ON THE TIME-VARYING WIENER FILTER*

For a new approach to designing the time-varying Wiener filter, the input is first divided into sections and then the time-varying filter is determined from the entire input and the desired output. The technique differs from the existing one in which the time-invariant filter is determined from each section. Hence, the main difference, between the proposed and the existing technique lies in the arrangement of input data. The proposed technique requires fewer computational operations and performs better than the time-invariant Wiener filter, as illustrated by numerical examples.     

Receiver Function Estimated by Wiener Filtering

Wiener filtering is used to estimate receiver function in a time-domain. With the vertical component of 3-component teleseismic P waveform as the input of a Wiener filter, receiver function as the filter response, and radial and tangential components as the expected output, receiver function is estimated by minimizing the error between expected and actual outputs. Receiver function can be obtained by solving the Toeplitz equation using the Leviuson algorithm. The non-singularity of the Toeplitz equation ensures the stability of Wiener Deconvolution. Both synthetic and observational seismogram checks show that Wiener Deconvolution is an effective time-domain method to estimate receiver function from teleseismic P waveform.     

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.     

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.     

用Wiener滤波方法提取台站接收函数

本文提出了一种在时间域用Wiener滤波方法提取台站接收函数的方法，用远震P波波形的垂直分量为输入，接收函数作为滤波因子，远震P波波形的径向和切向分量作为期望输出，通过期望输出与实际输出的均方误差达极小，来提取接收函数。接收函数的计算可归结为Toeplitz方程的求解，可以采用Levinson递推算法。Toeplitz方程的非奇异性保证了Wiener滤波反褶积方法的稳定性。合成地震图与观测地震图的检验表明，用Wiener滤波方法测定台站接收函数是一种有效的时间域反褶积方法。     

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.     

IDEAL PERFORMANCE CRITERIA FOR DECONVOLUTION OPERATORS*

A technique to evaluate an ideal performance of a deconvolution operator has been obtained by dividing the input trace into a number of sections. The error energy is seen to decrease with an increase in the number of sections. Numerical examples show that the error energy becomes zero following a relation between the number of sections, the length of input and the length of the filter.     

基于隐马尔可夫模型平滑估计的随机噪声压制方法

以地震勘探记录去噪为目标,本文提出了一种隐马尔可夫模型平滑估计方法.它是在基本隐马尔可夫模型滤波基础之上,运用信号检测环节将带噪信号段和无信号段加以区分,构建带噪地震记录的状态转移模型,在贝叶斯框架下,利用平滑密度函数进行状态估计,从而达到压制噪声的目的.数值模拟表明,无论对信噪比还是均方误差,隐马尔可夫模型平滑估计处理后的重构信号优于常规的维纳滤波所恢复信号.我们可以期待这种方法会成为实际地震记录噪声压制的有效手段.     

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.     

黄海地震前后的地磁日变化

本文采用多道滤波的方法,对江苏省内地磁台网记录到的地磁垂直分量日变化进行分析,结果表明,黄海地震发生前,特别是地震前三天,距震中最近的海安台原始数据和滤波器实际输出之间的日标准偏差,明显高于平常水平,发震后又恢复到正常水平.      

分离重磁区域场与局部场的维纳滤波器

本文从最佳线性滤波理论出发,对目前重磁资料数据处理中分离区域场与局部场的两种滤波器--匹配滤波和维纳滤波的频率响应特性作了分析比较,指出了匹配滤波只是一般维纳滤波的一个特例。将该两种滤波器与一般情况的维纳滤波器的误差作了对比,并通过简单的理论试例,说明它们的局限性和应用范围。     

Wiener optimal filtering of GRACE data

We present a spatial averaging method for Gravity Recovery and Climate Experiment (GRACE) gravity-field solutions based on the Wiener optimal filtering. The optimal filter is designed from the least-square minimization of the difference between the desired and filtered signals. It requires information about the power spectra of the desired gravitational signal and the contaminating noise, which is inferred from the average GRACE degree-power spectrum. We show that the signal decreases with increasing spherical harmonic degree j with approximately j^−b, where b = 1.5 for GRACE data investigations. This is termed the Second Kaula rule of thumb for temporal variations of the Earth’s gravity field. The degree power of the noise increases, in the logarithmic scale, linearly with increasing j. The Wiener optimal filter obtained for the signal model with b = 1.5 closely corresponds to a Gaussian filter with a spatial half width of 4° (∼440 km). We find that the filtered GRACE gravity signal is relatively insensitive to the exponent b of the signal model, which indicates the robustness of Wiener optimal filtering. This is demonstrated using the GFZ-GRACE gravity-field solution for April 2004.     

Suspended sediment balance for the mainstem of Changjiang (Yangtze River) in the period 1964–1985

Suspended sediment dynamics during the period 1964–1985 are examined along the mainstem of Changjiang (Yangtze River). The period represents a basin condition prior to major changes in land management policy and dam building on the river's mainstem. The downstream sediment dynamics reflect basin geology and topography and channel morphology. Sediment exchange within the mainstem was calculated by the development of reach sediment balances that reveal complex temporal and spatial patterns. There is relatively little sediment exchange in the upper, bedrock‐controlled reaches, with systematic increases in the downstream alluvial reaches. Degrading, transfer, and aggrading reaches were identified. Relations between input and output in all reaches were significant but no relation was found between sediment exchange and input/output. Comparison between ‘short‐term’ (22 years) and ‘long‐term’ (52 years) records demonstrates the importance of the record length in studying the suspended sediment dynamics in a large fluvial system. The longer record yielded better correlation and different trends than the shorter record. Sediment transfer (output/input ratio) changes downstream: the dominance of the upstream contributing area in sustaining the appearance of net degradation through most of the river system highlights the importance of reach length on characterisation of suspended sediment dynamics in large fluvial systems. Copyright © 2011 John Wiley & Sons, Ltd.     

位场向下延拓的改进迭代维纳滤波法

根据维纳滤波理论导出的位场向下延拓滤波器为最佳下延滤波器,但因其实现需要已知待求位场和噪声的功率谱而在实际应用中受到限制.针对该问题,本文首先提出一种基于位场径向平均功率谱的位场噪声水平估计方法,进而利用偏差准则求取正则化参数,实现位场正则化向下延拓;然后将位场正则化下延结果的功率谱作为待求位场功率谱的估计初值,采用带修正项的迭代维纳滤波方法来更新对待求位场功率谱的估计,最后提出本文的位场向下延拓改进迭代维纳滤波方法.基于理论重力模型数据及航磁实测数据进行了向下延拓对比试验,结果表明,改进迭代法具有较好的收敛性,且下延精度优于Tikhonov正则化法和递增型维纳滤波法.     

Study on eliminating noises in short-period data of geomagnetic intensity by using numerical multichannel predictive filtering

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Applications of Savitzky-Golay Filter for Seismic Random Noise Reduction

This article utilizes Savitzky–Golay (SG) filter to eliminate seismic random noise. This is a novel method for seismic random noise reduction in which SG filter adopts piecewise weighted polynomial via leastsquares estimation. Therefore, effective smoothing is achieved in extracting the original signal from noise environment while retaining the shape of the signal as close as possible to the original one. Although there are lots of classical methods such as Wiener filtering and wavelet denoising applied to eliminate seismic random noise, the SG filter outperforms them in approximating the true signal. SG filter will obtain a good tradeoff in waveform smoothing and valid signal preservation under suitable conditions. These are the appropriate window size and the polynomial degree. Through examples from synthetic seismic signals and field seismic data, we demonstrate the good performance of SG filter by comparing it with the Wiener filtering and wavelet denoising methods.     

TRANSFORMATION OF RESISTIVITY SOUNDING MEASUREMENTS OBTAINED IN ONE ELECTRODE CONFIGURATION TO ANOTHER CONFIGURATION BY MEANS OF DIGITAL LINEAR FILTERING*

The technique of digital linear filtering is used for transformation of apparent resistivity data from one electrode configuration into another. Usually filter spectra are determined via the discrete Fourier transforms of input and output functions: the filter characteristic is the quotient of the spectra of the output function and input function. In this paper, the transformation of the apparent resistivities is presented for four electrode configurations (Wenner, the two-electrode, Schlumberger, and dipole configurations). In our method, there is no need to use the discrete Fourier transform of the input and output functions in order to determine the filter spectrum for converting apparent resistivity in one electrode configuration to any other configuration. Sine responses for determination of the derivative of apparent resistivities are given in analytical form. If the filter spectrum for converting the apparent resistivity to the resistivity transform for one electrode configuration is known, the filter spectra for transforming the apparent resistivity to the resistivity transform for any electrode configurations can be calculated by using newly derived expressions.     

ADAPTIVE SIGNAL PROCESSING THROUGH STOCHASTIC APPROXIMATION*

One of the problems in signal processing is estimating the impulse response function of an unknown system. The well-known Wiener filter theory has been a powerful method in attacking this problem. In comparison, the use of stochastic approximation method as an adaptive signal processor is relatively new. This adaptive scheme can often be described by a recursive equation in which the estimated impulse response parameters are adjusted according to the gradient of a predetermined error function. This paper illustrates by means of simple examples the application of stochastic approximation method as a single-channel adaptive processor. Under some conditions the expected value of its weight sequence converges to the corresponding Wiener optimum filter when the least-mean-square error criterion is used.     

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.     

USE OF EXPECTED ERROR IN THE DESIGN OF LEAST-SQUARES OPTIMUM FILTERS*

The design of least-squares optimum filters is based upon minimizing a suitably defined error criterion. The expected value of this error is easily computable after the coefficients of the filter have been determined. When a particular filtering problem is specified, there are several parameters which are specifically not included in the optimization procedure. However, the magnitude of the expected error may be quite sensitive to these parameters. The examination of the relative values of the expected error for variations of these unspecified parameters may lead to a better definition of the filter problem. The parameters which are left unspecified by the general least-square filter definition include: 1. The addition of white noise to the signal autocorrelation to stabilize the filter behavior. 2. The specification of the shape of the desired output of the filter. 3. The specification of the lag between the desired output and the input. Examples are given showing the relationship between these parameters and the value of the expected error.