Proceedings of the Institute of Acoustics

 

Passive synthetic aperture sonar processing with a thin towed array

 

A. Mantouka, SEA Ltd, 17 Beckington Castle Corner, Somerset, BA11 6TA, UK
G. Verwey, SEA Ltd, 17 Beckington Castle Corner, Somerset, BA11 6TA, UK
C. Tucker, SEA Ltd, Pottington Business Park, Riverside Rd, Barnstaple, EX31 1LY, UK

 

1 INTRODUCTION

 

Passive Anti Submarine Warfare (ASW) operations benefit from an improved Signal-to-Noise Ratio (SNR) and bearing resolution at low frequencies. For the scenarios involving slow moving targets Passive Synthetic Aperture Sonar (PSAS) appears to offer these sonar performance enhancements by constructing a larger virtual aperture from hydrophone data during towing. An important prerequisite is the signal spatial and temporal coherency during aperture formation. Historical data on spatial coherence exhibit large spread, conservative values for the spatial coherency length ( 𝐿_𝑐 ) are 𝐿_𝑐 ~20-40 𝜆 for shallow and 𝐿_𝑐 ~80 𝜆 - 100 𝜆 for deeper waters respectively [1]. The temporal signal coherence is a property of the target and for this work stable signals are considered. In the past, the most successful PSAS processing appeared to be the Extended Towed Array Measurement (ETAM) method [2], which coherently combines acoustic signals arriving at a moving towed array by compensating the signal phases with a factor that corrects fluctuations experienced during the coherent integration time. Phase corrections are carried out by correlating overlapped array elements as outlined in [2], hence the processor is encountered with name ‘overlap correlator’. However, correlation tends to perform badly in noise and the essential problem in PSAS, i.e. to estimate accurately the phase correction factor from consecutive noisy measurements with positional inaccuracies, has remained. There have been attempts to improve the correlator performance, for example by increasing the correlation averages [3], but if the processing is fundamentally based on traditional Fourier correlation, limited improvement is achieved. A way to overcome the limitations of a traditional Fourier transform, is to apply a less conventional, optimization based, spectral estimation method (an example is presented in [7]). Here a sparse spectral estimation method has been applied with a Basis Pursuit Denoising (BPD) algorithm. The SNR of the received signals is enhanced prior to obtaining the phase correction factor from the computed Fourier coefficients. Details on the processing are given in section 2. The data was acquired during the Extra Large Uncrewed Underwater Vehicle (XLUUV) trial which took place in July 2022 under the Defence and Security Accelerator (DASA) funded project ‘Littoral Waters ASW and Acoustic Intelligence Gathering from XLUUV with the Krait Array, a Thin Line Towed Array’. The trial and results are discussed in section 3.

 

2 THEORY

 

Synthetic aperture techniques rely on methods of extending the sensor aperture beyond the physical aperture by assuming virtual hydrophone positions along the sensor track. A key aspect of the sonar processing is the determination of the receiver position during aperture formation. Typically, Inertial Motion Sensors (IMU) do not offer an accurate enough solution for this purpose and synthetic aperture sonars (and radars) deploy autofocusing methods to implement phase corrections. The ETAM algorithm with the ‘overlap correlator’ introduced by Stergiopoulos and Sullivan [2] is a kind of autofocusing method to determine the phase of the virtual hydrophones, where the cross-correlations of signals on multi-pairs of overlapped hydrophones are estimated and then the average value of the cross-correlation phase angles is taken as the estimation for the phase correction factor. For this paper, the signals have been denoised with the method described in section 2.1, followed by PSAS beamforming using a variation of the ETAM method of [2], with finer overlap averaging, as described in section 2.2.

 

2.1 Basis Pursuit Denoising

 

The Basis Pursuit algorithm (BP) is a fundamental optimisation technique encountered as a cornerstone in many modern applications such as compressive sensing and Artificial Intelligence algorithms. BP seeks to decompose a signal into individual time-frequency (t-f) atoms from a predefined dictionary. Time-frequency atoms are discrete-time elementary waveforms populating the dictionary. The dictionary is selected according to the application. The aim is to obtain a representation domain (i.e. dictionary) in which the signal components of interest can be efficiently represented. For narrow band stable tonals, the dictionary consists of the Fourier coefficients of the signal. The measured signals are represented as:

 

 

where 𝐴 is the dictionary, and 𝑛 is the noise. The sparse approximate decomposition of 𝑦 also known as basis pursuit denoising and is formulated as an l2- l1 optimisation problem:

 

 

Where the matrix 𝐴 satisfies the equation:

 

 

𝐴^𝐻 is the Hermitian of matrix 𝐴 , 𝑝 is a positive number and 𝐼 is the identity matrix. Equation (2) is a cost function. The parameter 𝜇 is the regularisation factor. It represents the weight which determines the desiredenergy ratio between the l1 sparsity component and least square error term (l2 norm term). The l1 norm as the regularization term tends to favour larger spectral values and sets the smaller ones equal to zero. For the analysis presented in this paper, the parameter 𝜇 was chosen such that, after application of soft thresholding, the additive noise is suppressed by eliminating the basis coefficients with small absolute values which are attributed to the noise. The method is detailed in [5] and assumes a priori knowledge of the variance of the additive white Gaussian noise ( 𝑛 ). The Basis Pursuit Denoising (BPD) problem, described by equation (2), is a convex problem which requires a fast iterative algorithm. The most efficient ‘off the shelf’ algorithm was found to be the Split Augmented Lagrangian Shrinkage Algorithm (SALSA), developed by Afonso et al. [4]. Their algorithm calls for the so-called ADMM parameter as a user defined step size parameter affecting the convergence speed of the algorithm. The results shown here were produced with a ADMM parameter value equal to 500 with 100 of iterations.

 

2.2 Beamformer description

 

 

Figure 1: Schematics of ETAM with K snapshots between 50 % overlaps

 

The PSAS algorithm implements a range focused beamformer in the frequency domain. It considers the array motion Doppler and implements bearing compensation between the overlapping cycles prior to forming the synthetic aperture, see Figure 1. For range focusing, the beamformer’s steering vector applies a second order approximation to the distance (r) between the target and the hydrophone location, the signal model for the steering vector is given for each range bin by:
 

 

The frequency entering the beamformer is corrected for the Doppler and the angle θ for the estimated bearing rate change due to the array motion. For the geometries considered here, the array is moving with a velocity u in the positive x direction of an x − y coordinate system. The direction of propagation of the signal is at angle θ with respect to the normal to the direction of motion of the receiver. The radial frequency of the received signal is Doppler shifted to the frequency:

 

 

The sign of the product 𝑢sin 𝜃 determines the sign of Doppler and depends on the direction of motion, i.e. positive when the array approaching the target. The PSAS results were compared with the Conventional Beam Former (CBF) results using the same steering vector (equation 4) and doppler correction (equation 5). The CBF was implemented in the frequency domain and for a like-with-like comparison the beamforming scheme and processing time was kept the same for both beamformers, i.e. the time taken to form the image with one virtual aperture for PSAS equals the CBF processing time. For all results presented in section 3 the PSAS aperture is double the physical aperture and is formed by two cycles of 50% overlap. The phase estimate resulted from averaging K=37 snapshot pairs between cycles.

 

3 XLUUV TRIAL

 

The trial was conducted under the UK Defence and Security Accelerator (DASA) program titled ‘Littoral Waters ASW and Acoustic Intelligence Gathering from XLUUV’ with industry participation from SEA Ltd, MSubs and Sonardyne. The main objective of this project was to demonstrate integration, deployment and operation of an XLUUV, ASW system utilizing the Krait Array thin line passive towed sensor (Figure 2) and associated Krait Sense processing. Successful integration results were obtained within the scope of the trial and are discussed in a separate paper of [6].

 

For this paper, data from specific runs of the trial were used outside the remit of the scope of the program. The purpose here is to investigate the possibility of inserting PSAS into the standard passive processing chain. Array data were processed offline using a PSAS prototype software and details are presented below.

 

Figure 2: On the left, photo of XLUUV, 9 m and 9 tonne platform developed by MSubs in collaboration with DSTL. On the right a photo of the Krait array

 

3.1 Description

 

The trial runs discussed here took place on the 21^st of July 2022, in the English Channel, approximately 10 km south of Plymouth; Figure 3 shows a map of the area.The water depth was approximately 50 m with almost constant sound speed (1507 ±1 m/s). During the day the sea state was low, however the acoustics of the area were challenging due to high levels of multipath and shipping in the area. The XLUUV was programmed to sail on a race track pattern and achieved during the runs a constant speed of 5.5 knots over ground at a depth of approximately 28 m. The acoustic source was deployed by another vessel in the direction broadside of the array, shown in Figure 4 (a) and (b). For the first run the source deployed at 20 m depth and at 1 km to 4 km from the XLUUV. For the second run the source was again deployed at 20 m depth and deployed at 6 km and then out to 8 km from the XLUUV. As seen in Figures 3 (a) and (b) there was shipping activity at ranges comparable to the sound source location.

 

Figure 3: Trials location

 

The sound source was a spherical source, which was programmed to transmit simultaneously tonals at 311 Hz, 429 Hz, 766 Hz, 857 Hz and 1608 Hz at an estimated source level of 120 dB-130 dB. A section of the Krait array aperture was used for data processing which was formed by a sub aperture of 56 hydrophones spaced at 0.75m. For this aperture the frequency of 1608 Hz is beyond the array design frequency and hence it was not considered further.

 

The physical array aperture is approximately 41 m with an estimated near field distance of less than 500 m, hence for the distances considered here a focus range beamformer is not necessary for this aperture size. However, forming a virtual array by doubling the size of this physical aperture extends the near field to approximately 2 km for the frequencies considered here. Hence for PSAS processing a range focused beamformer is required. To keep the processing uniform range focusing was applied to all datasets discussed in section 3.2.

 

3.2 Experimental results

 

All data were sampled at 5120 Hz and were processed in segments of 1024 samples with 50 % overlap and zero padding. Initially, a sample of data was processed with the array’s physical aperture by applying the frequency domain CBF in two ways a) with FFT spectral estimation and b) BPD spectral estimation. Results from this algorithmic experiment are shown in section 3.2.1, with the aim to demonstrate the effect of BPD on beamforming without PSAS processing.

 

The datasets of interest are those at 1km and 8km. The first case is the closest range where the range focusing and bearing rate effects are more pronounced. The second case is the longest range where the SNR is low and coherence effects are expected to be seen. Results from these runs are presented in section 3.2.2.

 

3.2.1 Conventional beamforming with FFT and BPD

A data set with the source at 1 km was selected to demonstrate the effect of BPD on frequency compression and denoising. A spectrogram of this data set is shown in Figure 5, where the four tonals of interest can be distinguished at 8° bearing. The spectra at this bearing are shown in Figure 6. The black line shows the spectra computed with the traditional FFT for a total time of approximately 22 seconds. The red line shows the same beam spectra after BPD with a regularization factor 𝜇= 0.15. The optimization scheme used in this work posed no constraint on the BPD solution, for example a constraint on the number of spectra. The optimization is based on the ‘best match’ of the time domain signal penalizing the spectra smaller amplitudes. The effect is, as shown in Figure 6, to have very little effect on the SNR of the lower frequencies and more significant effect on the SNR of higher frequencies under consideration. For this reason, the 311 Hz tonal is excluded from PSAS analysis. The BPD affects the frequency domain but not the beam domain, i.e. the beam patterns are the same for both conventional beamforming schemes using the array’s physical aperture, hence the same beampatterns are observed.

 

Figure 4: XLUUV and vessel tracks during the first and second run, plots (a) and (b) respectively. The upper plots show the XLUUV and support vessels tracks, whilst the lower plots have superimposed the AIS tracks of the shipping activity

 

3.2.2 Conventional beamforming with ETAM and BPD

 

For the same run described in section 3.2.1, with the source at 1km, the data was processed using the ETAM model in segments of 50% array overlap, i.e. the virtual array was formed by extending the array by 28 hydrophones (56/2) in two time steps. The total processing time remained 22 seconds. The beam patterns for the 429 Hz, 766 Hz and 857 Hz bins are shown in Figure 7 (a), (b) and (c) respectively. The effect of the aperture increase is noticeable for all frequencies, however at the lowest of the three the benefit is clearer, as the side lobes do not affect the picture. Here it has to be noted that the ETAM processing has the bearing rate correction between the 50% overlap cycles which comprise the total of approximately 22 seconds whilst the CBF with the physical array has no bearing rate corrections, for this reason the beam peaks do not align.

 

Figure 5: Spectrogram of 22 seconds using conventional FFT beamforming. The colour bar is in dB scale normalised with the maximum value of the spectrogram

 

Figure 6: Beam spectra for the 8 degrees bearing, shown in Figure 5

 

The next interesting run is the one with the source at 8 km. The 429 Hz tonal is between two shipping engine tonals, whilst the higher frequency ones could be discerned in space, the source tonals indicated with pink arrows in Figure 8. GPS data of the support vessel was used as ‘ground truth’ for the location of the source. The beampatterns for these three tonals are shown in Figure 9. In all cases the advantage of PSAS in terms of resolution is evident although the SNR is poor. The bearing resolution is increased in both beam and frequency space reducing the peak to a few pixels. The effect of BPD beamforming is shown in Figures 7(b): the upper figure shows the 429 Hz tonal with an engine to tonal and the lower figure shows the 766 Hz and 857 Hz tonals of the sound source. The improvement is not necessarily visual however it offers a better bearing accuracy if used as input to an autodetector. All spectrograms have been normalized with their own maximum value.

 

Figure 7: Beam patterns for the a) 429 Hz, b) 766 Hz bin and c) 857 Hz bin shown in Figure 5 (a) (b)

 

Figure 8: Spectrograms of approximately 22 seconds showing source at 8 km (pink arrows) and engine tonals (a) produced using FFT and (b) using ETAM BPD. The figures on the right zoom- in the regions where the source tonals can be discerned the upper plot also shows the engine tonal, indicated with black arrow on the FFT spectrogram. Colour bars in dB normalized with the spectrograms’ maximum.

 

4 CONCLUSIONS

 

This work demonstrated the potential of PSAS to operate in a shallow noisy environment, achieving a higher bearing resolution than the array when used conventionally with its physical aperture. In parallel the potential of the BPD to increase the frequency resolution was demonstrated within the context of conventional beamforming. The ETAM processing involved acoustic signals which were denoised at hydrophone level using the SALSA BPD algorithm. The results suggest that even a rudimentary denoising approach makes the PSAS algorithm less sensitive to noise as compared with the classical correlation techniques. Using a BPD algorithm for phase estimation, the ETAM has the potential to offer a performance advantage in shallow waters where longer arrays are difficult to deploy. Here we observed that for shallow water Carey’s [1] estimates are rather pessimistic and the environmental coherency exceeds the 40-45 λ. Hence a sonar performance advantage could be achieved with PSAS when compared with the classical physical aperture processing. Future work will include data processing from different environments and automation of the choice of the regularisation factor 𝜇 .

 

Figure 9: Beam patterns for the source at 8 km at the (a) 766 Hz bin and (b) at the 857 Hz bin, the beam patterns correspond to the detections indited by arrows in Figure 8.

 

5 ACKNOWLEDGEMENTS

 

The authors would like to acknowledge DASA for funding the program and DSTL for permitting dissemination of this work.

 

6 REFERENCES

 

1. W.M. Carey, ‘The determination of signal coherence length based on signal coherence and gain measurements in deep and shallow water’, J. Acoust. Soc. Am., Vol. 7, 831-837. (1998).
2. S. Stergiopoulos and E.J. Sullivan, ‘Extended towed array processing by an overlap correlator’, J. Acoust. Soc. Am., Vol. 86, 158-171. (1989).
3. S. Kim, D.H. Youn and C. Lee, ’Temporal Domain Processing for a Synthetic Aperture Array’, IEEE J.Ocean. Engin. Vol 27, 322-327, (2002).
4. V. M. Afonso, J.M. Bioucas-Dias and M.A. Figueiredo, ‘Fast image recovery using variable splitting and constrained optimization’, IEEE Transactions on Image Processing Vol 19(9), 2345–2356. (2010).
5. S.S. Chen, D.L. Donoho and M. A. Saunders, ’Atomic Decomposition by Basis Pursuit’, SIAM Review, Vol 43 (1),129–159. (2001).
6. C. Tucker, G. Verwey and M. Troughton, ‘Evolving Autonomous ASW: At sea experimentation and trials with an XLUUV and a thin line towed array’, UDT 2023.
7. C.Ming, et al, ’Passive Synthetic Aperture for direction-of arrival estimation using sparse Bayesian learning’, J. Acoust. Soc. Am., Vol. 153, 2061-2072. (2023)