Fast nearfield to farfield conversion algorithm for circular synthetic aperture sonar Open Access

Circular synthetic aperture sonar (CSAS, also known as acoustic reflection tomography) is a technique for creating a high resolution two-dimensional image of an object from acoustic scattering data at discrete angles over a full or partial circular aperture.^1–3 In the current examination backscattering data collected via a co-located source/receiver pair (hereafter called monostatic CSAS) is considered. Data may be obtained using a turntable with fixed transducer, scanning the transducer about a circular track, or using a vehicle mounted transducer. Time domain data from all captured aspect angles may be mapped to the image domain using a variety of techniques; see Ref. 1 for an overview.

Typical CSAS processing methods assume that data are acquired in the farfield of the object, given by⁴ 

for an object centered on the aperture. Here d_f is the farfield distance, D is the total length of the scattering object, and λ is the wavelength. For large objects or high frequency sonar systems this can impose an experimentally unrealistic requirement on the radius of the synthetic aperture, forcing data to be acquired in the nearfield. Failure to meet the farfield criteria leads to distortions in the resulting image, angular spreading of narrow glints, and migration in azimuthal angle of elastic features.

An algorithm is proposed that allows data to be taken in violation of the farfield condition and then converted to the farfield. The conversion technique presented in this paper has the less stringent experimental requirement that the aperture radius be larger than the maximum wavelength so as to eliminate evanescent components from the received wavefield. Related methods have been suggested for synthetic aperture radar⁵ and medical ultrasonic imaging.⁶ This paper will present the farfield conversion algorithm followed by its application to the results of a small scale laboratory experiment. The resulting changes in time, frequency, and image domains will be examined.

Backscattering data Ψ(θ, t) are recorded at discrete locations spaced equally in azimuth along a circular aperture of radius R. The resulting time-azimuth data are expanded into a partial wave series (PWS) in cylindrical coordinates with the origin at the center of the aperture.⁷ In the monochromatic case,

where N is the total number of samples, A_n are the partial wave coefficients, and $Hn(1)$(kR) are nth order Hankel functions of the first kind corresponding to outgoing waves.⁸ The ingoing $Hn(2)$ waves have been dropped and the e^−iωt time harmonic term is left as implied. The PWS is truncated at ±N/2 by the finite number of samples.

To eliminate spreading ambiguities in converting to 2-D, only the phase component of the Hankel functions is kept. Generalizing to a polychromatic system, the range independent partial wave coefficients are extracted using

where $ℑ(2)$ indicates the 2-D Fourier transform and the azimuthal wavenumber n is the Fourier conjugate variable to the azimuthal angle θ. Modification to the wavenumber-frequency relation must be made for monostatic applications. Sound propagation to and from the object must be included and thus the sound speed is effectively halved,⁹ 

Once the partial wave coefficients are obtained the Hankel functions are converted to their asymptotic forms¹⁰ 

The leading spreading factor will be suppressed, leaving only the phase component. Farfield data are given by

Suppression of the spreading component in both the exact and asymptotic forms of the Hankel functions allows conversion to the farfield without concern for the limitation to cylindrical spreading. The full conversion process is

where $ℑ−(2)$ is the 2-D inverse Fourier transform. The above function is cast so that the transform operation is in matrix form in (k, n) space.

Additional speed may be gained by applying a second asymptotic expansion. Expanding the Hankel function phase component ϕ_n(x) = arg[$Hn(1)$(x)] gives¹¹ 

Using this expansion for the initial partial wave series and then canceling the leading terms during conversion to the farfield gives the fast approximation

This formulation has the advantage of speed, allowing rapid implementation inside existing imaging algorithms and for application in iterative methods.

An aside will be made to examine sampling criteria. For monostatic systems, in order to avoid aliasing effects in either the farfield conversion or imaging processes, the number of samples must meet N ≥ int[4k_maxr′]. Here, r′ is the minimum radius of a circle enclosing the target and centered on the aperture origin. An object located off of the aperture center will increase the number of samples required. Hankel function azimuthal wavenumbers of |n| > 2kr′ are evanescent and decay exponentially away from the scatterer.⁷ 

To demonstrate the utility of the algorithm an experiment was performed on a solid 5:1 aluminum cylinder (38.1 × 190.5 mm or 1.5 in. × 7.5 in.) located in the water column of a freshwater tank (c ≈ 1486 m/s) in such a way as to minimize boundary effects. The cylinder was hung from two corners so that its axis was horizontal and was rotated through 360° azimuth using a rotating stage. The target was insonified by a fixed wide band (100–700 kHz) transducer placed along the cylinder meridian at a range of R = 0.964 m from the center of rotation. The resulting aperture's radius was considerably less than the farfield condition of d_f = 34.2 m but large enough to neglect evanescent effects. A short duration wide band insonifying pulse was used so as to give high spatial resolution.

Backscattering data were collected every 0.25° so as to meet sampling requirements. The resulting backscattering spectra were calibrated using backscattering data from a solid sphere of sufficient size that the specular reflection could be isolated.¹² 

Image reconstruction is performed using the Fourier transform method found in Refs. 1 and 2. Figure 1 shows the process; raw time domain data Ψ(θ, t) are Fourier transformed to give Ψ(θ,f) and then mapped from polar to Cartesian coordinates in k-space Ψ(k_x,k_y). Finally, a 2-D inverse Fourier transform places data in the image domain Ψ(x,y).

Figure 2 shows the results of image reconstruction on the raw data and on data after farfield conversion using Eq. (9). The original image is heavily distorted; the sides and end caps appear bowed and the corners are not flush. Features associated with elastic effects are also distorted. Conversion to the farfield straightens the sides and ends as well as makes the corners flush. The corrected image accurately reflects the size and shape of the target.

Certain elastic phenomena on the cylinder contribute to enhanced backscatter at intermediate azimuths. These include meridional¹² and face crossing¹³ Rayleigh-like waves, pictured in Fig. 1. The farfield coupling angle for these rays is given by θ_l = sin⁻¹(c/c_l), where c_l is the phase velocity of the surface wave and c is the fluid sound speed. For aluminum in water, this is predicted to be approximately¹⁴ θ_l = 30°. Scattering enhancements are thus expected near 30° and 60° azimuth. Such enhancements have been seen in field experiments and can dominate backscattering at these intermediate azimuths.¹⁴ 

Figure 3 shows the effect of farfield conversion on the acoustic color (backscattering spectrum versus azimuthal angle). In the raw nearfield data both meridional and face crossing scattering loci occur at angles differing from those predicted. After farfield conversion both features are observed nearer to the expected value. In addition, the bright feature at 0° azimuth corresponds to specular and elastic broadside scattering. The farfield angular width of the specular broadside main lobe is predicted³ to be θ_FF = 2 sin⁻¹(πc/ωL), where L is the length of the cylinder. This feature should be narrow in azimuth and become narrower at higher frequencies. In the nearfield data the broadside feature appears spread in angle and this angular width does not show strong frequency dependence. A simple geometric construction predicts that the nearfield angular width of the main specular lobe is

which is independent of frequency. The nearfield and farfield data in Fig. 3 are consistent with these estimates of the specular width.

The Hankel function based nearfield to farfield algorithm allows data to be gathered using smaller apertures and then converted into their farfield form. The algorithm preserves amplitudes by suppressing spreading. Due to the form of the algorithm, computation time required for conversion and imaging is short compared to alternative methods of recovering the farfield image such as back-propagation.

Many scattering theories for elastic targets utilize a farfield assumption. If an object is instead scanned in the nearfield, scattering loci may occur at azimuthal locations differing from predicted. Converting to the farfield removes this ambiguity and shifts backscattering loci closer to expected values, aiding manual analysis or use in automatic classifiers. In addition, current CSAS methods assume that data have been taken in the farfield. This process removes nearfield distortions from the resulting image. This may be important for automatic classification and iterative image based techniques such as autofocusing. Some processing procedures also rely on masking out portions of data in either the angle or image domains, both of which may be more easily understood in the farfield. Work of future interest includes cases where the transducer is outside the target's plane of rotation and in understanding how phase noise in the raw data affects the conversion.

Work supported by ONR.