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Leveraging Bayesian Optimization for Array Shape Self-Calibration in Underwater DoA Estimation

Xin Gui, Tianang Li, Changjia Wang, Bowen Han, Yunchuan Zhang, Zhengying Li

arXiv:2608.30488v1eess.SPphysics.optics

TL;DR

Flexible-array deformation makes DoA estimation and high-dimensional self-calibration difficult. The paper introduces BOGE, which combines a physics-informed parametric geometry model with Bayesian-optimization-guided hierarchical refinement. BOGE achieves the lowest mean geometric RMSE over most tested SNRs and reports 0.659 m geometric RMSE at 166 Hz on SWellEx-96, with comparable lake-trial localization and tracking performance.

  • Problem

    Flexible arrays undergo unpredictable deformation, while estimating individual element positions creates a high-dimensional calibration problem over long arrays.

  • Method

    BOGE combines a physics-informed parametric geometry model with Bayesian optimization for coarse estimation and numerical optimization for residual refinement.

  • Results

    BOGE achieved the lowest mean geometric RMSE over most tested SNRs and a geometric RMSE of 0.659 m at 166 Hz on SWellEx-96.

  • Takeaways & Limitations

    A lake trial found fixed-source localization and moving-target tracking performance comparable to the comparison methods.

Abstract

from arXiv · show

Flexible sensing arrays are commonly used in underwater acoustic networks, but suppressed by unpredictable geometric deformations. Existing array shape self-calibration methods often estimate individual element positions separately, leading to a high dimensional optimization problem over long arrays. To address this problem, this paper proposes a Bayesian Optimization-assisted Geometry Estimation (BOGE) strategy operating with a hierarchical optimization process and a physics-informed parametric model for array geometry correction. BOGE formulates array shape self-calibration as an optimization problem, where candidate geometries are evaluated by the noise subspace residual. We perform Bayesian optimization to configure the physics-informed parametric model and then refine the selected geometry through numerical optimization. Empirical results show that BOGE achieves lower mean geometric root mean square error (RMSE) than the benchmark methods across a wide range of noise levels. On the public SWellEx-96 dataset, BOGE achieves a geometric RMSE of $0.659$ meters at $166$ Hz. A lake trial further shows that BOGE provides fixed source localization and moving target tracking performance comparable to the comparison methods.

I. INTRODUCTION

Underwater flexible arrays suffer deformation and position errors that undermine DoA estimation and complicate self-calibration. BOGE addresses these challenges with a physics-informed parametric model and hierarchical Bayesian optimization, achieving strong geometric and localization results.

  • Dynamic factors such as currents, tides, and manufacturing imperfections deform flexible arrays, while small position errors can cause severe DoA errors.
  • Existing self-calibration methods face restrictive geometry assumptions, degraded features under strong noise, and limited reconstruction of complex deformations.
  • Nominally linear arrays can be spatially ambiguous under small position errors, so prior methods impose reference-element or global-rotation constraints for identifiability.
  • BOGE combines a physics-informed parametric geometry model with hierarchical Bayesian optimization to replace independent coordinate search with lower-dimensional estimation.A coarse deformation model is followed by low-dimensional residual refinement, without restricting the array to a single low-order shape.
  • At 166 Hz, BOGE obtains a geometric RMSE of 0.659 m on the North HLA and supports fixed-source localization and moving-source tracking in a lake trial.The moving source was tracked continuously for 60 seconds, with stable and reliable single-source DoA estimation performance reported.

A. Ideal Array Model

The ideal model represents a uniform linear array in two-dimensional space and derives its received signals through source directions, propagation delays, and steering vectors. Shape mismatch replaces nominal positions with actual displaced positions, introducing direction-dependent phase errors that degrade DoA estimation.

  • The ideal array is a 2D uniform linear array with M sensing elements, element spacing d, and positions aligned with a coordinate axis.
  • Far-field narrow-band sources are described by directions θ_k and unit propagation vectors, with negligible envelope variation across the array.
  • The received signal model uses propagation delays, steering vectors dependent on the element-position matrix R, and a steering matrix collecting all K sources.
  • Array model with shape mismatch: Actual element positions are modeled as nominal positions plus spatial displacements, so the array need not remain aligned with the coordinate axis.
  • Array model with shape mismatch: Distinct element displacements produce direction-dependent error steering vectors and an error matrix that enters the noisy received-signal model.
  • Array model with shape mismatch: Using the ideal steering matrix despite shape mismatch distorts the spatial spectrum and degrades DoA estimation.

C. DoA Performance with Array Shape Mismatch

Array shape mismatch breaks the orthogonality used by MUSIC, causing shifted, weakened, or broadened spatial-spectrum peaks. Calibration is therefore formulated with reference elements to recover absolute geometry and source direction.

  • MUSIC estimates DoA from orthogonality between signal and noise subspaces, but distorted steering vectors violate this relationship.
  • Shape mismatch can shift peak locations, reduce peak heights, and broaden main lobes, preventing the spatial spectrum from centering on the true direction.
  • A maximum positional error of 3 m produces a 3° DoA bias, while 10 m produces a 10° deviation for sinusoidal distortion on a nominal ULA.
  • Single-source observations are invariant to simultaneous array rotation and source-direction rotation, making reconstruction relative without references or structural constraints.
  • Known noncollinear first, middle, and last reference elements provide directional and positional references for absolute coordinates.
  • The calibration goal is to estimate unknown element coordinates and source direction from noisy observations and known reference-element coordinates.

III. ARRAY SHAPE CALIBRATION METHOD

The method models flexible-array deformations with a low-dimensional physics-informed parameterization and estimates geometry through coarse-to-fine optimization. Reference constraints and physical structure reduce ambiguity and search dimensionality, while residual modeling captures deviations from the coarse shape.

  • BOGE searches a low-dimensional physics-informed model for the global coarse shape, then optimizes residual coefficients to represent local deviations.The framework avoids direct optimization of every element coordinate.
  • For long arrays, overlap sub-arrays are aligned and spliced to reconstruct the full geometry.
  • Directly optimizing coordinates for an M-element array creates 2M variables, enlarging the nonconvex search space and computational complexity.
  • The array is modeled as a smooth continuous physical entity whose interelement distance varies within [dmin, dmax].The model reflects cable continuity, tensile strength, bending stiffness, slack, and stretching.
  • The shape vector η = [A, ω, ϕ, κ]T represents sinusoidal amplitude, spatial frequency, initial phase, and arc-like bending.
  • Global rotation is obtained by aligning the modeled end-to-end baseline with a known reference vector rather than treating rotation as a BO variable.
  • Reference-element constraints impose three scalar constraints on four shape parameters, leaving one feasible degree of freedom for acoustic observations.
  • When A = 0, the model becomes a circular arc independent of ω and ϕ, so identifiability concerns an effective geometry class rather than a unique four-parameter vector.

B. Objective Function Formulation

The paper constructs a data-driven objective from noise-subspace orthogonality and searches its constrained parameter space with Bayesian optimization. Feasible curves reduce the search to one-dimensional problems, while Gaussian-process surrogates and Expected Improvement guide sampling before geometry refinement and identifiability analysis.

  • B. Objective Function Formulation: Candidate geometries are evaluated by the minimum noise-subspace residual over an admissible angular scan, returning both residual and estimated direction.
  • B. Objective Function Formulation: With finite snapshots, the true geometry generally corresponds to a nearby sample minimum rather than an exact zero residual.
  • C. Phase I: BO-Driven Coarse Shape Estimation: The nonlinear, nonconvex coarse objective motivates Bayesian optimization, while reference constraints restrict the search to a lower-dimensional feasible set.
  • C. Phase I: BO-Driven Coarse Shape Estimation: When H(η) has rank three, numerical root finding and pseudo-arclength continuation construct feasible one-dimensional curves parameterized by normalized arclength.
  • C. Phase I: BO-Driven Coarse Shape Estimation: Expected Improvement selects new arclength samples by balancing predicted objective reduction against posterior uncertainty in the minimization problem.
  • C. Phase I: BO-Driven Coarse Shape Estimation: BO evaluates each feasible curve and selects the curve with the smallest observed objective before reconstructing the coarse parameters, geometry, and source direction.
  • C. Phase I: BO-Driven Coarse Shape Estimation: If shape perturbations cannot be canceled by direction perturbations, the true parameters form a strict isolated local minimum; global uniqueness requires injectivity and nonintersecting curve images.

D. Phase II: Fine Shape Refinement

Phase II refines the coarse Bayesian-optimization geometry using a low-dimensional residual model and constrained numerical optimization. The formulation addresses identifiability and long-array alignment through physical constraints, residual bases, and overlap fusion.

  • Fine refinement: The fine stage initializes numerical refinement from the coarse Bayesian-optimization geometry to place optimization in a locally informative region.Gradient-based optimization then refines residual coefficients within that localized region.
  • Residual model: Each residual-basis column represents a deformation mode across the array, mapping residual coefficients to two-dimensional coordinate corrections.The residual basis has p observable coefficients and reshapes 2M coordinates into an M × 2 correction matrix.
  • Residual model: Cubic B-splines generate smooth tangent- and normal-direction corrections while preserving the three reference-element coordinates.The reference corrections are fixed to zero.
  • Constraints: The admissible refinement set constrains coefficients, element displacements, and link-length changes to physically feasible deviations from Phase I.The displacement trust ratio is set to 0.08 in the current implementation.
  • Long-array alignment: For long arrays, the coarse-to-fine procedure runs on overlap sub-arrays, whose shared elements determine rigid transformations for global alignment and coordinate fusion.Adjacent sub-arrays are aligned using Kabsch alignment.
  • Identifiability: If the residual basis only approximates the true deformation, Phase 2 recovers an approximation rather than the exact geometry.Exact uniqueness requires the stated rank and modeling conditions.
  • Identifiability: With one source direction, elementwise refinement remains unidentifiable because fixing three references leaves 2M −6 degrees of freedom but provides at most M −3 constraints.Bounds and smoothness penalties can select a numerical solution but do not create data identifiability.

IV. SIMULATION RESULTS

The simulations evaluate BOGE against four calibration benchmarks for geometric reconstruction and resulting DoA estimation in 2D and 3D settings. They also test performance across SNRs and overlap ratios.

  • Simulation design: The simulation study compares BOGE with four methods while assessing geometric accuracy and DoA estimation in both 2D and 3D spaces.The study also examines performance across SNRs and the effect of overlap ratio.

A. Benchmark

The benchmark evaluation uses four independently reproduced calibration methods as comparators for BOGE under a common evaluator. The comparison includes ASMLM, WORKS, AS-TDE, and DAS-SC.

  • Comparison methods: Four representative baselines—ASMLM, WORKS, AS-TDE, and DAS-SC—are selected to evaluate BOGE.The methods use marginal-likelihood, robust Kalman-smoothing, time-delay, and distributed-acoustic-sensing calibration formulations, respectively.
  • Comparison methods: BOGE uses the authors’ implementation, while the four baselines are independently reproduced from their published equations.All methods are evaluated under a common evaluator.

B. Evaluation of Array Shape Calibration

The array-shape calibration evaluation tests BOGE on simulated 2D and 3D deformations representing current deflection, wave vibration, buoy suspension, and combined bending with axial rolling. The reconstructed geometries closely fit the simulated ground truths.

  • 2D evaluation: The 2D simulations use a 40-element, 1-meter-spaced flexible array with circular-arc and sinusoidal deformations.The circular arc represents steady-current deflection, while the sinusoid represents wave-induced vibration.
  • 2D evaluation: The calibrated 2D geometries fit the ground-truth shapes, and their MUSIC spectra closely match ideal spectra with sharp main lobes at −30°.The calibrated arrays eliminate the DoA estimation failure observed with the uncalibrated array.
  • 3D evaluation: The 3D simulations use a 100-element, 1-meter-spaced flexible array undergoing simultaneous bending and global axial rolling.The tested configurations are a buoy-suspended catenary and a tilted sinusoid with drooping and wave-induced vibration.
  • 3D evaluation: The calibrated 3D shapes fit the true geometries with high precision, attributed to the parametric model coupled with overlapped alignment.The evaluation includes sinusoidal and catenary array geometries.

C. Statistical Performance of BOGE

Monte Carlo evaluations show that BOGE maintains low geometry and DoA errors across noise levels, while 30% sub-array overlap balances accuracy and computational cost for long arrays.

  • Statistical evaluation: 200 Monte Carlo trials evaluate a 20-element 2D array across SNRs from −20 dB to 20 dB in 0.5 dB increments.Geometric RMSE and DoA MAE are reported for BOGE and benchmark methods.
  • Noise robustness: BOGE achieves the lowest mean geometry RMSE over most tested SNRs, reaching approximately 0.0737, 0.0145, and 0.00341 meters at −20, −10, and 0 dB.At high SNRs, WORKS and BOGE have similar geometry RMSEs, while BOGE has lower DoA MAE.
  • Noise robustness: AS-TDE has the largest low-SNR errors because its calibration signals are poor, whereas WORKS improves as multitone delay estimates become more reliable.DAS-SC shows stepped error reductions as source quality increases.
  • Long-array segmentation: At 10% and 20% overlap, weak coupling causes severe mismatch and DoA MAE above 13°, while 30% overlap reduces geometric RMSE to 0.2 meters and DoA MAE to nearly 0°.Further overlap increases provide only small precision gains while requiring more repeated processing.
  • Long-array segmentation: A 30% overlap ratio provides a favorable trade-off between estimation accuracy and computational cost for the 100-element array.The segmentation uses sliding windows with overlapping elements to reduce the dimensionality burden.

V. REAL-WORLD VALIDATION

Real-world validation on SWellEx-96 shows that BOGE reconstructs array geometry more accurately than the benchmarks, while the lake trial evaluates its practical localization and tracking use cases.

  • Validation scope: SWellEx-96 and a lake trial assess geometric reconstruction, fixed-source localization, and moving-source tracking in field data.ASMLM and WORKS assumptions are not fully satisfied in these settings, so real-world comparisons use BOGE, AS-TDE, and DAS-SC.
  • SWellEx-96 dataset: BOGE aligns more closely with the SWellEx-96 reference shape than the comparison methods.The North HLA contains 27 effective elements calibrated in 2D, with nonuniform spacing and invalid elements accounted for.

B. Lake Trial Setup

The lake trial tests BOGE on a long, buoy-suspended fiber-optic array for fixed-source localization and moving-target tracking, where absolute geometric ground truth is unavailable.

  • Lake Trial Setup: The lake array has 238 elements with 1-meter nominal spacing and is suspended 15 meters deep by ten unevenly spaced buoys.The deployment creates a multi-section catenary geometry with localized sagging sections.
  • Array Shape Calibration: BOGE reconstructs the multi-section catenary shape and captures the main localized sagging sections induced by the buoy deployment.Because absolute ground-truth geometry is unavailable, calibration accuracy is assessed through subsequent DoA performance.
  • Fixed-source localization: Fixed-source localization uses narrow-band MUSIC at 500 Hz for sources arriving from −30° and −5°.The normalized spectra compare BOGE with AS-TDE and DAS-SC.
  • Fixed-source localization: All three methods produce identifiable main lobes, while BOGE shows more concentrated peaks and lower sidelobe levels under the tested conditions.These spectra support reliable DoA estimation in the evaluated underwater environment.
  • Moving-target tracking: The moving target transitions from 0° to approximately −30° over 60 seconds, and all methods successfully capture its movement.BOGE produces a more focused main-lobe trajectory than DAS-SC and AS-TDE.
  • Moving-target tracking: BOGE achieves a 4.87° main-lobe width, compared with 8.98° for DAS-SC and 13.91° for AS-TDE, with similar noise levels.The reported tracking performance is comparable to the comparison methods under the tested conditions.
  • Conclusion: The conclusion reports comparable fixed-source localization and moving-target tracking performance on the 238-element lake array, with BOGE producing the narrowest moving-source main lobe.These results support calibration without cooperative sources.
  • Limitations and future work: Online calibration of towed arrays remains future work because BO and MUSIC evaluations must track geometry changes simultaneously within a short time.The planned extension also targets continuous 3D deformation caused by vessel maneuvers and currents.
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