消除地震前兆观测中干扰的一种DAI数字滤波器

本文介绍了一种DAI数字带通滤波器,这种滤波器能根据数据处理需要选择滤波参数。参数确定后,可计算出滤波器的频率响应,根据频率响应的好坏确定是否重新选择滤波参数,直到满意为止。滤波的误差可以在滤波器设计中给定,适当增加褶积的项数即滤波器的长度,可以使滤波的误差很小。本文用这种方法处理了地下水位观测资料,消去固体潮汐的影响。其效果比用潮汐理论值校正的方法消去潮汐影响更好。本文还用频移的方法将这种带通滤波器变成低通滤波器,用来消去视电阻率的年变化。直接用带通滤波提取水氡观测值中周期在半个月到两年的变化成分。这种滤波方法在以上应用中均得到了较好的效果。     

滤波原理和一般线性时间不变系统──地震学上用的数字信号处理的基本原理与实例(讲座1)

从滤波和系统理论的一些基本概念出发，介绍了简单的阻容滤波器原理，它的微分方程表达，频率响应函数，传递函数以及在离散数据处理中滤波器的差分方程表达。最后推广到对一般线性时间不变系统的处理。这些概念和原理是数字信号处理的基础。     

滤波原理和一般线性时间不变系统

从滤波和系统理论的一些基本概念出发，介绍了简单的阻容滤波器原理，它的微分方程表达，频率响应函数，传递函数以及在离散数据处理中滤波器的差分方程表达。最后推广到对一般线性时间不变系统的处理。这些概念和原理是数字信号处理的基础。     

利用小波分析重力的长期变化

运用小波滤波方法估算Chandler和周年项的潮汐因子.本文分析了四个台站(Brussels, Boulder, Membach以及Strasbourg)的观测记录,运用合成潮方法得到重力残差后,用Daubechies小波带通滤波器滤波残差,得到256～512 d时间尺度上的序列,根据标准差最小原则确定观测极潮周年和Chandler项的周期,然后利用最小二乘法估算它们的潮汐因子,同时给出未经模型改正的周年重力.由于高阶Daubechies小波构造的滤波器具有良好的频率响应,且能压制信号中的高阶异常成分,使滤波的信号更加光滑,因此计算结果具有更小的均方差,更加可靠.     

最佳数字滤波器及其应用

最佳数字滤波器是分离有给定周期的波或消除时间序列中短周期波动的有效数学工具。它可分成三种类型:低通滤波器、带通滤波器和高通滤波器。 本文提出一个计算正弦积分的新公式。在不同的地质领域内使用了最佳数字滤波器     

地震面波频散的数字计算——方法与试验

在108-乙型数字计算机上进行了地震面波频散处理方法的试验。包括:（1）参数固定的和可变的数值滤波;（2）测定群速度的“移动窗”和“多重滤波”技术;（3）测定相速度的互相关法。应用了快速傅氏变换计算两个台记录的地震面波的互相关函数。使用数字处理技术分析地震面波频散,除了提高计算速度和便于处理之外,还可扩大测量周期的范围至20-70秒,从此可作出对地壳参数性质的更可靠的测定。     

分离重磁区域场与局部场的维纳滤波器

本文从最佳线性滤波理论出发,对目前重磁资料数据处理中分离区域场与局部场的两种滤波器--匹配滤波和维纳滤波的频率响应特性作了分析比较,指出了匹配滤波只是一般维纳滤波的一个特例。将该两种滤波器与一般情况的维纳滤波器的误差作了对比,并通过简单的理论试例,说明它们的局限性和应用范围。     

EMD新技术在数字波形预处理中的初步应用

介绍了EMD技术,给出由该技术产生的固有模态函数重构数字地震波形信号处理中所用滤波器算法。然后利用该方法对实际波形信号进行了初步处理,处理结果表明,基于EMD的滤波技术具有许多其它分析手段所不具备的特点,是一种新的滤波方法,可用于数字地震波形信号的预处理。     

利用数字强震仪记录实时仿真地动速度

本文首先讨论了计算单自由度系统相对速度递归公式所存在的问题,然后根据递归方程不变性和中心差分,得到了新的计算单自由度系统相对速度地震反应的递归公式,分析了新公式的误差精度。在此基础上,利用地震仪原理,提出了由加速度记录实时恢复地动速度的计算方法。     

航空重力测量数据的小波滤波处理

构造三类连续小波函数对航空重力测量数据进行小波滤波处理. 三类连续小波函数分别用于对测量数据在某一空间尺度（或时间尺度）上的低通,一阶求导和二阶求导滤波. 着重介绍三类连续小波函数的构造原理与过程,并说明其相对于传统的数字滤波器的优势. 对系统的技术参数（滤波器窗口宽度参数δ 和尺度参数s）进行了调试实验. 实算结果显示了方法的可行性和有效性.     

二维最佳线性数字滤波器的设计原理

针对如何在干扰场的背景上区分出低缓异常,以及在位场的向下延拓一类计算中如何限制因误差的高频放大所导至的解的不稳定性等问题,本文探讨了在“最小二乘”意义下的最佳线性数字滤波器的设计原理,并将它转化为下述数学问题,即在L₂线性赋范函数空间中如何选取最佳滤波函数的问题。在空间域中直接解这个问题是十分复杂和困难的,我们发现在波数域中用变分法中的等周问题的解法直接选取最佳线性滤波器的传输函数（或波数响应）,则在数学方法上既简单又严格。这样选取的最佳线性滤波器的传输函数L（f,k）其表达式也很简单,即L（f,k）=|S_i（f,k）|²/{|S_i（f,k）|²+λ|N_i（f,k）|²}。式中,|S_i（f,k）|²及|N_i（f,k）|²分别代表滤波器输入端讯号和干扰的能谱（或功率谱）,f、k分别代表x、y方向上的波数,λ为大于零的常数。 对上述两类问题以及相关的两种最佳线性滤波器而言,L（f,k）的表达式是相同的,而区别仅在于其参变量λ的选取条件不同而已。 有了最佳线性滤波器的传输函数L（f,k）的理论公式,就可以在最小二乘的意义下分析和评价国内外所发表的解决上述两类问题的各种线性滤波方法,并能指出在不同的讯号与干扰条件下,在理论上线性滤波可能达到的最佳效果,从而为设计二维线性数字滤波器时,提供一个理论上的准则。 对位     

REALIZATION OF SHARP CUT-OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS*

Sharp cut-off frequency filtering is carried out in the discrete time domain on digital computers. A convolution of the digital filter impulse response with the sampled input yields the output. For practical reasons, the length of the filter inpulse response, corresponding to the number of filter coefficients, is limited, and consequently the resulting frequency characteristic will no longer be identical to that originally specified. This is analogous to synthesising some specified frequency characteristic with a finite number of resistive, capacitative and inductive components. In Part I of this paper, we examine the effect of approximating the sharp cut-off frequency characteristic best in a mean square sense by an impulse response of finite length. The resulting frequency characteristic corresponds to the truncated impulse response of the specified frequency characteristic. It has a cut-off slope proportional to, and a mean square error inversely proportional to, the length of the impulse response, and is a biassed odd function about the cut-off frequency point. Because of the Gibbs phenomenon for discontinuous functions, the resulting frequency characteristic will always have a maximum overshoot with respect to the specified characteristic of ± 9%, regardless of the length of the corresponding impulse response. Equal length truncated impulse responses of specified filters with different cut-off frequencies yield frequency characteristics which are almost identical about their respective cut-off points. Now on a log frequency scale (as against a linear frequency scale implied previously) such characteristics may be made almost identical about the respective cut-off points by having the truncated impulse responses composed of an equal number of zero crossings. Results for the low-pass filter are applicable to the high-pass and band-pass characteristics. In the latter case, the mean square error is double that for a single slope characteristic (low-pass or high-pass) and the slopes at both edges of the passband are approximately equal in magnitude to the length of the impulse response (linear frequency scale). Part II of this paper is concerned with reducing the ± 9% overshoot that results from the discontinuous nature of the sharp cut-off frequency characteristic and which is not dependent on the length of the truncated impulse response. The reduction is achieved, at the expense of the steepness of cut-off for the resulting frequency characteristic, by the use of functions which weight the truncated impulse response of the specified frequency characteristic. These functions are called apodising functions. Among other variables, the length of the truncated weighted impulse response will determine the amount of maximum overshoot since the effective frequency characteristic being approximated is no longer a discontinuous function. The digital realization of the finite length impulse responses of Parts I and II is discussed in Part III, together with the optimum partially specified digital filter approximation to the desired frequency characteristic.     

REALIZATION OF SHARP CUT-OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS

It was found in Part I of this paper that approximating the sharp cut-off frequency characteristic best in a mean square sense by an impulse response of finite length M produced a characteristic whose slope on a linear frequency scale was proportional to the length of impulse response, but whose maximum overshoot of ±9% was independent of this length (Gibbs' phenomenon). Weighting functions, based on frequency tapering or arbitrarily chosen, were used in Part II to modify the truncated impulse response of the sharp cut-off frequency characteristic, and thereby obtain a trade-off between the value of maximum overshoot and the sharpness of the resulting characteristic. These weighting functions, known as apodising functions, were dependent on the time-bandwidth product Mξ, where 2ξ, corresponded to the tapering range of frequencies. Part III now deals with digital filters where the number 2N–1 of coefficients is directly related to the finite length M of the continuous impulse response. The values of the filter coefficients are taken from the continuous impulse response at the sampling instants, and the resulting characteristic is approximately the same as that derived in Part II for the continuous finite length impulse response. Corresponding to known types of frequency tapering, we now specify a filter characteristic which is undefined in the tapering range, and determine the filter coefficients according to a mean square criterion over the rest of the frequency spectrum. The resulting characteristic is dependent on the time bandwidth product Mξ= (N–1/2)ξ up to a maximum value of 2, beyond which undesirable effects occur. This optimum partially specified characteristic is an improvement on the previous digital filters in terms of the trade-off ratio for values of maximum overshoot less than 1%. Similar to the previous optimum characteristic is the optimum partially specified weighted digital filter, where greater “emphasis is placed on reducing the value of maximum overshoot than of maximum undershoot”. Such characteristics are capable of providing better trade-off ratios than the other filters for maximum overshoots greater than 1/2%. However these filters have critical maximum numbers 2.N_C–1 of coefficients, beyond which the resulting characteristics have unsuitable shapes. This type of characteristic differs from the others in not being a biassed odd function about its cut-off frequency.     

论组合法滤波的数学物理概念

本文以傅里叶变换为基础,用类似数学滤波中的数学分析方法,比较系统地研究了组合法滤波的数学物理概念及方法的内在规律。可以把方向特性函数P（ω）看作是一种周期“波动”函数,而组合的加权分布函数h（x）则是其离散“频谱”。因此可以引用符合于频谱分析的物理概念来研究反演问题。可以看出,组合滤波的加权分布函数及方向特性分别相当于数学滤波中的滤波因子及频率响应;而数字滤波的移门法等又可以引伸到构成组合法带通滤波中去。二者在数学上有其共性,可以互相引用,在物理上又有各自的特殊性。研究结果给组合法以比较全面的认识,扩大了组合法应用的功能。     

REVIEW ON THE BASIC THEORETICAL ASSUMPTIONS IN SEISMIC DIGITAL FILTERING*

A review of the most significant mathematical properties of digital operators and an introduction to their important applications to seismic digital filtering is given. Basic definitions in the time-series field and the principles of digital filtering are introduced starting from the Z-transform domain. Predictive decomposition for stationary stochastic processes and inverse operators are also discussed. Applications of digital filtering to seismic signal concern the predictive deconvolution, characteristics of dispersive and recursive operators, matched filters, and multichannel operators. A brief discussion on frequency, wave number, and velocity filtering phylosophy is given at the end of the paper.     

OPTIMUM DIGITAL FILTERS FOR SIGNAL TO NOISE RATIO ENHANCEMENT*

One of the main objectives of seismic digital processing is the improvement of the signal-to-noise ratio in the recorded data. Wiener filters have been successfully applied in this capacity, but alternate filtering devices also merit our attention. Two such systems are the matched filter and the output energy filter. The former is better known to geophysicists as the crosscorrelation filter, and has seen widespread use for the processing of vibratory source data, while the latter is. much less familiar in seismic work. The matched filter is designed such that ideally the presence of a given signal is indicated by a single large deflection in the output. The output energy filter ideally reveals the presence of such a signal by producing a longer burst of energy in the time interval where the signal occurs. The received seismic trace is assumed to be an additive mixture of signal and noise. The shape of the signal must be known in order to design the matched filter, but only the autocorrelation function of this signal need be known to obtain the output energy filter. The derivation of these filters differs according to whether the noise is white or colored. In the former case the noise autocorrelation function consists of only a single spike at lag zero, while in the latter the shape of this noise autocorrelation function is arbitrary. We propose a novel version of the matched filter. Its memory function is given by the minimum-delay wavelet whose autocorrelation function is computed from selected gates of an actual seismic trace. For this reason explicit knowledge of the signal shape is not required for its design; nevertheless, its performance level is not much below that achievable with ordinary matched filters. We call this new filter the “mini-matched” filter. With digital computation in mind, the design criteria are formulated and optimized with time as a discrete variable. We illustrate the techniques with simple numerical examples, and discuss many of the interesting properties that these filters exhibit.     

PRINCIPLES OF DIGITAL WIENER FILTERING*

The theory of statistical communication provides an invaluable framework within which it is possible to formulate design criteria and actually obtain solutions for digital filters. These are then applicable in a wide range of geophysical problems. The basic model for the filtering process considered here consists of an input signal, a desired output signal, and an actual output signal. If one minimizes the energy or power existing in the difference between desired and actual filter outputs, it becomes possible to solve for the so-called optimum, or least squares filter, commonly known as the “Wiener” filter. In this paper we derive from basic principles the theory leading to such filters. The analysis is carried out in the time domain in discrete form. We propose a model of a seismic trace in terms of a statistical communication system. This model trace is the sum of a signal time series plus a noise time series. If we assume that estimates of the signal shape and of the noise autocorrelation are available, we may calculate Wiener filters which will attenuate the noise and sharpen the signal. The net result of these operations can then in general be expected to increase seismic resolution. We show a few numerical examples to illustrate the model's applicability to situations one might find in practice.     

对地震转换波的数字滤波处理

本文讨论运用低通滤波、偏振滤波、频率加强滤波、相关滤波及迭加处理等数字滤波方法,对地震转换波测深资料进行数字处理的方法及效果.数字滤波处理结果在一定程度上提高了记录的信噪比,为准确地进行相位对比和震相识别提供了可靠基础.      

TRANSFORMATION OF RESISTIVITY SOUNDING MEASUREMENTS OBTAINED IN ONE ELECTRODE CONFIGURATION TO ANOTHER CONFIGURATION BY MEANS OF DIGITAL LINEAR FILTERING*

The technique of digital linear filtering is used for transformation of apparent resistivity data from one electrode configuration into another. Usually filter spectra are determined via the discrete Fourier transforms of input and output functions: the filter characteristic is the quotient of the spectra of the output function and input function. In this paper, the transformation of the apparent resistivities is presented for four electrode configurations (Wenner, the two-electrode, Schlumberger, and dipole configurations). In our method, there is no need to use the discrete Fourier transform of the input and output functions in order to determine the filter spectrum for converting apparent resistivity in one electrode configuration to any other configuration. Sine responses for determination of the derivative of apparent resistivities are given in analytical form. If the filter spectrum for converting the apparent resistivity to the resistivity transform for one electrode configuration is known, the filter spectra for transforming the apparent resistivity to the resistivity transform for any electrode configurations can be calculated by using newly derived expressions.