Abstract
Acoustic emission is a non-destructive testing method where sensors monitor an area of a structure to detect and localize passive sources of elastic waves such as expanding cracks. Passive source localization methods based on times of arrival (TOAs) use TOAs estimated from the noisy signals received by the sensors to estimate the source position. In this work, we derive the probability distribution of TOAs assuming they were obtained by the fixed threshold technique—a popular low-complexity TOA estimation technique—and show that, if the sampling rate is high enough, TOAs can be approximated by a random variable distributed according to a mixture of Gaussian distributions, which reduces to a Gaussian in the low noise regime. The optimal source position estimator is derived assuming the parameters of the mixture are known, in which case its MSE matches the Cramér–Rao lower bound, and an algorithm to estimate the mixture parameters from noisy signals is presented. We also show that the fixed threshold technique produces biased time differences of arrival (TDOAs) and propose a modification of this method to remove the bias. The proposed source position estimator is validated in simulation using biased and unbiased TDOAs, performing better than other TOA-based passive source localization methods in most scenarios.
Highlights
Acoustic emission testing is a non-destructive testing method used to detect several kinds of faults in structures such as piping, bridges and aerospace structures
There are other TOA estimation methods that generate unbiased TDOAs, as methods based on Akaike information criterion (AIC) [8,31], but these methods present higher computational complexity than the fixed threshold method and our TOA estimation technique, whose complexity is dominated by the calculation of the energy of the signal
The proposed estimator has lower Mean Square Error (MSE) than the other estimators for very low variances, but it may perform worse than them if the variance is not very low, but low enough so that the TOA pdf is nearly Gaussian. This is because the MSE of JTOA, JTDOA and JCLS fall abruptly when the TOA distribution becomes approximately Gaussian, while the Maximum likelihood estimator (MLE) and the Gaussian mixture TOA estimator (GMTOA) estimator do not because they depend on the noisy parameters of the estimated Gaussian pdf
Summary
Acoustic emission testing is a non-destructive testing method used to detect several kinds of faults in structures such as piping, bridges and aerospace structures. TDOAs estimated by the fixed threshold method may present a large bias since signals reach sensors with different amplitudes due to attenuation (see Section 4 and [27]). To derive the optimal TOA-based source position estimator on a surface (that is, the estimator whose covariance matrix is the inverse of the Fisher information matrix) assuming TOAs follow a mixture of Gaussian distributions whose parameters are known given the source position. The optimal estimator is tested in simulation in several scenarios and compared with other passive source localization methods based on TOAs. To show that the optimal estimator reduces to one of the algorithms available in the literature if the TOA estimates are unbiased and the measurement noise is sufficiently small, in which case TOAs are approximately Gaussian-distributed. Vectors and matrices are written in boldface, and the derivative of the k-th element of a vector v with respect to x is denoted as vk,x
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