SUMMARY The spatial autocorrelation (SPAC) method has been applied to ambient seismic noise measured by arrays of translational seismometers for inverting phase-velocity dispersion curves of Rayleigh or Love waves for shallow S-wave velocity structure. Recently, it is becoming possible to observe wave spatial gradients such as strain, rotation and tilt owing to the development of dense seismic networks and improving measurement technologies. Therefore, it is desirable to extend the formulation of the SPAC method to strain, rotation and tilt. This study presents analytical expressions of cross-spectra and coherence of the strain, rotation and tilt components that are measured on the free surface. According to the results, both Rayleigh and Love waves contribute to most components of strains. The exceptions are the areal strain and the vertical axial strain (ezz) on the free surface that are affected by only Rayleigh waves. Only Rayleigh waves contribute to the tilts and rotations around the horizontal axes on the free surface, too. On the other hand, only Love waves contribute to the rotation around the vertical axis. Therefore, different kinds of wave spatial gradients are helpful to separate Rayleigh and Love waves correctly. For practical applications, the analytical expression for the radial axial strain (err) component will be applied directly to distributed acoustic sensing data measured with straight sections of a fibre-optic cable. On the other hand, dense observations of rotation and tilt may still be difficult to carry out at present. However, an application of analytical formulations in this study to arrays of at least several three-component rotational seismometers is attractive because it enables us to separately estimate the phase velocities of Rayleigh and Love waves.

中文翻译：

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将空间自相关 (SPAC) 方法的公式扩展到应变、旋转和倾斜

发明内容空间自相关（SPAC）方法已应用于由平移地震仪阵列测量的环境地震噪声，用于反转浅S波速度结构的瑞利波或洛夫波的相速度色散曲线。最近，由于密集地震台网的发展和测量技术的改进，观测波的空间梯度（如应变、旋转和倾斜）变得可能。因此，希望将 SPAC 方法的公式扩展到应变、旋转和倾斜。本研究提出了在自由表面上测量的应变、旋转和倾斜分量的交叉光谱和相干性的解析表达式。根据结果​​，瑞利波和洛夫波对应变的大部分成分都有贡献。自由表面上的面积应变和垂直轴向应变 (ezz) 除外，它们仅受瑞利波影响。只有瑞利波也有助于自由表面上围绕水平轴的倾斜和旋转。另一方面，只有爱波有助于围绕垂直轴的旋转。因此，不同种类的波空间梯度有助于正确分离瑞利波和洛夫波。对于实际应用，径向轴向应变 (err) 分量的解析表达式将直接应用于使用光纤电缆的直线段测量的分布式声学传感数据。另一方面，旋转和倾斜的密集观测目前可能仍难以开展。然而，