Theoretical development of the differential scattering decomposition for the 3D resistivity experiment

In any numerical solution of the DC resistivity experiment, care must be taken to deal with strong heterogeneity of electrical conductivity. In order to examine the importance of conductivity contrasts, we develop a scattering decomposition of the DC resistivity equation in the sparse differential domain as opposed to the traditional dense integral formulation of scattering‐type equations. We remove the singularity in the differential scattered series via separation of primary and secondary conductivity, thereby avoiding the need to address the singularity in a Green's function. The differential scattering series is observed to diverge for large conductivity contrasts and to converge for small contrasts. We derive a convergence criterion, in terms of matrix norms for the weak‐form finite‐volume equations, that accounts for both the magnitude and distribution of heterogeneity of electrical conductivity. We demonstrate the relationship between the differential scattering series and the Fréchet derivative of the electrical potential with respect to electrical conductivity, and we show how the development may be applied to the inverse problem. For linearization associated with the Fréchet derivative to be valid, the perturbation in electrical conductivity must be small as defined by the convergence of the scattered series. The differential scattering formulation also provides an efficient tool for gaining insight into charge accumulation across contrasts in electrical conductivity, and we present a derivation that equates accumulated surface charge density to the source of scattered potential.     

球坐标系下谱元法三维地震波场模拟

谱元法已成为区域性乃至大陆性尺度地震波场模拟的重要工具.对于区域或大陆尺度层析成像而言,地球曲率不可忽略,此时模拟地震波传播采用球坐标系更为合适.本文从球坐标系下弹性波动方程弱形式出发,基于球坐标系变分原理给出了球坐标系下求解三维地震波方程的谱元法.另一方面,计算Fréchet敏感核是进行全波形反演的关键,本文借助伴随原理,推导了全波走时层析成像三维Fréchet敏感核表达式.为了验证球坐标系下谱元法的精度,我们将数值模拟结果与normal mode方法得到的解析解在1-D PREM模型下进行了对比.同时,我们将此方法应用到华北克拉通区域,以期获得地球内部结构精确成像.基于3-D全球径向各向异性地幔模型S362ANI和3-D地壳模型Crust1.0,我们建立了华北克拉通初始3-D背景模型,并将数值模拟结果与实际观测台站记录波形资料进行对比分析,利用互相关方法提取走时残差,最后给出了Fréchet敏感核在3-D空间中的分布,这些工作为下一步开展球坐标系下三维大尺度全波形反演奠定了基础.

     

摄动求解Fréchet导数的电阻率层析成像方法

针对电阻率层析成像 ,比较了求解Fréchet导数的 4种方法 ,并提出摄动方法计算该导数 ,使得每一次迭代过程只计算一次正演。井间电阻率成像的数值模拟结果表明了该方法的稳定性和高效性 ,实际资料的处理证明了方法的有效性     

Electric Potential and Fréchet Derivatives for a Uniform Anisotropic Medium with a Tilted Axis of Symmetry

In this paper we develop analytic solutions for the electric potential, current density and Fréchet derivatives at any interior point within a 3-D transversely isotropic medium having a tilted axis of symmetry. The current electrode is assumed to be on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken perpendicular to strike. All three derivatives of the Green’s function with respect to longitudinal conductivity, transverse resistivity and dip angle of the symmetry axis (dG/dσ_l, dG/dσ_t, dG/dθ₀) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt angle. Such sensitivity patterns are useful in real-time experimental design as well as in the fast inversion of resistivity data collected over an anisotropic earth.     

Comparison of DC sensitivity patterns for anisotropic and isotropic media

Many rocks possess electrical properties with a clearly expressed anisotropy. The anisotropic character of the rocks is often overlooked in forming the Fréchet derivatives or sensitivity functions for parameter updating during the inversion of DC resistivity data. In this study we have compared the sensitivity patterns for an isotropic, homogeneous model with that for a transversely isotropic (i.e. anisotropic) model having a tilted axis of symmetry using a pole–pole array. The sensitivity functions are expressed in terms of the derivatives of the electric potential U with respect to the average conductivity σ_m (geometric mean of the longitudinal and transverse conductivities) and the coefficient of anisotropy λ. Results are plotted in both cross-section form and plan view for various dip and strike angles of the axis of symmetry. The derivative dU/dλ decreases more rapidly than the isotropic value dU/dσ, and shows pronounced asymmetry and weakening of magnitude with increasing dip of the plane of symmetry. The derivative dU/dσ_m also exhibits the asymmetric pattern (except for vertical and horizontal dip cases). The positive region between the electrodes only extends to a small depth compared to the isotropic derivative, even in the case of a vertical axis of symmetry (VTI medium). The ratio of this anisotropic derivative to the isotropic derivative, when plotted as a function of position and depth shows prominent differences in both the sign and the magnitude of the sensitivities, especially for steep dips and for strongly anisotropic rocks. The plot highlights the dangers of an isotropic assumption. Even for mildly anisotropic rocks (λ < 1.2) the possibility for error in interpretation is considerable. Combined borehole and surface measurements are needed to diagnose anisotropy. Further work is needed to design optimal electrode configurations in anisotropic situations.     

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.     

Some analytical solutions for sensitivity of well tests to variations in storativity and transmissivity

The theory of a pumping test or a slug test to measure aquifer transmissivity or storativity assumes that the aquifer properties are uniform around the well. The response of the drawdown to small spatial variations in aquifer properties in the volume of influence is determined by spatial weighting functions or Fréchet kernels, which in general are functions of space and time. The Fréchet kernels determine the effective “volume of influence” of the measurements at any time. Under the assumption that the well is a line sink we derive explicit analytical expressions for the Fréchet kernels for storativity and for transmissivity for both pumping and slug tests. We also derive the total sensitivity functions for uniform variations in storativity and transmissivity and show that they are the spatial integrals of the Fréchet kernels. We consider both the case of separate pumping and observation wells and also the radially symmetric case of observations made at the pumped or slugged well. The “volume of influence” is symmetric with respect to the pumping or slugged well and the observation well, and far from the well the contours of equal spatial sensitivity approach the shapes of ellipses with a well at each focus, rather than circles centered on the pumping well. We use the analytical solutions to investigate the nature of the singularities in the spatial sensitivity functions around the wells, which govern the importance of inhomogeneities close to the well or observation point.     

基于二维VTI拟声波程函方程的椭圆各向异性立体层析反演

基于二维VTI介质拟声波程函方程,应用射线扰动理论建立了该方程控制下的数据空间各参数对于模型空间各个参数之间的线性关系,从而获得二维VTI介质拟声波程函方程的各向异性立体层析核函数.考虑到拟声波近似程函方程中η参数与ε和δ存在强烈耦合,本文首先探讨椭圆各向异性情形,为二维拟声波程函方程椭圆各向异性立体层析算法奠定了理论基础.同时也为日后推广到非椭圆各向异性情况提供了一种获得高质量初始模型的可靠途径.理论数据算例证实了Fréchet核函数求取的正确性以及在此基础上设计的工作流程实现两参数反演的可行性.

     

Sensitivity Kernel for the Weighted Norm of the Frequency-Dependent Phase Correlation

Full-3D waveform tomography (F3DT) is often formulated as an optimization problem, in which an objective function defined in terms of the misfit between observed and model-predicted (i.e., synthetic) waveforms is minimized by varying the earth structure model from which the synthetic waveforms are calculated. Because of the large dimension of the model space and the computational cost for solving the 3D seismic wave equation, it is often mandatory to use Newton-type local optimization algorithms; in which case, spurious local optima in the objective function can prevent the global convergence of the descent algorithm if the initial estimate of the structure model is not close enough to the global optimum. By appropriate design of the objective function, it is possible to enlarge the attraction domain of the global optimum so that Newton-type local optimization algorithms can achieve global convergence. In this article, an objective function based on a weighted L ₂ norm of the frequency-dependent phase correlation between observed and synthetic waveforms is proposed and studied, and its full-3D Fréchet kernel is constructed using the adjoint state method. The relation between the proposed objective function and the conventional frequency-dependent group-delay is analyzed and illustrated using numerical examples. The methodology has been successfully applied on a set of ambient-noise Green’s function observations collected in northern California to derive a full-3D crustal structure model.     

Explicit expressions and numerical calculations for the Fréchet and second derivatives in 2.5D Helmholtz equation inversion

In order to perform resistivity imaging, seismic waveform tomography or sensitivity analysis of geophysical data, the Fréchet derivatives, and even the second derivatives of the data with respect to the model parameters, may be required. We develop a practical method to compute the relevant derivatives for 2.5D resistivity and 2.5D frequency-domain acoustic velocity inversion. Both geophysical inversions entail the solution of a 2.5D Helmholtz equation. First, using differential calculus and the Green's functions of the 2.5D Helmholtz equation, we strictly formulate the explicit expressions for the Fréchet and second derivatives, then apply the finite-element method to approximate the Green's functions of an arbitrary medium. Finally, we calculate the derivatives using the expressions and the numerical solutions of the Green's functions. Two model parametrization approaches, constant-point and constant-block, are suggested and the computational efficiencies are compared. Numerical examples of the derivatives for various electrode arrays in cross-hole resistivity imaging and for cross-hole seismic surveying are demonstrated. Two synthetic experiments of resistivity and acoustic velocity imaging are used to illustrate the method.     

Three‐dimensional inversion of towed streamer electromagnetic data

A towed streamer electromagnetic system capable of simultaneous seismic and electromagnetic data acquisition has recently been developed and tested in the North Sea. We introduce a 3D inversion methodology for towed streamer electromagnetic data that includes a moving sensitivity domain. Our implementation is based on the 3D integral equation method for computing responses and Fréchet derivatives and uses the re‐weighted regularized conjugate gradient method for minimizing the objective functional with focusing regularization. We present two model studies relevant to hydrocarbon exploration in the North Sea. First, we demonstrate the ability of a towed electromagnetic system to detect and characterize the Harding field, a medium‐sized North Sea hydrocarbon target. We compare our 3D inversion of towed streamer electromagnetic data with 3D inversion of conventional marine controlled‐source electromagnetic data and observe few differences between the recovered models. Second, we demonstrate the ability of a towed streamer electromagnetic system to detect and characterize the Peon discovery, which is representative of an infrastructure‐led shallow gas play in the North Sea. We also present an actual case study for the 3D inversion of towed streamer electromagnetic data from the Troll field in the North Sea and demonstrate our ability to image all the Troll West Oil and Gas Provinces and the Troll East Gas Province. We conclude that 3D inversion of data from the current generation of towed streamer electromagnetic systems can adequately recover hydrocarbon‐bearing formations to depths of approximately 2 km. We note that by obviating the need for ocean‐bottom receivers, the towed streamer electromagnetic system enables electromagnetic data to be acquired over very large areas in frontier and mature basins for higher acquisition rates and relatively lower cost than conventional marine controlled‐source electromagnetic methods.     

Reflection multi‐scale envelope inversion

Sufficient low‐frequency information is essential for full‐waveform inversion to get the global optimal solution. Multi‐scale envelope inversion was proposed using a new Fréchet derivative to invert the long‐wavelength component of the model by directly using the low‐frequency components contained in an envelope of seismic data. Although the new method can recover the main structure of the model, the inversion quality of the model bottom still needs to be improved. Reflection waveform inversion reduces the dependence of inversion on low‐frequency and long‐offset data by using travel‐time information in reflected waves. However, when the underground medium contains strong contrast or the initial model is far away from the true model, it is hard to get reliable reference reflectors for the generation of reflected waves. Here, we propose a combination inversion algorithm, i.e., reflection multi‐scale envelope inversion, to overcome the limitations of multi‐scale envelope inversion and reflection waveform inversion. First, wavefield decomposition was introduced into the multi‐scale envelope inversion to improve the inversion quality of the long‐wavelength components of the model. Then, after the initial model had been established to be accurate enough, migration and de‐migration were introduced to achieve multi‐scale reflection waveform inversion. The numerical results of the salt‐layer model and the SEG/EAGE salt model verified the validity of the proposed approach and its potential.     

二维电阻率成像研究

电阻率成像中最关键的问题就是获得雅可比偏导数矩阵。本文从二维微分方程的积分解出发推导了一种新的电阻率成像的雅可比偏导数矩阵,同时形成了成像方程。用内外迭代相结合的高斯塞德儿迭代方法解成像方程可以得到电阻率的分布图像。数值模拟结果表明该方法是有效和可靠的,尤其值得注意的是积分法电阻率成像方法初始模型可以采用均匀模型,减小了对初始模型的依赖。对用其它方法难以获得好的成像结果的单一高阻体,积分法也得到了较好的成像结果。河南商丘某野外资料结果表明,成像结果和实际地质情况吻合较好。     

Computing the Sensitivity Kernels for 2.5-D Seismic Waveform Inversion in Heterogeneous,Anisotropic Media

2.5-D modeling and inversion techniques are much closer to reality than the simple and traditional 2-D seismic wave modeling and inversion. The sensitivity kernels required in full waveform seismic tomographic inversion are the Fréchet derivatives of the displacement vector with respect to the independent anisotropic model parameters of the subsurface. They give the sensitivity of the seismograms to changes in the model parameters. This paper applies two methods, called ‘the perturbation method’ and ‘the matrix method’, to derive the sensitivity kernels for 2.5-D seismic waveform inversion. We show that the two methods yield the same explicit expressions for the Fréchet derivatives using a constant-block model parameterization, and are available for both the line-source (2-D) and the point-source (2.5-D) cases. The method involves two Green’s function vectors and their gradients, as well as the derivatives of the elastic modulus tensor with respect to the independent model parameters. The two Green’s function vectors are the responses of the displacement vector to the two directed unit vectors located at the source and geophone positions, respectively; they can be generally obtained by numerical methods. The gradients of the Green’s function vectors may be approximated in the same manner as the differential computations in the forward modeling. The derivatives of the elastic modulus tensor with respect to the independent model parameters can be obtained analytically, dependent on the class of medium anisotropy. Explicit expressions are given for two special cases—isotropic and tilted transversely isotropic (TTI) media. Numerical examples are given for the latter case, which involves five independent elastic moduli (or Thomsen parameters) plus one angle defining the symmetry axis.     

Practical techniques for 3D resistivity surveys and data inversion1

Techniques to reduce the time needed to carry out 3D resistivity surveys with a moderate number (25 to 100) of electrodes and the computing time required to interpret the data have been developed. The electrodes in a 3D survey are normally arranged in a square grid and the pole-pole array is used to make the potential measurements. The number of measurements required can be reduced to about one-third of the maximum possible number without seriously degrading the resolution of the resulting inversion model by making measurements along the horizontal, vertical and 45° diagonal rows of electrodes passing through the current electrode. The smoothness-constrained least-squares inversion method is used for the data interpretation. The computing time required by this technique can be greatly reduced by using a homogeneous half-space as the starting model so that the Jacobian matrix of partial derivatives can be calculated analytically. A quasi-Newton updating method is then used to estimate the partial derivatives for subsequent iterations. This inversion technique has been tested on synthetic and field data where a satisfactory model is obtained using a modest amount of computer time. On an 80486DX2/66 microcomputer, it takes about 20 minutes to invert the data from a 7 by 7 electrode survey grid. using the techniques described below, 3D resistivity surveys and data inversion can be carried out using commercially available field equipment and an inexpensive microcomputer.     

一阶Rytov近似有限频走时层析

传统的波动方程走时核函数（或走时Fréchet导数）多基于互相关时差测量方式及地震波场的一阶Born近似导出,其成立条件非常苛刻.然而,地震波走时与大尺度的速度结构具有良好的线性关系,对于小角度的前向散射波场,Rytov近似优于Born近似.因此,本文基于Rytov近似和互相关时差测量方式,导出了基于Rytov近似的有限频走时敏感度核函数的两种等价形式：频率积分和时间积分表达式.在此基础之上,本文提出了一种隐式矩阵向量乘方法,可以直接计算Hessian矩阵或者核函数与向量的乘积,而无需显式计算和存储核函数及Hessian矩阵.基于隐式矩阵向量乘方法,本文利用共轭梯度法求解法方程实现了一种高效的Gauss-Newton反演算法求解走时层析反问题.与传统的敏感度核函数反演方法相比,本文方法在每次迭代过程中,无需显式计算和存储核函数,极大降低了存储需求.与基于Born近似的伴随状态方法走时层析相比,本文方法具有准二阶的收敛速度,且适用范围更广.数值试验证明了本文方法的有效性.

     

Rapid 3-D Earthquake Location using a Hybrid Global–Local Inversion Approach

A novel hybrid approach to earthquake location is proposed which uses a combined coarse global search and fine local inversion with a minimum search routine. The method exploits the advantages of network ray tracing and robust formulation of the Fréchet derivatives to simultaneously update all sampled initial source parameters in the solution space to determine the best solution. Synthetic examples, involving a three-dimensional (3-D) complex velocity model and a challenging source–receiver layout, are used to demonstrate the advantages over direct grid search algorithms in terms of solution accuracy, computational efficiency, and sensitivity to noise. Therefore, this is a promising scheme for earthquake early warning, tsunami early warning, rapid hazard assessment, and emergency response after strong earthquake occurrence.     

Three-dimensional resistivity characterization of a coastal area: Application of Grounded Electrical-Source Airborne Transient Electromagnetic (GREATEM) survey data from Kujukuri Beach,Japan

An airborne electromagnetic (AEM) survey using the Grounded Electrical-Source Airborne Transient Electromagnetic (GREATEM) system was conducted over the Kujukuri coastal plain in southeast Japan to assess the system's ability to accurately describe the geological structure beneath shallow seawater. To obtain high-quality data with an optimized signal-to-noise ratio, a series of data processing techniques were used to obtain the final transient response curves from the field survey data. These steps included movement correction, coordinate transformation, the removal of local noise, data stacking, and signal portion extraction.We performed numerical forward modeling to generate a three-dimensional (3D) resistivity structure model from the GREATEM data. This model was developed from an initial one-dimensional (1D) resistivity structure that was also inverted from the GREATEM field survey data. We modified a 3D electromagnetic forward-modeling scheme based on a finite-difference staggered-grid method and used it to calculate the response of the 3D resistivity model along each survey line. We verified the model by examining the fit of the magnetic-transient responses between field data and the 3D forward-model computed data, the latter of which were convolved with the measured system responses of the corresponding data set.The inverted 3D resistivity structures showed that the GREATEM system has the capability to map resistivity structures as far as 800 m offshore and as deep as 300–350 m underground in coastal areas of relatively shallow seawater depth (5–10 m).     

VTI介质多参数联合走时层析成像方法

本文基于球谐展开群速度表达式计算走时关于各向异性参数的Fréchet核函数,利用共轭梯度法对两种参数化方法进行了VTI介质中多参数联合反演方法研究.经过理论分析和数值试验发现,与经典的Thomsen参数化方法相比,垂直慢度、水平慢度与动校正慢度的参数化方式更有利于VTI介质多参数联合走时层析反演.为了克服走时对ε参数的不敏感性,我们采用了两步法进行双参数反演,理论模型试验反演得到了与垂直速度精度相当的ε参数.可以将两步法扩展到三步法以同时反演各向异性介质中的三个参数,数值试验展示了该策略的应用潜力.