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.     

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.     

DETERMINISTIC ESTIMATION OF A WAVELET USING IMPEDANCE TYPE TECHNIQUE*

A new deterministic technique for estimating a wavelet suggested by Loewenthal and Jakubowicz requires measurement of both pressure and vertical particle velocity. Through construction of the impedance function a deterministic estimation of the wavelet and the reflectivity can be obtained. This idea is tested for a one-dimensional model. The test is carried out by forming a synthetic seismogram of both pressure and particle velocity and checking the formulas for obtaining the estimated wavelet under noisy conditions.     

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.     

ODD-DEPTH STRUCTURES FOR DETERMINISTIC DECONVOLUTION AND SEISMOGRAM TESTING*

In odd-depth structure the two-way traveltime to each boundary is constrained to be an odd integer. The odd-depth property of a model is exposed to possible refutation under a seismogram test. Test function is a simple transformation of a synthetic seismogram. For an odd-depth model the test function has identically the value 1. The testability of a synthetic seismogram over an odd-depth structure provides a method of deterministic deconvolution. There is no need of specialized assumptions, like the minimum-phase property, about the source wavelet. The deconvolution may be performed in the absence of the early segment of a seismogram.     

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.     

EXPERIMENTAL COMPARISON BETWEEN SPECTRAL RATIO AND RISE TIME TECHNIQUES FOR ATTENUATION MEASUREMENT*

Two techniques for the measurement of attenuation–spectral ratio and rise time techniques–were tested and compared in the laboratory. The spectral ratio technique proved to be reliable and easy to implement for intermediate values (5 < Q < 50) of attenuation. For low (Q > 50) and high attenuations, the spectral ratio technique is inaccurate. Calculating the rise time on simulated signals, we found a relation between rise time τ and the ratio travel-time to quality factor T/Q which could be approximated in intervals by the linear relation τ=τ+C*T/Q. The constants τ and C depend on the absolute value of T/Q and on the initial source signal. The rise time technique, performed on the first quarter period of the signal, enables high attenuations (Q < 5) to be measured. The determination of the relation between τ and T/Q is possible if one knows the initial source. We theoretically approximate this relation through a simulation using a realistic propagation model. With laboratory measurements made on Fontainebleau sandstone, we show that the rise time technique using the theoretical relation τ=τ(T/Q) gives comparable values of Q to those obtained from the spectral ratio technique. In borehole seismics, where it is often difficult to remove undesired signals, the rise time technique applied with the right (τ, T/Q) relation is the best method to use.     

COMPARISON BETWEEN SCHLUMBERGER ELECTRODE ARRANGEMENT AND SOME FOCUSED ELECTRODE ARRANGEMENTS IN RESISTIVITY PROFILES*

The Mining Geophysics Institute of the University of Rome has tested focused electrode arrangements. These have been used both in the field for resistivity profiles in investigations for buried cavities and in the laboratory for model tests in an electrolytic tank with cylindrical and parallelepipedal bodies. Anomalous potentials due to superficial noise, assumed to be a hemisphere having an equatorial plane coinciding with the earth surface, have been calculated for focused current electrode arrangements. Similar anomalies have also been calculated for the Schlumberger arrangement. Model tests in an electrolytic tank have also been carried out for a Schlumberger arrangement and parallelepipedal buried bodies. It is interesting to know whether the focused current arrangements give better results than the Schlumberger arrangement. Since it is not possible to compare directly the results of the two arrangements, the comparison is made by considering the ratio of the anomalies to the superficial noise (considered as hemisphere) for Schlumberger and focused arrangements. The results of the comparison show that, generally, the focused arrangements give better results even in the presence of superficial noise.     

HOMOMORPHIC FILTERING — THEORY AND PRACTICE*

The application of homomorphic filtering in marine seismic reflection work is investigated with the aims to achieve the estimation of the basic wavelet, the wavelet deconvolution and the elimination of multiples. Each of these deconvolution problems can be subdivided into two parts: The first problem is the detection of those parts in the cepstrum which ought to be suppressed in processing. The second part includes the actual filtering process and the problem of minimizing the random noise which generally is enhanced during the homomorphic procedure. The application of homomorphic filters to synthetic seismograms and air-gun measurements shows the possibilities for the practical application of the method as well as the critical parameters which determine the quality of the results. These parameters are:

• a) the signal-to-noise ratio (SNR) of the input data
• b) the window width and the cepstrum components for the separation of the individual parts
• c) the time invariance of the signal in the trace.

In the presence of random noise the power cepstrum is most efficient for the detection of wavelet arrival times. For wavelet estimation, overlapping signals can be detected with the power cepstrum up to a SNR of three. In comparison with this, the detection of long period multiples is much more complicated. While the exact determination of the water reverberation arrival times can be realized with the power cepstrum up to a multiples-to-primaries ratio of three to five, the detection of the internal multiples is generally not possible, since for these multiples this threshold value of detectibility and arrival time determination is generally not realized. For wavelet estimation, comb filtering of the complex cepstrum is most valuable. The wavelet estimation gives no problems up to a SNR of ten. Even in the presence of larger noise a reasonable estimation can be obtained up to a SNR of five by filtering the phase spectrum during the computation of the complex cepstrum. In contrast to this, the successful application of the method for the multiple reduction is confined to a SNR of ten, since the filtering of the phase spectrum for noise reduction cannot be applied. Even if the threshold results are empirical, they show the limits fór the successful application of the method.     

有限差分格式与物理量守恒性

为了成功地开展大气数值模拟,如何构造相应的离散格式是一项十分重要的工作.本文以大气球面浅水方程为例,从不同形式的微分方程组出发,分别构造了能量守恒格式、准辛格式和多守恒格式,并作相应的数值检验和比较,从中可以清楚地看到三类格式的优劣.这三类格式构造方法不同,所保持的物理守恒性不完全相同,数值计算时所需机时也不同,这些比较结果为实际工作的不同使用需求提供了选择依据.     

对波达波夫和Pride震电波方程组的对比分析

用Biot介质参数说明了波达波夫震电波方程组中弹性动力学 参数的含义，解释了第一类和第二类震电效应的意义，在忽略第一类震电效应条件下将该方 程组与Pride方程组进行比较，说明了二者在描述第二类震电效应方面的异同点. 同时指出 ：波达波夫方程组忽略了流体与固体的耦合质量；方程中的黏性耗散项丢掉了一个孔隙度因 子，依据该方程组计算出的弹性波和转换电场的幅度将偏大；边界条件之一存在错误，会影 响对波在界面上的反射透射规律的描述.     

SEISMIC SIGNAL DETECTION AND PARAMETER ESTIMATION*

In the mathematical theory of seismic signal detection and parameter estimation given, the seismic measurements are assumed to consist of a sum of signals corrupted by additive Gaussian white noise uncorrelated to the signals. Each signal is assumed to consist of a signal pulse multiplied by a space-dependent amplitude function and with a space-dependent arrival time. The signal pulse, amplitude, and arrival time are estimated by the method of maximum likelihood. For this signal-and-noise model, the maximum likelihood method is equivalent to the method of least squares which will be shown to correspond to using the signal energy as coherency measure. The semblance coefficient is equal to the signal energy divided by the measurement energy. For this signal model we get a more general form of the semblance coefficient which reduces to the usual expression in the case of a constant signal amplitude function. The signal pulse, amplitude, and arrival time can be estimated by a simple iterative algorithm. The effectiveness of the algorithm on seismic field data is demonstrated.     

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.     

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.     

A COMPARISON BETWEEN KIRCHHOFF AND GRT MIGRATION ON VSP DATA1

A modern approach to migration is to perform wavefield extrapolation, subject to an imaging condition. Correct wavefield extrapolation requires that the boundary conditions at the array of geophones satisfy the wave equation. A sufficient condition is to perform the survey with a single stationary source. Contrary to this condition, many VSPs are conducted in deviated wells, where the source is maintained vertically above the down-hole geophone at each well station. Such a survey fails to provide the boundary conditions theoretically necessary for wave-equation migration. A recently published inversion scheme, referred to as acoustic generalized Radon transform migration (GRT migration), was developed to handle any configuration of sources and geophones, including moving-source deviated-well VSP surveys. GRT migration may be viewed as a weighted version of the generalized Kirchhoff migration, derived in this paper from the exploding-reflector model. When a VSP-survey geometry has been specified, GRT migration can be expressed in terms of array parameters, and compared with the equivalent expression for Kirchhoff (wave-equation) migration. The differences between the two integrals are significant and their effect is demonstrated on VSP data.     

APPLICATION DU FILTRE DE KALMAN A LA NAVIGATION*

This paper is concerned with an application of the Kalman filter to integrated navigation systems. It is shown that this filter is well adapted to resetting the dead-reckoning position by means of a radio-hyperbolic system or of a Doppler sonar. The algorithm can also be used for resetting the dead-reckoning position by means of a satellite navigation system. The first part is briefly concerned with stochastic processes and system theory. The canonical equations for navigation problems are written. The second part is devoted to the Kalman filter. The third part discusses the application of the algorithm to navigation problems and presents some simulation results.     

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.     

COMPARISON OF TIME,FREQUENCY, AND PHASE MEASUREMENTS IN INDUCED POLARIZATION*

In comparing the similarity between time, frequency, and phase measurements as used in the induced polarization method of geophysical prospecting, parameters must first be defined and the basis for comparison established. The conditions for mathematical equivalence then follow in a natural sequence. Laboratory measurements are made on a variety of rock samples to indicate the type of comparison and correlation expected from field measurements. Results indicate that frequency domain and phase measurements are equivalent, but two frequency domain measurements are needed to produce the same amount of polarization information as a single phase measurement. Frequency and time domain measurements will be similar but never equivalent due to the basic differences in their respective parameter definitions.     

滇西北发震构造条件及其与鲜水河断裂带的对比

文章在分析滇西北发震构造条件的基础上,把它与鲜水河断裂带进行了对比。滇西北是一个伸展构造区,鲜水河断裂带为一平移剪切带。后者较前者断层面上正应力大、变形强烈、应变速率高。前者断层几何复杂,D?=1.56,后者断层几何简单,D?=1.08-1.2。震源深处变形环境前者者为低压中温,后为高压中温。由此造成地震活动性上的差异     

PARAMETER ESTIMATION AND FAULT DETECTION BY THREE-COMPONENT SEISMIC AND GEOELECTRICAL SURVEYS IN A COAL MINE*

Three-component seismic and geoelectrical in-mine surveys were carried out in Lyukobanya colliery near Miskolc, Hungary to determine the in situ petrophysical parameter distributions and to detect inhomogeneities in the coal seam. The seismic measurements comprise an underground vertical seismic profile, using body waves, and an in-seam seismic amplitude-depth distribution and transmission survey, using channel waves. The geoelectrical measurements are based on the drift- and seam-sounding method. Interval traveltime-, amplitude-, multiple-filter- and polarization analysis methods are applied to the seismic data. They lead to a five-layer model for the strata including the coal seam. The coal seam and two underlying beds act as a seismic waveguide. The layer sequence supports the propagation of both normal and leaky mode channel waves of the Love- and Rayleigh type. A calculation of the total reflected energy for each interface using Knott's energy coefficients shows that the velocity ranges of high reflection energy and of normal and leaky mode wavegroups coincide. The excitation of wavegroups strongly depends on the seismic source. A simultaneous inversion of a geoelectrical drift- and seam-sounding survey prevents misinterpretations of the seismic data by clearly identifying the low-velocity coal seam as a high-resistivity bed. Calculations of dispersion and sounding curves improve the resolution of the slowness and resistivity in each layer. Both diminished amplitudes and distortions in the polarization of transmission seismo-grams and decreasing resistivities in a geoelectrical pseudosection of the coal seam are related to an inhomogeneity. A calculation of synthetic seismograms for Love and Rayleigh channel waves with the finite-difference and the Alekseev-Mikhailenko method agrees well with the field data for the main features, i.e., particular arrivals in the wave train, wavegroups, velocities and symmetries or asymmetries. This in-mine experiment demonstrates that the simultaneous acquisition, processing and interpretation of seismic and geoelectrical data improve the lithological interpretation of petrophysical parameter distributions. Coal seam inhomogeneities can also be detected more reliably by the two independent surveys than by one alone.