POST-CRITICAL WAVELET ESTIMATION AND DECONVOLUTION1

Using synthetic data, it is demonstrated that the amplitude spectra of post-critical plane-wave components are stable and equal to the amplitude spectrum of the input wavelet (critical reflection theorem). Our analysis and physical explanation of the theorem are based only on amplitude versus offset arguments. The stability of the spectra in the post-critical region is directly related to a high amplitude post-critical reflection that dominates the trace in the slant-stack domain. The validity of the theorem for both the acoustic and elastic cases, its assumptions and limitations, are also examined with emphasis on applications for processing seismic reflection data. Based on the theorem, a deterministic procedure is developed (assuming minimum-phase properties) for wavelet estimation and subsequent deconvolution. We call this method Post-critical Deconvolution, which emphasizes reliance on post-critical reflection data. The performance of the method is shown with real data and the results are compared to those obtained with conventional deconvolution techniques.     

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.     

SOURCE ARRAY SCALING FOR WAVELET DECONVOLUTION*

A seismic source array is normally composed of elements spaced at distances less than a wavelength while the overall dimensions of the array are normally of the order of a wavelength. Consequently, unpredictable interaction effects occur between element and the shape of the far field wavelet, which is azimuth-dependent, can only be determined by measurements in the far field. Since such measurements are very often impossible to make, the shape of the wavelet—particularly its phase spectrum—is unknown. A theoretical design method for overcoming this problem is presented using two scaled arrays. The far field source wavelets from the source arrays have the same azimuth dependence at scaled frequencies, and the far field wavelets along any azimuth are related by a simple scaling law. Two independent seismograms are generated by the two scaled arrays for each pair of source-receiver locations, the source wavelets being related by the scaling law. The technique thus permits the far field waveform of an array to be determined in situations where it is impossible to measure it. Furthermore it permits the array design criteria to be changed: instead of sacrificing useful signal energy for the sake of the phase spectrum, the array may be designed to produce a wavelet with desired amplitude characteristics, without much regard for phase.     

ESTIMATION OF THE PRIMARY SEISMIC PULSE *

A seismic trace after application of suitable amplitude recovery may be treated as a stationary time-series. Such a trace, or a portion of it, is modelled by the expression where j represents trace number on the record, t is time, α_j is a time delay, α (t) is the seismic wavelet, s(t) is the reflection impulse response of the ground and n_j is uncorrelated noise. With the common assumption that s(t) is white, random, and stationary, estimates of the energy spectrum (or auto-correlation function) of the pulse α(t) are obtained by statistical analysis of the multitrace record. The time-domain pulse itself is then reconstituted under the assumption of minimum-phase. Three techniques for obtaining the phase spectrum have been evaluated: (A) use of the Hilbert transform, (B) Use of the z-transform, (C) a fast method based on inverting the least-squares inverse of the wavelets, i.e. inverting the normal time-domain deconvolution operator. Problems associated with these three methods are most acute when the z-transform of α(t) has zeroes on or near the unit circle. Such zeroes result from oversampling or from highly resonant wavelets. The behaviour of the three methods when the energy spectra are perturbed by measurement errors is studied. It is concluded that method (A) is the best of the three. Examples of reconstituted pulses are given which illustrate the variability from trace-to-trace, from shot-to-shot, and from one shot-point medium to another. There is reasonable agreement between the minimum-phase pulses obtained by this statistical analysis of operational records and those estimated from measurements close to the source. However, this comparison incorporates a “fudge-factor” since an allowance for absorption has to be made in order to attenuate the high frequencies present in the pulse measured close to the shot.     

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.     

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.     

WAVELET DECONVOLUTION USING A SOURCE SCALING LAW*

We present a new method for the extraction and removal of the source wavelet from the reflection seismogram. In contrast to all other methods currently in use, this one does not demand that there be any mathematically convenient relationship between the phase spectrum of the source wavelet and the phase spectrum of the earth impulse response. Instead, it requires a fundamental change in the field technique such that two different seismograms are now generated from each source-receiver pair: the source and receiver locations stay the same, but the source used to generate one seismogram is a scaled version of the source used to generate the other. A scaling law provides the relationship between the two source signatures and permits the earth impulse response to be extracted from the seismograms without any of the usual assumptions about phase. We derive the scaling law for point sources in an homogeneous isotropic medium. Next, we describe a method for the solution of the set of three simultaneous equations and test it rigorously using a variety of synthetic data and two types of synthetic source waveform: damped sine waves and non-minimum-phase air gun waveforms. Finally we demonstrate that this method is stable in the presence of noise.     

SEISMIC VELOCITY ESTIMATION*

The accuracy of the two most common arrival time functions used in seismic velocity estimation is investigated. It is shown that the hyperbolic arrival time function is more accurate than the parabolic arrival time function for a horizontally layered elastic medium. An upper bound on the difference between the two arrival time functions is given. A maximum-likehood detector for estimating the arrival time of the signals is given. For the signal-in-noise model that is used the maximum-likelihood detector is equivalent to a least-squares detector which corresponds to using the signal energy as coherency measure. The semblance coefficient corresponds to a normalized least-squares detector. The semblance coefficient is very similar to a filter performance measure that is used in least-squares filter design.     

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.     

WAVE SCATTERING DECONVOLUTION BY SEISMIC INVERSION*

We propose a wave scattering approach to the problem of deconvolution by the inversion of the reflection seismogram. Rather than using the least-squares approach, we study the full wave solution of the one-dimensional wave equation for deconvolution. Randomness of the reflectivity is not a necessary assumption in this method. Both the reflectivity and the section multiple train can be predicted from the boundary data (the reflection seismogram). This is in contrast to the usual statistical approach in which reflectivity is unpredictable and random, and the section multiple train is the only predictable component of the seismogram. The proposed scattering approach also differs from Claerbout's method based on the Kunetz equation. The coupled first-order hyperbolic wave equations have been obtained from the equation of motion and the law of elasticity. These equations have been transformed in terms of characteristics. A finite-difference numerical scheme for the downward continuation of the free-surface reflection seismogram has been developed. The discrete causal solutions for forward and inverse problems have been obtained. The computer algorithm recursively solves for the pressure and particle velocity response and the impedance log. The method accomplishes deconvolution and impedance log reconstruction. We have tested the method by computer model experiments and obtained satisfactory results using noise-free synthetic data. Further study is recommended for the method's application to real data.     

A STATISTICAL APPROACH TO THE EXTRACTION OF THE SEISMIC PROPAGATING WAVELET*

A model of the seismic trace is generally given as a convolution between the propagating wavelet and the reflectivity series of the earth and normally it is assumed that a white noise is added to the trace. The knowledge of the propagating wavelet is the basic point to estimate the reflectivity series from the seismic trace. In this paper a statistical method of wavelet extraction from several seismic traces, assuming the wavelet to be unique, is discussed. This method allows one to obtain the propagating wavelet without any classical limitative assumptions on the phase spectrum. Furthermore, a phase unwrapping method is suggested and some statistical properties of the phase spectrum of the reflectivity traces are examined.     

A NEW APPROACH TO VIBROSEIS DECONVOLUTION*

A method is proposed to obviate the shortcomings of conventional deconvolution approaches applied to vibroseis data. The vibroseis wavelet reduces the time domain resolution of the earth's impulse response by restricting its passband. The spectrum of the wavelet is assumed to be a “low quefrency”phenomenon, and hence it can be estimated by low cut cepstral filtering. The wavelet's amplitude spectrum can then be removed by spectral division. By using an approach which is consistent with the principle of maximum entropy, the undetermined portions of the seismogram's Fourier transform can be filled in by autoregressive prediction. The process of initially deconvolving in a restricted passband reduces the enhancement of noise contaminated parts of the spectrum, and the spectral extension scheme increases the time domain resolution of the process.     

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.     

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.     

沂沭带北段近期地震活动特征及地震危险性预测

主要利用《地震学分析预报方法程式指南》的方法,对沂沭带北段进行系统的计算分析研究。结果表明该区近期将无较大地震发生,b值外推缺ML4.3级地震。同时,利用灰色理论对该区未来地震危险性作出预测。     

新马德里地震带的岩石圈强度与板内形变

提出一个简单的假说来解释为什么在相对稳定的板块内部地区会存在高地震活动区与高构造形变区．首先,对于大多数板内地区而言,特别是大陆地盾地区与老的海洋盆地,下地壳与上地幔的温度相当低,那里的岩石相对坚硬在这些地区不可能发生明显的岩石圈变形,因为岩石图累积强度大大超过板块驱动力．相反,如果下地壳与上地幔温度相对较高,板块驱动力则主要由上地壳承受,因为下地壳与上地幔相对软弱在这种地区,由于岩石圈累积强度与板块驱动力大小相当,构造形变相对较快．本文将这种假说应用在位于美国中部的新马德里地震带与周围地区．地震带内部热流密度值约为60mw／m2,略高于本区背景热流密度值45mW／m2．计算得到的地温梯度与实验室结果所揭示的延性流动定律表明,在地震带内下地壳与上地幔相当软弱,板内应力主要由上地壳传递．那里的形变速率相对较高．与此相反,在周围地区热流值相对较低,岩石四累积强度大大超过板块驱动力,构造应力由地壳与上地幔共同承受热流值的大小和下地壳上地幔的受力状态是决定地震活动性在地震带内与周围地区强烈对比的主要因素．     

新马德里地震带的岩石圈强度与板内形变

提出一个简单的假说来解释为什么在相对稳定的板块内部地区会存在高地震活动区与高构造形变区．首先,对于大多数板内地区而言,特别是大陆地盾地区与老的海洋盆地,下地壳与上地幔的温度相当低,那里的岩石相对坚硬在这些地区不可能发生明显的岩石圈变形,因为岩石图累积强度大大超过板块驱动力．相反,如果下地壳与上地幔温度相对较高,板块驱动力则主要由上地壳承受,因为下地壳与上地幔相对软弱在这种地区,由于岩石圈累积强度与板块驱动力大小相当,构造形变相对较快．本文将这种假说应用在位于美国中部的新马德里地震带与周围地区．地震带内部热流密度值约为60mw／m2,略高于本区背景热流密度值45mW／m2．计算得到的地温梯度与实验室结果所揭示的延性流动定律表明,在地震带内下地壳与上地幔相当软弱,板内应力主要由上地壳传递．那里的形变速率相对较高．与此相反,在周围地区热流值相对较低,岩石四累积强度大大超过板块驱动力,构造应力由地壳与上地幔共同承受热流值的大小和下地壳上地幔的受力状态是决定地震活动性在地震带内与周围地区强烈对比的主要因素．     

Lp模反褶积及其在地震勘探中的应用

在文献[1,2]的基础上,本文提出一种矩阵逆迭代和增强强反射的自适应算法,方法简明,使用方便。理论模拟和实际资料处理结果说明该法是很有效的。     

Lp模反褶积及其在地震勘探中的应用

在文献[1,2]的基础上,本文提出一种矩阵逆迭代和增强强反射的自适应算法,方法简明,使用方便。理论模拟和实际资料处理结果说明该法是很有效的。     

MAXIMUM-LIKELIHOOD ESTIMATION OF SEISMIC IMPULSE RESPONSES*

A seismic trace is assumed to consist of a known signal pulse convolved with a reflection coefficient series plus a moving average noise process (colored noise). Multiple reflections and reverberations are assumed to be removed from the trace by conventional means. The method of maximum likelihood (ML) is used to estimate the reflection coefficients and the unknown noise parameters. If the reflection coefficients are known from well logs, the seismic pulse and the noise parameters can be estimated. The maximum likelihood estimation problem is reduced to a nonlinear least-squares problem. When the further assumption is made that the noise is white, the method of maximum likelihood is equivalent to the method of least squares (LS). In that case the sampling rate should be chosen approximately equal to the Nyquist rate of the trace. Statistical and numerical properties of the ML- and the LS-estimates are discussed briefly. Synthetic data examples demonstrate that the ML-method gives better resolution and improved numerical stability compared to the LS-method. A real data example shows the ML- and LS-method applied to stacked seismic data. The results are compared with reflection coefficients obtained from well log data.