Complex coherence function represented by discrete formula on the basis of Lamb's problem for Rayleigh waves : An application for new interpretation of the spatial autocorrelation method
Complex coherence function represented by discrete formula on the basis of Lamb's problem for Rayleigh waves : An application for new interpretation of the spatial autocorrelation method
复制标题
基于瑞利波兰姆问题的离散公式表示的复相干函数：空间自相关方法新解释的应用
DOI：
10.3124/segj.58.137
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
T. Matsuoka
中科院分区：
文献类型：
--
作者：
H. Shiraishi;T. Matsuoka
空间自相关（SPAC）方法需要一个特殊的圆形阵列，其中几个观测点在周长上等间距。为了寻找一种比SPAC方法排列限制更少的阵列观测新方法的可能性，我们提出了在任意位置的几个观测点上测量的瑞利波的复相干函数（CCF）公式。该公式是在Lamb问题解析解的基础上推导出来的，目的是研究波源与观测点的关系。该公式是由第一类零阶贝塞尔函数J0 (kr) （k：波数，r：阵列半径）和具有高阶贝塞尔函数的无穷级数组成的简单离散表示。在SPAC方法中，将ccf（实部）方向平均的SPAC系数取为J0 (kr)，计算瑞利波（波数k）的相速度。我们首先研究了CCF值与远离几个观测点的波源之间的关系，发现随着kr的增加，CCF值会随方向发生强烈的变化，并且这种方向性主要是由CCF公式中无穷级数的变化引起的。此外，我们将该公式应用于SPAC方法，揭示了方向平均的机理以及SPAC方法需要在圆上等距放置传感器的特殊圆形阵列的原因。结果表明：1)对CCFs进行方向平均后，无穷级数的值减小到可以忽略不计的程度，使得SPAC系数可以近似为J0 （kr）。2)通过对无穷级数值为零条件的逆分析，发现不仅在通常的SPAC阵列中满足该条件，而且在一些由非等间距的观测点组成的额外圆形阵列中也满足该条件。这一结果表明，使用一种新的算法来获得J0 (kr)，而不需要SPAC方法中的方向平均运算，从而实现对观测点排列限制较少的阵列设计是可能的。
The spatial autocorrelation (SPAC) method requires a special circular array where several observation points are equally spaced on the circumference. In order to look for the possibility of developping a new method of array observation with fewer restriction of arrangement than the SPAC method, we proposed a formula of the complex coherence function (CCF) of the Rayleigh wave measured on a couple of observation points located at any place. This formula was derived on the basis of an analytical solution of Lamb's problem, aiming to study the relation between wave source and observation point. The formula was given as simple discrete representation consisting of the Bessel function of the first kind of zero order J0 (kr) (k: wavenumber, r: radius of array) and an infinite series with higher-order Bessel functions. In the SPAC method, by regarding the SPAC coefficient from directional average of CCFs (real part) as J0 (kr), phase velocities of Rayleigh waves (wave number k) are calculated.We first studied the relationship between the values of CCF and wave sources located far from a couple of observation points, and found that the values of CCF strongly varies depending on the direction with increase in kr, and also found that such directional properties were mainly caused by the variation of the infinite series in the formula of CCF. Furthermore, we applied the formula to the SPAC method for revealing the mechanism of the directional average and the reason why the SPAC method requires the special circular array with sensors equally spaced on a circle. The results are summarized as follows: 1) The values of the infinite series gets lower enough to be negligible after the directional average of CCFs, so that the SPAC coefficient can be approximated to J0 (kr). 2) From the inverse analysis on the condition that the values of the infinite series is equal to zero, it was found that the condition was satisfied not only in the usual SPAC arrays but also in some extra circular arrays consisting of observation points not equally spaced on the circumference.This result suggests the possibility of array design with fewer restriction of arrangement of observation points, using a new algorithm for obtaining J0 (kr) without the operation of directional average used in the SPAC method.