Distorted wave Born approximation (DWBA) for weak scatterers

Calculates backscatter for fluid-like weak scatterers using the distorted wave Born approximation (DWBA). Frequencies in Hz; sound speed in m/s; density in kg/m^3. Material properties may be provided as contrasts or absolute values (contrasts derived relative to seawater).

Usage

This model is accessed via:


target_strength(
  ...,
  model="dwba",
  sound_speed_sw,
  density_sw
)

Arguments

sound_speed_sw: Seawater sound speed ($m~s^{-1}$).
density_sw: Seawater density ($kg~m^{-3}$).

Theory

The DWBA approach is derived under the weak scattering assumption, meaning that the differences in compressibility ($\kappa$) and density ($\rho$) between the scatterer and surrounding fluid are sufficiently small to linearize the acoustic scattering problem. This linearization allows the scattered field to be expressed as an integral over the scatterer volume.

The far-field backscattering amplitude is given by:

$$ f_{bs} = \frac{k_1}{4\pi} \int \int \int\limits_v (\gamma_\kappa - \gamma_\rho) e^{2i k_2 \cdot r_v} dv $$

where

$$ \gamma_\kappa = \frac{\kappa_2 - \kappa_1}{\kappa_1}, \quad \gamma_\rho = \frac{\rho_2 - \rho_1}{\rho_2}, $$

and

$$ \kappa = (\rho c^2)^{-1}. $$

where $c$ is the sound speed (m s^-1). For elongated bodies, the volume integral reduces to a line intergral along the body axis:

$$ f_{bs} = \frac{k_1}{4} \int_{r_{pos}} (\gamma_\kappa - \gamma_\rho) e^{2i k_2 \cdot r_{pos}} \frac{\text{J}_1(2 k_2 a \cos \beta_{tilt})}{\cos \beta_{tilt}} |dr_{pos}|, $$

where $a$ is the local radius and $\beta_{tilt}$ the local tilt angle. The cylindrical Bessel function of the first kind of order 1, $\text{J}_1$, accounts for the circular cross-section of each segment. The wavenumber $k_2$ is evaluated inside the scatterer.

Assumptions

The DWBA assumes that the scatterer is weakly inhomogenous where the material property contrasts for the interior ($c_2$, $\rho_2$) and surrounding fluid ($c_1$, $\rho_1$) where:

$$ g = \frac{\rho_2}{\rho_1} \approx 1, \quad h = \frac{c_2}{c_1} \approx 1. $$

In practice, $c_2$ and $\rho_2$ within 5% of $c_1$ and $\rho_1$, respectively, can be considered to be sufficent for the weak scattering assumption whereby:

$$ |h - 1| \le 0.05, \quad |g - 1| \le 0.05. $$

This model also assumes that the scatterer's body has no sharp edges or irregularities, and its cross-section is symmetric around the longitudinal axis. This allows the scattering integral to be reduced to a line integral along the body axis in the first place, simplifying the computation of phase contributions from different segments. Moreover, this enables the body to be discretized into along-axis segments that can approximate arbitrary body shapes. Since the body is axisymmetric and smooth, this further allows the DWBA to be applied for arbitrary orientation angles without additional correction terms.

The DWBA provides a first-order approximation that neglects multiple scattering within the body whereby secondary interactions between different parts of the body are ignored. This is valid for weakly scattering objects where the amplitude of scattered waves is small and is consistent with the Born approximation in wave physics. This model also disregards any elastic or shelled effects, treating scatterers are being purely fluid-like. Consequently, the lack of internal elasticity means there is no support for shear waves or resonances due to solid boundaries. Mathematically, this means that only the contrasts in compressibility and density between the scatterer and the surrounding medium contribute to the scattered field.

Implementation

The model extracts geometric and acoustic parameters from the input object, constructs rotation matrices and wavenumber matrices, and evaluates the integral numerically for each frequency of interest. The DWBA is computationally efficient and handles elongated, weakly scattering targets such as zooplankton and small fish.

References

Morse, P.M., and Ingard, K.U. (1968). Theoretical Acoustics. Princeton University Press.

Stanton, T.K., Chu, D., and Wiebe, P.H. (1998). Sound scattering by several zooplankton groups. II. Scattering models. The Journal of the Acoustical Society of America, 103, 236-253.