Kirchhoff Ray Mode (KRM) scattering model

Computes the far-field acoustic scattering amplitude and derived quantities for fish and similar elongated scatterers using the Kirchhoff–Ray Mode (KRM) approximation described by Clay and Horne (1994). The KRM model is widely used in fisheries acoustics for estimating target strength, particularly for fish with gas-filled swimbladders.

The model represents the fish body and swimbladder as a series of contiguous cylindrical elements and computes the coherent sum of their scattered fields. Depending on frequency and element size, scattering from each segment is evaluated using either a low-frequency modal solution or a high-frequency Kirchhoff (ray-based) approximation.

Usage

This model is accessed via:


target_strength(
  ...,
  model="krm",
  sound_speed_sw,
  density_sw,
  krm_variant = "lowcontrast"
)

Arguments

sound_speed_sw: Seawater sound speed ($m~s^{-1}$).
density_sw: Seawater density ($kg~m^{-3}$).
krm_variant: Swimbladder-medium convention for combined body-plus-bladder targets. Use "lowcontrast" for the low-contrast approximation $k_B \approx k$ in both swimbladder regimes, "body_embedded" for the literal body-embedded interpretation of Clay and Horne (1994), or "mixed" for the intermediate convention in which the high-$ka$ swimbladder term uses the body medium but the low-$ka$ breathing-mode term uses the low-contrast approximation. This argument is ignored for fluid-only body targets.

Scatterer representation

The scatterer must provide body and swimbladder geometry discretized into axial segments. Each segment is described by its longitudinal position and cross-sectional dimensions. Typical geometric quantities include $x(j)$, $z_U(j)$, $z_L(j)$, and radius $a(j)$ for both the body and the swimbladder.

The incident plane wave is defined by angle $\theta$, and coordinates are transformed into a rotated system according to:

$$ u(j) = x(j) \sin \theta - z(j) \cos \theta, $$ $$ v(j) = x(j) \cos \theta + z(j) \sin \theta, $$ $$ \Delta u_j = [x(j+1) - x(j)] \sin \theta, $$ $$ a_j = [w(j) + w(j+1)] / 4. $$

Segment lengths and effective radii are computed from adjacent points and used to evaluate the scattering contribution from each element.

Theory

The total backscattering amplitude is computed as the coherent sum of all scatterer segments:

$$ \mathcal{L} = \sum\limits_{b=1}^{N_b} \mathcal{L}_{B,b} + \sum\limits_{s=1}^{N_s} \mathcal{L}_{SB,s}, $$

where $\mathcal{L}_{B,b}$ is the contribution from the $b$-th body segment and $\mathcal{L}_{SB,s}$ is from the $s$-th swimbladder segment. For body segments:

$$ \mathcal{L}_{B,b} \approx -i \frac{\mathcal{R}_{wb}}{2\sqrt{\pi}} (k a_b)^{1/2} \Delta u_b \left[ e^{-i2k v_{U,b}} - \mathcal{T}_{wb}\mathcal{T}_{bw} e^{-i2k v_{U,b} + i 2 k_B (v_{U,b} - v_{L,b}) + i \psi_{B,b}} \right], $$

where $\mathcal{R}_{wb}$ and $\mathcal{T}_{wb}$ are reflection and transmission coefficients at the water-body interface, $k$ is the wavenumber in water, and $k_B$ in the body. The quantities $v_{U,b}$ and $v_{L,b}$ are the rotated upper- and lower-surface coordinates of the $b$-th segment, $\Delta u_b = [x(b+1)-x(b)] \sin\theta$, and $\psi_{B,b}$ is an empirical phase correction.

The scattering from the swimbladder depends on the dimensionless frequency parameter $ka_e$, $a_e$ is the equivalent swimbladder radius. For low frequencies ($ka < 0.15$), the swimbladder is treated as a finite-length gas-filled cylinder, and the scattering is dominated by the first cylindrical mode (breathing mode):

$$ L_M(ka)|_{m=0} = e^{i(\chi - \pi/4)} \frac{L_e}{\pi} \frac{\sin \Delta}{\Delta} b_0, $$ $$ b_0 = -\frac{1}{1 + i C_0}, $$

where $C_0$ is a mode coefficient determined by the material properties and boundary conditions of the swimbladder. For higher frequencies $ka \ge 0.15$), the Kirchhoff-ray approximation is used:

$$ \mathcal{L}_{SB,s} \approx -i \frac{\mathcal{R}_{bc} \mathcal{T}_{wb}\mathcal{T}_{bw}}{2\sqrt{\pi}} A_{SB,s} [(k a_s + 1)\sin\theta]^{1/2} \Delta u_s e^{-i (2 k_B v_s + \psi_{p,s})}. $$

Here, $a_s$ and $v_s$ are the averaged radius and longitudinal position for the segment, $\mathcal{R}_{bc}$ is the reflection coefficient at the body-cylinder interface, and $A_{SB,s}$ and $\psi_{p,s}$ are empirical amplitude and phase adjustments, respectively.

For body-plus-swimbladder targets, the default coherent KRM combines the two complex scattering lengths as:

$$ f_{bs}^{(\mathrm{coh})} = f_{body} + f_{bladder}, $$

which gives:

$$ \sigma_{bs}^{(\mathrm{coh})} = \left|f_{body} + f_{bladder}\right|^2. $$

Swimbladder medium conventions

Clay and Horne (1994) derive the swimbladder term for a gas-filled inclusion embedded in body tissue, but also note that the low body-water contrast can justify the approximation $k_B \approx k$. For combined body-plus- bladder targets, acousticTS exposes that choice through krm_variant:

"body_embedded": Use the body medium for both the high-$ka$ Kirchhoff swimbladder term and the low-$ka$ breathing-mode term. This is the most literal interpretation of the body-embedded swimbladder geometry in Clay and Horne (1994).
"mixed": Use the body medium for the high-$ka$ swimbladder term but the water approximation for the low-$ka$ breathing-mode term. This follows the later mixed convention discussed in the fisheries-acoustics literature.
"lowcontrast": Use the low-contrast approximation $k_B \approx k$ for both swimbladder regimes, i.e. evaluate both the high-$ka$ and low-$ka$ swimbladder terms with the external-medium wavenumber rather than the body-medium wavenumber.

These named variants make the swimbladder-medium assumption explicit while keeping the public API tied to the scientific interpretation rather than to implementation-specific knob names.

Assumptions and limitations

The KRM formulation assumes that the scatterer is smooth, elongated, and approximately axisymmetric, such that both the Kirchhoff (ray) approximation and low-order cylindrical mode solutions are valid representations of the local scattering physics. The body andswimbladder are discretized into short axial segments, each with its own local height and width. While the segments are treated as locally cylindrical for the purpose of computing the scattered field, their dimensions may vary independently along the body axis. The total scattered field is obtained as a coherent sum of the contributions from these segments, which implicitly assumes that geometric and material properties vary gradually relative to the acoustic wavelength.

The model further assumes that scattering is dominated by first-order interactions between the incident field and each local element. Multiple internal reflections and higher-order multiple scattering between different segments are neglected. For the fish body, the material is treated as fluid-like, with no explicit treatment of elastic shear waves or bending modes. As a result, elastic shell effects, bone resonance, and complex internal structural scattering are not represented explicitly, except insofar as they may be approximated through effective reflection or transmission coefficients.

The Kirchhoff approximation employed in the high-frequency regime assumes that the local radius of curvature of each segment is large compared to the acoustic wavelength, and that the incident wave interacts primarily with the illuminated surface. Accuracy degrades for end-on incidence, sharp geometric discontinuities, or highly concave regions, where shadowing and diffraction effects become significant. The low-frequency mode solution used for small values of $ka$ is limited to the lowest-order cylindrical modes and does not capture higher-order resonances that may arise for more complex internal geometries.

Orientation dependence is treated deterministically through the incident angle $\theta$, and the model does not account for stochastic body deformations, posture changes, or dynamic swimbladder shape variations. Surface roughness may be included through empirical attenuation of the reflection coefficient, but this treatment is phenomenological and does not represent scattering from discrete roughness elements. Consequently, the KRM is best suited for predicting mean or orientation-specific target strength for smooth-bodied organisms, and its accuracy decreases when applied to organisms with highly irregular shapes, strong elastic contrasts, or complex internal skeletal structures.

Fluid-only and alternative scattering configurations

Although the KRM formulation is most commonly applied to fish with gas-filled swimbladders, the model can also be used to compute target strength using only the fluid-like scattering contribution of the body.

In this configuration, the swimbladder scattering term is omitted and the total scattered field is obtained solely from the fluid contrast between the animal body and the surrounding medium. This mode of operation is appropriate for organisms that lack swimbladders (e.g. many invertebrates, elasmobranchs, or larval fish), or when swimbladder geometry is unknown or intentionally excluded.

More generally, the swimbladder component in the KRM framework may be replaced or supplemented by alternative internal scattering features, provided they can be represented geometrically and assigned appropriate acoustic boundary conditions. Examples include rigid or elastic skeletal structures (e.g. vertebral columns) or other localized impedance contrasts within an otherwise fluid-like body.

In all cases, the total target strength is computed from the coherent sum of the selected scattering components. Users are responsible for ensuring that the assumed boundary conditions and material contrasts are consistent with the biological structure being modeled.

Implementation

The implementation extracts geometric and acoustic parameters from the input object, transforms coordinates to the rotated reference frame, evaluates the appropriate modal or ray-based scattering expression for each segment, and coherently sums the contributions to obtain the total scattering amplitude and target strength.

References

Clay, C.S. (1991). Low resolution acoustic scattering models: Fluid-filled cylinders and fish with swimbladders. The Journal of the Acoustical Society of America, 89: 2168-2179.

Clay, C.S. (1992). Composite ray-mode approximations for backscattered sound from gas-filled cylinders and swimbladders.The Journal of the Acoustical Society of America, 92: 2173-2180.

Clay, C.S., and Horne, J.K. (1994). Acoustic models of fish: The Atlantic cod (Gadus morhua). The Journal of the Acoustical Society of America, 96: 1661-1668.