Distributed ToA Localization of Acoustic Sources with Unknown Time of Emission via Operator Splitting

I. INTRODUCTION

• GNSS-denied, non-cooperative settings provide reception timestamps but not the source’s emission time, leaving it to be eliminated through TDoA preprocessing or estimated jointly with position.These settings include marine-life tracking, search-and-rescue, and maritime security.Source: §I · p. 1
• TDoA differences couple receiver pairs, complicating distributed estimation because no receiver can evaluate its residual from its own data alone.The paper notes that this structure limits standard schemes that rely on local estimates reconciled with neighbors.Source: §I · pp. 1–2
• The paper retains ToA measurements and treats emission time as an unknown, enabling local measurement models to contribute to a cooperative global optimization.DADMM enforces agreement on shared variables through consensus constraints and dual updates exchanged between neighboring agents.Source: §I · p. 2
• DADMM distributes the estimation process across networked agents, unlike prior uses of ADMM primarily as a centralized computational tool.The method provides closed-form local updates without nested numerical optimization at each iteration.Source: §I · p. 2
• The paper establishes convergence properties for a smoothed measurement-model approximation and identifies sufficient penalty-parameter conditions for DADMM.Source: §I · p. 2

II. PROBLEM STATEMENT

• The formulation considers one target; distinguishable multiple targets can be handled independently, yielding separate single-target estimation problems.The paper gives frequency content and other signal features as examples of ways targets may be distinguished.Source: §II · p. 2

A. Network Model

• The receiver network is an undirected, connected graph with known neighborhood topology and receiver positions in a common reference frame.Source: A. Network Model · p. 2

B. Measurement Model

• Each receiver measures a noisy ToA value; the model includes unknown emission time and target position, known acoustic propagation velocity, and independent Gaussian noise.The target position is represented in either two or three spatial dimensions.Source: B. Measurement Model · p. 2

C. Objective

• The objective is decentralized target-position estimation from ToA measurements, assuming synchronized receiver clocks and at least three receivers in 2D or four in 3D.Source: C. Objective · pp. 2–3

III. LOCALIZATION PROBLEM FORMULATION

• TDoA preprocessing forms pairwise arrival-time differences and range differences, but introduces correlated noise, extra communication, and a challenging hyperbolic model.The paper instead uses the ToA formulation to avoid explicit TDoA preprocessing and obtain closed-form local updates.Source: §III · p. 3

• Each receiver has a local state containing its target-position and emission-time estimates, while edge variables impose agreement between neighboring states.The distributed formulation assigns dual variables to the agreement constraints.Source: §IV · p. 3, Figure 2 · p. 3
• DADMM updates each receiver’s local state using its local cost and neighboring-edge terms, followed by an edge-variable update.Source: §IV · p. 3
• Reparameterizing local position as p_i = s_i + r_i n_i yields a closed-form local minimizer instead of requiring a numerical solve at each iteration.Here r_i ≥ 0 and ∥n_i∥ = 1.Source: §IV · p. 3
• The nonconvex objective motivates a cold-start initialization; for later measurements, reusing the previous estimate as a warm start can reduce ADMM iterations.The cold start uses the local receiver centroid, with a small offset if that centroid coincides with the receiver.Source: §IV · pp. 3–4
• Algorithm 1 summarizes initialization, local subproblem and dual-variable updates, and stopping criteria for the distributed procedure.Figure 3 illustrates the data flow for receiver i.Source: §IV · p. 4, Figure 3 · p. 4

• The original objective is nonconvex and nondifferentiable at a receiver position, so the analysis uses a smoothed approximation of the measurement model.The relaxation parameter ε smooths behavior near s_i; the analysis interprets √ε as a minimum radius around s_i within which the target is not expected to appear.Source: §V · p. 4
• Smoothing makes the objective continuously differentiable on the analysis region, with block-wise Lipschitz bounds for position and emission time.The analysis also defines a strong-convexity modulus for each receiver’s local smoothed objective.Source: §V · p. 4
• Under the stated assumptions and condition (18), all limit points of the ADMM iterates are first-order stationary points of the smoothed problem.Source: §V · p. 5
• The convergence result does not quantify the effect of graph topology on convergence rate.The paper notes that denser connected graphs may lead to faster convergence in practice.Source: §V · p. 5

A. Simulation Setup

• All simulations use a two-dimensional scene, with receiver and target positions in R2, and a fixed sparse communication graph connecting eight receivers; the sound speed is v = 1500 m/s.Figure 4 depicts the receiver geometry and estimated target position in the representative single-target experiment.Source: A. Simulation Setup · p. 5, Figure 4 · p. 5

B. Convergence for Single Target Realization

• The node-wise estimates progressively agree and converge to a common solution in the representative single-target experiment.Figure 5 plots emission-time estimates at the top and target-position estimates at the bottom.Source: B. Convergence for Single Target… · p. 5, Figure 5 · p. 5

C. Accuracy over Random Target Positions

• Across 100 randomly sampled target positions, estimates remain close to the true targets, while farther targets take more DADMM iterations to converge.Figure 6 encodes iteration count through circle radius.Source: Figure 6 · p. 5, C. Accuracy over Random Target Positions · p. 5

D. Sensitivity to Measurement Noise

• The distributed RMSE follows the overall trend of the centralized benchmark and CRLB reference, and tighter stopping thresholds improve accuracy at the cost of more iterations.Figure 7 compares three stopping-threshold settings over varying measurement-noise levels.Source: Figure 7 · p. 6, D. Sensitivity to Measurement Noise · p. 6

VII. CONCLUSIONS AND FUTURE WORK

• The approach provides closed-form local updates and convergence conditions for a smoothed measurement model; future work considers distributed Moving Horizon Estimation.The paper notes that this tracking method has a problem structure closely related to the present formulation.Source: §VII · p. 6
• The numerical results show consensus and accurate localization across target positions, with performance comparable to a centralized solver and a similar trend to the CRLB reference.Stopping thresholds and penalty parameters illustrate trade-offs among accuracy, convergence speed, and numerical stability.Source: §VII · p. 6

E. Penalty Parameters Analysis

• Small penalty parameters can produce oscillatory or weakly damped behavior and may prevent consensus within the iteration budget, while overly large penalties slow convergence.The emission-time and position estimates are examined with different penalty parameters in Figure 8.Source: Figure 8 · p. 6, E. Penalty Parameters Analysis · p. 6

A. Proximal step for x[k+1]

• The derivation reparameterizes target position as p = s + rn, with r ≥ 0 and a unit direction n, and defines d from the average vector and receiver position.This reduces the local position representation to a receiver-relative radius and direction.Source: A. Proximal step for x[k+1] · p. 7
• For fixed radius and emission time, the direction minimizing the objective aligns with d when d is nonzero, yielding a reduced optimization problem.The derivation substitutes this direction into the reparameterized objective.Source: A. Proximal step for x[k+1] · p. 7
• The reduced problem is strictly convex quadratic in emission time and radius, and its unconstrained minimizer follows from a linear system.When the unconstrained radius is nonnegative, it is the unique constrained minimizer; otherwise, the constrained minimizer lies at r = 0.Source: A. Proximal step for x[k+1] · p. 7
• For d ≠ 0, the resulting solution gives the update formulas used in the distributed procedure, with the nominal case having r* ≥ 0.The paper notes that initialization ensures d ≠ 0 except in a degenerate centroid case.Source: A. Proximal step for x[k+1] · p. 7

B. Lipschitz Constants for the ToA Residual Function

• For twice-continuously-differentiable f, Lipschitz-differentiability on Ω is stated using a positive constant L; ∥·∥ denotes the spectral norm.Source: B. Lipschitz Constants for the ToA… · p. 7
• The Hessian is block-structured, with time, position, and mixed derivatives used in the constant derivations.The section considers the smoothed objective for a single agent.Source: B. Lipschitz Constants for the ToA… · p. 7
• The derivation imposes a trust-region condition, with k a confidence multiplier scaled by the measurement-noise standard deviation σ.Source: B. Lipschitz Constants for the ToA… · p. 7
• On Ω, f is Lipschitz-differentiable with block-wise constants Lt and Lp.The section derives these constants for the time and position blocks.Source: B. Lipschitz Constants for the ToA… · p. 7