WAVELET ESTIMATION USING THE MULTITAPER METHOD1

An accurate estimate of the seismic wavelet on a seismic section is extremely important for interpretation of fine details on the section and for estimation of acoustic impedance. In the absence of well-control, the recognized best approach to wavelet estimation is to use the technique of multiple coherence analysis to estimate the coherent signal and its amplitude spectrum, and thence construct the seismic wavelet under the minimum-phase assumption. The construction of the minimum-phase wavelet is critically dependent on the decay of the spectrum at the low-frequency end. Traditional methods of cross-spectral estimation, such as frequency smoothing using a Papoulis window, suffer from substantial side-lobe leakage in the areas of the spectrum where there is a large change of power over a relatively small frequency range. The low-frequency end of the seismic spectrum (less than 4 Hz) decays rapidly to zero. Side-lobe leakage causes poor estimates of the low-frequency decay, resulting in degraded wavelet estimates. Thomson's multitaper method of cross-spectral estimation which suffers little from side-lobe leakage is applied here, and compared with the result of using frequency smoothing with the Papoulis window. The multitaper method seems much less prone to estimating spuriously high coherences at very low frequencies. The wavelet estimated by the multitaper approach from the data used here is equivalent to imposing a low-frequency roll-off of some 48 dB/oct (below 3.91 Hz) on the amplitude spectrum. Using Papoulis smoothing the equivalent roll-off is only about 36 dB/oct. Thus the multitaper method gives a low-frequency decay rate of the amplitude spectrum which is some 4 times greater than for Papoulis smoothing. It also gives more consistent results across the section. Furthermore, the wavelet obtained using the multi-taper method and seismic data only (with no reference to well data) has more attractive physical characteristics when compared with a wavelet extracted using well data, than does an estimate using traditional smoothing.     

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.     

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.     

Lp-NORM DECONVOLUTION1

The purpose of deconvolution is to retrieve the reflectivity from seismic data. To do this requires an estimate of the seismic wavelet, which in some techniques is estimated simultaneously with the reflectivity, and in others is assumed known. The most popular deconvolution technique is inverse filtering. It has the property that the deconvolved reflectivity is band-limited. Band-limitation implies that reflectors are not sharply resolved, which can lead to serious interpretation problems in detailed delineation. To overcome the adverse effects of band-limitation, various alternatives for inverse filtering have been proposed. One class of alternatives is L_p-norm deconvolution, L₁norm deconvolution being the best-known of this class. We show that for an exact convolutional forward model and statistically independent reflectivity and additive noise, the maximum likelihood estimate of the reflectivity can be obtained by L_p-norm deconvolution for a range of multivariate probability density functions of the reflectivity and the noise. The L_∞-norm corresponds to a uniform distribution, the L₂-norm to a Gaussian distribution, the L₁-norm to an exponential distribution and the L₀-norm to a variable that is sparsely distributed. For instance, if we assume sparse and spiky reflectivity and Gaussian noise with zero mean, the L_p-norm deconvolution problem is solved best by minimizing the L₀-norm of the reflectivity and the L₂-norm of the noise. However, the L₀-norm is difficult to implement in an algorithm. From a practical point of view, the frequency-domain mixed-norm method that minimizes the L₁norm of the reflectivity and the L₂-norm of the noise is the best alternative. L_p-norm deconvolution can be stated in both time and frequency-domain. We show that both approaches are only equivalent for the case when the noise is minimized with the L₂-norm. Finally, some L_p-norm deconvolution methods are compared on synthetic and field data. For the practical examples, the wide range of possible L_p-norm deconvolution methods is narrowed down to three methods with p= 1 and/or 2. Given the assumptions of sparsely distributed reflectivity and Gaussian noise, we conclude that the mixed L₁norm (reflectivity) L₂-norm (noise) performs best. However, the problems inherent to single-trace deconvolution techniques, for example the problem of generating spurious events, remain. For practical application, a greater problem is that only the main, well-separated events are properly resolved.     

自适应Kalman滤波反褶积的快速实现方法

提出了以二进小波变换为基础的自适应Kalman滤波反褶积(AKFD)新方法,针对该方法的计算复杂程度,提出了一种快速实现方法.二进小波变换的AKFD抛弃了传统预测反褶积对信号平稳性的假设,克服了提高分辨率而信噪比明显降低的问题,具有很好的抗噪性能.在小波域进行的AKFD在压制假反射以及提高分辨率方面比时间域的AKFD好,克服了在时域内进行AKFD抬升低频成分的缺陷.利用二维地震数据的局部平稳性的假设提出了快速实现方法,通过分段求取自适应预测算子,分别于横向及纵向采用样条插值的方法进行插值,来减少求取自适应预测算子的计算量,达到快速实现的目的.经过大量实验表明计算速度提高数百倍,仍能保持原来的计算效果.     

自适应Kalman滤波反褶积的快速实现方法

提出了以二进小波变换为基础的自适应Kalman滤波反褶积(AKFD)新方法，针对该方法的计算复杂程度，提出了一种快速实现方法.二进小波变换的AKFD抛弃了传统预测反褶积对信号平稳性的假设，克服了提高分辨率而信噪比明显降低的问题，具有很好的抗噪性能.在小波域进行的AKFD在压制假反射以及提高分辨率方面比时间域的AKFD好，克服了在时域内进行AKFD抬升低频成分的缺陷.利用二维地震数据的局部平稳性的假设提出了快速实现方法，通过分段求取自适应预测算子，分别于横向及纵向采用样条插值的方法进行插值，来减少求取自适应预测算子的计算量，达到快速实现的目的.经过大量实验表明计算速度提高数百倍，仍能保持原来的计算效果.     

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.     

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。     

基于连续小波变换的大地电磁信号谱估计方法

在基于连续小波变换的大地电磁信号谱估计方法中 ,通过引入整体平均、小波系数收缩和显著性检验等统计技术 ,以提高谱估计的精度 .文中同时讨论了连续小波变换中各种参数的选取问题 ,给出了Morlet小波函数中尺度与傅里叶频率之间转换的经验公式 ,并给出了谱估计的具体算法 .结果表明 ,本文方法可有效压制较强的白噪声和局部相关噪声 .与FFT谱估计方法相比 ,该方法大大降低了对信号记录长度的要求 ,因而对大地电磁信号的处理有实际意义 .     

基于连续小波变换的大地电磁信号谱估计方法

在基于连续小波变换的大地电磁信号谱估计方法中 ,通过引入整体平均、小波系数收缩和显著性检验等统计技术 ,以提高谱估计的精度 .文中同时讨论了连续小波变换中各种参数的选取问题 ,给出了Morlet小波函数中尺度与傅里叶频率之间转换的经验公式 ,并给出了谱估计的具体算法 .结果表明 ,本文方法可有效压制较强的白噪声和局部相关噪声 .与FFT谱估计方法相比 ,该方法大大降低了对信号记录长度的要求 ,因而对大地电磁信号的处理有实际意义 .     

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。     

PROTEE A SELF-ADAPTING DECONVOLUTION AND DERINGING PROGRAM*

Deconvolution and deringing are well known subjects and it is not necessary to state again their objectives nor the basical methods used to reach them. Let us just remember that, generally, among many others, the two following assumptions are made for simplification purposes:

• —for deconvolution, it is assumed that the recorded seismic signal is constant, meaning that its shape is the same all along the time interval during which the trace is to be deconvolved;
• —for de-ringing, it is assumed that the ringing period is constant and that the intensity of the ringing phenomenon is independant of the time.

With these two assumptions, a single constant operator can be applied for deconvolving, deringing or both. In most cases, the time variations of the signal or of the ringing are small enough and the error resulting of the application of a constant operator is acceptable. It results into a slight increase of the noise level or into a small residual ringing in the processed trace. When this noise or the residual ringing are too important, the assumption of a constant signal and ringing period must be rejected. This is the case that is examined here according to the following steps:

• —short definition of the problem;
• —fast evaluation of some possible solutions;
• —the selected solution: resulting approximations and how to obviate them, computing method and a remark about the operators;
• —theoretical example: the efficiency of the process used is evaluated on data in which the results aimed at are known; the influence of the selection of numerical values to be assigned to the parameters is examined;
• —real cases: comparison of results obtained with the Protee process and with more conventional processes assuming a time invariance or including a weighted composition of several conventional processes each with a different operator.

     

PRINCIPLES AND APPLICATION OF MAXIMUM KURTOSIS PHASE ESTIMATION1

Methods of minimum entropy deconvolution (MED) try to take advantage of the non-Gaussian distribution of primary reflectivities in the design of deconvolution operators. Of these, Wiggins’(1978) original method performs as well as any in practice. However, we present examples to show that it does not provide a reliable means of deconvolving seismic data: its operators are not stable and, instead of whitening the data, they often band-pass filter it severely. The method could be more appropriately called maximum kurtosis deconvolution since the varimax norm it employs is really an estimate of kurtosis. Its poor performance is explained in terms of the relation between the kurtosis of a noisy band-limited seismic trace and the kurtosis of the underlying reflectivity sequence, and between the estimation errors in a maximum kurtosis operator and the data and design parameters. The scheme put forward by Fourmann in 1984, whereby the data are corrected by the phase rotation that maximizes their kurtosis, is a more practical method. This preserves the main attraction of MED, its potential for phase control, and leaves trace whitening and noise control to proven conventional methods. The correction can be determined without actually applying a whole series of phase shifts to the data. The application of the method is illustrated by means of practical and synthetic examples, and summarized by rules derived from theory. In particular, the signal-dominated bandwidth must exceed a threshold for the method to work at all and estimation of the phase correction requires a considerable amount of data. Kurtosis can estimate phase better than other norms that are misleadingly declared to be more efficient by theory based on full-band, noise-free data.     

自然伽马测井曲线的分析与反褶积校正

本文探讨系统响应函数对自然伽马测井曲线所造成的影响及反褶积校正的方法,从而使受围岩影响而降低了幅值的薄层曲线恢复原来的异常,改善了厚层曲线在界面处的陡度,以利于分层定厚、地层对比和进一步作各种定量的数字解释。     

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.     

CORRECTING FOR COLOURED PRIMARY REFLECTIVITY IN DECONVOLUTION1

Statistical deconvolution, as it is usually applied on a routine basis, designs an operator from the trace autocorrelation to compress the wavelet which is convolved with the reflectivity sequence. Under the assumption of a white reflectivity sequence (and a minimum-delay wavelet) this simple approach is valid. However, if the reflectivity is distinctly non-white, then the deconvolution will confuse the contributions to the trace spectral shape of the wavelet and reflectivity. Given logs from a nearby well, a simple two-parameter model may be used to describe the power spectral shape of the reflection coefficients derived from the broadband synthetic. This modelling is attractive in that structure in the smoothed spectrum which is consistent with random effects is not built into the model. The two parameters are used to compute simple inverse- and forward-correcting filters, which can be applied before and after the design and implementation of the standard predictive deconvolution operators. For whitening deconvolution, application of the inverse filter prior to deconvolution is unnecessary, provided the minimum-delay version of the forward filter is used. Application of the technique to seismic data shows the correction procedure to be fast and cheap and case histories display subtle, but important, differences between the conventionally deconvolved sections and those produced by incorporating the correction procedure into the processing sequence. It is concluded that, even with a moderate amount of non-whiteness, the corrected section can show appreciably better resolution than the conventionally processed section.     

自适应的最小平方反褶积及在地震勘探中的应用

本文给出了一种自适应的最小平方反褶积方法（ALSD法）,其基本原理是用滤波器的输出来修正愿望输出,从而使结果得到改善。从迭代格式来讲,它是极小熵反褶积方法的推广。本文还从理论上讨论了自适应函数的选取方法。通过人工模拟地震记录数据及真实地震剖面计算的结果,显示出ALSD法是一种计算量省且效果好的方法。对小相位子波,滤波效果与最小平方预测反褶积相当;对混合相位子波,仍具有近于小相位时的效果。     

小波变换及其应用

小波变换是近年来发展起来的一门新的数学分支.它适宜于分析研究信号的局部性质,对于图象数据压缩、奇性检测、非卷积型算子的化简及取样插值定理等方面都有着重要的应用,在地震勘探、大气湍流、语声合成、图象处理等许多领域中都有着广泛应用的前景.本文介绍了小波变换及其正交基的基本概念,并对它的一些重要应用作了概括的介绍.     

自适应的最小平方反褶积及在地震勘探中的应用

本文给出了一种自适应的最小平方反褶积方法（ALSD法）,其基本原理是用滤波器的输出来修正愿望输出,从而使结果得到改善。从迭代格式来讲,它是极小熵反褶积方法的推广。本文还从理论上讨论了自适应函数的选取方法。通过人工模拟地震记录数据及真实地震剖面计算的结果,显示出ALSD法是一种计算量省且效果好的方法。对小相位子波,滤波效果与最小平方预测反褶积相当;对混合相位子波,仍具有近于小相位时的效果。