hydrac.model.scattering

Scattering

Scattering package computes the scattering models (form function, total scattering cross section, …) based on the model presets defined by the user.

Below is a quick introduction to the concept of scattering models. More details can be found in Medwin & Clay 1997 and Morse & Ingaard 1987.

In the current problem, we consider an acoustic pressure field of amplitude \(P_{i}\) emitted by a source, and propagating away from that source. In presence of an inhomogeniety such as a particle, the acoustic wave will be scattered in all directions. The amplitude of the scattered pressure field is \(P_{s}\). Considering the particle as the source for the scattered wave,:math:P_{s} can be expressed in terms of \(P_{i}\).

At large distance from a source, the amplitude of an acoustic wave decreases as \(\frac{1}{r}\) by spherical spreading, and \(10^{-\frac{\alpha_{w}r}{20}}\) by water absorption. The amplitude of the scattered wave thus writes :

\[P_s = \frac{1}{r}P_{i}|L(\theta,\phi,f)|10^{-\frac{\alpha_{w}r}{20}}\]

with

\[|L(\theta,\phi,f)|^2=\Delta_s(\theta,\phi,f)\]

where \(|L(\theta,\phi,f)|\) and \(\Delta_s(\theta,\phi,f)\) are called the scattering length (in m) and differential cross-section (in m²), and represents the amount of energy scattered by the particle in all space compared to the incident energy (\(\theta\) and \(\phi\) are respectively the polar and colatitude angles in spherical coordinate system). \(\sigma_{bs}=\Delta_s(0,0,f)\) is called the backscattering cross-section (differential cross section evaluated in the backscattered direction). Other useful and widespread notations are the form function \(f=\frac{2}{a}\sqrt(\sigma_{bs})\) and the total scattering cross section \(\chi\) is the integral of \(\Delta_s(\theta,\phi,f)\) over all space.

These complex functions vary with the size and shape of the particle, the carrier frequency of the acoustic wave, the angle of orientation in the incident pressure field, or the acoustic impedence ratios between the inhomogeneity and surrounding fluid (density ratio, compressibility ratio). They decribe the scattering properties for one single particle. Most of the time, they are made function of the wave number and the characteristic length of the particle (ex. radius for a sphere, semi-minor axis for a prolate spheroid…) \(ka\).

Using active acoustic instruments the pressure amplitude backscattered by all the particles present in the insonified volume can be measured. In this sense, modeling these functions is of utmost importance for the interpretation of the backscatter pressure/intensity.

These function do possess exact analytical solution for canonical shapes such as sphere and cylinders (Medwin & Clay 1997). But when considering arbitrarily or irregularly shaped particles (ex. zooplankton), the backscattering cross-section modeling becomes intrincate and several approximations need to be made, leading to new theoretical formulations (ex. Stanton et al. 1998) or empirical-based models (ex. Thorne & Hanes 2002).

Below is an example of the scattering length for a rigid sphere for different ka values, normalized by the root mean square of the geometric cross section of the sphere (\(\sqrt(\pi a^2)\)):

Here is a plot containing multiple backscatter models that can be computed using the present package. The complete list of models that can be computed with hydrac is given in the package hydrac.model.particle, along with the useful bibliography:

The package contains several model generation schemes : empirical, theoretical, arbitrary integral models… The following tools allow the computation of a fairly wide variety of scattering models routinely used in fisheries acoustics and sediment transport monitoring :

scattering	Scattering calculation (hydrac.model.scattering.scattering)
dwba	Distorted Wave Born Approximation model (hydrac.model.scattering.dwba)
emp	Empirically based model (hydrac.model.scattering.emp)
thq	Thoery based model (hydrac.model.scattering.thq)
hp	High-pass models (hydrac.model.scattering.hp)