SUMMARY

We build a model of discretization errors, known as directional aliasing, to theoretically evaluate how biases in the microtremor spatial autocorrelation (SPAC) coefficient, or the real part of the SPAC spectrum of microtremor analysis, are related to the magnitudes of the imaginary part when a seismic array of only two sensors is used. By using this model, we investigate the potential utility of the imaginary spectrum component as an indicator of applicability of the two-sensor SPAC method to the field of microtremors generated at an observation site. Field data of microtremors from compact seismic arrays (1–15 m) are used to test the model. It is found that, when the imaginary components are very large in magnitude (where the threshold depends on the rk, the array radius times the wavenumber), the field of microtremors is dominated by waves arriving from a single direction parallel to the array axis and the SPAC coefficients tend to be underestimated in small rk ranges (i.e. rk < 3.8; the range considered throughout this study). In this study, which is based on the observations of 400 microtremor arrays, the underestimates seldom exceeded 30 per cent. The SPAC coefficient estimates could be corrected in that case by using information on the imaginary part. When the imaginary components are very modest in magnitude, by contrast, there are two possible scenarios: either (i) the waves are arriving predominantly from a single direction perpendicular to the array axis and the SPAC coefficients are wildly overestimated (i.e. there was a small percentage of low-quality data, with relative errors exceeding +50 per cent, based on the observed data analyses) or (ii) the wavefield is close to isotropic and the SPAC coefficients are unbiased (i.e. 70–90 per cent of all observed data fell within the relative error range of ±20 per cent). It is difficult in that case to have SPAC coefficient estimates corrected by using information on the imaginary part alone.

中文翻译：

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双传感器微震SPAC方法：虚频谱分量的潜在效用

概要

我们建立了离散化误差模型，称为定向混叠，从理论上评估了微震空间自相关（SPAC）系数或微震分析的SPAC频谱实部中的偏差如何与虚部的大小相关。仅使用两个传感器的地震阵列。通过使用该模型，我们调查了虚频谱分量的潜在效用，作为双传感器SPAC方法对观察点产生的微震领域的适用性的指标。来自紧凑地震阵列（1-15 m）的微震的现场数据用于测试模型。发现，当虚分量非常大时（阈值取决于rk，阵列半径乘以波数），微震的领域由平行于阵列轴的单一方向的波所主导，在较小的rk范围（即rk <3.8；整个研究中考虑的范围）内，SPAC系数往往被低估。在这项基于400个微震阵列的观测的研究中，低估很少超过30％。在这种情况下，可以通过使用虚部上的信息来校正SPAC系数估计。相比之下，当虚部的幅值非常适中时，有两种可能的情况：（i）波主要从垂直于阵列轴的单个方向入射，并且SPAC系数被高估了（即，相对误差超过+ 50％的低质量数据的百分比，基于观测数据的分析）或（ii）波场接近各向同性，并且SPAC系数是无偏的（即，所有观测数据的70-90％都落在±20％的相对误差范围内）。在那种情况下，仅通过使用虚部上的信息来校正SPAC系数估计是困难的。