THE PHASE PROPERTY OF A FINITE REALIZATION OF RANDOM NOISE AND ITS INFLUENCE ON DECONVOLUTION *

A finite realization of a discrete random noise process may be considered as a one-sided energy signal. Its phase property can then be described by means of the center position. The samples of such a realization are the components of a random signal vector and the center position is therefore a random variable. A statistical analysis shows that the expected value of the center position equals half the time duration of the realization. This implies that the Z-transform of the realization may be expected to have an equal number of poles and zeros inside and outside the unit circle. The standard deviation from the expected value of the center position is shown to depend on the time duration of the realization and on the autocorrelation of the process. It follows that, for processes that can be described by the convolution of a white series and a disturbance wavelet, the center position is independent of the phase property of the wavelet. A conclusion based on these results is that the homomorphic technique of wavelet estimation through cepstrum stacking must give questionable outcomes. Another conclusion is that the super-position of a realization of random noise on a minimum phase wavelet will in general give a mixed phase resulting signal. It is pointed out that schemes for the derivation of deconvolution filters do not take account of this phenomenon.     

AUTOMATIC DECONVOLUTION OF GRAVIMETRIC ANOMALIES*

Existing techniques of deconvolution of gravity anomalies are principally based on upward and downward continuation of measured fields. It can be shown that a unique set of linear filters, depending only on geometrical parameters, relates density distribution at a given depth to gravity measured on the surface. A method to compute the filter coefficients is developed. Very accurate reconstitution of theoretical models of intricate shape, prove the validity of the linear relationship. One of these sets of linear filters is applied to a field case of underground quarries.     

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.     

ADAPTIVE PREDICTIVE DECONVOLUTION OF SEISMIC DATA*

The Wiener prediction filter has been an effective tool for accomplishing dereverberation when the input data are stationary. For non-stationary data, however, the performance of the Wiener filter is often unsatisfactory. This is not surprising since it is derived under the stationarity assumption. Dereverberation of nonstationary seismic data is here accomplished with a difference equation model having time-varying coefficients. These time-varying coefficients are in turn expanded in terms of orthogonal functions. The kernels of these orthogonal functions are then determined according to the adaptive algorithm of Nagumo and Noda. It is demonstrated that the present adaptive predictive deconvolution method, which combines the time-varying difference equation model with the adaptive method of Nagumo and Noda, is a powerful tool for removing both the long- and short-period reverberations. Several examples using both synthetic and field data illustrate the application of adaptive predictive deconvolution. The results of applying the Wiener prediction filter and the adaptive predictive deconvolution on nonstationary data indicate that the adaptive method is much more effective in removing multiples. Furthermore, the criteria for selecting various input parameters are discussed. It has been found that the output trace from the adaptive predictive deconvolution is rather sensitive to some input parameters, and that the prediction distance is by far the most influential parameter.     

MIGRATION IN TERMS OF SPATIAL DECONVOLUTION*

The relationship between two finite-difference schemes (15° and 40°) and the Kirchhoff summation approach is discussed by using closed form solutions of Claerbout's approximate versions of the wave equation. Forward extrapolation is presented as a spatial convolution procedure for each frequency component. It is shown that downward extrapolation can be considered as a wavelet deconvolution procedure, the spatial wavelet being given by the wave theory. Using this concept, a three-dimensional model for seismic data is proposed. The advantages of downward extrapolation in the space-frequency domain are discussed. Finally, it is derived that spatial sampling imposes an upper limit on the aperture and a lower limit on the extrapolation step.     

AUTOMATIC THREE-LAYER,THREE-DIMENSIONAL DECONVOLUTION OF THE PAYS DE BRAY ANTICLINE*

Gravity data have been transformed into a three-layer, three-dimensional model by using an automatic procedure based on linear filtering. The Bouguer anomaly is first transformed by linear filtering into density variations located between two planes 1100 and 2500 m deep. These densities are then transformed into thicknesses with a constant density contrast of 0.4 g/cm³ with two geological constraints for the second and third interface: —minimum at 2500 m depth; —maximum below a variable limit given by geology. This gives the contact between the second and third layer. Differences between measured and computed gravity are then applied by a similar procedure to a layer located between depths of 0 and 500 m, giving the contact between the first and second layer. Interesting secondary anticlines and transverse faults are shown by various structural maps.     

USE OF THE SEISMIC DYNAMIC DECONVOLUTION ALGORITHM IN DIRECT RESISTIVITY INTERPRETATION*

Seismic dynamic deconvolution is the mathematical basis on which a degree of unification in different prospecting methods is possible, relative to the parameter identification in horizontally stratified media. There is a basic structure which has some immediate applications to the inversion of resistivity data and possibly to other problems. For resistivity soundings there exists a key equation which is parallel to the energy conservation law in the theory of synthetic seismograms.     

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.     

DIRECTIONAL DECONVOLUTION OF MARINE SEISMIC REFLECTION DATA: NORTH SEA EXAMPLE1

Directional deconvolution of the signature from a marine seismic source array may be achieved in combination with prestack migration or dip moveout (DMO) processing. The benefit is demonstrated using an example profile from the southern North Sea. In particular, shallow, dipping reflectors have improved continuity and frequency content. The method could be extended to 3D data to remove both in-line and cross-line directivity effects.     

A MODIFIED GEOELECTRICAL PROCEDURE USING POLARDIPOLE ARRAYS—AN EXAMPLE OF APPLICATION TO DEEP EXPLORATION *

Dipole soundings are more sensitive to noise caused by lateral and superficial inhomogeneities than Schlumberger soundings. However, the former are preferable for deep explorations in view of the relatively short cables required. The simple solution of carrying out the field work by means of dipole spreads, and to transform the dipole resistivity diagrams into Schlumberger ones by means of proper formulae would be valid only for smooth and regular curves; but often, owing to the presence of lateral noises, the dipole data show a considerable scatter. For such cases a “continuous dipole sounding” method is proposed for which all successive dipoles are contiguous, so that all parts of the profiles are covered and interpolation is not necessary. Obviously the moving dipoles have lengths proportional to their distances, so that they appear equal in the usual bilogarithmic scale. It follows that only polar-dipole arrays may be used. The transition from a dipole to the corresponding Schlumberger apparent resistivity diagram requires an integration constant which is not unequivocally determined. Therefore, the solution is not unique, but all possible derived Schlumberger diagrams have a common part. Similarly, they have some common interpretative results, which may be referred to the original dipole diagram obtained in the field. A special measurement technique is required since the dipole-dipole voltages to be determined are noticeably smaller than the Schlumberger ones. This is true also because dipole soundings are used for great depths and for long distances between the two dipoles.     

分形脉冲反褶积方法

解地震反演问题的脉冲反褶积方法是基于反射系数白噪和子波为最小相位的假设下提出的．近几年的研究证明反射系数并不都是白噪,而是某种分形噪声,如果用一类分形反褶积方法,则将地震反演问题化为难以求解的非线性方程组．本文用反射系数的分形性质,推导出一个更为简单易解的线性方程组,称为分形脉冲反褶积．数值计算表明,本文的方法是有效的．     

分形脉冲反褶积方法

解地震反演问题的脉冲反褶积方法是基于反射系数白噪和子波为最小相位的假设下提出的．近几年的研究证明反射系数并不都是白噪,而是某种分形噪声,如果用一类分形反褶积方法,则将地震反演问题化为难以求解的非线性方程组．本文用反射系数的分形性质,推导出一个更为简单易解的线性方程组,称为分形脉冲反褶积．数值计算表明,本文的方法是有效的．     

RESOLUTION OF RESPONSES DUE TO CONDUCTIVE OVERBURDEN AND ORE-BODY THROUGH TIME-DOMAIN ELECTROMAGNETIC MEASUREMENTS—A FIELD EXAMPLE*

A technique is suggested to separate the effect of conductive overburden from the recorded time-domain electromagnetic anomalies. Such a separation leads one to construct a relatively undistorted picture of the conductor underlying a conductive overburden. A parameter τ (with dimensions of time) is introduced to characterize various conductors. To illustrate the applicability of the technique, four ‘Input’ (Mark V) anomalies obtained over the Cuddapah District, A.P., India have been analyzed and different values of the time-parameter τ corresponding to the overburden and the mineralized zones are obtained.     

A FIELD EXAMPLE OF THE USE OF ANISOTROPY PARAMETERS DERIVED FROM RESISTIVITY SOUNDINGS*

In areas where steep dips are encountered conventional practice in resistivity work has involved orienting arrays favourably in relation to the geological strike. In concealed conditions, however, the geological strike may not be known; moreover, strike may change with depth. Considerable advantage is to be gained, in such circumstances, by the use of crossed square arrays in that these yield orientationally insensitive resistivity measurements and also allow strike determinations and measurements of the effective vertical anisotropy. Two traverses of crossed square array observations are presented, together with one deeper sounding. The results show that, in favourable circumstances, reliable data on concealed strike directions can be obtained, and that the anisotropy findings greatly assist the subsequent interpretation. Model results pertinent to the field material are presented and discussed.     

AN EXAMPLE OF THE APPLICATION OF AUTO-CORRELATION AND SPECTRAL ANALYSIS IN MINING EXPLORATION *

This paper deals with a case study on the application of auto-correlation and spectral analysis methods to mineral assay data taken along seven drives and two winzes concerning the McTaggart's (West) lode of Kolar gold fields. The general model conceived in this study is: measured (estimated) ore grade at jth location =true value + error at jth location. Employing an auto-correlation function of the form C e^{- a} |^k| for signals, the respective contributions of signals and noise to the total logarithmic variances for the data along drives and winzes are analysed. Smoothed spectral density estimates and their 90% confidence limits are computed for the series of data.     

波速异常的进一步研究和问题讨论(一)

本文首先清理了近年来我国一些中强地震前后波速比异常概况。在21个震例中,有7次震前提出过不同程度的预报意见,12次是震后总结发现异常的,2次震后总结未发现异常。其次,讨论了利用天然地震资料探索地震波速度异常的几个问题:(1)现有较密的区域电信传输台网需提高观测精度,使得较大的前兆异常量不致于被误差所掩盖;(2)使用多台和达法探索波速比异常时必须注意其方向性效应,而且,若能把单台波速比法与多台波速比法平行对照使用,效果可能更好些;(3)要注意远台与近台出现异常的早晚与反应的程度;(4)高值点反映地震前兆的信息较小,而低值却可能携带某些前兆信息;(5)不会有固定不变的异常区,应该考虑到异常区的变迁。