Phase-compensated distorted wave Born approximation
Source:vignettes/pcdwba/pcdwba-theory.Rmd
pcdwba-theory.RmdIntroduction
The phase-compensated distorted wave Born approximation
(PCDWBA) applies a weak-scattering, slender-body kernel
along a curved centerline (Chu and Ye 1999). Its distinguishing
term is the two-way phase evaluated on the curved geometry.
It remains a first-order Born approximation, not an exact curved-boundary solution.
Medium indices follow Notation and symbols. The weak-contrast derivation and straight-centerline limit are given in DWBA theory.
Weak-scattering starting point
Born contrast term
PCDWBA inherits its material-contrast physics from the
ordinary distorted wave Born approximation. For a fluid-like body in
seawater, the density and sound-speed contrasts are:
g_{21} = \frac{\rho_2}{\rho_1}, \qquad h_{21} = \frac{c_2}{c_1},
where \rho_1, c_1 are the surrounding-fluid density and sound speed and \rho_2, c_2 are the corresponding body properties.
The standard fluid-like contrast factor is then:
C_{21} = \frac{1 - g_{21} h_{21}^2}{g_{21} h_{21}^2} - \frac{g_{21} - 1}{g_{21}}.
The approximation is most defensible when g_{21} and h_{21} remain close to unity, so that the total field inside the body remains close to the distorted incident field.
Volume-integral viewpoint
In the general Born picture, the scattered field is written as a volume integral over the target:
p^{\mathrm{sca}}(\mathbf{r}) \propto \int_V C_{21}(\mathbf{r}') \, G_1(\mathbf{r},\mathbf{r}') \, p^{\mathrm{ref}}(\mathbf{r}') \, dV',
where G_1 is the Green’s function of the exterior medium and p^{\mathrm{ref}} is the chosen reference field. For slender bodies, this three-dimensional integral is reduced by separating the local cross-sectional response from the phase accumulated along the body axis.
Curved centerline geometry
Centerline parameterization
Let s denote arc length along the body centerline and let \mathbf{r}_c(s) denote the corresponding centerline position. The local body radius is a(s), and the local tangent angle is \beta(s).
For a uniformly bent body, the centerline follows a circle of radius \rho_c and curvature \kappa=1/\rho_c. A taper changes a(s) but not the form of the centerline phase.
Local cylindrical reduction
At each centerline location, the body is approximated locally by a circular cylinder with radius a(s). The azimuthal integration over that local cross-section produces the same Bessel-type factor that appears in slender-body Born models:
\frac{J_1\!\left(2 k_2 a(s)\, \chi(s)\right)} {2 k_2 a(s)\, \chi(s)},
Here J_1 is the cylindrical Bessel function of the first kind, k_2 = \omega / c_2 = k_1 / h_{21} is the body wavenumber, and \chi(s) is the local projection factor set by the incident and receive directions relative to the local tangent.
For monostatic backscatter, this projection reduces to a local cosine term involving the body orientation and tangent angle.
Phase-compensated line integral
General curved-body form
After the local cross-sectional reduction, the scattered field becomes a centerline integral. In monostatic form, the backscattering amplitude can be written schematically as:
f_{\mathrm{bs}} \propto \int_{-L/2}^{L/2} C_{21}(s) \left(k_2 a(s)\right)^2 \frac{J_1\!\left(2 k_2 a(s)\, \chi(s)\right)} {2 k_2 a(s)\, \chi(s)} \exp\!\left[ 2 i k_2 \hat{\mathbf{q}}\cdot \mathbf{r}_c(s) \right] ds,
where \hat{\mathbf{q}} denotes the backscatter direction.
The phase-compensation factor is:
\exp\!\left[ 2 i k_2 \hat{\mathbf{q}}\cdot \mathbf{r}_c(s) \right].
The actual curved centerline replaces the straight-axis phase approximation.
Discrete form
In segmented form, the same model appears as:
f_{\mathrm{bs}} \propto \sum_j C_{21,j} \left(k_2 a_j\right)^2 \frac{J_1\!\left(2 k_2 a_j \chi_j\right)} {2 k_2 a_j \chi_j} \exp\!\left[ 2 i k_2 \hat{\mathbf{q}}\cdot \mathbf{r}_{c,j} \right] \Delta s_j,
where the index j labels body segments and \Delta s_j is the local centerline spacing.
This form accommodates a varying radius, curvature, and segment spacing.
Relation to straight DWBA
If the centerline becomes straight, then \mathbf{r}_c(s) reduces to a linear function
of s and the phase-compensated
expression collapses to the ordinary straight-body
DWBA.
Thus the two models share their local contrast kernel and differ in geometric phase bookkeeping.
Backscatter and target strength
Once the complex backscattering amplitude has been assembled from the curved centerline sum, the monostatic outputs are (MacLennan et al. 2002):
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}\right|^2, \qquad \mathrm{TS} = 10 \log_{10}\left( \frac{\sigma_{\mathrm{bs}}}{1\ \mathrm{m}^2}\right).
No new reporting convention is introduced by the curvature correction. The change is entirely in the underlying phase-sensitive amplitude.
Assumptions and regime
PCDWBA rests on the following assumptions:
- weak fluid-like material contrast,
- slender-body reduction to a local cylindrical kernel,
- single scattering,
- curvature enters through centerline phase rather than through a new exact cross-sectional boundary solve,
- the target is described meaningfully by a centerline and local radius profile.
These assumptions make PCDWBA a curved-axis extension of DWBA. Its additional physics is the geometry-dependent coherent phase.