Klauder wavelet removal before vibroseis deconvolution

The spiking deconvolution of a field seismic trace requires that the seismic wavelet on the trace be minimum phase. On a dynamite trace, the component wavelets due to the effects of recording instruments, coupling, attenuation, ghosts, reverberations and other types of multiple reflection are minimum phase. The seismic wavelet is the convolution of the component wavelets. As a result, the seismic wavelet on a dynamite trace is minimum phase and thus can be removed by spiking deconvolution. However, on a correlated vibroseis trace, the seismic wavelet is the convolution of the zero-phase Klauder wavelet with the component minimum-phase wavelets. Thus the seismic wavelet occurring on a correlated vibroseis trace does not meet the minimum-phase requirement necessary for spiking deconvolution, and the final result of deconvolution is less than optimal. Over the years, this problem has been investigated and various methods of correction have been introduced. In essence, the existing methods of vibroseis deconvolution make use of a correction that converts (on the correlated trace) the Klauder wavelet into its minimum-phase counterpart. The seismic wavelet, which is the convolution of the minimum-phase counterpart with the component minimum-phase wavelets, is then removed by spiking deconvolution. This means that spiking deconvolution removes both the constructed minimum-phase Klauder counterpart and the component minimum-phase wavelets. Here, a new method is proposed: instead of being converted to minimum phase, the Klauder wavelet is removed directly. The spiking deconvolution can then proceed unimpeded as in the case of a dynamite record. These results also hold for gap predictive deconvolution because gap deconvolution is a special case of spiking deconvolution in which the deconvolved trace is smoothed by the front part of the minimum-phase wavelet that was removed.     

Bayesian frequency-domain blind deconvolution of ground-penetrating radar data

Enhancing the resolution and accuracy of surface ground-penetrating radar (GPR) reflection data by inverse filtering to recover a zero-phased band-limited reflectivity image requires a deconvolution technique that takes the mixed-phase character of the embedded wavelet into account. In contrast, standard stochastic deconvolution techniques assume that the wavelet is minimum phase and, hence, often meet with limited success when applied to GPR data. We present a new general-purpose blind deconvolution algorithm for mixed-phase wavelet estimation and deconvolution that (1) uses the parametrization of a mixed-phase wavelet as the convolution of the wavelet's minimum-phase equivalent with a dispersive all-pass filter, (2) includes prior information about the wavelet to be estimated in a Bayesian framework, and (3) relies on the assumption of a sparse reflectivity. Solving the normal equations using the data autocorrelation function provides an inverse filter that optimally removes the minimum-phase equivalent of the wavelet from the data, which leaves traces with a balanced amplitude spectrum but distorted phase. To compensate for the remaining phase errors, we invert in the frequency domain for an all-pass filter thereby taking advantage of the fact that the action of the all-pass filter is exclusively contained in its phase spectrum. A key element of our algorithm and a novelty in blind deconvolution is the inclusion of prior information that allows resolving ambiguities in polarity and timing that cannot be resolved using the sparseness measure alone. We employ a global inversion approach for non-linear optimization to find the all-pass filter phase values for each signal frequency. We tested the robustness and reliability of our algorithm on synthetic data with different wavelets, 1-D reflectivity models of different complexity, varying levels of added noise, and different types of prior information. When applied to realistic synthetic 2-D data and 2-D field data, we obtain images with increased temporal resolution compared to the results of standard processing.     

Nonstationary deconvolution using maximum kurtosis optimization

Deconvolution is an essential step for high-resolution imaging in seismic data processing. The frequency and phase of the seismic wavelet change through time during wave propagation as a consequence of seismic absorption. Therefore, wavelet estimation is the most vital step of deconvolution, which plays the main role in seismic processing and inversion. Gabor deconvolution is an effective method to eliminate attenuation effects. Since Gabor transform does not prepare the information about the phase, minimum-phase assumption is usually supposed to estimate the phase of the wavelet. This manner does not return the optimum response where the source wavelet would be dominantly a mixed phase. We used the kurtosis maximization algorithm to estimate the phase of the wavelet. First, we removed the attenuation effect in the Gabor domain and computed the amplitude spectrum of the source wavelet; then, we rotated the seismic trace with a constant phase to reach the maximum kurtosis. This procedure was repeated in moving windows to obtain the time-varying phase changes. After that, the propagating wavelet was generated to solve the inversion problem of the convolutional model. We showed that the assumption of minimum phase does not reflect a suitable response in the case of mixed-phase wavelets. Application of this algorithm on synthetic and real data shows that subtle reflectivity information could be recovered and vertical seismic resolution is significantly improved.     

基于物理小波的频谱分解方法及应用研究

在地震资料频谱分解中,采用匹配地震子波的物理小波,依据地震信号的特征,用振幅、能量衰减率、能量延迟时间及地震子波的中心频率等四类参数构造基本小波,把地震信号分解在小波域,高频分量能够得到精细的刻画.本文以物理小波变换为工具, 给出了该变换中的核函数的选择方法,进而提出了基于物理小波变换的频谱成像方法.我们将此方法用于海上某油田河流相储层的描述,并与常规软件中的小波变换频谱成像结果进行了对比, 结果表明,本文提出的方法更能精细地刻画地质事件.     

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.     

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.     

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.     

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.     

基于带状混合矩阵ICA实现地震盲反褶积

基于对地震反褶积本质上是一个盲过程的认识,引入高阶统计学盲源分离技术——独立分量分析（ICA）实现地震盲反褶积.在无噪声假设条件下,利用地震记录时间延迟矩阵和地震子波带状褶积矩阵,将地震褶积模型转化为一般线性混合ICA模型,采用FastICA算法,将带状性质作为先验信息,实现所谓带状ICA算法（B\|ICA）,得到个数与子波算子长度相等的多个估计反射系数序列和估计子波序列,最后利用褶积模型提供的附加信息从中优选出最佳的反射系数序列及相应的地震子波.模型数据和实际二维地震道数值算例表明：对于统计性反褶积,在不对反射系数作高斯白噪假设,不对子波作最小相位假设的所谓“全盲”条件下,基于ICA方法（反射系数非高斯分布,地震子波非最小相位）可以较好解决地震盲反褶积问题,是基于二阶统计特性的地震信号统计性反褶积方法的提升,具有可行性和应用前景.     

Perfect zerophase sections,fact or fiction?

The desired result of an optimum seismic data processing sequence, is a broad band zerophase section, i.e. a bandpassed version of the actual reflectivity function. However, a lot of socalled zerophase-sections still carry a significant phase-error, which is due to unrealistic assumptions in the processing stream in terms of the design of standard processes as for example deconvolution. The two major issues here are the color of the reflectivity series and the misuse of prewhitening. If not properly handled they lead to a phase- and amplitude spectrum bias in the final section, preventing it from being zerophase. Whereas the reflectivity bias leads to a phase error of 50 to 90 deg, the prewhitening bias results in a phase error, which is directly proportional to the logarithm of the actual prewhitening factor.Therefore, if the spike deconvolution process is applied in a time-variant manner, as a consequence a time-variant and usually frequency dependent phase error is introduced! In this article we have made an effort to include sufficient detail to facilitate a clear understanding of the problems involved.The standard processing flow should have a minimum-delay transform and spike deconvolution prestack, followed by a zerophase transform poststack, where the residual wavelet is assumed to be minimum phase.     

Ricker‐compliant deconvolution

Ricker‐compliant deconvolution spikes at the center lobe of the Ricker wavelet. It enables deconvolution to preserve and enhance seismogram polarities. Expressing the phase spectrum as a function of lag, it works by suppressing the phase at small lags. A by‐product of this decon is a pseudo‐unitary (very clean) debubble filter where bubbles are lifted off the data while onset waveforms (usually Ricker) are untouched.     

基于保角变换的双谱域相位谱估计方法及其在地震子波估计中的应用(英文)

地震子波估计是地震资料处理与解释中的重要环节,它的准确与否直接关系到反褶积及反演等结果的好坏。高阶谱(双谱和三谱)地震子波估计方法是一类重要的、新兴的子波估计方法,然而基于高阶谱的地震子波估计往往因为高阶相位谱卷绕的原因,导致子波相位谱求解产生偏差,进而影响了混合相位子波估计的效果。针对这一问题,本文在双谱域提出了一种基于保角变换的相位谱求解方法。通过缩小傅里叶相位谱的取值范围,有效避免了双谱相位发生卷绕的情况,从而消除了原相位谱估计中双谱相位卷绕的影响。该方法与最小二乘法相位谱估计相结合,构成了基于保角变换的最小二乘地震子波相位谱估计方法,并与最小二乘地震子波振幅谱估计方法一起,应用到了地震资料混合相位子波估计中。理论模型和实际资料验证了该方法的有效性。同时本文将双谱域地震子波相位谱估计中保角变换的思想推广到三谱域地震子波相位谱估计中。     

基于高阶统计的非最小相位地震子波恢复

利用高阶统计包含信号的相位信息特性,并基于信号的四阶累积量及其四阶谱,提出一种地震信号的非最小相位子波的估计方法.在任意高斯噪声环境下,对地震子波进行最小相位和最大相位谱分解,两部分信息完全可以从四阶谱中恢复.计算机数值模拟实验证实了方法的有效性.     

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.     

反卷积在人工震源数据处理中的应用

震源到接收台站之间的地层响应函数能够反映地下介质信息。对地震波传播过程中的卷积模型进行推导:记录信号是众多震源子波经过时移加权叠加的结果;通过反卷积方法可去除震源子波信息,提取震源到接收台站之间的地层响应函数;地层响应函数中第一个突跳值对应的时间即为P波走时。在河北赤城—张北地区进行人工震源实验,通过反卷积计算得到该地区地层响应函数剖面图,得出P波波速约6 km/s。利用人工震源系统还可以对地下介质波速变化进行长期动态监测,对地震预测具有一定意义。     

Non-minimum phase wavelet estimation by non-linear optimization of all-pass operators

Convolution of a minimum‐phase wavelet with an all‐pass wavelet provides a means of varying the phase of the minimum‐phase wavelet without affecting its amplitude spectrum. This observation leads to a parametrization of a mixed‐phase wavelet being obtained in terms of a minimum‐phase wavelet and an all‐pass operator. The Wiener–Levinson algorithm allows the minimum‐phase wavelet to be estimated from the data. It is known that the fourth‐order cumulant preserves the phase information of the wavelet, provided that the underlying reflectivity sequence is a non‐Gaussian, independent and identically distributed process. This property is used to estimate the all‐pass operator from the data that have been whitened by the deconvolution of the estimated minimum‐phase wavelet. Wavelet estimation based on a cumulant‐matching technique is dependent on the bandwidth‐to‐central‐frequency ratio of the data. For the cumulants to be sensitive to the phase signatures, it is imperative that the ratio of bandwidth to central frequency is at least greater than one, and preferably close to two. Pre‐whitening of the data with the estimated minimum‐phase wavelet helps to increase the bandwidth, resulting in a more favourable bandwidth‐to‐central‐frequency ratio. The proposed technique makes use of this property to estimate the all‐pass wavelet from the prewhitened data. The paper also compares the results obtained from both prewhitened and non‐whitened data. The results show that the use of prewhitened data leads to a significant improvement in the estimation of the mixed‐phase wavelet when the data are severely band‐limited. The proposed algorithm was further tested on real data, followed by a test involving the introduction of a 90°‐phase‐rotated wavelet and then recovery of the wavelet. The test was successful.     

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.     

Review of vibroseis data acquisition and processing for better amplitudes: adjusting the sweep and deconvolving for the time-derivative of the true groundforce

The goal of vibroseis data acquisition and processing is to produce seismic reflection data with a known spatially-invariant wavelet, preferably zero phase, such that any variations in the data can be attributed to variations in geology. In current practice the vibrator control system is designed to make the estimated groundforce equal to the sweep and the resulting particle velocity data are cross-correlated with the sweep. Since the downgoing far-field particle velocity signal is proportional to the time-derivative of the groundforce, it makes more sense to cross-correlate with the time-derivative of the sweep. It also follows that the ideal amplitude spectrum of the groundforce should be inversely proportional to frequency. Because of non-linearities in the vibrator, bending of the baseplate and variable coupling of the baseplate to the ground, the true groundforce is not equal to the pre-determined sweep and varies not only from vibrator point to vibrator point but also from sweep to sweep at each vibrator point. To achieve the goal of a spatially-invariant wavelet, these variations should be removed by signature deconvolution, converting the wavelet to a much shorter zero-phase wavelet but with the same bandwidth and signal-to-noise ratio as the original data. This can be done only if the true groundforce is known. The principle may be applied to an array of vibrators by employing pulse coding techniques and separating responses to individual vibrators in the frequency domain. Various approaches to improve the estimate of the true groundforce have been proposed or are under development; current methods are at best approximate.     

VIBROSEIS DECONVOLUTION*

Vibroseis deconvolution can be performed either before or after correlation. As regards the deconvolution before correlation, the Vibroseis deconvolution operator can be described as convolution of a spike deconvolution operator with a minimum-phase filter operator with bandpass properties. As regards the deconvolution after correlation, the deconvolution operator can be shown to be the convolution of three operators: spike deconvolution operator and two-fold convolution with a minimum phase operator. Time-varying Vibroseis deconvolution can particularly well be described and performed after correlation.