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1.
R. Lehmann 《Journal of Geodesy》2000,74(3-4):327-334
The definition and connection of vertical datums in geodetic height networks is a fundamental problem in geodesy. Today,
the standard approach to solve it is based on the joint processing of terrestrial and satellite geodetic data. It is generalized
to cases where the coverage with terrestrial data may change from region to region, typically across coastlines. The principal
difficulty is that such problems, so-called altimetry–gravimetry boundary-value problems (AGPs), do not admit analytical solutions
such as Stokes' integral. A numerical solution strategy for the free-datum problem is presented. Analysis of AGPs in spherical
and constant radius approximation shows that two of them are mathematically well-posed problems, while the classical AGP-I
may be ill posed in special situations.
Received: 2 December 1998 / Accepted: 30 November 1999 相似文献
2.
A new method for calculating analytical solar radiation pressure models for GNSS spacecraft has been developed. The method
simulates the flux of light from the Sun using a pixel array. The method can cope with a high level of complexity in the spacecraft
structure and models effects due to reflected light. Models have been calculated and tested for the Russhar global navigation
satellite system GLONASS IIv spacecraft. Results are presented using numerical integration of the force model and long-arc
satellite laser ranging (SLR) analysis. The integrated trajectory differs from a precise orbit calculated using a network
of global tracking stations by circa 2 m root mean square over a 160 000-km arc. The observed − computed residuals for the
400-day SLR arc are circa 28 mm.
Received: 23 December 1999 / Accepted: 28 August 2000 相似文献
3.
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
4.
The value of the Earth's rotational angular velocity determined from observations is given in the GCRS (geocentric celestial
reference system) defined by Resolution B1 of the 2000 IAU General Assembly. The same quantity derived from dynamical theories
of the Earth's rotation, such as SMART97, is referred to the␣DGRSC (dynamically non-rotating ecliptical reference system).
The relativistic theory of reference systems (RSs) enables unambiguous general-relativity relations between these quantities
to be derived.
Received: 9 November 2000 / Accepted: 9 July 2001 相似文献
5.
Regularization of geopotential determination from satellite data by variance components 总被引:11,自引:18,他引:11
Different types of present or future satellite data have to be combined by applying appropriate weighting for the determination
of the gravity field of the Earth, for instance GPS observations for CHAMP with satellite to satellite tracking for the coming
mission GRACE as well as gradiometer measurements for GOCE. In addition, the estimate of the geopotential has to be smoothed
or regularized because of the inversion problem. It is proposed to solve these two tasks by Bayesian inference on variance
components. The estimates of the variance components are computed by a stochastic estimator of the traces of matrices connected
with the inverse of the matrix of normal equations, thus leading to a new method for determining variance components for large
linear systems. The posterior density function for the variance components, weighting factors and regularization parameters
are given in order to compute the confidence intervals for these quantities. Test computations with simulated gradiometer
observations for GOCE and satellite to satellite tracking for GRACE show the validity of the approach.
Received: 5 June 2001 / Accepted: 28 November 2001 相似文献
6.
A solution to the downward continuation effect on the geoid determined by Stokes' formula 总被引:2,自引:1,他引:2
L.E. Sjöberg 《Journal of Geodesy》2003,77(1-2):94-100
The analytical continuation of the surface gravity anomaly to sea level is a necessary correction in the application of Stokes'
formula for geoid estimation. This process is frequently performed by the inversion of Poisson's integral formula for a sphere.
Unfortunately, this integral equation corresponds to an improperly posed problem, and the solution is both numerically unstable,
unless it is well smoothed, and tedious to compute. A solution that avoids the intermediate step of downward continuation
of the gravity anomaly is presented. Instead the effect on the geoid as provided by Stokes' formula is studied directly. The
practical solution is partly presented in terms of a truncated Taylor series and partly as a truncated series of spherical
harmonics. Some simple numerical estimates show that the solution mostly meets the requests of a 1-cm geoid model, but the
truncation error of the far zone must be studied more precisely for high altitudes of the computation point. In addition,
it should be emphasized that the derived solution is more computer efficient than the detour by Poisson's integral.
Received: 6 February 2002 / Accepted: 18 November 2002
Acknowledgements. Jonas ?gren carried out the numerical calculations and gave some critical and constructive remarks on a draft version of
the paper. This support is cordially acknowledged. Also, the thorough work performed by one unknown reviewer is very much
appreciated. 相似文献
7.
A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions 总被引:11,自引:7,他引:11
Spherical harmonic expansions form partial sums of fully normalised associated Legendre functions (ALFs). However, when evaluated
increasingly close to the poles, the ultra-high degree and order (e.g. 2700) ALFs range over thousands of orders of magnitude.
This causes existing recursion techniques for computing values of individual ALFs and their derivatives to fail. A common
solution in geodesy is to evaluate these expansions using Clenshaw's method, which does not compute individual ALFs or their
derivatives. Straightforward numerical principles govern the stability of this technique. Elementary algebra is employed to
illustrate how these principles are implemented in Clenshaw's method. It is also demonstrated how existing recursion algorithms
for computing ALFs and their first derivatives are easily modified to incorporate these same numerical principles. These modified
recursions yield scaled ALFs and first derivatives, which can then be combined using Horner's scheme to compute partial sums,
complete to degree and order 2700, for all latitudes (except at the poles for first derivatives). This exceeds any previously
published result. Numerical tests suggest that this new approach is at least as precise and efficient as Clenshaw's method.
However, the principal strength of the new techniques lies in their simplicity of formulation and implementation, since this
quality should simplify the task of extending the approach to other uses, such as spherical harmonic analysis.
Received: 30 June 2000 / Accepted: 12 June 2001 相似文献
8.
The global positioning system (GPS) model is distinctive in the way that the unknown parameters are not only real-valued,
the baseline coordinates, but also integers, the phase ambiguities. The GPS model therefore leads to a mixed integer–real-valued
estimation problem. Common solutions are the float solution, which ignores the ambiguities being integers, or the fixed solution,
where the ambiguities are estimated as integers and then are fixed. Confidence regions, so-called HPD (highest posterior density)
regions, for the GPS baselines are derived by Bayesian statistics. They take care of the integer character of the phase ambiguities
but still consider them as unknown parameters. Estimating these confidence regions leads to a numerical integration problem
which is solved by Monte Carlo methods. This is computationally expensive so that approximations of the confidence regions
are also developed. In an example it is shown that for a high confidence level the confidence region consists of more than
one region.
Received: 1 February 2001 / Accepted: 18 July 2001 相似文献
9.
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
10.
H. Nahavandchi 《Journal of Geodesy》2000,74(6):488-496
The direct topographical correction is composed of both local effects and long-wavelength contributions. This implies that
the classical integral formula for determining the direct effect may have some numerical problems in representing these different
signals. On the other hand, a representation by a set of harmonic coefficients of the topography to, say, degree and order
360 will omit significant short-wavelength signals. A new formula is derived by combining the classical formula and a set
of spherical harmonics. Finally, the results of this solution are compared with the Moritz topographical correction in a test
area.
Received: 27 July 1998 / Accepted: 29 March 2000 相似文献
11.
Downward continuation and geoid determination based on band-limited airborne gravity data 总被引:4,自引:3,他引:4
The downward continuation of the harmonic disturbing gravity potential, derived at flight level from discrete observations
of airborne gravity by the spherical Hotine integral, to the geoid is discussed. The initial-boundary-value approach, based
on both the direct and inverse solution to Dirichlet's problem of potential theory, is used. Evaluation of the discretized
Fredholm integral equation of the first kind and its inverse is numerically tested using synthetic airborne gravity data.
Characteristics of the synthetic gravity data correspond to typical airborne data used for geoid determination today and in
the foreseeable future: discrete gravity observations at a mean flight height of 2 to 6 km above mean sea level with minimum
spatial resolution of 2.5 arcmin and a noise level of 1.5 mGal. Numerical results for both approaches are presented and discussed.
The direct approach can successfully be used for the downward continuation of airborne potential without any numerical instabilities
associated with the inverse approach. In addition to these two-step approaches, a one-step procedure is also discussed. This
procedure is based on a direct relationship between gravity disturbances at flight level and the disturbing gravity potential
at sea level. This procedure provided the best results in terms of accuracy, stability and numerical efficiency. As a general
result, numerically stable downward continuation of airborne gravity data can be seen as another advantage of airborne gravimetry
in the field of geoid determination.
Received: 6 June 2001 / Accepted: 3 January 2002 相似文献
12.
Gravitational perturbation theory for intersatellite tracking 总被引:7,自引:0,他引:7
M. K. Cheng 《Journal of Geodesy》2002,76(3):169-185
An analytical gravitational perturbation theory for the intersatellite tracking range and range-rate measurement between
two satellites is developed. The satellite-to-satellite tracking (SST) range data measure the difference between the position
perturbations of two satellites along the direction of the intersatellite range. The SST range-rate data measure the difference
between the velocity perturbations along the direction of the intersatellite range, and the difference of the position perturbation
along the direction perpendicular to the intersatellite range (cross-range). The SST range and range rate depend on different
orbital excitations for mapping the gravity field. For the Gravity Recovery and Climate Experiment (GRACE), approximately
97% of the geopotential coefficient pairs produce perturbations with a root-mean-square larger than 1 m on the range and 0.1
m/sec on the range rate based on the EGM96 gravity field truncated at degree and order 140. Results in this study showed that
ocean tides produce significant perturbations in the range and range-rate measurements. An ocean tide field with a higher
degree and order (>70) is required to model the ocean tide perturbations on the intersatellite range and range-rate measurement.
Received: 17 May 2000 / Accepted: 3 September 2001 相似文献
13.
A synthetic Earth for use in geodesy 总被引:1,自引:0,他引:1
R. Haagmans 《Journal of Geodesy》2000,74(7-8):503-511
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
14.
Random simulation and GPS decorrelation 总被引:13,自引:1,他引:13
Peiliang Xu 《Journal of Geodesy》2001,75(7-8):408-423
(i) A random simulation approach is proposed, which is at the centre of a numerical comparison of the performances of different
GPS decorrelation methods. The most significant advantage of the approach is that it does not depend on nor favour any particular
satellite–receiver geometry and weighting system. (ii) An inverse integer Cholesky decorrelation method is proposed, which
will be shown to out-perform the integer Gaussian decorrelation and the Lenstra, Lenstra and Lovász (LLL) algorithm, and thus
indicates that the integer Gaussian decorrelation is not the best decorrelation technique and that further improvement is
possible. (iii) The performance study of the LLL algorithm is the first of its kind and the results have shown that the algorithm
can indeed be used for decorrelation, but that it performs worse than the integer Gaussian decorrelation and the inverse integer
Cholesky decorrelation. (iv) Simulations have also shown that no decorrelation techniques available to date can guarantee
a smaller condition number, especially in the case of high dimension, although reducing the condition number is the goal of
decorrelation.
Received: 26 April 2000 / Accepted: 5 March 2001 相似文献
15.
The performance of the L-curve criterion and of the generalized cross-validation (GCV) method for the Tikhonov regularization
of the ill-conditioned normal equations associated with the determination of the gravity field from satellite gravity gradiometry
is investigated. Special attention is devoted to the computation of the corner point of the L-curve, to the numerically efficient
computation of the trace term in the GCV target function, and to the choice of the norm of the residuals, which is important
for the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) in the presence of colored observation noise. The
trace term in the GCV target function is estimated using an unbiased minimum-variance stochastic estimator. The performance
analysis is based on a simulation of gravity gradients along a 60-day repeat circular orbit and a gravity field recovery complete
up to degree and order 300. Randomized GCV yields the optimal regularization parameter in all the simulations if the colored
noise is properly taken into account. Moreover, it seems to be quite robust against the choice of the norm of the residuals.
It performs much better than the L-curve criterion, which always yields over-smooth solutions. The numerical costs for randomized
GCV are limited provided that a reasonable first guess of the regularization parameter can be found.
Received: 17 May 2001 / Accepted: 17 January 2002 相似文献
16.
J. Mäkinen 《Journal of Geodesy》2002,76(6-7):317-322
A bound is established for the Euclidean norm of the difference between the best linear unbiased estimator and any linear
unbiased estimator in the general linear model. The bound involves the spectral norm of the difference between the dispersion
matrices of the two estimators, and the residual sum of squares, all evaluated at the assumed model, but is independent of
the provenance of the observation vector at hand. The bound, a straightforward consequence of first principles in Gauss–Markov
theory, generalizes previous results on the difference between the best linear unbiased estimator and the ordinary least-squares
estimator. In a numerical example from repeated precise levelling, the bound is used to analyse the sensitivity of estimates
of vertical motion to the choice of estimator.
Received: 9 September 1999 / Accepted: 15 March 2002 相似文献
17.
Since the beginning of the International Global Navigation Satellite System (GLONASS) Experiment, IGEX, in October 1998,
the Center for Orbit Determination in Europe (CODE) has acted as an analysis center providing precise GLONASS orbits on a
regular basis. In CODE's IGEX routine analysis the Global Positioning System (GPS) orbits and Earth rotation parameters are
introduced as known quantities into the GLONASS processing. A new approach is studied, where data from the IGEX network are
combined with GPS observations from the International GPS Service (IGS) network and all parameters (GPS and GLONASS orbits,
Earth rotation parameters, and site coordinates) are estimated in one processing step. The influence of different solar radiation
pressure parameterizations on the GLONASS orbits is studied using different parameter subsets of the extended CODE orbit model.
Parameterization with three constant terms in the three orthogonal directions, D, Y, and X (D = direction satellite–Sun, Y = direction of the satellite's solar panel axis), and two periodic terms in the X-direction, proves to be adequate for GLONASS satellites. As a result of the processing it is found that the solar radiation
pressure effect for the GLONASS satellites is significantly different in the Y-direction from that for the GPS satellites, and an extensive analysis is carried out to investigate the effect in detail.
SLR observations from the ILRS network are used as an independent check on the quality of the GLONASS orbital solutions. Both
processing aspects, combining the two networks and changing the orbit parameterization, significantly improve the quality
of the determined GLONASS orbits compared to the orbits stemming from CODE's IGEX routine processing.
Received: 10 May 2000 / Accepted: 9 October 2000 相似文献
18.
L. E. Sjöberg 《Journal of Geodesy》2002,76(2):115-120
The problems of intersection on the sphere and ellipsoid are studied. On the sphere, the problem of intersection along great
circles is explicitly solved. On the ellipsoid, each of the problems of intersection along arcs of constant azimuth, normal
sections and geodesic lines is solved without any limitation on arc length. In the last case the solution is based on the
Newton–Raphson method of iteration including numerical integration.
Received: 11 April 2001 / Accepted: 3 September 2001 相似文献
19.
W. Prószyñski 《Journal of Geodesy》2000,74(7-8):581-589
A specific subclass of Gauss–Markov models has been defined as containing the models for which the disturbance/response matrix,
determined under the assumption of uncorrelated observations, consists of independent diagonal blocks. A proposed modification
of reliability assessment procedure for such models is presented By the appropriate reduction of a given full covariance matrix
for the observations, the proposal allows the assessment to be made in the resulting model which, in contrast to the initial
model, is free from outlier-hiding effects of the type not occurring in ordinary models. The theoretical findings are demonstrated
using simple numerical examples. All the proofs supporting the proposal are gathered in Appendixes. The proposal, which is
not without its own weak points, is an attempt to associate the reliability assessment in specific Gauss–Markov models with
effective outlier detection.
Received: 23 June 1998 / Accepted: 5 July 2000 相似文献
20.
The single- and dual-satellite crossover (SSC and DSC) residuals between and among Geosat, TOPEX/Poseidon (T/P), and ERS
1 or 2 have been used for various purposes, applied in geodesy for gravity field accuracy assessments and determination as
well as in oceanography. The theory is presented and various examples are given of certain combinations of SSC and DSC that test for residual altimetry data errors, mostly of non-gravitational origin, of the order of a few centimeters.
There are four types of basic DSCs and 12 independent combinations of them in pairs which have been found useful in the present
work. These are defined in terms of the `mean' and `variable' components of a satellite's geopotential orbit error from Rosborough's
1st-order analytical theory. The remaining small errors, after all altimeter data corrections are applied and the relative
offset of coordinate frames between altimetry missions removed, are statistically evaluated by means of the Student distribution.
The remaining signal of `non-gravitational' origin can in some cases be attributed to the main ocean currents which were not
accounted for among the media or sea-surface corrections. In future, they may be resolved by a long-term global circulation
model. Experience with two current models, neither of which are found either to cover the most critical missions (Geosat &
TOPEX/Poseidon) or to have the accuracy and resolution necessary to account for the strongest anomalies found across them,
is described. In other cases, the residual signal is due to errors in tides, altimeter delay corrections or El Ni?o. (Various
examples of these are also presented.) Tests of the combinations of the JGM 3-based DSC residuals show that overall the long-term
data now available are well suited for a gravity field inversion refining JGM 3 for low- and resonant-order geopotential harmonics
whose signatures are clearly seen in the basic DSC and SSC sets.
Received: 15 January 1999 / Accepted: 9 September 1999 相似文献