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.     

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.     

Recursive evaluation of interaction forces of unbounded soil in the time domain from dynamic-stiffness coefficients in the frequency domain

The interaction forces of the linear unbounded soil in a non-linear soil-structure-interaction analysis can be calculated recursively, starting directly from the dynamic-stiffness coefficients in the frequency domain. Two possibilities of choosing a recursive equation are discussed.

• (i) The recursive equation in the frequency domain. For each frequency, the interaction force at a specific time station is expressed as a function of the corresponding interaction force at the previous time station and of the displacements at the current time station and at the two most recent past time stations. This recursive evaluation of the convolution integral. which can be derived using the z-transformation, is rigorous. By using interpolation in the frequency domain, an approximate procedure results, which leads to a significant reduction in computational effort.
• (ii) The recursive equation in the time domain. By approximating the dynamic-stiffness coefficients as the ratios of two polynomials in frequency using a curve-fitting technique based on the least-squares method and by applying the partial-fraction expansion and using the z-transformation, the recursive coefficients can be determined explicitly. Alternatively, the ratio of two polynomials can also be transformed to an ordinary differential equation together with the initial conditions.

The recursive equations using interpolation in the frequency domain and based on a ratio of two polynomials lead to a reduction in the computational effort of one and up to three orders of magnitude, respectively.     

Accurate and efficient shot-gather dip moveout processing in the log-stretch domain1

An accurate analytical expression for shot-gather dip-moveout (DMO) in the timespace log-stretch domain has until now not been published. We present a simpler, alternative derivation of the exact DMO relationships of Black et al. which correctly take account of the repositioning of the midpoint. A new computationally efficient frequency-wavenumber (F-K) DMO operator for shot profiles is then derived, based on these DMO relationships in the time-space log-stretch domain. The newly derived DMO operator is, unlike most other log-stretch DMO operators) accurate for the full range of reflector dips. Along with other schemes which are performed in the log-stretch domain, it offers considerable time savings over conventional DMO processing. We have compared numerically the impulse response of the new operator with those of a number of other shot-gather DMO operators, and found it to be superior and well match to the theoretical elliptical DMO response.     

三维共偏移距叠前地震数据反假频射线束形成偏移成像

对稀疏/非规则采样或者低信噪比数据,射线束提取困难并伴随有假频产生,对叠加剖面和道集造成严重干扰.为了提升射线束偏移在稀疏和低信噪比地震数据采集中的成像效果,本文提出基于三角滤波的局部倾斜叠加波束形成偏移假频压制方法.射线束偏移首先将地震数据划分为超道集,经过部分NMO后转化为以射线束中心定义的共偏移距数据,倾斜叠加和反假频操作均在局部共中心点坐标上实现.时间域倾斜叠加是对地震数据的时移累加操作,三角低通滤波同样可以在时间域完成,在对地震数据进行因果和反因果积分后,亦为地震数据的时移累加.因此,三角低通滤波与倾斜叠加可在时间域结合同时完成,避免了频域滤波的正反傅里叶变换.本文在反假频公式中加入权重系数,用以对反假频的程度进行控制,达到分辨率和噪声压制的最佳折衷.以某海上三维实际数据为例,文中展示了反假频射线束形成对偏移叠加剖面和共成像点偏移距道集中的噪声进行了有效压制.     

Incremental time—space finite strip method for non-linear structural vibrations

Based on an incremental Hamilton's principle a versatile and systematic computer method for analysing non-linear structural vibrations is developed in this paper. The essence of the proposed method can be regarded as an incremental harmonic balance method associated with a finite strip procedure in the time-space domain. Only linearized equations in terms of frequency increment, amplitude increments, etc. have to be formulated and solved in each incremental step. This method is applicable to highly non-linear problems and may be generalized to related non-linear periodic structural motions such as dynamic stability, flutter and some motions of a rotating body, etc. To show the effectiveness and versatility of this method, a typical time-space finite strip for beam problems is worked out and examples for a wide variety of vibration problems including free and forced vibrations, super- and sub-harmonic resonances, and complicated phenomena such as internal resonance are computed. Comparisons with previous results are also made.     

The Wilson–Burg method of spectral factorization with application to helical filtering

Spectral factorization is a computational procedure for constructing minimum-phase (stable inverse) filters required for recursive inverse filtering. We present a novel method of spectral factorization. The method iteratively constructs an approximation of the minimum-phase filter with the given autocorrelation by repeated forward and inverse filtering and rearranging of the terms. This procedure is especially efficient in the multidimensional case, where the inverse recursive filtering is enabled by the helix transform. To exemplify a practical application of the proposed method, we consider the problem of smooth two-dimensional data regularization. Splines in tension are smooth interpolation surfaces whose behaviour in unconstrained regions is controlled by the tension parameter. We show that such surfaces can be efficiently constructed with recursive filter preconditioning and we introduce a family of corresponding two-dimensional minimum-phase filters. The filters are created by spectral factorization on a helix.     

Three-parameter Radon transform in layered transversely isotropic media

Radon transform is a powerful tool with many applications in different stages of seismic data processing, because of its capability to focus seismic events in the transform domain. Three-parameter Radon transform can optimally focus and separate different seismic events, if its basis functions accurately match the events. In anisotropic media, the conventional hyperbolic or shifted hyperbolic basis functions lose their accuracy and cannot preserve data fidelity, especially at large offsets. To address this issue, we propose an accurate traveltime approximation for transversely isotropic media with vertical symmetry axis, and derive two versions of Radon basis functions, time-variant and time-invariant. A time-variant basis function can be used in time domain Radon transform algorithms while a time-invariant version can be used in, generally more efficient, frequency domain algorithms. Comparing the time-variant and time-invariant Radon transform by the proposed basis functions, the time-invariant version can better focus different seismic events; it is also more accurate, especially in presence of vertical heterogeneity. However, the proposed time-invariant basis functions are suitable for a specific type of layered anisotropic media, known as factorized media. We test the proposed methods and illustrate successful applications of them for trace interpolation and coherent noise attenuation.     

Stability and identification for rational approximation of frequency response function of unbounded soil

Exact representation of unbounded soil contains the single output–single input relationship between force and displacement in the physical or transformed space. This relationship is a global convolution integral in the time domain. Rational approximation to its frequency response function (frequency‐domain convolution kernel) in the frequency domain, which is then realized into the time domain as a lumped‐parameter model or recursive formula, is an effective method to obtain the temporally local representation of unbounded soil. Stability and identification for the rational approximation are studied in this paper. A necessary and sufficient stability condition is presented based on the stability theory of linear system. A parameter identification method is further developed by directly solving a nonlinear least‐squares fitting problem using the hybrid genetic‐simplex optimization algorithm, in which the proposed stability condition as constraint is enforced by the penalty function method. The stability is thus guaranteed a priori. The infrequent and undesirable resonance phenomenon in stable system is also discussed. The proposed stability condition and identification method are verified by several dynamic soil–structure‐interaction examples. Copyright © 2009 John Wiley & Sons, Ltd.     

瞬变电磁虚拟波场二阶Born近似成像算法

依据瞬变电磁扩散场与虚拟波动场之间的数学表达式,通过波场反变换得到虚拟波场,进而借用地震勘探上成熟的成像方法定位地质界面的位置以及形态.本文首先将时间域波动方程转换为频率域波动方程,将总波动场分为背景场和散射场、总波速分为背景速度和扰动速度,然后运用Green定理得到了散射场关于地质界面速度扰动量的表达式,并使用Bor...     

Numerical viscoelastic modelling by the spectral Laguerre method

Seismic wave propagation in a viscoelastic media can be described by a system of integro-differential equations. The solution of such equations requires special methods when using finite-difference techniques in the time domain. In the frequency domain, the integral terms are represented by complex elastic parameters. This paper presents an efficient algorithm for viscoelastic modelling based on the integral Laguerre transform for the approximation of temporal derivatives and for the calculation of convolution integrals. For the calculation of spatial derivatives, it is possible to use various methods: finite-difference and finite-element techniques, spectral and pseudo-spectral methods. We then obtain a system of algebraic equations with a matrix independent of the parameter m , i.e. the degree of the Laguerre polynomials. In this case, only the right-hand side of the system has recurrent dependence on the parameter m , which is an analogue of the temporal frequency in the frequency domain. The obtained system with a large number of right-hand sides can be solved using fast methods, where the matrix is transformed only once, as opposed to the frequency-domain approach, when the matrix is transformed for each temporal frequency.     

Inversion-based data-driven time-space domain random noise attenuation method

Conventional time-space domain and frequency-space domain prediction filtering methods assume that seismic data consists of two parts, signal and random noise. That is, the so-called additive noise model. However, when estimating random noise, it is assumed that random noise can be predicted from the seismic data by convolving with a prediction error filter. That is, the source-noise model. Model inconsistencies, before and after denoising, compromise the noise attenuation and signal-preservation performances of prediction filtering methods. Therefore, this study presents an inversion-based time-space domain random noise attenuation method to overcome the model inconsistencies. In this method, a prediction error filter (PEF), is first estimated from seismic data; the filter characterizes the predictability of the seismic data and adaptively describes the seismic data’s space structure. After calculating PEF, it can be applied as a regularized constraint in the inversion process for seismic signal from noisy data. Unlike conventional random noise attenuation methods, the proposed method solves a seismic data inversion problem using regularization constraint; this overcomes the model inconsistency of the prediction filtering method. The proposed method was tested on both synthetic and real seismic data, and results from the prediction filtering method and the proposed method are compared. The testing demonstrated that the proposed method suppresses noise effectively and provides better signal-preservation performance.     

An improved numerical solution for the diffusion equation

This paper is concerned with the application of the two-dimensional Galerkin approximation to the modelling of the dynamic groundwater flow equation. Attention will be paid to a problem almost universally encountered in the modelling of these problems: the singularities caused by the presence of sources and/or sinks in the domain of interest. One method to improve the solution of the dynamic flow equation near boreholes, is to subtract the singular behaviour of flow with a time-dependent analytic solution for near-boreholes zones. By subtracting the singularities in this way, the resulting smooth problem may be solved by the Galerkin method and the result can be combined with the analytic solution. The result is an improved solution near boreholes without the need to use a refined mesh and thus a saving in computer storage and time.     

基于分数阶拉普拉斯算子解耦的黏声介质地震正演模拟与逆时偏移

时间域常Q黏声波方程,由于含分数阶时间导数项,数值求解需要大量内存,计算效率低,不利于地震偏移的实施.通过一系列近似,可将该方程简化为介质频散效应和衰减效应解耦的分数阶拉普拉斯算子黏声波方程,数值求解内存需求少,计算效率高.本文采用交错网格有限差分逼近时间导数,改进的伪谱法计算空间导数,PML吸收边界去除边界反射,对该方程进行数值离散和地震正演模拟,开展地震数据的黏声介质逆时偏移,实现波场逆时延拓过程中同时完成频散校正和衰减补偿.改善深层构造的成像精度,数值结果表明,基于分数阶拉普拉斯算子解耦的黏声介质地震正演模拟与逆时偏移可大幅度提高地震模拟计算效率,偏移剖面明显优于常规声波偏移剖面,极大改善深层构造的成像品质.     

单程波算子积分解的象征表示

单程波波场延拓算子在地震偏移成像中有重要应用.单程波波场延拓算子按其实现方式可分为Kirchhoff积分、空间隐式有限差分和Fourier变换方法,他们代表了算子的不同表示方法,当截断使用这些方法时会得到不同的精度.象征表示对这些方法的导出和精度分析有重要作用.算子作用于正弦波函数所得函数称为算子的象征.算子的象征是褶积算子Fourier变换的推广.Fourier变换方法则直接用象征函数的可分表示求出.空间隐式有限差分则可以用象征函数的Padè近似或部分分式导出.单程波算子在深度域的积分称为单程波算子积分解.本文推导了单程波算子积分解的象征表达式,给出了算子象征的代数运算的头几阶表达式,这些表达式还未在前人文献中发现.Kirchhoff积分所需格林函数可以通过象征函数和鞍点法导出.基于积分解的象征表达式给出了非对称走时公式,对改善Kirchhoff积分的聚焦性能有重要意义.     

Time‐domain viscoelastic analysis of earth structures

In this paper two causal models that approximate the nearly frequency‐independent cyclic behaviour of soils are analysed in detail. The study was motivated by the need to conduct time‐domain viscoelastic analysis on soil structures without adopting the ad hoc assumption of Rayleigh damping. First, the causal hysteretic model is introduced in which its imaginary part is frequency independent the same way that is the imaginary part of the popular non‐causal constant hysteretic model. The adoption of an imaginary part that is frequency independent even at the zero‐frequency limit, in conjunction with the condition that the proposed model should be causal, yields a real part that is frequency dependent and singular at zero frequency. The paper shows that the causal hysteretic model, although pathological at the static limit, is the mathematical connection between the non‐causal constant hysteretic model and the physically realizable Biot model. The mathematical structure of the two causal models is examined and it is shown that the causal hysteretic model is precisely the high‐frequency limit of the Biot model. Although both models have a closed‐form time‐domain representation, only the Biot model is suitable for a time‐domain viscoelastic analysis with commercially available computer software. The paper demonstrates that the simplest, causal and physically realizable linear hysteretic model that can approximate the cyclic behaviour of soil is the Biot model. The proposed study elucidates how the dynamic analysis of soil structures can be conducted rigorously in terms of the viscoelastic properties of the soil material and not with the ad hoc Rayleigh damping approach which occasionally has been criticized that tends to overdamp the higher vibration modes. The study concludes that under pulse‐type motions the Rayleigh damping approximation tends to overestimate displacements because of the inappropriate viscous type of dissipation that is imposed. Under longer motions that induce several cycles, the concept of equivalent viscous damping is more appropriate and the Rayleigh damping approximation results to a response that is comparable to the response computed with a rigorous time‐domain viscoelastic finite element analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Recursive evaluation of interaction forces of unbounded soil in the time domain

The interaction forces representing the contribution of the linear unbounded soil to the equations of motion of a nonlinear soil-structure-interaction analysis are specified in the form of convolution integrals. They can be evaluated recursively in the time domain. In this procedure, the forces at a specific time are computed from the displacements at the same time and from the most recent forces and most recent past displacements. It is, in principle, only approximate. When the dynamic-stiffness coefficients can be expressed as the ratios of two polynomials in frequency, the appropriately chosen recursive equations are exact. Two possibilities of choosing a recursive equation are discussed.

• (i) The impulse-invariant method, where the unknown recursive coefficients are calculated by solving a system of equations which are established by equating the rigorous and recursive formulations for a discretized unit impulse displacement.
• (ii) In the segment approach, the dynamic-stiffness coefficients in the time domain are interpolated piecewise. Applying the z-transformation analytically then results in an explicit recursive equation without solving a system of equations.

The recursive evaluation of the convolution integrals in the time domain leads to a dramatic reduction in the computational effort up to two and three orders of magnitude and in the storage requirement. This makes the time-domain analysis using the substructure method computationally competitive with the corresponding direct (non-recursive) frequency-domain procedure of determining the complex response which is, however, applicable only to a linear (total) system.     

高速列车-桥梁竖向随机振动的时域分析方法

提出时间相关多维有色噪声形式的轨道不平顺激励下列车-桥梁耦合系统协方差响应的时域递推方法。用白噪声滤波法生成轨道不平顺有色噪声过程,在宽频带内识别滤波器参数以同时实现滤波成型和波长截断功能。提出基于高阶Pade近似的累次时滞系统,以实现列车多轮对下轨道不平顺激励的大时滞再现;再结合成型滤波器构造列车下轨道不平顺激励的一致白噪声模型。建立列车-桥梁垂向振动的状态方程,将其与激励模型联立得到一致白噪声激励下的列车-桥梁扩阶状态方程。将方差递推法推广到时变系统,求解列车-桥梁系统的随机振动。分析结果与Monte Carlo模拟法符合良好,表明了方法的正确性。     

Modified state–space procedures for pseudodynamic testing

The existing on‐line numerical integration algorithms are derived from the Newmark method, which is based on an approximation of derivatives in the differential equation. The state–space procedure (SSP), based on an interpolation of the discrete excitation signals for piecewise convolution integral, has been confirmed as more reliable than the Newmark method in terms of numerical accuracy and stability. In an attempt to enhance the pseudodynamic test, this study presents an on‐line integration algorithm (referred to as the OS–SSP method) via an integration of the state–space procedure with Nakashima's operator‐splitting concept. Numerical stability and accuracy assessment of the proposed algorithm in addition to the explicit Newmark method and the OS method were investigated via an eigenvalue, frequency‐domain and time‐domain analysis. Of the on‐line integration algorithms investigated, the OS–SSP method is demonstrated as the most accurate method with an acceptable stability (although not unconditionally stable) characteristic. Therefore, the OS–SSP method is the most desirable method for pseudodynamic testing if the numerical stability criterion (Δt/T⩽0.5) is ensured for every vibration mode involved. Copyright © 2001 John Wiley & Sons, Ltd.