﻿ 基于分数阶小波域GSM模型的地震信号随机噪声压制方法
 地球物理学报  2018, Vol. 61 Issue (7): 2989-2997 PDF

1. 合肥工业大学数学学院, 合肥 230009;
2. 合肥工业大学计算机与信息学院, 合肥 230009

Random noise attenuation method of seismic signal based on the fractional order wavelet domain GSM model
WANG JinJu1, LI Qing1, XU XiaoHong2, CAO Li1
1. School of Mathematics, Hefei University of Technology, Hefei 230009, China;
2. School of Computer and Information, Hefei University of Technology, Hefei 230009, China
Abstract: Random noise is a kind of interference wave, which reduces the signal-to-noise ratio of the seismic signal. It also affects the subsequent data processing and analysis of the seismic signal. According to the differences of the effective signal and the random noise, this paper puts forward a new method combining the fractional order B spline wavelet transform with Gaussian Scale Mixture model to attenuate the random noise of the seismic signal. Firstly, the seismic signal is transformed into the optimal factional wavelet time-frequency domain using the fractional order B spline wavelet. Gaussian Scale Mixture model is set up for each sub-band coefficients. Bayesian method is used to estimate the wavelet coefficients of the original seismic signal. Finally, the denoised seismic signal can be reconstructed using the fractional order B spline wavelet inverse transform. Through experiments on synthetic records and the field seismic signal, the results demonstrate that the proposed method can attenuate random noise of the seismic signal effectively.
Key words: Seismic signal    Random noise    Fractional order B spline wavelet    Gaussian Scale Mixture model
0 引言

1 分数阶B样条小波

Unser and Blu(2000)定义了α阶因果分数B样条函数：

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

 图 1 分数阶B样条小波分解和合成示意图 Fig. 1 Schematic diagram of fractional order B spline wavelet decomposition and synthesis
2 分数阶小波域GSM模型降噪方法 2.1 降噪方法

 (8)

 (9)

 (10)

 (11)

n=1, 2, …, NX, m=1, 2, …, NY, σ为噪声标准差，利用中值估计方法(Donoho，1994)来估计：

CW已知，就可以得到随机向量Y的协方差矩阵CY，求出高斯随机向量U的协方差矩阵CU.因为CY可以由CY|ZZ取数学期望得到

 (12)

E{Z}=1，则

 (13)

 (14)

 (15)

 (16)

 (17)

 (18)

 (19)

 (20)

2.2 算法实现步骤

Step 1：对含噪地震信号进行分数阶B样条小波变换.

Step 2：对每个子带小波系数，根据2.1节介绍的方法，(a)估计出噪声系数邻域协方差矩阵CW；(b)估计出含噪地震信号的小波系数邻域的协方差矩阵CY；(c)根据式CU=CY-CW求出CU；(d)计算ΛM.

Step3:对每个邻域，对式(14)积分范围内的每个Z，(a)用式(18)计算E{Xc|Y, Z}，(b)用式(20)计算p(Y|Z)；(c)用式(19)计算p(Z|Y)；

Step 4：用式(14)计算E{Xc|Y}得到小波系数的估计值，从而得到源地震信号的小波系数估计值

Step5：对源地震信号的小波系数估计值进行分数阶B样条小波逆变换重构得到降噪后的地震信号.

3 仿真实验与分析

 (21)

 (22)

 图 2 含噪合成地震信号以及它的一层分数阶B样条小波分解系数 (a)含噪地震信号；(b) α=-0.4的低频系数和高频系数；(c) α=0.01的低频系数和高频系数；(d) α=1.5的低频系数和高频系数. Fig. 2 The noisy synthetic seismic signal and the one layer fractional B spline wavelet transform coefficients (a) The noisy seismic signal; (b) The low frequency and high frequency coefficients for α=-0.4; (c) The low frequency and high frequency coefficients for α=0.01; (d) The low frequency and high frequency coefficients for α=1.5.

 图 3 参数α对SNR的影响 (a)、(b)、(c)表示噪声标准差分别为50，60，70时参数α对SNR的影响. Fig. 3 The influence of parameter α on SNR (a)、(b)、(c) The influence of parameter α on SNR with standard deviation equals to 50, 60, 70.

 图 4 合成地震信号及降噪结果 (a)源地震信号；(b)含噪地震信号；(c)小波域GSM模型降噪后的信号；(d)小波域GSM模型降噪的差剖面；(e)小波软阈值降噪后的信号；(f)小波软阈值降噪的差剖面；(g)小波硬阈值降噪后的信号；(h)小波硬阈值降噪的差剖面；(i)本文方法降噪后的信号；(j)本文方法降噪的差剖面. Fig. 4 The synthetic seismic signal and denoised results (a) The ideal seismic signal; (b) The noisy seismic signal; (c) The denoised signal using Gaussian Scale Mixture model in wavelet domain; (d) The differential profile using Gaussian Scale Mixture model in wavelet domain; (e) The denoised signal using soft-threshold in wavelet domain; (f) The differential profile using soft-threshold in wavelet domain; (g) The denoised signal using hard-threshold in wavelet domain; (h) The differential profile using hard-threshold in wavelet domain; (i) The denoised signal using our method; (j) The differential profile using our method.

 图 5 单道地震信号降噪结果及对应的振幅谱 (a)源地震信号、小波域GSM模型和本文方法降噪结果的局部放大比较图；(b)地震信号及其降噪后信号对应的振幅谱；(c)振幅谱的局部放大图. Fig. 5 The denoised results of single channel seismic signal and corresponding amplitude spectrum (a) The local enlarged figure of the ideal seismic signal、the denoised seismic signal using the Gaussian Scale Mixture model in wavelet domain and our method; (b) The amplitude spectrums of the seismic signal and the denoised seismic signal; (c) The local enlarged figure of the amplitude spectrum.

4 实际地震信号的降噪

 图 6 实际地震信号的降噪结果 (a)实际地震信号；(b)小波域GSM模型降噪后的地震信号；(c)小波域GSM模型降噪的差剖面；(d)小波软阈值降噪后的信号；(e)小波软阈值降噪的差剖面；(f)小波硬阈值降噪后的信号；(g)小波硬阈值降噪的差剖面；(h)本文方法降噪后的地震信号；(i)本文方法降噪的差剖面. Fig. 6 The denoised results of the field seismic signal (a) The field seismic signal; (b) The denoised seismic signal using the Gaussian scale mixture model in wavelet domain; (c) The differential profile using the Gaussian scale mixture model in wavelet domain; (d) The denoised signal using soft-threshold in wavelet domain; (e) The differential profile using soft-threshold in wavelet domain; (f) The denoised signal using hard-threshold in wavelet domain; (g) The differential profile using hard-threshold in wavelet domain; (h) The denoised seismic signal using our method; (i) The differential profile using our method.

5 结论

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