Stochastic distorted wave Born approximation (SDWBA) for weak scatterers

Calculates the far-field scattering amplitude and related quantities for fluid-like, weak scatterers using the stochastic distorted wave Born approximation (SDWBA), as described by Demer and Conti (2003). The SDWBA extends the deterministic DWBA by incorporating stochastic phase variability to account for unresolved structural complexity and dynamic variability in biological scatterers.

Usage

This model is accessed via:


target_strength(
  ...,
  model = "sdwba",
  n_iterations,
  n_segments_init,
  phase_sd_init,
  length_init,
  frequency_init,
  sound_speed_sw,
  density_sw
)

Arguments

n_iterations: Number of stochastic realizations for averaging target strength predictions.
n_segments_init: Reference number of body segments.
phase_sd_init: Reference phase deviation (radians).
length_init: Reference body length (m).
frequency_init: Reference frequency (Hz).
sound_speed_sw: Seawater sound speed ($m~s^{-1}$).
density_sw: Seawater density ($kg~m^{-3}$).

Theory

The SDWBA is derived under the weak scattering assumption, where the differences in compressibility ($\kappa$) and density ($\rho$) between the scatterer and the surrounding fluid are small enough to linearize the acoustic scattering problem (see DWBA). in this regime, multiple scattering within the body is neglected, and the total scattered field is approximated as the coherent sum of first-order contributions from individual body segments.

The key extension introduced by the SDWBA is the inclusion of stochastic phase variability to represent unresolved morphological complexity, internal inhomogeneity, and dynamic effects such as body flexure and orientation variability. The linear scattering coefficient is written as:

$$ f_{bs}(\theta) = \sum\limits_{j=1}^N f_{bs}^{(j)}(\theta) \exp(i \varphi_j), $$

where $N$ is the number of body segments, $f_{bs}^{(j)}$ is the contribution from segment $j$, and $\varphi_j$ is a random phase perturbation drawn independently for each segment.

The phase perturbations are assumed to follow a zero-mean Gaussian distribution with variance related to the effective signal-to-noise ratio (SNR) of the scattering process. The minimum expected phase variance due to noise is given by:

$$ \mathbb{V}(\varphi_j) = \frac{1}{2 \mathrm{SNR}}, $$

though in practice larger variances are used to account for additional physical sources of phase decorrelation not explicitly modeled.

The expected backscattering cross-section is obtained by ensemble averaging over multiple stochastic realizations: $$ \langle \sigma_{bs}(\theta) \rangle = \mathbb{E}\!\left[ \left| f_{bs}(\theta) \right|^2 \right] \approx \frac{1}{M} \sum_{m=1}^{M} \left| f_{bs}^{(m)}(\theta) \right|^2, $$

and the expected target strength, $\mathbb{E}[TS(\theta)]$, is computed from this mean.

To ensure consistency across frequencies and body sizes, the SDWBA enforces scale invariance by preserving the product of the phase standard deviation, $\mathrm{sd}_\varphi$, and frequency, $f$:

$$ \mathrm{sd}_{\varphi}(f)\, f = \mathrm{sd}_{\varphi_0}\, f_0, $$

and by scaling the number of segments to maintain constant spatial resolution relative to acoustic wavelength:

$$ N(f, L) = N_0 \frac{f L}{f_0 L_0}. $$

The phase standard deviation at arbitrary frequency and length is then:

$$ \mathrm{sd}_{\varphi}(f, L) = \mathrm{sd}_{\varphi_0} \frac{N_0 L}{N(f, L) L_0}. $$

These scaling relationships ensure that stochastic decorrelation effects remain physically consistent across different acoustic and geometric regimes.

Implementation

The implementation extracts geometric and acoustic parameters from the input object, constructs the required rotation and wavenumber matrices, and evaluates the DWBA contribution for each segment. For each stochastic realization, random phase perturbations are applied, and the resulting backscattering amplitudes are averaged over all realizations to estimate the expected target strength.

References

Conti, D.A., and Conti, S.G. (2006). Improved parameterization of the SDWBA for estimating krill target strength. ICES Journal of Marine Science, 63: 928-935.

Demer, D.A., and Conti, S.G. (2003). Reconciling theoretical versus empirical target strengths of krill: effects of phase variability on the distorted-wave Born approximation. ICES Journal of Marine Science, 60: 429-434.

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.