EL DERECHO A LA PROTECCIÓN DE LOS INTERESES DEL

Since the TS is a stochastic measure dependent on a variety of parameters (Blaxter and Batty 1990; MacLennan 1990; Ona 1990, 2001, 2003; Hazen and Horne 2003), the presently used Equation 1.11 is a convenient if not accurate description to derive mean TS from a single metric - the fish length. McClatchie et al. (1996) used an analysis of variance (ANOVA) to combine data from different studies and attempted to analyse discrepancies in results between TS estimation methods. They found significant effects of species, freshwater vs. marine, swimbladdered vs. non- swimbladdered and dead vs. alive fish on the relationship between maximum dorsal aspect TS and fish length. Additionally, quadratic dependence of TS on fish length (i.e. equation 1.10, where m = 20) was found to be the exception rather than the rule (McClatchie et al. 1996; McClatchie et al. 2003), with data usually widely scattered around the slope (Simmonds and MacLennan 2005). Considerable variation in TS estimates based on empirical measurements, as observed in herring (c.f. Table 1.2), led to an increased interest in developing theoretical scattering models of fish to aid the interpretation of experimental results (Horne and Clay 1998). These models are generally based on representing different body components of the fish (most importantly the swimbladder) as a function of their shape, orientation and acoustic properties.

Early models dealt with the representation of the swimbladder as a gas-filled simple geometric shape such as a sphere (Andreeva 1964; Haslett 1965; Love 1978) or a finite cylinder (Do and Surti 1990; Clay 1992). Nero et al. (2004) used Love’s (1978) model to estimate the swimbladder volume of herring from low frequency acoustic measurements near the resonance region (1.5 - 5 kHz). More recently, Gorska and Ona (2003a, 2003b) have attempted to approximate the herring swimbladder using a prolate spheroid to more realistically represent its true shape. The scattering theory of such simple geometric shapes is well known and acoustic equations can be solved to give exact results for the backscattering strength (Horne and Clay 1998). More sophisticated models can make use of the true arbitrary shape of scattering objects such as the swimbladder. Such approaches usually involve

mapping of the object surface from physical replica reconstructions (Foote 1985) or radiographic imaging techniques like computer tomography (CT, Horne et al. 2000) or magnetic resonance imaging (MRI) scanners (Peña and Foote 2008).

1.4.1. Approximation solutions

1.4.1.1. Deformed cylinder model

Several approximation solutions have been applied to calculate backscatter for prolate spheroids and more realistic morphological shapes of various fish body components. Stanton (1988, 1989) described a method where the target is modelled as a cylinder whose radius varies along the axis. Total backscatter is then derived as the combined contributions from a modal series of single narrow sections of the cylinder along the axis. This so-called modal series-based deformed cylinder model (MSB-DCM) has been applied by Gorska and Ona (2003a, 2003b) to calculate herring backscatter by representing both the swimbladder and the fish body as respective gas-filled and fluid-filled prolate spheroids. They attempted to determine the depth and frequency dependence of the herring swimbladder by fitting different swimbladder contraction factors and compared the results to Ona’s (2003) empirical in situ TS data set. They found that the decrease in mean backscatter with increasing depth could be explained by the compression of the herring swimbladder. Also, the model fitted the empirical data best if the swimbladder length-contraction was less than the width (or height)- contraction.

1.4.1.2. Kirchhoff ray-mode approximation

A different model, the Kirchhoff ray-mode approximation (KRM), calculates backscatter of objects by approximating them as a set of short cylinders along the main axis (Clay and Horne 1994; Horne and Jech 1999). The model uses the Helmholtz-Kirchhoff integral (Foote and Traynor 1988), which assumes that the reflection at every point of the surface is the same as the reflection of an infinite plane wave from an infinitive tangential interface (Medwin and Clay 1998). The total backscatter is determined by summation of the combined contributions of each

cylinder element. Morphological dimensions of fish components (height and width), such as the swimbladder, are usually determined by radiological imaging using X-ray techniques (Clay and Horne 1994; Horne et al. 2000; Hazen and Horne 2004). So far, herring backscatter has not been modelled by the KRM, however, Horne and Jech (1999) used the method to calculate backscatter of a simulated population of another clupeoid, the threadfin shad at multiple frequencies. They found that theoretical backscatter amplitudes varied unpredictably among frequencies between 38 and 420 kHz. Hazen and Horne (2003) and Horne (2003) used the KRM method to evaluate the effects of factors affecting fish TS, such as length, frequency, tilt angle, ontogeny, physiology and behaviour. Even though the KRM is relatively simple to compute it has a distinct disadvantage in that it is inaccurate at low frequencies and high tilt angles (Simmonds and MacLennan 2005).

1.4.1.3. Boundary element method

Similar to the KRM model, the boundary element method (BEM) can calculate backscatter from arbitrarily shaped surfaces (Francis 1993). It takes account of energy diffracted into the shadow zone within the scattering body, which is not true for the KRM (Simmonds and MacLennan 2005). It also enables backscatter modelling of three-dimensional structures with discrete interior inclusions that may have different acoustic properties (Foote and Francis 2002). Like the KRM, the BEM has so far not been applied to model backscatter of herring. Nevertheless, both these models are promising since they can be applied to accurate representations of fish body parts, and therefore provide a distinct advantage over models based on simple geometric shapes. Foote and Francis (2002) used both the KRM and BEM method to model gadoid TS and found the results to be in close agreement. In a further study on gadoids, Francis and Foote (2003) attempted to quantify the depth-dependence of TS caused by increased mass density of the swimbladder gas assuming a constant volume. They modelled the swimbladder backscatter of depth-adapted pollack and saithe (Pollachius virens) and found that the mean TS did not change significantly

with depth. Even though the orientation dependence of TS was found to increase with depth they concluded that there were no implications for echo counting applications.

1.4.1.4. Distorted wave Born approximation

An approach that can be applied to weak scattering bodies of any shape is the distorted wave Born approximation (DWBA; Morse and Ingard 1968). Backscattering is expressed as a three-dimensional integral over the body volume. The model is valid at all acoustic frequencies and object orientations, however, secondary scattering and absorption within the body are ignored. Because of its abilities, the DWBA has so far been widely used to model backscatter of plankton (McGehee et al. 1998; Stanton et al. 1998; Lavery et al. 2002), but could equally well be applied to weak scattering components of fish, such as the fish flesh.