地球物理学报  2014, Vol. 57 Issue (3): 789-799 PDF

1. 中国科学院大学, 北京 100049;
2. 中国科学院地质与地球物理研究所, 北京 100029;
3. 中国地质科学院, 北京 100037

Multiple sinusoidal tapers method to estimate receiver function
SI Shao-Kun1,2, TIAN Xiao-Bo2, ZHANG Hong-Shuang3, TENG Ji-Wen2
1. University of Chinese Academy of Sciences, Beijing 100049, China;
2. Institutes of Geophysics, Chinese Academy of Sciences, Beijing 100029, China;
3. Chinese Academy of Geological Sciences, Beijing 100037, China
Abstract: Receiver function method is important in researching the crust and upper mantle structures, and a stable and reliable extracting method is the basic issue. Frequency domain deconvolution is a commonly used method for extracting the receiver functions. While cutting a section of the continuous data with a single taper, it will reduce the saturation and produce spectral leakage. So a series of orthogonal tapers are brought into multiple sampling, which ensure the signal saturation and reduce spectral leakage. The multiple sinusoidal tapers are a series of orthogonal window functions, which have the minimum bias. Besides, it is convenient to apply because they have a simple analytic form. We apply some sets of sinusoidal tapers to calculate receiver function. This procedure prevents the spectra form having very small values due to spectral leakage. Moreover, it stabilizes the spectral division by adding the pre-event noise power spectra to the vertical power spectra in the denominator. Its application to the observational data indicates that the method in this paper is effective to estimate receiver functions with high stability and resolution in frequency-domain.
Key words: Receiver function     Multiple sinusoidal tapers     Deconvolution in frequency-domain
1 引言

2 原理与方法 2.1 接收函数频率域提取

DV(ω)、DR(ω)和DT(ω) 分别是垂直分量、径向分量和切向分量的频率域表示， V(ω) 是 DV(ω) 的复共轭，c为控制谱振幅水准量的常数， ΦSS(ω) 意味着凡是小于水准量的谱振幅都毫无例外地用“水准量”来代替，为控制高斯频谱带宽的常数.

2.2 离散椭球窗

2.2.1 椭球序列的求取

2.2.2 离散椭球序列特性

 图 1 离散椭球窗在时域与频域的形态(a) 时间带宽积NW为2.5时它的前四个Slepian窗函数在时域形态；(b) 前四个多正弦窗在时域的形态； (c)、(d)、(e)和(f) 中实线和虚线分别表示椭球窗和正弦窗函数在频域的形态. Fig.1 Shapes of discrete prolate tapers in time-domain and frequency-domain (a) The four lowest-order Slepian tapers (time-bandwidth NW=2.5) in time-domain; (b) The four lowest-order sinusoidal taper in time-domain; (c)、(d)、(e) and (f) show the Slepian tapers (solid line) and sinusoidal tapers (dash line) in frequency-domain respectively.

2.3 多正弦数据窗

2.4 多正弦窗提取任意时间长度接收函数

D(k)V(ω)、D(k)R(ω)和D(k)T(ω)分别表示径向、垂向和切向分量通过第k阶正弦窗得到的频率域形式； (k)V(ω) 为D(k)V(ω)的共轭； N(k)V(ω) 为垂向分量信号前的噪声估计；K表示多正弦窗的最大阶数；α为控制高斯频谱带宽的常数.

3 资料检验

 图 2 多正弦窗接收函数提取方法的模型检验 (a) 实际远震记录的垂向分量；(b) 理论的接收函数；(c) 合成的远震记录径向分量；(d) 本文方法(灰线)与 传统频率域( Helmberger and Wiggins (1971))方法(黑线)提取的接收函数. Fig.2 The model test of the multiple sinusoidal tapers to calculate receiver function (a) Vertical component of the field teleseismic event; (b) The theoretical reciver function; (c) Radial component of teleseismic event synthesized from (a) and (b); (d) Receiver functions estimated by the method of this paper(grey line) and Helmberger and Wiggins (1971) (black line).

 图 3 台站TLY所处位置以及记录的地震事件分布(a)显示台站所在的位置，红色三角形代表台站TLY；(b)显示了地震事件相对于台站的方位，圆圈代表所记录的地震事件，圆圈大小代表震级，颜色越深代表震源越深. Fig.3 The position of the station TLY and the seismic event recorded (a) shows the position of the station TLY and the red triangle represents the station. (b) shows the positions of earthquakes, circles represent the earthquakes and their sizes show the earthquake magnitude. The darker the circle, the deeper the focus of the earthquake.

 图 4 (实例一)三种方法对台站TLY实际数据提取的接收函数(a)实际数据垂向分量，南北分量和东西分量；(b)、(c)、(d)分别表示用传统频率域反褶积方法、多椭球窗以及多正弦窗方法提取的接收函数. Fig.4 (Example 1) Receiver functions from the station TLY using three methods(a) show vertical、north-south and east-west component respectively; (b)、(c)、(d) show receiver functions estimated by traditional deconvolution in frequency-domain、multiple prolate tapers and multiple sinusoidal tapers methods respectively.

 图 5 (实例二)三种方法对台站TLY实际数据提取的接收函数(a)实际数据垂向分量、南北分量和东西分量；(b)、(c)、(d)分别表示用传统频率域反褶积方法、多椭球窗以及多正弦窗方法提取的接收函数. Fig.5 (Example 2) Receiver functions from the station TLY using three methods(a) show vertical, north-south and east-west component respectively; (b)、(c)、(d) show receiver functions estimated by traditional deconvolution in frequency-domain, multiple prolate tapers and multiple sinusoidal tapers methods respectively.

 图 6 不同的正弦窗个数对接收函数的影响 (a)实际数据的垂向分量、南北分量以及东西分量；(b) 用不同的正弦窗个数所得的接收函数. 绿色、红色和蓝色分别表示所采用的正弦窗个数为3、6以及10个. Fig.6 Effect on the receiver function using different number of the sinusoidal tapers (a) shows vertical, north-south and east-west component; (b) shows receiver function estimated using different number of the sinusoidal tapers. Green、red and blue show the number of the sinusoidal tapers is 3、6 and 10 respectively.

 图 7 三种方法对台站TLY实际数据提取的接收函数叠加以及拾取莫霍转换波与直达P波的振幅比 (a)、(b)和(c)分别表示用传统频率域反褶积、多椭球窗和多正弦窗方法对震中距在40°~50°，反方位角在100°~110°内的19个地震记录计算的接收函数；(d)对三种方法计算的19条接收函数拾取的莫霍转换波与直达P波的振幅比，其中红色五角星、蓝色圆圈和绿色三角分别对应多正 弦窗、多椭球窗方法和传统频率域方法的结果； (e)所有台站TLY提取的接收函数的叠加； (f)表示对图(e)中黑色线框标记的区域进行放大. Fig.7 Stack of the receiver function from station TLY with three methods and the ratio of the converted wave at Moho to the direct P wave (a)、(b)、(c) represent 19 receiver functions within epicentral distance 40°~50°, reverse azimuth 100°~110°, estimated by traditional deconvolution in frequency-domain, multiple prolate tapers and multiple sinusoidal tapers methods respectively. (d) show the ratio of the converted wave at Moho to the direct P wave from 19 receive function estimated by three methods. Red star, blue circle and green triangle represent the results obtained by multiple sinusoidal tapers method, multiple prolate tapers method and traditional deconvolution in frequency-domain; (e) shows the stack all the receiver function from the station TLY; (f) shows the enlargement of the area delimited by black line in (e).

 图 8 用Bootstrap方法抽样叠加100次拾取的上地幔间断面转换波与直达P波振幅比的统计结果 (a) 410 km间断面；(b) 660 km间断面.其中灰白色、灰色和黑色分别代表传统频率域反褶积、多椭球窗和多正弦窗方法的结果. Fig.8 Distribution of the ratio of converted wave at the upper discontinuities to the direct P wave, which obtained 100 stack using bootstrap method (a) 410 km discontinuity；(b) 660 km discontinuity. Off-white, gray and white represent result from traditional disconvolution in frequency-domain, multiple prolate tapers and multiple sinusoidal tapers method.
4 结论