Dense 3D residual moveout analysis as a tool for HTI parameter estimation

Three‐dimensional residual moveout analysis is the basic step in velocity model refinement. The analysis is generally carried out using horizontal and/or vertical semblances defined on a sparse set of in‐lines or cross‐lines with densely sampled source–receiver offsets. An alternative approach, which we call dense residual moveout analysis (DRMA), is to use all the bins of a three‐dimensional survey but sparsely sampled offsets. The proposed technique is very fast and provides unbiased and statistically efficient estimates of the residual moveout. Indeed, for the sparsest possible offset distribution, when only near‐ and far‐angle stacks are used, the variance of the residual moveout estimate is only 1.4 times larger than the variance of the least‐squares estimate obtained using all offsets. The high performance of DRMA makes it a useful tool for many applications, of which azimuthal velocity analysis is considered here. For a horizontal transverse isotropy (HTI) model, a deterministic procedure is proposed to define, at every point of residual moveout estimation, the azimuthal angle of the HTI axis of symmetry, the Thomsen anisotropy coefficients, and the interval (or root‐mean‐square) velocities in both the HTI isotropy and symmetry planes. The procedure is not restricted by DRMA assumptions; for example, it is also applicable to semblance‐based residual moveout estimates. The high resolution of the technique is illustrated by azimuthal velocity analysis over an oilfield in West Siberia.     

Seismic inversion with generalized Radon transform based on local second-order approximation of scattered field in acoustic media

Sound velocity inversion problem based on scattering theory is formulated in terms of a nonlinear integral equation associated with scattered field. Because of its nonlinearity, in practice, linearization algorisms (Born/single scattering approximation) are widely used to obtain an approximate inversion solution. However, the linearized strategy is not congruent with seismic wave propagation mechanics in strong perturbation (heterogeneous) medium. In order to partially dispense with the weak perturbation assumption of the Born approximation, we present a new approach from the following two steps: firstly, to handle the forward scattering by taking into account the second-order Born approximation, which is related to generalized Radon transform (GRT) about quadratic scattering potential; then to derive a nonlinear quadratic inversion formula by resorting to inverse GRT. In our formulation, there is a significant quadratic term regarding scattering potential, and it can provide an amplitude correction for inversion results beyond standard linear inversion. The numerical experiments demonstrate that the linear single scattering inversion is only good in amplitude for relative velocity perturbation ( \( \delta_{c}/c_{0} \) ) of background media up to 10 %, and its inversion errors are unacceptable for the perturbation beyond 10 %. In contrast, the quadratic inversion can give more accurate amplitude-preserved recovery for the perturbation up to 40 %. Our inversion scheme is able to manage double scattering effects by estimating a transmission factor from an integral over a small area, and therefore, only a small portion of computational time is added to the original linear migration/inversion process.     

2.5D modelling, inversion and angle migration in anisotropic elastic media

2.5D modelling approximates 3D wave propagation in the dip‐direction of a 2D geological model. Attention is restricted to raypaths for waves propagating in a plane. In this way, fast inversion or migration can be performed. For velocity analysis, this reduction of the problem is particularly useful. We review 2.5D modelling for Born volume scattering and Born–Helmholtz surface scattering. The amplitudes are corrected for 3D wave propagation, taking into account both in‐plane and out‐of‐plane geometrical spreading. We also derive some new inversion/migration results. An AVA‐compensated migration routine is presented that is simplified compared with earlier results. This formula can be used to create common‐image gathers for use in velocity analysis by studying the residual moveout. We also give a migration formula for the energy‐flux‐normalized plane‐wave reflection coefficient that models large contrast in the medium parameters not treated by the Born and the Born–Helmholtz equation results. All results are derived using the generalized Radon transform (GRT) directly in the natural coordinate system characterized by scattering angle and migration dip. Consequently, no Jacobians are needed in their calculation. Inversion and migration in an orthorhombic medium or a transversely isotropic (TI) medium with tilted symmetry axis are the lowest symmetries for practical purposes (symmetry axis is in the plane). We give an analysis, using derived methods, of the parameters for these two types of media used in velocity analysis, inversion and migration. The kinematics of the two media involve the same parameters, hence there is no distinction when carrying out velocity analysis. The in‐plane scattering coefficient, used in the inversion and migration, also depends on the same parameters for both media. The out‐of‐plane geometrical spreading, necessary for amplitude‐preserving computations, for the TI medium is dependent on the same parameters that govern in‐plane kinematics. For orthorhombic media, information on additional parameters is required that is not needed for in‐plane kinematics and the scattering coefficients. Resolution analysis of the scattering coefficient suggests that direct inversion by GRT yields unreliable parameter estimates. A more practical approach to inversion is amplitude‐preserving migration followed by AVA analysis. SYMBOLS AND NOTATION A list of symbols and notation is given in Appendix D .     

The effectiveness of a pseudo-inverse extended Born operator to handle lateral heterogeneity for imaging and velocity analysis applications

Wave equation–based migration velocity analysis techniques aim to construct a kinematically accurate velocity model for imaging or as an initial model for full waveform inversion applications. The most popular wave equation–based migration velocity analysis method is differential semblance optimization, where the velocity model is iteratively updated by minimizing the unfocused energy in an extended image volume. However, differential semblance optimization suffers from artefacts, courtesy of the adjoint operator used in imaging, leading to poor convergence. Recent findings show that true amplitude imaging plays a significant role in enhancing the differential semblance optimization's gradient and reducing the artefacts. Here, we focus on a pseudo-inverse operator to the horizontally extended Born as a true amplitude imaging operator. For laterally inhomogeneous models, the operator required a derivative with respect to a vertical shift. Extending the image vertically to evaluate such a derivative is costly and impractical. The inverse operator can be simplified in laterally homogeneous models. We derive an extension of the approach to apply the full inverse formula and evaluate the derivative efficiently. We simplified the implementation by applying the derivative to the imaging condition and utilize the relationship between the source and receiver wavefields and the vertical shift. Specifically, we verify the effectiveness of the approach using the Marmousi model and show that the term required for the lateral inhomogeneity treatment has a relatively small impact on the results for many cases. We then apply the operator in differential semblance optimization and invert for an accurate macro-velocity model, which can serve as an initial velocity model for full waveform inversion.     

Normal moveout velocity for pure‐mode and converted waves in layered orthorhombic medium

We study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi‐compressional, quasi‐shear, and converted waves in one‐dimensional multi‐layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface‐offset azimuth and applying our derived relationship between the surface‐offset azimuth and phase‐velocity azimuth, we obtain the normal moveout velocity versus the phase‐velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time‐domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local‐angle‐domain common image gathers. The angle and azimuth parameters of the local‐angle‐domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψ_phs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi‐layer structure is based on the fact that, for a one‐dimensional model assumption, the horizontal slowness and the azimuth of the phase velocity ψ_phs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface‐offset azimuth and phase‐velocity azimuth domains.     

PARAMETRIZATION OF GRT INVERSION FOR ACOUSTIC AND P–P SCATTERING1

The generalized Radon transform (GRT) inversion contains an explicit relationship between seismic amplitude variations, the reflection angle and the physical parameters which can be used to describe the earth efficiently for inversion purposes. Using this relationship, we have derived parametrizations for acoustic and P–P scattering so that the variations in seismic amplitude with reflection angle for each parameter are sufficiently independent. These parametrizations show that small offset and large offset amplitudes are related to different physical parameters. In the case of acoustic scattering, the small-offset amplitudes are related to impedance variations while large-offset amplitudes are related to velocity variations. A similar result has been established for P–P scattering. The Born approximation (which is used to derive the GRT inversion) does not correctly predict the amplitude due to velocity variations at large offsets, and thus the inversion of velocity is not as satisfactory as the inversion of impedance.     

一种倾角时差校正和叠前成象方法

本文从测量射线参数出发进行反向射线追踪,导出倾角时差校正（DMO）的公式。经过DMO后,可以从一组等炮检距剖面得出共分角线点道集。用于对这些道集进行叠加的速度值与界面倾角无关。对经过DMO的资料的等时切片进行叠前成象（PSI）,就可以把分布在圆上的绕射能量沿圆弧加起来,并放在圆弧上对应于最大炮检距的位置。经过这两种处理,再应用标准的速度分析和叠加方法,就可得出偏移后的剖面。这两种处理均与速度无关。最后用物理模型试验说明了DMO和PSI的效果是好的。     

波动方程CT技术在地震数据处理中的应用

本文从畸变的Born近似的微扰技术出发,给出了利用广义Radon变换和Fourier积分算子的理论反演介质间断性的原理.将声学的广义Radon变换与经典Radon变换进行类比,近似地导出了声学广义Radon变换的反演公式. 本文对于反射地震学的情况,提出了一种拟线性化方法,考虑了成象点的一次散射场,从某种程度上减少了Born近似对弱散射的苛求. 利用同一模型的理论记录和物理实验记录的反演计算结果对提出的方法进行了验证,并讨论了进一步提高成象精度的方法.     

Moveout approximation for horizontal transversely isotropic and vertical transversely isotropic layered medium. Part II: effective model‡

We use residual moveouts measured along continuous full azimuth reflection angle gathers, in order to obtain effective horizontal transversely isotropic model parameters. The angle gathers are generated through a special angle domain imaging system, for a wide range of reflection angles and full range of phase velocity azimuths. The estimation of the effective model parameters is performed in two stages. First, the background horizontal transversely isotropic (HTI)/vertical transversely isotropic (VTI) layered model is used, along with the values of reflection angles, for converting the measured residual moveouts (or traveltime errors) into azimuthally dependent normal moveout (NMO) velocities. Then we apply a digital Fourier transform to convert the NMO velocities into azimuthal wavenumber domain, in order to obtain the effective HTI model parameters: vertical time, vertical compression velocity, Thomsen parameter delta and the azimuth of the medium axis of symmetry. The method also provides a reliability criterion of the HTI assumption. The criterion shows whether the medium possesses the HTI type of symmetry, or whether the azimuthal dependence of the residual traveltime indicates to a more complex azimuthal anisotropy. The effective model used in this approach is defined for a 1D structure with a set of HTI, VTI and isotropic layers (with at least one HTI layer). We describe and analyse the reduction of a multi‐layer structure into an equivalent effective HTI model. The equivalent model yields the same NMO velocity and the same offset azimuth on the Earth's surface as the original layered structure, for any azimuth of the phase velocity. The effective model approximates the kinematics of an HTI/VTI layered structure using only a few parameters. Under the hyperbolic approximation, the proposed effective model is exact.     

Wide-angle linear forward modelling of synthetic seismograms

We address the issue of linearity and scale dependence in forward modelling of seismic data from well logs, for large ray parameters, wide angles or large offsets. We present a forward model, within the context of seismic‐to‐well matching, that is linearized in the elastic properties of the earth. This model preserves linearity at large ray parameters and can handle fine‐layering effects such as induced anisotropy. Starting from a low‐contrast small‐ray‐parameter model, we extend it to a large‐ray‐parameter model by fully linearizing the elastic‐property contrasts. Overall linearity of the forward model is extended by partitioning the compressional‐wave and shear‐wave velocity fields into two fundamental scales: a kinematic scale that governs wavefield propagation effects and a dynamic scale that governs wavefield scattering effects. This analysis reveals that the standard practice in forward modelling of strongly filtering the ratio of compressional‐wave velocity to shear‐wave velocity is well founded in the underlying physics. The partitioning of the velocity fields also leads naturally to forward modelling that accounts fully for stretch effects, to resolution of the angle‐of‐incidence versus ray‐parameter dichotomy in seismic‐amplitude analysis, and to full accounting for induced anisotropy and dispersion effects due to fine‐layering of isotropic media. With the onset of routine long‐offset acquisition and the compelling need to optimize asset management in order to maximize reserve recovery, this forward model recognizes the physics of seismic wave propagation and enables a more complete exploitation of amplitude information in pre‐critical seismic data.     

Linearized rays,amplitude and inversion

In this paper, ray theoretical amplitudes and travel times are calculated in slightly perturbed velocity models using perturbation analysis. Also, test inversions using travel time and amplitude are computed. The pertubation method is tested using a 3-D velocity model for NORSAR having velocity variations up to 8.0 percent. The perturbed amplitudes are found to be in excellent agreement with the calculated ray amplitudes. Velocity inversions based on travel time and amplitude are next investigated. Perturbation analysis using linearized ray equations is efficiently used to compute amplitude derivatives with respect to model parameters. The trial linearized inversions use smaller velocity variations of 1.7 percent to avoid possible effects due to ray shift, even though the perturbation analysis is valid for larger variations. The trial 2-D inversion results show that linearized amplitude inversions are complementary and not redundant to travel time inversions, even in smoothly varying models.     

Moveout approximation for horizontal transversely isotropic and vertical transversely isotropic layered medium. Part I: 1D ray propagation‡

Anisotropy in subsurface geological models is primarily caused by two factors: sedimentation in shale/sand layers and fractures. The sedimentation factor is mainly modelled by vertical transverse isotropy (VTI), whereas the fractures are modelled by a horizontal transversely isotropic medium (HTI). In this paper we study hyperbolic and non‐hyperbolic normal reflection moveout for a package of HTI/VTI layers, considering arbitrary azimuthal orientation of the symmetry axis at each HTI layer. We consider a local 1D medium, whose properties change vertically, with flat interfaces between the layers. In this case, the horizontal slowness is preserved; thus, the azimuth of the phase velocity is the same for all layers of the package. In general, however, the azimuth of the ray velocity differs from the azimuth of the phase velocity. The ray azimuth depends on the layer properties and may be different for each layer. In this case, the use of the Dix equation requires projection of the moveout velocity of each layer on the phase plane. We derive an accurate equation for hyperbolic and high‐order terms of the normal moveout, relating the traveltime to the surface offset, or alternatively, to the subsurface reflection angle. We relate the azimuth of the surface offset to its magnitude (or to the reflection angle), considering short and long offsets. We compare the derived approximations with analytical ray tracing.     

Kirchhoff叠前时间偏移角度道集

提出三维Kirchhoff叠前时间偏移角度域共像点道集的改进算法,克服传统角度求取算法局限,可计算相对倾斜地层法线入射角;与Kirchhoff直射线叠前时间偏移求角度算法相比,本文方法考虑射线弯曲效应,包含层速度,角度范围加大,更接近真实入射角;计算走时采取弯曲射线或者适应线性横向变速介质的非对称走时等算法,角度道集在大角度处得到拉平;采用相对保幅的权因子以及覆盖次数校正技术,有利于叠前AVA反演.模型测试结果表明：叠前时间偏移角度道集,相对CMP、CRP所转化角度道集,更准确反应AVA效应;实际三维数据测试表明本文方法可以提供品质优良的角度道集,适用于AVA分析、反演,提高叠前反演分辨率.     

基于Gabor-Daubechies小波束叠前深度偏移的角度域共成像道集

传统炮检距域共像集（CIG）在复杂介质中因波传播的多路径而存在反射体位置不确定的问题. 角度域CIG由于克服了这一缺陷而逐步成为速度分析、AVA以及振幅保真偏移成像等研究的主要手段. 以波动理论为基础的地震偏移成像方法的发展为获得高质量的角度域CIG提供了可靠的实现途径. 其中,基于波场局域化分解和传播的小波束域波场延拓和偏移成像方法,因其波场分解基本函数和传播算子在空间和方向上的双重局域特性,而成为角度相关分析研究的有效工具. 本文在采用Gabor Daubechies框架分解的小波束叠前角度域偏移成像基础上,利用不同的叠加方法由局部角度域像矩阵得到了反射角域CIG（CRAIG）和倾角域CIG（CDAIG）. 以SEG EAGE二维盐体模型为例,通过对CRAIG和CDAIG的对比,探讨了这两种角度域CIG的特点及其在地震偏移成像中的潜在应用.     

三维倾斜界面PS转换波CMP道集时距及参数估计

在PS转换波资料处理过程中,往往需要联合P波资料提供相应的模型.在实际应用中存在P波和PS转换波层位对比困难.本文仅利用PS转换波数据,通过三维倾斜界面PS转换波CMP道集精确时距关系推导了近似时距解析表达式;分析了PS波的精确与近似时距关系随测线方位、界面倾角与倾向的变化规律及其拟合误差;并讨论了近似时距关系的三个时距参数随方位的变化特征;理论上给出描述时距的三维倾斜界面倾角、倾向、深度、纵波速度和横波速度这5个独立参数的估计方法,并通过理论模拟数据证明了该方法的可行性.     

Automatic cross-well tomography: an application of the differential semblance optimization to two real examples

An automatic tomography algorithm, based on differential semblance optimization (DSO), has been used to invert real cross-well seismic data for the background velocity. The method relies on the first-arrival transmitted waves. Given a background velocity model, the traveltimes between the sources and the receivers are computed, then semblance panels are created by back-propagating the data traces. If the velocity model is correct all the first-arrival transmitted waves will be aligned in the semblance panels. The DSO method consists of finding the background velocity by minimizing the L ₂-norm of the difference between adjacent back-propagated traces. Thanks to the good behaviour of this DSO cost function about the solution, a local (gradient) optimization can be performed. This provides a relatively fast algorithm when ray tracing and analytic computation of the gradient are used.
Unfortunately the method fails in the presence of caustics in the data. However, this difficulty can be circumvented by applying suitable masks to the data. This approach is first applied to a synthetic example then to two real data sets: the McElroy data set recorded in West Texas and the NIMR data set recorded in Oman. The results are quite encouraging and similar to those obtained with classical tomography.     

Velocity model updating in prestack Kirchhoff time migration for PS converted waves: Part I – Theory

A velocity model updating approach is developed based on moveout analysis of the diffraction curve of PS converted waves in prestack Kirchhoff time migration. The diffraction curve can be expressed as a product of two factors: one factor depending on the PS converted‐wave velocity only, and the other factor depending on all parameters. The velocity‐dependent factor represents the hyperbolic behaviour of the moveout and the other is a scale factor that represents the non‐hyperbolic behaviour of the moveout. This non‐hyperbolic behaviour of the moveout can be corrected in prestack Kirchhoff time migration to form an inverse normal‐moveout common‐image‐point gather in which only the hyperbolic moveout is retained. This hyperbolic moveout is the moveout that would be obtained in an isotropic equivalent medium. A hyperbolic velocity is then estimated from this gather by applying hyperbolic moveout analysis. Theoretical analysis shows that for any given initial velocity, the estimated hyperbolic velocity converges by an iterative procedure to the optimal velocity if the velocity ratio is optimal or to a value closer to the optimal velocity if the velocity ratio is not optimal. The velocity ratio (V_P/V_S) has little effect on the estimation of the velocity. Applying this technique to a synthetic seismic data set confirms the theoretical findings. This work provides a practical method to obtain the velocity model for prestack Kirchhoff time migration.     

COMPUTATION OF THE NORMAL MOVEOUT VELOCITY IN 3D LATERALLY INHOMOGENEOUS MEDIA WITH CURVED INTERFACES*

For a 3D velocity model of curved first order interfaces and layer velocities which are arbitrary smooth functions of the space coordinates, the normal moveout (NMO)-velocity can be computed by numerically integrating a system of first order ordinary differential equations for a hypothetical wavefront that originates at the normal incidence point of the normal ray and moves up along the ray to the common mid-point of the common datum point (CDP) profile.     

Migration velocity analysis using pre‐stack wave fields

Using both image and data domains to perform velocity inversion can help us resolve the long and short wavelength components of the velocity model, usually in that order. This translates to integrating migration velocity analysis into full waveform inversion. The migration velocity analysis part of the inversion often requires computing extended images, which is expensive when using conventional methods. As a result, we use pre‐stack wavefield (the double‐square‐root formulation) extrapolation, which includes the extended information (subsurface offsets) naturally, to make the process far more efficient and stable. The combination of the forward and adjoint pre‐stack wavefields provides us with update options that can be easily conditioned to improve convergence. We specifically use a modified differential semblance operator to split the extended image into a residual part for classic differential semblance operator updates and the image (Born) modelling part, which provides reflections for higher resolution information. In our implementation, we invert for the velocity and the image simultaneously through a dual objective function. Applications to synthetic examples demonstrate the features of the approach.