最大熵反褶积原理与最小相位化滤波器设计方法

本文阐明了最大熵反褶积(MEDEC)的理论根据,并证明:仅当地层脉冲响应为白噪序列和子波为最小相位时,MEDEC预测滤波器的逆为子波的最小相型,地震序列预测滤波输出的误差序列等于地层脉冲响应的估值;同时还分析了它的误差来源;模拟实例证实了以上分析.阐明克莱鲍特(Claerbout)将伯格(Burg)递推中的偏相关系数看成层反射系数的解释,只适应于Goupilau型地层中深部震源子波为白噪的模型,而不能把他的解释误解为MEDEC的理论根据.作者提出一种使褶积序列中子波最小相位化的滤波器G(z)的求法:将G(z)=S(z~(-1))/Sm(z~(-1))用升幂长除展开,即提G(z)的时间响应g(n),这是一种精确算法而非统计估计.这里定义S(n)为子波,S_m(n)为它的最小相型,S(z)=(s(n)),S_m(z)=z(S_m(n)).作者还导出一个多项式长除的快速递推公式,它不仅可以求解上述最小相位化滤波器,还可以以求有理分式型系统或信号z变换的反变换.

THE PRINCIPLE OF THE MAXIMUM ENTROPY DECONVOLUTION AND THE DESIGN OF THE MINIMUM-PHASING FILTER

In this paper, the fundamentals of the maximum entropy deconvolu-tion (MEDEC) has been expounded by the author theoretically. It has been proved that only when the impulse response of the earth is in a white noise sequence and the wavelet is in its minimum phase, the inverse of the predicting filter of the MEDEC can thus be in the minimum phase type and the error sequence of the predicated filtering output of the seismic sequence can be equal to the estimated values of the impulse response of the earth. The origin of the error was analysed and was proved by modelling.Claerbout holds that the partial correlation coefficients in Burg recursive calculation could be considered as the reflection coefficients of the earth. But this only fit for the case when the formation is in Gou-pilan pattern and when the wavelet of the deep source is in white noise. Hence, we shouldn't hold the explanation made by Claerbout to be the theoretical fundamental of MEDEC.A method for calculating the filter G(z) was suggested. Such a filter can make the wavelet of the convolution sequence turn to be minimum phase. By dint of the raising power long division, the formula G(z) = S(z^-1)/S_m(z^-1) could be extended and the time response g(n) of G(z) could be derived. This is an accuracy calculation but not an estimation. In the formula given above, the s(n) was defined as the wavelet, the S_m(n) was in its minimum phase pattern, S(z) = Z(S(n), S_m(z) =Z(S_m(n).Besides, a fast recursive formula for polynomial long divi sion was derived, it can be used not only to solve the minimum phase filter, but it can be used to calculate the rational fraction type system or do the inversion of signal Z-transformation.