Introduction
The body-backbone fish model (BBFM) represents a
swimbladder-less fish as two explicit contributors: a weakly scattering
flesh body and an elastic backbone. This separation is motivated by
measurements and models in which skeletal scattering cannot be absorbed
reliably into a homogeneous flesh contrast (Gorska et al. 2005; Stanton et al. 1998).
BBFM is a coherent component model rather than an exact three-medium boundary solution:
- the flesh body is treated as a weakly scattering fluid-like region,
- the backbone is treated as an elastic cylindrical structure, and
- the two terms are embedded into one body-fixed frame through a phase translation before their complex amplitudes are summed.
The complex amplitudes are combined before forming backscattering cross-section.
Medium indices and target-strength quantities follow Notation and symbols. The component approximations are developed on the DWBA theory and ECMS theory pages.
Geometry and medium indexing
The family uses the shared package convention: medium 1
is the surrounding seawater, medium 2 is the flesh body,
and medium 3 is the backbone.
The exterior seawater wavenumber is:
k_1 = \frac{\omega}{c_1},
where c_1 is the seawater sound speed. The flesh density and sound-speed contrasts are therefore:
g_{21} = \frac{\rho_2}{\rho_1}, \qquad h_{21} = \frac{c_2}{c_1}.
For the backbone, the model keeps the absolute elastic properties explicit:
\rho_3, \qquad c_{L,3}, \qquad c_{T,3},
where c_{L,3} and c_{T,3} are the longitudinal and transverse wave speeds of the elastic backbone.
The backbone term is a seawater-referenced elastic-cylinder surrogate. It is not the exact solution for an elastic region 3 embedded in flesh region 2.
Flesh-body contribution
Weak-fluid assumption
The flesh component follows the distorted-wave Born approximation. Material contrasts relative to seawater must be weak enough for first-order scattering, while the phase is retained across the extended body (Chu and Ye 1999).
Using the same contrast notation as the DWBA theory
page, the compressibility and density perturbations are:
\gamma_{\kappa,21} = \frac{\kappa_2 - \kappa_1}{\kappa_1}, \qquad \gamma_{\rho,21} = \frac{\rho_2 - \rho_1}{\rho_2}.
Schematic backscattering amplitude
At the volume-integral level, the flesh contribution may be written schematically as:
f_{\mathrm{bs}}^{(2)} = \frac{k_1^2}{4\pi} \iiint_{V_2} \left( \gamma_{\kappa,21} - \gamma_{\rho,21}\cos^2\beta \right) \exp\!\left(2 i \mathbf{k}_2\cdot \mathbf{r}\right) \, dV.
Here V_2 is the flesh-body volume, \mathbf{k}_2 is the distorted interior propagation vector, and \beta is the local angle between propagation direction and body tangent.
For an elongated axisymmetric body, analytic transverse integration reduces this volume expression to the one-dimensional DWBA body integral.
Backbone contribution
Elastic-cylinder surrogate
The backbone is represented as a finite elastic cylinder rather than
as another weak-fluid inclusion. Its local physics therefore follows the
same elastic cylinder modal logic used by ECMS.
The elastic interior supports both longitudinal and transverse waves, with wavenumbers:
k_{L,3} = \frac{\omega}{c_{L,3}}, \qquad k_{T,3} = \frac{\omega}{c_{T,3}}.
For each cylindrical modal order m, the elastic boundary conditions produce an order-dependent phase shift \eta_m. The finite-cylinder backscattering amplitude may then be written schematically as:
f_{\mathrm{bs}}^{(3)} = \frac{L_3}{\pi} \frac{\sin(k_1 L_3 \cos\theta_3)} {k_1 L_3 \cos\theta_3} \sum_{m=0}^{\infty} (-1)^m \epsilon_m \sin\eta_m e^{-i\eta_m}.
Here L_3 is backbone length, \theta_3 is the backbone incidence angle, \epsilon_m is the usual Neumann factor, and \eta_m collects the elastic-cylinder boundary-condition physics.
The backbone is therefore an elastic structure with longitudinal-to-transverse wave conversion, not another weak contrast perturbation.
Spatial placement in the body frame
The flesh and backbone amplitudes cannot be added meaningfully unless they are referred to the same spatial frame. The family uses the body-fixed coordinate system for that purpose.
If the representative backbone position is \mathbf{r}_c, then the backbone amplitude is translated into the body frame by the monostatic two-way phase factor:
\exp\!\left(2 i k_1 \hat{\mathbf{q}}_{\mathrm{bs}}\cdot\mathbf{r}_c\right).
where \hat{\mathbf{q}}_{\mathrm{bs}} is the backscatter direction.
In the axisymmetric body-frame convention used here, that projection becomes:
\hat{\mathbf{q}}_{\mathrm{bs}}\cdot\mathbf{r}_c = x_c\cos\theta + z_c\sin\theta.
with (x_c, z_c) the backbone centroid and \theta the stored body angle.
This translation prevents the two components from being treated as if they scattered from the same point.
Coherent composite amplitude
Once the flesh and backbone terms are available in a common frame, the total backscattering amplitude is:
f_{\mathrm{bs}}^{(\mathrm{BBFM})} = f_{\mathrm{bs}}^{(2)} + f_{\mathrm{bs}}^{(3)} \exp\!\left(2 i k_1 \hat{\mathbf{q}}_{\mathrm{bs}}\cdot\mathbf{r}_c\right).
Adding the amplitudes before squaring retains flesh-backbone interference.
Cross-section and interference structure
The linear backscattering cross-section is:
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}^{(\mathrm{BBFM})}\right|^2,
and the target strength is (MacLennan et al. 2002):
\mathrm{TS} = 10 \log_{10}\left(\frac{\sigma_{\mathrm{bs}}}{1\ \mathrm{m}^2}\right).
Expanding the squared magnitude makes the composite physics explicit:
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}^{(2)}\right|^2 + \left|f_{\mathrm{bs}}^{(3)}\right|^2 + 2\,\Re\!\left\{ f_{\mathrm{bs}}^{(2)} \overline{f_{\mathrm{bs}}^{(3)}} \exp\!\left( -2 i k_1 \hat{\mathbf{q}}_{\mathrm{bs}}\cdot\mathbf{r}_c \right) \right\}.
The third term is the interference term. It is frequency-dependent
and position-dependent, and it is the reason the composite
TS does not reduce to a simple sum of the flesh and
backbone TS curves.
What the family does and does not solve
Included physics
BBFM explicitly includes:
- a weak-fluid flesh-body contribution,
- an elastic backbone contribution,
- coherent interference between those two components through a shared body frame.
Excluded physics
BBFM does not yet solve:
- a true embedded elastic-cylinder-in-flesh transmission problem,
- repeated rescattering between flesh and backbone,
- shadowing or blockage of one component by the other,
- anatomical variability in backbone placement across an ensemble.
These omissions distinguish the component model from a fully coupled composite-wave solution.
Why this family is still useful
BBFM is useful when flesh is weakly scattering but the backbone is much stiffer than the surrounding tissue. Keeping the components separate exposes their individual amplitudes and their frequency-dependent interference.