This article is devoted to the problem of determining the coordinates of wave field sources using phases of complex Fourier transforms of signals recorded by the sensor array. Phase-based source localization algorithms have an important reliability property: they provide the quality of determining the source coordinates, which weakly depends on the statistical characteristics of random noise affecting the array sensors. For this reason, phase algorithms are widely used to localize wave radiation sources in radio and acoustic applications and, more recently, in seismology. Naturally, preference should be given to phase-based source localization algorithms that provide the smallest average errors in estimating source coordinates. Such algorithms can be constructed and analyzed using methods of mathematical statistics. In this article, we synthesized a phase-based statistically optimal (PSO) algorithm, which is a modification of the asymptotically efficient (ASE) statistical algorithm for estimating the coordinates of microseismic sources proposed in Kushnir et al. (Int J Geomath 4(2):201–225, 2013. https://doi.org/10.1007/s13137-013-0049-6 ). Unlike the ASE algorithm, the PSO algorithm does not require additional observations of “pure” noise and is resistant to the statistical properties of random noise. The article shows that the PSO algorithm is an extended version of the well-known phase-based SRP-PHAT algorithm (Brandstein and Ward in Microphone arrays signal processing techniques and applications, Springer, Berlin, 2001, Chapter 8; Zhang et al. in IEEE Trans Multimedia 10(3):538–548, 2008), which is widely used in acoustic applications. In this article, we describe Monte-Carlo simulation, the purpose of which is to compare the accuracy of determining the coordinates of micro-seismic sources provided by various phase-based algorithms. This simulation showed that the proposed PSO algorithm provides much better source positioning accuracy than the traditional SRP-PHAT algorithm. It is also shown that the PSO algorithm provides the source positioning accuracy, which is almost the same as for the phase-based source location algorithm recently synthesized by the maximum likelihood (ML) method in Varypaev and Kushnir (Int J Geomath 9(2):335–358, 2018. https://doi.org/10.1007/s13137-018-0108-0 ). However, the PSO algorithm has much higher computational efficiency than the ML algorithm, which allows it to be used for online processing of array data.

中文翻译：

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用于波场源定位的相位对齐算法的统计综合

本文致力于解决使用传感器阵列记录的信号的复傅立叶变换相位来确定波场源坐标的问题。基于相位的源定位算法具有重要的可靠性特性：它们提供确定源坐标的质量，而其质量弱依赖于影响阵列传感器的随机噪声的统计特性。出于这个原因，相位算法被广泛用于在无线电和声学应用以及最近的地震学中定位波辐射源。自然，应该优先考虑基于相位的源定位算法，这些算法在估计源坐标时提供最小的平均误差。可以使用数理统计方法构建和分析此类算法。在本文中，我们合成了一种基于相位的统计优化 (PSO) 算法，该算法是对 Kushnir 等人提出的用于估计微震源坐标的渐近有效 (ASE) 统计算法的修改。(Int J Geomath 4(2):201–225, 2013。https://doi.org/10.1007/s13137-013-0049-6)。与 ASE 算法不同，PSO 算法不需要额外观察“纯”噪声，并且能够抵抗随机噪声的统计特性。文章表明 PSO 算法是著名的基于相位的 SRP-PHAT 算法的扩展版本（Brandstein 和 Ward in Microphone arrays signal processing technology and Applications, Springer, Berlin, 2001, Chapter 8; Zhang et al. in IEEE Trans Multimedia 10(3):538–548, 2008)，广泛用于声学应用。在本文中，我们描述了蒙特卡罗模拟，其目的是比较各种基于相位的算法提供的微震源坐标确定的准确性。该仿真表明，所提出的 PSO 算法比传统的 SRP-PHAT 算法提供了更好的源定位精度。还表明，PSO 算法提供的源定位精度与最近在 Varypaev 和 Kushnir 中通过最大似然 (ML) 方法合成的基于相位的源定位算法几乎相同 (Int J Geomath 9(2) :335–358, 2018. https://doi.org/10.1007/s13137-018-0108-0)。但是PSO算法的计算效率比ML算法高很多，这使得它可以用于数组数据的在线处理。其目的是比较各种基于相位的算法提供的确定微震源坐标的准确性。该仿真表明，所提出的 PSO 算法比传统的 SRP-PHAT 算法提供了更好的源定位精度。还表明，PSO 算法提供的源定位精度与最近在 Varypaev 和 Kushnir 中通过最大似然 (ML) 方法合成的基于相位的源定位算法几乎相同 (Int J Geomath 9(2) :335–358, 2018. https://doi.org/10.1007/s13137-018-0108-0)。但是PSO算法的计算效率比ML算法高很多，这使得它可以用于数组数据的在线处理。其目的是比较各种基于相位的算法提供的确定微震源坐标的准确性。该仿真表明，所提出的 PSO 算法比传统的 SRP-PHAT 算法提供了更好的源定位精度。还表明，PSO 算法提供的源定位精度与最近在 Varypaev 和 Kushnir 中通过最大似然 (ML) 方法合成的基于相位的源定位算法几乎相同 (Int J Geomath 9(2) :335–358, 2018. https://doi.org/10.1007/s13137-018-0108-0)。但是PSO算法的计算效率比ML算法高很多，这使得它可以用于数组数据的在线处理。