旋转椭球面上的应变与转动张量表达

以旋转椭球体面上某点为原点建立一个大地坐标单位活动坐标架. 通过平移, 使活动坐标架的原点与以椭球中心为原点的笛卡尔单位标架的原点相重合. 然后再通过两次标架旋转, 使活动坐标架与笛卡尔单位标架完全重合. 本文给出了使两个单位标架相重合的转换关系式, 以及该点位移在两个单位标架中的坐标转换式; 在此基础上, 考虑该点的位移及活动坐标架皆为该点大地坐标的函数, 经复杂推导, 分别给出了该点位移向量的微分在大地坐标系中的分量以及该点分别沿坐标曲线的弧微分表达式, 继而导出了该点的位移梯度矩阵; 最后推导出了椭球坐标系的应变张量与转动张量表达式, 并对转动张量的几何含义进行了较详细的解释, 且采用曲面理论对球面与椭球面的应变张量间的内在关系进行了讨论.      

曲线坐标系下的块体运动与应变模型研究

本文建立了顾及地球扁率和局部切标架随点变化特性的椭球坐标系下的刚体运动模型和块体运动与应变模型,以及球坐标系下顾及局部切标架随点变化特性的严密的块体运动与应变模型,分析了球坐标系下块体运动与应变模型及椭球坐标系下的块体运动与应变模型间的差异;通过计算具体讨论了地球扁率和曲线坐标系的局部切标架随点变化特性对欧拉矢量与应变张量的影响.结果表明：地球扁率对刚体欧拉矢量和应变参数的影响甚小,具体计算时可以不予考虑,但曲线坐标系的局部切标架随点变化特性对两者的影响较大,在建模过程中需要顾及,常用的Savage模型需要修正.     

An optimized method to transform the Cartesian to geodetic coordinates on a triaxial ellipsoid

Studia Geophysica et Geodaetica - A general triaxial ellipsoid is suitable to represent the reference surface of the celestial bodies. The transformation from the Cartesian to geodetic coordinates...     

大地坐标系应变张量表达及其与地心直角坐标系的相互转换

推导并给出了不同正交曲线坐标系应变张量和转动张量的普适表达和在旋转椭球坐标系下的应变张量表达,给出了不同正交曲线坐标系之间应变张量转换的普适表达式,以及在大地坐标系与地心直角坐标系这两种坐标系之间应变张量矩阵相互转换的具体表达,可供实际研究工作应用。     

Increasing numerical efficiency of iterative solution for total least-squares in datum transformations

Cartesian coordinate transformation between two erroneous coordinate systems is considered within the Errors-In-Variables (EIV) model. The adjustment of this model is usually called the total Least-Squares (LS). There are many iterative algorithms given in geodetic literature for this adjustment. They give equivalent results for the same example and for the same user-defined convergence error tolerance. However, their convergence speed and stability are affected adversely if the coefficient matrix of the normal equations in the iterative solution is ill-conditioned. The well-known numerical techniques, such as regularization, shifting-scaling of the variables in the model, etc., for fixing this problem are not applied easily to the complicated equations of these algorithms. The EIV model for coordinate transformations can be considered as the nonlinear Gauss-Helmert (GH) model. The (weighted) standard LS adjustment of the iteratively linearized GH model yields the (weighted) total LS solution. It is uncomplicated to use the above-mentioned numerical techniques in this LS adjustment procedure. In this contribution, it is shown how properly diminished coordinate systems can be used in the iterative solution of this adjustment. Although its equations are mainly studied herein for 3D similarity transformation with differential rotations, they can be derived for other kinds of coordinate transformations as shown in the study. The convergence properties of the algorithms established based on the LS adjustment of the GH model are studied considering numerical examples. These examples show that using the diminished coordinates for both systems increases the numerical efficiency of the iterative solution for total LS in geodetic datum transformation: the corresponding algorithm working with the diminished coordinates converges much faster with an error of at least 10^-5 times smaller than the one working with the original coordinates.     

Two modified algorithms to transform Cartesian to geodetic coordinates on a triaxial ellipsoid

The paper presents modified Lin and Wang??s (1995) and Hedgley??s (1976) algorithms to tackle the problem of transforming Cartesian to geodetic coordinates on a triaxial ellipsoid. Originally, the methods were developed for an ellipsoid of revolution but due to their universality, they may be adapted to the more complicated problem stated on a triaxial ellipsoid what is in fact done in this work. Two modified methods are compared to the vector method recently introduced by Feltens. The modified methods turn out to be more accurate and faster than the algorithm presented by Feltens.     

利用地球重力位模型计算重力和重力梯度

高阶高精度地球重力场模型具有广泛的用途。本文利用地球重力位模型计算重力和重力梯度在应用中很有实用阶值,同时也是计算重力场其它量的关键。利用伪局部笛卡尔坐标与球坐标的关系计算了重力与重力梯度在伪局部笛卡尔坐标系下的分量;利用张量变换的原理给出了已知重力与重力梯度在某一坐标系下的分量求它们在另一坐标系下分量的方法,并具体给出了重力与重力梯度在局部笛卡儿坐标系下的分量计算公式,同时还给出计算重力场五参量与垂线偏差的计算公式,本研究推进了地球重力场的可视化进程。     

椭球坐标系下多震相地震射线追踪

地震射线追踪方法技术在地震学领域有着较为广泛的应用,然而大多数算法建立在直角坐标系或球坐标系下,实际地球并非完美的球体,而是两极略扁的椭球体,因此,球坐标系下计算结果与真实情况存在一定误差.传统的做法一般是在球坐标系下进行计算,而后进行椭球校正.本文提出了一种直接在椭球体模型中采用分区多步最短路径算法进行多震相地震射线追踪的方法技术,实现了椭球坐标系下多震相地震波射线路径追踪和走时计算.与解析解的对比表明:该算法具有较高的计算精度,适用于任意形状的椭球体,且不需要进行额外的走时校正.数值模拟结果表明,计算所得P波和PcP反射波的走时与AK135走时表的误差小于0.1 s.当震中距较大时,使用球对称模型和椭球体模型计算所得的走时差异显著,说明采用椭球坐标系的必要性.     

Closed-form transformation between geodetic and ellipsoidal coordinates

We present formulas for direct closed-form transformation between geodetic coordinates (φ, λ, h) and ellipsoidal coordinates (β, λ, u) for any oblate ellipsoid of revolution. These will be useful for those dealing with ellipsoidal representations of the Earth’s gravity field or other oblate ellipsoidal figures. The numerical stability of the transformations for near-polar and near-equatorial regions is also considered.     

Computation of strain and rotation tensor as well as their uncertainties for small arrays in spherical coordinate system

Based on Taylor series expansion and strain components expressions of elastic mechanics, we derive formulae of strain and rotation tensor for small arrays in spherical coordinates system. By linearization process of the formulae, we also derive expressions of strain components and Euler vector uncertainties respectively for subnets using the law of error propagation. Taking GPS velocity field in Sichuan-Yunnan area as an example, we compute dilation rate and maximum shear strain rate field using the above procedure, and their characteristics are preliminarily car- ried on. Limits of the strain model for small array are also discussed. We make detailed explanations on small array method and the choice of small arrays. How to set weights of GPS observations are further discussed. Moreover relationship between strain and radius of GPS subnets is also analyzed.     

Effect of common point selection on coordinate transformation parameter determination

The use of satellite positioning techniques commonly requires a transformation from a Conventional Terrestrial coordinate system to a Geodetic coordinate system, or vice versa. For such a transformation, the main problem is the determination of transformation parameters between these coordinate systems. The transformation parameters are estimated by a least-squares process using “common” points, i.e., those points whose coordinates are known in both systems. Therefore, the precision of so estimated transformation parameters is closely related to certain characteristics of the common points. In this contribution, we have formulated some theoretical relations between the transformation parameters and the number and the distribution of common points, and corroborated the theoretical results numerically, using a simulated geodetic network.     

Die Ersatzkugel in geodätischen Berechnungen

Summary For precise geodetic computations over larger distances the reference surface of an ellipsoid of rotation should be used. However it is often replaced by a sphere of an adequate radius. The formulae are derived from figures which usually represent the conditions in a cross-section of the ellipsoid and the reference sphere through the normal plane. Equation (9) is given for the differences s of the length of the ellipse arc of the normal section and the corresponding arc of the circle with radius R. Also Eq. (19) is given for the distance d between the ellipse of the normal section and the circle (at the end point). Both equations are applied for various radii of the reference sphere. Table 1 shows the values s, Tab. 2 and Fig. 2 give the d-values for chosen lengths. It was found that especially the distance between the ellipsoid and the sphere need not always be negligible.     

用应变地震观测求解震源矩张量的基本原理

震源机制解,即对地震矩张量的推断,对于地震研究具有至关重要的意义.应变地震观测是张量观测,与摆式地震仪的位移矢量观测不同,可以为地震研究提供新的数据源.本文讨论用应变地震观测求解震源矩张量的基本原理.在距离震源足够远的地方,地震波可以看成平面波,其性质决定于震源矩张量.假设平面地震波的应变张量可以由震源矩张量通过坐标变换计算得到,就可以通过观测应变地震波求解震源机制.这个假设至少对于双力偶震源机制是成立的.由此可以证明,在理想的无限介质中,只要有两个以上不同地点的应变地震波观测,就可以解出震源矩张量.这为解决震源机制问题提供了新的方法.目前的地震矩张量求解方法需要两方面的条件:或者需要很多观测点(例如体波反演),或者需要长周期地震波资料(例如面波反演).这些方法只适用于分析比较大的地震.对于小震,因为通常其震中周围不会有足够多的摆式地震仪观测点观测到其地震波,而地震波周期又短,难以利用传统方法给出可靠的震源机制解,所以只需少数观测点就能求解震源矩张量的新方法特别有意义.用应变地震观测求解震源机制,可以给出更为精确的结果.     

福建沿海、台湾海峡GPS观测分析及地球动力学特征研究

利用 3期GPS联测结果所获得的福建沿海地壳水平运动信息 ,采用ITRF94全球框架为基础的GPS测站地壳运动模型及其处理软件 ,对所获得的观测数据进行处理和精度分析。得到福建省高精度的GPS测站大地坐标、边长及其位移矢量 ,其精度达到 1 7×10 - 8。计算了福建地壳运动速率、主应变率 ,东西与南北向线应变率、面应变率、剪应变率、大地转动率和最大剪应变率等值线并给出了它们的分布图象。根据多年形变和现今GPS观测资料 ,分析福建地壳垂直运动与水平运动 ,显示区域应力场优势分布特征。最后 ,对福建沿海及台湾海峡地壳动力学特征作了初步的探讨     

DARCY'S COEFFICIENT OF PERMEABILITY AS SYMMETRIC TENSOR OF SECOND RANK

Synopsis

The dynamic equation of motion that governs the laminar flow of water through soils is the empirical equation of Darcy. According to Darcy's equation the velocity of the flowing water is proportional to the hydraulic gradient under which the water is flowing, with the constant of proportionality being the coefficient of permeability. The interesting question arising is whether or not the coefficient of permeability is a scalar quantity (having only a magnitude) or a vector (having both magnitude and direction). It is proved, in the present paper, that the permeability coefficient is neither a scalar nor a vector but a symmetric tensor of second rank. The fact that the permeability tensor is symmetric gives rise to great simplifications and permits a simple graphical construction of the tensor ellipsoid. Having the tensor ellipsoid, the determination of the direction at which the water will flow under a known imposed hydraulic gradient can be found graphically. In case of isotropic soils (the permeability coefficient has the same value along any direction) the ellipsoid reduces to a sphere and the tensor becomes a scalar. In the general case of anisotropic soils the permeability tensor is an entity with nine elements, six of which are independent representing pure extension or contraction along the three principal coordinate axes, thus transforming the permeability sphere into an ellipsoid and vice versa. It should be noted that in anisotropic soils the only directions along which the flow takes place in the direction of the hydraulic gradient are those of the principal axes of the tensor ellipsoid.

Permeability tests were conducted on anisotropic sandstone samples taken at different directions with respect to rectangular coordinates. The permeability coefficient values plotted on a two-dimensional polar coordinate graph paper give rise to an ellipse substantiating therefore the tensor concept of the permeability coefficient. The graphical construction of the tensor ellipse and the use of it in order to obtain the direction of flow by knowing the direction of the hydraulic gradient is also shown.     

Damage Tensor Analysis on Regional Seismic Status

INTRODUCTIONThe introductiontothe damage variable was made originally by Kachanov (1958) and Rabotnov(1963) .As a factor of deterioration in material or rock,damage develops with the changing of loadand environment . The damage variable describes the characteristics of material in the stage frommicrofissure generationto macrofracture formation.Grady and Kipp (1980) loaded oil shale at a constant strainrate andits damage degreeincreasedrapidly. Whenthe load reachedits utmost ,collapse occ…     

岩石裂纹扩展微观机制声发射定量反演

岩石受载内部微裂纹扩展及其震源机制反演有助于认识宏观裂纹扩展过程的非线性断裂力学行为.借助声发射监测手段,本文建立了仅涉及微裂纹张开/闭合和剪切滑移的位移不连续震源模型,通过各位置处传感器耦合质量标定及点源远场P波矩张量反演获得了含预制裂纹砂岩受载过程的震源机制解及时变响应特征,在全局坐标系下分析了微裂纹的三种断裂力学行为.结果表明:在位移不连续模型中,震源矩张量特征值与试样泊松比之间必须满足特定约束条件,该约束条件下的优化问题可采用拉格朗日乘子和Levenberg-Marquardt迭代法求解;受载砂岩裂纹扩展过程中,声发射震源以剪切滑移机制占优,微裂纹空间取向及运动方向与试样宏观主裂纹夹角平均值分别为40.9°和17.7°;对微裂纹体积分解表明岩体微观破裂机制以沿X方向I型张开为主,而沿Y方向的II型断裂滑移方向与试样全局变形方向相一致,由于试样内部晶粒分布非均质性造成了少量沿Z方向的III型平面外微裂纹滑移行为;受载砂岩裂纹扩展过程中微裂纹模式角与震源极性值变化趋势一致、利用震源震级评估的局部应力降值与实验观测结果相吻合,这两者均表明了位移不连续模型在震源机制定量反演中的适用性.     

任意空间取向TI弹性张量解析表述

本文理论给出任意空间取向TI(ATI)四阶弹性张量的解析表述,其以VTI弹性常数及其简单组合为系数,包括各向同性项、TI对称轴方向矢量分量的二次项和四次项,其中TI对称轴方向矢量可以在固定坐标系定义, 也可以相对三维倾斜界面甚至相对波传播方向.相比四阶张量变换法和Bond变换法,ATI弹性张量能简洁而透明地为本构关系和波动方程提供四阶张量的所有元素. ATI弹性张量能为诸多方面的理论研究提供支撑.     

Tangent Vectors of Isochron Rays and Velocity Rays Expressed in Global Cartesian Coordinates

This paper provides Cartesian expressions for tangent vectors of isochron rays and velocity rays previously derived in so-called isochron-orthonormal coordinates. The Cartesian expressions are simpler and easier to implement than the expressions in isochron-orthonormal coordinates. It is shown that the expressions in both coordinate systems are equivalent.     

磁偶极子梯度张量的几何不变量及其应用

磁梯度张量系统姿态的变化将影响梯度场测量和数据解释的精度,使得具有坐标变换不变性特点的张量不变量成为磁梯度张量数据解释的研究热点.本文在对磁偶极子产生的磁梯度张量进行特征值分析的基础上得到了:测量点与磁偶极子位置形成的位置矢量、磁偶极子磁矩矢量与绝对值最小的特征值对应的特征向量垂直;位置矢量和磁矩矢量与最大及最小特征值对应的特征向量共面,且两矢量间的夹角可由磁梯度张量矩阵的特征值表示.最后,将本文所得磁偶极子梯度张量的几何不变量用于磁性目标的跟踪中,取得了较好的实时跟踪效果.