USE OF EXPECTED ERROR IN THE DESIGN OF LEAST-SQUARES OPTIMUM FILTERS*

The design of least-squares optimum filters is based upon minimizing a suitably defined error criterion. The expected value of this error is easily computable after the coefficients of the filter have been determined. When a particular filtering problem is specified, there are several parameters which are specifically not included in the optimization procedure. However, the magnitude of the expected error may be quite sensitive to these parameters. The examination of the relative values of the expected error for variations of these unspecified parameters may lead to a better definition of the filter problem. The parameters which are left unspecified by the general least-square filter definition include: 1. The addition of white noise to the signal autocorrelation to stabilize the filter behavior. 2. The specification of the shape of the desired output of the filter. 3. The specification of the lag between the desired output and the input. Examples are given showing the relationship between these parameters and the value of the expected error.     

DETERMINATION OF RESISTIVITY SOUNDING FILTERS BY THE WIENER-HOPF LEAST-SQUARES METHOD*

It has been found that the Wiener-Hopf least-squares method is a very successful tool for the determination of resistivity sounding filters. The values of the individual filter coefficients differ quite appreciably from those obtained by the Ghosh procedure. These differences in the filter coefficients, however, have only a negligible effect on the output of the filter. It seems that these differences in the coefficients correspond to a filter function of a rather narrow frequency band around the Nyquist frequency, which is only very weakly present in the input and output functions.     

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.     

THE TRANSIENT ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE IN AN IMPERFECTLY CONDUCTING HALF-SPACE*

The presence of a conducting environment about a spherical ore body must be considered when calculating the transient electromagnetic response of the ore body due to a step current flowing in a large circular loop at the earth's surface. Failure to do this can easily lead to errors in excess of 10% in numerical calculations. Moreover, there is only a limited time interval in which the response of the spherical conductor is easily seen. In a poorly conducting ground the resonance response of the sphere is the first to be excited. Later, however, the non-resonance or wave-type response is excited. These waves destructively interfere and finally the response of the sphere decays with time as t^?7/2. For a range of times and depths the best loop for detecting the sphere has about the same radius as the sphere.     

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.     

THE PERFORMANCE OF OPTIMUM STACKING FILTERS IN SUPPRESSING UNCORRELATED NOISE*

Optimum stacking filters based on estimates of trace signal-to-uncorrelated noise ratios are assessed and compared in performance with conventional straight stacking. It is shown that for the trace durations and signal bandwidths normally encountered in seismic reflection data the errors in estimating signal/noise ratios largely counteract the theoretical advantages of the optimum filter. The more specific the filter (e.g. the more frequency components included in its design) the more this is true. Even for a simple weighted stack independent of frequency, the performance is likely to be better than a straight (equal weights) stack only for relatively high signal/noise ratios, when the performance is not critical anyway.     

FILTER DESIGN FOR SUPPRESSION OF SURFACE MULTIPLES IN A NON-NORMAL INCIDENCE SEISMOGRAM*

In this paper we design a non-linear filter to suppress surface multiples in a non-normal incidence plane wave seismogram of a horizontally-layered elastic (or acoustic) earth model. Our filter is optimal in a least-squares sense and is very efficient in suppressing surface multiples, especially for small incidence angles. The design is based on an extension of earlier work by Mendel on normal incidence Bremmer series decomposition to the case of non-normal incidence, and relies heavily on work by Aminzadeh on a non-normal incidence state space model.     

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.     

ON THE INTERPRETATION OF RESISTIVITY SOUNDINGS BY THE LEAST-SQUARES METHOD*

An interactive least-squares method for the interpretation of VES curves was proposed by Johansen (1977). The method permits one to select some parameters (thicknesses and/or resistivities of individual layers) and to change the rest in such a way that the interpreted model approaches the measured data. This note suggests a modification of Johansen's method, in which not only the individual parameters can be selected but also linear combinations of parameters—in particular, the sum of thicknesses of several layers.     

COMPARISON OF FIVE LEAST-SQUARES INVERSION TECHNIQUES IN RESISTIVITY SOUNDING*

A brief history of the development of the inverse problem in resistivity sounding is presented with the development of the equations governing the least-squares inverse. Five algorithms for finding the minimum of the least-square problem are described and their speed of convergence is compared on data from two planar earth models. Of the five algorithms studied, the ridge-regression algorithm required the fewest numbers of forward problem evaluations to reach a desired minimum. Solution space statistics, including (1) parameter-standard errors, (2) parameter correlation coefficients, (3) model parameter eigenvectors, and (4) data eigenvectors are discussed. The type of weighting applied to the data affects these statistical parameters. Weighting the data by taking log₁₀ of the observed and calculated values is comparable to weighting by the inverse of a constant data error. The most reliable parameter standard errors are obtained by weighting by the inverse of observed data errors. All other solution statistics, such as dataparameter eigenvector pairs, have more physical significance when inverse data error weighting is used.     

多次透射边界模拟成层介质中瞬态波动的精度分析

本文将对多次透射边界在瞬态波动模拟中的精度分析推广到介质中存在多个物理波速的情形，阐明了它模拟成层介质中瞬态波动的适用性，指出多次透射边界的误差随着透射阶数的增加而迅速减小，二阶透射边界已具有足够高的模拟精度。     

THE QUALITY OF SOME TWO-DIMENSIONAL FILTERS IN SEPARATING REGIONAL AND LOCAL GRAVITY ANOMALIES *

The separation of regional from local gravity anomalies by means of the application of two-dimensional linear filters is analyzed. It was found that optimization of the filter in the least squares sense leads to filters that produce strong localized concentrations of the error, which may erroneously be interpreted as anomalies. For this reason the maximum absolute value of the error is a more important criterion for the quality of the filter than the root mean square error. This maximum absolute error is minimized by the minimax filter. Intermediate filters are derived which give a transition zone which comes appreciably closer to that of the optimal filter at only a small price in terms of increase of the maximum absolute error.     

INFLUENCE OF A CONDUCTING SHIELD IN THE ONE-LOOP VERSION OF THE TRANSIENT PULSE INDUCTION METHOD*

The transient response of a conductive shell-shell model in the one-loop version was obtained analytically. The results indicate that four zones, namely early, late early, intermediate, and late zone can be identified in the total transient characteristic of the model. In case the measurements are carried out in the late early zone, a conductive target appears as a resistive one. It is suggested that the optimum time of measurement should be so selected as to fall in the intermediate zone.     

DIURNAL CORRECTIONS TO MAGNETIC SURVEYS—AN ASSESSMENT OF ERRORS*

The comparison of a group of continuous total field stations in Ireland with the record of the total field from Valentia Observatory enables the errors which would be involved in using the Observatory as a diurnal correction base to be examined. For the most part, for a single correction, these errors lie between ± 2 and ± 6 gammas. However, they have three notable characteristics: (i) they are not closely dependent on the magnetic ‘noisiness’ of the day, (ii) they are only marginally reduced by the application of station-base time differences and (iii), they are strongly related to differences between the amplitude of the daily magnetic variation at the station and at the base. A permanent geographical basis for the latter differences enables a system for the estimation of the errors to be proposed.     

AN EXAMPLE OF THE APPLICATION OF THE RULES OF COMPOSITION IN RESISTIVITY INTERPRETATION*

The application of approximate rules, whereby apparent resistivity space sections for two dimensional structures can be composited from spaces derived for elementary features is extended to a complex example drawn from a field survey over a fluorite mineral vein. A quantitative solution for the observed resistivity space is presented and the computational sequence involved in matching the observed space is given in detail. The interpreted results are examined in relation to the known geology, supplemented by the results of excavation, and to model tests conducted using a tank analogue. The example also illustrates how successive compositions can be employed in estimating the form of resistivity space in a relatively complex situation.     

PROGRESS IN THE DIRECT INTERPRETATION OF RESISTIVITY SOUNDINGS: AN ALGORITHM*

An algorithm is presented for the direct interpretation of resistivity sounding data. The algorithm is based on the method of successive reductions to lower boundary plane of the resistivity transform function. A novel aspect of the algorithm is that error limits are assigned to the initial values of the resistivity transform, and these error limits are carried through in all the subsequent computations. The width of the error range is then used as the basis for assigning weight factors in the final computation of thicknesses and resistivities of the layers. The errors in the resistivity transform derived from the solution given by the algorithm are usually not more than twice as large as those in the original data.     

INFLUENCE OF VELOCITY ERRORS ON THE FOCUSING ASPECTS OF MIGRATION*

In migration procedures, the velocity profile of the subsurface is the most important input information. Since, in general, this information is only approximately known, errors in the migration output due to errors in the velocity input occur in all practical applications. In migration, velocity errors and depth errors can be interchanged. This interchange property is perfect in the paraxial approximation. From this result it follows that migration with incorrect velocities may still yield correctly migrated data if the imaging principle is modified. This attractive property can be used in the stripping version of migration (recursive migration).     

花岗岩非稳态流变实验

脆塑性转化带对于研究岩石圈变形、断层强度和变形机制以及强震的孕育和发生具有重要意义.文中采用汶川地震震源区彭灌杂岩中具有代表性的细粒花岗岩样品,在固体压力介质三轴实验系统上开展了高温高压非稳态流变实验研究.实验设计模拟了汶川地震区地壳10～30km深度的实际温度和压力,温度为190～490℃,压力为250～750MPa...     

有源带通滤波器的设计、调试与应用

所述的有源带通滤波器是采用现代网络理论，通过查图表反复计算数值调试并利用多项式综合滤波器的设计理论得到的设计结果，从频率响应特性曲线和电路结构可见，它完全满足了设计时所提供的各项技术指标的要求。文中详细地讨论了由单元电路到全极点多反馈滤波网络电路设计时的理论根据，各支路参数实际的计算和调试方法。这种电路结构简单合理，调试和维修十分方便，代替了无源滤波器中的大型电感和电容，有利于设备小型化。     

空间域滤波在求二维重磁接触面参数中的应用

1975年、1976年格林和斯坦利发表了求二维接触面参数的一个方法。本文试图在他们工作的基础上,用空间域滤波来代替格林和斯特利用差商求水平导数和用希尔伯特变换求垂直导数的方法。文中给出了应用空间域滤波计算重、磁异常水平和垂直导数的方法。模型试验证实了理论上的推断。与差商和希尔伯特变换方法相比,优点在于它既能同时滤掉高频干扰,又能使求得的垂直导数和水平导数具有相同的精度,进而提高反演结果的精确度。