SOME IMPROVEMENTS IN THE INTERPRETATION OF VERTICAL ELECTRIC SOUNDING CURVES*

The process of VES interpretation is discussed, including the following points. (a) Preliminary interpretation by means of master curves. It is shown that the positions of the auxiliary points K and Q depend on the resistivity of the substratum. The interpretation is improved if the auxiliary points are determined separately for each master curve. (b) The individual parts of the measured curves are shifted in overlapping MN electrode positions so that the total sum of squares of the shifts is minimal. (c) The ambiguity may be reduced by means of supplementary information or assumptions on the resistivities. Fixing the resistivities is not always possible because discrepancies may arise between the ground measurement and the well-logging data. The simultaneous interpretation of several VES curves is recommended assuming constant resistivities. This assumption may be subsequently verified by means of the F-test. (d) A nonlinear algorithm is proposed for the determination of confidence intervals. As the multi-dimensional confidence intervals are often very complicated, it is recommended to construct only one-dimensional confidence intervals for the estimable parametric functions. (e) A ‘double-least-squares’ optimization technique is presented. The optimization is performed on the estimable parametric functions, and the individual parameters are determined so that the solution remains near the initial guess. This technique is faster than the singular value decomposition.     

电阻率测深的数字解释

本文主要介绍了应用积分变换的方法和采样定理将视电阻率ρ_s曲线作线性滤波,得出一新的电阻率转换函数T′曲线,然后,以层参数（各层的电阻率和厚度）算得的T用最优化数值方法在DJS-6型电子计算机上与其进行自动拟合,以达到解释电测深曲线的目的。 文中简述了戈什（Ghosh）提出的对ρ_s作线性滤波的原理,介绍了与国外不同的取样间距和滤波系数的确定以及阻尼最小二乘法和变尺度最优化法的计算框图和应用,最后附有实例和简要的讨论。     

电阻率测深的数字解释

本文主要介绍了应用积分变换的方法和采样定理将视电阻率ρs曲线作线性滤波,得出一新的电阻率转换函数T′曲线,然后,以层参数（各层的电阻率和厚度）算得的T用最优化数值方法在DJS-6型电子计算机上与其进行自动拟合,以达到解释电测深曲线的目的。 文中简述了戈什（Ghosh）提出的对ρs作线性滤波的原理,介绍了与国外不同的取样间距和滤波系数的确定以及阻尼最小二乘法和变尺度最优化法的计算框图和应用,最后附有实例和简要的讨论。     

SELF-MATCHING DECONVOLUTION IN THE FREQUENCY DOMAIN *

In a previous paper the author showed how, by computing an inverse filter in the frequency domain, an automatic compromise could be made between the conflicting requirements to spike a wavelet and to keep the attendant noise amplification within bounds. This paper extends the technique to take account of errors in the estimated shape of the wavelet defined to the deconvolution process. The drastic effects which such errors can have if they are ignored are demonstrated. A novel form of filter–called the “self-matching filter”–is defined which allows the user to limit not only the noise amplification but also the sensitivity of the filter to random uncertainties in the estimated wavelet. This is achieved by whitening the spectrum only within automatically selected pass bands whilst suppressing other noise-dominated or uncertainly defined frequency components. Conventional Wiener filtering is shown to be a special case of this more general filter, namely one in which the wavelet uncertainty is completely ignored. The type of phase spectrum which the output pulse should be designed to possess (e.g. zero phase or minimum phase) is briefly discussed.     

CALCULATION OF SYNTHETIC REFLECTION SEISMOGRAMS IN THE FREQUENCY DOMAIN *

Calculation of reflection seismograms in the frequency domain makes it possible to treat absorption correctly, i.e. makes it possible to include dispersion (= causality) and frequency dependent reflection coefficients. In the paper it is shown how such a calculation can be done in a reasonably efficient manner.     

ERROR PROPAGATION AND UNCERTAINTY IN THE INTERPRETATION OF RESISTIVITY SOUNDING DATA*

An analysis is made of the propagation of the measuring error in the different stages of the interpretation by the linear filter and reducing method. This analysis leads to an understanding of the range of possible values of the layer parameters and of the nature of the relation between them. It is shown that this relation is not always adequately described by the equivalence expressions of Maillet.     

ANALYSIS OF LEAST SQUARES SMOOTHING OPERATORS IN THE FREQUENCY DOMAIN*

The process of data smoothing by least squares operators with real and symmetric coefficients is shown to be equivalent to linear digital filtering. Study of the frequency response functions of the least squares operators shows that these operators are essentially low-pass filters. The frequency response functions are characterized by the presence of a main lobe centered about the zero frequency together with a number of minor lobes of smaller amplitudes at higher frequencies. The predominant band of frequencies which is passed by a least squares operator occurs within the main lobe of the frequency response function. The filtering characteristics of a least squares operator are dependent on the degree of the polynomial and the number of data points. Generally speaking, the higher the degree of the polynomial the broader the pass band of the filter will be. But the effect of the number of data points on the pass band is exactly opposite. Several figures showing the plots of the half-power frequency, the quarter-power frequency, and the half-width of the main lobe for a large number of least squares operators with different combinations of the degree of the polynomial and the number of data points are presented in this paper. These figures will be helpful to those people who wish to make a selection of least squares operators suitable for their purposes.     

ON THE TRANSFORMATION OF DIPOLE TO SCHLUMBERGER SOUNDING CURVES *

A method is described to transform a dipole sounding curve, obtained with any one of the common dipole arrays over a horizontally layered earth, to the form of a Schlumberger sounding curve. Starting from the general expression which relates the dipole apparent resistivity to the Schlumberger apparent resistivity and its derivative with respect to the spacing, it is possible with some approximations to derive an easy numerical computation procedure in order to perform the transformation. The applicability of the method is discussed briefly.     

COMPARATIVE ANALYSIS OF TIME DOMAIN AND FREQUENCY DOMAIN IN THE INDUCED POLARIZATION PROSPECTING METHOD*

The polarization content of a medium, in both the time and frequency domains, can be described by parameters which differ in inherent physical meaning and their practical significance. For real situations, general expressions for the apparent parameters, previously determined for both domains, exist for the general case of soundings on a horizontally multi-layered earth. The comparative analysis of these expressions, here restricted to the simple case of a two-layered earth, shows that the theoretical sounding curves of the frequency-domain are different from those of the time-domain. In particular, for every resistivity or chargeability contrasts examined, the apparent frequency-effect curve lies always over the corresponding apparent chargeability curve, but both curves reach the same asymptotical values for shortest and largest spacings. The important conclusions which can be drawn from this result is that both techniques are suitable to investigate subsoil polarizability anomalies. However, from a practical point of view, it is more convenient to adopt the frequency-domain technique when the polarizability increases with depth, while, on the contrary, the time-domain technique is more efficacious when the polarizability decreases with depth.     

EQUIVALENCE IN ELECTROMAGNETIC (FREQUENCY) SOUNDING*

An analysis of the equivalence phenomenon, analogous to that encountered in resistivity sounding, has been made for electromagnetic (frequency) sounding with the following systems: horizontal coplanar coils, perpendicular coils, vertical coplanar coils, and vertical coaxial coils. Cases analyzed are three-layer H and K type resistivity distributions. The theoretical responses for the EM sounding systems have been computed by the digital linear filter method using short filters. An analysis has been made of the resolution by the four EM systems considered f the equivalence effect. It is concluded that from the equivalence point of view electromagnetic (frequency) sounding has relatively better resolution compared to resistivity sounding for the H type three-layer cases. For the K type three-layer cases electromagnetic (frequency) sounding provides a serious problem in resolution similar to that encountered in resistivity sounding. Empirical relationships have been established for the H and K type cases respectively in order to numerically quantify the equivalence involved in electromagnetic (frequency) sounding.     

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.     

AN APPROXIMATE METHOD OF RESISTIVITY SOUNDING INTERPRETATION*

An approximate method of interpretation of resistivity sounding is presented, which may be described as a very crude manner of application of the exact direct interpretation method. The accuracy of this method is fairly low, the errors being in the order of 25%. The method is very fast in application and well suited for application to multilayer cases. The main advantage of the method is that it is in close and clear relation to the exact theory.     

鄂尔多斯块体西南定边-大罗山大地电磁剖面解释

鄂尔多斯块体定边—大罗山段大地电磁结果表明，鄂尔多斯块体内部电性结构简单，成层性好，上地幔第一高导层顶面埋深基本在１０９ｋｍ左右，相当平坦，块体内部不存在壳内高导层。而靠近大罗山处，上地幔第一高导层向上隆起，隆起最高处距地表９２ｋｍ，此处测点壳内有电性分层，但由于断裂带的切割使测点之间的壳内分层不易连接，形成不连续层。分析认为该处为深大断裂带，是鄂尔多斯块体的西部边界     

频率域线源近区(过渡区)测深理论初步研究

对频率域无限长线源近区(过渡区)测深的理论基础进行了探索性研究。通过计算均匀半空间、层状介质等多个模型的线源响应,进而定义并计算全区视电阻率。结果发现,利用水平电场、垂直磁场分别计算的全区视电阻率均能较好地反映地下电性结构的变化。把水平磁场与水平电场联合通过二分搜索算法计算的全区视电阻率也能较好地反映地下电性结构的变化。在近区通过单分量定义的全区视电阻率与一维MT曲线吻合得很好,因此可以用成熟的MT反演技术来进行线源近区电磁资料的反演,从而把有源的问题转化到无源的问题。这初步表明频率域线源近区(过渡区)测深是可行的,但还有一些问题需要解决。     

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 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).     

CENTRAL FREQUENCY SOUNDING IN SHALLOW ENGINEERING AND HYDRO-GEOLOGICAL PROBLEMS*

The paper deals with the early stages of development of a convenient form of electromagnetic induction method of sounding referred to as ‘Central Frequency Sounding’ and abbreviated as CFS. The method is introduced as a rapid and useful technique for investigation of shallow engineering and hydro-geological problems. Sets of theoretical two-layer master curves, suitable for interpretation of field data involving measurement of the vertical magnetic component of the field induced at the center of a loop placed on a two-layer earth, have been presented. The approximate but reasonably accurate solutions for a two-layer earth of any arbitrary resistivity contrast have been considered for the purpose and expressed in a form suitable for computation. The computed results have been presented in sets of curves useful for interpretation of field data.     

THE USE OF TWO-ELECTRODE AND SCHLUMBERGER FILTERS FOR COMPUTING RESISTIVITY AND EM SOUNDING CURVES*

Different sets of filter coefficients for the linear filter technique for the computations of resistivity and EM sounding curves are evaluated for several electrode and coil configurations. Instead of this procedure, the two-electrode filter can be used for computations of Wenner, Schlumberger, and dipole—dipole apparent resistivity model curves by defining convolutional expressions which contain the new input functions in terms of the resistivity transform function. Similarly, the Schlumberger filter performs the computations of dipole—dipole apparent resistivity model curves. The Wenner, Schlumberger, and dipole—dipole filter functions are defined in terms of the two-electrode filter using the new convolutional expressions. A relationship between the Schlumberger and dipole—dipole filter functions is given. The above arguments are adopted for the computations of EM sounding curves. It is shown that the EM filter for the horizontal coplanar loop system (which is identical to the two-electrode filter) performs the computations of the mutual coupling ratios for perpendicular, vertical coplanar, and vertical coaxial loop systems. In the same way, the Schlumberger filter can be used to compute vertical coaxial sounding curves. The corresponding input functions are defined in terms of the EM kernel for all convolutional expressions presented. After these considerations, integral expressions of the mutual coupling ratios involving zero-order Bessel function are derived. The mutual coupling ratio for the vertical coaxial loop system is given in the same form as the mutual coupling ratio for the vertical coplanar loop system.     

声波方程频率域有限元参数反演

推导出频率域有限元声波正演方程,为了消除边界反射,将Clayton-Engquist旁轴波动方程吸收边界条件引入频率域,并对有限元刚度矩阵和质量矩阵进行压缩存储,利用广义共轭梯度法求解有限元方程获得正演解.在此基础上,推导出在某一频率下波场数据残差δU与单元物性参数修改量δλ之间关系的Jacobi矩阵,反演方法允许利用地面二维炮集全波场资料与给出初始模型参数的正演值的差值δU,迭代求得δλ.由于计算机内存的限制,方法计算不允许有过多数目的未知数个数,因此还提出了对同一介质物性单元的Jacobi矩阵元素进行压缩组装的措施,从而使反演的未知量个数减少,结合采用共轭梯度迭代法,使得只需利用有效波频段的少数一些频率即可进行迭代反演.正演和反演理论模型的数值模拟结果表明方法是有效的.     

声波方程频率域有限元参数反演

推导出频率域有限元声波正演方程,为了消除边界反射,将Clayton-Engquist旁轴波动方程吸收边界条件引入频率域,并对有限元刚度矩阵和质量矩阵进行压缩存储,利用广义共轭梯度法求解有限元方程获得正演解.在此基础上,推导出在某一频率下波场数据残差δU与单元物性参数修改量δλ之间关系的Jacobi矩阵,反演方法允许利用地面二维炮集全波场资料与给出初始模型参数的正演值的差值δU,迭代求得δλ.由于计算机内存的限制,方法计算不允许有过多数目的未知数个数,因此还提出了对同一介质物性单元的Jacobi矩阵元素进行压缩组装的措施,从而使反演的未知量个数减少,结合采用共轭梯度迭代法,使得只需利用有效波频段的少数一些频率即可进行迭代反演.正演和反演理论模型的数值模拟结果表明方法是有效的.