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.     

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.     

WAVELET SHAPING AND NOISE REDUCTION *

Approximate deconvolution by means of Wiener filters has become standard practice in seismic data-processing. It is well-known that addition of a certain percentage of noise energy to the autocorrelation of the signal wavelet leads to a filter that does not increase, or even reduces, the noise level on the seismogram. This noise addition will, in general, cause a minimum phase signal to become mixed phase. A technique is presented for the calculation of the optimum-lag shaping filter for a contaminated signal wavelet. The advantages of this method over the more conventional approach are that it needs less arithmetic operations and that it automatically gives the filter with the optimum combination of shaping performance and noise reduction.     

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.     

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.     

An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method

Wiener deconvolution is generally used to improve resolution of the seismic sections, although it has several important assumptions. I propose a new method named Gold deconvolution to obtain Earth’s sparse-spike reflectivity series. The method uses a recursive approach and requires the source waveform to be known, which is termed as Deterministic Gold deconvolution. In the case of the unknown wavelet, it is estimated from seismic data and the process is then termed as Statistical Gold deconvolution. In addition to the minimum phase, Gold deconvolution method also works for zero and mixed phase wavelets even on the noisy seismic data. The proposed method makes no assumption on the phase of the input wavelet, however, it needs the following assumptions to produce satisfactory results: (1) source waveform is known, if not, it should be estimated from seismic data, (2) source wavelet is stationary at least within a specified time gate, (3) input seismic data is zero offset and does not contain multiples, and (4) Earth consists of sparse spike reflectivity series. When applied in small time and space windows, the Gold deconvolution algorithm overcomes nonstationarity of the input wavelet. The algorithm uses several thousands of iterations, and generally a higher number of iterations produces better results. Since the wavelet is extracted from the seismogram itself for the Statistical Gold deconvolution case, the Gold deconvolution algorithm should be applied via constant-length windows both in time and space directions to overcome the nonstationarity of the wavelet in the input seismograms. The method can be extended into a two-dimensional case to obtain time-and-space dependent reflectivity, although I use one-dimensional Gold deconvolution in a trace-by-trace basis. The method is effective in areas where small-scale bright spots exist and it can also be used to locate thin reservoirs. Since the method produces better results for the Deterministic Gold deconvolution case, it can be used for the deterministic deconvolution of the data sets with known source waveforms such as land Vibroseis records and marine CHIRP systems.     

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.     

Sparsity‐enhanced wavelet deconvolution

We propose a three‐step bandwidth enhancing wavelet deconvolution process, combining linear inverse filtering and non‐linear reflectivity construction based on a sparseness assumption. The first step is conventional Wiener deconvolution. The second step consists of further spectral whitening outside the spectral bandwidth of the residual wavelet after Wiener deconvolution, i.e., the wavelet resulting from application of the Wiener deconvolution filter to the original wavelet, which usually is not a perfect spike due to band limitations of the original wavelet. We specifically propose a zero‐phase filtered sparse‐spike deconvolution as the second step to recover the reflectivity dominantly outside of the bandwidth of the residual wavelet after Wiener deconvolution. The filter applied to the sparse‐spike deconvolution result is proportional to the deviation of the amplitude spectrum of the residual wavelet from unity, i.e., it is of higher amplitude; the closer the amplitude spectrum of the residual wavelet is to zero, but of very low amplitude, the closer it is to unity. The third step consists of summation of the data from the two first steps, basically adding gradually the contribution from the sparse‐spike deconvolution result at those frequencies at which the residual wavelet after Wiener deconvolution has small amplitudes. We propose to call this technique “sparsity‐enhanced wavelet deconvolution”. We demonstrate the technique on real data with the deconvolution of the (normal‐incidence) source side sea‐surface ghost of marine towed streamer data. We also present the extension of the proposed technique to time‐varying wavelet deconvolution.     

OPTIMUM DIGITAL FILTERS FOR SIGNAL TO NOISE RATIO ENHANCEMENT*

One of the main objectives of seismic digital processing is the improvement of the signal-to-noise ratio in the recorded data. Wiener filters have been successfully applied in this capacity, but alternate filtering devices also merit our attention. Two such systems are the matched filter and the output energy filter. The former is better known to geophysicists as the crosscorrelation filter, and has seen widespread use for the processing of vibratory source data, while the latter is. much less familiar in seismic work. The matched filter is designed such that ideally the presence of a given signal is indicated by a single large deflection in the output. The output energy filter ideally reveals the presence of such a signal by producing a longer burst of energy in the time interval where the signal occurs. The received seismic trace is assumed to be an additive mixture of signal and noise. The shape of the signal must be known in order to design the matched filter, but only the autocorrelation function of this signal need be known to obtain the output energy filter. The derivation of these filters differs according to whether the noise is white or colored. In the former case the noise autocorrelation function consists of only a single spike at lag zero, while in the latter the shape of this noise autocorrelation function is arbitrary. We propose a novel version of the matched filter. Its memory function is given by the minimum-delay wavelet whose autocorrelation function is computed from selected gates of an actual seismic trace. For this reason explicit knowledge of the signal shape is not required for its design; nevertheless, its performance level is not much below that achievable with ordinary matched filters. We call this new filter the “mini-matched” filter. With digital computation in mind, the design criteria are formulated and optimized with time as a discrete variable. We illustrate the techniques with simple numerical examples, and discuss many of the interesting properties that these filters exhibit.     

非稳态地震稀疏约束反褶积研究（英文）

传统Robinson褶积模型主要受缚于三种不合理的假设,即白噪反射系数、最小相位地震子波与稳态假设,而现代反射系数反演方法（如稀疏约束反褶积等）均在前两个假设上寻求突破的同时却忽视了一个重要事实：实际地震信号具有典型的非稳态特征,这直接冲击着反射系数反演中地震子波不随时间变化的这一基础性假设。本文首先通过实际反射系数测试证实,非稳态效应造成重要信息无法得到有效展现,且对深层影响尤为严重。为校正非稳态影响,本文从描述非稳态方面具有普适性的非稳态褶积模型出发,借助对数域的衰减曲线指导检测非稳态影响并以此实现对非稳态均衡与校正。与常规不同,本文利用对数域Gabor反褶积仅移除非稳态影响,而将分离震源子波和反射系数的任务交给具有更符合实际条件的稀疏约束反褶积处理,因此结合两种反褶积技术即可有效解决非稳态特征影响,又能避免反射系数和地震子波理想化假设的不利影响。海上地震资料的应用实际表明,校正非稳态影响有助于恢复更丰富的反射系数信息,使得与地质沉积和构造相关的细节特征得到更加清晰的展现。     

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.     

Estimation of an optimal mixed-phase inverse filter

Inverse filtering is applied to seismic data to remove the effect of the wavelet and to obtain an estimate of the reflectivity series. In many cases the wavelet is not known, and only an estimate of its autocorrelation function (ACF) can be computed. Solving the Yule-Walker equations gives the inverse filter which corresponds to a minimum-delay wavelet. When the wavelet is mixed delay, this inverse filter produces a poor result.
By solving the extended Yule-Walker equations with the ACF of lag α on the main diagonal of the filter equations, it is possible to decompose the inverse filter into a finite-length filter convolved with an infinite-length filter. In a previous paper we proposed a mixed-delay inverse filter where the finite-length filter is maximum delay and the infinite-length filter is minimum delay.
Here, we refine this technique by analysing the roots of the Z -transform polynomial of the finite-length filter. By varying the number of roots which are placed inside the unit circle of the mixed-delay inverse filter, at most 2 ^α different filters are obtained. Applying each filter to a small data set (say a CMP gather), we choose the optimal filter to be the one for which the output has the largest L ^p -norm, with p =5. This is done for increasing values of α to obtain a final optimal filter. From this optimal filter it is easy to construct the inverse wavelet which may be used as an estimate of the seismic wavelet.
The new procedure has been applied to a synthetic wavelet and to an airgun wavelet to test its performance, and also to verify that the reconstructed wavelet is close to the original wavelet. The algorithm has also been applied to prestack marine seismic data, resulting in an improved stacked section compared with the one obtained by using a minimum-delay filter.     

SEQUENTIAL WIENER DECONVOLUTION TO IMPROVE SEISMIC RESOLUTION1

Two different techniques for performing time-variable Wiener deconvolution are compared using stacked seismic data. The conventional technique involves the empirical division of the data into a number of gates and the determination of time-invariant deconvolution filters for each gate. In the second technique, the deconvolution filter is recomputed after each time increment from a fixed-length data gate sliding along the trace. This scheme has the advantage that no a priori segmentation of the data is needed.     

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.     

A COMPARISON BETWEEN WIENER FILTERING,KALMAN FILTERING,AND DETERMINISTIC LEAST SQUARES ESTIMATION*

The least squares estimation procedures used in different disciplines can be classified in four categories:

• a. Wiener filtering,
• b. b. Autoregressive estimation,
• c. c. Kalman filtering,
• d. d. Recursive least squares estimation.

The recursive least squares estimator is the time average form of the Kalman filter. Likewise, the autoregressive estimator is the time average form of the Wiener filter. Both the Kalman and the Wiener filters use ensemble averages and can basically be constructed without having a particular measurement realisation available. It follows that seismic deconvolution should be based either on autoregression theory or on recursive least squares estimation theory rather than on the normally used Wiener or Kalman theory. A consequence of this change is the need to apply significance tests on the filter coefficients. The recursive least squares estimation theory is particularly suitable for solving the time variant deconvolution problem.     

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.     

基于广义高斯分布的地震盲反褶积方法研究

Bussgang算法是针对褶积盲源分离问题提出的,本文将其用于地震盲反褶积处理.由于广义高斯概率密度函数具有逼近任意概率密度函数的能力,从反射系数序列的统计特征出发,引入广义高斯分布来体现反射系数序列超高斯分布特征.依据反射系数序列的统计特征和Bussgang算法原理,建立以Kullback-Leibler距离为非高斯性度量的目标函数,并导出算法中涉及到的无记忆非线性函数,最终实现了地震盲反褶积.模型试算和实际资料处理结果表明,该方法能较好地适应非最小相位系统,能够同时实现地震子波和反射系数估计,有效地提高地震资料分辨率.     

SOME FADS AND FALLACIES IN SEISMIC DATA ANALYSIS1

In many branches of science, techniques designed for use in one context are used in other contexts, often with the belief that results which hold in the former will also hold or be relevant in the latter. Practical limitations are frequently overlooked or ignored. Three techniques used in seismic data analysis are often misused or their limitations poorly understood: (1) maximum entropy spectral analysis; (2) the role of goodness-of-fit and the real meaning of a wavelet estimate; (3) the use of multiple confidence intervals. It is demonstrated that in practice maximum entropy spectral estimates depend on a data-dependent smoothing window with unpleasant properties, which can result in poor spectral estimates for seismic data. Secondly, it is pointed out that the level of smoothing needed to give least errors in a wavelet estimate will not give rise to the best goodness-of-fit between the seismic trace and the wavelet estimate convolved with the broadband synthetic. Even if the smoothing used corresponds to near-minimum errors in the wavelet, the actual noise realization on the seismic data can cause important perturbations in residual wavelets following wavelet deconvolution. Finally the computation of multiple confidence intervals (e.g. at several spatial positions) is considered. Suppose a nominal, say 90%, confidence interval is calculated at each location. The confidence attaching to the simultaneous use of the confidence intervals is not then 90%. Methods do exist for working out suitable confidence levels. This is illustrated using porosity maps computed using conditional simulation.     

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.     

SIGNAL ADJUSTMENT OF VIBROSEIS AND IMPULSIVE SOURCE DATA*

In certain areas continuous Vibroseis profiling is not possible due to varying terrain conditions. Impulsive sources can be used to maintain continuous coverage. While this technique keeps the coverage at the desired level, for the processing of the actual data there is the problem of using different sources resulting in different source wavelets. In addition, the effect of the free surface is different for these two energy sources. The approach to these problems consists of a minimum-phase transformation of the two-sided Vibroseis data by removal of the anticipation component of the autocorrelation of the filtered sweep and a minimum-phase transformation of the impulsive source data by replacement of the recording filter operator with its minimum-phase correspondent. Therefore, after this transformation, both datasets show causal wavelets and a conventional deconvolution (spike or predictive) may be used. After stacking, a zero-phase transformation can be performed resulting in traces well suited for computing pseudo-acoustic impedance logs or for application of complex seismic trace analysis. The solution is also applicable to pure Vibroseis data, thereby eliminating the need for a special Vibroseis deconvolution. The processing steps described above are demonstrated on synthetic and actual data. The transformation operators used are two-sided recursive (TSR) shaping filters. After application of the above adjustment procedure, remaining signal distortions can be removed by modifying only the phase spectrum or both the amplitude and phase spectra. It can be shown that an arbitrary distortion defined in the frequency domain, i.e., a distortion of the amplitude and phase spectrum, is noticeable in the time section as a two-sided signal.