首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献，搜索用时 15 毫秒
1.
 When standard boundary element methods (BEM) are used in order to solve the linearized vector Molodensky problem we are confronted with two problems: (1) the absence of O(|x|−2) terms in the decay condition is not taken into account, since the single-layer ansatz, which is commonly used as representation of the disturbing potential, is of the order O(|x|−1) as x→∞. This implies that the standard theory of Galerkin BEM is not applicable since the injectivity of the integral operator fails; (2) the N×N stiffness matrix is dense, with N typically of the order 105. Without fast algorithms, which provide suitable approximations to the stiffness matrix by a sparse one with O(N(logN) s ), s≥0, non-zero elements, high-resolution global gravity field recovery is not feasible. Solutions to both problems are proposed. (1) A proper variational formulation taking the decay condition into account is based on some closed subspace of co-dimension 3 of the space of square integrable functions on the boundary surface. Instead of imposing the constraints directly on the boundary element trial space, they are incorporated into a variational formulation by penalization with a Lagrange multiplier. The conforming discretization yields an augmented linear system of equations of dimension N+3×N+3. The penalty term guarantees the well-posedness of the problem, and gives precise information about the incompatibility of the data. (2) Since the upper left submatrix of dimension N×N of the augmented system is the stiffness matrix of the standard BEM, the approach allows all techniques to be used to generate sparse approximations to the stiffness matrix, such as wavelets, fast multipole methods, panel clustering etc., without any modification. A combination of panel clustering and fast multipole method is used in order to solve the augmented linear system of equations in O(N) operations. The method is based on an approximation of the kernel function of the integral operator by a degenerate kernel in the far field, which is provided by a multipole expansion of the kernel function. Numerical experiments show that the fast algorithm is superior to the standard BEM algorithm in terms of CPU time by about three orders of magnitude for N=65 538 unknowns. Similar holds for the storage requirements. About 30 iterations are necessary in order to solve the linear system of equations using the generalized minimum residual method (GMRES). The number of iterations is almost independent of the number of unknowns, which indicates good conditioning of the system matrix. Received: 16 October 1999 / Accepted: 28 February 2001  相似文献   

2.
 Simplified techniques for high-degree spherical harmonic synthesis are extended to include gravitational potential second derivatives with respect to latitude. Received: 23 July 2001 / Accepted: 12 April 2002 Acknowledgement. The authors would like to thank Christian Tscherning for recommending Laplace's equation as an accuracy test. Our use of Legendre's differential equation, as the most direct means for extending our simplified synthesis methods to second-order derivatives, was a direct result of this suggestion. Correspondence to: S. A. Holmes  相似文献   

3.
 This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N 0 evaluated from Stokes's formula and the ellipsoidal correction N 1, makes the relative geoidal height error decrease from O(e 2) to O(e 4), which can be neglected for most practical purposes. The ellipsoidal correction N 1 is expressed as a sum of an integral about the spherical geoidal height N 0 and a simple analytical function of N 0 and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N 1 is done in an area where the spherical geoidal height N 0 has already been evaluated, it is also more effective than the solution of Martinec and Grafarend. Received: 27 January 1999 / Accepted: 4 October 1999  相似文献   

4.
 The structure of normal matrices occurring in the problem of weighted least-squares spherical harmonic analysis of measurements scattered on a sphere with random noises is investigated. Efficient algorithms for the formation of the normal matrices are derived using fundamental relations inherent to the products of two surface spherical harmonic functions. The whole elements of a normal matrix complete to spherical harmonic degree L are recursively obtained from its first row or first column extended to degree 2L with only O(L 4) computational operations. Applications of the algorithms to the formation of surface normal matrices from geoid undulations and surface gravity anomalies are discussed in connection with the high-degree geopotential modeling. Received: 22 March 1999 / Accepted: 23 December 1999  相似文献   

5.
 A fast spherical harmonic approach enables the computation of gravitational or magnetic potential created by a non-uniform shell of material bounded by uneven topographies. The resulting field can be evaluated outside or inside the sphere, assuming that density of the shell varies with latitude, longitude, and radial distance. To simplify, the density (or magnetization) source inside the sphere is assumed to be the product of a surface function and a power series expansion of the radial distance. This formalism is applied to compute the gravity signal of a steady, dry atmosphere. It provides geoid/gravity maps at sea level as well as satellite altitude. Results of this application agree closely with those of earlier studies, where the atmosphere contribution to the Earth's gravity field was determined using more time-consuming methods. Received: 14 August 2000 / Accepted: 19 March 2001  相似文献   

6.
 Equations expressing the covariances between spherical harmonic coefficients and linear functionals applied on the anomalous gravity potential, T, are derived. The functionals are the evaluation functionals, and those associated with first- and second-order derivatives of T. These equations form the basis for the prediction of spherical harmonic coefficients using least-squares collocation (LSC). The equations were implemented in the GRAVSOFT program GEOCOL. Initially, tests using EGM96 were performed using global and regional sets of geoid heights, gravity anomalies and second-order vertical gravity gradients at ground level and at altitude. The global tests confirm that coefficients may be estimated consistently using LSC while the error estimates are much too large for the lower-order coefficients. The validity of an error estimate calculated using LSC with an isotropic covariance function is based on a hypothesis that the coefficients of a specific degree all belong to the same normal distribution. However, the coefficients of lower degree do not fulfil this, and this seems to be the reason for the too-pessimistic error estimates. In order to test this the coefficients of EGM96 were perturbed, so that the pertubations for a specific degree all belonged to a normal distribution with the variance equal to the mean error variance of the coefficients. The pertubations were used to generate residual geoid heights, gravity anomalies and second-order vertical gravity gradients. These data were then used to calculate estimates of the perturbed coefficients as well as error estimates of the quantities, which now have a very good agreement with the errors computed from the simulated observed minus calculated coefficients. Tests with regionally distributed data showed that long-wavelength information is lost, but also that it seems to be recovered for specific coefficients depending on where the data are located. Received: 3 February 2000 / Accepted: 23 October 2000  相似文献   

7.
 A potential-type Molodensky telluroid based upon a minimum-distance mapping is derived. With respect to a reference potential of Somigliana–Pizzetti type which relates to the World Geodetic Datum 2000, it is shown that a point-wise minimum-distance mapping of the topographical surface of the Earth onto the telluroid surface, constrained to the gauge W(P)=u(p), leads to a system of four nonlinear normal equations. These normal equations are solved by a fast Newton–Raphson iteration. Received: 7 February 2000 / Accepted: 23 October 2001  相似文献   

8.
Green's function solution to spherical gradiometric boundary-value problems   总被引：1，自引：1，他引：1  
 Three independent gradiometric boundary-value problems (BVPs) with three types of gradiometric data, {Γ rr }, {Γ r θ,Γ r λ} and {Γθθ−Γλλ,Γθλ}, prescribed on a sphere are solved to determine the gravitational potential on and outside the sphere. The existence and uniqueness conditions on the solutions are formulated showing that the zero- and the first-degree spherical harmonics are to be removed from {Γ r θ,Γ r λ} and {Γθθ−Γλλ,Γθλ}, respectively. The solutions to the gradiometric BVPs are presented in terms of Green's functions, which are expressed in both spectral and closed spatial forms. The logarithmic singularity of the Green's function at the point ψ=0 is investigated for the component Γ rr . The other two Green's functions are finite at this point. Comparisons to the paper by van Gelderen and Rummel [Journal of Geodesy (2001) 75: 1–11] show that the presented solution refines the former solution. Received: 3 October 2001 / Accepted: 4 October 2002  相似文献   

9.
 The recovery of a full set of gravity field parameters from satellite gravity gradiometry (SGG) is a huge numerical and computational task. In practice, parallel computing has to be applied to estimate the more than 90 000 harmonic coefficients parameterizing the Earth's gravity field up to a maximum spherical harmonic degree of 300. Three independent solution strategies (preconditioned conjugate gradient method, semi-analytic approach, and distributed non-approximative adjustment), which are based on different concepts, are assessed and compared both theoretically and on the basis of a realistic-as-possible numerical simulation regarding the accuracy of the results, as well as the computational effort. Special concern is given to the correct treatment of the coloured noise characteristics of the gradiometer. The numerical simulations show that the three methods deliver nearly identical results—even in the case of large data gaps in the observation time series. The newly proposed distributed non-approximative adjustment approach, which is the only one of the three methods that solves the inverse problem in a strict sense, also turns out to be a feasible method for practical applications. Received: 17 December 2001 / Accepted: 17 July 2002 Acknowledgments. We would like to thank Prof. W.-D. Schuh, Institute of Theoretical Geodesy, University of Bonn, for providing us with the serial version of the PCGMA algorithm, which forms the basis for the parallel PCGMA package developed at our institute. This study was partially performed in the course of the GOCE project `From E?tv?s to mGal+', funded by the European Space Agency (ESA) under contract No. 14287/00/NL/DC. Correspondence to: R. Pail  相似文献   

10.
A synthetic Earth for use in geodesy   总被引：1，自引：0，他引：1  
 A synthetic Earth and its gravity field that can be represented at different resolutions for testing and comparing existing and new methods used for global gravity-field determination are created. Both the boundary and boundary values of the gravity potential can be generated. The approach chosen also allows observables to be generated at aircraft flight height or at satellite altitude. The generation of the synthetic Earth shape (SES) and gravity-field quantities is based upon spherical harmonic expansions of the isostatically compensated equivalent rock topography and the EGM96 global geopotential model. Spherical harmonic models are developed for both the synthetic Earth topography (SET) and the synthetic Earth potential (SEP) up to degree and order 2160 corresponding to a 5′×5′ resolution. Various sets of SET, SES and SEP with boundary geometry and boundary values at different resolutions can be generated using low-pass filters applied to the expansions. The representation is achieved in point sets based upon refined triangulation of a octahedral geometry projected onto the chosen reference ellipsoid. The filter cut-offs relate to the sampling pattern in order to avoid aliasing effects. Examples of the SET and its gravity field are shown for a resolution with a Nyquist sampling rate of 8.27 degrees. Received: 6 August 1999 / Accepted: 26 April 2000  相似文献   

11.
 Two iterative vector methods for computing geodetic coordinates (φ, h) from rectangular coordinates (x, y, z) are presented. The methods are conceptually simple, work without modification at any latitude and are easy to program. Geodetic latitude and height can be calculated to acceptable precision in one iteration over the height range from −106 to +109 m. Received: 13 December 2000 / Accepted: 13 July 2001  相似文献   

12.
New solutions for the geodetic coordinate transformation   总被引：5，自引：2，他引：5  
 The Cartesian-to-geodetic-coordinate transformation is approached from a new perspective. Existence and uniqueness of geodetic representation are presented, along with a clear geometric picture of the problem and the role of the ellipse evolute. A new solution is found with a Newton-method iteration in the reduced latitude; this solution is proved to work for all points in space. Care is given to error propagation when calculating the geodetic latitude and height. Received: 9 August 2001 / Accepted: 27 March 2002 Acknowledgments. The author would like to thank the Clifford W.␣Tompson scholarship fund, Dr. Brian DeFacio, the University of Missouri College of Arts &Sciences, and the United States Air Force. He also thanks a reviewer for suggesting and providing a prototype MATLAB code. A MATLAB program for the iterative sequence is presented at the end of the paper (Appendix A).  相似文献   

13.
 The new GFZ/GRGS gravity field models GRIM5-S1 and GRIM5-C1, currently used as initial models for the CHAMP mission, have been compared with other recent models (JGM 3, EGM 96) for radial orbit accuracy (by means of latitude lumped coefficients) in computations on altimetry satellite orbits. The bases for accuracy judgements are multi-year averages of crossover sea height differences from Geosat and ERS 1/2 missions. This radially sensitive data is fully independent of the data used to develop these gravity models. There is good agreement between the observed differences in all of the world's oceans and projections of the same errors from the scaled covariance matrix of their harmonic geopotential coefficients. It was found that the tentative scale factor of five for the formal standard deviations of the harmonic coefficients of the new GRIM fields is justified, i.e. the accuracy estimates, provided together with the GRIM geopotential coefficients, are realistic. Received: 20 February 2001 / Accepted: 24 October 2001  相似文献   

14.
The topographic potential and the direct topographic effect on the geoid are presented as surface integrals, and the direct gravity effect is derived as a rigorous surface integral on the unit sphere. By Taylor-expanding the integrals at sea level with respect to topographic elevation (H) the power series of the effects is derived to arbitrary orders. This study is primarily limited to terms of order H 2. The limitations of the various effects in the frequently used planar approximations are demonstrated. In contrast, it is shown that the spherical approximation to power H 2 leads to a combined topographic effect on the geoid (direct plus indirect effect) proportional to H˜2 (where terms of degrees 0 and 1 are missing) of the order of several metres, while the combined topographic effect on the height anomaly vanishes, implying that current frequent efforts to determine the direct effect to this order are not needed. The last result is in total agreement with Bjerhammar's method in physical geodesy. It is shown that the most frequently applied remove–restore technique of topographic masses in the application of Stokes' formula suffers from significant errors both in the terrain correction C (representing the sum of the direct topographic effect on gravity anomaly and the effect of continuing the anomaly to sea level) and in the term t (mainly representing the indirect effect on the geoidal or quasi-geoidal height). Received: 18 August 1998 / Accepted: 4 October 1999  相似文献   

15.
Efficient gravity field recovery from GOCE gravity gradient observations   总被引：4，自引：2，他引：2  
 An efficient algorithm is proposed for gravity field recovery from Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite gravity gradient observations. The mathematical model is formulated in the time domain, which allows the inclusion of realistic observational noise models. The algorithm combines the iterative solution of the normal equations, using a Richardson-type iteration scheme, with the fast computation of the right-hand side of the normal equations in each iteration step by a suitable approximation of the design matrix. The convergence of the iteration is investigated, error estimates are provided, and the unbiasedness of the method is proved. It is also shown that the method does not converge to the solution of the normal equations. The performance of the approach for white noise and coloured noise is demonstrated along a simulated GOCE orbit up to spherical harmonic degree and order 180. The results also indicate that the approximation error may be neglected. Received: 30 November 1999 / Accepted: 31 May 2000  相似文献   

16.
 The traditional remove-restore technique for geoid computation suffers from two main drawbacks. The first is the assumption of an isostatic hypothesis to compute the compensation masses. The second is the double consideration of the effect of the topographic–isostatic masses within the data window through removing the reference field and the terrain reduction process. To overcome the first disadvantage, the seismic Moho depths, representing, more or less, the actual compensating masses, have been used with variable density anomalies computed by employing the topographic–isostatic mass balance principle. In order to avoid the double consideration of the effect of the topographic–isostatic masses within the data window, the effect of these masses for the used fixed data window, in terms of potential coefficients, has been subtracted from the reference field, yielding an adapted reference field. This adapted reference field has been used for the remove–restore technique. The necessary harmonic analysis of the topographic–isostatic potential using seismic Moho depths with variable density anomalies is given. A wide comparison among geoids computed by the adapted reference field with both the Airy–Heiskanen isostatic model and seismic Moho depths with variable density anomaly and a geoid computed by the traditional remove–restore technique is made. The results show that using seismic Moho depths with variable density anomaly along with the adapted reference field gives the best relative geoid accuracy compared to the GPS/levelling geoid. Received: 3 October 2001 / Accepted: 20 September 2002 Correspondence to: H.A. Abd-Elmotaal  相似文献   

17.
 The Somigliana–Pizzetti gravity field (the International gravity formula), namely the gravity field of the level ellipsoid (the International Reference Ellipsoid), is derived to the sub-nanoGal accuracy level in order to fulfil the demands of modern gravimetry (absolute gravimeters, super conducting gravimeters, atomic gravimeters). Equations (53), (54) and (59) summarise Somigliana–Pizzetti gravity Γ(φ,u) as a function of Jacobi spheroidal latitude φ and height u to the order ?(10−10 Gal), and Γ(B,H) as a function of Gauss (surface normal) ellipsoidal latitude B and height H to the order ?(10−10 Gal) as determined by GPS (`global problem solver'). Within the test area of the state of Baden-Württemberg, Somigliana–Pizzetti gravity disturbances of an average of 25.452 mGal were produced. Computer programs for an operational application of the new international gravity formula with (L,B,H) or (λ,φ,u) coordinate inputs to a sub-nanoGal level of accuracy are available on the Internet. Received: 23 June 2000 / Accepted: 2 January 2001  相似文献   

18.
 A methodology for precise determination of the fundamental geodetic parameter w 0, the potential value of the Gauss–Listing geoid, as well as its time derivative 0, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w 0 and 0 values (w 0=62 636 855.75 ± 0.21 m2/s2, 0=−0.0099±0.00079 m2/s2 per year, w 0/&γmacr;=6 379 781.502 m,0/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m2/s2) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1) the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums of countries around the Baltic Sea. Received: 14 August 2000 / Accepted: 19 June 2001  相似文献   

19.
 The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients n,m . In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients n,m e are transformed to the unknown n,m . The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients. Received: 12 December 2000 / Accepted: 21 June 2001  相似文献   

20.
 On the Earth and in its neighborhood, spherical harmonic analysis and synthesis are standard mathematical procedures for scalar, vector and tensor fields. However, with the advent of multiresolution applications, additional considerations about convolution filtering with decimation and dilation are required. As global applications often imply discrete observations on regular grids, computational challenges arise and conflicting claims about spherical harmonic transforms have recently appeared in the literature. Following an overview of general multiresolution analysis and synthesis, spherical harmonic transforms are discussed for discrete global computations. For the necessary multi-rate filtering operations, spherical convolutions along with decimations and dilations are discussed, with practical examples of applications. Concluding remarks are then included for general applications, with some discussion of the computational complexity involved and the ongoing investigations in research centers. Received: 13 November 2000 / Accepted: 12 June 2001  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号