广义WKB法在波场模拟中的应用研究

针对地震勘探中的波场模拟和反演困难等问题,本文从二维波动方程出发,采用Fourier变换得到了赫姆霍兹方程,并创造性地提出了一种全新的近似解法－广义WKB法;该法可以更加近似赫姆霍兹方程,且它的解中包含WKB的近似值.数值模拟显示了WKB法与广义WKB求解的异同.结果表明：广义WKB法是WKB法的一种推广,在求解非线性波数问题时具有更好的优势.     

单程波最优化可分近似偏移方法精度分析(英文)

单程波算子的可分近似形式是关于空间变量和波数变量的函数,这使得快速傅里叶变换的利用成为可能,从而极大提高了计算效率。从函数近似的角度来看,最优可分近似方法与其他几种可分近似方法一样都具有单程波算子可分表达的形式,但最优可分近似方法是其中唯一的函数整体近似的方法,这就造成了该方法在相位误差曲线、脉冲响应、模型偏移结果上表现出与其他可分近似方法不一样的特征:该方法精度较高,且随着阶数的增大,对速度变化的敏感度降低。     

Errors due to the anisotropic-common-ray approximation of the coupling ray theory

The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common-ray approximation applied. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. We derive the equations for estimating the travel-time errors due to the anisotropic-common-ray (and also isotropic-common-ray) approximation of the coupling ray theory. The errors of the common-ray approximations are calculated along the anisotropic common rays in smooth velocity models without interfaces. The derivation is based on the general equations for the second-order perturbations of travel time.     

Mass lumping models of the linear diffusion equation

The nodal domain integration method is used to develop a numerical model of the linear diffusion equation. The nodal domain integration approach is shown to represent an infinity of finite element mass matrix lumping schemes including the Galerkin and subdomain integration versions of the weighted residual method and an integrated finite difference method. Neumann, Dirichlet and mixed boundary conditions are accommodated analogous to the Galerkin finite element method. In order to reduce the overall integrated approximation relative error, a mass matrix lumping formulation is developed which is based on the Crank-Nicolson time advancement approximation. The optimum mass lumping factors are found to be strongly related to the model timestep size.     

快速拟线性近似方法及三维频谱激电反演研究

Zhdanov提出的拟线性(QL)近似方法有效解决了积分方程计算速度慢、占用内存多的缺陷,但因为三维频谱激电(SIP)换源频繁,每次换源都需要重新计算剖分单元的并矢格林函数和一次场,所以用该方法模拟三维SIP的速度仍然缓慢.本文根据一次场及格林函数的空间对称性,提出了一种适用于多源电磁法的快速QL近似正演方法.在此基础上,实现了Cole-Cole参数范围约束的三维SIP共轭梯度反演方法.理论试算结果表明:快速QL近似方法适用于三维SIP正演模拟,计算速度较QL近似方法有了很大提高;Cole-Cole参数范围约束的三维SIP共轭梯度反演方法对零频电阻率、极化率等参数的反演效果良好,将异常区域剖分成2000块时迭代一次只需约0.2s的时间,占用内存大约为140 MB.     

Errors Due to the Common Ray Approximations of the Coupling Ray Theory

The common ray approximation considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common ray approximation.We derive the equations for estimating the travel-time errors due to the isotropic and anisotropic common ray approximations of the coupling ray theory. These equations represent the main result of the paper. The derivation is based on the general equations for the second-order perturbations of travel time. The accuracy of the anisotropic common ray approximation can be studied along the isotropic common rays, without tracing the anisotropic common rays.The derived equations are numerically tested in three 1-D models of differing degree of anisotropy. The first-order and second-order perturbation expansions of travel time from the isotropic common rays to anisotropic-ray-theory rays are compared with the anisotropic-ray-theory travel times. The errors due to the isotropic common ray approximation and due to the anisotropic common ray approximation are estimated. In the numerical example, the errors of the anisotropic common ray approximation are considerably smaller than the errors of the isotropic common ray approximation.The effect of the isotropic common ray approximation on the coupling-ray-theory synthetic seismograms is demonstrated graphically. For comparison, the effects of the quasi-isotropic projection of the Green tensor, of the quasi-isotropic approximation of the Christoffel matrix, and of the quasi-isotropic perturbation of travel times on the coupling-ray-theory synthetic seismograms are also shown. The projection of the travel-time errors on the relative errors of the time-harmonic Green tensor is briefly presented.     

Approximation approach to the SRM based on root decomposition in the simulation of spatially varying ground motions

The spectral representation method(SRM) is widely used to simulate spatially varying ground motions.This study focuses on the approximation approach to the SRM based on root decomposition,which can improve the efficiency of the simulation.The accuracy of the approximation approach may be affected by three factors: matrix for decomposition,distribution of frequency interpolation nodes and elements for interpolation.The influence of these factors on the accuracy of this approach is examined and the following conclusions are drawn.The SRM based on the root decomposition of the lagged coherency matrix exhibits greater accuracy than the SRM based on the root decomposition of the cross spectral matrix.The equal energy distribution of frequency interpolation nodes proposed in this study is more effective than the counter pith with an equal spacing.Elements for interpolation do not have much of an effect on the accuracy,so interpolation of the elements of the decomposed matrix is recommended because it is less complicated from a computational efficiency perspective.     

Interpolation of the coupling-ray-theory Green function within ray cells

The coupling–ray–theory tensor Green function for electromagnetic waves or elastic S waves is frequency dependent, and is usually calculated for many frequencies. This frequency dependence represents no problem in calculating the Green function, but may pose a significant challenge in storing the Green function at the nodes of dense grids, typical for applications such as the Born approximation or non–linear source determination. Storing the Green function at the nodes of dense grids for too many frequencies may be impractical or even unrealistic. We have already proposed the approximation of the coupling–ray–theory tensor Green function, in the vicinity of a given prevailing frequency, by two coupling–ray–theory dyadic Green functions described by their coupling–ray–theory travel times and their coupling–ray–theory amplitudes. The above mentioned prevailing–frequency approximation of the coupling ray theory enables us to interpolate the coupling–ray–theory dyadic Green functions within ray cells, and to calculate them at the nodes of dense grids. For the interpolation within ray cells, we need to separate the pairs of prevailing–frequency coupling–ray–theory dyadic Green functions so that both the first Green function and the second Green function are continuous along rays and within ray cells. We describe the current progress in this field and outline the basic algorithms. The proposed method is equally applicable to both electromagnetic waves and elastic S waves. We demonstrate the preliminary numerical results using the coupling–ray–theory travel times of elastic S waves.     

Shear stress on the base of a lithospheric plate

This paper is a review of the theoretical and observational evidence bearing on the magnitude of the shear stress which acts on the base of a lithospheric plate. Estimates based on the viscosity of the upper mantle do not yield useful limits. Arguments based on the thermal stability of the upper mantle indicate that the basal shear stress is no larger than a few bars. An indirect measurement of the rheology and shear stress can be made by studying the diffusion of stress and displacement following a large decoupling earthquake. When applied to the 1965 Rat Island Earthquake, this method yields a basal shear stress of about 2 bars. These results indicate that for small plates the forces produced by basal shear stress are probably small in comparison with forces acting on plate boundaries. To a first approximation, the smaller plates act as if they were decoupled from the mantle below. These stress estimates lead to a model in which the motion of the smaller lithospheric plates is governed almost entirely by the forces acting on their edges. Forces due to basal shear stress may be comparable to forces acting on the edges of large lithospheric plates. Thus, complete decoupling may not be a good approximation for such plates.     

Ellipsoidal area mean gravity anomalies — precise computation of gravity anomaly reference fields for remove-compute-restore geoid determination

Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available as area mean values.     

An approximate method for determining the stability of two-scale flow in the mantle

Summary An approximate method of calculating the stability of two-dimensional convection rolls to cross-roll disturbances is evaluated by comparison with the results from exact analyses and good agreement is obtained. In particular, investigation of the second mode of the disturbance provides a qualitative estimate of the terms excluded in making the approximation. We conclude that this approximate method, sensibly used, gives a good indication of the stability of rolls to cross-roll disturbances.Application of this method is made to convection rolls heated partially or wholly from within, and to rolls in the presence of a longitudinal shear flow. The approximation indicates that, at high Rayleigh Numbers, growth rates are underestimated and so the amount of shear calculated to stabilise longitudinal rolls is a lower bound. Our results suggest that two-dimensional rolls are unlikely to be a stable flow pattern beneath even the fastest moving plates.     

斯托克司函数逼近及截断误差估计

本文对高度异常及垂线偏差的截断误差进行了估计,导出了高阶截断系数的近似表达式。这些截断系数是振幅逐渐衰减的正弦函数,而且其振幅与斯托克司函数在界圆φ_0处的值密切相关。经分析认为,采用莫洛金斯基的最小平方逼近方法,将可使截断误差的数量级大为降低,值得在实际中采用。为了进一步提高截断系数的收敛速度,建议在最小平方逼近的基础上,附加上界圆φ_0处的边界条件,这样将较单纯的逼近为优。为此提出两种实施的方法:利用拉格朗日的条件极值和利用样条函数逼近。     

地球物理反演的拟线性近似方法综述

以地球物理反演的非线性及其线性化方法为背景，系统地介绍了拟线性近似的基本思想及其在地球物理反演中的应用，在数学类比的基础上，提出了将拟线性近似方法应用到直流位场反演的研究中的思想框架，通过分析，得出了拟线性近似方法可以在地球物理反演研究中广泛应用的结论。     

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.     

基于入射角的两项流体阻抗反演方法

本文在孔隙弹性介质理论的指导下,基于入射角AVO近似方程推导了包含Russell流体项的两项AVO近似方程和相应的弹性阻抗方程,通过分析可知其精度符合反演要求.在贝叶斯理论框架下,建立了包含正则化约束的弹性阻抗反演方法,在此基础上直接提取Russell流体项.该方法可在缺少大角度叠前地震资料的情况下进行叠前直接反演得到流体因子,减少传统方法带来的累积误差.模型试算表明,该方法具有较好的准确度和稳定性.实际工区应用取得了良好的效果,表明该方法有实用性.     

有限频率线性理论的波恩近似佯谬

对有限频率层析成像线性理论的波恩近似问题进行梳理, 用数值方法统计分析其适用范围, 结果表明波恩近似要求最大速度扰动不超过1%; 然后对相关走时一阶近似进行统计分析, 结果表明它也只适用于最大速度扰动在1%以内的情形. 然而, 结合波恩近似和相关走时一阶近似而得到的有限频率线性理论, 其适用的速度扰动范围最大可达10%. 这个表面上的逻辑悖论, 称为“波恩近似佯谬”. 此佯谬是由于不恰当地使用波恩近似造成的. 本文摒弃波恩近似, 使用泛函的Fréchet微分和隐函数定理推导得到有限频率线性理论, 圆满解释了波恩近似佯谬. 由于有限频率非线性理论早已摒弃了波恩近似, 因此波恩近似概念在有限频率层析成像理论中完全没有必要.      

Stationary phase approximation in the ambient noise method revisited

The method of extracting Green's function between stations from cross correlation has proven to be effective theoretically and experimentally.It has been widely applied to surface wave tomography of the crust and upmost mantle.However,there are still controversies about why this method works.Snieder employed stationary phase approximation in evaluating contribution to cross correlation function from scatterers in the whole space,and concluded that it is the constructive interference of waves emitted by the scatterers near the receiver line that leads to the emergence of Green's function.His derivation demonstrates that cross correlation function is just the convolution of noise power spectrum and the Green's function.However,his derivation ignores influence from the two stationary points at infinities,therefore it may fail when attenuation is absent.In order to obtain accurate noise-correlation function due to scatters over the whole space,we compute the total contribution with numerical integration in polar coordinates.Our numerical computation of cross correlation function indicates that the incomplete stationary phase approximation introduces remarkable errors to the cross correlation function,in both amplitude and phase,when the frequency is low with reasonable quality factor Q.Our results argue that the distance between stations has to be beyond several wavelengths in order to reduce the influence of this inaccuracy on the applications of ambient noise method,and only the station pairs whose distances are above several (5) wavelengths can be used.     

对直流电法测井几何因子的评论

本文通过对直流电法测井应用数值解方法和应用几何因子方法所得结果的比较,说明对直流电法测井应用几何因子近似,效果不好。     

井间电磁成像的迭代反演算法

提出一种基于逐次逼近解法的选代反演算法,对并间地层电导率的构造进行成像．该方法用一阶Ｂｏｒｎ近似将积分方程线性化,得到对电导率分布的初始估计,在选代反演中用高阶Ｂｏｒｎ近似对井间地层电导率构造进行更精确的估计．应用该方法还可以对井间电子率分布进行二次成像,从而使成像分辨率更为准确．数值计算结果表明,这种迭代反演算法与基于Ｂｏｒｎ近似、二阶Ｂｏｒｎ近似和扩展Ｂｏｒｎ近似的反演算法相比提高了成像分辨率,且计算效率相当．     

基于直射线近似的一步法基准面校正方法研究

地形基准面校正算子（Topographic Datuming Operator,以下简称TDO）是一种基于直射线近似得到的基准面延拓算子.TDO可以视为是两步法波动方程基准面校正与常规静校正之间的一种过渡算法,该方法的最大特点在于它可以基于共炮点道集将炮点和检波点同时延拓到给定的水平基准面,因此相对于常规的两步法叠前波动方程基准面校正,TDO方法可以认为是一种更为高效的一步法基准面延拓方法.本文基于理论与实际数据论证了上述观点.