Data-Driven Nonlinear TDOA for Accurate Source Localization in Complex Signal Dynamics

TL;DR

This work addresses locating a propagation source in non-homogeneous, dynamic media where the speed of propagation is unknown or spatially varying (e.g., AFib rotors, wildfires, tsunamis). It introduces a data-driven nonlinear TDOA (NTDOA) framework and an intermediate modified TDOA (mTDOA) that jointly estimate the source location $r_0$, start time $t_0$, and the spatially varying speed $c(\cdot)$. The NTDOA model decomposes $c$ into spatial and angular components using $f(R)$ and $g(\theta)$ with low-order expansions, enabling robust localization with a small number of anchors; initialization from mTDOA improves convergence. Across simulations and real satellite data, NTDOA consistently outperforms classical TDOA in terms of mean absolute error and radial accuracy, enabling better speed and direction forecasting of subsequent propagation.

Abstract

The complex and dynamic propagation of oscillations and waves is often triggered by sources at unknown locations. Accurate source localization enables the elimination of the rotor core in atrial fibrillation (AFib) as an effective treatment for such severe cardiac disorder; it also finds potential use in locating the spreading source in natural disasters such as forest fires and tsunamis. However, existing approaches such as time of arrival (TOA) and time difference of arrival (TDOA) do not yield accurate localization results since they tacitly assume a constant signal propagation speed whereas realistic propagation is often non-static and heterogeneous. In this paper, we develop a nonlinear TDOA (NTDOA) approach which utilizes observational data from various positions to jointly learn the propagation speed at different angles and distances as well as the location of the source itself. Through examples of simulating the complex dynamics of electrical signals along the surface of the heart and satellite imagery from forest fires and tsunamis, we show that with a small handful of measurements, NTDOA, as a data-driven approach, can successfully locate the spreading source, leading also to better forecasting of the speed and direction of subsequent propagation.

Figures (6)

• Figure 1: A cardiac wave is simulated using a modified FitzHugh–Nagumo model you2017demonstration. Each sub-figure shows an evolution of the system, with the rotor model being visible in the stripe identified by the white arrow.
• Figure 2: (a) The source (star in the middle) starts transmitting at an unknown time, $t_{0}$. The speed of propagation of the signal is fixed at all points. For Case-I (Section \ref{['ssec:isotropic model']}), it is assumed that the speed of propagation is known, and for Case-II (Section \ref{['ssec:mTDOA']}), it is unknown. The signal is received at the anchors at times $t_{l}$, with $t_{3}$ = $t_{4}$, since they are at the same distance from the source. In the inverse problem, with the anchor locations and times $t_{l}, l \neq 0$ known, we estimate the location of the source, and the time $t_{0}$, when the source begins transmission. Additionally, for Case-II, the speed of propagation is also estimated. (b) The source (star in the middle) starts transmitting at time $t_{0}$. The speed of propagation of the signal can vary across the medium. The signal is received at the anchors at times $t_{l}$. In this case, it is not required that $t_{1}$ = $t_{4}$, even though they are at the same distance from the source since the speed of propagation can vary between the source and each anchor. In the inverse problem, with the anchor locations and times $t_{l}, l \neq 0$ known, we estimate the source location, the speed of propagation at each point, and $t_{0}$.
• Figure 3: Signal propagation patterns in realistic scenarios. Each sub-figure (top to bottom) shows an evolution of the three dynamic processes. (L-R) AFib (modified FHN), Creek wildfire NASAsafford20222020, Tonga tsunami omira2022global. The Source of propagation is marked with a white hexagram in each figure.
• Figure 4: Localization Results using NTDOA: (Error Quantification) To estimate the accuracy of the algorithm, the number of estimates within a circle of a given radius are counted. The percentage of estimates for each radius value is plotted. The faster the curve reaches 100%, indicates a more accurate the estimator. The algorithm is tested with 50 randomly placed anchors, repeated 1000 times, and the results are averaged. The results show that NTDOA performs better compared to other methods.
• Figure 5: Localization error results using NTDOA: (Error Quantification) To quantify the accuracy of the algorithm, we measured the mean absolute error (MAE) under varying conditions. We simulated scenarios with 10 to 50 anchors in the search space, repeating each simulation 1,000 times and averaging the results. Our findings demonstrate that NTDOA consistently achieved the lowest MAE compared to other methods.