Phase-compensated distorted wave Born approximation
Source:vignettes/pcdwba/pcdwba-theory.Rmd
pcdwba-theory.RmdIntroduction
The phase-compensated distorted wave Born approximation
(PCDWBA) extends the ordinary DWBA to
elongated bodies whose centerlines are curved rather than straight (Chu and Ye 1999; Stanton 1989). It preserves the
same weak-scattering local kernel used for slender fluid-like bodies,
but it replaces the straight-body phase bookkeeping by a
centerline-dependent phase that follows the bent geometry
explicitly.
The family therefore sits conceptually between two extremes: it is more physical than treating a curved body as if it were straight, but it remains a Born-type perturbation model rather than a full curved-body boundary-value solution.
The notation below follows the shared package conventions: medium
1 is the surrounding seawater and medium 2 is
the weakly scattering body.
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 canonical body, the centerline is wrapped around
an osculating circle of radius \rho_c,
with curvature \kappa = 1/\rho_c. A
taper function may also scale the local radius toward the ends, but that
taper only changes the local cross-sectional factor; it does not alter
the phase logic that motivates PCDWBA.
Why curvature matters
In a straight-body DWBA, the two-way phase factor
depends only on the axial position along a straight line. Once the body
bends, two points with the same arc-length separation no longer have the
same projection onto the incident or receive directions. That means a
curved target cannot be described correctly by the same straight-axis
phase term unless the curvature is negligible.
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(\frac{k_2 a(s)}{1}\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_1 \hat{\mathbf{q}}\cdot \mathbf{r}_c(s) \right] ds,
where \hat{\mathbf{q}} denotes the backscatter direction.
The crucial ingredient is the phase factor:
\exp\!\left[ 2 i k_1 \hat{\mathbf{q}}\cdot \mathbf{r}_c(s) \right].
This is what makes the model phase compensated: the phase is evaluated on the actual curved centerline rather than on a fictitiously straight axis.
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_1 \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 is the form most directly useful for irregular profiles, because it treats curvature, taper, and segment spacing in the same centerline-aligned framework.
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.
That limiting behavior is important physically: DWBA is
the straight-axis weak-scattering limit, while PCDWBA is
the curved-axis weak-scattering extension.
The two models therefore differ in phase bookkeeping, not in the local cross-sectional contrast physics.
Backscatter and target strength
Once the complex backscattering amplitude has been assembled from the curved centerline sum, the standard monostatic outputs are (MacLennan, Fernandes, and Dalen 2002; Urick 1983; Simmonds and MacLennan 2005):
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}\right|^2, \qquad \mathrm{TS} = 10 \log_{10}\left(\sigma_{\mathrm{bs}}\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.
That is why PCDWBA should be read as a curved-body
extension of DWBA, not as a new exact geometry family. Its
main purpose is to keep the part of the physics that DWBA
misses first when the body bends: the geometry-dependent coherent
phase.