ω-k FILTER DESIGN*

Two-dimensional band-pass filters can be constructed by a simple extension of the theory of one-dimensional band-pass filters. Similarly to the one-dimensional analogue the shape of the two-dimensional filter is important in determining its effectiveness. The band-pass filter formulation can be further refined so that the filter will concentrate its rejection energies in certain areas of the ω, k plane. Such band-pass, band-reject filters are found by solving a set of simultaneous equations.     

A FREQUENCY DOMAIN MAPPING APPROXIMATION OF MOVEOUT FILTERS WITH APPLICATIONS*

Wavenumber aliasing is the main limitation of conventional optimum least-squares linear moveout filters: it prevents adequate reject domain weighting for efficient coherent noise rejection. A general frequency domain multichannel filter design technique based on a one-to-one mapping method between two-dimensional (2D) space and one-dimensional (1D) space is presented. The 2D desired response is mapped to the 1D frequency axis after a suitable sorting of the coefficients. A min-max or Tchebycheff approximation to the desired response is obtained in the 1D frequency domain and mapped back to the 2D frequency domain. The algorithm is suitable for multiband 2D filter design. No aliasing damage is inherent in the linear moveout filters designed using this technique because the approximation is done in the frequency-wavenumber (f, k)-domain. Linear moveout filters designed by using the present coefficient mapping technique achieve better pass domain approximations than the corresponding conventional least-squares filters. Compatible reject domain approximations can be obtained from suitable mappings of the origin coefficient of the desired (f k)-response to the 1D frequency axis. The (fk)-responses of linear moveout filters designed by using the new technique show equi-ripple behavior. Synthetic and real data applications show that the present technique is superior to the optimum least-squares filters and straight stacking in recovering and enhancing the signal events with relatively high residual statics. Their outputs also show higher resolution than those of the optimum least-squares filters.     

数字滤波器在高台地震台地震记录日常分析中的应用

简要介绍了滤波器的一般原理与Butterworth低通滤波器、高通滤波器和带通滤波器在高台地震台日常工作中的应用。     

A FILTER,DELAY AND SPREAD TECHNIQUE FOR 3D DMO1

The calculation of dip moveout involves spreading the amplitudes of each input trace along the source-receiver axis followed by stacking the results into a 3D zero-offset data cube. The offset-traveltime (x–t) domain integral implementation of the DMO operator is very efficient in terms of computation time but suffers from operator aliasing. The log-stretch approach, using a logarithmic transformation of the time axis to force the DMO operator to be time invariant, can avoid operator aliasing by direct implementation in the frequency-wavenumber (f–k) domain. An alternative technique for log-stretch DMO corrections using the anti-aliasing filters of the f–k approach in the x-log t domain will be presented. Conventionally, the 2D filter representing the DMO operator is designed and applied in the f–k domain. The new technique uses a 2D convolution filter acting in single input/multiple output trace mode. Each single input trace is passed through several 1D filters to create the overall DMO response of that trace. The resulting traces can be stacked directly in the 3D data cube. The single trace filters are the result of a filter design technique reducing the 2D problem to several ID problems. These filters can be decomposed into a pure time-delay and a low-pass filter, representing the kinematic and dynamic behaviour of the DMO operator. The low-pass filters avoid any incidental operator aliasing. Different types of low-pass filters can be used to achieve different amplitude-versus-offset characteristics of the DMO operator.     

Body-wave dispersion: Measurement and interpretation

Generalizing previous studies on short-period data, it is shown that body-wave dispersion can be measured from broad-band records of earthquakes of moderate magnitude. The method is based on the direct measurement of the arrival time of the frequency components of a seismic wave, and the arrival time is defined by its expectation value. The frequency components of the signal are obtained through a narrow band-pass filtering process. Previous to any interpretation, a correction of the arrival time for instrument response and group delay of the filter is needed. In the first step, body-wave dispersion is related to an absorption band to account for intrinsic attenuation, and thereafter we generalize this interpretation by considering a cascade of filters to account for medium parameters (attenuation and a layered crust) and source parameters (source time function and finiteness of fault). An inversion scheme to obtain the filter parameters can be devised by following, in a formal way, the same procedure as for the case of surface wave dispersion.     

PLANE-WAVE CONSTRAINTS IN 2D FILTER DESIGN1

A method is presented for developing and/or evaluating 2D filters applied to seismic data. The approach used is to express linear 2D filtering operations in the space-frequency (x–ω) domain. Correction filters are then determined using plane-wave constraints. For example, requiring a vertically propagating plane wave to be unaffected by migration necessitates application of a half-derivative correction in Kirchhoff migration. The same approach allows determination of the region of time-offset space where half-derivative corrections are correct in x–t domain dip moveout. Finally, an x–ω domain dip filter is derived using the constraint that a plane wave be attenuated as its dip increases. This filter has the advantage that it is significantly faster than f–k domain dip filtering and can be used on irregularly spaced data. This latter property also allows the filter to be used for interpolation of irregular data onto a regular grid.     

极低频电磁观测数据噪声处理方法研究

“十五”期间,在全国各地震台站共安装了约12套ELF极低频电磁观测仪器,从记录到的观测资料来看,所有台站均不同程度地受到工频50 Hz及其谐波的干扰,同时还有一些不必要的信息存在,因此,对ELF观测数据进行滤波处理显得尤其重要。该文用MATLAB编程软件设计了FIR数字滤波器,运用理论数据和ELF实测数据对滤波器的性能进行了验证,分析了ELF时间序列经过带通滤波和陷波处理后的效果,发现其对台站ELF观测数据的处理有一定的应用价值。     

基于递归高通滤波的经验模态分解及其在地震信号分析中的应用

将地震信号分解成包含频谱互不重叠的单主周期的分量有利于地震信号的分析.分析了经验模态分解(EMD)中模态混叠的内在原因和已有的解决方法,梳理了解决模态混叠的思路框架,进而提出了一种新的基于输入递归高通滤波的EMD算法.首先用递归高通滤波器将信号预分解成频率由高到低的多个分量,实现信号的等价带通滤波,再用EMD对各带通分量按频率高低逐级递归筛分,获得完备的经验模态分量.通过合成信号和地震信号的仿真实验表明,该算法较好地克服了模态混叠,获得了频谱互不重叠的单主周期分量,并成功用于震相分离和分析,为地震信号分析提供了一种新思路.     

基于RBF神经网络的地震与爆破识别技术

利用天然地震震源和人工爆破震源之间信号能量分布的差异,结合RBF神经网络技术,对2类事件进行分类,具体步骤如下:使用8个带通滤波器对事件波形进行滤波,并划分为4个波形段:P波、P波尾波、S波和S波尾波,分别计算每个滤波器信道和波形段的能量特征值,以所得32个特征参数作为输入向量,利用RBF神经网络,对地震和爆破事件进行分类识别。结果表明,基于RBF神经网络的地震事件识别方法,识别率为88.1%,具有较高的准确性,可作为地震与爆破事件识别的一个重要依据。     

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.     

Filter formulation and wavefield separation of cross-well seismic data

Multichannel filtering to obtain wavefield separation has been used in seismic processing for decades and has become an essential component in VSP and cross-well reflection imaging. The need for good multichannel wavefield separation filters is acute in borehole seismic imaging techniques such as VSP and cross-well reflection imaging, where strong interfering arrivals such as tube waves, shear conversions, multiples, direct arrivals and guided waves can overlap temporally with desired arrivals. We investigate the effects of preprocessing (alignment and equalization) on the quality of cross-well reflection imaging wavefield separation and we show that the choice of the multichannel filter and filter parameters is critical to the wavefield separation of cross-well data (median filters, f–k pie-slice filters, eigenvector filters). We show that spatial aliasing creates situations where the application of purely spatial filters (median filters) will create notches in the frequency spectrum of the desired reflection arrival. Eigenvector filters allow us to work past the limits of aliasing, but these kinds of filter are strongly dependent on the ratio of undesired to desired signal amplitude. On the basis of these observations, we developed a new type of multichannel filter that combined the best characteristics of spatial filters and eigenvector filters. We call this filter a ‘constrained eigenvector filter’. We use two real data sets of cross-well seismic experiments with small and large well spacing to evaluate the effects of these factors on the quality of cross-well wavefield separation. We apply median filters, f–k pie-slice filters and constrained eigenvector filters in multiple domains available for these data sets (common-source, common-receiver, common-offset and common-midpoint gathers). We show that the results of applying the constrained eigenvector filter to the entire cross-well data set are superior to both the spatial and standard eigenvector filter results.     

Frequency‐space wavefield extrapolation using infinite impulse response digital filters: is it feasible?

The purpose of this paper is to study the possibility of performing practically stable and efficient frequency‐space (f?x) wavefield extrapolation for the application of seismic imaging and datuming via infinite impulse response (IIR) filters. The model reduction control theory was adopted to design such IIR f?x extrapolation filters. The model reduction theory reduces the order of a given order system which, in this case, involves reducing a finite impulse response (FIR) f?x extrapolation filter system into an IIR f?x extrapolation filter system. This theory relies on decomposing the states of the given filter system into strong and weakly coupled sub‐systems, and then eliminating the weakly coupled states via singular value decomposition of the Hankel and the impulse response Gramian matrices. Simulation results indicate that IIR f?x filters can be obtained, which are stable from an IIR filter design point of view. Simulations also indicate that stable seismic impulse responses and synthetics can be obtained with a reduced system model order and, hence, less computational efforts with respect to the number of complex multiplications and additions per output sample. It is hoped that this study will open new possibilities for researchers to reconsider designing IIR f?x explicit depth extrapolation filters due to their expected computational savings and wavenumber response accuracy, when compared to the FIR f?x explicit depth extrapolation filters.     

Median filter behaviour with seismic data1

Median filters may be used with seismic data to attenuate coherent wavefields. An example is the attenuation of the downgoing wavefield in VSP data processing. The filter is applied across the traces in the ‘direction’ of the wavefield. The final result is given by subtracting the filtered version of the record from the original record. This method of median filtering may be called ‘median filtering operated in subtraction’. The method may be extended by automatically estimating the slowness of coherent wavefields on a record. The filter is then applied in a time- and-space varying manner across the record on the basis of the slowness values at each point on the record. Median filters are non-linear and hence their behaviour is more difficult to determine than linear filters. However, there are a number of methods that may be used to analyse median filter behaviour: (1) pseudo-transfer functions to specific time series; (2) the response of median filters to simple seismic models; and (3) the response of median filters to steps that simulate terminating wavefields, such as faults on stacked data. These simple methods provide an intuitive insight into the behaviour of these filters, as well as providing a semiquantitative measurement of performance. The performance degradation of median filters in the presence of trace-to-trace variations in amplitude is shown to be similar to that of linear filters. The performance of median filters (in terms of signal distortion) applied obliquely across a record may be improved by low-pass filtering (in the t-dimension). The response of median filters to steps is shown to be affected by background noise levels. The distortion of steps introduced by median filters approaches the distortion of steps introduced by the corresponding linear filter for high levels of noise.     

Bias aware Kalman filters: Comparison and improvements

This paper reviews two different approaches that have been proposed to tackle the problems of model bias with the Kalman filter: the use of a colored noise model and the implementation of a separate bias filter. Both filters are implemented with and without feedback of the bias into the model state. The colored noise filter formulation is extended to correct both time correlated and uncorrelated model error components. A more stable version of the separate filter without feedback is presented. The filters are implemented in an ensemble framework using Latin hypercube sampling. The techniques are illustrated on a simple one-dimensional groundwater problem. The results show that the presented filters outperform the standard Kalman filter and that the implementations with bias feedback work in more general conditions than the implementations without feedback.     

ZERO—PHASE FORWARD FILTERS FOR RESISTIVITY SOUNDING*

Forward filters to transform the apparent resistivity function over a layered half-space into the resistivity transform have been derived for a number of sample intervals. The filters have no apparent Gibbs' oscillations and hence require no phase shift. In addition, the end points of the filter were modified to compensate for truncation. The filters were tested on simulated ascending and descending two-layer cases. As expected, “dense” filters with sample spacing of In (10)/6 or smaller performed very well. However, even “sparse” filters with spacing of In (10)/2 and a total of nine coefficients have peak errors of less than 5% for p₁:p₂ ratios of 10^–6 to 10⁶. If a peak error of 5.5% is acceptable, then an even sparser filter with only seven coefficients at a spacing of 3 In (10)/5 may be used.     

中强地震前中小地震速度谱的变化特征

从Ｄｕｄａ和Ｎｏｒｔｍａｎｎ提出的谱震级定义出发,利用江苏地区遥测台网的５个台站记录到的５４次地震的数字化垂向位移波形记录,采用１３组０．５倍频程的滤波器,计算了各次地震的Ｐ波和及Ｓ波在没中心周期处的平均速度谱值,研究了地震的Ｐ波或Ｓ波速度谱的拐角周期及综合因子ｆ值的时间变化特征。结果显示：中强地震发生之前１年或更短的时间内所发生的中小地震,其速度谱综合因子ｆ值随时间的变化具有由高值向低值的变化趋     

An algorithm for interpolation in the pyramid domain‡

With the pyramid transform, 2D dip spectra can be characterized by 1D prediction‐error filters (pefs) and 3D dip spectra by 2D pefs. These filters, contrary to pefs estimated in the frequency‐space domain (ω, x) , are frequency independent. Therefore, one pef can be used to interpolate all frequencies. Similarly, one pef can be computed from all frequencies, thus yielding robust estimation of the filter in the presence of noise. This transform takes data from the frequency‐space domain (ω, x) to data in a frequency‐velocity domain (ω, u=ω·x) using a simple mapping procedure that leaves locations in the pyramid domain empty. Missing data in (ω, x) ‐space create even more empty bins in (ω, u) ‐space. We propose a multi‐stage least‐squares approach where both unknown pefs and missing data are estimated. This approach is tested on synthetic and field data examples where aliasing and irregular spacing are present.     

REALIZATION OF SHARP CUT-OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS

It was found in Part I of this paper that approximating the sharp cut-off frequency characteristic best in a mean square sense by an impulse response of finite length M produced a characteristic whose slope on a linear frequency scale was proportional to the length of impulse response, but whose maximum overshoot of ±9% was independent of this length (Gibbs' phenomenon). Weighting functions, based on frequency tapering or arbitrarily chosen, were used in Part II to modify the truncated impulse response of the sharp cut-off frequency characteristic, and thereby obtain a trade-off between the value of maximum overshoot and the sharpness of the resulting characteristic. These weighting functions, known as apodising functions, were dependent on the time-bandwidth product Mξ, where 2ξ, corresponded to the tapering range of frequencies. Part III now deals with digital filters where the number 2N–1 of coefficients is directly related to the finite length M of the continuous impulse response. The values of the filter coefficients are taken from the continuous impulse response at the sampling instants, and the resulting characteristic is approximately the same as that derived in Part II for the continuous finite length impulse response. Corresponding to known types of frequency tapering, we now specify a filter characteristic which is undefined in the tapering range, and determine the filter coefficients according to a mean square criterion over the rest of the frequency spectrum. The resulting characteristic is dependent on the time bandwidth product Mξ= (N–1/2)ξ up to a maximum value of 2, beyond which undesirable effects occur. This optimum partially specified characteristic is an improvement on the previous digital filters in terms of the trade-off ratio for values of maximum overshoot less than 1%. Similar to the previous optimum characteristic is the optimum partially specified weighted digital filter, where greater “emphasis is placed on reducing the value of maximum overshoot than of maximum undershoot”. Such characteristics are capable of providing better trade-off ratios than the other filters for maximum overshoots greater than 1/2%. However these filters have critical maximum numbers 2.N_C–1 of coefficients, beyond which the resulting characteristics have unsuitable shapes. This type of characteristic differs from the others in not being a biassed odd function about its cut-off frequency.     

数值模式中地形滤波处理及水平扩散对降雨预报的影响

为克服高分辨率模拟中,对于具有陡峭山峰及深谷的区域,存在不真实降雨场预报问题,本文引入数字滤波器及水平扩散方案分别对地形及计算噪音进行处理.滤波器由不同的一维高阶低通隐式正切滤波器耦合而成,能选择性地过滤由地形坡度所引起的不同尺度的噪音.水平扩散方案是将一个通量受地形限制的线性四阶单调水平扩散项加到预报方程,去控制由数值扩散、非线性不稳定及不连续物理过程等引起的小尺度噪音.试验结果表明：地形滤波处理及水平扩散方案能消除山区降雨预报量集中在山顶,而同时山谷和背风面又无雨的现象.因而,降雨分布更真实.     

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.