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Improved Approximation of Sensor Network Performance for Seabed Acoustic Sensors

Mingyu Kim, Daniel J. Stilwell, Harun Yetkin, Jorge Jimenez

TL;DR

The paper tackles accurate estimation of the void probability for seabed acoustic sensor networks tasked with detecting Poisson-distributed targets. It introduces a variance-aware approximation by applying a second-order Taylor expansion to $\mathbb{E}_{\lambda}[e^{-X}]$, yielding $\mathbb{E}_{\lambda}[e^{-X}] \approx e^{-{\mathbb{E}_{\lambda}[X]}} \left(1 + \tfrac{1}{2}\sigma_X^2\right)$, where $X$ encodes undetected targets and $\sigma_X^2$ captures intensity uncertainty from a log-Gaussian Cox process (LGCP). The mean/variance of the intensity are estimated with INLA/inlabru, and a greedy sensor-placement approach optimizes the variance-aware objective without added computational cost. Theoretical analysis shows the new approximation provides a tighter bound on the gap to the true void probability than the Jensen-based bound, and numerical experiments on Hampton Roads ship-traffic data demonstrate reduced approximation error and improved placement decisions. Collectively, the work enhances real-time maritime surveillance capabilities by delivering more accurate, variance-aware void-probability estimates for sensor networks.

Abstract

Sensor locations to detect Poisson-distributed targets, such as seabed sensors that detect shipping traffic, can be selected to maximize the so-called void probability, which is the probability of detecting all targets. Because evaluation of void probability is computationally expensive, we propose a new approximation of void probability that can greatly reduce the computational cost of selecting locations for a network of sensors. We build upon prior work that approximates void probability using Jensen's inequality. Our new approach better accommodates uncertainty in the (Poisson) target model and yields a sharper error bound. The proposed method is evaluated using historical ship traffic data from the Hampton Roads Channel, Virginia, demonstrating a reduction in the approximation error compared to the previous approach. The results validate the effectiveness of the improved approximation for maritime surveillance applications.

Improved Approximation of Sensor Network Performance for Seabed Acoustic Sensors

TL;DR

The paper tackles accurate estimation of the void probability for seabed acoustic sensor networks tasked with detecting Poisson-distributed targets. It introduces a variance-aware approximation by applying a second-order Taylor expansion to , yielding , where encodes undetected targets and captures intensity uncertainty from a log-Gaussian Cox process (LGCP). The mean/variance of the intensity are estimated with INLA/inlabru, and a greedy sensor-placement approach optimizes the variance-aware objective without added computational cost. Theoretical analysis shows the new approximation provides a tighter bound on the gap to the true void probability than the Jensen-based bound, and numerical experiments on Hampton Roads ship-traffic data demonstrate reduced approximation error and improved placement decisions. Collectively, the work enhances real-time maritime surveillance capabilities by delivering more accurate, variance-aware void-probability estimates for sensor networks.

Abstract

Sensor locations to detect Poisson-distributed targets, such as seabed sensors that detect shipping traffic, can be selected to maximize the so-called void probability, which is the probability of detecting all targets. Because evaluation of void probability is computationally expensive, we propose a new approximation of void probability that can greatly reduce the computational cost of selecting locations for a network of sensors. We build upon prior work that approximates void probability using Jensen's inequality. Our new approach better accommodates uncertainty in the (Poisson) target model and yields a sharper error bound. The proposed method is evaluated using historical ship traffic data from the Hampton Roads Channel, Virginia, demonstrating a reduction in the approximation error compared to the previous approach. The results validate the effectiveness of the improved approximation for maritime surveillance applications.
Paper Structure (14 sections, 1 theorem, 29 equations, 4 figures)

This paper contains 14 sections, 1 theorem, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation: Void probability approximation incorporating both mean and variance
  3. Proposed void probability approximation
  4. Searching for Optimal Sensor Locations
  5. Gap analysis: true void probability and our proposed void probability approximation
  6. Bound of Gap between the Void Probability and the Proposed Void Probability Approximation
  7. Numerical Result
  8. Estimating Mean and Variance of Ship Arrival Intensity function
  9. 1-D Sensor Model
  10. Sensor Placement Algorithm for Maximizing the Improved Void Probability Approximation
  11. Conclusion
  12. Proof of Lemma 1 (positivity of $\max(|J|) - \max(|\Tilde{J}|)$)
  13. Verification of \ref{['eq: cond1']}
  14. Verification of \ref{['eq: cond2']}

Key Result

Lemma 1

Let $\Omega$ be a positive random variable with expectation $\mathbb{E}_{\lambda}[\Omega]>0$ and variance $\sigma_{\Omega}^2>0$. Then, the following inequality holds where $J(\Omega)$ and $\Tilde{J}(\Omega)$ denote the approximation errors in estimating the void probability $\mathbb{E}_{\lambda}[e^{-\Omega}]$ using $e^{-\mathbb{E}_{\lambda}[\Omega]}$ and $e^{-\mathbb{E}_{\lambda}[\Omega]} \left(1

Figures (4)

  • Figure 1: Heatmap of ship traffic data with a line segment $O$ near Hampton Roads Channel, Virginia, USA marinecadastre.gov
  • Figure 2: Estimated mean and variance of the ship arrival intensity functions along the line segment $O$ in Fig. \ref{['fig:1']}
  • Figure 3: Estimated void probability (blue dashed curve) and void probability approximations (red and black dotted curves) with greedily selected sensor locations
  • Figure 4: Void probability difference from the estimated void probability: 1) the lower bound of Jensen's inequality (black dotted curve) and 2) proposed method (red curve)

Theorems & Definitions (1)

  • Lemma 1