Common-Reflection-Surface Stack for Converted Waves

The finite-offset (FO) common-reflection-surface (CRS) stack has been shown to be able to handle not only P-P or S-S but also arbitrarily converted reflections. It can provide different stack sections such as common-offset (CO), common-midpoint (CMP) and common-shot (CS) sections with significantly increased signal-to-noise ratio from the multi-coverage pre-stack seismic data in a data-driven way. It is our purpose in this paper to demonstrate the performance of the FO CRS stack on data involving converted waves in inhomogeneous layered media. In order to do this we apply the FO CRS stack for common-offset to a synthetic seismic data set involving P-P as well as P-S converted primary reflections. We show that the FO CRS stack yields convincing improvement of the image quality in the presence of noisy data and successfully extracts kinematic wavefield attributes useful for further analyses. The extracted emergence angle information is used to achieve a complete separation of the wavefield into its P-P and P-S wave components, given the FO CRS stacked horizontal and vertical component sections.     

A new parameterization for generalized moveout approximation,based on three rays

A simple and accurate traveltime approximation is important in many applications in seismic data processing, inversion and modelling stages. Generalized moveout approximation is an explicit equation that approximates reflection traveltimes in general two-dimensional models. Definition of its five parameters can be done from properties of finite offset rays, for general models, or by explicit calculation from model properties, for specific models. Two versions of classical finite-offset parameterization for this approximation use traveltime and traveltime derivatives of two rays to define five parameters, which makes them asymmetrical. Using a third ray, we propose a balance between the number of rays and the order of traveltime derivatives. Our tests using different models also show the higher accuracy of the proposed method. For acoustic transversely isotropic media with a vertical symmetry axis, we calculate a new moveout approximation in the generalized moveout approximation functional form, which is explicitly defined by three independent parameters of zero-offset two-way time, normal moveout velocity and anellipticity parameter. Our test shows that the maximum error of the proposed transversely isotropic moveout approximation is about 1/6 to 1/8 of that of the moveout approximation that had been reported as the most accurate approximation in these media. The higher accuracy is the result of a novel parameterization that do not add any computational complexity. We show a simple example of its application on synthetic seismic data.     

Seismoelectric numerical simulation in 2D vertical transverse isotropic poroelastic medium

Seismoelectric coupling in an electric isotropic and elastic anisotropic medium is developed using a primary–secondary formulation. The anisotropy is of vertical transverse isotropic type and concerns only the poroelastic parameters. Based on our finite difference time domain algorithm, we solve the seismoelectric response to an explosive source. The seismic wavefields are computed as the primary field. The electric field is then obtained as a secondary field by solving the Poisson equation for the electric potential. To test our numerical algorithm, we compared our seismoelectric numerical results with analytical results obtained from Pride's equation. The comparison shows that the numerical solution gives a good approximation to the analytical solution. We then simulate the seismoelectric wavefields in different models. Simulated results show that four types of seismic waves are generated in anisotropic poroelastic medium. These are the fast and slow longitudinal waves and two separable transverse waves. All of these seismic waves generate coseismic electric fields in a homogenous anisotropic poroelastic medium. The tortuosity has an effect on the propagation of the slow longitudinal wave. The snapshot of the slow longitudinal wave has an oval shape when the tortuosity is anisotropic, whereas it has a circular shape when the tortuosity is isotropic. In terms of the Thomsen parameters, the radiation anisotropy of the fast longitudinal wave is more sensitive to the value of ε, while the radiation anisotropy of the transverse wave is more sensitive to the value of δ.     

基于walkaway VSP下行P波的TTI各向异性参数反演

本文基于弱各向异性（WA）介质的正反演公式和qP波的坐标变换,推导了利用qP波反演任意倾斜对称轴的横向各向同性（TTI）介质的各向异性参数和对称轴方向的公式.理论和数值实验表明,利用2个相互正交的变井源距垂直地震剖面（walkaway VSP）可以完全确定钻井中TTI介质qP波的3个WA参数和对称轴的2个方向参数.我们完成了几个由不同数量剖面组成的walkaway VSP模拟实验,使用TTI模型和一般各向异性模型对模拟数据进行了反演,证明了反演公式的正确性和可靠性.使用这些公式,对来自Java Sea的由3条剖面组成的walkaway VSP观测数据进行了各向异性反演,获得了钻井中接收点处介质的WA参数.     

S-wave in 2D acoustic transversely isotropic media with a tilted symmetry axis

In an acoustic transversely isotropic medium, there are two waves that propagate. One is the P-wave and another one is the S-wave (also known as S-wave artefact). This paper is devoted to analyse the S-wave in two-dimensional acoustic transversely isotropic media with a tilted symmetry axis. We derive the S-wave slowness surface and traveltime function in a homogeneous acoustic transversely isotropic medium with a tilted symmetry axis. The S-wave traveltime approximations in acoustic transversely isotropic media with a tilted symmetry axis can be mapped from the counterparts for acoustic transversely isotropic media with a vertical symmetry axis. We consider a layered two-dimensional acoustic transversely isotropic medium with a tilted symmetry axis to analyse the S-wave moveout. We also illustrate the behaviour of the moveout for reflected S-wave and converted waves.     

On anelliptic approximations for qP velocities in VTI media

A unified approach to approximating phase and group velocities of qP seismic waves in a transversely isotropic medium with vertical axis of symmetry (VTI) is developed. While the exact phase‐velocity expressions involve four independent parameters to characterize the elastic medium, the proposed approximate expressions use only three parameters. This makes them more convenient for use in surface seismic experiments, where the estimation of all four parameters is problematic. The three‐parameter phase‐velocity approximation coincides with the previously published ‘acoustic’ approximation of Alkhalifah. The group‐velocity approximation is new and noticeably more accurate than some of the previously published approximations. An application of the group‐velocity approximation for finite‐difference computation of traveltimes is shown.     

叠前纵波和转换波地震资料Q值提取及反Q滤波(英文)

纵波和转换波联合的多波地震勘探技术是解决复杂油气勘探的有效技术,提高转换波的分辨率是其关键问题之一。影响转换波分辨率的主要原因是地层对地震波的吸收,有效地计算地层Q值、消除地层吸收对转换波传播的影响,是提高转换波分辨率的关键。本文提出了从叠前转换波道集中估算横波Q值的方法,并利用沿射线路径的波场延拓,将一种稳定有效的反Q滤波方法应用到叠前共炮点纵波和转换波道集的衰减补偿中。模型资料结果表明,本文提出的估算转换横波Q值的方法精度较高;模型资料和实际资料的吸收补偿结果表明,此稳定全反Q滤波能有效地提高叠前纵波和转换波资料的分辨率。     

Modelling and migration with orthogonal isochron rays

For increasing time values, isochrons can be regarded as expanding wavefronts and their perpendicular lines as the associated orthogonal isochron rays. The speed of the isochron movement depends on the medium velocity and the source-receiver position. We introduce the term equivalent-velocity to refer to the speed of isochron movement. In the particular case of zero-offset data, the equivalent velocity is half of the medium velocity. We use the concepts of orthogonal isochron-rays and equivalent velocity to extend the application of the exploding reflector model to non-zero offset imaging problems. In particular, we employ these concepts to extend the use of zero-offset wave-equation algorithms for modelling and imaging common-offset sections. In our imaging approach, the common-offset migration is implemented as a trace-by-trace algorithm in three steps: equivalent velocity computation, data conditioning for zero-offset migration and zero-offset wave-equation migration. We apply this methodology for modelling and imaging synthetic common-offset sections using two kinds of algorithms: finite-difference and split-step wavefield extrapolation. We also illustrate the isochron-ray imaging methodology with a field-data example and compare the results with conventional common-offset Kirchhoff migration. This methodology is attractive because it permits depth migration of common-offset sections or just pieces of that by using wave-equation algorithms, it extends the use of robust zero-offset algorithms, it presents favourable features for parallel processing, it permits the creation of hybrid migration algorithms and it is appropriate for migration velocity analysis.     

各向异性介质的矢量波场分离与应用

针对裂缝各向异性介质,本文提出一种非正交假设下的矢量波场分离方法.本文首先对多分量地震勘探中常见的波型泄漏现象进行了数学描述,提出在纵、横波波场分离的同时应该考虑恢复纵、横波的矢量振幅.为了对裂缝方位角与各向异性系数进行定量预测,本文将矢量波场分离拆分成三个步骤来实施：第一步,用Z、R两分量的仿射坐标系变换分离ZR平面内的P波投影与SV波;第二步,用ZR平面内的P波投影与T分量的仿射坐标系变换分离P波与SH波;第三步,用纯净的SV波与SH波的成像剖面分离快慢横波,并预测裂缝发育参数.模型数据与实际数据的试验结果表明,本文提出的纵、横波波场分离方法能够获得完整的矢量振幅信息,并提供裂缝预测的精度.     

Common-reflection-surface (CRS) stack for common offset

We provide a data-driven macro-model-independent stacking technique that migrates 2D prestack multicoverage data into a common-offset (CO) section. We call this new process the CO common-reflection-surface (CRS) stack. It can be viewed as the generalization of the zero-offset (ZO) CRS stack, by which 2D multicoverage data are stacked into a well-simulated ZO section. The CO CRS stack formula can be tailored to stack P-P, S-S reflections as well as P-S or S-P converted reflections. We point out some potential applications of the five kinematic data-derived attributes obtained by the CO CRS stack for each stack value. These include (i) the determination of the geometrical spreading factor for reflections, which plays an important role in the construction of the true-amplitude CO section, and (ii) the separation of the diffractions from reflection events. As a by-product of formulating the CO CRS stack formula, we have also derived a formula to perform a data-driven prestack time migration.     

Multiwave velocity analysis based on Gaussian beam prestack depth migration

Prestack depth migration of multicomponent seismic data improves the imaging accuracy of subsurface complex geological structures. An accurate velocity field is critical to accurate imaging. Gaussian beam migration was used to perform multicomponent migration velocity analysis of PP- and PS-waves. First, PP- and PS-wave Gaussian beam prestack depth migration algorithms that operate on common-offset gathers are presented to extract offset-domain common-image gathers of PP- and PS-waves. Second, based on the residual moveout equation, the migration velocity fields of P- and S-waves are updated. Depth matching is used to ensure that the depth of the target layers in the PP- and PS-wave migration profiles are consistent, and high-precision P- and S-wave velocities are obtained. Finally, synthetic and field seismic data suggest that the method can be used effectively in multiwave migration velocity analysis.     

Ground-roll attenuation using modified common-offset–common-reflection-surface stacking

We modified the common-offset–common-reflection-surface (COCRS) method to attenuate ground roll, the coherent noise typically generated by a low-velocity, low-frequency, and high-amplitude Rayleigh wave. The COCRS operator is based on hyperbolas, thus it fits events with hyperbolic traveltimes such as reflection events in prestack data. Conversely, ground roll is linear in the common-midpoint (CMP) and common-shot gathers and can be distinguished and attenuated by the COCRS operator. Thus, we search for the dip and curvature of the reflections in the common-shot gathers prior to the common-offset section. Because it is desirable to minimize the damage to the reflection amplitudes, we only stack the multicoverage data in the ground-roll areas. Searching the CS gathers before the CO section is another modification of the conventional COCRS stacking. We tested the proposed method using synthetic and real data sets from western Iran. The results of the ground-roll attenuation with the proposed method were compared with results of the f–k filtering and conventional COCRS stacking after f–k filtering. The results show that the proposed method attenuates the aliased and nonaliased ground roll better than the f–k filtering and conventional CRS stacking. However, the computation time was higher than other common methods such as f–k filtering.     

Common-offset common-reflection-surface attributes estimation with differential evolution

Common-reflection surface is a method to describe the shape of seismic events, typically the slopes (dip) and curvature portions (traveltime). The most systematic approach to estimate the common-reflection surface traveltime attributes is to employ a sequence of single-variable search procedures, inheriting the advantage of a low computational cost, but also the disadvantage of a poor estimation quality. A search strategy where the common-reflection surface attributes are globally estimated in a single stage may yield more accurate estimates. In this paper, we propose to use the bio-inspired global optimization algorithm differential evolution to estimate all the two-dimensional common-offset common-reflection surface attributes simultaneously. The differential evolution algorithm can provide accurate estimates for the common-reflection surface traveltime attributes, with the benefit of having a small set of input parameters to be configured. We apply the differential evolution algorithm to estimate the two-dimensional common-reflection surface attributes in the synthetic Marmousi data set, contaminated by noise, and in a land field data with a small fold. By analysing the stacked and coherence sections, we could see that the differential evolution based common-offset common-reflection surface approach presented significant signal-to-noise ratio enhancement.     

Errors Due to the Common Ray Approximations of the Coupling Ray Theory

The common ray approximation considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common ray approximation.We derive the equations for estimating the travel-time errors due to the isotropic and anisotropic common ray approximations of the coupling ray theory. These equations represent the main result of the paper. The derivation is based on the general equations for the second-order perturbations of travel time. The accuracy of the anisotropic common ray approximation can be studied along the isotropic common rays, without tracing the anisotropic common rays.The derived equations are numerically tested in three 1-D models of differing degree of anisotropy. The first-order and second-order perturbation expansions of travel time from the isotropic common rays to anisotropic-ray-theory rays are compared with the anisotropic-ray-theory travel times. The errors due to the isotropic common ray approximation and due to the anisotropic common ray approximation are estimated. In the numerical example, the errors of the anisotropic common ray approximation are considerably smaller than the errors of the isotropic common ray approximation.The effect of the isotropic common ray approximation on the coupling-ray-theory synthetic seismograms is demonstrated graphically. For comparison, the effects of the quasi-isotropic projection of the Green tensor, of the quasi-isotropic approximation of the Christoffel matrix, and of the quasi-isotropic perturbation of travel times on the coupling-ray-theory synthetic seismograms are also shown. The projection of the travel-time errors on the relative errors of the time-harmonic Green tensor is briefly presented.     

TTI介质的交错网格伪P波正演方法

研究了三维弱各向异性近似下,利用伪P波(伪纵波)模拟弹性波场P分量在倾斜对称轴的横向各向同性(TTI)介质中的传播过程,并对比了分别基于弹性Hooke定律、弹性波投影和运动学色散方程所建立的三种二阶差分伪P波方程的正演特点.目前这些伪P波方程数值计算主要采用规则网格差分,但是规则网格在TTI模拟中有低效率、低精度以及不稳定的缺点.为了提高计算的精度,本文构建出相应方程的交错网格有限差分格式.通过对比伪P波方程在三维TTI介质中不同的数值模拟的表达形式,本文认为基于色散方程所建立的伪P波方程在模拟弹性波中P波传播的过程中具有最小的噪声.本文分析不同的各向同性对称轴空间角度的频散特征,并引入适当的横波速度维持计算的稳定.二维模型算例表明,本文提出的交错网格正演算法可以得到稳定光滑的伪P波正演波场.使用本文交错网格算法对二维BP TTI模型的逆时偏移也具有较稳定的偏移结果.     

Seismic imaging of complex structures with the CO-CDS stack method

Various seismic imaging methods are introduced to resolve some of the possible ambiguities of seismic interpretation in complex structures. Reducing dependency of imaging techniques on velocity or using diffraction energy for imaging more structural details are the main topics of the imaging research. In this study, we try to improve the seismic image quality in semi-complex structures by combining the common reflection surface (CRS) method with a diffraction based scheme in the common-offset domain. Previously introduced partial CRS and common offset CRS methods exhibited reliable performance in imaging complex media. Here, we were looking for stable and efficient solutions, preserving advantages of the previous methods. Herewith, the proposed operator fits better to diffractions than to reflections. Therefore, we call it the commonoffset common diffraction surface stack (CO CDS). In a previous study, improvement of the quality of seismic image by the CRS method was achieved by combination of the CDS method with the partial CRS. This resulted in the introduction of the partial CDS. Initially, in this study, the common-offset CRS traveltime equation was modified to the common-offset CDS. The hypothetical shot reflector experiment in the CRS method was changed to shot diffraction point experiment. In the introduced operator, two wavefront curvatures, observed at receivers positions, are set equal in order to satisfy the diffraction condition. In the proposed method, we search for accurate attribute sets for each considered offset individually, and then form a new operator by four coherent attributes. Application of the common- offset CDS method on synthetic and field data shows more details of the geological structures with higher quality, while preserving continuity of reflection events. The proposed method is, however, more expensive than the partial and common offset CRS for large dataset.     

Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

Existing and commonly used in industry nowadays, closed‐form approximations for a P‐wave reflection coefficient in transversely isotropic media are restricted to cases of a vertical and a horizontal transverse isotropy. However, field observations confirm the widespread presence of rock beds and fracture sets tilted with respect to a reflection boundary. These situations can be described by means of the transverse isotropy with an arbitrary orientation of the symmetry axis, known as tilted transversely isotropic media. In order to study the influence of the anisotropy parameters and the orientation of the symmetry axis on P‐wave reflection amplitudes, a linearised 3D P‐wave reflection coefficient at a planar weak‐contrast interface separating two weakly anisotropic tilted tranversely isotropic half‐spaces is derived. The approximation is a function of the incidence phase angle, the anisotropy parameters, and symmetry axes tilt and azimuth angles in both media above and below the interface. The expression takes the form of the well‐known amplitude‐versus‐offset “Shuey‐type” equation and confirms that the influence of the tilt and the azimuth of the symmetry axis on the P‐wave reflection coefficient even for a weakly anisotropic medium is strong and cannot be neglected. There are no assumptions made on the symmetry‐axis orientation angles in both half‐spaces above and below the interface. The proposed approximation can be used for inversion for the model parameters, including the orientation of the symmetry axes. Obtained amplitude‐versus‐offset attributes converge to well‐known approximations for vertical and horizontal transverse isotropic media derived by Rüger in corresponding limits. Comparison with numerical solution demonstrates good accuracy.     

Application of a conceptual method for separating runoff components in daily hydrographs in Kimakia forest catchments,Kenya

This paper highlights the use of a conceptual method for separating runoff components in daily hydrographs, contrary to the traditionally used graphical method of separation. In the conceptual method, the components, viz. surface flow, interflow and baseflow, are regarded as high, medium and low frequency signals and their separation is done using the principle of a recursive digital filter commonly used in signal analysis and processing. It requires estimates of the direct runoff (β_d) and surface runoff (β_s) filter parameters which are obtained by a least‐squares procedure involving baseflow and interflow indices based on graphical and recursive digital filter estimation techniques. The method thus circumvents the subjective element associated with the graphical procedure of hydrograph separation, in which case the eye approximation and/or one's skill at plotting is the prime basis for the whole analysis. The analysis based on three forest catchments in Kimakia, Kenya, East Africa, revealed that β_d=K_b and β_s=K_i , where K_b and K_i are the baseflow and interflow recession constants. Copyright © 1999 John Wiley & Sons, Ltd.     

Interval anisotropic parameter estimation from walkaway vertical seismic profiling data

This paper presents a new explicit method for the estimation of layered vertical transverse isotropic (VTI) anisotropic parameters from walkaway VSP data. This method is based on Dix‐type normal moveout (NMO) inversion. To estimate interval anisotropic parameters above a receiver array, the method uses time arrivals of surface‐related double‐reflected downgoing waves. A three‐term NMO approximation function is used to estimate NMO velocity and a non‐hyperbolic parameter. Assuming the vertical velocity is known from zero‐offset VSP data, Dix‐type inversion is applied to estimate the layered Thomsen anisotropic parameters ?, δ above the receivers array. Model results show reasonable accuracy for estimates through Dix‐type inversion. Results also show that in many cases we can neglect the influence of the velocity gradient on anisotropy estimates. First breaks are used to estimate anisotropic parameters within the walkaway receiver interval. Analytical uncertainty analysis is performed to NMO parameter estimates. Its conclusions are confirmed by modelling.