DWBA/SDWBA model flavor and resolving cross-software disagreement

[brandynlucca]

DWBA/SDWBA model flavor and resolving cross-software disagreement

Hi @gavinmacaulay and @jmjech (and anyone else contributing to echoSMs), I've recently been doing some cross-implementation comparisons for various models (e.g., #68) and have gotten to the SDWBA and DWBA. I've run into a few discrepancies/differences in DWBA/SDWBA results from echoSMs and other software, as well as published equations/formulations, and I'd appreciate y'all's perspective below. This grew into more material than I expected, so I've put the main questions first and linked the supporting comparisons in collapsible sections. I welcome any and all corrections where I've misunderstood a convention or missed relevant context. Thank you for any time you give any of these questions.

---

The echoSMs package states that it uses the Stanton et al. (1998) and Demer and Conti (2003) models for DWBA and SDWBA, respectively. Its predictions differ from other implementations and the outputs do not appear to coincide with the description by Demer and Conti (2003). Some or all of this may be a simple matter of clarifying conventions and the documentation, but nonetheless, there seem to be inconsistencies as well with the description between the Jech et al. (2015) paper as well. This discussion concerns the formulation echoSMs should implement, including its integration rule, phase statistics, averaging, and whether to use the phase definitions from Demer and Conti (2003; 2004), Conti and Demer (2006), and/or some other notation.

Questions for discussion

---

Important

1. DWBA integration in echoSMs.

echoSMs evaluates a nodal disc sum, assigning full adjacent-interval lengths to the endpoints. ^ref1 The interval-based implementations integrate linearly interpolated positions and radii. Increasing echoSMs from 15 to 700 nodes reduces the discrepancy for the same cylinder. ^ref2

• For profiles whose first and last points mark the physical ends, should echoSMs use half-interval endpoint weights or offer interval integration?

• If the existing weights instead assume disc-center coordinates, how should users specify the physical extent to compare identical shapes?

The documentation should explain how the nodal sum approximates Stanton et al. (1998) Eq. (5), including whether the supplied positions represent disc centers or physical boundaries and how endpoint weights are assigned.

---

Important

2. SDWBA averaging in echoSMs.

echoSMs averages backscattering magnitude before converting to TS. ^ref3 Demer and Conti (2003), the cited SDWBA reference, average backscattering cross sections.

• Should echoSMs average squared magnitude to reproduce that definition, or is there a documented reason for retaining mean magnitude?

---

Important

3. The number of independent phase elements.

echoSMs assigns one independent phase to each node, giving 15 phase elements for the 14-interval cylinder and 700 for the refined profile. Refinement therefore changes both the integration accuracy and the stochastic model. ^ref4

• Should echoSMs retain one phase per node or support the cylinder-interval phase elements described by Demer and Conti (2003)?

• How should the phase-element count and its calibration be specified independently of numerical integration accuracy?

The documentation should distinguish the numerical integration from the definition and calibration of independent phase elements. At fixed geometry, frequency and phase standard deviation, approximately uniform refinement reduces the incoherent contribution to expected power roughly as $1/N$, where $N$ denotes the number of independent phase elements. ^ref5 This can bring predictions closer to those of SDWBA_TS and the underlying Jech et al. (2015) (S)DWBA benchmarks in some cases, but does not establish equivalence between their phase definitions.

---

Important

4. Clarifying which SDWBA parameterization and flavor.

echoSMs keeps the supplied node count and phase standard deviation fixed across frequency and length. Conti and Demer (2006) introduce coupled scaling of the cylinder count and phase deviation using $N_0,f_0,L_0,\mathrm{sd}_{\varphi_0}$.

• Should echoSMs offer that parameterization alongside the cited 2003 formulation?

• If so, should it follow the published $L/L_0$ dependence, also stated by Calise and Skaret (2011), despite its omission from the SDWBA_TS source code's count and phase expressions? ^ref6

---

Important

5. Phase convention and Jech et al. (2015) benchmark agreement.

echoSMs applies $\varphi_j=\mathrm{sd}_\varphi \mathcal{Z}_j$, consistent with a Gaussian phase standard deviation of $\mathrm{sd}_\varphi$. SDWBA_TS applies $\varphi_j=\mathrm{sd}_\varphi^2\mathcal{Z}_j$. The SWFSC applet also squares the phase parameter and uses four independent random angles, one in each sine or cosine term. ^ref7 The cited Gaussian phase formulation specifies neither operation, yet SDWBA_TS and the SWFSC applet agree more closely with the benchmarked SDWBA values in the spheroid example from Jech et al. (2015). ^ref8

• Which phase convention, phase-element count, and averaging statistic generated the benchmarked SDWBA values from Jech et al. (2015)?

• What derivation or calibration would justify incorporating either alternative into echoSMs, and should any option intended to reproduce a particular software package be distinguished from the published formulation? ^ref9

Investigation

For reference, the comparisons use common shape dimensions, material properties, orientation, and surrounding medium. The coordinate-based implementations received identical initial position and radius vectors. SDWBA retains each implementation's own resampling and phase rules; however, shared arguments are parameterized identically (e.g., initial phase deviation). The centerline of shapes was offset to be 1 mm above the origin.

Note

Investigation parameterization table

Input	38.35 mm cylinder	Jech et al. (2015) cases
Dimensions	Length 38.35 mm, radius 1 mm	Sphere radius 10 mm, spheroid semiaxes 70/10 mm, cylinder length/radius 70/10 mm
Water sound speed, density	1456 m/s, 1025 kg/m³	1477.4 m/s, 1026.8 kg/m³
Density contrast $g$, sound-speed contrast $h$	1.0357, 1.0279	1.002, 1.002
Orientation	$\frac{\pi}{2}$ rad	$\frac{\pi}{2}$ rad
$f_0$, $L_0$, $N_0$ (SDWBA)	120 kHz, 38.35 mm, 14	120 kHz, 38.35 mm, 14
$\mathrm{sd}_{\varphi_0}$	$\sqrt{0.5}$ rad	$\sqrt{2}/32$ rad
SDWBA realizations	1,000 per case	100 per case

DWBA implementations

The DWBA was used for the first copmarison to isolate whether downstream inconsistencies in SDWBA predictions were due to phase definitions or other structural components.

Note

Software DWBA implementation differences table

Software	Language	DWBA integration	Local angle definition
McGehee et al. (1998), Appendix A	MATLAB	Interval integration with linear position and radius interpolation.	Interval tangent.
SDWBA_TS	MATLAB	Adaptive interval integration with linear position and radius interpolation.	Interval tangent.
echoSMs	Python	Nodal disc sum with full adjacent-distance weights at endpoints and half-sums internally.	Supplied nodal tangents.
SDWBA.jl	Julia	Adaptive interval integration with linear position and radius interpolation.	Position vector.
acousticTS	R	Adaptive interval integration with linear position and radius interpolation.	Interval tangent.
SWFSC web applet	HTML/JavaScript	Separate real and imaginary interval integrals with linear position and radius interpolation.	Interval tangent.

Warning

One additional note: the exactly end-on Bessel limit for SDWBA_TS is $ka$, whereas McGehee et al. (1998), acousticTS, and SDWBA.jl use $\frac{ka}{h}$. Due to how echoSMs handles the local angle definitions, it was parameterized with the same number of cylinders as the models, and then re-ran using 700 nodes to evaluate null frequency precision.

Note

DWBA incidence angles and frequency spectra

Incidence angle

Left: 70/10 mm spheroid. Right: 70/10 mm cylinder. Frequency: 38 kHz. Broadside: $\frac{\pi}{2}$.

acousticTS, SDWBA_TS, McGehee et al. (1998), and the SWFSC applet generally track closely. echoSMs and SDWBA.jl show different off-broadside behavior despite receiving the same initial profiles. The additional 700-node echoSMs curve shows its sensitivity to spatial discretization. These are deterministic differences, independent of stochastic phase handling.

Frequency

Left: 75°. Center: 90°. Right: 105°. Cylinder length/radius: 38.35/1 mm. Frequency: 38–400 kHz.

acousticTS, SDWBA_TS, McGehee et al. (1998), and the SWFSC applet once again generally track closely. echoSMs shifts minima through its nodal integration. Increasing its resolution reduces the difference from the four former models at 120 kHz broadside from 0.60 to 0.01 dB. SDWBA.jl develops a different high-frequency response. It is worth noting that shifting the position vector's center line onto the origin makes the DWBA results from SDWBA.jl agree with the former four models within $10^{-12}$ dB. With the 1 mm offset, differences reach ~ 23.45 dB. The SDWBA.jl local-angle calculation depends on the absolute position, although translation should not alter physical TS. Agreement on an origin-aligned shape profile conceals this problem.

SDWBA implementations

Let $\mathscr{f}_\mathrm{bs}(j)$ denote the complex DWBA backscattering amplitude of component $j$, and let $\mathcal{Z}_j\overset{\mathrm{iid}}{\sim}\mathcal{N}(0,1)$ denote independent standard normal draws, each with mean zero and variance one. The rescaled phase parameter is $\mathrm{sd}_\varphi(f,L)$. Reference inputs are $N_0,f_0,L_0,\mathrm{sd}_{\varphi_0}$. The count column below describes the 38.35 mm cylinder: its 15 boundary points define 14 intervals.

Note

SDWBA implementation differences table

Software	Stochastic phase	$N_0$	Segment rescaling	Phase rescaling	Resampling
SDWBA_TS	$\mathscr{f}_\mathrm{bs}(j)\,\exp\left[i\,\mathrm{sd}_\varphi(f,L)^2\mathcal{Z}_j\right]$	14 intervals	$N(f,\,L) = \max(N_0,\, \left\lceil \frac{N_0f}{f_0} \right\rceil)$	$\mathrm{sd}_\varphi(f,\,L) = \mathrm{sd}_{\varphi_0} \frac{N_0}{N}$	Subdivides supplied intervals using a spline centerline and inherited radii
echoSMs	$\mathscr{f}_\mathrm{bs}(j)\exp\,\left[i\,\mathrm{sd}_\varphi(f,L)\mathcal{Z}_j\right]$	15 nodes	Fixed	$\mathrm{sd}_\varphi(f,L)=\mathrm{sd}_{\varphi_0}$	Uses supplied positions and radii
echoSMs (700 nodes)	$\mathscr{f}_\mathrm{bs}(j)\exp\,\left[i\,\mathrm{sd}_\varphi(f,L)\mathcal{Z}_j\right]$	700 nodes	Fixed	$\mathrm{sd}_\varphi(f,L)=\mathrm{sd}_{\varphi_0}$	Subdivides the supplied profile while retaining its original vertices
SDWBA.jl	$\mathscr{f}_\mathrm{bs}(j)\exp\,\left[i\,\mathrm{sd}_\varphi(f,L)\mathcal{Z}_j\right]$	14 intervals	Fixed	$\mathrm{sd}_\varphi(f,L)=\mathrm{sd}_{\varphi_0}$	Uses supplied positions and radii with optional arclength resampling disabled
acousticTS	$\mathscr{f}_\mathrm{bs}(j)\exp\,\left[i\,\mathrm{sd}_\varphi(f,L)\mathcal{Z}_j\right]$	14 intervals	$N(f,\,L) = \max \left( N_0,\, \left\lceil \frac{N_0 f L}{f_0 L_0} \right\rceil \right)$	$\mathrm{sd}_\varphi(f,\,L) = \mathrm{sd}_{\varphi_0} \left( \frac{N_0 L}{N(f,\,L)L_0} \right)$	Repartitions supplied intervals using a spline centerline and inherited radii
SWFSC applet	Separate real/imaginary updates below	14 intervals	$N(f,\,L) = \left\lceil \frac{N_0f}{f_0} \right\rceil$	$\mathrm{sd}_\varphi(f,\,L) = \mathrm{sd}_{\varphi_0} \sqrt{\rule{0pt}{12pt} \frac{N(f,\,L)L_0}{N_0 L}}$	Uses evenly spaced axial points and calculates radii from the selected shape equation

Warning

echoSMs assigns independent phases to nodes, giving 15 phase elements for the same 14-interval input and 700 in the additional curve. Both echoSMs variants use the same $\mathrm{sd}_{\varphi_0}$ and realization count. Increasing the node count changes both the integration resolution and the stochastic discretization. SDWBA_TS and acousticTS retain radii from the supplied samples when repartitioning this profile.

Note

Phase deviation definitions

acousticTS, echoSMs, and SDWBA.jl apply $\varphi_j=\mathrm{sd}_\varphi(f,L)\mathcal{Z}_j$, with variance $\mathrm{sd}_\varphi(f,L)^2$. SDWBA_TS applies $\varphi_j=\mathrm{sd}_\varphi(f,L)^2\mathcal{Z}_j$, with variance $\mathrm{sd}_\varphi(f,L)^4$. The Gaussian-averaged interference factor $\exp[-\mathrm{sd}_\varphi(f,L)^2]$ follows from averaging phases with standard deviation $\mathrm{sd}_\varphi(f,L)$. ^{mean power} For the SWFSC applet, a prime denotes the phase-perturbed component amplitude. Its update is:

$\displaystyle \mathrm{Re}\mathscr{f}'_\mathrm{bs}(j) = \mathrm{Re}\mathscr{f}_\mathrm{bs}(j)\cos\,\left[\mathrm{sd}_\varphi(f,L)^2\mathcal{Z}_{j1}\right] - \mathrm{Im}\mathscr{f}_\mathrm{bs}(j)\sin\,\left[\mathrm{sd}_\varphi(f,L)^2\mathcal{Z}_{j2}\right], $

$\displaystyle \mathrm{Im}\mathscr{f}'_\mathrm{bs}(j) =\mathrm{Im}\mathscr{f}_\mathrm{bs}(j)\cos\,\left[\mathrm{sd}_\varphi(f,L)^2\mathcal{Z}_{j3}\right] +\mathrm{Re}\mathscr{f}_\mathrm{bs}(j)\sin\,\left[\mathrm{sd}_\varphi(f,L)^2\mathcal{Z}_{j4}\right],$

where $\mathcal{Z}_{jk}\overset{\mathrm{iid}}{\sim}\mathcal{N}(0,1)$ for $k = 1, 2, 3, 4$, independent across components and terms. The applet draws a different angle for each sine/cosine term, allowing an individual component's magnitude to change. A single phase rotation would preserve that magnitude.

Averaging operations also differs. With $\mathscr{f}'_\mathrm{bs}(j)$ denoting each implementation's phase-perturbed component, echoSMs returns $20\log_{10}|\langle \sum_j\mathscr{f}'_\mathrm{bs}(j) \rangle|$, while the other ensemble curves use $10\log_{10} \langle |\sum_j\mathscr{f}'_\mathrm{bs}(j)|^2 \rangle$, where $\langle \cdot \rangle$ corresponds to the average across realizations.

Caution

Attempting to derive the squared phase definition used by SDWBA_TS

A note on the phase definition by SDWBA_TS: squaring the phase deviation allows for much stronger agreement with the benchmark values, particularly for the Jech et al. (2015) prolate spheroids. However, Calise and Skaret (2011) describe Gaussian phases by their standard deviation without specifying this extra square. It is quite possible I am missing something here or not thinking about this correctly. With that being said, the only thing that immediately comes to mind that justifies this squared standard deviation are correlated phases... so below I took a stab at that to see if it could potentially explain it. Crucial: the construction below does not establish why the implemnentation squares the parameter.

For fixed scattering contributions, $\mathscr{f}_\mathrm{bs}$, TS depends on the phase differences between fixed copmonents. A random phase shared by every component rotates their sum without changing its magnitude. For jointly Gaussian, zero-mean phases:

$\displaystyle \mathbb{E} \left[e^{i(\varphi_j - \varphi_\ell)}\right ] = \exp \left[- \frac{1}{2} \mathrm{Var}(\varphi_j - \varphi_\ell) \right].$

Colosi and Morozov (200), Eqs. (28)-(29) use this dependence on phase-difference variance to describe coherence between ocean acoustic modes. This supports the general relationship between covariance and interference, although their study does not address krill scattering.

To illustrate that connection, suppose every component has phase standard deviation $\mathrm{sd}_\varphi$ and the same pairwise correlation $\rho_\varphi$, with $0 \leq \rho_\varphi \leq 1$. These phases can be represented as:

$\displaystyle \varphi_j = \underbrace{\mathrm{sd}_\varphi \sqrt{\rho_\varphi}\mathcal{Z}_0}_{\text{shared phase}} + \underbrace{\mathrm{sd}_\varphi \sqrt{1-\rho_\varphi}\mathcal{Z}_j}_{\text{independent phase}}.$

where $\mathcal{Z}_0, \mathcal{Z}_1, \ldots$ are independent standard normal draws. The independent contribution affecting TS has standard deviation $\mathrm{sd}_\varphi \sqrt{1 - \rho_\varphi}$, giving, for distinct components:

$\displaystyle \mathbb{E} \left[e^{i(\varphi_j - \varphi_\ell)} \right ] = \exp \left[ - \mathrm{sd}_\varphi^2 (1 - \rho_\varphi) \right].$

This correlation therefore reduces the effective perturbation, but does not by itself produce a quadratic dependence. Taht would require the independent contribution to grow faster than the shared contribution as the underlying motion increases. One possible mechanism is shared displacement producing a first-order phase change while local ending produces a second-order change.

The position-dependent backscatter phase in Stanton et al. (1998), Eq. (5) illustrates this to some degree. Consider two components separated by a fixed distance $d$, initially aligned with the incident wave. Tilting their separation through a small angle $\theta$ changes their relative phase by:

$\displaystyle \Delta \varphi = 2 k_2 d(\cos \theta - 1) \approx -k_2 d \theta^2.$

While this is an illustrative expansion of the geometric phase, it is not a mechanism established by that paper for the squared phase deviation used by SDWBA_TS. But if shared displacement and local bending both grow with the same underlying motion amplitude, shared phase variability could grow linearly while differential variability grows quadratically. At small amplitudes, the shared contribution would dominate the total phase variance. If the local residuals are treated as independent, this would yield approximately $1 - \rho_\varphi \propto \mathrm{sd}_\varphi^2$. That is the proposed connection between the geomeetric dependence and changing phase correlation.

The proposed mechanism could make the independent phase standard deviation proportional to $\mathrm{sd}_\varphi^2$, but it does not determine the proportionality constant. To reproduce the implementation exactly, the Gaussian construction must assume:

$\displaystyle \mathrm{sd}_\varphi \sqrt{1 - \rho_\varphi} = \mathrm{sd}_\varphi^2 \quad \Longrightarrow \quad \rho_\varphi=1-\mathrm{sd}_\varphi^2 \qquad (\mathrm{sd}_\varphi>0).$

At $\mathrm{sd}_\varphi=\sqrt{2}/32$, this requires $\rho_\varphi \approx 0.998$, so approximately 99.8% of each component's phase variance is shared. This remains an assumed construction, however. The geometric exaxmple neither determines its coefficient nor establishes independent Gaussian residuals. Its quadratic behavior also depends on the initial alignment, and squared Gaussian angular fluctuations are themselves non-Gaussian. It supplies a plausible physical route to quadratic differential phase, but does not establish why SDWBA_TS squares its Gaussian phase-deviation parameter.

Note

Phase scaling

70/10 mm spheroid. Top left: rescaled phase parameter. Top right: actual random-angle standard deviation. Bottom left: native SDWBA minus each implementation's zero-phase result on its own integration grid at 38 kHz. Bottom right: expected mean power using identical component amplitudes on the common 100-interval profile and the three applied phase standard deviations. Colors identify the phase prescriptions. Line types distinguish coherent, incoherent and total power.

The SWFSC applet's rescaled phase parameter increases as $\sqrt f$. SDWBA_TS's decreases approximately as $\frac{1}{f}$ above its count floor, $N_0$. Squaring these parameters gives applied angle standard deviations proportional to $f$ and $f^{-2}$, apart from integer-count steps. The different operations for computing rescaled $\mathrm{sd}_\varphi(f,L)$ and the actual random-angle standard deviation yield different results:

Implementation, 38 kHz	Rescaled parameter $\mathrm{sd}_\varphi(f,L)$	Actual random-angle standard deviation (rad)
acousticTS	0.139082	0.139082
SDWBA_TS	0.044194	0.001953
SWFSC applet	0.024912	0.000621 per angle

For fixed component amplitudes and independent Gaussian phase angles with a common variance $\mathrm{Var}(\varphi_j)$:

$\displaystyle \mathbb{E}\left|\sum_j\mathscr{f}'_\mathrm{bs}(j)\right|^2 = \underbrace{e^{-\mathrm{Var}(\varphi_j)}\left|\sum_j\mathscr{f}_\mathrm{bs}(j)\right|^2}_{\text{coherent contribution}} +\underbrace{\left(1-e^{-\mathrm{Var}(\varphi_j)}\right)\sum_j\left|\mathscr{f}_\mathrm{bs}(j)\right|^2}_{\text{incoherent contribution}},$

where the variance is $\mathrm{sd}_\varphi(f,L)^2$ for acousticTS, echoSMs and SDWBA.jl, and $\mathrm{sd}_\varphi(f,L)^4$ for SDWBA_TS. The same mean-power identity holds for the SWFSC applet's four-angle update using each angle's variance $\mathrm{sd}_\varphi(f,L)^4$, although its realization distribution differs. The SWFSC applet and SDWBA_TS's small applied angles preserve coherent cancellation here, keeping results close to the DWBA outputs. acousticTS's larger applied standard deviation raises the incoherent contribution until it dominates deep off-broadside minima, resulting with a much higher effective TS prediction floor for off-incidence angles. This contribution varies with incidence through $\mathscr{f}_\mathrm{bs}(j)$, without an angular threshold. Assessing these differences requires reconciling the applied phase distributions with the published standard deviation and calibration.

Note

SDWBA frequency spectra and small phase deviations

Frequency

Left: 75°. Center: 90°. Right: 105°. Same cylinder, materials and frequency range as the DWBA spectra.

Differences are most pronounced near off-broadside minima, where phase perturbations weaken cancellation between component amplitudes. The SWFSC applet's 75° and 90° spectra are distinct. A shared reference phase SD does not ensure the same effective phase distribution after the operations in the table.

Zero and small phase deviations

At zero phase deviation, SDWBA and DWBA agree to numerical precision across the tested angles and frequencies when both use the same shape discretization. Changing that discretization can change TS even at zero phase. For the spheroid at 38 kHz, a reference phase standard deviation of $\mathrm{sd}_{\varphi_0}=0.001$ rad gives $\mathrm{sd}_\varphi=0.003147$ rad under acousticTS's frequency and length scaling. TS changes by less than 0.0001 dB at broadside and increases by 2.275 dB at the 26° minimum. Small phase perturbations weaken destructive interference, so their relative effect is greatest near deep minima.

Jech et al. (2015) reference comparisons

The figures use the archived columns accompanying Jech et al. (2015). Modal series is Benchmark, DWBA (Jech et al., 2015) is Gorska_DWBA_TS, and SDWBA (Jech et al., 2015) is Demer_SDWBA_TS. DWBA and SDWBA panels contain their respective model column. For the finite cylinder, the modal-series reference is the paper's DCM approximation, applicable near broadside at approximately 70–90°. At the spheroid's 38 kHz/26° minimum, the archived SDWBA value is −137.41 dB. acousticTS gives −105.00, SDWBA_TS −135.17, SWFSC applet −136.95, SDWBA.jl −115.70, and echoSMs −115.38 dB, or −123.48 dB with 700 nodes. acousticTS's exact Gaussian expectation is −105.01 dB, confirming the discrepancy persists without Monte Carlo noise. Its effective phase standard deviation is 0.139082 rad. SDWBA.jl and both echoSMs variants retain 0.044194 rad despite sharing the expression $\mathrm{sd}_\varphi(f,L)\mathcal{Z}_j$.

Note

All comparisons: DWBA on the left, SDWBA on the right

Figure 3(D): sphere, frequency.

Figure 5(D): spheroid, frequency through 200 kHz.

Figure 6(D): spheroid, frequency through 400 kHz.

Figure 7(D): spheroid, incidence at 38 kHz.

Figure 8(D): cylinder, frequency.

Figure 9(D): cylinder, 70–90° incidence at 38 kHz.

Figure 10(D): cylinder, 0–90° incidence at 38 kHz.