Proceedings of the Institute of Acoustics

 

An investigation into optimal acquisition geometries for repeat-pass synthetic aperture sonar bathymetric mapping

 

BW Thomas, Department of Mechanical Engineering, University of Bath, Bath, UK
AJ Hunter, Department of Mechanical Engineering, University of Bath, Bath, UK

 

 

1 INTRODUCTION

 

Synthetic aperture sonar (SAS) systems are able to generate two-dimensional images of the seafloor with resolution on order of a centimetre. This high resolution is achieved by coherent processing of the acoustic data collected from multiple pings along the path travelled by the sensor.

 

It is desirable to achieve the same resolution in height for 3D mapping since 3D shape information is invariant to imaging geometry and may therefore improve the probability of classification of underwater objects. Traditionally this is achieved by interferometry, which uses two vertically separated arrays mounted on the vehicle. However, due to size constraints on typical autonomous underwater vehicles (AUVs), these arrays have a vertical separation limited to the order of several tens of wavelengths. This gives a resolution in the vertical direction on the order of tens of centimetres, which is significantly poorer than the 2D image resolution of a typical SAS.

 

It has been shown in the synthetic aperture radar (SAR) literature¹ that combining data from multiple passes using tomographic processing can improve resolution in elevation compared to interferometry between only two passes. Successful coherent co-registration of SAS images has been shown with timescales varying from hours to a few days². Furthermore, tomographic imaging has been successfully demonstrated from a circular SAS system performing 9 passes of an interferometric array both at multiple heights and radii³. In recent work, repeat-pass interferograms were demonstrated over multiple passes from the CMRE MUSCLE AUV⁴, suggesting that repeat-pass SAS tomography is possible with this hardware.

 

Carrying out repeated passes for high-resolution 3D tomography is time-consuming. Therefore, we wish to choose the optimal number of passes to achieve a desired resolution. To this end, we have used analytical and numerical models to simulate the expected height resolution as a function of the number of passes and the repeat-pass geometry. We intend that this information will be used to inform the planning of future 3D mapping missions.

 

 

2 3D TOMOGRAPHIC PROCESSING

 

2.1 Forward model

 

In order to perform tomographic processing, multiple passes are made over a scene from slightly different positions, as illustrated in Figure 1. In the following, ‘𝑠’ is the elevation axis, ‘𝑟’ is the range and ‘𝑏_𝑛’ is the orthogonal component of the baseline between some chosen reference pass and the 𝑛^𝑡ℎ pass. We make the assumption that for the range of elevations of interest, the elevation axis may be assumed straight.

 

                                                                                              

 

Figure 1: - Repeat-pass imaging geometry
Figure 2: – Accumulation of images

 

Tomographic processing is performed on the 2D images obtained by any standard SAS focussing method. Thus, a 3 dimensional stack 𝑰 of 𝑁 images 𝑰_𝑛 is available for processing. The value 𝑔 𝑛 of each pixel in these complex-valued images is given by the coherent sum of all back-scattering contributions along the elevation direction s. This complex backscattering function is termed 𝛾(𝑠) .

                                                                      

 

where 𝑒^{𝜑𝑚(𝑛)} is a phase term associated with the propagation velocity in the medium, 𝜆 is the wavelength, 𝐼_𝑠 is the extent of the scene in the elevation direction, 𝑑(𝑠,𝑡_𝑛) is the deformation affecting the scatterer at elevation s, and 𝑤_𝑛 is a noise term, accounting for thermal noise and decorrelation.

 

In the following it is assumed that the medium propagation term is compensated for in a pre-processing image registration step, so the first term may be discounted. If the assumption of simultaneous passes is made such that there is no deformation term, then the tomographic equation is reached⁵:

                                                                 

 

This equation is identical to the Fourier transform operator, allowing the problem to be posed as one of spectral estimation. 𝛾(𝑠) may be seen as the spatial signal, with 𝑔_𝑛 being a spatial spectrum sampled with spatial frequencies  𝜉_𝑛 = −2𝑏_𝑛/(𝜆𝑟).  Discretising equation 2, and assuming the scattering is distributed over small elevation intervals within the range of elevations of interest,

                                                                 

 

where 𝒈 collects the values of a particular pixel over N passes as shown in figure 2, 𝒂(𝑠) is the steering vector representing the tomographic model; for pass 𝑛 , 𝒂(𝑠)_𝒏 = 𝑒 ^{−𝑗2𝜋𝜉_𝑛𝒔} /√𝑁 . The noise in each pixel is collected in the vector 𝒘.

 

The problem may then be phrased as a matrix multiplication, where 𝑨 collects the steering vectors for all passes over the range of elevations of interest, and 𝜸(𝑠) is the vector representing the complex back-scattering function over that range.

                                                                   

                                                              

 

 

2.2 Inverse model

 

The desired backscattering function 𝜸(𝑠) may then be estimated by approximate inversion of the matrix 𝑨(𝑠), as in Equation 6. Since 𝑨(𝑠) is in general not a square matrix, in the classical beamforming methodology the Hermitian operator is used. For wide-band systems, the resolution may be improved by performing the tomographic processing at a range of frequencies, and computing the mean back-scattering function 𝜸̂(𝑠), as in Equation 7.

                                                                 

 

Under the assumption that only one dominant scatterer is present in each pixel stack, the elevation 𝑧̂ of that pixel is approximated as the location of the maximum of its estimated backscattering function. The height of the peak is termed the ‘multi-acquisition coherence’ and gives a measure of the degree to which the estimate may be trusted. Finally, a 3-pixel square median filter is applied to the height estimates, which acts to reject outliers. Various alternative methods of spectral estimation have been presented in the literature, including SVD, CAPON, compressive sensing and MUSIC.

 

 

3 METHOD FOR PREDICTING HEIGHT RESOLUTION

 

3.1 Analytical model

 

Model pixel stack data 𝒈 were generated using the expression in Equation 9, and were processed tomographically to give the ideal likelihood functions in the absence of noise and decorrelation effects. In this expression, 𝑧 is a delta function, modelling a single dominant scatterer occuring in each range-azimuth cell

                                                        

 

 

3.2 Numerical Model

 

Simulated data were generated by a ray-tracing point-scatterer simulator, with 161 regularly spaced simulated passes over an identical flat scene consisting of 1125 pixels, with a maximum pass separation of 8m. Additional white noise at -30dB was added to all images. Each pass is directly above the previous pass, such that the vehicle only moves vertically between passes. Each pass generates two images, one from each of the upper and lower arrays, where the arrays are permanently separated by a distance of 19 wavelengths. By choosing sets of passes which are linearly spaced, and performing the tomographic processing on this subset of images, a histogram of height estimates may be generated and used as a probability density function 𝑃(𝑧), for that pass distribution. The median height error and a confidence interval may be computed for each 𝑃(𝑧), and the 95 percentile is used as a comparison metric. Linear repeat-pass distributions are then compared in terms of attainable resolution.

 

 

4 RESULTS

 

                                                                                      

 

Figure 3: Example likelihood function from modelled data (left), Collated likelihood functions from simulated data (right). Both plots are for 4 equally spaced passes, with a total baseline of 2.5m.

 

The above figure shows the height likelihood functions after tomographic processing at the center frequency of the system (300kHz), and the mean of the result from 11 equally spaced frequencies across the 60kHz bandwidth of the system. The left plots in the figure represent the model data, and the right plots collect the likelihood functions of all pixels, for 4 equally spaced passes with a total baseline span of 2.5m. Baseline span is defined as the maximum distance between passes perpendicular to the look angle of the reference array, and is synonomous with maximum baseline. For the single frequency case, the height estimate distribution is flat, and therefore the height estimates are very poor. Over multiple frequencies, the model data and simulated data are seen to follow broadly the same shape, however the high frequency oscillations shown in the model are not seen in the simulated data due to the additive image noise. The spread of the likelihood function in the simulated case is also rather large. A tukey window with 𝛼= 0.25 has been applied as a prior on the likelihood functions.

 

Figure 4 presents the median and 95% confidence line of the height error PDF 𝑃(𝑧) for each linearly spaced repeat-pass array. Incresing the spacing between passes results in an increase in the total baseline span. This maximum baseline defines the Rayleigh limit for the repeat-pass array, and explains the width of the main-lobe of the likelihood function, The optimal baseline is chosen as the baseline span 0.25m lower than the smallest baseline span which causes a significant increase in the 95 percentile line, which is caused by to grating lobes of the repeat-pass array. The chosen baselines are marked by a green marker. This additional spacing gives an AUV some tolerance in the required heave accuracy.

                                                                                                

 

Figure 4: - Height error distributions as a function of maximum baseline, for multiple passes.

 

As expected, the height resolution improves with increasing numbers of repeated passes. The required baseline for these values of optimal resolution appears to increase approximately linearly with number of passes while the improvement in resolution per repeated pass diminishes with a greater number of passes. This suggests that there is little utility in performing more than around 5 linearly spaced passes. This data is summarised in Figure 5.

                                                                                                             

 

Figure 5: - Optimal baseline span for N passes (left), elevation resolution distributions at optimal baseline (right).

 

 

5 CONCLUSIONS AND FURTHER WORK

 

The method described in this work predicts a median pixel height error which is comparable to the 2D resolution of a typical SAS system, when at least 4 passes are performed at regular intervals and the optimum spacing between repeated passes is chosen. The method may be used to model any arbitrary repeat-pass geometry, and further work will investigate different repeat-pass strategies. In contrast to the spaceborne synthetic aperture radar case, SAS operators are able to choose the heights of repeated passes with good precision either ahead of time and on the fly. Therefore there is an opportunity to use improved repeat-pass strategies which minimise the ratio of attained resolution to number of passes, and therefore time.

 

The pixel height resolution is clearly a function of the spectral estimation method used. Further work should include processing using SVD⁶, CAPON, MUSIC and compressive sensing⁷ techniques. Also, since the sea-floor is known to be largely continuous, the use of smoothing operators acting on the individual pixel likelihood functions (rather than the absolute height estimates) is expected to increase the attainable resolution further.

 

 

6 REFERENCES

 

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