Acoustic scattering by an arbitrarily shaped body: an application of the boundary-element method

Abstract

The boundary-element method (BEM) is applied to calculate target strength (TS) and extinction cross-section of a scatterer. The scattering amplitudes of four types of prolate spheroid, namely, the vacant, rigid, liquid-filled, and gas-filled, are calculated using the BEM. Comparison between the results of the BEM and the prolate-spheroid model (PSM), which is a numerical calculation using spheroidal-wave functions, shows excellent agreement for the vacant, rigid, and liquid-filled spheroids. The TS of a gas-filled prolate spheroid, calculated by the BEM, shows resonance at a low frequency. The lowest resonant frequency, the peak ratio, and the quality factor (Q) of this resonance are compared with those given by the T-matrix and an analytical solution. Again the results show excellent agreement.

Introduction

Wideband methods for fisheries-acoustics surveys are being widely used to provide biological information, such as species, body shape, and the target strength (TS) of fish. A scattering model is used to estimate the scattering amplitude for parameters, such as frequency, tilt angle, and the various shapes of fish. Because fish-species identification is a typical inverse problem, the corresponding forward problem, namely, the scattering characteristics of an arbitrarily shaped body, has to be resolved (Grroetsch, 1993). The relatively new X-ray computed-tomography technique allows precise measurement of both fish-body and swimbladder shape. A model that can accurately predict the scattering amplitude of a target of arbitrary shape will therefore play an important part in interpreting the acoustic data obtained by wideband methods.

The boundary-element method (BEM) has certain advantages as a scattering model for various shapes of marine animals. First, the boundary conditions can be treated on curved surfaces and, second, acoustic problems, such as scattering amplitudes, can be analysed in the infinite domain because the acoustic field is used only on the target surface (Wu, 2000). The BEM is less useful, however, in the case of inhomogeneous scatterers. The physical properties of the scatterer have to be homogeneous, or partially homogeneous, to be suitable for analysis with the BEM.

The studies of Foote and Francis (2002) and Francis and Foote (1998) are relevant in the use of the BEM for TS calculation. In these studies confirming the accuracy of the BEM for a spherical void and a rigid sphere, the BEM was applied to calculate the TS of pollack. Both studies compared the TS obtained by the Kirchhoff approximation, the BEM, and experimental results and showed that the method is not a great improvement on the earlier Kirchhoff approximation results, which agreed with the measured TS patterns. Furthermore, the authors noted that the acoustic field on the swimbladder surface, calculated by the BEM, could be used to analyse the auditory function of the swimbladder. Because many fish species have a connection between the auditory system and the swimbladder, it is believed that the swimbladder amplifies the acoustic signal to the auditory system and that this signal can be calculated using the BEM. Theoretical studies to estimate the extinction cross-section have been conducted by several authors (Ye, 1996; Chu and Ye, 1999; Gorska and Chu, 2001) in studying the attenuation of sound energy caused by marine organisms. They used an analytical expression based on the optical theorem, which relates the extinction cross-section with the imaginary part of the forward-scattering amplitude. On the other hand, the BEM also allows calculation of the forward-scattering amplitude. Hence, the cited works on attenuation suggest another application for the BEM, namely, estimation of the extinction cross-section based on the optical theorem with the forward-scattering amplitude evaluated numerically using the technique.

In this paper, we present the BEM method for calculating the frequency dependence of the TS of an arbitrarily shaped body at any incident angle considering four types, especially, the vacant, rigid, liquid-filled, and gas-filled prolate spheroids. We also show how to calculate the extinction cross-section, σ, by using the BEM with the optical theorem. The accuracy of the BEM is confirmed by comparison with the prolate-spheroid model (PSM), which is a numerical calculation using spheroidal-wave functions (Furusawa, 1988), in the case of the vacant, rigid, and liquid-filled prolate spheroids. The validity of the BEM for the gas-filled prolate spheroid, which has a low-frequency resonance, is confirmed by comparison with calculations using the T-matrix method (Feuillade and Werby, 1994), and the analytical solution for a gas-filled prolate spheroid described by Ye (1997).

BEM formulation for vacant, rigid, liquid-filled, and gas-filled scatterers

The BEM is a numerical method for solving the wave equation in the presence of a scatterer that can be described as a surface with homogeneous physical properties. The basis of the BEM is the Helmholtz integral equation. Essentially, the BEM determines both the sound pressure and the normal component of the sound velocity on the scatterer surface (Wu, 2000). Three types of boundary condition are used: for a vacant scatterer, the pressure-release boundary condition is applied; for a rigid scatterer, the immovable-boundary condition is applied; for liquid-filled and gas-filled scatterers, the acoustic pressure and the normal component of the sound velocity are continuous between the exterior and interior sides of the surface of the scatterer.

There is a major difficulty with the Helmholtz integral equation when the frequency of the incident wave is near a characteristic frequency of the scatterer. In this case, pseudo-resonances can occur at the corresponding characteristic frequencies even when the exterior scattering problem has no resonance feature. To overcome this problem, we used the combined Helmholtz integral equation formulation (CHIEF), a technique devised by Schenck (1968). This method supplements the original Helmholtz formulation with additional equations, which are also Helmholtz integral equations, applied to several points within the scatterer. The points at which additional constraints are made are called the CHIEF points. We generated the CHIEF points randomly within the scatterer, avoiding any highly symmetric positions, since these are likely to be on nodes of the eigen function. We used CHIEF points between three and 10, depending on the behaviour of the calculated results.

Following the BEM treatment of Wu (2000), the acoustic pressure P(x′) and the normal component of the acoustic velocity v_n(x′) on the exterior surface of the scatterer are calculated. The scattering amplitude, as observed by an echosounder, is then calculated from the acoustic field on the surface. If the unit vector normal to the surface is n, and the scattered-wave vector is k_out, then the scattering amplitude f(k_in,k_out) is:

(1)

for the vacant, rigid, liquid-filled, and gas-filled scatterers. If the scattered-wave vector is chosen as k_out = −k_in, Equation (1) gives the backscattering amplitude. If the scattered-wave vector is chosen as k_out = k_in, Equation (1) gives the forward-scattering amplitude.

The TS, which is a key parameter in fisheries acoustics, is the absolute value of the scattering amplitude expressed in decibels:

(2)

The extinction cross-section σ of any scatterer that determines the attenuation of acoustic waves within a fish school or a cloud of plankton is calculated by the optical theorem. The σ is proportional to the imaginary part of the forward-scattering amplitude, which is obtained from the BEM:

(3)

where k is the norm of the incident-wave vector (Landau and Lifshitz, 1977).

If the acoustic field on the surface of the scatterer can be obtained without solving the full wave equations, one can immediately calculate the scattering amplitude using Equation (1). The Kirchhoff approximation is a case in which the acoustic field on the scatterer surface is estimated by physical intuition. For an impenetrable scatterer, such as a vacant or rigid object, at high frequencies, the surface of the object S is considered in two parts, i.e. the illuminated area S_i and the shadow area S_s. In the shadow area S_s, both the acoustic pressure P(x′) and the normal component of the acoustic velocity v_n(x′) are reasonably assumed to be zero, because the acoustic field is negligible there. In the illuminated area S_i, the scattering on the local curved surface can be treated as s reflection from the tangent plane at the point where the incident wave is scattered. By applying this physical picture to Equation (1), the backscattering amplitude by the Kirchhoff approximation is derived as:

(4)

where R is the reflection coefficient, R=−1 for a vacant object and R=1 for a rigid object (Foote, 1985). The BEM results for a vacant or rigid object should therefore converge asymptotically with those of the Kirchhoff approximation at high frequency. Equation (4) was used to check the BEM results for a vacant or rigid object at high frequency.

Results and discussion

We chose a prolate spheroid, with major radius a = 0.025 m and minor radius b = 0.005 m, as the test scatterer for our BEM calculation. This is a convenient choice, because we can check the validity of our program by comparison with the results of previous studies on prolate spheroids. The spheroidal surface was divided into 1960 triangular elements with 982 nodes for this numerical calculation. The same structure was used for all the following results. Sound velocity c = 1500 m s⁻¹ and mass density ρ = 1000 kg m⁻³ were assumed for the exterior fluid. In this study, we were mainly interested in the TS and σ of the prolate spheroids. Three incident angles, broadside (0°), oblique (45°), and end-on (90°), were considered.

Vacant prolate spheroid

The normalized TS of a vacant prolate spheroid by the BEM and PSM, which is a numerical calculation using spheroidal-wave functions (Furusawa, 1988), is shown in Figure 1a. TS is normalized to that at high-frequency limit for broadside incidence calculated by the Kirchhoff approximation (Figure 1). Frequency is normalized to ka, where k is the norm of the incident-wave vector. The solid lines show the TS by the BEM at broadside, oblique, and end-on incidence. The crosses show the TS by the PSM at broadside, oblique, and end-on incidence. Ten CHIEF points generated randomly within the prolate spheroid were used in this calculation. The results of the BEM and PSM show excellent agreement within the frequency range 0.1<ka<10.0. At the low-frequency limit, the TS was isotropic, and approached a finite value. The maximum frequency we used in the BEM was about ka=20. At ka=20, the TS differences between the BEM and the optical limit of the Kirchhoff approximation were −0.1 dB (broadside), −1.9 dB (oblique), and −3.6 dB (end-on). TS by the BEM at broadside incidence therefore approached the optical limit of the Kirchhoff approximation faster than that at end-on incidence.

Figure 1

TS of vacant, rigid, liquid-filled and gas-filled prolate spheroids calculated by the BEM and the PSM normalized to the high-frequency TS limit at broadside incidence (calculated by the Kirchhoff approximation). Frequency is the non-dimensional ka. The TS oscillations above ka=1 are due to interference between (b) the direct reflection from the front surface of the rigid spheroid and the Frantz wave, and (c) the reflections from the front and bottom surfaces of the liquid-filled spheroid. In (d), the peak TS shows the low-frequency resonance of the gas-filled spheroid.

The normalized extinction cross-section σ of the vacant prolate spheroid is shown in Figure 2a. At the high-frequency limit, the σ of a vacant or rigid scatterer is estimated as twice the projected area of the scatterer on the plane normal to the incident-wave vector (Landau and Lifshitz, 1987). Thus, in Figure 2, the vertical axis is normalized by the high-frequency limit of the extinction cross-section at broadside incidence, calculated by the optical theorem. At the low-frequency limit, σ approached a finite value, without showing the traditional Rayleigh frequency dependence. The normalized extinction cross-section σ is thus expected to approach 0.427 at broadside, 0.308 at oblique, and 0.085 at end-on incidence. However, the normalized σ from the BEM was not close to the high-frequency limits at ka=15, as seen in Figure 2a.

Figure 2

Extinction cross-sections of vacant, rigid, liquid-filled, and gas-filled prolate spheroids calculated by the BEM and the PSM, normalized by “2πab”, which is the high-frequency limit of σ for a vacant spheroid at broadside incidence. (a) The normalized σ of vacant spheroid versus ka. Contrary to the backscattering by rigid and liquid-filled spheroids, σ shows no oscillation due to interference as shown in (b) and (c). In (d), the peak σ at low frequency indicates the resonance of the gas-filled spheroid.

Rigid prolate spheroid

The normalized TS of the rigid prolate spheroid by both the BEM and the PSM is shown in Figure 1b. Ten CHIEF points were used in this case. The BEM and the PSM show excellent agreement within the frequency range 0.1<ka<10.0. At the low-frequency limit, the TS follows the Rayleigh frequency dependence. At frequencies ka>1, the TS oscillates in the frequency space because of the interference between the direct reflection from the front surface of the scatterer and the Frantz wave, which travels along the boundary of the scatterer but within the surrounding fluid (Medwin and Clay, 1998). The peak values of the TS by the BEM were close to those of the Kirchhoff approximation. The maximum frequency of the BEM in our study was about ka=15.

The normalized extinction cross-section of the rigid prolate spheroid is shown in Figure 2b. At the low-frequency limit, σ is expected to show the Rayleigh (ka)⁴ frequency dependence. Although σ according to the BEM decreased significantly at low frequency, it did not follow the Rayleigh law. The low-frequency behaviour of the extinction cross-section is subsequently discussed.

Liquid-filled prolate spheroid

The normalized TS of the liquid-filled prolate spheroid by both the BEM and the PSM is shown in Figure 1c. We chose the sound velocity and density in the liquid-filled prolate spheroid to be 5% higher than those of the exterior fluid: c′ = 1575 m s⁻¹ and ρ′ = 1050 kg m⁻³. Three CHIEF points were used in this calculation. The BEM and the PSM show excellent agreement in the frequency range 0.1<ka<10.0. At the low-frequency limit, the TS follows the Rayleigh frequency dependence. For ka>1, the TS oscillates in the frequency domain because of the interference between the reflections from the front and bottom surfaces (Medwin and Clay, 1998). The maximum frequency to which the BEM is applicable is about ka=15, which is less than that of the vacant prolate spheroid.

The normalized extinction cross-section of the liquid-filled prolate spheroid is shown in Figure 2c. At the low-frequency limit, σ is expected to follow the Rayleigh frequency dependence. However, the BEM results for σ do not show the expected (ka)⁴ frequency dependence. This low-frequency behaviour of the extinction cross-section is again discussed subsequently.

Gas-filled prolate spheroid

The normalized TS of the gas-filled prolate spheroid by the BEM is shown in Figure 1d. The normalization rules and symbols are as for the vacant prolate spheroid, except that the horizontal axis in Figure 1d is log scaled. We assumed the following sound velocity and density values for the gas in the scatterer: c′ = 330 m s⁻¹ and ρ′ = 1.26 kg m⁻³. No CHIEF point was used in this calculation. At the low-frequency limit, the TS follows the Rayleigh frequency dependence. At ka=0.047 (458 Hz), the TS exhibits a resonance. For ka<1, the TS was isotropic, while for ka>1, the TS was anisotropic. The TS curve at frequencies ka>1 is almost the same as that of the vacant prolate spheroid.

The normalized extinction cross-section of the gas-filled prolate spheroid is shown in Figure 2d. At very low frequencies, σ is expected to follow the Rayleigh frequency dependence. However, the BEM results for σ do not show the expected (ka)⁴ frequency dependence. This low-frequency behaviour of the extinction cross-section is subsequently discussed. The extinction cross-section exhibits a resonance at ka=0.047 (458 Hz).

In Figures 1d and 2d, only the BEM predictions of TS and σ are shown because we did not apply the PSM to a gas-filled prolate spheroid. To confirm the validity of the TS predicted by the BEM, we calculated the resonance behaviour at broadside and end-on incidence for gas-filled prolate spheroids with aspect ratios a/b=1, 2, 4, 8, 10, 16, and 20. All the spheroids had the same volume, equal to that of a prolate spheroid with major radius a = 0.025 m and minor radius b = 0.005 m. We then compared our BEM calculations with the T-matrix results of Feuillade and Werby (1994) and the results of an analytical approximation reported by Ye (1997), which is only valid at low frequency. Figure 3a shows the resonance structure calculated by the BEM at broadside incidence. The vertical axis is normalized to the resonance peak of the sphere with aspect ratio a/b=1. The horizontal axis is normalized to ka_es, where a_es is the radius of the equivalent sphere having the same volume as the prolate spheroids. As shown in Figure 3a, the higher the aspect ratio, the higher is the resonant frequency, the lower is the resonance peak, and the broader is the resonance width. In Figure 3b–d, the ratios of the peaks, the quality factor (Q), and the resonant frequencies calculated by the BEM are compared with those of Feuillade and Werby (1994) and Ye (1997). In Figure 3b, the ratios of the peaks by the BEM showed isotropic scattering and good agreement with that of Ye (1997). However, the ratios of the peaks by T-matrix were significantly different between end-on and broadside incidences. The quality factor in Figure 3c is defined as the ratio of the resonant frequency to the frequency difference between the two points, 3 dB below the resonance peak. As regards the resonant frequencies, there was good agreement between all three methods (Figure 3d).

Figure 3

The resonance features of gas-filled spheroids with various aspect ratios. (a) The backscattering amplitude normalized to f_max, the peak for aspect ratio a/b=1. (b) A comparison of peak amplitudes, normalized to f_max, with the results of Ye (1997) and the T-matrix approach (Feuillade and Werby, 1994). The BEM and Ye (1997) show isotropic scattering at resonance; the T-matrix approach is anisotropic. (c) Resonance quality factor (Q) versus the aspect ratio. (d) Resonance frequency versus the aspect ratio.

We also show how to calculate the depth-dependence of resonance as another application of the BEM to the gas-filled prolate spheroid. The effect of a depth change can be included in the BEM formulation by modifying the sound velocity and the mass density of the gas-filled scatterer as follows:

(5)

where z is the depth of the scatterer and P₀ is the atmospheric pressure at the sea surface. We assume implicitly in Equation (5) that the thermodynamics of the gas-filled prolate spheroid are iso-thermal and iso-volumic when the depth increases. The resonance structure of the gas-filled prolate spheroid at broadside incidence is shown in Figure 4a for z=0, 50, 100, 200, and 400 m. The deeper the scatterer, the higher is the resonant frequency and the broader is the width of the resonance. In Figure 4b–d, the ratios of the peaks, the Q, and the resonant frequencies calculated by the BEM are compared with those of Ye (1997). Good agreement is seen in all the cases.

Figure 4

The resonance features of gas-filled spheroids at various depths and broadside incidence. (a) Backscattering amplitude normalized to f_max, the value at zero depth. (b) BEM resonance peaks relative to f_max compared with the results of Ye (1997). (c) Resonance quality factor (Q) versus depth. (d) Resonance frequency versus depth.

Completely transparent prolate spheroid

The numerical error in the BEM may be estimated by applying it to an ideal, acoustically transparent scatterer, one in which the sound velocity and mass density are the same as those of the exterior fluid. There is no scattering from a completely transparent target; however, the BEM predicts small but finite values for the TS and extinction cross-section. This is probably due to the inaccuracy of the numerical integration over the finite, triangular elements; thus, the finite TS and extinction cross-section are indicators of the numerical error in the BEM.

In Figure 5a, the normalized TS of the completely transparent prolate spheroid calculated by the BEM is shown at broadside, oblique, and end-on incidence. No CHIEF point was used in this calculation. At frequencies ka>10, the TS exhibits pseudo-resonances. In the frequency range 0.1<ka<10, the TS of the completely transparent prolate spheroid is less than that of the liquid-filled prolate spheroid. At ka<0.1, there is a plateau on the TS curve. This has the potential to cause errors in the BEM solutions for rigid and liquid-filled scatterers, because their TS follow the Rayleigh (ka)⁴ law at low frequencies. In Figure 5b, the normalized extinction cross-sections of the completely transparent prolate spheroid are shown at each incident angle. The σ shows the same frequency dependence as the normalized TS, but is much larger. This explains why the BEM extinction cross-section of the rigid, gas-filled, and fluid-filled prolate spheroids did not follow Rayleigh's law at low frequencies. When the true σ decreases below the plateau level, the errors become the dominant term in the BEM calculation. In the case of the TS, however, the numerical errors are small enough not to deviate from the Rayleigh frequency dependence.

Figure 5

BEM calculations for a completely transparent prolate spheroid. (a) TS normalized to the high-frequency limit at broadside incidence. Frequency is the non-dimensional ka. (a) The normalized TS of vacant spheroid versus ka. (b) Extinction cross-sections normalized by “2πab”, the high-frequency limit of σ for a vacant spheroid at broadside incidence. There should be no scattering, but the BEM indicates small but finite values. These results are indicators of the numerical error in the BEM formulation.

Future work

As a pre-processing technique for the BEM, the automatic triangulation of the surfaces of both the fish body and swimbladder using tomograms obtained by X-ray CT, or MRI looks promising. If the idea of constructing the surface triangulation, required by the BEM from tomograms of fish, is successfully developed then the result would be a new method for estimating the TS, consisting of the BEM and the pre-processing scheme. This would not require the use of acoustic tanks and would have wide application.

As an improvement to the present BEM, we suggest that there is some potential in multi-domain models, which could, for example, treat the surfaces of the swimbladder and the fish body simultaneously.

We thank Dr T. W. Wu for his textbook and programs, from which we made our BEM program. We also thank Dr M. Furusawa for his program of the PSM, with which we calculated the TS for vacant, rigid, and liquid-filled prolate spheroids. K. Oda is thanked for his help and patience with our activity. We thank Dr D. Chu for his useful comments. The numerical calculation was conducted by the Ministry of Agriculture, Forestry and Fisheries Research Network.