Permissions for Reuse

Passive detection with a baffled cylindrical array can potentially be improved at low frequencies by exploiting signal diffraction around the baffle. A model based on infinite rigid cylinder scattering suggests that large gains in signal-to-noise ratio are potentially available to adaptive beamformers if the sensor arc is widened to include sensors in the acoustic shadow. However, elastic scatter effects become increasingly important as frequency decreases, so the gains obtained in practice are unknown. The gains in detection performance are examined in this letter by analyzing data recorded at sea from a platform-mounted sonar array.

Contents

• Introduction
• Signal model and ABF
• SNR and detection index
• Bearing-time record
• Conclusion
• REFERENCES
• FIGURES

Passive cylindrical sonar arrays are operated over the widest possible frequency range to exploit all available acoustic energy. But it is a challenge to obtain acceptable performance at low frequencies due to high levels of ambient and platform-generated noises and poor bearing resolution.¹ Adaptive beamformers (ABFs) are important for this purpose. In addition to providing improved bearing resolution and side-lobe suppression, ABFs can potentially provide a higher array gain than conventional beamformers if the noise has a high degree of spatial correlation.² This is the case for ambient noise at low frequencies.¹ Typically, the beamformer processes an arc of sensors mounted on a cylindrical metal baffle. An arc smaller than 180° is typically used because sensors in the acoustic shadow of the baffle have low signal-to-noise ratio (SNR) and make little contribution to the output over most of the frequency range. At low frequencies, however, significant signal energy is diffracted around the baffle, and detection may be improved by using a larger arc.

Because ABF is sensitive to steering vector errors,² the ability to exploit these additional gains depends on accurate modeling of the signal in the acoustic shadow region. The usual approach is to treat the baffle as a rigid scatterer.^3,4 Meyer,⁵ Teutsch and Kellermann,⁶ Teutsch,⁷ and Bertilone et al.⁸ analyzed beamforming, detection, localization, and array gain for arrays mounted on rigid baffles. However, the rigid model is inadequate at low frequencies where the elastic properties of the materials are important.⁴ Unfortunately it is difficult to develop a more accurate model, as it requires detailed analysis of the baffle, its mounting to the platform, and scattering from other parts of the platform. Despite the limitations of the infinite rigid cylinder model, it is of interest to examine the gains achievable in practice when it is incorporated into an ABF with ultra-wide arc. This letter presents results obtained using data recorded at sea from a platform-mounted array, as the arc is extended into the acoustic shadow by increasing the number of processed sensors while keeping sensor spacing fixed.

The array is mounted at the front of a platform and has Q staves uniformly spaced on a circle of radius r, surrounding a cylindrical metal baffle of radius ar. Each stave is a line of omnidirectional phones parallel to the cylinder axis. To simplify the discussion, we analyze data where the phones in each stave have been summed with zero time-delay. Thus the array can be viewed as a baffled circular array of directional sensors, in which each sensor is a line array steered to broadside. Signals are assumed to be plane-waves at zero-elevation, i.e., perpendicular to the cylinder axis. The beamformer processes MQ sensors that lie on an arc that swings around with the steering direction. In practice, the acoustic signals from distant sources often exhibit small deviations from zero-elevation, leading to phase errors, but these errors are minor at the low frequencies considered in this letter.

Modeling the baffle as an infinite rigid cylinder, and ignoring scatter from other parts of the platform, we introduce cylindrical coordinates (r,,z) with z-axis at the center of the baffle. If a plane wave signal of unit amplitude and wavenumber k=2f/c arrives normal to the z-axis from azimuth =0°, then the complex acoustic field outside the baffle is³

Here H and J_n are Hankel and Bessel functions of the first kind, respectively, ₀=1 and _n=2,  n1. If the array is steered to =0° using sensors at (r,₁)(r,_M), then the steering vector is =[(r,₁)(r,_M)]^T, where T denotes transpose. For frequency-domain beamforming,² complex sensor outputs at frequency f are obtained by fast Fourier transformation (FFT) and placed in an M×1 vector X. Doing this for I snapshots of data, we form

as an estimate the cross-spectral matrix (CSM), where H denotes conjugate transpose. Output power is

where v is the weight vector. We use a widely studied ABF, the minimum power distortionless response beamformer with sample matrix inverse² (MPDR SMI), for which

The effect of baffle scatter on the amplitude and phase of has been discussed elsewhere.⁸

The top row of Fig. 1 shows power vs bearing near a contact, computed from trials data using MPDR SMI with constructed using the infinite rigid cylinder model. The data were processed using FFTs with 32 Hz bins and Hann windowing. The CSM was estimated using 96 snapshots with 50% overlap. No diagonal loading² was used. Outputs are shown for arcs of sizes 112.5°, 180°, and 225° at normalized frequencies ka=3.1,4.5,6.1. We observe a large improvement in bearing resolution and noise suppression as the arc is extended from 112.5° to 180°, but also a noticeable improvement as the arc is extended into the acoustic shadow from 180° to 225°. Bearing resolution for ABF is improved as the arc extends beyond 180° because it is not determined solely by the physical aperture and frequency.

Figure 1.

We analyze the change in output SNR as the size of the arc is increased from 112.5° to , SNR()=SNR()−SNR(112.5°). Here SNR=10 log₁₀[(P_S+N−P_N)/P_N], where P_S+N is the mean output power when signal and noise are present, and P_N is the mean output power when noise-only is present. The dashed curves in the middle row of Fig. 1 show predictions of SNR using an array gain model that treats the baffle as an infinite rigid cylinder, and the noise field as a superposition of independent plane-waves with surface dipole power distribution.⁸ Results are shown for optimum processing to maximize SNR.² The surface dipole model is often used to represent ambient noise in deep water originating from wave action at the surface;¹ the noise arrives from above the array (i.e., at elevations 0°90°) with power per unit steradian proportional to sin . The modeling suggests that large increases in SNR are potentially available. At ka=4.5, for example, SNR is increased by almost 4 dB as the arc increases from 112.5° to 180°, and by a further 3 dB when it increases from 180° to 315°. Note that the modeling requires inversion of a noise CSM and could not be obtained for all arcs at the lowest frequencies due to ill-conditioning of the matrix.

To examine the gains obtained in practice, we computed 12 estimates of SNR for each arc, by processing consecutive segments of the same data recording used in the top row of Fig. 1, using the same processing parameters. SNR was estimated by replacing P_S+N by the power at the contact bearing, and P_N by the average power over a window of bearings that excluded the signal. The circles in the middle row of Fig. 1 show the mean value of the estimates, and error bars indicate upper and lower quartiles so that 50% of the estimates lie within the indicated bounds. Note that if was, in fact, the true steering vector, and if the sample CSM was the true CSM, then MPDR SMI would have a mean output that achieves the maximum SNR when steered to a solitary signal in noise.² In fact, for arcs up to 247.5° or 270° we do indeed find that the experimental SNRs are in broad agreement with the model prediction. However, SNR drops away as the arc is extended further. We find a SNR increase of 3–4 dB when the arc increases from 112.5° to 180°, and up to 2 dB of additional gain when it is increased from 180° to a value between 225° and 270°. The drop-off in SNR occurs because ABF is sensitive to steering vector errors,² and these errors grow as sensors deep inside the acoustic shadow are included in the processing. The errors are most likely due to a combination of elastic scatter effects, and scattering from other parts of the platform. ABF can be made more robust by diagonally loading² the CSM in Eq. (5), but we found that this did not allow larger arcs to be utilized.

The bottom row of Fig. 1 shows the change in detection index⁹ (DI) as the arc is increased from 112.5°, DI()=DI()−DI(112.5°). Here DI=20 log₁₀[(P_S+N−P_N)/_N], where _N is the standard deviation of the output power when noise-only is present. DI is more directly related to detection performance than SNR because it explicitly accounts for fluctuation in the background noise. Capon and Goodman¹⁰ showed that _N=P_N/(I−M+1) if certain statistical properties of the noise field are applicable, and this leads to DI=2SNR+10 log₁₀(I−M+1). Dashed curves show predictions obtained by applying this formula with the surface dipole noise model. Circles with error bars show experimental results obtained by applying ABF to the data and replacing _N by the sample standard deviation of power in a window of bearings that excludes the signal. At the lower frequencies the model is in good agreement with the data for arcs up to the optimum size, but at the higher frequency the model and data diverge almost immediately. The cause of the rapid divergence is unclear, but in this case there is little to be gained by extending the arc into the shadow zone. At the two lower frequencies we find large increases in DI of 7–9 dB as the arc is increased from 112.5° to 180°, and an additional increase of more than 2 dB as the arc is increased to its optimum size. To put these numbers into context, we utilize the receiver operating characteristic P_d=erfc[erfc⁻¹(P_f)−10^DI/20] for signal detection from Gaussian distribution theory⁹ when the signal is sufficiently weak that the noise-only variance approximates the signal plus noise variance. Here P_d and P_f are the detection and false alarm probabilities, respectively, and erfc is the complementary error function. Assuming that this approximates the true detection statistics when the number of snapshots is sufficiently large, then a DI of 9.8 dB allows a signal to be detected 50% of the time at P_f=0.1%. An additional 2 dB raises P_d to 79%, while a reduction of 2 dB lowers it to 26%.

The gains in a multiple contact scenario are illustrated in Fig. 2, which shows broadband power displayed as a bearing-time record, computed from trials data using ABF with the same parameters as used in Fig. 1. Background noise was equalized⁹ before power was summed over the band ka=3–6. Horizontal scans were normalized to a maximum of 0 dB. The image at top left shows the output for a 112.5° arc. Two contacts are observed: a strong contact at left and a weaker contact near the center. The contact at left is broad, suggesting that it might be comprised of multiple contacts that cannot be spatially resolved. The image at top right shows the output for a 180° arc. The contact at left can now be resolved into a strong contact and nearby weak contact. A fourth contact at far right can only just be discerned. The bottom image shows the output for a 247.5° arc. There are noticeable improvements in the quality of the tracks, which are primarily due to increases in signal to interference plus noise ratio for the weak signals.

Figure 2.

We have demonstrated that low frequency sonar detection with a cylindrical array can be improved by exploiting signal diffraction around the baffle. ABF was applied to data recorded at sea from a platform-mounted array, using a steering vector constructed from the infinite rigid cylinder model. Best results were obtained using sensor arcs in the range 225°–270°, depending on the frequency, but further increases caused a drop-off in performance due to growing steering vector errors. Additional gains could be accessed if a better signal model was available or if an experimentally measured steering vector was used.

References and links

1. W. S. Burdic, Underwater Acoustic System Analysis, 2nd ed. (Prentice-Hall, Englewood Cliffs, NJ, 1991). first citation in article
   
2. H. L. Van Trees, Optimum Array Processing (Wiley-Interscience, New York, 2002), Pt. IV. first citation in article
   
3. E. A. Skelton and J. H. James, Theoretical Acoustics of Underwater Structures (Imperial College Press, London, 1997). first citation in article
   
4. M. C. Junger and D. Feit, Sound, Structures, and Their Interaction, 2nd ed. (Acoustical Society of America, Melville, NY, 1986). first citation in article
   
5. J. Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects,” J. Acoust. Soc. Am. 109, 185–193 (2001). [ISI] first citation in article
   
6. H. Teutsch and W. Kellermann, “Acoustic source detection and localization based on wavefield decomposition using circular microphone arrays,” J. Acoust. Soc. Am. 120, 2724–2736 (2006). [MEDLINE] first citation in article
   
7. H. Teutsch, Modal Array Signal Processing: Principles and Applications of Acoustic Wavefield Decomposition (Springer, Berlin, 2007). first citation in article
   
8. D. C. Bertilone, D. S. Killeen, and C. Bao, “Array gain for a cylindrical array with baffle scatter effects,” J. Acoust. Soc. Am. 122, 2679–2685 (2007). [MEDLINE] first citation in article
   
9. R. O. Nielsen, Sonar Signal Processing (Artech House, Boston, MA, 1991). first citation in article
   
10. J. Capon and N. R. Goodman, “Probability distributions for estimators of the frequency-wavenumber spectrum,” Proc. IEEE 58, 1785–1786 (1970). [Inspec] first citation in article
    

Full figure

Fig. 1. Top: Power vs bearing near a contact, from applying ABF to trial data using arcs of sizes 112.5° (solid), 180° (dashed), and 225° (dash-dot). Middle: Change in output SNR as a function of arc size. Circles with error bars show experimental results using ABF, and dashed curves show model predictions. Bottom: Change in DI as a function of arc size. Circles with error bars show experimental results using ABF, and dashed curves show model predictions. First citation in article

Full figure

Fig. 2. Broadband bearing-time records for data processed using ABF with arcs of various sizes. Four contacts can be distinguished. First citation in article

Up: Issue Table of Contents
Go to: Previous Article | Next Article
Other formats: HTML (smaller files) | PDF ( kB)