Preconditioned non‐linear conjugate gradient method for frequency domain full‐waveform seismic inversion

We present preconditioned non‐linear conjugate gradient algorithms as alternatives to the Gauss‐Newton method for frequency domain full‐waveform seismic inversion. We designed two preconditioning operators. For the first preconditioner, we introduce the inverse of an approximate sparse Hessian matrix. The approximate Hessian matrix, which is highly sparse, is constructed by judiciously truncating the Gauss‐Newton Hessian matrix based on examining the auto‐correlation and cross‐correlation of the Jacobian matrix. As the second preconditioner, we employ the approximation of the inverse of the Gauss‐Newton Hessian matrix. This preconditioner is constructed by terminating the iteration process of the conjugate gradient least‐squares method, which is used for inverting the Hessian matrix before it converges. In our preconditioned non‐linear conjugate gradient algorithms, the step‐length along the search direction, which is a crucial factor for the convergence, is carefully chosen to maximize the reduction of the cost function after each iteration. The numerical simulation results show that by including a very limited number of non‐zero elements in the approximate Hessian, the first preconditioned non‐linear conjugate gradient algorithm is able to yield comparable inversion results to the Gauss‐Newton method while maintaining the efficiency of the un‐preconditioned non‐linear conjugate gradient method. The only extra cost is the computation of the inverse of the approximate sparse Hessian matrix, which is less expensive than the computation of a forward simulation of one source at one frequency of operation. The second preconditioned non‐linear conjugate gradient algorithm also significantly saves the computational expense in comparison with the Gauss‐Newton method while maintaining the Gauss‐Newton reconstruction quality. However, this second preconditioned non‐linear conjugate gradient algorithm is more expensive than the first one.     

Anisotropic elastic full‐waveform inversion of walkaway vertical seismic profiling data from the Arabian Gulf

Borehole seismic addresses the need for high‐resolution images and elastic parameters of the subsurface. Full‐waveform inversion of vertical seismic profile data is a promising technology with the potential to recover quantitative information about elastic properties of the medium. Full‐waveform inversion has the capability to process the entire wavefield and to address the wave propagation effects contained in the borehole data—multi‐component measurements; anisotropic effects; compressional and shear waves; and transmitted, converted, and reflected waves and multiples. Full‐waveform inversion, therefore, has the potential to provide a more accurate result compared with conventional processing methods. We present a feasibility study with results of the application of high‐frequency (up to 60 Hz) anisotropic elastic full‐waveform inversion to a walkaway vertical seismic profile data from the Arabian Gulf. Full‐waveform inversion has reproduced the majority of the wave events and recovered a geologically plausible layered model with physically meaningful values of the medium.     

Monte-Carlo Bayesian look-ahead inversion of walkaway vertical seismic profiles

We describe a method to invert a walkaway vertical seismic profile (VSP) and predict elastic properties (P‐wave velocity, S‐wave velocity and density) in a layered model looking ahead of the deepest receiver. Starting from Bayes's rule, we define a posterior distribution of layered models that combines prior information (on the overall variability of and correlations among the elastic properties observed in well logs) with information provided by the VSP data. This posterior distribution of layered models is sampled by a Monte‐Carlo method. The sampled layered models agree with prior information and fit the VSP data, and their overall variability defines the uncertainty in the predicted elastic properties. We apply this technique first to a zero‐offset VSP data set, and show that uncertainty in the long‐wavelength P‐wave velocity structure results in a sizable uncertainty in the predicted elastic properties. We then use walkaway VSP data, which contain information on the long‐wavelength P‐wave velocity (in the reflection moveout) and on S‐wave velocity and density contrasts (in the change of reflectivity with offset). The uncertainty of the look‐ahead prediction is considerably decreased compared with the zero‐offset VSP, and the predicted elastic properties are in good agreement with well‐log measurements.     

Analysis of different parameterisations of waveform inversion of compressional body waves in an elastic transverse isotropic Earth with a vertical axis of symmetry

In a multi‐parameter waveform inversion, the choice of the parameterisation influences the results and their interpretations because leakages and the tradeoff between parameters can cause artefacts. We review the parameterisation selection when the inversion focuses on the recovery of the intermediate‐to‐long wavenumbers of the compressional velocities from the compressional body (P) waves. Assuming a transverse isotropic medium with a vertical axis of symmetry and weak anisotropy, analytical formulas for the radiation patterns are developed to quantify the tradeoff between the shear velocity and the anisotropic parameters and the effects of setting to zero the shear velocity in the acoustic approach. Because, in an anisotropic medium, the radiation patterns depend on the angle of the incident wave with respect to the vertical axis, two particular patterns are discussed: a transmission pattern when the ingoing and outgoing slowness vectors are parallel and a reflection pattern when the ingoing and outgoing slowness vectors satisfy Snell's law. When the inversion aims at recovering the long‐to‐intermediate wavenumbers of the compressional velocities from the P‐waves, we propose to base the parameterisation choice on the transmission patterns. Since the P‐wave events in surface seismic data do not constrain the background (smooth) vertical velocity due to the velocity/depth ambiguity, the preferred parameterisation contains a parameter that has a transmission pattern concentrated along the vertical axis. This parameter can be fixed during the inversion which reduces the size of the model space. The review of several parameterisations shows that the vertical velocity, the Thomsen parameter δ, or the Thomsen parameter ε have a transmission pattern along the vertical axis depending on the parameterisation choice. The review of the reflection patterns of those selected parameterisations should be done in the elastic context. Indeed, when reflection data are also inverted, there are potential leakages of the shear parameter at intermediate angles when we carry out acoustic inversion.     

Cauchy prior distribution-based AVO elastic parameter estimation via weakly nonlinear waveform inversion

Cauchy priori distribution-based Bayesian AVO reflectivity inversion may lead to sparse estimates that are sensitive to large reflectivities. For the inversion, the computation of the covariance matrix and regularized terms requires prior estimation of model parameters, which makes the iterative inversion weakly nonlinear. At the same time, the relations among the model parameters are assumed linear. Furthermore, the reflectivities, the results of the inversion, or the elastic parameters with cumulative error recovered by integrating reflectivities are not well suited for detecting hydrocarbons and fuids. In contrast, in Bayesian linear AVO inversion, the elastic parameters can be directly extracted from prestack seismic data without linear assumptions for the model parameters. Considering the advantages of the abovementioned methods, the Bayesian AVO reflectivity inversion process is modified and Cauchy distribution is explored as a prior probability distribution and the time-variant covariance is also considered. Finally, we propose a new method for the weakly nonlinear AVO waveform inversion. Furthermore, the linear assumptions are abandoned and elastic parameters, such as P-wave velocity, S-wave velocity, and density, can be directly recovered from seismic data especially for interfaces with large reflectivities. Numerical analysis demonstrates that all the elastic parameters can be estimated from prestack seismic data even when the signal-to-noise ratio of the seismic data is low.     

Joint AVO inversion, wavelet estimation and noise-level estimation using a spatially coupled hierarchical Bayesian model

The main objective of the AVO inversion is to obtain posterior distributions for P-wave velocity, S-wave velocity and density from specified prior distributions, seismic data and well-log data. The inversion problem also involves estimation of a seismic wavelet and the seismic-noise level. The noise model is represented by a zero mean Gaussian distribution specified by a covariance matrix. A method for joint AVO inversion, wavelet estimation and estimation of the noise level is developed in a Bayesian framework. The stochastic model includes uncertainty of both the elastic parameters, the wavelet, and the seismic and well-log data. The posterior distribution is explored by Markov-chain Monte-Carlo simulation using the Gibbs' sampler algorithm. The inversion algorithm has been tested on a seismic line from the Heidrun Field with two wells located on the line. The use of a coloured seismic-noise model resulted in about 10% lower uncertainties for the P-wave velocity, S-wave velocity and density compared with a white-noise model. The uncertainty of the estimated wavelet is low. In the Heidrun example, the effect of including uncertainty of the wavelet and the noise level was marginal with respect to the AVO inversion results.     

场地土层模型参数的地震动记录反演方法

考虑场地土层不同物理参数的综合反演,改进了水平与竖向谱比(Horizontal-to-Vertical Spectral Ratio,简称HVSR)混合全局优化反演方法,进一步通过土层的S、P波波速、厚度、泊松比、密度和S、P波品质因子等土层参数反演的敏感性分析,形成了可同时反演场地土层厚度及剪切波速的混合全局优化反演方法.以美国GVDA和日本IWTH27竖向强震动观测台阵为例,分别以理论HVSR及加速度观测记录获得的HVSR曲线为目标,反演获得了场地浅层速度结构,并与观测台阵场地钻孔揭示的土层模型进行比较,验证了发展的反演方法的合理性和适用性.本文研究表明,基于地震加速度记录的HVSR全局优化反演方法是获取场地土层速度结构的一种有效的途径.     

Site characterization using Gauss–Newton inversion of 2-D full seismic waveform in the time domain

Site characterization for design of deep foundations is very crucial, as unanticipated site conditions still represent significant problems and disputes occur during construction. Traditional surface-based geophysical methods, which use wave velocity dispersion or first-arrival times, have been widely used recently to assess spatial variation; however they cannot well characterize reverse profiles or buried low-velocity zones. For better characterization of these challenging site conditions, a full waveform inversion based on Gauss–Newton method is presented. The inversion scheme is based on a finite-difference solution of the 2-D elastic wave equation in the time domain. The strength of this approach is the ability to generate all possible wave types of seismic wavefields that are then compared with observed data to infer complex subsurface properties. Virtual sources and reciprocity of wavefields are used for calculation of partial derivative wavefields to reduce computer time. Cross convolution between observed and estimated wavefields are also employed to allow the technique to be independent of the source signatures. The capability of the presented technique is tested with both synthetic and real experimental data sets. The inversion results from synthetic data show the ability of characterizing anomalies of low- and high-velocity zones, and the inversion results from real data are generally consistent with SPT N-value, including the identification of a buried low-velocity layer.     

Linearized seismic waveform inversion using a multiple re-weighted least-squares method with QR preconditioning

Linearized inversion methods such as Gauss‐Newton and multiple re‐weighted least‐squares are iterative processes in which an update in the current model is computed as a function of data misfit and the gradient of data with respect to model parameters. The main advantage of those methods is their ability to refine the model parameters although they have a high computational cost for seismic inversion. In the Gauss‐Newton method a system of equations, corresponding to the sensitivity matrix, is solved in the least‐squares sense at each iteration, while in the multiple re‐weighted least‐squares method many systems are solved using the same sensitivity matrix. The sensitivity matrix arising from these methods is usually not sparse, thus limiting the use of standard preconditioners in the solution of the linearized systems. For reduction of the computational cost of the linearized inversion methods, we propose the use of preconditioners based on a partial orthogonalization of the columns of the sensitivity matrix. The new approach collapses a band of co‐diagonals of the normal equations matrix into the main diagonal, being equivalent to computing the least‐squares solution starting from a partial solution of the linear system. The preconditioning is driven by a bandwidth L which can be interpreted as the distance for which the correlation between model parameters is relevant. To illustrate the benefit of the proposed approach to the reduction of the computational cost of the inversion we apply the multiple re‐weighted least‐squares method to the 2D acoustic seismic waveform inversion problem. We verify the reduction in the number of iterations in the conjugate'gradient algorithm as the bandwidth of the preconditioners increases. This effect reduces the total computational cost of inversion as well.     

基于贝叶斯理论的叠前多波联合反演弹性模量方法

AVO反演可以获得地层岩性和流体信息,而叠前反演问题都是高维的和非适定的,因此获得可靠稳定的解对叠前反演至关重要. 本文给出了一种基于贝叶斯理论的纵波和转换波联合反演密度比和模量比的方法. 鉴于剪切模量比、体积模量比可以更好地指示油气,基于岩石物理中速度比与模量比之间的关系,将此关系式代入Zoeppritz方程的近似形式Aki-Richards公式中,得到与模量比有关的反射系数近似公式. 联合纵波和转换波,利用最小二乘准则构建目标函数,最终反演出密度比、剪切模量比、体积模量比三个参数. 在反演过程中引入贝叶斯理论,假定先验信息服从高斯分布,待求参数服从改进的Cauchy分布,并去除待求参数之间的相关性. 利用模型数据和实际数据对本文方法进行测试,并与常规的单独利用纵波数据来反演方法进行比较,结果表明联合反演稳定性更好、精度更高、抗噪音能力更强,验证了本文方法的可行性和有效性.     

弹性波阻抗在时移地震中的应用分析

常规声波波阻抗由于缺乏横波的信息,使得它对于流体的变化很不敏感,而弹性波阻抗反演,除了与纵波速度,密度有关之外,还与横波速度、入射角度有关,由于考虑了AVO(Amplitude Variation with Offset)效应,使得对流体和岩性的预测能力增强.本文将弹性波阻抗反演应用于时移地震中,采用AVA(Amplitude Variation with Angle)约束稀疏脉冲反演来进行弹性波阻抗反演,得到纵横波阻抗等弹性参数,通过交会图分析,将弹性参数转换为油藏参数,如预测含水饱和度的变化,通过本方法可增强对储层流体的预测,提高储层管理能力.     

Frequency‐domain acoustic‐elastic coupled waveform inversion using the Gauss‐Newton conjugate gradient method

We developed a frequency‐domain acoustic‐elastic coupled waveform inversion based on the Gauss‐Newton conjugate gradient method. Despite the use of a high‐performance computer system and a state‐of‐the‐art parallel computation algorithm, it remained computationally prohibitive to calculate the approximate Hessian explicitly for a large‐scale inverse problem. Therefore, we adopted the conjugate gradient least‐squares algorithm, which is frequently used for geophysical inverse problems, to implement the Gauss‐Newton method so that the approximate Hessian is calculated implicitly. Thus, there was no need to store the Hessian matrix. By simultaneously back‐propagating multi‐components consisting of the pressure and displacements, we could efficiently extract information on the subsurface structures. To verify our algorithm, we applied it to synthetic data sets generated from the Marmousi‐2 model and the modified SEG/EAGE salt model. We also extended our algorithm to the ocean‐bottom cable environment and verified it using ocean‐bottom cable data generated from the Marmousi‐2 model. With the assumption of a hard seafloor, we recovered both the P‐wave velocity of complicated subsurface structures as well as the S‐wave velocity. Although the inversion of the S‐wave velocity is not feasible for the high Poisson's ratios used to simulate a soft seafloor, several strategies exist to treat this problem. Our example using multi‐component data showed some promise in mitigating the soft seafloor effect. However, this issue still remains open.     

基于快速非支配排序遗传算法的VTI介质多分量叠前联合反演

在VTI介质中,由于引入了各向异性参数使得多分量多参数地震反演问题的非线性程度显著增加,因此采用传统的权重加权法构建单目标函数进行反演得到的反演结果往往并不理想.本文以反射率法为基础,结合快速非支配排序遗传算法研究了一种VTI介质的多分量叠前联合反演方法.该方法以反射率法为正演方程,应用互相关原理构建PP波和PSV波的多目标函数,进而采用快速非支配排序遗传算法全局寻优获得VTI介质的厚度、纵横波速度、密度和各向异性参数等多个参数.在正演的过程中,反射率法可以考虑几何扩散、吸收衰减、透射损失、多次波以及纵横波旅行时不匹配等地震波传播效应,更能精确地描述地震波在地下地层中的真实传播情况;在反演的过程中,快速非支配排序遗传算法可以在不引入权重系数的条件下同时优化多个目标函数,获得联合反演问题的Pareto最优解,既不添加权重系数影响又充分利用多分量地震数据.模型测试结果验证了该反演方法的有效性和可行性.     

利用联合反演技术进行反射地震的波速成象

本文介绍了根据反射地震数据进行波速成象的一种方法,其基础为多种反演技术的综合。由于要求的波速图象C（x,z）具有间断性,除利用走时数据T（x,t）外,在地层比较水平的情况下,还利用了均方根速度V（x,t）和统计子波W（t）的数据来成象。计算机层析成象过程分为三步:首先重做速度分析,取得与初次反射走时一致的均方根速度数据;然后用反射走时与均方根速度联合反演对应分析道的层速度和界面深度;最后由联合反演结果和反射面走时求波速图象函数的数字化版。文中还给出了波速成象方法在我国西北某沉积盆地上的应用及验证结果。     

Kinematical characteristics of the factorized velocity model

The factorized velocity model that incorporates both vertical heterogeneity and constant anisotropy is one of the complicated analytical models used in seismic data processing and interpretation. In this paper, I derive the analytic equations for offset, traveltime and relative geometrical spreading for the quasi‐compressional (qP‐) waves that can be used for modelling and inversion of the traveltime parameters. I show that the presence of anelliptic anisotropy usually dominates over the vertical heterogeneity with respect to the non‐hyperbolicity of the factorized velocity model.     

Field and synthetic experiments for virtual source crosswell tomography in vertical wells: Perth Basin,Western Australia

It is common for at least one monitoring well to be located proximally to a production well. This presents the possibility of applying crosswell technologies to resolve a range of earth properties between the wells. We present both field and synthetic examples of dual well walk-away vertical seismic profiling in vertical wells and show how the direct arrivals from a virtual source may be used to create velocity images between the wells. The synthetic experiments highlight the potential of virtual source crosswell tomography where large numbers of closely spaced receivers can be deployed in multiple wells. The field experiment is completed in two monitoring wells at an aquifer storage and recovery site near Perth, Western Australia. For this site, the crosswell velocity distribution recovered from inversion of travel times between in-hole virtual sources and receivers is highly consistent with what is expected from sonic logging and detailed zero-offset vertical seismic profiling. When compared to conventional walkaway vertical seismic profiling, the only additional effort required to complete dual-well walkaway vertical seismic profiling is the deployment of seismic sensors in the second well. The significant advantage of virtual source crosswell tomography is realised where strong near surface heterogeneity results in large travel time statics.     

基于阈值约束的协克里金法联合反演重力与重力梯度数据

常规协克里金方法反演重力或重力梯度数据具有抗噪性好、加入先验信息容易等优点,其反演的地下密度分布能够识别异常体中心位置,还原异常体基本形态,但反演图像光滑,分辨率低,这是由于常规方法估计的密度协方差矩阵全局发散、平稳.为了通过协克里金方法获得聚焦的密度分布需要改善密度协方差矩阵的性质.首先,本文推导了理论密度协方差公式,其性质表明,当理论模型聚焦分布时,其密度协方差矩阵是非平稳且聚焦分布的.为了打破常规协方差矩阵全局平稳、发散的特征,本文设置密度阈值处理协方差矩阵,通过不断更新协方差矩阵来迭代实现协克里金反演,最终得到相对聚焦的反演结果.用本文方法处理重力与重力梯度数据恢复两种密度模型,均得到了与正演模型匹配的反演结果;再将方法运用于文顿盐丘的实际测量重力与重力梯度数据,反演结果与已知的地质情况匹配较好.     

Geostatistical seismic Amplitude‐versus‐angle inversion

Seismic inversion plays an important role in reservoir modelling and characterisation due to its potential for assessing the spatial distribution of the sub‐surface petro‐elastic properties. Seismic amplitude‐versus‐angle inversion methodologies allow to retrieve P‐wave and S‐wave velocities and density individually allowing a better characterisation of existing litho‐fluid facies. We present an iterative geostatistical seismic amplitude‐versus‐angle inversion algorithm that inverts pre‐stack seismic data, sorted by angle gather, directly for: density; P‐wave; and S‐wave velocity models. The proposed iterative geostatistical inverse procedure is based on the use of stochastic sequential simulation and co‐simulation algorithms as the perturbation technique of the model parametre space; and the use of a genetic algorithm as a global optimiser to make the simulated elastic models converge from iteration to iteration. All the elastic models simulated during the iterative procedure honour the marginal prior distributions of P‐wave velocity, S‐wave velocity and density estimated from the available well‐log data, and the corresponding joint distributions between density versus P‐wave velocity and P‐wave versus S‐wave velocity. We successfully tested and implemented the proposed inversion procedure on a pre‐stack synthetic dataset, built from a real reservoir, and on a real pre‐stack seismic dataset acquired over a deep‐water gas reservoir. In both cases the results show a good convergence between real and synthetic seismic and reliable high‐resolution elastic sub‐surface Earth models.     

用平面波延拓方程进行地震数据的叠前速度反演

本文讨论地震勘探数据的叠前速度反演方法及其在海洋地震勘探数据上的反演试验.反演主要的计算步骤是:1.采用Fourier-Hankel变换把球面波分解为平面谐波;2.用平面谐波的延拓方程将上行波与下行波同时向下延拓,并计算每一层底部的反射系数和下一层的波阻抗;3.用最小二乘法从波阻抗中确定该层的声波速度.重复第2步与第3步,直到某一预定深度时结束.通过反演试验,对地震振幅比例的改变,子波变形,以及第1层速度和密度的误差对反演方法的稳定性及其精度的影响进行了分析.还通过实际海洋地震勘探数据的反演试验,对这一方法在地震勘探中的应用前景作了论述.     

Accelerating the T‐matrix approach to seismic full‐waveform inversion by domain decomposition

A seismic variant of the distorted Born iterative inversion method, which is commonly used in electromagnetic and acoustic (medical) imaging, has been recently developed on the basis of the T‐matrix approach of multiple scattering theory. The distorted Born iterative method is consistent with the Gauss–Newton method, but its implementation is different, and there are potentially significant computational advantages of using the T‐matrix approach in this context. It has been shown that the computational cost associated with the updating of the background medium Green functions after each iteration can be reduced via the use of various linearisation or quasi‐linearisation techniques. However, these techniques for reducing the computational cost may not work well in the presence of strong contrasts. To deal with this, we have now developed a domain decomposition method, which allows one to decompose the seismic velocity model into an arbitrary number of heterogeneous domains that can be treated separately and in parallel. The new domain decomposition method is based on the concept of a scattering‐path matrix, which is well known in solid‐state physics. If the seismic model consists of different domains that are well separated (e.g., different reservoirs within a sedimentary basin), then the scattering‐path matrix formulation can be used to derive approximations that are sufficiently accurate but far more speedy and much less memory demanding because they ignore the interaction between different domains. However, we show here that one can also use the scattering‐path matrix formulation to calculate the overall T‐matrix for a large model exactly without any approximations at a computational cost that is significantly smaller than the cost associated with an exact formal matrix inversion solution. This is because we have derived exact analytical results for the special case of two interacting domains and combined them with Strassen's formulas for fast recursive matrix inversion. To illustrate the fact that we have accelerated the T‐matrix approach to full‐waveform inversion by domain decomposition, we perform a series of numerical experiments based on synthetic data associated with a complex salt model and a simpler two‐dimensional model that can be naturally decomposed into separate upper and lower domains. If the domain decomposition method is combined with an additional layer of multi‐scale regularisation (based on spatial smoothing of the sensitivity matrix and the data residual vector along the receiver line) beyond standard sequential frequency inversion, then one apparently can also obtain stable inversion results in the absence of ultra‐low frequencies and reduced computation times.