Systematic Uncertainty Quantification of First-Polarity-Based Moment Tensor Inversion Due to Sparse Coverage of Sensor Arrays in Laboratory Acoustic Emission Monitoring

Moment tensor inversion (MTI) can be utilized in Acoustic Emission (AE) monitoring to determine source mechanisms of AE events and hence provide critical information on failure mechanisms of microcracks in laboratory rock tests. However, true amplitudes of motion may not be extracted from AE recordings due to several sensor unknowns, including the extent of sensor-specimen coupling, the transfer function of the sensor, and the uncertain effects of non-vertical incidence at piezoelectric sensors. Hence, amplitude or waveform-based moment-tensor inversion methods may not be practical for many AE monitoring of laboratory tests, and MTI will have to resort to methods based solely on first-P polarity which may have significant non-uniqueness, and hence uncertainty, due to the sparse coverage of first-P motion data on the focal sphere by typical AE sensor arrays. In this study, we explore the uncertainty of MTI based on P polarity as a result of the sparse coverage of the sensor array in laboratory AE monitoring through a set of synthetic examples. In a unit 3D homogeneous isotropic cubic rock sample, 18 selected moment tensors are placed at each of 6 arbitrarily selected source locations to represent typical tensile, shearing and compressional source types. The corresponding first P motion polarity data for 8 different designs of sensor arrays with varying number of sensors and coverage of the focal sphere are computed. Then, for each case, a grid search method is applied to determine the moment tensor solutions that can exactly match the recorded synthetic polarity data and estimate their non-uniqueness and uncertainty. Our synthetic examples show that with a limited number of first P motion data utilized (e.g., 6 and 8), a large number of moment-tensor solutions can generate polarities that exactly fit with the observations, making it difficult to separate different source types. On the other hand, with reliable first motion polarities recorded on at least 18 sensors evenly distributed on the sample, the non-uniqueness in moment-tensor solutions can be significantly reduced. The cumulative probability for uncertainty ≤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le$$\end{document} 5% increases from 0.4492 and 0.5358 for 6 and 8 first P motion polarities to 0.8960 for 18 polarity data according to our examples. This estimation of moment-tensor uncertainties helps improve sensor array designs, and for practical AE tests, it can be also used to provide the lower bound estimate of MTI uncertainty due to sensor array distribution.