ON WIENER FILTER AND MAXIMUM ENTROPY METHOD FOR MULTICHANNEL COMPLEX SYSTEMS*

The Wiener filtering scheme is obtained for a multichannel complex system utilizing the “block-Toeplitz” property of autocorrelation matrix. A numerical example is given to elucidate the application of the filter design. The study also outlines Burg's maximum entropy method to include the multi-channel complex realm.     

THE KALMAN FILTER AS A PREDICTION ERROR FILTER*

In mathematical statistical filtering the deconvolution problem can be solved by two different methods:

• 1 by inverse filtering
• 2 by calculating the prediction error.

Both methods are well known in the theory of Wiener filters. If, however, the generating process of the signal is known and can be described by a set of linear first order differential equations, then the Kalman filter can also be used to solve the deconvolution problem. In the case of the inverse filtering method this was shown by Bayless and Brigham (1970). But, while their method can only be used if the original signal is a colored random process, this paper shows that in the case of a white process the prediction error filtering method is a more appropriate approach. The method is extremely efficient and simple. This can be demonstrated by an example which maybe of special interest for seismic exploration.     

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.     

A NOTE ON THE LINEAR FILTER METHOD OF INTERPRETING RESISTIVITY SOUNDING DATA*

In the linear filter method of interpreting resistivity sounding data, as developed by Ghosh (1971), it appears that the filter function in the x-domain approaches an oscillating function for both large positive and large negative abscissa values. In the present note the reason for this oscillating behaviour is derived, and a possible practical application is indicated.     

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。     

A NOTE ON THE LINEAR FILTER METHOD OF COMPUTING ELECTROMAGNETIC SOUNDING CURVES *

The values of the filter coefficients used for the computation of electromagnetic sounding curves are studied in conjunction with the values of the input function to the filter, or the range of values which the input function may assume, and the filters are broken off at such a place that the error in the sum of the products of filter coefficient and input function does not exceed a prescribed value. This analysis is carried out for the horizontal coils system, the perpendicular coils system, and the vertical coplanar coils system. The lengths of the filters so derived depend on the layer parameters, the frequency and the coil spacing. Even in the most unfavourable cases the filters are shorter than the filters published by Koefoed, Ghosh, and Polman (1972).     

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。     

二维Wiener滤波器及在物探中应用的一些问题

本文结合二维随机场函数z（x,y）及其统计特征、平稳性和各态遍历性的概念,导出了二维Wiener滤波器的两个波数响应Hopt（u,v）和 opt（u,v）,并详细地讨论了导出它们的条件。为了正确地应用Wiener滤波技术,我们提供了检验物探观测数据的平稳性和各态遍历性的简单而适用的方法。最后还指出,当应用Wiener滤波的不相关技术时,将在区域异常的谱函数中引入一个非线性畸变因子,这个区域异常是从具有局部干扰的观测数据中提取的。于是,所提取的区域异常必然会产生某种程度的失真。     

AN OPTIMIZED DIGITAL FILTER FOR THE FOURIER TRANSFORM*

Short filters for calculating Hankel-transformations, with special attention to the d.c.-sounding problem, have been published in recent years. These filters, with a typical length of less than 25 coefficients, have made it possible to implement, e.g., VES-interpretation programs on microcomputers and 3-D electric and electromagnetic modeling programs on minicomputers. Initially the performance of the short filters was rather poor, but with the introduction of short optimized filters there has been a considerable improvement in the accuracy. An optimization procedure is applied to design a 20-point filter for the Fourier sine-transformation. This filter may be useful in electromagnetic prospecting theory, e.g., in the calculation of the electric and magnetic field from a line source.     

二维Wiener滤波器及在物探中应用的一些问题

本文结合二维随机场函数z（x,y）及其统计特征、平稳性和各态遍历性的概念,导出了二维Wiener滤波器的两个波数响应H_opt（u,v）和 _opt（u,v）,并详细地讨论了导出它们的条件。为了正确地应用Wiener滤波技术,我们提供了检验物探观测数据的平稳性和各态遍历性的简单而适用的方法。最后还指出,当应用Wiener滤波的不相关技术时,将在区域异常的谱函数中引入一个非线性畸变因子,这个区域异常是从具有局部干扰的观测数据中提取的。于是,所提取的区域异常必然会产生某种程度的失真。     

THE DETERMINATION OF DIGITAL WIENER FILTERS FROM AN ILL-CONDITIONED SYSTEM OF NORMAL EQUATIONS*

The system of normal equations associated with the discrete Wiener filter is sometimes ill-conditioned. The purpose of this paper is to show that in such cases the solutions obtained vary drastically with the particular choice of an algorithm and of the computer used for its implementation. A review of the basic mathematical theory behind an ill-conditioned matrix is first presented. Numerical examples are then given to illustrate that the solutions of the normal equations are sensitive to the word length of a given computer. Finally, two possible remedies are described: (1) The well-known method of prewhitening and (2) the use of the conjugate-gradient algorithm for solving the normal equations.     

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.     

THE APPLICATION OF LINEAR FILTER THEORY TO THE DIRECT INTERPRETATION OF GEOELECTRICAL RESISTIVITY SOUNDING MEASUREMENTS*

Koefoed has given practical procedures of obtaining the layer parameters directly from the apparent resistivity sounding measurements by using the raised kernel function H(λ) as the intermediate step. However, it is felt that the first step of his method—namely the derivation of the H curve from the apparent resistivity curve—is relatively lengthy. In this paper a method is proposed of determining the resistivity transform T(λ), a function directly related to H(λ), from the resistivity field curve. It is shown that the apparent resistivity and the resistivity transform functions are linearily related to each other such that the principle of linear electric filter theory could be applied to obtain the latter from the former. Separate sets of filter coefficients have been worked out for the Schlumberger and the Wenner form of field procedures. The practical process of deriving the T curve simply amounts to running a weighted average of the sampled apparent resistivity field data with the pre-determined coefficients. The whole process could be graphically performed within an quarter of an hour with an accuracy of about 2%.     

A COMPARISON OF STATISTICAL AND DETERMINISTIC WIENER DECONVOLUTION OF DEEP-TOW SEISMIC DATA*

Wiener ‘spiking’ deconvolution of seismic traces in the absence of a known source wavelet relies upon the use of digital filters, which are optimum in a least-squares error sense only if the wavelet to be deconvolved is minimum phase. In the marine environment in particular this condition is frequently violated, since bubble pulse oscillations result in source signatures which deviate significantly from minimum phase. The degree to which the deconvolution is impaired by such violation is generally difficult to assess, since without a measured source signature there is no optimally deconvolved trace with which the spiked trace may be compared. A recently developed near-bottom seismic profiler used in conjunction with a surface air gun source produces traces which contain the far-field source signature as the first arrival. Knowledge of this characteristic wavelet permits the design of two-sided Wiener spiking and shaping filters which can be used to accurately deconvolve the remainder of the trace. In this paper the performance of such optimum-lag filters is compared with that of the zero-lag (one-sided) operators which can be evaluated from the reflected arrival sequence alone by assuming a minimum phase source wavelet. Results indicate that the use of zero-lag operators on traces containing non-minimum phase wavelets introduces significant quantities of noise energy into the seismic record. Signal to noise ratios may however be preserved or even increased during deconvolution by the use of optimum-lag spiking or shaping filters. A debubbling technique involving matched filtering of the trace with the source wavelet followed by optimum-lag Wiener deconvolution did not give a higher quality result than can be obtained simply by the application of a suitably chosen Wiener shaping filter. However, cross correlation of an optimum-lag spike filtered trace with the known ‘actual output’ of the filter when presented with the source signature is found to enhance signal-to-noise ratio whilst maintaining improved resolution.     

湟源地震台数字地震记录的环境干扰分析及滤波器制作

对湟源地震台周边各种因素所致的地震记录干扰情况进行了分析,认为干扰源主要有采石场爆破、灌溉水渠、小河、车辆干扰及各种天气及固有干扰等。根据这些干扰源的频谱特性,采用MATLAB语言,设计、编制了不同类型的滤波器,在震相识别上发挥了很大作用。     

APPLICATION OF WIENER FILTERS TO TRANSFORMATIONS OF GRAVITY AND MAGNETIC FIELDS*

Consideration of the spectral representation of gravity and magnetic fields shows the field to be the result of the convolution of factors depending on the parameters of the field. Wiener filters, calculated using model transformations, provide an optimum method for altering these factors and hence effecting field transformations.     

THE OPTIMUM GATE LENGTH FOR THE TIME-VARYING DECONVOLUTION OPERATOR1

The time-varying deconvolution operator designed by dividing time-varying sequence has been extended to include an optimal division of the input data. A numerical example illustrates that the error energy is less in the case of optimally divided input in comparison with arbitrary division.     

DIGITAL LINEAR FILTER FOR COMPUTING TYPE CURVES FOR THE TWO-ELECTRODE SYSTEM OF RESISTIVITY SOUNDING*

In this paper a technique for computing type curves for the two-electrode resistivity soundings is presented. It is shown that the apparent resistivity due to the system can be represented by a convolution integral. Thus, it is possible to apply the principle of digital linear filtering and compute the desired type-curves. The filter function required for the purpose is found to be identical with that used to compute the EM sounding curves for the two coplanar horizontal loop system. It is further shown that from the two-electrode apparent-resistivity expression one can easily derive the apparent resistivities for other configurations. A comparison of depths of investigation for various systems reveals that the two-electrode system has greater depth of investigation than other conventional systems. This is also supported by the field example presented in the end, which illustrates the relative performance of the two-electrode system vis-a-vis the Wenner system.     

ON THE PROPERTIES OF THE RECIPROCAL GEOELECTRIC SECTION*

The interpretation of vertical electrical sounding data can be facilitated by the application of the reciprocal geoelectric section. If an apparent resistivity field curve has a descending right end, the apparent resistivity curve of the reciprocal geoelectric section can be obtained by the application of linear filter theory; from this the total transverse resistance of the geoelectric section can be calculated without having to interpret the field curve. In addition, Orellana's auxiliary point method can now be extended to interpret three and four layer apparent resistivity curves of all types. This paper summarizes the properties of the resistivity transform curve, the apparent resistivity curve, and the apparent resistivity curve of the reciprocal geoelectric section, with several new applications.     

MORE ON THE LATEROLOG* DEVICE**

Basic Laterolog devices employ a measure-current focusing technique to provide well logs of formation resistivity that feature sharp vertical definition combined with a depth of investigation much improved over those of normals of comparable spacings. To develop the concept of measure-current focusing, a disk-electrode arrangement analogous to a guard-ring capacitor is considered first, leading finally to consideration of a real sonde in a cylindrical borehole. In the latter, auxiliary currents are adjusted to maintain substantially at zero the vertical potential gradients above and below the central measure-current electrode; this has the effect of focusing the measure-current beam. However, prediction of sonde responses on the basis of current patterns can in some cases be misleading. Indeed, for a given Laterolog array, one may use a (reciprocalmode) current pattern radically different from the one defined by the basic (direct-mode) measure-current focusing scheme and still obtain the same resistivity measurement. This feature is illustrated for Laterolog 3 by use of transfer impedances, and demonstrated for certain more general cases using the reciprocity theorem. Comparisons of Laterolog responses across typical thin beds, both non-invaded and invaded, with those of normal devices of a comparable range of spacings illustrate the improved vertical resolution obtained with Laterolog devices. The examples also show the different magnitudes of the auxiliary currents (thenceforth designated “bucking”- or “focusing” currents) used in the direct and reciprocal modes and illustrate how these currents must continuously vary in order to maintain the focused condition as the sonde moves through the bed. Finally, briefly considering possible applications of Laterolog-type arrays in surface prospecting, it is shown that a one-dimensional array laid out on the surface does not provide the same degree of measure-current focusing as is obtained in a borehole. Some improvement is obtained with a combination of two crossed linear arrays. Much more is obtained using a circular array.