O ct 2 00 1 Comment on “ Multiple scattering in a reflecting cavity : Application to fish counting in a tank [ J . Acoust . Soc . Am . 109 , 2587-2597 ( 2001 ) ] ”

In a recent paper [1], de Rosny and Roux proposed a method for counting fish in a tank, an important issue for the fisheries community. As far as we know, this is the first published report on counting fish in a tank. We recognize that the fish information may have been detected by the method. However, in its present form of data analysis, we believe that the foundations for the analysis are questionable. As a matter of fact, their approach has failed to provide reasonable scattering properties of fish. The reasons are elaborated below. (1) The multiple scattering in a reflecting cavity. The basis for the fish counting method in [1] is described by Fig. 1 of [1]. The analysis relies on the treatment of the multiple scattering picture in this figure. In the context, the authors argued that the effect of the reflecting boundaries of a tank can be regarded as mirrors, and thus acoustic scattering in such a system is regarded as a medium of scatterers and their images without the boundaries. The multiple reflections make the system equivalent to a system of infinite size. The first ambiguity in the later development is that effects from the images of the acoustic sources are not considered in [1]. Although the images of the sources are shown in Fig. 1, their possible effects are not discussed anywhere in [1]. In addition, the reflection from the water surface is not taken into account either. The presence of the water surface also acts as a mirror, giving rise to interactions between the scatterers and their images due to this mirror. It is worth pointing out that the water surface acts as a pressure release plane, which has different effects from the presumably total-reflecting side boundaries of the tank. The second ambiguity lies in the way how the multiple scattering among the scatterers and the images in Fig. 1 is accounted for, a crucial problem in the analysis. In [1], the authors wrote down two formulas in Eqs. (1) and (2) for total and coherent intensity as a result of the multiple scattering; here we note that in [1] Eqs. (1) and (2) have been mistakenly regarded as the coherent intensity and incoherent intensity respectively the detailed information about the definitions of these quantities can be referred to [2]. With reference to [2], it is ready to verify that both Eqs. (1) and (2) are actually the results of the multiple scattering when wave propagates in a cloud of completely independent random scatterers (Refer to, e. g. Chapter 14 in [2]). In the system such as many scatterers in a reflecting cavity, the scatterers and their images are not independent scatterers. Thus Eqs. (1) and (2) are not applicable to the present case. There is a rich body of literature on acoustic scattering by bodies in the presence of boundaries (e. g. [3–6]). Correlated scatterers can give rise to interference between multiply scattered waves that is absent in a group of independent scatterers. Take the simplest case of an object near a boundary as example. The multiple scattering from a scatterer and its image differs from the scattering from two actual scatterers. This can be easily inferred from, for example, Ref. [6]. (2) The mathematical derivation. The results of Ref. [1] rely mainly on its Eq. (4). In writing down this equation, the authors made the assertion “Eq. (3) can be written (as Eq. (4)) by changing from a space variable to a time variable”, then Eq. (4) is obtained. But no details were given. The derivation from Eq. (3) to Eq. (4) is mathematically unclear. Eq. (3) denotes how wave is attenuated after travelling a distance, but Eq. (4) represents the time evolution of backscattered signals. The former is a forward propagating process, while the later is a backscattering process. The link between the two is not obvious. In fact, the time evolution of backscattered signals has been detailed in Chapter 5 of [2]. It is not evident how the time series of the ratio between the coherent and the total backscattered signals can be represented by a simple equation like Eq. (4). (3) The experimental results. The method of [1] yields doubtful scattering properties of fish. The main goal of [1] is to obtain accurate total scattering cross section of fish. Two fish species are measured: zebra fish with length about 1 cm at 400 kHz and 35 cm long striped bass at 12.8 kHz. Take the bass fish as the example. According to [1], the total scattering cross section of the strip bass is 4πR with R ∼ 3.8 cm, leading to σs ∼ 181 cm . This value is much larger than what would be expected for a fish of 35 cm length. The reasons follow. First, the total scattering cross section is a computable quantity. According to the optical theorem [2], the total cross section can be calculated from the imaginary part of the forward scattering function of a scatterer