Comparison of methods to model the gravitational gradients from topographic data bases

A number of methods have been developed over the last few decades to model the gravitational gradients using digital elevation data. All methods are based on second-order derivatives of the Newtonian mass integral for the gravitational potential. Foremost are algorithms that divide the topographic masses into prisms or more general polyhedra and sum the corresponding gradient contributions. Other methods are designed for computational speed and make use of the fast Fourier transform (FFT), require a regular rectangular grid of data, and yield gradients on the entire grid, but only at constant altitude. We add to these the ordinary numerical integration (in horizontal coordinates) of the gradient integrals. In total we compare two prism, two FFT and two ordinary numerical integration methods using 1" elevation data in two topographic regimes (rough and moderate terrain). Prism methods depend on the type of finite elements that are generated with the elevation data; in particular, alternative triangulations can yield significant differences in the gradients (up to tens of Eötvös). The FFT methods depend on a series development of the topographic heights, requiring terms up to 14th order in rough terrain; and, one popular method has significant bias errors (e.g. 13 Eötvös in the vertical–vertical gradient) embedded in its practical realization. The straightforward numerical integrations, whether on a rectangular or triangulated grid, yield sub-Eötvös differences in the gradients when compared to the other methods (except near the edges of the integration area) and they are as efficient computationally as the finite element methods.     

Traction image method for irregular free surface boundaries in finite difference seismic wave simulation

In this study, we propose a new numerical method, named as Traction Image method, to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography. In this algorithm, the computational domain is discretized by boundary-conforming grids, in which the irregular surface is transformed into a 'flat' surface in computational space. Thus, the artefact of staircase approximation to arbitrarily irregular surface can be avoided. Such boundary-conforming gridding is equivalent to a curvilinear coordinate system, in which the first-order partial differential velocity-stress equations are numerically updated by an optimized high-order non-staggered finite difference scheme, that is, DRP/opt MacCormack scheme. To satisfy the free surface boundary conditions, we extend the Stress Image method for planar surface to Traction Image method for arbitrarily irregular surface by antisymmetrically setting the values of normal traction on the grid points above the free surface. This Traction Image method can be efficiently implemented. To validate this new method, we perform numerical tests to several complex models by comparing our results with those computed by other independent accurate methods. Although some of the testing examples have extremely sloped topography, all tested results show an excellent agreement between our results and those from the reference solutions, confirming the validity of our method for modelling seismic waves in the heterogeneous media with arbitrary shape topography. Numerical tests also demonstrate the efficiency of this method. We find about 10 grid points per shortest wavelength is enough to maintain the global accuracy of the simulation. Although the current study is for 2-D P-SV problem, it can be easily extended to 3-D problem.     

A traveltime reciprocity discrepancy in the Podvin & Lecomte time3d finite difference algorithm

We have found that the extensively used finite difference scheme time3d produces time fields which are 'asymmetric' in the sense that traveltimes computed to the right of the source are faster than traveltimes computed to the left. All finite difference schemes will, as they are approximations to the wave equation, to some extent fail to obey reciprocity perfectly. We show, however, that the errors in time3d may be significant—and unnecessarily large. An asymmetry in the point source initialization has been identified, and after correction time3d produces time fields with an improved reciprocity.     

Fast computation of Hankel Transform using orthonormal exponential approximation of complex kernel function

The computation of electromagnetic (EM) fields, for 1-D layered earth model, requires evaluation of Hankel Transform (HT) of the EM kernel function. The digital filtering is the most widely used technique to evaluate HT integrals. However, it has some obvious shortcomings. We present an alternative scheme, based on an orthonormal exponential approximation of the kernel function, for evaluating HT integrals. This approximation of the kernel function was chosen because the analytical solution of HT of an exponential function is readily available in literature. This expansion reduces the integral to a simple algebraic sum. The implementation of such a scheme requires that the weights and the exponents of the exponential function be estimated. The exponents were estimated through a guided search algorithm while the weights were obtained using Marquardt matrix inversion method. The algorithm was tested on analytical HT pairs available in literature. The results are compared with those obtained using the digital filtering technique with Anderson filters. The field curves for four types (A-, K-, H-and Q-type) of 3-layer earth models are generated using the present scheme and compared with the corresponding curves obtained using the Anderson sc heme. It is concluded that the present scheme is more accurate than the Anderson scheme     

基于有限差分法的抗滑桩计算机辅助设计

基于地基系数“m-m”法、“m-k”法、“k-k”法的原理,考虑桩顶和桩底边界条件以及桩在滑动面处位移、转角、弯矩和剪力的连续条件,可解得桩身各节点的位移和内力,提出了进行抗滑桩全桩内力计算的有限差分法。根据差分方程并用VB6.0编制了实用的计算程序,既可避免繁琐的查表计算,提高计算速度,又可提高计算精度,直观生动,真正实现了人机交互,在界面的引导下,设计人员可完成全部计算,并绘出内力图形和抗滑桩截面配筋图,使设计更方便快捷,该软件可以极大地提高生产效率,降低工程造价,从而实现抗滑桩的优化设计。最后采用上述方法对某滑坡的悬臂抗滑桩进行了设计与计算。     

燕山水库坝基防渗墙优化设计

因现场资料不足或分析手段落后等原因,当前进行水库坝基防渗墙设计,往往存在过于保守的倾向。以河南省燕山水库为例,基于优化设计的思路,采用目前较为成熟的二维有限单元法,对不同防渗条件下的坝基渗流场分别进行了模拟,并将计算所得比降、流量与允许比降、允许流量做了对比,从定量的角度提出了既安全、又经济的防渗方案:防渗墙厚度取0.8m,深度取打入破碎带5m。