On the estimation of a multi-resolution representation of the gravity field based on spherical harmonics and wavelets

The gravitational potential of the Earth is usually modeled by means of a series expansion in terms of spherical harmonics. However, the computation of the series coefficients requires preferably homogeneous distributed global data sets. Since one of the most important features of wavelet functions is the ability to localize both in the spatial and in the frequency domain, regional and local structures may be modeled by means of a spherical wavelet expansion. In general, applying wavelet theory a given input data set is decomposed into a certain number of frequency-dependent detail signals, which can be interpreted as the building blocks of a multi-resolution representation. On the other hand, there is no doubt that the low-frequency part of the geopotential can be modeled appropriately by means of spherical harmonics. Hence, the main idea of this paper is to derive a combined model consisting of an expansion in spherical harmonics for the low-frequency part and an expansion in spherical wavelets for the remaining medium and high-frequency parts of the gravity field. Furthermore, an appropriate parameter estimation procedure is outlined to solve for the unknown model coefficients.     

COMPUTING THE KERNEL FUNCTION IN RESISTIVITY SOUNDING WITH AN ARBITRARY ELECTRODE CONFIGURATION*

A power series expansion can be used to obtain the kernel from apparent resistivities for an arbitrary electrode configuration. Three types of function are most appropriate for this purpose. The expansion coefficients can be by a least-squares method. In this case, ortho-normalization of the functions is of great advantage. An example with the Wenner configuration is given.     

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.     

Space-time modeling of the main magnetic field by combined methods of spherical harmonic analysis and natural orthogonal components

A simple method for obtaining a space-time model of the main magnetic field from the high-precision satellite survey data is described. At the first stage, the CHAMP satellite data for one-day interval are expanded into the spherical harmonics with constant coefficients. This yields a set of daily mean spherical harmonic models (DMSHM) over the survey interval of a few years. At the second stage, the coefficients of this set are used as source data for expansion into the natural orthogonal components (NOC). It is shown that the terms of the NOC series decrease rapidly, and the accuracy of the space-time model of the main geomagnetic field over the time interval under discussion is not worse than the accuracy of the models obtained by traditional methods.     

一种新的三维速度结构反演模型参数化方法

本文针对三维速度反演问题中的一个重要步骤--模型的参数化作了探讨和研究,提出了一种新的参数化方法.同以往采用的矩形分块或划分网格点的方法不同,将待求扰动场按其空间频率展开,反演各阶频率系数.从三维傅氏级数理论出发,导出了一个简明且便于计算的三维空间频谱展开公式,并从理论上分析了在空间域内离散模型和空间频率域内离散模型的不同之处. 理论分析表明,在空间域内采用过参数化离散模型来反演,会在所得到的解中混入人为的虚假结构,从而使反演解发生畸变.采用在频率域内离散模型的频谱参数化方法反演,可以有效地避免这一点.频谱参数化方法也使得我们可以过参数化（over-parameterization）离散模型,便于最大程度地从资料中提取正确信息,对模型的分辨程度完全取决于观测系统的分辨能力.这种方法在得到稳定的反演解的同时,还可得到观测系统在空间三个方向上的分辨率.数值模拟实验结果证实了以上结论.     

弹性波逆散射微扰论

本文将普遍声逆散射微扰论应用于弹性波层析成像问题,在Born变换下推出了以旋转角为补偿参数的各阶微扰重建公式,实现了对非均匀各向同性散射体内3个参数（质量密度ρ和两个Lamé系数λ,μ）的同时重建. 对于层析成像问题,在弹性波的传播过程中P波与SV波有耦合,但它们不会和SH波发生耦合,于是可以得到3个形式相对简单的标量方程. 在Born变换下,在散射波中引入微扰参数,将散射体的3个参数分别按该微扰参数展开,然后利用二维自由空间的Green函数分别得到散射的P波、SV波和SH波的积分表示. 最后,经一维傅氏变换后,得到Born变换下散射体3个参数的各阶微扰重建公式.     

用波函数展开法求解界面圆孔的SH波散射问题

研究了ＳＨ波绕界面圆孔散射的波函数展开理论。由入射,反射和透射波组成的自由波场与孔的散射场叠加成总波场。将定义于两个半平面的散射场按一定方式延拓到全平面。通过Ｈａｎｋｅｌ－Ｆｏｕｒｉｅｒ型积分来满足界面与孔缘处的边界条件,得到了确定特定系数的封闭代数方程组。给出了具体算例,计算了孔缘处的动应力系数分布。     

DYNAMIC RESPONSE VARIABILITY OF STRUCTURES WITH UNCERTAIN PROPERTIES

A modal-based analysis of the dynamic response variability of multiple degree-of-freedom linear structures with uncertain parameters subjected to either deterministic or stochastic excitations is considered. A probabilistic methodology is presented in which random variables with specified probability distributions are used to quantify the parameter uncertainties. The uncertainty in the response due to uncertainties in the structural modelling and loading is quantified by various probabilistic measures such as mean, variance and coefficient of excess. The computation of these probabilistic measures is addressed. A series expansion involving orthogonal polynomials in terms of the system parameters is first used to model the response variability of each contributing mode. Linear equations for the coefficients of each series expansion are derived using the weighted residual method. Mode superposition is then used to derive analytical expressions for the variability and statistics of the uncertain response in terms of the coefficients of the series expansions for all contributing modes. A primary–secondary system and a ten-story building subjected to deterministic and stochastic loads are used to demonstrate the methodology, as well as evaluate its performance by comparing it to existing methods, including the computationally cost-efficient perturbation method.     

Broadband Analytical Solution to the Scattering of Plane SV Waves by Circular Cylindrical Canyons

Based on Fourier-Bessel series expansion of wave functions,an analytical solution to 2-D scattering ofincident plane SV waves by circular cylindrical canyons with variable depthto-width ratios is deduced in this paper. Unlike other analytical solutions,this paper uses the asymptotic behavior of the cylindrical function to directly define the undetermined coefficients of scattered waves,thus,avoiding solving linear equation systems and the related numerical computation problems under high-frequency incident waves,thereby broadening the applicable frequency range of analytical solutions. Through comparison with existing analytical solutions,the correctness of this solution is demonstrated. Finally, the incident plane SV wave scattering effect under circular cylindrical canyons in wider frequency bands is explored.     

各向异性半空间上方垂直磁偶极子电磁场的球面波展开式

文中推导出单轴各向异性媒质半空间上方垂直磁偶极源电磁场的精确解析表达式. 应用圆柱波函数的球面波展开式和超几何函数理论，场分量中的索末菲尔德型积分被表示成快速、绝对收敛的球面波函数系展开式；展开系数是以物性参数为复宗量的勒让德多项式. 该展开式数学物理意义明显，并且不受场点和源点的位置、媒质的物性参数和频率等条件的限制. 利用本文的结果可十分方便地计算和分析任意场点处的电磁场分布.     

GOCE Quick-Look Gravity Solution: Application of the Semianalytic Approach in the Case of Data Gaps and Non-Repeat Orbits

The satellite mission GOCE (Gravity Field and Steady-State Ocean Circulation Explorer), the first Core Mission of the Earth Explorer Programme funded by ESA (European Space Agency), is dedicated to the precise modelling of the Earth's gravity field, with its launch planned for 2006. The mathematical models for parameterizing the Earth's gravity field are based on a series expansion into spherical harmonics, yielding a huge number of unknown coefficients. Their computation leads to the solution of very large normal equation systems. An efficient way to handle these equation systems is the so-called semianalytic or lumped coefficients approach, which theoretically requires an uninterrupted, continuous time series of observations, recorded along an exact circular repeat orbit. In this paper the consequences of violating these conditions are analyzed. The effects of an interrupted observation stream onto the estimated spherical harmonic coefficients are demonstrated, and an iterative strategy, which reduces the negative influence depending on the characteristics of the data gaps, is proposed. Additionally, the impact of an imperfectly closing orbit (non-repeat orbit) on the gravity field model is analyzed, and a strategy to minimize the corresponding errors is presented. The applicability of the semianalytic approach also to a joint inversion of satellite-to-satellite tracking data in high-low mode (hl-SST) and satellite gravity gradiometry (SGG) observations is demonstrated, where the analysis of the former component is based on the energy conservation law. Several realistic case studies prove that the semianalytic approach is a feasible tool to generate quick-look gravity solutions, i.e. fast coefficient estimates using only partial data sets. This quick-look analysis shall be able to detect potential distortions of statistical significance (e.g. systematic errors) in the input data, and to give a fast feedback to the GOCE mission control.     

Group modeling of monthly runoff hydrographs

The possibility of group simulation of monthly runoff hydrograph by canonical expansion method is demonstrated. The simulation procedure takes into account the coefficients of autocorrelation and mutual correlation and enables one to reproduce the ratios between parameters C _ν and C _s for monthly time intervals. The model preserves the similarity between within-year distributions of the measured and simulated series.     

Numerical solution of the complete equations for nearly horizontal flows

The complete mathematical model for the non steady hydrodynamic circulation due to wind, waves and density gradients in a coastal area or a lake is solved by the combined application of the weighted residuals—Galerkin method and the finite difference method. The computation of u and v velocity components in x, y, z, t space is achieved through expansion in series of base functions times undetermined coefficients over the depth. The computation of these coefficients giving the vertical variation of the velocity is done by the Galerkin method. The rapid convergence of this procedure permits a quick and economic evaluation of 3-D flows on a 2-D grid (in x, y space). The capabilities of the method were demonstrated by applying it to a tidal flow in an estuary.     

Response analysis of hysteretic multi-storey frames under earthquake excitation

Local incremental stiffness relations are formulated for a class of elasto-plastic beam elements. The earthquake acceleration is modelled as a filtered white noise process. The Itǒ differential equations of the integrated system made up of the structural system and the excitation process are then formulated. Instead of the original system an equivalent nonlinear system is considered, in which the drift vector is given by a series expansion of order n ≧ 1, where n = 1 represents the well-known case of equivalent linearization. Only components of the drift vector representing the non-analytieal constitutive equations are replaced by a polynomial expansion. The coefficients of this expansion are determined from a least mean square criterion, and are sequentially updated. Especially an equivalent system with a cubic expansion to the drift vector is investigated. The hierarchy of statistical moment equations is closed by a cumulant neglect closure scheme. The method has been applied to a two-storey frame. The results are compared to those of numerical simulation, and provide substantial improvements compared to equivalent linearization.     

Scattering and reflection of SH waves around a slope on an elastic wedged space

The scattering and reflection of SH waves by a slope on an elastic wedged space is investigated. A series solution is obtained by using the wave function expansion method. The slope on a wedged space is divided into two subregions by an artificial, auxiliary circular arc. The wave fields with unknown complex coefficients within each sub-region are derived. Applying Graf addition theorem, the scattered waves in the sub-regions are expressed in a global coordinate system. Fourier transform is adopted to derive a consistent form of standing waves in the inner region using the orthogonality of the cosine functions. The boundary-valued problem is solved by stress and displacement continuity along the artificial, auxiliary arc to obtain the unknown complex coefficients. Parametric studies are next performed to investigate how the topography from the slope on the wedged space will affect the scattering and diffraction, and hence the amplification and de-amplification of the SH waves. Numerical results show that the surface motions on the slope of the wedged space is influenced greatly by the topography. Amplification of the surface motions near the slope vertex is significant. The corresponding phases along the wedged space surfaces are consistent with the direction that the SH waves are propagating.     

海洋潮汐对重力潮汐观测的影响

本文论述了海潮对重力固体潮观测的影响,把整个褶积计算分为近洋区及远洋区两部分,其中近洋区采用格林函数进行数值积分的方法,远洋区则利用全球海潮潮高的球函数展开式。着重讨论了远洋区的实施方案,导出了必要的工作公式,并就加速远洋区级数收敛问题作了探讨。利用导得的公式,计算了M₂海洋潮波对中国境内重力观测的影响。     

Optimization and capacity expansion of a water distribution system

This paper develops an iterative procedure for capacity expansion studies for water distribution systems. We propose a methodology to analyze an existing water distribution system and identify the potential bottlenecks in the system. Based on the results, capacity expansion alternatives are proposed and evaluated for improving the efficiency of water supply. The methodology includes a network flow based optimization model, four evaluation indices, and a series of evaluation steps. We first use a directed graph to configure the water distribution system into a network. The network flow based model optimizes the water distribution in the system so that different expansion alternatives can be evaluated on a comparable basis. This model lends itself to linear programming (LP) and can be easily solved by a standard LP code. The results from the evaluation tool help to identify the bottlenecks in the water distribution system and provide capacity expansion alternatives. A useful complementary tool for decision making is composed of a series of evaluation steps with the bottleneck findings, capacity expansion alternatives, and the evaluation of results. We apply the proposed methodology to the Tou-Qian River Basin, located in the northern region of Taiwan, to demonstrate its applicability in optimization and capacity expansion studies.     

Using orthogonal Legendre polynomials to parameterize global geophysical optimizations: Applications to seismic-petrophysical inversion and 1D elastic full-waveform inversion

We use Legendre polynomials to reparameterize geophysical inversions solved through a particle swarm optimization. The subsurface model is expanded into series of Legendre polynomials that are used as basis functions. In this framework, the unknown parameters become the series of expansion coefficients associated with each polynomial. The aim of this peculiar parameterization is threefold: efficiently decreasing the number of unknowns, inherently imposing a 1D spatial correlation to the recovered subsurface model and searching for maximally decoupled parameters. The proposed approach is applied to two highly non-linear geophysical optimization problems: seismic-petrophysical inversion and 1D elastic full-waveform inversion. In this work, with the aim to maintain the discussion at a simple level, we limit the attention to synthetic seismic data. This strategy allows us to draw general conclusions about the suitability of this peculiar parameterization for solving geophysical problems. The results demonstrate that the proposed approach ensures fast convergence rates together with accurate and stable final model predictions. In particular, the proposed parameterization reveals to be effective in reducing the ill conditioning of the optimization problem and in circumventing the so-called curse-of-dimensionality issue. We also demonstrate that the implemented algorithm greatly outperforms the outcomes of the more standard approach to global inversion in which each subsurface parameter is considered as an independent unknown.     

Systematic lumped‐parameter models for foundations based on polynomial‐fraction approximation

Based on the approximation by polynomial‐fraction, a series of systematic lumped‐parameter models are developed in this paper for efficiently representing the dynamic behaviour of unbounded soil. Concise formulation is first employed to represent the dynamic flexibility function of foundation with a ratio of two polynomials. By defining an appropriate quadratic error function, the optimal coefficients of the polynomials can be directly solved from a system of linear equations. Through performing partial‐fraction expansion on this polynomial‐fraction and designing two basic discrete‐element models corresponding to the partial fractions, systematic lumped‐parameter models can be conveniently established by connecting these basic units in series. Since the systematic lumped‐parameter models are configured without introducing any mass, the foundation input motion can be directly applied to these models for their applications to the analysis of seismic excitation. The effectiveness of these new models is strictly validated by successfully simulating a semi‐infinite bar on an elastic foundation. Subsequently, these models are applied for representing the dynamic stiffness functions for different types of foundation. Comparison of the new models with the other existing lumped‐parameter models is also made to illustrate their advantages in requiring fewer parameters and featuring a more systematic expansion. Copyright © 2002 John Wiley & Sons, Ltd.     

修正辛格式有限元法的地震波场模拟

三角网格有限元法具有网格剖分的灵活性,能有效模拟地震波在复杂介质中的传播.但传统有限元法用于地震波场模拟时计算效率较低,消耗较大计算资源.本文采用改进的核矩阵存储(IKMS)策略以提高有限元法的计算效率,该方法不用组合总体刚度矩阵,且相比于常规有限元法节省成倍的内存.对于时间离散,将有限元离散后的地震波运动方程变换至Hamilton体系,在显式二阶辛Runge-Kutta-Nystr9m(RKN)格式的基础之上加入额外空间离散算子构造修正辛差分格式,通过Taylor展开式得到具有四阶时间精度时间格式,且辛系数全为正数.本文从理论上分析了时空改进方法相比传统辛-有限元方法在频散压制、稳定性提升等方面的优势.数值算例进一步证实本方法具有内存消耗少、稳定性强和数值频散弱等优点.