Interferometric interpolation of sparse marine data

We present the theory and numerical results for interferometrically interpolating 2D and 3D marine surface seismic profiles data. For the interpolation of seismic data we use the combination of a recorded Green's function and a model‐based Green's function for a water‐layer model. Synthetic (2D and 3D) and field (2D) results show that the seismic data with sparse receiver intervals can be accurately interpolated to smaller intervals using multiples in the data. An up‐ and downgoing separation of both recorded and model‐based Green's functions can help in minimizing artefacts in a virtual shot gather. If the up‐ and downgoing separation is not possible, noticeable artefacts will be generated in the virtual shot gather. As a partial remedy we iteratively use a non‐stationary 1D multi‐channel matching filter with the interpolated data. Results suggest that a sparse marine seismic survey can yield more information about reflectors if traces are interpolated by interferometry. Comparing our results to those of f‐k interpolation shows that the synthetic example gives comparable results while the field example shows better interpolation quality for the interferometric method.     

Global-scale seismic interferometry: theory and numerical examples

Progress in the imaging of the mantle and core is partially limited by the sparse distribution of natural sources; the earthquake hypocenters are mainly along the active lithospheric plate boundaries. This problem can be approached with seismic interferometry. In recent years, there has been considerable progress in the development of seismic interferometric techniques. The term seismic interferometry refers to the principle of generating new seismic responses by cross‐correlating seismic observations at different receiver locations. The application of interferometric techniques on a global scale could create sources at locations where no earthquakes occur. In this way, yet unknown responses would become available for the application of travel‐time tomography and surface‐wave dispersion studies. The retrieval of a dense‐enough sampling of source gathers would largely benefit the application of reflection imaging. We derive new elastodynamic representation integrals for global‐scale seismic interferometry. The relations are different from other seismic interferometry relations for transient sources, in the sense that they are suited for a rotating closed system like the Earth. We use a correlation of an observed response with a response to which free‐surface multiple elimination has been applied to account for the closed system. Despite the fact that the rotation of the Earth breaks source‐receiver reciprocity, the seismic interferometry relations are shown to be valid. The Coriolis force is included without the need to evaluate an extra term. We synthesize global‐scale earthquake responses and use them to illustrate the acoustic versions of the new interferometric relations. When the sampling of real source locations is dense enough, then both the responses with and without free‐surface multiples are retrieved. When we do not take into account the responses from the sources in the direct neighborhood of the seismic interferometry‐constructed source location, the response with free‐surface multiples can still be retrieved. Even when only responses from sources at a certain range of epicentral distances are available, some events in the Green's function between two receiver locations can still be retrieved. The retrieved responses are not perfect, but the artefacts can largely be ascribed to numerical errors. The reconstruction of internal events – the response as if there was a source and a receiver on (major) contrasts within the model – could possibly be of use for imaging. With modelling it is possible to discover in which region of the correlation panel stationary phases occur that contribute to the retrieval of events. This knowledge opens up a new way of filtering out undesired events and of discovering whether specific events could be retrieved with a given source‐receiver configuration.     

Effect of uneven noise source and/or station distribution on estimating the azimuth anisotropy of surface waves

With the development of the dense array, the surface wave velocity and azimuthal anisotropy under the array can be directly obtained by beamforming the noise cross-correlation functions (NCFs). However, the retrieval of the Green's function by cross-correlating the seismic noise requires that the noise source has a uniform distribution. For the case with uneven noise source, the azimuthal dependence on the sources in the expression for the spatial coherence function, which corresponds to the NCF in the time domain, has the same form as the azimuthal dependence of the surface wave velocity in weakly anisotropic media. Therefore, the uneven noise source will affect the surface wave anisotropy extraction. In this study, three passive seismic methods, i.e., beamforming, SPAC (spatial autocorrelation), and NCF, are compared to demonstrate that an uneven source distribution and uneven station distribution have equivalent effects on the outcome from each method. A beamforming method is proposed to directly extract the velocity and azimuthal anisotropy of surface waves. The effect of uneven noise source and/or station distribution on estimating the azimuth anisotropy of surface waves was investigated using data from the ChinArray Phase II. A method for correcting the apparent anisotropy in beamforming results caused by an uneven station distribution is suggested.     

Application of the virtual refraction to near‐surface characterization at the Boise Hydrogeophysical Research Site‡

Seismic interferometry is a relatively new technique to estimate the Green's function between receivers. Spurious energy, not part of the true Green's function, is produced because assumptions are commonly violated when applying seismic interferometry to field data. Instead of attempting to suppress all spurious energy, we show how spurious energy associated with refractions contains information about the subsurface in field data collected at the Boise Hydrogeophysical Research Site. By forming a virtual shot record we suppress uncorrelated noise and produce a virtual refraction that intercepts zero offset at zero time. These two features make the virtual refraction easy to pick, providing an estimate of refractor velocity. To obtain the physical parameters of the layer above the refractor we analyse the cross‐correlation of wavefields recorded at two receivers for all sources. A stationary‐phase point associated with the correlation between the reflected wave and refracted wave from the interface identifies the critical offset. By combining information from the virtual shot record, the correlation gather and the real shot record we determine the seismic velocities of the unsaturated and saturated sands, as well as the variable relative depth to the water‐table. Finally, we discuss how this method can be extended to more complex geologic models.     

Directional‐oriented wavefield imaging: a new wave‐based subsurface illumination imaging condition for reverse time migration

The key objective of an imaging algorithm is to produce accurate and high‐resolution images of the subsurface geology. However, significant wavefield distortions occur due to wave propagation through complex structures and irregular acquisition geometries causing uneven wavefield illumination at the target. Therefore, conventional imaging conditions are unable to correctly compensate for variable illumination effects. We propose a generalised wave‐based imaging condition, which incorporates a weighting function based on energy illumination at each subsurface reflection and azimuth angles. Our proposed imaging kernel, named as the directional‐oriented wavefield imaging, compensates for illumination effects produced by possible surface obstructions during acquisition, sparse geometries employed in the field, and complex velocity models. An integral part of the directional‐oriented wavefield imaging condition is a methodology for applying down‐going/up‐going wavefield decomposition to both source and receiver extrapolated wavefields. This type of wavefield decomposition eliminates low‐frequency artefacts and scattering noise caused by the two‐way wave equation and can facilitate the robust estimation for energy fluxes of wavefields required for the seismic illumination analysis. Then, based on the estimation of the respective wavefield propagation vectors and associated directions, we evaluate the illumination energy for each subsurface location as a function of image depth point and subsurface azimuth and reflection angles. Thus, the final directional‐oriented wavefield imaging kernel is a cross‐correlation of the decomposed source and receiver wavefields weighted by the illuminated energy estimated at each depth location. The application of the directional‐oriented wavefield imaging condition can be employed during the generation of both depth‐stacked images and azimuth–reflection angle‐domain common image gathers. Numerical examples using synthetic and real data demonstrate that the new imaging condition can properly image complex wave paths and produce high‐fidelity depth sections.     

Virtual signals combination – phase analysis for 2D and 3D data representation (acoustic medium)

We formulate the Kirchhoff‐Helmholtz representation theory for the combination of seismic interferometry signals synthesized by cross‐correlation and by cross‐convolution in acoustic media. The approach estimates the phase of the virtual reflections from the boundary encompassing a volume of interest and subtracts these virtual reflections from the total seismic‐interferometry wavefield. The reliability of the combination result, relevant for seismic exploration, depends on the stationary‐phase and local completeness in partial coverage regions. The analysis shows the differences in the phase of the corresponding seismic interferometry (by cross‐correlation) and virtual reflector (by cross‐convolution) signals obtained by 2D and 3D formulations, with synthetic examples performed to remove water layer multiples in ocean bottom seismic (OBS) acoustic data.     

地震干涉测量法近地表散射波分离技术

针对山地地震勘探数据低信噪比问题,近地表散射波分离意义显得尤为突出,地震干涉测量法为此提供了一种技术手段.本文将地震干涉测量理论和散射理论结合起来,导出了近地表散射波地震干涉测量表达式,分为互相关型和褶积型表达式,它们由实际波场和背景波场干涉测量构成.根据近地表散射波分离理论,结合陆上地震勘探实际观测系统,采用褶积和反褶积混合型地震干涉测量配置,用实际地震资料展示了近地表散射波分离技术的应用效果.经过理论分析和砾石区实际资料试验,表明地震干涉测量不仅能分离测线上散射源产生的散射波,而且能分离部分侧面散射波.该技术的优点在于它适应于起伏地形和不均匀近地表结构,并且不需要起伏地形和近地表速度信息.为了从实际资料中消除近地表散射波,本文采用多道匹配滤波自适应减法,在砾石区见到较好效果.     

Calibration of cross-line components for sea-bed 4C acquisition systems

Faithful recording of the elastic wavefield at the sea‐bed is required for quantitative applications of 4C seismic. The accuracy of the recorded vectorial wavefield depends on factors that vary from deployment to deployment. This paper focuses on one such factor: the interaction of the acquisition system with the sea‐bed, which is referred to here as coupling. We show, using multi‐azimuth data recorded with a cable‐based sea‐bed acquisition system, whose sensor housing is cylindrically shaped and with the in‐line geophone fixed to the cable, that coupling depends on the propagation direction and wave type (P‐ or S‐waves) of the incident wavefield. We show that coupling is more critical for S‐waves than for P‐waves. Detection of inconsistent coupling using both P‐ and S‐waves is therefore mandatory. A data‐driven processing method to compensate for the frequency‐dependent coupling response of the cross‐line geophone is derived. Its application to field data verifies the effectiveness of the method.     

基于平面波模型重访地震背景噪声互相关及空间自相关(SPAC)

自Aki(1957)提出微震的空间自相关(SPatial AutoCorelation,SPAC)技术以来,SPAC技术一直独立发展,并在工程地震领域获得了广泛应用.近20年来,地震干涉(Seisimic Interferometry,SI)在多个领域引起人们的关注,该技术的核心思想是连续地震背景噪声的互相关函数(Noise Crosscorrelation Function,NCF)可以重建系统的格林函数(Green's Function,GF),对该技术的回溯性研究建立了SPAC和NCF的关系:它们是对同一物理现象的不同描述,SPAC在频率域中描述随机平稳噪声的空间相干,NCF在时间域中描述扩散场的互相关.理论上SAPC和NCF技术要求背景噪声源均匀分布,这样的噪声场可以用平面波叠加来模拟.本文基于平面波模型重访地震背景噪声的互相关和空间自相关技术,从单色平面波的互相关表示出发,对地震背景噪声互相关及空间自相关技术进行评述,试图使这些概念更易于理解.与之前众多研究地震干涉技术的理论相比,本文尤其关注以下几点:(1)基于简单的平面波模型,给出不同维度下,源或台站对方位均匀分布时,平面波互相关对入射波的方位平均和台阵对的方位平均结果,并对格林函数GF和时域互相关函数NCF的关系进行总结.(2)给出声源和(或)交叉台站方位分布不均匀时的互相关表示,指出这种非均匀性对方位的依赖关系,与弱各向异性介质中面波速度的方位依赖关系类似,因此,非均匀源的影响在反演时可能会映射到面波方位各向异性结果中.(3)互相关运算中,哪一个台站是虚拟源.NCF包含因果性和非因果性两部分,NCF的非对称性通常用于研究噪声源的方位分布,但由于源和接收的互易关系,及对互相关运算的不同定义和不同的傅里叶变换习惯,哪一个台站是虚拟源在目前的文献中并不明确.(4)方位平均和时间平均的关系.在SPAC处理中,需要对不同方位分布的台站对进行方位平均,本文从理论上说明,单个平面波入射时,交叉台站互相关系数对台站对的方位平均,等价于单个台站对互相关系数对入射波的时间平均.(5)几种特定分布非均匀噪声源的SPAC表示.包括单独的因果性噪声源和非因果性噪声源给出的互相关函数表示,及由此带来的相移问题.(6)利用SPAC、NCF和面波GF之间的关系,给出交叉分量的空间自相关系数表示.(7)衰减介质的空间相干表示.虽然利用地震干涉技术研究介质衰减在理论上仍然存在一些争议,但人们正试图研究从连续背景噪声记录中提取介质衰减的可能性.本文基于平面波模型,给出了不同坐标选择下,衰减介质的空间相干表示,这种表达的不同,指示了由地震干涉技术提取介质衰减的困难.与众多研究地震干涉的理论相比,比如稳相近似理论、互易定理、时间反转声学等,本文主要考虑均匀介质,不涉及非均匀介质的散射,从最简单的平面波模型,理解背景噪声重建系统格林函数这一地震干涉的核心思想和相应的基本概念.     

Internal multiple attenuation by Kirchhoff extrapolation

Surface‐related multiple elimination is the leading methodology for surface multiple removal. This data‐driven approach can be extended to interbed multiple prediction at the expense of a huge increase of the computational burden. This cost makes model‐driven methods still attractive, especially for the three dimensional case. In this paper we present a methodology that extends Kirchhoff wavefield extrapolation to interbed multiple prediction. In Kirchhoff wavefield extrapolation for surface multiple prediction a single round trip to an interpreted reflector is added to the recorded data. Here we show that interbed multiples generated between two interpreted reflectors can be predicted by applying the Kirchhoff wavefield extrapolation operator twice. In the first extrapolation step Kirchhoff wavefield extrapolation propagates the data backward in time to simulate a round trip to the shallower reflector. In the second extrapolation step Kirchhoff wavefield extrapolation propagates the data forward in time to simulate a round trip to the deeper reflector. In the Kirchhoff extrapolation kernel we use asymptotic Green's functions. The prediction of multiples via Kirchhoff wavefield extrapolation is possibly sped up by computing the required traveltimes via a shifted hyperbola approximation. The effectiveness of the method is demonstrated by results on both synthetic and field data sets.     

用改进的经验格林函数方法模拟唐山地震动

建筑物的抗震设防需要尽可能地掌握未来大地震强震动记录信息,但大地震强震动记录的匮乏阻碍了抗震设防实践的发展。经验格林函数方法作为模拟地震动的主要方法,可以提供可靠的大地震强震动记录,但也存在着许多问题,如缺乏对大地震断层滑动分布不均匀的描述、用经验确定小震数目、模拟方法受到大小地震相似条件的限制等。文中对上述经验格林函数方法存在的问题进行了研究,改进的经验格林函数方法,有效地解决了上述问题。并用其对唐山大地震进行了模拟,并把模拟的地震动时程和反应谱与实际记录相比较,发现用改进方法模拟的地震动加速度反应谱比用未改进方法模拟结果更接近实际的地震动记录加速度反应谱。由此说明改进的经验格林函数可更准确的模拟地震动。     

A Discrete Wavenumber Boundary Element Method for study of the 3-D response 2-D scatterers

A mathematical formulation of the 2·5D elastodynamic scattering problem is presented and validated. The formulation is a straightforward extension of the Discrete Wave number Boundary Integral Equation Method (DWBIEM) originally proposed by Kawase¹ for 2D scattering problems and subsequently extended to the 3D problem by Kim and Papageorgiou.² It is demonstrated that the Green's function which is appropriate for a boundary formulation of the 2·5D elastodynamic scattering problem is the one corresponding to a unit force moving on a straight line with constant velocity. Such a Green's function is derived in the present study. The formulation may be used to study the wavefields in models of sedimentary deposits (e.g. valleys) or topography (e.g. canyons or ridges) with a 2D variation in structure but obliquely incident plane waves. The advantage of a 2·5D formulation is that it provides the means for calculations of 3D wavefields in scattering problems by requiring a storage comparable to that of the corresponding 2D calculations. © 1998 John Wiley & Sons, Ltd.     

A model for the 3D kinematic interaction analysis of pile groups in layered soils

The paper presents a numerical model for the analysis of the soil–structure kinematic interaction of single piles and pile groups embedded in layered soil deposits during seismic actions. A finite element model is considered for the pile group and the soil is assumed to be a Winkler‐type medium. The pile–soil–pile interaction and the radiation problem are accounted for by means of elastodynamic Green's functions. Condensation of the problem permits a consistent and straightforward derivation of both the impedance functions and the foundation input motion, which are necessary to perform the inertial soil–structure interaction analyses. The model proposed allows calculating the internal forces induced by soil–pile and pile‐to‐pile interactions. Comparisons with data available in literature are made to study the convergence and validate the model. An application to a realistic pile foundation is given to demonstrate the potential of the model to catch the dynamic behaviour of the soil–foundation system and the stress resultants in each pile. Copyright © 2009 John Wiley & Sons, Ltd.     

SEISMOGRAM SYNTHESIS AND RECOMPRESSION OF DISPERSIVE IN-SEAM SEISMIC MULTIMODE DATA USING A NORMAL-MODE SUPERPOSITION APPROACH

In spite of a geometrical rotation into radial and transverse parts, two- or three-component in-seam seismic data used for underground fault detection often suffer from the problem of overmoding ‘noise’. Special recompression filters are required to remove this multimode dispersion so that conventional reflection seismic data processing methods, e.g. CMP stacking techniques, can be applied afterwards. A normal-mode superposition approach is used to design such multimode recompression filters. Based on the determination of the Green's function in the far-field, the normal-mode superposition approach is usually used for the computation of synthetic single- and multi-mode (transmission) seismograms for vertically layered media. From the filter theory's point of view these Green's functions can be considered as dispersion filters which are convolved with a source wavelet to produce the synthetic seismograms. Thus, the design of multimode recompression filters can be reduced to a determination of the inverse of the Green's function. Two methods are introduced to derive these inverse filters. The first operates in the frequency domain and is based on the amplitude and phase spectrum of the Green's function. The second starts with the Green's function in the time domain and calculates two-sided recursive filters. To test the performance of the normal-mode superposition approach for in-seam seismic problems, it is first compared and applied to synthetic finite-difference seismograms of the Love-type which include a complete solution of the wave equation. It becomes obvious that in the case of one and two superposing normal modes, the synthetic Love seam-wave seismograms based on the normal-mode superposition approach agree exactly with the finite-difference data if the travel distance exceeds two dominant wavelengths. Similarly, the application of the one- and two-mode recompression filters to the finite-difference data results in an almost perfect reconstruction of the source wavelet already two dominant wavelengths away from the source. Subsequently, based on the dispersion analysis of an in-seam seismic transmission survey, the normal-mode superposition approach is used both to compute one- and multi-mode synthetic seismograms and to apply one- and multimode recompression filters to the field data. The comparison of the one- and two-mode synthetic seismograms with the in-seam seismic transmission data reveals that arrival times, duration and shape of the wavegroups and their relative excitation strengths could well be modelled by the normal-mode superposition approach. The one-mode recompressions of the transmission seismograms result in non-dispersive wavelets whose temporal resolution and signal-to-noise ratio could clearly be improved. The simultaneous two-mode recompressions of the underground transmission data show that, probably due to band-limitation, the dispersion characteristics of the single modes could not be evaluated sufficiently accurately from the field data in the high-frequency range. Additional techniques which overcome the problem of band-limitation by modelling all of the enclosed single-mode dispersion characteristics up to the Nyquist frequency will be mandatory for future multimode applications.     

Green's function for three-dimensional elastic wave equation with a moving point source on the free surface with applications

High-speed train seismology has come into being recently. This new kind of seismology uses a high-speed train as a repeatable moving seismic source. Therefore, Green's function for a moving source is needed to make theoretical studies of the high-speed train seismology. Green's function for three-dimensional elastic wave equation with a moving point source on the free surface is derived. It involves a line integral of the Green's function for a fixed point source with different positions and corresponding time delays. We give a rigorous mathematical proof of this Green's function. According to the principle of linear superposition, we have also obtained the Green's function for a group of moving sources which can be regarded as a model of a traveling high-speed train. Based on a temporal convolution, an analytical formula for other moving sources is also given. In terms of a moving Gaussian source, we deal with the issue of numerical calculations of the analytical formula. Applications to modelling of a traveling high-speed train are presented. We have considered both the land case and the bridge case for a traveling high-speed train. The theoretical seismograms show different waveform features for these two cases.     

Source Time Function of Seismic Events at Rudna Copper Mine, Poland

—?The empirical Green's function deconvolution technique is used to retrieve the source time functions from the records of P waves of seven seismic events that occurred at the Rudna copper mine in 1996 and were located in the middle of the underground network. Their moment magnitudes ranged from 2.1 to 2.9. The records of smaller events from the same area and with similar source mechanism, with moment magnitudes ranging from 1.5 to 2.0, were accepted as empirical Green's functions. The relative source time functions were successfully retrieved at a number of stations for six events. Directivity effects, implying unilateral rupture propagation, were observed in five cases. The azimuth of rupture propagation direction and the rupture velocity were estimated from the distribution of pulse widths and pulse maximum amplitudes as a function of the cosine of station azimuths. The rupture propagated approximately either from south to north or from north to south. The rupture velocity was low, ranging from 0.25 to 0.54 of the shear-wave velocity. The source dimensions, represented by the fault length, were also small in comparison with those estimated in the frequency domain and ranged from 80 to 250?m.     

基于时空域自适应高阶有限差分的声波叠前逆时偏移

叠前逆时偏移在理论上是现行偏移方法中最为精确的一种成像方法,其实现过程中的核心步骤之一是波动方程的波场延拓,而波场延拓的本质是求解波动方程,所以精确、快速地求解波动方程对逆时偏移至关重要.本文采用一种基于时空域频散关系的有限差分方法来求解声波方程,分析其频散和稳定性,实现波场数值模拟,并将分析和模拟结果与传统有限差分法进行对比.分析结果和模型数值模拟结果都表明时空域有限差分法模拟精度更高、稳定性更好.将时空域高阶有限差分法应用到叠前逆时偏移波场延拓的方程求解中,然后再利用归一化互相关成像条件成像,理论模型数据偏移处理获得了精度更高的成像.同时,在逆时偏移波场延拓的实现中,采用自适应变长度的空间差分算子求解空间导数的有限差分策略,在不影响数值模拟和成像精度的前提下,有效地提高了计算效率.