-way sediment penetration of more than 50m. A current challenge is to be able to mount such an array system on, for instance, a glider, and to cope with the effects of vertical motion on the necessary time integration.

 

 

It has been demonstrated experimentally that the same approach can be applied to detection of a target. In the usual geometry in the vertical plane the target must be at a greater depth than the array but otherwise can be offset from the array axis. To understand what geometries are possible one need only consider the travel time differences from noise source to the array directly and via the target. In principle a target could also be detected in a more general geometry in the horizontal plane using, say, a horizontal planar array, but this has not yet been experimentally verified. In both the target detection and the sub-bottom profiling case one can also make use of the correlation peak amplitude.

 

 

It was shown that the CSDM naturally separates into a Toeplitz and a Hankel part, and furthermore these two can be extracted. This leads to three variants of CBF processing, each with a different computation time and rate of convergence. The first of these simply evaluates < v † Cw > using matrix algebra; the second evaluates it through the separate terms  v^†x> and <w^†x>^† ; the third evaluates T and H then sums them to form C. In contrast ABF processing improves performance by reducing interference from shipping, but the necessary inversion of the CSDM requires either full rank or application of palliatives such as diagonal loading. A formula was given for the rank, and it was shown that the CSDM is always rank deficient in any acoustic application where the frequency is less than the design frequency, so some palliatives are always required.

 

 

In bad weather (2004) the array, though well submerged, oscillated vertically which interferes with the correlation process and smudges the impulse response. Smaller arrays mounted on gliders need longer integration times but at the same time have to cope with the implied vertical motion of the glide. Even with a low vertical drift speed component the 'up' and 'down' time series are stretched and shrunk differently which can be seen as a Doppler problem. Knowing the motion beforehand the stretch/shrink can be corrected out enabling integration times that are longer than the oscillation period, but otherwise this question is ongoing.

 

 

8 ACKNOWLEDGEMENTS

 

 

The author thanks the Captain and crew of the NRV Alliance and CMRE engineering staff for facilitating collection of noise data during the three cited BOUNDARY experiments, and more recently CMRE for supplying an Emeritus office and other facilities after his retirement. He also thanks Steve Robinson, NPL and Peter Dobbins for suggesting this talk.

 

 

9 REFERENCES

 

 

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