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1.
Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances 总被引：1，自引：1，他引：0
Quantities like tropospheric zenith delays or station coordinates are repeatedly measured at permanent VLBI or GPS stations
so that time series for the quantities at each station are obtained. The covariances of these quantities can be estimated
in a multivariate linear model. The covariances are needed for computing uncertainties of results derived from these quantities.
The covariance matrix for many permanent stations becomes large, the need for simplifying it may therefore arise under the
condition that the uncertainties of derived results still agree. This is accomplished by assuming that the different time
series of a quantity like the station height for each permanent station can be combined to obtain one time series. The covariance
matrix then follows from the estimates of the auto- and cross-covariance functions of the combined time series. A further
approximation is found, if compactly supported covariance functions are fitted to an estimated autocovariance function in
order to obtain a covariance matrix which is representative of different kinds of measurements. The simplification of a covariance
matrix estimated in a multivariate model is investigated here for the coordinates of points of a grid measured repeatedly
by a laserscanner. The approximations are checked by determining the uncertainty of the sum of distances to the points of
the grid. To obtain a realistic value for this uncertainty, the covariances of the measured coordinates have to be considered.
Three different setups of measurements are analyzed and a covariance matrix is found which is representative for all three
setups. Covariance matrices for the measurements of laserscanners can therefore be determined in advance without estimating
them for each application. 相似文献
2.
卫星重力径向梯度数据的最小二乘配置调和分析 总被引：3，自引：2，他引：1
本文深入研究了利用卫星重力梯度径向分量确定地球引力场位系数的最小二乘配置(LSC)调和分析方法。首先论述了最小二乘配置法的原理,推导了扰动引力梯度观测量与球谐系数之间的协方差和自协方差矩阵,在扰动引力梯度观测数据为等经差规则网格数据的情况下,引力位与扰动引力梯度之间的协方差矩阵具有分块Toeplitz循环阵的结构,有效的利用FFT变换技术将其降阶;研究利用截断奇异值分解法（TSVD）解决协方差阵的病态性问题;最后得到了引力梯度径向分量的最小二乘配置调和分析的完整计算公式。模拟试算结果表明,基于TSVD的最小二乘配置调和分析方法,能够以较高的精度还原全球重力场,验证了本文算法的有效性和实用性。 相似文献
3.
六边形全球离散格网系统是支持多源地球空间信息融合处理的优选解决方案,相关研究已引起学术界广泛关注。相较于完整的全球格网,局部不规则区域格网的应用需求更加广泛,相关生成算法是当前六边形全球离散格网系统研究的重要问题之一。将二十面体相邻三角面组合为菱形逻辑结构,在此基础上,提出一种局部区域多孔径六边形格网系统生成算法。首先,分析格网剖分类型,建立离散整数坐标系,描述多孔径六边形格网单元空间位置;其次,将局部区域分解到球面二十面体的菱形逻辑结构上创建子区域;然后,根据子区域边界设计外接最小菱形遍历算法,剔除与目标区域无关单元;最后,遍历最小菱形,生成多孔径局部格网。对比实验结果表明,提出的算法具有灵活性好、效率高等优点,生成的多孔径六边形格网用于栅格数据组织,可显著减少数据量,具有较好的应用潜力。 相似文献
4.
5.
Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis 总被引：5，自引：3，他引：2
Collocation is widely used in physical geodesy. Its application requires to solve systems with a dimension equal to the number
of observations, causing numerical problems when many observations are available. To overcome this drawback, tailored step-wise
techniques are usually applied. An example of these step-wise techniques is the space-wise approach to the GOCE mission data
processing. The original idea of this approach was to implement a two-step procedure, which consists of first predicting gridded
values at satellite altitude by collocation and then deriving the geo-potential spherical harmonic coefficients by numerical
integration. The idea was generalized to a multi-step iterative procedure by introducing a time-wise Wiener filter to reduce
the highly correlated observation noise. Recent studies have shown how to optimize the original two-step procedure, while
the theoretical optimization of the full multi-step procedure is investigated in this work. An iterative operator is derived
so that the final estimated spherical harmonic coefficients are optimal with respect to the Wiener–Kolmogorov principle, as
if they were estimated by a direct collocation. The logical scheme used to derive this optimal operator can be applied not
only in the case of the space-wise approach but, in general, for any case of step-wise collocation. Several numerical tests
based on simulated realistic GOCE data are performed. The results show that adding a pre-processing time-wise filter to the
two-step procedure of data gridding and spherical harmonic analysis is useful, in the sense that the accuracy of the estimated
geo-potential coefficients is improved. This happens because, in its practical implementation, the gridding is made by collocation
over local patches of data, while the observation noise has a time-correlation so long that it cannot be treated inside the
patch size. Therefore, the multi-step operator, which is in theory equivalent to the two-step operator and to the direct collocation,
is in practice superior thanks to the time-wise filter that reduces the noise correlation before the gridding. The criteria
for the choice of this filter are investigated numerically. 相似文献
6.
The differencing technique is useful in global positioning system (GPS) positioning when two or more GPS receivers collect simultaneous observables from common satellites at each epoch, and all carrier-phase observables have the same normal distribution. An analytical probability distribution of the single-, double-, triple- and multi-difference GPS observables is obtained. This analytical model, called ISO2002, has a good matrix structure, in which I indicates the number of receivers, S indicates the number of observed satellites, and O indicates the number of epochs. The variance–covariance matrix can be expressed as the Kronecker product of several small matrices, so its inverse is equal to the Kronecker product of the inverses of these sub-matrices. Moreover, these small matrices are circulant or symmetric diagonal Toeplitz matrices, so their inverses have analytical solutions. The analytical model ISO2002 proposed to compute the inverse variance–covariance matrix is shown to be very effective. 相似文献
7.
采用最小二乘方法解算超高阶重力场模型,不可避免会遇到大型矩阵的计算,直接求解是难以实现的。本文从重力场模型的基本观测方程出发,利用正余弦函数和面球谐因子的正交性,分析系数矩阵及法矩阵的特点,在法矩阵块对角化的基础上,利用系数矩阵求解法矩阵时“次m”递增的特点,对法矩阵求解方程进行约化、对Legendre函数的计算和存储方式进行了设计,结合缔合Legendre函数关于赤道的对称性,解决了大型矩阵存储及计算效率低下的难题,实现了超高阶重力场模型最小二乘方法的小存储、高效率的解算。通过试验模拟,改进后的方法相比传统块对角方法效率提高300倍,利用此方法可以在普通PC机上快速、高精度地解算2160阶次超高阶重力场模型,算法精度相比数值积分方法至少提高了5个数量级,并且在一定程度上可以评估原始观测数据的精度。 相似文献
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9.
给出了全球重力场数据插值处理中常用的三对角、五对角对称Toeplitz矩阵逆元素的解析计算表达式。它避免了逆矩阵计算中需要调用三角函数的缺陷，只需要进行简单的幂次运算，从而大大提高了计算速度，为等距B-样条插值等应用领域拓展了算法，具有潜在的实用意义。 相似文献
10.
The M-estimator for the 3D symmetric Helmert coordinate transformation problem is developed. Small-angle rotation assumption is abandoned. The direction cosine matrix or the quaternion is used to represent the rotation. The \(3 \times 1\) multiplicative error vector is defined to represent the rotation estimation error. An analytical solution can be employed to provide the initial approximate for iteration, if the outliers are not large. The iteration is carried out using the iterative reweighted least-squares scheme. In each iteration after the first one, the measurement equation is linearized using the available parameter estimates, the reweighting matrix is constructed using the residuals obtained in the previous iteration, and then the parameter estimates with their variance-covariance matrix are calculated. The influence functions of a single pseudo-measurement on the least-squares estimator and on the M-estimator are derived to theoretically show the robustness. In the solution process, the parameter is rescaled in order to improve the numerical stability. Monte Carlo experiments are conducted to check the developed method. Different cases to investigate whether the assumed stochastic model is correct are considered. The results with the simulated data slightly deviating from the true model are used to show the developed method’s statistical efficacy at the assumed stochastic model, its robustness against the deviations from the assumed stochastic model, and the validity of the estimated variance-covariance matrix no matter whether the assumed stochastic model is correct or not. 相似文献
11.
Erik de Min 《Journal of Geodesy》1995,69(4):223-232
Summary Basically two different evaluation methods are available to compute geoid heights from residual gravity anomalies in the inner zone: numerical integration and least squares collocation.If collocation is not applied to a global gravity data set, as is usually the case in practice, its result will not be equal to the numerical integration result. However, the cross covariance function between geoid heights and gravity anomalies can be adapted such that the geoid contribution is computed only from a small gravity area up to a certain distance
o from the computation point. Using this modification, identical results are obtained as from numerical integration.Applying this modification makes the results less dependent on the covariance function used. The difference between numerical integration and collocation is mainly caused by the implicitly extrapolated residual gravity anomaly values, outside the original data area. This extrapolated signal depends very much on the covariance function used, while the interpolated values within the original data area depend much less on it.As a sort of by-product, this modified collocation formula also leads to a new combination technique of numerical integration and collocation, in which the optimizing practical properties of both methods are fully exploited.Numerical examples are added as illustration. 相似文献
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13.
The least squares collocation algorithm for estimating gravity anomalies from geodetic data is shown to be an application
of the well known regression equations which provide the mean and covariance of a random vector (gravity anomalies) given
a realization of a correlated random vector (geodetic data). It is also shown that the collocation solution for gravity anomalies
is equivalent to the conventional least-squares-Stokes' function solution when the conventional solution utilizes properly
weighted zero a priori estimates. The mathematical and physical assumptions underlying the least squares collocation estimator
are described. 相似文献
14.
The fixed gravimetric boundary-value problem (FGBVP) represents an exterior oblique derivative problem for the Laplace equation. Terrestrial gravimetric measurements located by precise satellite positioning yield oblique derivative boundary conditions in the form of surface gravity disturbances. In this paper, we discuss the boundary element method (BEM) applied to the linearized FGBVP. In spite of previous BEM approaches in geodesy, we use the so-called direct BEM formulation, where a weak formulation is derived through the method of weighted residuals. The collocation technique with linear basis functions is applied for deriving the linear system of equations from the arising boundary integral equations. The nonstationary iterative biconjugate gradient stabilized method is used to solve the large-scale linear system of equations. The standard MPI (message passing interface) subroutines are implemented in order to perform parallel computations. The proposed approach gives a numerical solution at collocation points directly on the Earth’s surface (on a fixed boundary). Numerical experiments deal with (i) global gravity field modelling using synthetic data (surface gravity disturbances generated from a global geopotential model (GGM)) (ii) local gravity field modelling in Slovakia using observed gravity data. In order to extend computations, the memory requirements are reduced using elimination of the far-zone effects by incorporating GGM or a coarse global numerical solution obtained by BEM. Statistical characteristics of residuals between numerical solutions and GGM confirm the reliability of the approach and indicate accuracy of numerical solutions for the global models. A local refinement in Slovakia results in a local (national) quasigeoid model, which when compared with GPS-levelling data, does not make a large improvement on existing remove-restore-based models. 相似文献
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16.
地形改正等地球重力场元量的FFT算法(平面二维、球面近似二维、球面单带)研究已基本成熟,并广泛应用于科研与工程实践。但是FFT算法也有其固有的缺陷,即存在着频谱混迭、泄漏等现象。本文利用离散卷积分具有Toeplitz矩阵的性质,将其拓展为循环Toeplitz与拓展观测数据矢量的乘积,利用循环Toeplitz的Fourier表示,实现离散卷积分的快速算法。该算法与离散求和等价,当数据量较大时,可明显地提高计算速度。 相似文献
17.
This letter proposes a new model for the second-order statistics of spatial texture in synthetic aperture radar images. The autocovariance function is locally approximated by a two-dimensional anisotropic Gaussian kernel (AGK) to characterize texture by its local orientation and anisotropy. The estimation of texture parameters at a given scale is based on the gradient structure tensor operator and does not require the explicit computation of the autocovariance. Finally, a new filter called AGK minimum mean square error (MMSE) that takes into account this spatial information is introduced and compared with the refined MMSE filter. The proposed filter has better performance in terms of texture preservation and structure enhancement 相似文献
18.
误差向量的方差-协方差阵是一般对称正定矩阵下的附不等式约束加权整体最小二乘平差模型,研究了其参数估计和精度评定问题。首先,将残差平方和极小化函数在整体最小二乘准则下转化为只包含模型参数的目标函数,同时将所有的不等式约束表示成一个等价的凝聚约束函数,并运用乘子罚函数策略将不等式约束加权整体最小二乘平差问题转化为相应的无约束最优化问题,并用BFGS方法求解。然后,将误差方程和约束函数线性展开,推导了最优解和观测量间的近似线性函数关系,运用方差-协方差传播律得到了最优解的近似方差。最后,用数值实例验证了方法的有效性和可行性。 相似文献
19.
格网系统是一种多分辨率栅格数据结构，在多尺度地理空间数据组织、处理和分析中应用广泛。六边形格网系统具有重要几何特性，相关研究已引起学术界广泛关注，格网层次关系描述及计算是研究难点之一。本文引入复进制数理论，通过间隔层次格网单元隶属关系，建立平面四孔六边形格网系统数学模型，据此提出等效编码方案，定义编码运算并归纳运算规则，设计编码索引、编码与笛卡儿坐标互换算法。对比试验结果表明，本文提出的编码方案与同类成果相比具有结构对称性，可显著提高编码操作效率，具有实际应用潜力。 相似文献
20.
Arne Bjerhammar 《Journal of Geodesy》1982,56(4):312-328
The boundary value problem in physical geodesy is nowadays mostly presented with the use of an advanced stochastic model by
Krarup-Moritz. This model includes a primary Gauss-Markov model and an adjoining Wiener-Hopf model. Degenerations of the Wiener-Hopf
section are found in thesingular auto-covariance matrix of the residuals. The non-singular inverse of the auto-covariance matrix of the signal is proved to be a generalized inverse of the singular
auto-covariance matrix of the residuals. The joint model is given a non-stochastic evaluation for a case with spherical external
surface (using a non-singular inverse). These findings will not prevent a successful application of the model, which has important
merits, specially when using suitablea priori values for the stochastic parameters in the covariance functions.
A method for quadratic unbiased estimation ofa priori variances is presented in an introductory section. It is meant to be of value when using a solution of the boundary value problem with
the collocation technique based on the classical Gauss-Markov solution. (Bjerhammar (1963).) 相似文献