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1.
In this paper, a new formula for evaluating the truncation coefficientQ n is derived from recurrence relations of Legendre polynomials. The present formula has been conveniently processed by an electronic computer, providing the value ofQ n up to a degreen=49 which are exactly equal to those of Paul (1973).  相似文献   

2.
A recursive method is derived for computing the Molodenskii truncation error coefficients at altitude for the altitude-generalized Stokes integral. Furthermore, the Cook truncation error coefficients at altitude corresponding to the generalized Vening-Meinesz integral are derived in terms of the Molodenskii coefficients. Also, the gravity disturbance truncation error coefficients at altitude are related to the Molodenskii coefficients. By combining these results, it is shown how the truncation error for the complete gravity disturbance vector at altitude may be computed recursively.  相似文献   

3.
美国海洋卫星测高仪的出现,使应用Hotine积分确定海洋大地水准面成为现实。本文通过对Hotine积分及垂线偏差的计算公式进行改进,较好地改善了求和项的收敛性,减小了截断误差影响,并提出了利用Hotine函数和重力异常确定海洋大地水准面的方法。 实际计算表明:海洋重力大地水准面的精度在1米以内;卫星测高大地水准面间存在0.5米系统差;它和海底地形有一定的相关性,能较好地反映出海底地形的宏观特性。  相似文献   

4.
Summary The system of normal equations for the adjustment of a free network is a singular one. Therefore, a number of coordinates has to be fixed according to the matrix. The mean square errors and the error ellipses of such an adjustment are dependent on this choice. This paper gives a simple, direct method for the adjustment of free networks, where no coordinates need to be fixed. This is done by minimizing not only the sum of the squares of the weighted errorsV T PV=minimun but also the Euclidean norm of the vectorX and of the covariance matrixQ X T X=minimum trace (Q)=minimum This last condition is crucial for geodetic problems of this type.  相似文献   

5.
Recently much work has been done concerning the behavior of the truncation errors of the integral formulas of Stokes and Vening Meinesz. In our paper we examine the theoretical foundations of truncation error behavior.  相似文献   

6.
Based on sine functions for fitting horizontal velocity errors in an inertial survey system and a step or slope function approximation of the vertical deflection between stops, a positioning accuracy of 10 m (CEP) can be achieved with a car-borne system – a modified -21 aircraft navigation system, with a 10 min travel period between Zero Velocity Update Times (ZUPTs) and a total 1.5 hour survey time. A rough estimation of the vertical deflection can also be expected in real time out of the approach. Also a forward-backward-forward run is presented which demonstrates determination of the gravity anomaly. Received 10 March 1995; Accepted 8 July 1996  相似文献   

7.
基于$\frac{{{{\bar{P}}}_{nm}}\left( \cos \theta \right)}{\sin \theta }\left( m>0 \right)$的非奇异递推公式,给出了基于球坐标的引力矢量和垂线偏差非奇异计算公式;针对极点λ可任意取值引起的地方指北坐标系方向的不确定性问题,证明了引力矢量在转换到地心直角坐标系后不随λ的变化而变化,即与λ的取值无关。最终的数值计算结果表明,直角坐标系下的非奇异计算公式与本文提出的球坐标下的非奇异计算公式所得计算结果绝对值差异小于10-16m/s2,证明了该非奇异公式的正确性。最后总结了所有引力位球函数一阶导、二阶导非奇异性计算的一般思路。  相似文献   

8.
A recurrence relation is presented for the smoothing function, βn, which is used in geodesy to relate spherical harmonics to their mean values over circular areas (caps). The proposed formula does not require the computation of the Legendre's polynomials. Moreover, it is numerically more stable than the formulas ofPellinen (1969) andMeissl (1971).  相似文献   

9.
ABSTRACT

Digitizing the land surface temperature (Ts) and surface soil moisture (mv) is essential for developing the intelligent Digital Earth. Here, we developed a two parameter physical-based passive microwave remote sensing model for jointly retrieving Ts and mv using the dual-polarized Tb of Aqua satellite advanced microwave scanning radiometer (AMSR-E) C-band (6.9 GHz) based on the simplified radiative transfer equation. Validation using in situ Ts and mv in southern China showed the average root mean square errors (RMSE) of Ts and mv retrievals reach 2.42 K (R2 = 0.61, n = 351) and 0.025 g cm?3 (R2 = 0.68, n = 663), respectively. The results were also validated using global in situ Ts (n = 2362) and mv (n = 1657) of International Soil Moisture Network. The corresponding RMSE are 3.44 k (R2 = 0.86) and 0.039 g cm?3 (R2 = 0.83), respectively. The monthly variations of model-derived Ts and mv are highly consistent with those of the Moderate Resolution Imaging Spectroradiometer Ts (R2 = 0.57; RMSE = 2.91 k) and ECV_SM mv (R2 = 0.51; RMSE = 0.045 g cm?3), respectively. Overall, this paper indicates an effective way to jointly modeling Ts and mv using passive microwave remote sensing.  相似文献   

10.
The formulas for the determination of the coefficients of the spherical harmonic expansion of the disturbing potential of the earth are defined for data given on a sphere. In order to determine the spherical harmonic coefficients, the gravity anomalies have to be analytically downward continued from the earth's surface to a sphere—at least to the ellipsoid. The goal of this paper is to continue the gravity anomalies from the earth's surface downward to the ellipsoid using recent elevation models. The basic method for the downward continuation is the gradient solution (theg 1 term). The terrain correction has also been computed because of the role it can play as a correction term when calculating harmonic coefficients from surface gravity data. Theg 1 term and the terrain correction were expanded into the spherical harmonics up to180 th order. The corrections (theg 1 term and the terrain correction) have the order of about 2% of theRMS value of degree variance of the disturbing potential per degree. The influences of theg 1 term and the terrain correction on the geoid take the order of 1 meter (RMS value of corrections of the geoid undulation) and on the deflections of the vertical is of the order 0.1″ (RMS value of correction of the deflections of the vertical).  相似文献   

11.
Construction of anisotropic covariance functions using Riesz-representers   总被引：1，自引：1，他引：0  
A reproducing-kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R 0, having a reproducing kernel K 0(P,Q) is considered (P, Q, and later P n being points in the set of harmonicity). The degree variances of this kernel will be denoted σ0 n . The set of Riesz representers associated with the evaluation functionals (or gravity functionals) related to distinct points P n ,n = 1,…,N, on a two-dimensional surface surrounding the bounding sphere, will be linearly independent. These functions are used to define a new N-dimensional RKHS with kernel (a n >0)
If the points all are located on a concentric sphere with radius R 1>R 0, and form an ε-net covering the sphere, and a n are suitable area elements (depending on N), then this kernel will converge towards an isotropic kernel with degree variances
Consequently, if K N (P,Q) is required to represent an isotropic covariance function of the Earth's gravity potential, COV(P,Q), σ0 n can be selected so that σ n becomes equal to the empirical degree variances. If the points are chosen at varying radial distances R n >R 0, then an anisotropic kernel, or equivalent covariance function representation, can be constructed. If the points are located in a bounded region, the kernel may be used to modify the original kernel
Values of anisotropic covariance functions constructed based on these ideas are calculated, and some initial ideas are presented on how to select the points P n . Received: 24 September 1998 / Accepted: 10 March 1999  相似文献   

12.
精密三角高程测量严密计算的理论研究与初步实验   总被引：10，自引：0，他引：10  
姜晨光 《四川测绘》1996,19(3):125-128
本文根据水准面、似大地水准面、参考椭球面、高程异常、垂线偏差与正常高之间的关系，推导出了新的普通三角高程测量及光电测距三角高程测量的精密计算公式，并对新公式进行了初步的实验验证。  相似文献   

13.
The investigations refer to the compartment method by using mean terrestrial free air anomalies only. Three main error influences of remote areas (distance from the fixed point >9°) on height anomalies and deflections of the vertical are being regarded:
  1. The prediction errors of mean terrestrial free air anomalies have the greatest influence and amount to about ±0″.2 in each component for deflections of the vertical and to ±3 m for height anomalies;
  2. The error of the compartment method, which originates from converting the integral formulas of Stokes and Vening-Meinesz into summation formulas, can be neglected if the anomalies for points and gravity profiles are compiled to 5°×5° mean values.
  3. The influences of the mean gravimetric correction terms of Arnold—estimated for important mountains of the Earth by means of an approximate formula—on height anomalies may amount to 1–2 m and on deflections of the vertical to 0″0.5–0″.1, and, therefore, they have to be taken into account for exact calculations.
The computations of errors are carried out using a global covariance function of point free air anomalies.  相似文献   

14.
The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y−E Y = (X−E X ) · Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, E Y and E X . Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem by using the nonlinear Euler–Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative algorithm, modify it, and compare it with a new ‘closed form’ solution that is based on the singular-value decomposition. For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing the columns in the coefficient matrix are investigated. This case study illuminates the issue of “symmetry” in the treatment of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335–342). The differences between the standard least-squares and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion matrix of the estimated parameters.  相似文献   

15.
Traditionally, the evaluation of geoidal height by Stokes formula and the vertical deflection by Vening-Meinesz one, and the estimation of the influence of neglecting the distant zone on computing the geoidal height and the vertical deflection were done by taking the inner zone as a spherical cap. It is not very convenient from the point of view of modern numerical methods such as fast Fourier and Hartley transforms where the inner zone is not a spherical cap, but a spherical trapezoid. So, we generalized the known formulas for evaluating the geoidal height and the vertical deflection for an integration area of arbitrary shape. The corresponding formulas for computing the effects of neglecting the distant zone have been derived. Some issues on computation techniques have been investigated. As an example, the case where the inner zone is modeled as a spherical trapezoid was given special attention, and practical computations were performed.  相似文献   

16.
The calculation of topographic (and iso- static) reductions is one of the most time-consuming operations in gravity field modelling. For this calculation, the topographic surface of the Earth is often divided with respect to geographical or map-grid lines, and the topographic heights are averaged over the respective grid elements. The bodies bounded by surfaces of constant (ellipsoidal) heights and geographical grid lines are denoted as tesseroids. Usually these ellipsoidal (or spherical) tesseroids are replaced by “equivalent” vertical rectangular prisms of the same mass. This approximation is motivated by the fact that the volume integrals for the calculation of the potential and its derivatives can be exactly solved for rectangular prisms, but not for the tesseroids. In this paper, an approximate solution of the spherical tesseroid integrals is provided based on series expansions including third-order terms. By choosing the geometrical centre of the tesseroid as the Taylor expansion point, the number of non-vanishing series terms can be greatly reduced. The zero-order term is equivalent to the point-mass formula. Test computations show the high numerical efficiency of the tesseroid method versus the prism approach, both regarding computation time and accuracy. Since the approximation errors due to the truncation of the Taylor series decrease very quickly with increasing distance of the tesseroid from the computation point, only the elements in the direct vicinity of the computation point have to be separately evaluated, e.g. by the prism formulas. The results are also compared with the point-mass formula. Further potential refinements of the tesseroid approach, such as considering ellipsoidal tesseroids, are indicated.  相似文献   

17.
重力异常和垂线偏差是测高卫星非常重要的产品。二者的精度指标对于未来的测高卫星方案设计至关重要。本文利用球谐函数来对重力异常和垂线偏差的精度指标进行讨论,首先从理论上推导了重力异常和垂线偏差误差的近似匹配关系,然后通过6个超高阶重力场模型验证了有关结论的正确性。数值试验表明:垂线偏差误差和重力异常误差满足近似的比例关系,即若垂线偏差各方位向等精度测量,且假定精度均为1μrad,则所对应的重力异常精度约为1.4mGal;反之,若重力异常的精度为1mGal,则所对应的垂线偏差的精度约为0.7μrad。  相似文献   

18.
The use of the fast Fourier transform algorithm in the evaluation of the Molodensky series terms is demonstrated in this paper. The solution by analytical continuation to point level has been reformulated to obtain convolution integrals in planar approximation which can be efficiently evaluated in the frequency domain. Preliminary results show that the solution by Faye anomalies is not sufficient for highly accurate deflections of the vertical and height anomalies. The Molodensky solution up to at least the second-order term must be carried out. Part of the unrecovered deflection and height anomaly signal appears to be due to density variations, verifying the essential role of density modelling. A remove-restore technique for the terrain effects can improve the convergence of the series and minimize the interpolation errors. Paper presented at theI Hotine-Marussi Symposium on Mathematical Geodesy, Rome, June 3–6, 1985.  相似文献   

19.
Comparisons of gravimetric and astrogeodetic deflections of the vertical in the Australian region indicate that the former are affected by position dependent systematic errors, even after orientation onto the Australian Geodetic Datum. These are probably due to errors in the predicted mean anomalies for gravimetrically unsurveyed oceanic regions to the east, south and west of the continent. Deflection component residuals (astrogeodetic minus oriented gravimetric) at 83 control stations are made the observables in a set of observation equations, based on the Vening Meinesz equations, from which pseudocorrections to the mean anomalies for a set of arbitrarily selected surface elements are computed. These pseudocorrections compensate for prediction errors in much larger unsurveyed regions. Their effects on individual deflection components are calculated using the Vening Meinesz equations. Statistical tests indicate that pseudocorrections computed for four large offshore elements and six smaller elements in unsurveyed areas produce corrections to the gravimetric deflections which make the ξ and η components in seconds of arc consistent with normally distributed populations N (0.00, 0.702).  相似文献   

20.
《测量评论》2013,45(94):349-361
Abstract

A recent investigation into the flatness of Multiplex diapositive slides has shown that flatness errors occur ranging from 0 to 0·03–0·04 mm. referred to the flat projector stage. In a first attempt to ascertain the effects of errors of this kind on Multiplex bridges, the flatness values of two sets of nine diapositives each were measured using a simple interferometric method and the vertical (wants of correspondence) and horizontal parallaxes introduced by these flatness errors were subjected to computational bridging. The resulting height errors at the end of the two strips proved to be of noticeable size, as large as +0·6 and ?0·9 mm. respectively. Indicative as these figures may already be, it seems useful to abstract the investigation from the vagaries of the individual case and to put it on a more general footing.  相似文献   

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