Full waveform inversion using oriented time‐domain imaging method for vertical transverse isotropic media

Full waveform inversion for reflection events is limited by its linearised update requirements given by a process equivalent to migration. Unless the background velocity model is reasonably accurate, the resulting gradient can have an inaccurate update direction leading the inversion to converge what we refer to as local minima of the objective function. In our approach, we consider mild lateral variation in the model and, thus, use a gradient given by the oriented time‐domain imaging method. Specifically, we apply the oriented time‐domain imaging on the data residual to obtain the geometrical features of the velocity perturbation. After updating the model in the time domain, we convert the perturbation from the time domain to depth using the average velocity. Considering density is constant, we can expand the conventional 1D impedance inversion method to two‐dimensional or three‐dimensional velocity inversion within the process of full waveform inversion. This method is not only capable of inverting for velocity, but it is also capable of retrieving anisotropic parameters relying on linearised representations of the reflection response. To eliminate the crosstalk artifacts between different parameters, we utilise what we consider being an optimal parametrisation for this step. To do so, we extend the prestack time‐domain migration image in incident angle dimension to incorporate angular dependence needed by the multiparameter inversion. For simple models, this approach provides an efficient and stable way to do full waveform inversion or modified seismic inversion and makes the anisotropic inversion more practicable. The proposed method still needs kinematically accurate initial models since it only recovers the high‐wavenumber part as conventional full waveform inversion method does. Results on synthetic data of isotropic and anisotropic cases illustrate the benefits and limitations of this method.     

Perturbation‐based moveout approximations in anisotropic media

The moveout approximations play an important role in seismic data processing. The standard hyperbolic moveout approximation is based on an elliptical background model with two velocities: vertical and normal moveout. We propose a new set of moveout approximations based on a perturbation series in terms of anellipticity parameters using the alternative elliptical background model defined by vertical and horizontal velocities. We start with a transversely isotropic medium with a vertical symmetry axis. Then, we extend this approach to a homogeneous orthorhombic medium. To define the perturbation coefficients for a new background, we solve the eikonal equation with horizontal velocities in transversely isotropic medium with a vertical symmetry axis and orthorhombic media. To stabilise the perturbation series and improve the accuracy, the Shanks transform is applied for all the cases. We select different parameterisations for both velocities and anellipticity parameters for an orthorhombic model. From the comparison in traveltime error, the new moveout approximations result in better accuracy comparing with the standard perturbation‐based methods and other approximations.     

Optimal implicit staggered‐grid finite‐difference schemes based on the sampling approximation method for seismic modelling

We propose new implicit staggered‐grid finite‐difference schemes with optimal coefficients based on the sampling approximation method to improve the numerical solution accuracy for seismic modelling. We first derive the optimized implicit staggered‐grid finite‐difference coefficients of arbitrary even‐order accuracy for the first‐order spatial derivatives using the plane‐wave theory and the direct sampling approximation method. Then, the implicit staggered‐grid finite‐difference coefficients based on sampling approximation, which can widen the range of wavenumber with great accuracy, are used to solve the first‐order spatial derivatives. By comparing the numerical dispersion of the implicit staggered‐grid finite‐difference schemes based on sampling approximation, Taylor series expansion, and least squares, we find that the optimal implicit staggered‐grid finite‐difference scheme based on sampling approximation achieves greater precision than that based on Taylor series expansion over a wider range of wavenumbers, although it has similar accuracy to that based on least squares. Finally, we apply the implicit staggered‐grid finite difference based on sampling approximation to numerical modelling. The modelling results demonstrate that the new optimal method can efficiently suppress numerical dispersion and lead to greater accuracy compared with the implicit staggered‐grid finite difference based on Taylor series expansion. In addition, the results also indicate the computational cost of the implicit staggered‐grid finite difference based on sampling approximation is almost the same as the implicit staggered‐grid finite difference based on Taylor series expansion.     

带地形的可控源电磁法2.5维各向异性介质正演研究

目前,对于可控源电磁法各向异性介质2.5维问题,主要采用一次场、二次场分离的方法消除场源奇异性并降低截断边界对计算区域的影响.该方法数值计算精度高,但是很难适用于复杂地形条件下的数值模拟.针对复杂地形问题,基于总场的有限元方法表现出一定的优越性,然而,这种方法存在场源奇异性问题和截断边界问题.本文采用基于总场计算的方法对带地形的可控源电磁法2.5维各向异性介质进行模拟研究,推导了考虑电导率和介电常数各向异性的2.5维控制方程;引入网格加密-收缩算法降低场源奇异性的影响范围,提升数值计算效率;引入行波分解边界条件降低截断边界的影响;提出任意采样反傅里叶变换方法,快速、高精度地计算出空间域电磁场分量.理论模型数值算例中：首先,验证了本文算法的有效性;其次,对任意各向异性倾角产生的可控源电磁响应规律进行研究;最后,采用山丘模型对各向异性介质电磁场的响应规律进行了模拟和分析.

     

Theory of equivalent staggered‐grid schemes: application to rotated and standard grids in anisotropic media

The previous finite‐difference numerical schemes designed for direct application to second‐order elastic wave equations in terms of displacement components are strongly dependent on Poisson's ratio. This fact makes theses schemes useless for modelling in offshore regions or even in onshore regions where there is a high Poisson's ratio material. As is well known, the use of staggered‐grid formulations solves this drawback. The most common staggered‐grid algorithms apply central‐difference operators to the first‐order velocity–stress wave equations. They have been one of the most successfully applied numerical algorithms for seismic modelling, although these schemes require more computational memory than those mentioned based on second‐order wave equations. The goal of the present paper is to develop a general theory that enables one to formulate equivalent staggered‐grid schemes for direct application to hyperbolic second‐order wave equations. All the theory necessary to formulate these schemes is presented in detail, including issues regarding source application, providing a general method to construct staggered‐grid formulations to a wide range of cases. Afterwards, the equivalent staggered‐grid theory is applied to anisotropic elastic wave equations in terms of only velocity components (or similar displacements) for two important cases: general anisotropic media and vertical transverse isotropy media using, respectively, the rotated and the standard staggered‐grid configurations. For sake of simplicity, we present the schemes in terms of velocities in the second‐ and fourth‐order spatial approximations, with second‐order approximation in time for 2D media. However, the theory developed is general and can be applied to any set of second‐order equations (in terms of only displacement, velocity, or even stress components), using any staggered‐grid configuration with any spatial approximation order in 2D or 3D cases. Some of these equivalent staggered‐grid schemes require less computer memory than the corresponding standard staggered‐grid formulation, although the programming is more evolved. As will be shown in theory and practice, with numerical examples, the equivalent staggered‐grid schemes produce results equivalent to corresponding standard staggered‐grid schemes with computational advantages. Finally, it is important to emphasize that the equivalent staggered‐grid theory is general and can be applied to other modelling contexts, e.g., in electrodynamical and poroelastic wave propagation problems in a systematic and simple way.     

Ray–Kirchhoff multicomponent borehole seismic modelling in 3D heterogeneous, anisotropic media

Kirchhoff–Helmholtz (KH) theory is extended to synthesize two-way elastic wave propagation in 3D laterally heterogeneous, anisotropic media. I have developed and tested numerically a specialized algorithm for the generation of three-component synthetic seismograms in multi-layered isotropic and transversely isotropic (TI) media with dipping interfaces and tilted axes of symmetry. This algorithm can be applied to vertical seismic profile (VSP) geometries and works well when the receiver is located near the reflector interface. It is superior to ray methods in predicting elliptical polarization effects observed on radial and transverse components. The algorithm is used to study converted-wave propagation for determining fracture-related shear-wave anisotropy in realistic reservoir models. Results show that all wavefront attributes are strongly affected by the anisotropy. However, it is necessary to resolve a trade-off between the effects of fractures and formation dip prior to converted-wave interpretation. These results provide some assurance that the present scheme is sufficiently versatile to handle shear wave behaviour due to various generalized rays propagating in complex geological models.     

Eigenrays in 3D heterogeneous anisotropic media,Part I: Kinematics

We present a new ray bending approach, referred to as the Eigenray method, for solving two‐point boundary‐value kinematic and dynamic ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where conventional initial‐value ray shooting methods, followed by numerical convergence techniques, become challenging. The kinematic ray bending solution corresponds to the vanishing first traveltime variation, leading to a stationary path between two fixed endpoints (Fermat's principle), and is governed by the nonlinear second‐order Euler–Lagrange equation. The solution is based on a finite‐element approach, applying the weak formulation that reduces the Euler–Lagrange second‐order ordinary differential equation to the first‐order weighted‐residual nonlinear algebraic equation set. For the kinematic finite‐element problem, the degrees of freedom are discretized nodal locations and directions along the ray trajectory, where the values between the nodes are accurately and naturally defined with the Hermite polynomial interpolation. The target function to be minimized includes two essential penalty (constraint) terms, related to the distribution of the nodes along the path and to the normalization of the ray direction. We distinguish between two target functions triggered by the two possible types of stationary rays: a minimum traveltime and a saddle‐point solution (due to caustics). The minimization process involves the computation of the global (all‐node) traveltime gradient vector and the traveltime Hessian matrix. The traveltime Hessian is used for the minimization process, analysing the type of the stationary ray, and for computing the geometric spreading of the entire resolved stationary ray path. The latter, however, is not a replacement for the dynamic ray tracing solution, since it does not deliver the geometric spreading for intermediate points along the ray, nor the analysis of caustics. Finally, we demonstrate the efficiency and accuracy of the proposed method along three canonical examples.     

Eigenrays in 3D heterogeneous anisotropic media,Part II: Dynamics

This paper is the second in a sequel of two papers and dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in the first paper. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e. the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point–source and plane‐wave. For the proposed point–source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in the first paper, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyse these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach.     

基于截断牛顿法的VTI介质声波多参数全波形反演

不同类别参数间的相互耦合使多参数地震全波形反演的非线性程度显著增加, 地震波速度与各向异性参数取值数量级的巨大差异也会使反演问题的性态变差.合理使用Hessian逆算子可以减弱这两类问题对反演的影响, 提高多参数反演的精度, 而截断牛顿法是一种可以比较准确地估计 Hessian 逆算子的优化方法.本文采用截断牛顿法在时间域进行了VTI介质的声波双参数同时反演的研究.不同模型的反演试验表明, 在VTI介质声波双参数同时反演中, 截断牛顿法比有限内存 BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno, L-BFGS)法能更准确地估计 Hessian 逆算子, 进而较好地平衡两类不同参数的同时更新, 得到了比较精确的反演结果.     

Elastic wave‐vector decomposition in heterogeneous anisotropic media

The goal of wave‐mode separation and wave‐vector decomposition is to separate a full elastic wavefield into three wavefields with each corresponding to a different wave mode. This allows elastic reverse‐time migration to handle each wave mode independently. Several of the previously proposed methods to accomplish this task require the knowledge of the polarisation vectors of all three wave modes in a given anisotropic medium. We propose a wave‐vector decomposition method where the wavefield is decomposed in the wavenumber domain via the analytical decomposition operator with improved computational efficiency using low‐rank approximations. The method is applicable for general heterogeneous anisotropic media. To apply the proposed method in low‐symmetry anisotropic media such as orthorhombic, monoclinic, and triclinic, we define the two S modes by sorting them based on their phase velocities (S1 and S2), which are defined everywhere except at the singularities. The singularities can be located using an analytical condition derived from the exact phase‐velocity expressions for S waves. This condition defines a weight function, which can be applied to attenuate the planar artefacts caused by the local discontinuity of polarisation vectors at the singularities. The amplitude information lost because of weighting can be recovered using the technique of local signal–noise orthogonalisation. Numerical examples show that the proposed approach provides an effective decomposition method for all wave modes in heterogeneous, strongly anisotropic media.     

Travel time computations using a compact eikonal equation for vertical transverse isotropic media

Eikonal solvers often have stability problems if the velocity model is mildly heterogeneous. We derive a stable and compact form of the eikonal equation for P‐wave propagation in vertical transverse isotropic media. The obtained formulation is more compact than other formulations and therefore computationally attractive. We implemented ray shooting for this new equation through a Hamiltonian formalism. Ray tracing based on this new equation is tested on both simple as well as more realistic mildly heterogeneous velocity models. We show through examples that the new equation gives travel times that coincide with the travel time picks from wave equation modelling for anisotropic wave propagation.     

Determining the focal mechanisms and depths of relatively small earthquakes using a few stations by full‐waveform modelling

Determining the focal mechanism of earthquakes helps us to better define faults and understand the stress regime. This technique can be helpful in the oil and gas industry where it can be applied to microseismic events. The objective of this paper is to find double couple focal mechanisms, excluding scalar seismic moments, and the depths of small earthquakes using data from relatively few local stations. This objective is met by generating three‐component synthetic seismograms to match the observed normalized velocity seismograms. We first calculate Green's functions given an initial estimate of the earthquake's hypocentre, the locations of the seismic recording stations and a 1D velocity model of the region for a series of depths. Then, we calculate the moment tensor for different combinations of strikes, dips and rakes for each depth. These moment tensors are combined with the Green's functions and then convolved with a source time function to produce synthetic seismograms. We use a grid search to find the synthetic seismogram with the largest objective function that best fits all three components of the observed velocity seismogram. These parameters define the focal mechanism solution of an earthquake. We tested the method using three earthquakes in Southern California with moment magnitudes of 5.0, 5.1 and 4.4 using the frequency range 0.1–2.0 Hz. The source mechanisms of the events were determined independently using data from a multitude of stations. Our results obtained, from as few as three stations, generally match those obtained by the Southern California Earthquake Data Center. The main advantage of this method is that we use relatively high‐frequency full‐waveforms, including those from short‐period instruments, which makes it possible to find the focal mechanism and depth of earthquakes using as few as three stations when the velocity structure is known.     

任意各向异性介质中三维可控源音频大地电磁正演模拟

在一些地层层理发育的地区,地下介质存在显著的电各向异性,此时基于各向同性模型解释含各向异性效应的可控源音频大地电磁（CSAMT）测深观测数据会导致错误的结果.本文通过引入3×3的对称正定张量表征电导率各向异性,采用非结构四面体网格和矢量有限元方法离散电场满足的矢量Helmholtz方程,并将电磁场源等效为系列电偶极子,实现任意各向异性介质中CSAMT高效数值模拟.本文首先通过层状各向异性模型检验三维有限元算法的精度和有效性,进一步建立三维地电模型研究异常体各向异性和围岩各向异性对CSAMT响应的影响,最后使用视电阻率极性图来识别各向异性电导率主轴方向.数值模拟结果表明,各向异性电导率对CSAMT视电阻率幅值及分布规律都有很大影响,视电阻率极性图能够很好地识别各向异性主轴方向.

     

Extended reflectivity method for modelling the propagation of diffusive–viscous wave in dip‐layered media

The reflectivity method plays an important role in seismic modelling. It has been used to model different types of waves propagating in elastic and anelastic media. The diffusive–viscous wave equation was proposed to investigate the relationship between frequency dependence of reflections and fluid saturation. It is also used to describe the attenuation property of seismic wave in a fluid‐saturated medium. The attenuation of diffusive–viscous wave is mainly characterised by the effective attenuation parameters in the equation. Thus, it is essential to obtain those parameters and further characterise the features of the diffusive–viscous wave. In this work, we use inversion method to obtain the effective attenuation parameters through quality factor to investigate the characteristics of diffusive–viscous wave by comparing with those of the viscoacoustic wave. Then, the reflection/transmission coefficients in a dip plane‐layered medium are studied through coordinate transform and plane‐wave theory. Consequently, the reflectivity method is extended to compute seismograms of diffusive–viscous wave in a dip plane multi‐layered medium. Finally, we present two models to simulate the propagation of diffusive–viscous wave in a dip plane multi‐layered medium by comparing the results with those in a viscoacoustic medium. The numerical results demonstrate the validity of our extension of reflectivity method to the diffusive–viscous medium. The numerical examples in both time domain and time–frequency domain show that the reflections from a dip plane interface have significant phase shift and amplitude change compared with the results of horizontal plane interface due to the differences in reflection/transmission coefficients. Moreover, the modelling results show strong attenuation and phase shift in the diffusive–viscous wave compared to those of the viscoacoustic wave.     

Full waveform inversion of SH‐ and Love‐wave data in near‐surface prospecting

We develop a two‐dimensional full waveform inversion approach for the simultaneous determination of S‐wave velocity and density models from SH ‐ and Love‐wave data. We illustrate the advantages of the SH/Love full waveform inversion with a simple synthetic example and demonstrate the method's applicability to a near‐surface dataset, recorded in the village ?achtice in Northwestern Slovakia. Goal of the survey was to map remains of historical building foundations in a highly heterogeneous subsurface. The seismic survey comprises two parallel SH‐profiles with maximum offsets of 24 m and covers a frequency range from 5 Hz to 80 Hz with high signal‐to‐noise ratio well suited for full waveform inversion. Using the Wiechert–Herglotz method, we determined a one‐dimensional gradient velocity model as a starting model for full waveform inversion. The two‐dimensional waveform inversion approach uses the global correlation norm as objective function in combination with a sequential inversion of low‐pass filtered field data. This mitigates the non‐linearity of the multi‐parameter inverse problem. Test computations show that the influence of visco‐elastic effects on the waveform inversion result is rather small. Further tests using a mono‐parameter shear modulus inversion reveal that the inversion of the density model has no significant impact on the final data fit. The final full waveform inversion S‐wave velocity and density models show a prominent low‐velocity weathering layer. Below this layer, the subsurface is highly heterogeneous. Minimum anomaly sizes correspond to approximately half of the dominant Love‐wavelength. The results demonstrate the ability of two‐dimensional SH waveform inversion to image shallow small‐scale soil structure. However, they do not show any evidence of foundation walls.     

An adaptive free‐surface expression for three‐dimensional finite‐difference frequency‐domain modelling of elastic wave

Finite‐difference frequency‐domain modelling of seismic wave propagation is attractive for its efficient solution of multisource problems, and this is crucial for full‐waveform inversion and seismic imaging, especially in the three‐dimensional seismic problem. However, implementing the free surface in the finite‐difference method is nontrivial. Based on an average medium method and the limit theorem, we present an adaptive free‐surface expression to describe the behaviour of wavefields at the free surface, and no extra work for the free‐surface boundary condition is needed. Essentially, the proposed free‐surface expression is a modification of density and constitutive relation at the free surface. In comparison with a direct difference approximate method of the free‐surface boundary condition, this adaptive free‐surface expression can produce more accurate and stable results for a broad range of Poisson's ratio. In addition, this expression has a good performance in handling the lateral variation of Poisson's ratio adaptively and without instability.     

Frequency‐domain waveform modelling and inversion for coupled media using a symmetric impedance matrix

To simulate the seismic signals that are obtained in a marine environment, a coupled system of both acoustic and elastic wave equations is solved. The acoustic wave equation for the fluid region simulates the pressure field while minimizing the number of degrees of freedom of the impedance matrix, and the elastic wave equation for the solid region simulates several elastic events, such as shear waves and surface waves. Moreover, by combining this coupled approach with the waveform inversion technique, the elastic properties of the earth can be inverted using the pressure data obtained from the acoustic region. However, in contrast to the pure acoustic and elastic cases, the complex impedance matrix for the coupled media does not have a symmetric form because of the boundary (continuity) condition at the interface between the acoustic and elastic elements. In this study, we propose a manipulation scheme that makes the complex impedance matrix for acoustic–elastic coupled media to take a symmetric form. Using the proposed symmetric matrix, forward and backward wavefields are identical to those generated by the conventional approach; thus, we do not lose any accuracy in the waveform inversion results. However, to solve the modified symmetric matrix, LDLT factorization is used instead of LU factorization for a matrix of the same size; this method can mitigate issues related to severe memory insufficiency and long computation times, particularly for large‐scale problems.     

层状各向异性地层中三维感应测井响应快速计算及资料处理

本文采用广义反射系数法推导了水平层状各向异性地层中电磁场的积分解析解,并利用快速汉克尔变换技术实现了三维感应仪器测井响应的快速计算.三维感应测井响应与地层水平电导率、垂直电导率和井斜角及仪器方位角同时有关,单一分量的测井曲线不能满足资料解释的需要.通过对仪器测量分量响应特征的考察,本文提出了一种基于组合量测井曲线的资料直观解释方法.数值模拟显示,交叉分量相关组合量可准确划分地层纵向边界,并可直观识别各向异性层;与单独分量相比,主分量相关组合量提高了纵向分辨率、减弱了与地层电导率参数的非线性关系.     

Pseudo‐spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media

Staggering grid is a very effective way to reduce the Nyquist errors and to suppress the non‐causal ringing artefacts in the pseudo‐spectral solution of first‐order elastic wave equations. However, the straightforward use of a staggered‐grid pseudo‐spectral method is problematic for simulating wave propagation when the anisotropy level is greater than orthorhombic or when the anisotropic symmetries are not aligned with the computational grids. Inspired by the idea of rotated staggered‐grid finite‐difference method, we propose a modified pseudo‐spectral method for wave propagation in arbitrary anisotropic media. Compared with an existing remedy of staggered‐grid pseudo‐spectral method based on stiffness matrix decomposition and a possible alternative using the Lebedev grids, the rotated staggered‐grid‐based pseudo‐spectral method possesses the best balance between the mitigation of artefacts and efficiency. A 2D example on a transversely isotropic model with tilted symmetry axis verifies its effectiveness to suppress the ringing artefacts. Two 3D examples of increasing anisotropy levels demonstrate that the rotated staggered‐grid‐based pseudo‐spectral method can successfully simulate complex wavefields in such anisotropic formations.     

The variable projection method for waveform inversion with an unknown source function

This paper compares three alternative algorithms for simultaneously estimating a source wavelet at the same time as an earth model in full‐waveform inversion: (i) simultaneous descent, (ii) alternating descent and (iii) descent with the variable projection method. The latter is a technique for solving separable least‐squares problems that is well‐known in the applied mathematics literature. When applied to full‐waveform inversion, it involves making the source wavelet an implicit function of the earth model via a least‐squares filter‐estimation process. Since the source wavelet becomes purely a function of medium parameters, it no longer needs to be treated as a separate unknown in the inversion. Essentially, the predicted data are projected onto the measured data in a least‐squares sense at every function evaluation, making use of the fact that the filter estimation problem is trivial when compared to the full‐waveform inversion problem. Numerical tests on a simple 1D model indicate that the variable projection method gives the best result; actually producing results in quality that are very similar to control experiments with a known, correct wavelet.