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.     

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.     

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）=|Si（f,k）|2/{|Si（f,k）|2+λ|Ni（f,k）|2}。式中,|Si（f,k）|2及|Ni（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     

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     

OPTIMUM DIGITAL FILTERING AND INVERSE FILTERING IN THE FREQUENCY DOMAIN*

Two distinct filters are developed in the frequency domain which represent an attempt to increase the resolution of fine structure contained in the signal whilst keeping the expected filtered noise energy within reasonable bounds. A parameter termed the White Noise Amplification is defined and used together with a measure of the deconvolved pulse width in order to provide a more complete characterisation of the filters. Each of the two main types of frequency domain filters discussed varies in properties with respect to a single adjustable parameter. This may be contrasted with a time domain Wiener filter which in general has three variables: length, delay and an adjustable noise parameter or weight. The direct frequency domain analogue of the Wiener filter is termed a gamma-Fourier filter, and is shown to have properties which span the range from those of a spiking filter with zero least square error at one extreme, to those of a matched filter at the other extreme of its variable parameter's range. The second type of filter considered—termed the modulated Gaussian filter—is similarly shown to be a perfect spiking filter at one extreme of its parameter range, but adopts the properties of an output energy filter at the other extreme.     

TRANSFORMATION OF DIPOLE RESISTIVITY SOUNDING MEASUREMENTS OVER LAYERED EARTH BY LINEAR DIGITAL FILTERING *

The technique of linear digital filtering as developed for the direct interpretation of Schlumberger and Wenner soundings (Ghosh 1971) has been applied here for the derivation of the resistivity transform function from the field dipole measurements as the first step in directly interpreting dipole data. Filter coefficients for this transformation have been worked out for the radial-polar, perpendicular and parallel (30°) arrays of dipole sounding. The procedure combines speed with accuracy.     

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.      

THE ANALYSIS AND DESIGN OF TWO-DIMENSIONAL FILTERS FOR TWO-DIMENSIONAL DATA *

Some of the methods such as regional removal and second derivative calculations which can be used to outline anomalies on potential data maps can be thought of as a filtering operation. The analysis and design of such two-dimensional filters by means of direct and inverse two-dimensional Fourier transforms have been considered. An analysis of several published sets of second derivative coefficient sets indicates that, in general, they are not a good approximation to the theoretical second derivative filter. Alternate methods of designing regional removal and second derivative filters are discussed. The properties of various two-dimensional filters are further illustrated by means of maps obtained from the convolution of several of these filters with a set of observed field data. These maps show the large changes in anomaly shape which can result from the inclusion or rejection of various wavelength components.     

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.     

LINEAR FILTERING OF VLF DIP-ANGLE MEASUREMENTS*

The suggested interpretation technique is based on discrete linear filtering of VLF data. The output of the described filtering results is expressed in terms of an equivalent current density at a specified depth that would cause the measured magnetic field. The most practical six-point filter gives an accuracy of 8%. The filter is an extension of the commonly used Fraser filter to process VLF dip-angle data. Filtering the same data set for various depths gives an idea about the change of current density with depth. Areas with high current-density correspond to good conductors. The conductor dip can also be determined. The use of the technique is illustrated on theoretical and field examples. In all cases a good correlation with original models and other types of geophysical measurements was obtained. As shown in the last example, the filtering technique is also applicable in interpretation of other electromagnetic methods.     

TIME-VARYING PREDICTION FILTERING BY MEANS OF UPDATING*

In contrast to the conventional deconvolution technique (Wiener-Levinson), the spike-, predictive-, and gap-deconvolution is realized with the help of an adaptive updating technique of the prediction operator. As the prediction operator will be updated from sample to sample, this procedure can be used for time variant deconvolution. Updating formulae discussed are the adaptive updating formula and the sequential algorithm for the sequential estimation technique. This updating technique is illustrated using both synthetic and real seismic data.     

RAPID NUMERICAL EVALUATION OF TWO-DIMENSIONAL GRAVITY PROFILES*

A precise method of gravity interpretation is described which uses tables for the attraction of rectangular prisms in all positions relative to a given point. Hence the attraction of any body or combination of bodies may be assessed without complicated mathematical procedures.     

TWO-DIMENSIONAL POWER SPECTRAL ANALYSIS OF AEROMAGNETIC FIELDS *

A study of probability distribution function and spectrum of the Airborne Total Intensity Map for an area of 3000 Squ. miles covered by the Bundelkhand granite in India has revealed a close correlation between the Gaussian or non-Gaussian nature of distribution and the shape of unite value contour on the spectrum plot. Further studies of radial spectrum plots suggest that the surface magnetic sources are mainly responsible for the non-Gaussian character of the distribution function and as well as for irregular spectrum shape. The histogram of depth values for magnetic sources obtained from radial spectrum plots show three horizons, namely (i) surface (ii) 410 m(1400′) and (iii) 1000 m (3400′). Based on these informations blocks with ferromagnetic composition at the surface can be identified from those which are non magnetic at the surface.     

THE DETERMINATION OF FILTER COEFFICIENTS FOR THE COMPUTATION OF STANDARD CURVES FOR DIPOLE RESISTIVITY SOUNDING OVER LAYERED EARTH BY LINEAR DIGITAL FILTERING *

The technique of linear digital filtering developed for the computation of standard curves for conventional resistivity and electromagnetic depth soundings is applied to the determination of filter coefficients for the computation of dipole curves from the resistivity transform function by convolution. In designing the filter function from which the coefficients are derived, a sampling interval shorter than the one used in the earlier work on resistivity sounding is found to be necessary. The performance of the filter sets is tested and found to be highly accurate. The method is also simple and very fast in application.     

DIRECT INTERPRETATION OF RESISTIVITY DATA OVER TWO-DIMENSIONAL STRUCTURES*

A direct interpretation scheme is developed which is capable of determining most of the geological features of a ground which can be assumed to be two dimensional in structure. This scheme extends the earlier work of Pekeris (1940) and Koefoed (1968) to the case where the basal layer of a ground is undulating. It also has a limited use for finding the parameters of a dipping dyke in the lower medium. Though the top and dip of the dyke can be determined, this is not true for the thickness.     

ANALYSIS OF GRAVITY ANOMALIES OF TWO-DIMENSIONAL FAULTS USING FOURIER TRANSFORMS*

The Fourier transform formula for a two-dimensional fault truncating a horizontal bed at an arbitrary angle of inclination is derived. The amplitude spectrum of the Fourier transform is found to give information about the depth to the top of the upper part of the faulted bed and the inclination of the fault-plane. Under suitable conditions the thickness and the displacement of the bed involved can be obtained. With actual field data, these transforms can be obtained at discrete points by a Fourier analysis of the gravity anomaly. A field example from the Logan fault area near Montreal, Que., Canada, is given.     

根据地下水资料的数字滤波探讨唐山大震的孕育过程

采用纯振幅、点阻、带通数字滤波器,处理了距唐山180公里范围内13口深井水位的大震前六年连续观测资料。结果表明,大震前存在井水水位的异常下降,这种异常下降首先发生在距大震震中100公里处,并且以每月5-6公里的速度分别向震中和离开震中的方向传播。地下水异常下降的时间-距离关系与该区里小震发生的时间-距离关系较一致。据地下水异常下降结果,我们提出唐山大震孕育过程可以分为三个阶段:1972年-1974年底,1975年-1976年4月,1976年4月-大震发生。     

根据地下水资料的数字滤波探讨唐山大震的孕育过程

采用纯振幅、点阻、带通数字滤波器,处理了距唐山180公里范围内13口深井水位的大震前六年连续观测资料。结果表明,大震前存在井水水位的异常下降,这种异常下降首先发生在距大震震中100公里处,并且以每月5—6公里的速度分别向震中和离开震中的方向传播。地下水异常下降的时间—距离关系与该区里小震发生的时间—距离关系较一致。据地下水异常下降结果,我们提出唐山大震孕育过程可以分为三个阶段:1972年—1974年底,1975年—1976年4月,1976年4月—大震发生。     

DIGITAL PROCESSING OF REFRACTION DATA STUDY OF FIRST ARRIVALS*

It has been necessary to resort to the use of “long-line” refraction marine operations in certain areas where it proved impossible to eliminate singing from reflection records despite the number and variety of programs at our disposal for this purpose. Experience has shown that manual processing of offshore refraction records takes a disproportionate length of time in comparison with the surveys themselves, and this is incompatible with the requirements for choosing the site of an exploration well. It thus became necessary to find an “industrial approach” to the solution of this processing problem. It was apparent that automatic picking could also facilitate the interpretation of land refraction data, and that in the case of both marine and land work the interpretations would be more accurate when factors were taken into account which could not be considered when working without the aid of a digital computer. For these reasons a set of programs was developed for automatic picking and interpretation of refracted arrivals. The picking itself consists in searching for the maximum values of the normalized cross-correlation functions of the traces with a “model” trace. The first results thus supplied are: “picked” times, intercept times, maximum values of the correlations, and the values of the tie constants between overlapping spreads. Next, the construction of the relative intercept time curves is performed; a statistical analysis of these curves then allows the determination of the offset distance. From these elements, ⊙ either the delay time curve is produced, after ensuring correct reciprocal times by means of additional minor corrections. Such work is carried out in order to enable the geophysicist to gain a sound idea of the quality of the interpretation. To assist in this aim, part of the trace on both sides of the pick is plotted on the final documents. Valid groupings of several traces involving the same amount of refraction data are thus possible. ⊙ or the refractor depth is constructed with the wavefront method, making use of the relative intercept times. Such a procedure, which is normally applied to first breaks, can also be used for later arrivals exhibiting slight interference and should represent an important step towards processing secondary arrivals with high interference. The development of this package, in response to a need which is shared by both SNPA and CGG, is the result of the joint efforts of the Geophysical Group of SNPA's Pau Research Center and CGG's Technical and Scientific Departments.