High-pass approximation (HPA) scattering model

Computes the far-field scattering amplitude and related quantities spherical and elongated scatterers using a high-pass approximation model. The high-pass model provides a simplified analytical expression for the backscattering cross-section that is valid for all values of $ka$, where $k$ is the acoustic wavenumber and $a$ is a characteristic dimension of the scatterer. The model is named after the analogy to a two-pole high-pass filter in electrical circuit theory, where the frequency response resembles the backscattering behavior as a function of $ka$. The high-pass model is computationally efficient and captures the essential physics of acoustic scattering from weakly scattering bodies, including spheres, prolate spheroids, and straight or bent cylinders. Two implementations are available: the Johnson (1977) formulation for fluid spheres, and the Stanton (1989) generalization for spheres, prolate spheroids, and cylinders.

Usage

This model is accessed via:


target_strength(
  ...,
  model = "HPA",
  method,
  deviation_fun,
  null_fun,
  sound_speed_sw,
  density_sw
)

Arguments

method: Method for computing the high-pass model. One of "johnson" (Johnson 1977 formulation for fluid spheres) or "stanton" (Stanton 1989 formulation for spheres, prolate spheroids, and cylinders).
deviation_fun: Function or scalar specifying the expected deviation in the phase of the scattered wave due to shape complexity, flexure, or other factors. Expressed as a function of $ka$, or as a scalar constant. Default is 1.
null_fun: Function or scalar specifying the expected reduction in scattering amplitude due to nulls in the scattering pattern. Expressed as a function of $ka$, or as a scalar constant. Default is 1.
sound_speed_sw: Seawater sound speed ($m~s^{-1}$).
density_sw: Seawater density ($kg~m^{-3}$).

Theory

The high-pass model is based on the observation that the backscattering cross-section of a scatterer can be approximated by a simple analytical expression that captures the low-frequency Rayleigh scattering regime (where $\sigma_{bs} \propto (ka)^4$) and the high-frequency geometric scattering regime (where $\sigma_{bs}$ approaches a constant). The transition between these regimes is governed by an auxiliary material property parameter, $\alpha$, which depends on the density contrast $g$ and sound speed contrast $h$ between the scatterer and the surrounding medium. For spheres and prolate spheroids, the auxiliary parameter is given by

$$ \alpha_{\pi s} = \frac{1 - g h^2}{3 g h^2} + \frac{1 - g}{1 + 2g} $$

and for cylinders and prolate spheroids (in the cylindrical limit), the auxiliary parameter is

$$ \alpha_{\pi c} = \frac{1 - g h^2}{2 g h^2} + \frac{1 - g}{1 + g} $$

where $g$ is the density contrast (target to medium) and $h$ is the sound speed contrast (target to medium). The reflection coefficient at the scatterer surface is given by

$$ \mathcal{R} = \frac{gh - 1}{gh + 1} $$

The Johnson (1977) high-pass model for a fluid sphere is expressed as

$$ \sigma_{bs} = \frac{a^2 (ka)^4 \alpha_{\pi s}^2}{1 + \frac{3}{2} (ka)^4} $$

where $a$ is the spherical radius. This formulation is valid for all $ka$ and provides a good approximation for fluid spheres. The Stanton (1989) high-pass model generalizes this approach to spheres, prolate spheroids, and cylinders, and incorporates empirical terms to account for nulls in the scattering pattern and deviations due to shape complexity. For a sphere, the Stanton formulation is

$$ \sigma_{bs} = \frac{ a^2 (ka)^4 \alpha_{\pi s}^2 \mathcal{G} }{ 1 + \frac{4(ka)^4 \alpha_{\pi s}^2}{\mathcal{R}^2 \mathcal{F}} } $$

where $\mathcal{G}$ is a null function that accounts for reductions in scattering amplitude at certain frequencies, and $\mathcal{F}$ is a deviation function that accounts for phase variability. For a prolate spheroid, the Stanton formulation is

$$ \sigma_{bs} = \frac{ \frac{1}{9} L^2 (ka)^4 \alpha_{\pi c}^2 \mathcal{G} }{ 1 + \frac{\frac{16}{9}(ka)^4 \alpha_{\pi c}^2} {\mathcal{R}^2 \mathcal{F}} } $$

where $L$ is the length of the prolate spheroid. For a straight cylinder at angle $\theta$, the Stanton formulation is

$$ \sigma_{bs} = \frac{ \frac{1}{4} L^2 (Ka)^4 \alpha_{\pi c}^2 s^2 \mathcal{G} }{ 1 + \frac{\pi (Ka)^3 \alpha_{\pi c}^2}{\mathcal{R}^2 \mathcal{F}} } $$

where $K = k \sin \theta$, $a$ is the cylindrical radius, and $s = \sin(kL \cos \theta) / (kL \cos \theta)$ accounts for the finite length and orientation of the cylinder. For a bent cylinder with radius of curvature $\rho_c$ relative to the cylinder length $L$, the Stanton formulation is

$$ \sigma_{bs} = \frac{ \frac{1}{4} L^2 (ka)^4 \alpha_{\pi c}^2 \mathcal{H}^2 \mathcal{G} }{ 1 + \frac{ L^2 (ka)^4 \alpha_{\pi c}^2 \mathcal{H}^2 }{ \rho_c a \mathcal{R}^2 \mathcal{F} } } $$

where $\mathcal{H} = \frac{1}{2} + \frac{1}{2} (\rho_c / L) \sin(L / \rho_c)$ is an effective length factor that accounts for the curvature. The empirical functions $\mathcal{F}$ and $\mathcal{G}$ are specified by the user as functions of $ka$, and can be used to fit the model to measured or numerically simulated scattering data. In all cases, the wavenumber satisfies $kr \gg 1$ and $L \ll 2\sqrt{r \lambda}$, where $r$ is the distance from the scatterer to the receiver and $\lambda$ is the acoustic wavelength, ensuring that the far-field and elongated object assumptions are valid.

References

Johnson, R.K. (1977). Sound scattering from a fluid sphere revisited. The Journal of the Acoustical Society of America, 61: 375-377.

Johnson, R.K. (1978). Erratum: Sound scattering from a fluid sphere revisited. The Journal of the Acoustical Society of America, 63: 626.

Stanton, T.K. (1989). Simple approximate formulas for backscattering of sound by spherical and elongated objects. The Journal of the Acoustical Society of America, 86: 1499-1510.