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。     

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。     

位场向下延拓组合滤波器的设计和应用

一 重、磁异常向下延拓能将水平方向迭加的异常分开,并能突出异常细节,从而丰富异常的解释内容,提高异常解释的可靠性.本文从对引起向下延拓不稳定的各种因素的讨论入手,结合对一些向下延拓方法在频率域的分析,提出了一种与已知资料频谱特征相“匹配”的组合滤波器。经理论模型和实例考核,说明该滤波器既能保证向下延拓有较高的精度,又有较强的抗干扰能力。     

位场向下延拓组合滤波器的设计和应用

一 重、磁异常向下延拓能将水平方向迭加的异常分开,并能突出异常细节,从而丰富异常的解释内容,提高异常解释的可靠性.本文从对引起向下延拓不稳定的各种因素的讨论入手,结合对一些向下延拓方法在频率域的分析,提出了一种与已知资料频谱特征相“匹配”的组合滤波器。经理论模型和实例考核,说明该滤波器既能保证向下延拓有较高的精度,又有较强的抗干扰能力。     

BASIC PRINCIPLES OF TWO-DIMENSIONAL DIGITAL FILTERING *

The geophysicist involved in the analysis of two-dimensional data should have an understanding of the two-dimensional finite Fourier transform and the mechanics of two-dimensional filtering. Frequency aliasing must be considered when working with sampled data. In two dimensions it is advantageous to consider aliasing in terms of the overlap of the repeating spectra inherent in the finite Fourier transform. Two-dimensional filtering can be performed as a transient convolution in the space domain, as cyclic convolution utilizing the frequency domain or as the multiplication of polynomials using the z-transform. If the “edge” effects are removed, the results of the three methods are identical.     

根据数字滤波器的频率响应优选数字处理方法

本文根据数字滤波器压制干扰的原理,指出利用数字滤波器的频率响应、优选数字处理方法和确定最佳滤波参数的具体原则。对于那些未给出频率响应函数或周期选择性函数的数字处理方法,可以直接根据计算方法的权系数算出它们的频率响应。计算递归滤波器的频率响应式子,也可以用来计算其它非递归滤波器的频率响应。本文利用重力固体潮汐分析中计算中心点的零点飘移值若干方法的系数和它们的周期选择性函数,分别计算了它们的频率响应,从两者的数值上比较,结果是一致的。本文还计算了目前大家常用的一些数字处理方法,如五日均值、一阶差分、二阶差分、多点数字平滑等方法的频率响应值。并分析了它们的处理效果和方法的局限性。还根据最佳数字滤波器和DAI数字滤波器的频率响应曲线,说明选择数字滤波器和确定最佳滤波参数的具体方法。     

A DIGITAL LINEAR FILTER FOR RESISTIVITY SOUNDING WITH A GENERALIZED ELECTRODE ARRAY*

This paper presents a digital linear filter which maps composite resistivity transforms to apparent resistivities for any four—electrode array over a horizontally layered earth. A filter is provided for each of three sampling rates; the choice of filter will depend on resistivity contrasts and computational facilities. Two methods of filter design are compared. The Wiener-Hopf least-squares method is preferable for low sampling rate filters. The Fourier transform method is more successful in producing a filter with a high sampling rate which can handle resistivity contrasts of 100 000: 1.     

COMPUTATION OF TYPE CURVES FOR ELECTROMAGNETIC DEPTH SOUNDING WITH A HORIZONTAL TRANSMITTING COIL BY MEANS OF A DIGITAL LINEAR FILTER*

The computation method described in this paper is based on the existence of a linear relationship between the mutual coupling ratio and the kernel function in the integral expression for it. Accordingly, the mutual coupling ratio can be determined by first computing sample values of the kernel function and then subjecting these to a digital linear filter. In the present paper the appropriate sampling distance is determined and the values of the digital filter coefficients are computed, both for electromagnetic sounding with horizontal coils and for electromagnetic sounding with perpendicular coils.     

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.     

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.     

数字滤波技术在电测井中的应用

本文采用数字滤波技术,研究了电阻率测井的理论曲线计算和Ra曲线的环境影响校正,并对影响数字滤波效果的因素进行了分析。结果表明,用数字滤波新技术进行环境校正,提高了校正质量,比传统的图版校正方法有明显优点。文中采用数字滤波方法绘制的Ra理论曲线,证明所推导的数学模型是正确的。     

数字滤波技术在电测井中的应用

本文采用数字滤波技术,研究了电阻率测井的理论曲线计算和R_a曲线的环境影响校正,并对影响数字滤波效果的因素进行了分析。结果表明,用数字滤波新技术进行环境校正,提高了校正质量,比传统的图版校正方法有明显优点。文中采用数字滤波方法绘制的R_a理论曲线,证明所推导的数学模型是正确的。     

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.