遗传算法反演地球等离子体层离子密度分布

本文介绍了采用一维遗传算法从地球等离子体层极紫外图像反演地球等离子体层He⁺密度的原理.首先采用通量管近似和磁偶极近似将三维问题转化为一维问题.通过引入权矩阵,将极紫外光强积分离散为求和函数,再采用一维实数编码遗传算法反演得到磁赤道面等离子体层He⁺密度,最后通过磁力线追迹得到三维密度分布.算法采用动态全球核心等离子体模式模拟的密度和光强分布作为初始输入参数,并通过遗传算法得到相应密度分布.反演结果表明,等离子体层密度相对误差在8%以内,光强相对误差趋于0,算法有效可行.本文研究为中国探月二期工程中月基极紫外图像反演奠定了基础.     

Stability analysis of SDOF real‐time hybrid testing systems with explicit integration algorithms and actuator delay

Real‐time hybrid testing is a method that combines experimental substructure(s) representing component(s) of a structure with a numerical model of the remaining part of the structure. These substructures are combined with the integration algorithm for the test and the servo‐hydraulic actuator to form the real‐time hybrid testing system. The inherent dynamics of the servo‐hydraulic actuator used in real‐time hybrid testing will give rise to a time delay, which may result in a degradation of accuracy of the test, and possibly render the system to become unstable. To acquire a better understanding of the stability of a real‐time hybrid test with actuator delay, a stability analysis procedure for single‐degree‐of‐freedom structures is presented that includes both the actuator delay and an explicit integration algorithm. The actuator delay is modeled by a discrete transfer function and combined with a discrete transfer function representing the integration algorithm to form a closed‐loop transfer function for the real‐time hybrid testing system. The stability of the system is investigated by examining the poles of the closed‐loop transfer function. The effect of actuator delay on the stability of a real‐time hybrid test is shown to be dependent on the structural parameters as well as the form of the integration algorithm. The stability analysis results can have a significant difference compared with the solution from the delay differential equation, thereby illustrating the need to include the integration algorithm in the stability analysis of a real‐time hybrid testing system. Copyright © 2007 John Wiley & Sons, Ltd.     

AN ACCURATE SCHEME FOR SEISMIC FORWARD MODELLING*

A new time integration technique for use in forward modelling programmes is introduced. The technique presents an alternative to second-order temporal differencing. It is based on a Chebyshev expansion of the formal evolution operator to the spatially discretized wave equation. The computational effort in forward modelling based on the new technique is about the same as in methods based on temporal differencing. However, machine accuracy can be obtained. The implementation of the technique to solve the acoustic wave equation in two spatial dimensions is described. Finally, an example of using the technique to solve a problem of wave propagation in a single layer is presented.     

黏弹性VTI介质频率空间域准P波正演模拟

有限差分方法是波场数值模拟的一个重要方法,时间域有限差分计算方法因按时间片递推计算,每个时间片的舍入误差会累积到下一片中,当时间片较多,最终会导致累积误差太大.而频率域计算是按频率片对空间网格进行整体求解方程组,其计算误差分配到了每个网格点上,并且各个频率片之间是独立计算的,因此不存在累计误差,而且在频率-空间域更易于...     

A simple error estimator and adaptive time stepping procedure for dynamic analysis

A simple local error estimator is presented for time integration schemes in dynamic analysis. This error estimator involves only a small computational cost. The time step size is adaptively adjusted so that the local error at each time step is within a prescribed accuracy. It is found that the estimator performs well under various circumstances and provides an economical adaptive process. Attempts to estimate the global time integration error are also reported.     

A primitive pseudo wave equation formulation for solving the harmonic shallow water equations

A finite element method formulation for solving the harmonic shallow water equations in their primitive or unmodified form is developed and analysed. The scheme, referred to as the Primitive Pseudo Wave Equation Formulation (PPWE), involves developing a weak weighted residual form of the continuity equation and furthermore forming a pseudo wave equation by substituting the discretized form of the momentum equation into the discretized form of the continuity equation. The final set of equations to be solved, the pseudo wave equation and the primitive momentum equations, deceptively resemble the discretized equations of the wave equation formulation of Lynch and Gray. Despite this resemblance, Fourier analysis indicates that the PPWE scheme is still fundamentally primitive.However, application of the PPWE scheme to a set of stringent test problems results in very good solutions with well controlled nodal oscillations. It is shown that this low degree of spurious oscillations is due to the treatment of the boundary conditions such that elevation is taken as an essential condition and normal flux is taken as a natural condition. This particular boundary condition treatment is suggested by the formation of the pseudo wave equation. Furthermore, even though the equation re-arrangement does not in itself effect the solutions, it does make the scheme much more efficient.     

A nearly analytic symplectic partitioned Runge–Kutta method based on a locally one-dimensional technique for solving two-dimensional acoustic wave equations

In this paper, we develop a new nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique for numerically solving two-dimensional acoustic wave equations. We first split two-dimensional acoustic wave equation into the local one-dimensional equations and transform each of the split equations into a Hamiltonian system. Then, we use both a nearly analytic discrete operator and a central difference operator to approximate the high-order spatial differential operators, which implies the symmetry of the discretized spatial differential operators, and we employ the partitioned second-order symplectic Runge–Kutta method to numerically solve the resulted semi-discrete Hamiltonian ordinary differential equations, which results in fully discretized scheme is symplectic unlike conventional nearly analytic symplectic partitioned Runge–Kutta methods. Theoretical analyses show that the nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique exhibits great higher stability limits and less numerical dispersion than the nearly analytic symplectic partitioned Runge–Kutta method. Numerical experiments are conducted to verify advantages of the nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique, such as their computational efficiency, stability, numerical dispersion and long-term calculation capability.     

Direct BEM for high-resolution global gravity field modelling

The paper presents a high-resolution global gravity field modelling by the boundary element method (BEM). A direct BEM formulation for the Laplace equation is applied to get a numerical solution of the linearized fixed gravimetric boundary-value problem. The numerical scheme uses the collocation method with linear basis functions. It involves a discretization of the complicated Earth’s surface, which is considered as a fixed boundary. Here 3D positions of collocation points are simulated from the DNSC08 mean sea surface at oceans and from the SRTM30PLUS_V5.0 global topography model added to EGM96 on lands. High-performance computations together with an elimination of the far zones’ interactions allow a very refined integration over the all Earth’s surface with a resolution up to 0.1 deg. Inaccuracy of the approximate coarse solutions used for the elimination of the far zones’ interactions leads to a long-wavelength error surface included in the obtained numerical solution. This paper introduces an iterative procedure how to reduce such long-wavelength error surface. Surface gravity disturbances as oblique derivative boundary conditions are generated from the EGM2008 geopotential model. Numerical experiments demonstrate how the iterative procedure tends to the final numerical solutions that are converging to EGM2008. Finally the input surface gravity disturbances at oceans are replaced by real data obtained from the DNSC08 altimetryderived gravity data. The ITG-GRACE03S satellite geopotential model up to degree 180 is used to eliminate far zones’ interactions. The final high-resolution global gravity field model with the resolution 0.1 deg is compared with EGM2008.     

Adaptive local discontinuous Galerkin approximation to Richards’ equation

We propose a spatially and temporally adaptive solution to Richards’ equation based upon a local discontinuous Galerkin approximation in space and a high-order, backward difference method in time. We cast our approach in terms of a general, decoupled adaption algorithm based upon operators. We define non-unique instances of all operators to result in an adaption method from within the general class of methods that is defined. We formally decouple the spatial adaption from the temporal adaption using a method of lines approach and limit the temporal truncation error so that the total error is dominated by the spatial component. We use a multiple grid approach to guide adaption and support the data structures. Spatial adaption decisions are based upon error and regularity indicators, which are economical to compute. The resultant methods are compared for two test problems. The results show that the proposed adaption methods are superior to methods that adapt only in time and that in cases in which the problem has sufficient smoothness, adapting the order of the elements in addition to the grid spacing can further improve the efficiency of this robust solution approach.     

Assimilation of near-surface temperature using extended Kalman filter

In order to improve the soil temperature profile predictions in land-surface models, an assimilation scheme using the extended Kalman filter is developed. This formulation is based on the discretized diffusion equation of heat transfer through the soil column. The scheme is designed to incorporate the knowledge of the uncertainties in both the model and the measurement. Model uncertainty is estimated by quantifying the model drift from observations when the model is initialized using the observed values. Furthermore, the initial error covariance has a significant influence on the performance of the assimilation scheme. It is shown that an inaccurate initial value for the error covariance can actually diminish the predictive capabilities of the model. When an appropriate initial error covariance is specified, using the top layer soil temperature observations in the assimilation scheme allows for improved predictive capabilities in lower layers. Observations at 30 min intervals have a significant effect on the model predictions in the lower layers. Assimilation of observations at 24 h intervals also has an effect on the lower layer predictive capability of the model, albeit more slowly than the 30 min assimilation scenario.     

Analysis of floodplain inundation using 2D nonlinear diffusive wave equation solved with splitting technique

In the paper a solution of two-dimensional (2D) nonlinear diffusive wave equation in a partially dry and wet domain is considered. The splitting technique which allows to reduce 2D problem into the sequence of one-dimensional (1D) problems is applied. The obtained 1D equations with regard to x and y are spatially discretized using the modified finite element method with the linear shape functions. The applied modification referring to the procedure of spatial integration leads to a more general algorithm involving a weighting parameter. Time integration is carried out using a two-level difference scheme with the weighting parameter as well. The resulting tri-diagonal systems of nonlinear algebraic equations are solved using the Picard iterative method. For particular sets of the weighting parameters, the proposed method takes the form of a standard finite element method and various schemes of the finite difference method. On the other hand, for the linear version of the governing equation, the proper values of the weighting parameters ensure an approximation of 3rd order. Since the diffusive wave equation can be solved no matter whether the area is dry or wet, the numerical computations can be carried out over entire domain of solution without distinguishing a current position of the shoreline which is obtained as a result of solution.     

一种全局优化的隐式交错网格有限差分算法及其在弹性波数值模拟中的应用

在数值模拟中,隐式有限差分具有较高的精度和稳定性.然而,传统隐式有限差分算法大多由于需要求解大型矩阵方程而存在计算效率偏低的局限性.本文针对一阶速度-应力弹性波方程,构建了一种优化隐式交错网格有限差分格式,然后将改进格式由时间-空间域转换为时间-波数域,利用二范数原理建立目标函数,再利用模拟退火法求取优化系数.通过对均匀模型以及复杂介质模型进行一阶速度-应力弹性波方程数值模拟所得单炮记录、波场快照分析表明:这种优化隐式交错网格差分算法与传统的几种显式和隐式交错网格有限差分算法相比不但降低了计算量,而且能有效的压制网格频散,使弹性波数值模拟的精度得到有效的提高.     

Wave‐equation migration velocity analysis with time‐shift imaging

Wave‐equation migration velocity analysis is a technique designed to extract and update velocity information from migrated images. The velocity model is updated through the process of optimizing the coherence of images migrated with the known background velocity model. The capacity for handling multi‐pathing of the technique makes it appropriate in complex subsurface regions characterized by strong velocity variation. Wave‐equation migration velocity analysis operates by establishing a linear relation between a slowness perturbation and a corresponding image perturbation. The linear relationship and the corresponding linearized operator are derived from conventional extrapolation operators and the linearized operator inherits the main properties of frequency‐domain wavefield extrapolation. A key step in the implementation is to design an appropriate procedure for constructing an image perturbation relative to a reference image that represents the difference between the current image and a true, or more correct image of the subsurface geology. The target of the inversion is to minimize such an image perturbation by optimizing the velocity model. Using time‐shift common‐image gathers, one can characterize the imperfections of migrated images by defining the focusing error as the shift of the focus of reflections along the time‐shift axis. The focusing error is then transformed into an image perturbation by focusing analysis under the linear approximation. As the focusing error is caused by the incorrect velocity model, the resulting image perturbation can be considered as a mapping of the velocity model error in the image space. Such an approach for constructing the image perturbation is computationally efficient and simple to implement. The technique also provides a new alternative for using focusing information in wavefield‐based velocity model building. Synthetic examples demonstrate the successful application of our method to a layered model and a subsalt velocity update problem.     

Solution of dynamic response of SDOF system using piecewise Lagrange polynomial

As an extension of the procedure in which an arbitrary dynamic loading is approximated by piecewise linear segments, the second‐ and third‐degree piecewise Lagrangian interpolating polynomials are employed to approximate an arbitrary dynamic loading in the Duhamel integral for the solution of dynamic response of a SDOF system. The related formulae are derived. The proposed method offers computational advantage over the traditional step‐by‐step solution techniques for comparable accuracy, and far better accuracy than the piecewise linear approximation procedure for comparable time interval when the loading cannot be represented by straight‐line segments. Copyright © 2001 John Wiley & Sons, Ltd.     

New digital linear filters for Hankel J0 and J1 transforms

The numerical evaluation of certain integral transforms is required for the interpretation of some geophysical exploration data. Digital linear filter operators are widely used for carrying out such numerical integration. It is known that the method of Wiener–Hopf minimization of the error can be used to design very efficient, short digital linear filter operators for this purpose. We have found that, with appropriate modifications, this method can also be used to design longer filters. Two filters for the Hankel J₀ transform (61-point and 120-point operators), and two for the Hankel J₁ transform (47-point and 140-point operators) have been designed. For these transforms, the new filters give much lower errors compared to all other known filters of comparable, or somewhat longer, size. The new filter operators and some results of comparative performance tests with known integral transforms are presented. These filters would find widespread application in many numerical evaluation problems in geophysics.     

The meaning of oscillations in unit hydrograph S-curves

ABSTRACT

The Duhamel superposition integral is used to obtain some exact solutions for unit hydrograph applications. These equations and numerical examples are used to show that oscillations will occur in an S-curve when the time step is less than the excess rainfall duration if the measured hydrograph differs from a hydrograph that would be obtained by solving a linear differential equation with time-independent coefficients. The implications of this result with regard to the calculation of the instantaneous unit hydrograph (IUH) are discussed.     

单程波算子积分解的象征表示

单程波波场延拓算子在地震偏移成像中有重要应用.单程波波场延拓算子按其实现方式可分为Kirchhoff积分、空间隐式有限差分和Fourier变换方法,他们代表了算子的不同表示方法,当截断使用这些方法时会得到不同的精度.象征表示对这些方法的导出和精度分析有重要作用.算子作用于正弦波函数所得函数称为算子的象征.算子的象征是褶积算子Fourier变换的推广.Fourier变换方法则直接用象征函数的可分表示求出.空间隐式有限差分则可以用象征函数的Padè近似或部分分式导出.单程波算子在深度域的积分称为单程波算子积分解.本文推导了单程波算子积分解的象征表达式,给出了算子象征的代数运算的头几阶表达式,这些表达式还未在前人文献中发现.Kirchhoff积分所需格林函数可以通过象征函数和鞍点法导出.基于积分解的象征表达式给出了非对称走时公式,对改善Kirchhoff积分的聚焦性能有重要意义.     

Recursive evaluation of interaction forces of unbounded soil in the time domain

The interaction forces representing the contribution of the linear unbounded soil to the equations of motion of a nonlinear soil-structure-interaction analysis are specified in the form of convolution integrals. They can be evaluated recursively in the time domain. In this procedure, the forces at a specific time are computed from the displacements at the same time and from the most recent forces and most recent past displacements. It is, in principle, only approximate. When the dynamic-stiffness coefficients can be expressed as the ratios of two polynomials in frequency, the appropriately chosen recursive equations are exact. Two possibilities of choosing a recursive equation are discussed.

• (i) The impulse-invariant method, where the unknown recursive coefficients are calculated by solving a system of equations which are established by equating the rigorous and recursive formulations for a discretized unit impulse displacement.
• (ii) In the segment approach, the dynamic-stiffness coefficients in the time domain are interpolated piecewise. Applying the z-transformation analytically then results in an explicit recursive equation without solving a system of equations.

The recursive evaluation of the convolution integrals in the time domain leads to a dramatic reduction in the computational effort up to two and three orders of magnitude and in the storage requirement. This makes the time-domain analysis using the substructure method computationally competitive with the corresponding direct (non-recursive) frequency-domain procedure of determining the complex response which is, however, applicable only to a linear (total) system.     

Extensions of nonlinear error propagation analysis for explicit pseudodynamic testing

Two important extensions of a technique to perform a nonlinear error propagation analysis for an explicit pseudodynamic algorithm (Chang, 2003) are presented. One extends the stability study from a given time step to a complete step-by-step integration procedure. It is analytically proven that ensuring stability conditions in each time step leads to a stable computation of the entire step-by-step integration procedure. The other extension shows that the nonlinear error propagation results, which are derived for a nonlinear single degree of freedom (SDOF) system, can be applied to a nonlinear multiple degree of freedom (MDOF) system. This application is dependent upon the determination of the natural frequencies of the system in each time step, since all the numerical properties and error propagation properties in the time step are closely related to these frequencies. The results are derived from the step degree of nonlinearity. An instantaneous degree of nonlinearity is introduced to replace the step degree of nonlinearity and is shown to be easier to use in practice. The extensions can be also applied to the results derived from a SDOF system based on the instantaneous degree of nonlinearity, and hence a time step might be appropriately chosen to perform a pseudodynamic test prior to testing.     

On the accuracy of an implicit algorithm for pseudodynamic tests

The error-propagation characteristics of an implicit time integration algorithm in pseudodynamic testing are examined. It is shown that the implicit algorithm is superior to explicit integration algorithms in terms of experimental error amplification. The influence of systematic experimental errors is studied and methods for controlling these errors are examined. In spite of the fact that the implicit algorithm is unconditionally stable, it is shown that the integration time interval in a pseudodynamic test is limited by the calibration range of the electronic hardware as well as the degree of participation of the higher modes. Furthermore, the tolerance for experimental errors decreases as the integration time interval increases.