﻿ 地动噪声RMS值计算方法的适用性及误差概率分析
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 大地测量与地球动力学  2019, Vol. 39 Issue (5): 447-451  DOI: 10.14075/j.jgg.2019.05.002

### 引用本文

LI Xiaojun, YUAN Songyong, YANG Dake, et al. Applicability and Error Probability Analysis for the Calculation Method of RMS of Background Noise[J]. Journal of Geodesy and Geodynamics, 2019, 39(5): 447-451.

### Foundation support

The Spark Program of Earthquake Technology of Hebei Earthquake Agency, No.DZ20160621074; Advanced Programs for 2016 of Ministry of Human Resources and Social Security; Innovative Research Team of Hebei Earthquake Agency for Seismograph's Calibration, Valuation and Selection of Observation Sites.

### Corresponding author

YUAN Songyong, senior engineer, majors in the seismic observation technology, E-mail: ysy1@vip.sina.com.

### 第一作者简介

LI Xiaojun, senior engineer, majors in seismograph's calibration and valuation, E-mail:lxj@eq-he.ac.cn.

### 文章历史

1. 河北省地震局，石家庄市槐中路262号，050021;
2. 中国地震局地球物理研究所，北京市民族大学南路5号，100081;
3. 广东省地震局，广州市先烈中路81号，510070;
4. 中国地震局地震观测技术研究院，北京市民族大学南路5号，100081

1 地震信号与地动噪声的RMS表征

2 仿真数据合成

 $\begin{array}{l} {S_{{\rm{sub}}}}\left( {i, j} \right) = A\left( {i, j} \right) \times \\ \sin \left( {2\pi f\left( j \right)t + {\Delta _t}\left( {i, j} \right)} \right) \end{array}$ (1)

3 仿真数据分析

 ${S_{{\rm{sub}}-{\rm{T}}}}\left( {i, k} \right) = \sum\limits_{j = {N_{k-l}}}^{{N_{k-h}}} {{S_{{\rm{sub}}}}\left( {i, j} \right)}$ (2)

3.1 滤波后计算RMS值

RMSmnk-NF(m, i, n, k)为使用FIR滤波器对ST(i)滤波后，使用每个波形后50 s数据计算得到的RMS值，m为滤波器窗口函数的编号，i为波形编号，n为滤波器阶数, k为子频带编号, NF表示窄带滤波器。

 $\begin{array}{l} {\rm{RM}}{{\rm{S}}_{\_{\rm{error}}\_{\rm{ratio}}}}(m, i, n, k) = \\ \frac{{{\rm{RM}}{{\rm{S}}_{mnk-{\rm{NF}}}}(m, i, n, k)-{\rm{RM}}{{\rm{S}}_R}(i, k)}}{{{\rm{RM}}{{\rm{S}}_R}(i, k)}} \times 100 \end{array}$ (3)

 图 1 使用不同阶数的hann及rectwin窗口函数滤波器滤波后计算的RMS值误差概率统计 Fig. 1 The error probability statistics for the RMS calculated after filtered with FIR filter designed in hann and rectwin window with different orders

 图 2 使用16种滤波器窗口函数在不同滤波器阶数下对波形滤波后计算的RMS值误差在±1%及±5%以内的概率统计 Fig. 2 Theprobability statistics for the error within ±1% and ±5% of RMS calculated after filtered with filter designed in 16 different windows with different orders
3.2 PSD值反算RMS值

PSDRMS_imjgk是由PSD反算的RMS值，其中i表示波形编号，m表示滤波器的窗口函数类型编号，j代表窗口长度占原数据长度的百分比，g代表相邻两段数据的重叠率，k代表子频带编号。

 ${\rm{PS}}{{\rm{D}}_{2{\rm{RMS}}\_ijk\_1}} = \frac{{N\_{\rm{PS}}{{\rm{D}}_{2{\rm{RMS}}\_mjg\_1}}}}{{N\_{\rm{PS}}{{\rm{D}}_{2{\rm{RMS}}\_mjg\_{\rm{T}}}}}}$ (4)
 ${\rm{PS}}{{\rm{D}}_{2{\rm{RMS}}\_ijk\_5}} = \frac{{N\_{\rm{PS}}{{\rm{D}}_{2{\rm{RMS}}\_mjg\_5}}}}{{N\_{\rm{PS}}{{\rm{D}}_{2{\rm{RMS}}\_mjg\_{\rm{T}}}}}}$ (5)

 图 3 应用flattopwin窗口函数在不同窗口长度和重叠率参数组合下计算的PSD反算RMS值误差±1%及±5%以内的概率统计 Fig. 3 The probability statistics for the error within ±1% and ±5% of RMS calculated from PSD calculated with different window length and different overlap ratio under flattopwin window

 图 4 应用turkeywin窗口函数在不同窗口长度和重叠率参数组合下计算的PSD反算RMS值误差±1%及±5%以内的概率统计 Fig. 4 The probability statistics for the error within ±1% and ±5% of RMS calculated from PSD calculated with different window length and different overlap ratio under turkeywin window

4 结语

 [1] Bormann P. New Manual of Seismological Observatory Practice [R]. 2002 (0) [2] Aki K, Richards P G. Quantitative Seismology[M]. Sausalito: University Science Books, 2002 (0) [3] Aki K, Richards P G. Quantitative Seismology-Theory and Methods[M]. Freeman and Company, 1980 (0) [4] Fix J E. Ambient Earth Motion in the Period Range from 0.1 to 2 560 sec[J]. Bull Seism Soc Am, 1972, 62: 1753-1760 (0) [5] Melton B S. The Sensitivity and Dynamic Range of Inertial Seismographs[J]. Rev Geophys Space Phys, 1978, 14: 393-116 (0) [6] Milivojevic Z.Digital Filter Design[EB/OL]. https://learn.mikroe.com/ebooks/digitalfilter (0) [7] Welch P D. The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging over Short, Modified Periodograms[J]. IEEE Trans Audio Electroacoust, 1967, AU-15: 70-73 (0) [8] Oppenheim A V, Schafer R W. Digital Signal Processing[M]. New Jersey: Prentice-Hall Press, 1975 (0) [9] Ringler A T, Hutt C R, Evans J R, et al. A Comparison of Seismic Instrument Noise Coherence Analysis Techniques[J]. Bull Seismol Soc Am, 2011, 101: 558-567 DOI:10.1785/0120100182 (0) [10] Carter G C, Knapp C H, Nuttall AH. Estimation of the Magnitude-Squared Coherence Function via Overlapped Fast Fourier Transform Processing[J]. IEEE Transactions Audio and Electroacoustics, 1973, AU-21(4): 337-344 (0)
Applicability and Error Probability Analysis for the Calculation Method of RMS of Background Noise
LI Xiaojun1,4     YUAN Songyong2,4     YANG Dake2,4     XIE Jianbo3,4     MA Jiemei2,4     LI Dongsheng1
1. Hebei Earthquake Agency, 262 Huaizhong Road, Shijiazhuang 050021, China;
2. Institute of Geophysics, CEA, 5 South-Minzudaxue Road, Beijing 100081, China;
3. Guangdong Earthquake Agency, 81 Mid-Xianlie Road, Guangzhou 510070, China;
4. E-Institute of Earthquake Observation Technology, 5 South-Minzudaxue Road, Beijing 100081, China
Abstract: Combined with error probability statistics, this paper compares two main methods for calculating RMS value and gives the distribution of error probability. The results show that the result error for RMS calculated from waveform filtered by FIR bandpass filter is mostly dependent on the types of window function and filter order, with filter order playing a more import role. In addition, the error probability of RMS calculated from PSD depends on the parameter combinations of window function type, window length, and overlap ratio related to the pwelch function, mainly on the window length, and will reach a steady status with a certain number of sample waveforms. The distribution of error probability indicates that the second method is better than the first.
Key words: root mean square; power spectrum density; FIR filter; distribution for error probability; parameter combination