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.     

波阻抗映射法外推

首先介绍了自适应滤波的方法和波阻抗映射法外推的基本原理．在小反射系数的假设条件下推导了对数波阻抗和地震道之间近似为线性映射的关系,应用自适应滤波的方法可以实现这种映射．实际资料的试算结果表明,该方法能够提供分辨率明显高于常规方法的波阻抗或层速度剖面,并与测井资料较吻合．因此,它作为测井约束反演的补充手段,是一种可用于开发地震的新方法．最后讨论了映射法波阻抗反演的局限性．     

INVERSE THEORY APPLIED TO MULTI-SOURCE CROSS-HOLE TOMOGRAPHY.

Frequency-domain methods are well suited to the imaging of wide-aperture cross-hole data. However, although the combination of the frequency domain with the wavenumber domain has facilitated the development of rapid algorithms, such as diffraction tomography, this has also required linearization with respect to homogeneous reference media. This restriction, and association restrictions on source-receiver geometries, are overcome by applying inverse techniques that operate in the frequency-space domain. In order to incorporate the rigorous modelling technique of finite differences into the inverse procedure a nonlinear approach is used. To reduce computational costs the method of finite differences is applied directly to the frequency-domain wave equation. The use of high speed, high capacity vector computers allow the resultant finite-difference equations to be factored in-place. In this way wavefields can be computed for additional source positions at minimal extra cost, allowing inversions to be generated using data from a very large number of source positions. Synthetic studies show that where weak scatter approximations are valid, diffraction tomography performs slightly better than a single iteration of non-linear inversion. However, if the background velocities increase systematically with depth, diffraction tomography is ineffective whereas non-linear inversion yields useful images from one frequency component of the data after a single iteration. Further synthetic studies indicate the efficacy of the method in the time-lapse monitoring of injection fluids in tertiary hydrocarbon recovery projects.     

TRACE INTERPOLATION BY SLANT-STACK MIGRATION1

The slant-stack migration formula based on the Radon transform is studied with respect to the depth step Δz of wavefield extrapolation. It can be viewed as a generalized trace-interpolation procedure including wave extrapolation with an arbitrary step Δz. For Δz= 0 the formula yields the familiar plane-wave decomposition, while for Δz > 0 it provides a robust tool for migration transformation of spatially undersampled wavefields. Using the stationary phase method, it is shown that the slant-stack migration formula degenerates into the Rayleigh-Sommerfeld integral in the far-field approximation. Consequently, even a narrow slant-stack gather applied before the diffraction stack can significantly improve the representation of noisy data in the wavefield extrapolation process. The theory is applied to synthetic and field data to perform trace interpolation and dip reject filtration. The data examples presented prove that the Radon interpolator works well in the dip range, including waves with mutual stepouts smaller than half the dominant period.     

频率多尺度全波形速度反演

以二维声波方程为模型,在时间域深入研究了全波形速度反演.全波形反演要解一个非线性的最小二乘问题,是一个极小化模拟数据与已知数据之间残量的过程.针对全波形反演易陷入局部极值的困难,本文提出了基于不同尺度的频率数据的"逐级反演"策略,即先基于低频尺度的波场信息进行反演,得出一个合理的初始模型,然后再利用其他不同尺度频率的波场进行反演,并且用前一尺度的迭代反演结果作为下一尺度反演的初始模型,这样逐级进行反演.文中详细阐述和推导了理论方法及公式,包括有限差分正演模拟、速度模型修正、梯度计算和算法描述,并以Marmousi复杂构造模型为例,进行了MPI并行全波形反演数值计算,得到了较好的反演结果,验证了方法的有效性和稳健性.     

Joint inversion of Rayleigh‐wave dispersion data and vertical electric sounding data: synthetic tests on characteristic sub‐surface models

In the traditional inversion of the Rayleigh dispersion curve, layer thickness, which is the second most sensitive parameter of modelling the Rayleigh dispersion curve, is usually assumed as correct and is used as fixed a priori information. Because the knowledge of the layer thickness is typically not precise, the use of such a priori information may result in the traditional Rayleigh dispersion curve inversions getting trapped in some local minima and may show results that are far from the real solution. In this study, we try to avoid this issue by using a joint inversion of the Rayleigh dispersion curve data with vertical electric sounding data, where we use the common‐layer thickness to couple the two methods. The key idea of the proposed joint inversion scheme is to combine methods in one joint Jacobian matrix and to invert for layer S‐wave velocity, resistivity, and layer thickness as an additional parameter, in contrast with a traditional Rayleigh dispersion curve inversion. The proposed joint inversion approach is tested with noise‐free and Gaussian noise data on six characteristic, synthetic sub‐surface models: a model with a typical dispersion; a low‐velocity, half‐space model; a model with particularly stiff and soft layers, respectively; and a model reproduced from the stiff and soft layers for different layer‐resistivity propagation. In the joint inversion process, the non‐linear damped least squares method is used together with the singular value decomposition approach to find a proper damping value for each iteration. The proposed joint inversion scheme tests many damping values, and it chooses the one that best approximates the observed data in the current iteration. The quality of the joint inversion is checked with the relative distance measure. In addition, a sensitivity analysis is performed for the typical dispersive sub‐surface model to illustrate the benefits of the proposed joint scheme. The results of synthetic models revealed that the combination of the Rayleigh dispersion curve and vertical electric sounding methods in a joint scheme allows to provide reliable sub‐surface models even in complex and challenging situations and without using any a priori information.     

计算台站接收函数的最大熵谱反褶积方法

提出一种在时间域采用最大熵谱反褶积提取台站接收函数的方法，以最大熵作为自相关函数和互相关函数的递推准则，利用Toeplitz方程及Levinson递推算法，得到预测误差滤波系数的递推公式，从而计算台站接收函数．外推运算过程中，反射系数总是小于1，保证了最大熵谱反褶积的稳定性．时窗外数据熵极大提高了接收函数的分辨率．合成地震图与实测地震图的检验表明, 最大熵谱反褶积是一种在时间域测定台站接收函数的有效方法．      

基于波场分离的弹性波逆时偏移

尽管叠前逆时偏移成像精度高,但仅针对单一纵波的成像也可能形成地下介质成像盲区,由于基于弹性波方程的逆时偏移成像可形成多波模式的成像数据,因此弹性波逆时偏移成像可提供更为丰富的地下构造信息.本文依据各向同性介质的一阶速度-应力方程组构建震源和检波点矢量波场,再利用Helmholtz分解提取纯纵波和纯横波波场,使用震源归一化的互相关成像条件获得纯波成像,避免了直接使用坐标分量成像而引起的纵横波串扰问题.针对转换波成像的极性反转问题,文中提出一种共炮域极性校正方法.为有效节约存储成本,也提出一种适用于弹性波逆时偏移的震源波场逆时重建方法,在震源波场正传过程中,仅保存PML边界内若干层的速度分量波场,进而逆时重建出所有分量的震源波场.本文分别对地堑模型和Marmousi2模型进行了弹性波逆时偏移成像测试,结果表明:所提出的共炮域极性校正方法正确有效,基于波场分离的弹性波逆时偏移成像的纯波数据能够对复杂地下构造准确成像.     

Constrained tomography of realistic velocity models in microseismic monitoring using calibration shots

The knowledge of the velocity model in microseismic jobs is critical to achieving statistically reliable microseismic event locations. The design of microseismic networks and the limited sources for calibration do not allow for a full tomographic inversion. We propose optimizing a priori velocity models using a few active shots and a non‐linear inversion, suitable to poorly constrained systems. The considered models can be described by several layers with different P‐ and S‐wave velocities. The velocities may be constant or have 3D gradients; the layer interfaces may be simple dipping planes or more complex 3D surfaces. In this process the P‐ and S‐ wave arrival times and polarizations measured on the seismograms constitute the observed data set. They are used to estimate two misfit functions: i) one based on the measurement residuals and ii) one based on the inaccuracy of the source relocation. These two functions are minimized thanks to a simulated annealing scheme, which decreases the risk of converging to a local solution within the velocity model. The case study used to illustrate this methodology highlights the ability of this technique to constrain a velocity model with dipping layers. This was performed by jointly using sixteen perforation shots recorded during a multi‐stage fracturing operation from a single string of 3C‐receivers. This decreased the location inaccuracies and the residuals by a factor of six. In addition, the retrieved layer dip was consistent with the pseudo‐horizontal trajectories of the wells and the background information provided by the customer. Finally, the theoretical position of each calibration shot was contained in the uncertainty domain of the relocation of each shot. In contrast, single‐stage inversions provided different velocity models that were neither consistent between each other nor with the well trajectories. This example showed that it is essential to perform a multi‐stage inversion to derive a better updated velocity model.     

利用矩阵分解理论的双程波方程叠前深度偏移方法

叠前深度偏移理论及方法一直是地震数据成像中研究的热点问题.业界对单程波叠前深度偏移方法和逆时深度偏移开展了深入的研究,但对双程波方程波场深度延拓理论及成像方法的研究还鲜有报道.本文以地表记录的波场值为基础,利用单程波传播算子估计波场对深度的偏导数,为在深度域求解双程波方程提供充分的边界条件,并提出利用矩阵分解理论实现双程波方程的波场深度外推.通过对强速度变化介质中传播波场的计算,与传统的单程波偏移方法相比,本文提出的偏移方法计算的波场与常规有限差分技术计算的波场相一致,证明了本方法计算的准确性.通过对SEAM模型的成像,在相同的成像参数下,与传统的单程波偏移算法和逆时深度偏移算法方法相比,本文提出的偏移方法能够提供更少的虚假成像和更清晰的成像结果.本文所提偏移算法具有深度偏移和双程波偏移的双重特色,推动和发展了双程波叠前深度偏移的理论和实践.     

Impedance matrices for circular foundation embedded in layered medium

A numerical scheme is developed in the paper for calculating torsional, vertical, horizontal, coupling and rocking impedances in frequency domain for axial-symmetric foundations embedded in layered media. In the scheme, the whole soil domain is divided into interior and exterior domains. For the exterior domain, the analytic solutions with unknown coefficients are obtained by solving three-dimensional (3D) wave equations in cylindrical coordinates satisfying homogeneous boundary conditions. For the interior domain, the analytical solutions are also obtained by solving the same 3D wave equations satisfying the homogeneous boundary conditions and the prescribed boundary conditions. The prescribed conditions are the interaction tractions at the interfaces between embedded foundation and surrounding soil. The interaction tractions are assumed to be piecewise linear. The piecewise linear tractions at the bottom surface of foundation will be decomposed into a series of Bessel functions which can be easily fitted into the general solutions of wave equations in cylindrical coordinates. After all the analytic solutions with unknown coefficients for both interior and exterior domains are found, the variational principle is employed using the continuity conditions (both displacements and stresses) at the interfaces between interior and exterior domains, interior domain and foundation, and exterior domain and foundation to find impedance functions.     

Post-stack impedance blocky inversion based on analytic solution of viscous acoustic wave equation

The conventional impedance inversion method ignores the attenuation effect, transmission loss and inter-layer multiple waves; the smooth-like regularization approach makes the corresponding impedance solution excessively smooth. Both fundamentally limit the resolution of impedance result and lead to the inadequate ability of boundary characterization. Therefore, a post-stack impedance blocky inversion method based on the analytic solution of viscous acoustic equation is proposed. Based on the derived recursive formula of reflections, the 1D viscous acoustic wave equation is solved analytically to obtain zero-offset full-wave field response. Applying chain rule, the analytical expression of the Fréchet derivative is derived for gradient-descent non-linear inversion. Combined with smooth constraints, the blocky constraints can be introduced into the Bayesian inference framework to obtain stable and well-defined inversion results. According to the above theory, we firstly use model data to analyse the influence of incompleteness of forward method on seismic response, and further verify the effectiveness of the proposed method. Then the Q-value sensitivity analysis of seismic trace is carried out to reduce the difficulty of Q-value estimation. Finally, the real data from Lower Congo Basin in West Africa indicate that the proposed approach provide the high-resolution and well-defined impedance result. As a supplement and development of linear impedance inversion method, the non-linear viscous inversion could recover more realistic and reliable impedance profiles.     

GENERALIZED KUNETZ-TYPE EQUATIONS*

A Kunetz equation is often used as the starting point in the development of solutions for the inversion of one-dimensional, noise free, normal incident seismograms, for which |r_o|= 1. In this paper we demonstrate a need for a Kunetz-type equation in which filtered signals can be used, so that noise effects can be reduced. We then show that an infinite number of Kunetz-type equations exist for the lossless wave equation in layered media. Finally, we show that it is indeed valid to formulate and solve the inverse problem using filtered signals.     

Simulation and analysis of point-source surface wave fields

The complexity of near surface intensifies the diversity of seismic wave fields, which makes study on near surface wavefields important in many aspects. The strong absorption of low velocity layer can affect the resolution of seismic data, and free boundary can cause surface wave. Considering the above problems, we focus on the Rayleigh wavefields simulation using finite-difference wave equation of higher-order staggered grids and PML boundary conditions. Free boundary, buried source and overlying low velocity layer are taken into consideration and point explosion source is adopted. Through some numerical simulation with different parameters, we quantitatively analyze relationship between wave intensity and source depth, as well as the energy variation with propagation and obtain some practical knowledge and conclusions.     

High-order kernels for Riemannian wavefield extrapolation

Riemannian wavefield extrapolation is a technique for one‐way extrapolation of acoustic waves. Riemannian wavefield extrapolation generalizes wavefield extrapolation by downward continuation by considering coordinate systems different from conventional Cartesian ones. Coordinate systems can conform with the extrapolated wavefield, with the velocity model or with the acquisition geometry. When coordinate systems conform with the propagated wavefield, extrapolation can be done accurately using low‐order kernels. However, in complex media or in cases where the coordinate systems do not conform with the propagating wavefields, low order kernels are not accurate enough and need to be replaced by more accurate, higher‐order kernels. Since Riemannian wavefield extrapolation is based on factorization of an acoustic wave‐equation, higher‐order kernels can be constructed using methods analogous to the one employed for factorization of the acoustic wave‐equation in Cartesian coordinates. Thus, we can construct space‐domain finite‐differences as well as mixed‐domain techniques for extrapolation. High‐order Riemannian wavefield extrapolation kernels improve the accuracy of extrapolation, particularly when the Riemannian coordinate systems does not closely match the general direction of wave propagation.     

不依赖源子波的跨孔雷达时间域波形反演

波形反演是近年来较热门的反演方法,其分辨率可以达到亚波长级别.在波形反演的实际应用中,源子波的估计十分重要.传统方法使用反褶积来估计源子波并随着反演过程更新,该方法在合成数据波形反演中效果较好,但在实际数据反演过程中存在一系列的问题.由于实际数据信噪比较低,在源子波估计过程中需要大量的人为干涉,且结果并不一定可靠.本文使用一种基于褶积波场的新型目标函数,令反演过程不再依赖源子波.详细推导了针对跨孔雷达波形反演的梯度及步长公式,实现介电常数和电导率的同步反演.针对一个合成数据模型同时反演介电常数和电导率,结果表明该方法能够反演出亚波长尺寸异常体的形状和位置.接着,将该方法应用到两组实际数据中,并与基于估计源子波的时间域波形反演结果进行比较.结果表明不依赖源子波的时间域波形反演结果分辨率更高,也更准确.     

地震转换波对叠前同时反演的影响研究

叠前同时反演是油气探测的一种有效工具.其理论基础是平面P波Zoeppritz方程计算的反射系数的近似,是入射角的函数.叠前同时反演可以利用三项或两项Fatti方程进行反演分析.本文针对实际油田的测井数据,利用反射率法模拟了仅包含P波一次反射记录,包含P波一次反射和P波层间多次波记录以及全波场地震记录,再利用叠前同时反演对合成地震记录进行反演研究.研究结果表明,在大偏移距处P波主要反射受到其它模式波的污染,从而影响了叠前同时反演结果的精度.对于薄互层介质当转换波影响严重时,使用小角度数据的两项AVO反演比使用大角度数据的三项AVO反演更合理可靠.     

Data acquisition and pre-processing required for simultaneous P-SV inversion

The goal of seismic reflection surveys is the derivation of petrophysical subsurface parameters from surface measurements. Today's well established technique in data acquisition, as well as processing terms, is based on the acoustic approximation to the real world's wave propagation. In recent years a lot of work has been done to extend the technique to the elastic approximation. There was especially an important trend towards elastic inversion techniques operating on plane-wave seismograms, called simultaneous P-SV inversion (or short P-SV inversion) within this paper. Being still under investigation, some important aspects of P-SV inversion concerning data acquisition as well as pre-processing, should be pointed out. To fit the assumptions of P-SV inversion schemes, at least a two-dimensional picture of the reflected wavefield with vertical and in-line horizontal receivers has to be recorded. Moreover, the theoretical work done suggests that in addition to a survey with a compressional wave source, a second survey should be done using sources radiating vertically polarized shear waves, is needed. Finally, proper slant stacking must be performed to get plane-wave seismograms. The P/S separated plane-wave seismograms are then well prepared for feeding into the inversion algorithms. P/S separated planewave seismograms are then well prepared for feeding into the inversion algorithm.s In this paper, a tutorial overview of the data acquisition and pre-processing in accordance with the P-SV inversion philosophy is given and illustrated using synthetic seismograms. A judgement on the feasibility of the P-SV inversion philosophy must be left to ongoing research.     

Wavefield interpolation in 3D large‐step Fourier wavefield extrapolation

Extrapolating wavefields and imaging at each depth during three‐dimensional recursive wave‐equation migration is a time‐consuming endeavor. For efficiency, most commercial techniques extrapolate wavefields through thick slabs followed by wavefield interpolation within each thick slab. In this article, we develop this strategy by associating more efficient interpolators with a Fourier‐transform‐related wavefield extrapolation method. First, we formulate a three‐dimensional first‐order separation‐of‐variables screen propagator for large‐step wavefield extrapolation, which allows for wide‐angle propagations in highly contrasting media. This propagator significantly improves the performance of the split‐step Fourier method in dealing with significant lateral heterogeneities at the cost of only one more fast Fourier transform in each thick slab. We then extend the two‐dimensional Kirchhoff and Born–Kirchhoff local wavefield interpolators to three‐dimensional cases for each slab. The three‐dimensional Kirchhoff interpolator is based on the traditional Kirchhoff formula and applies to moderate lateral velocity variations, whereas the three‐dimensional Born–Kirchhoff interpolator is derived from the Lippmann–Schwinger integral equation under the Born approximation and is adapted to highly laterally varying media. Numerical examples on the three‐dimensional salt model of the Society of Exploration Geophysicists/European Association of Geoscientists demonstrate that three‐dimensional first‐order separation‐of‐variables screen propagator Born–Kirchhoff depth migration using thick‐slab wavefield extrapolation plus thin‐slab interpolation tolerates a considerable depth‐step size of up to 72 ms, eventually resulting in an efficiency improvement of nearly 80% without obvious loss of imaging accuracy. Although the proposed three‐dimensional interpolators are presented with one‐way Fourier extrapolation methods, they can be extended for applications to general migration methods.