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1.
The Center for Orbit Determination in Europe (CODE) has been involved in the processing of combined GPS/GLONASS data during the International GLONASS Experiment (IGEX). The resulting precise orbits were analyzed using the program SORBDT. Introducing one satellites positions as pseudo-observations, the program is capable of fitting orbital arcs through these positions using an orbit improvement procedure based on the numerical integration of the satellites orbit and its partial derivative with respect to the orbit parameters. For this study, the program was enhanced to estimate selected parameters of the Earths gravity field. The orbital periods of the GPS satellites are —in contrast to those of the GLONASS satellites – 2:1 commensurable (P
Sid:P
GPS) with the rotation period of the Earth. Therefore, resonance effects of the satellite motion with terms of the geopotential occur and they influence the estimation of these parameters. A sensitivity study of the GPS and GLONASS orbits with respect to the geopotential coefficients reveals that the correlations between different geopotential coefficients and the correlations of geopotential coefficients with other orbit parameters, in particular with solar radiation pressure parameters, are the crucial issues in this context. The estimation of the resonant geopotential terms is, in the case of GPS, hindered by correlations with the simultaneously estimated radiation pressure parameters. In the GLONASS case, arc lengths of several days allow the decorrelation of the two parameter types. The formal errors of the estimates based on the GLONASS orbits are a factor of 5 to 10 smaller for all resonant terms.
AcknowledgmentsThe authors would like to thank all the organizations involved in the IGS and the IGEX campaign, in particular those operating an IGS or IGEX observation site and providing the indispensable data for precise orbit determination. 相似文献
2.
J. Klokočník Ch. Reigber P. Schwintzer C. A. Wagner J. Kostelecký 《Journal of Geodesy》2002,76(4):189-198
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
3.
Exploring gravity field determination from orbit perturbations of the European Gravity Mission GOCE 总被引:5,自引:0,他引:5
A comparison was made between two methods for gravity field recovery from orbit perturbations that can be derived from global
positioning system satellite-to-satellite tracking observations of the future European gravity field mission GOCE (Gravity
Field and Steady-State Ocean Circulation Explorer). The first method is based on the analytical linear orbit perturbation
theory that leads under certain conditions to a block-diagonal normal matrix for the gravity unknowns, significantly reducing
the required computation time. The second method makes use of numerical integration to derive the observation equations, leading
to a full set of normal equations requiring powerful computer facilities. Simulations were carried out for gravity field recovery
experiments up to spherical harmonic degree and order 80 from 10 days of observation. It was found that the first method leads
to large approximation errors as soon as the maximum degree surpasses the first resonance orders and great care has to be
taken with modeling resonance orbit perturbations, thereby loosing the block-diagonal structure. The second method proved
to be successful, provided a proper division of the data period into orbital arcs that are not too long.
Received: 28 April 2000 / Accepted: 6 November 2000 相似文献
4.
The identification of mean semi-major axes (suitably defined) for satellite orbits to satisfy a variety of requirements for
geodesy, geophysics and oceanography, in terms of repeat orbits (with orbital resonances), is investigated. Various options
for the definition of semi-major axis, from the viewpoint of satellite dynamics, are described. Simple simulations of the
expected resonant changes in inclination are presented, and tools for the analysis of orbit resonances to extract certain
lumped harmonic coefficients of the geopotential (e.g. from the very precise CHAMP orbit) are resurrected. Finally, a preliminary
example of the 46th-order resonance analysis possible for CHAMP, based on the mean orbital elements produced by GFZ (GeoForschungs
Zentrum) for ephemeris prediction, is presented.
Received: 10 July 2001 / Accepted: 17 July 2002
Correspondence to: J. Klokočník at Ondřejov Observatory
Acknowledgements. We thank Prof. Dr. Ch. Reigber, Dr. P. Schwintzer, Dr. T. Gruber and Dr. R. K?nig from GFZ Potsdam for various consultations
and discussions, and for the CHAMP two-line mean elements. This investigation was performed under the aegis of CEDR (Center
for Earth's Dynamics Research, Prague-Ondřejov); it has been supported by project LN00A005 (provided by the Ministry of Education
of the Czech Republic) and by grant A 3004 of the Grant Agency of the Academy of Sciences of the Czech Republic. 相似文献
5.
The results from 14 satellite orbit analyses, two of which are new objects, are used to determine individual tesseral harmonic coefficients of 30th-order and even degree. Six C, S pairs are evaluated by solving the equations using a modified least-squares technique. The results are compared with comprehensive geopotential models. The recent models GRIM4-C1, GEM-T3 and JGM-2 emerge well from such tests and are generally closest to the resonance values. A tentative solution is found for four pairs of harmonic coefficients of 30th-order and odd degree. 相似文献
6.
This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84(3):165–178, 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first- and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first- and second-order derivatives in terms of the geopotential coefficients. These equations are converted into recurrent relations from which the coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission. 相似文献
7.
G. Beutler W. Gurtner M. Rothacher T. Schildknecht I. Bauersima 《Journal of Geodesy》1986,60(3):205-220
Summary Carrier phase measurements are potentially the most precise observations available from theGPS satellite system, the formal precision being of the order of one centimeter per observation. If the so called double differences
are used as the basic observable, the analysis is relatively simple, since satellite- and receiver-clocks may be represented
by basic models. We investigate the feasibility of double difference phase observations for orbit determination using the
material of the 1985 High Precision Baseline Test, where the coordinates of the so called fiducial points (Haystack, Ft. Davis
Richmond and Mojave) are held fixed.TI-4100 andAFGL-receiver observations were used in the same orbit determination process.
Although no surface weather data had been available to us, the orbit quality seems to be of the order of0.1 ppm. When we use these orbits to estimate the coordinates of the five “non-fiducial points” Owens Valley, Hat Creek Mammoth Lake,
Austin and Dahlgren we get a repeatability of the order of5 cm for latitude and longitude and10 cm for height, if the observations of the first four days of the campaign are compared to those of the second four days.
If we use our orbits estimated withTI andAFGL observations to process the Mojave—Owens Valley baseline (length245 km) measured by the twoSERIES-X receivers, we obtain day to day repeatabilities of1.6 cm (0.06 ppm) in length,2 cm (0.08 ppm) in latitude,4 cm (0.16 ppm) in longitude and7 cm (0.29 ppm) in height.
Since there are indications that regional networks will be realized in the near future, the results presented here should
encourage the realization of regional high precision orbit determination services. 相似文献
8.
ERS-1 radial positioning using the JGM-2 and JGM-3 gravity fields is assessed by analysing dual crossovers with TOPEX/Poseidon,
neither field containing ERS-1 data. This method allows a more complete recovery of ERS-1 radial orbit error, specifically
of the previously unattainable mean geographical error. The global analysis shows that the theoretical error derived from
the JGM-2 covariance matrix is realistic and that JGM-3 represents a slight improvement, at least at the inclination of ERS-1.
A latitudinal-based study in the southern ocean indicates possible weaknesses in both fields, notably for low and resonant
geopotential orders m. A refinement of JGM-2, RGM-2, is undertaken through inclusion of ERS-1 and STELLA laser tracking and ERS-1 altimetry, reducing
several of its deficiencies.
Received: 14 May 1996 / Accepted: 17 February 1997 相似文献
9.
The differential equations which generate a general conformal mapping of a two-dimensional Riemann manifold found by Korn
and Lichtenstein are reviewed. The Korn–Lichtenstein equations subject to the integrability conditions of type vectorial Laplace–Beltrami
equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the
function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and ellipsoidal latitude. The related
coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein
equations which directly generates Gau?–Krüger conformal coordinates as well as the Universal Transverse Mercator Projection
(UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference
generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials
up to degree five with coefficients listed in closed form.
Received: 3 June 1997 / Accepted: 17 November 1997 相似文献
10.
Computation of spherical harmonic coefficients and their error estimates using least-squares collocation 总被引:4,自引:0,他引:4
C. C. Tscherning 《Journal of Geodesy》2001,75(1):12-18
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
11.
Spherical harmonic expansions of the geopotential are frequently used for modelling the earth’s gravity field. Degree and
order of recently available models go up to 360, corresponding to a resolution of about50 km. Thus, the high degree potential coefficients can be verified nowadays even by locally distributed sets of terrestrial gravity
anomalies. These verifications are important when combining the short wavelength model impact, e.g. for regional geoid determinations
by means of collocation solutions. A method based on integral formulae is presented, enabling the improvement of geopotential
models with respect to non-global distributed gravity anomalies. To illustrate the foregoing, geoid computations are carried
out for the area of Iran, introducing theGPM2 geopotential model in combination with available regional gravity data. The accuracy of the geoid determination is estimated
from a comparison with Doppler and levelling data to ±1.4m. 相似文献
12.
M. S. Petrovskaya 《Journal of Geodesy》1979,53(3):259-271
Summary The geopotential on and outside the earth is represented as a series in surface harmonics. The principal terms in it correspond
to the solid harmonics of the external potential expansion with the coefficients being Stokes’ constantsC
nm
andS
nm
. The additional terms which occur near the earth’s surface due to its non-sphericity and topography are expressed in terms
of Stokes’ constants too. This allows performing downward continuation of the potential derived from satellite observations.
In the boundary condition which correlates Stokes’ constants and the surface gravity anomalies there occur additional terms
due to the earth’s non-sphericity and topography. They are expressed in terms of Stokes’ constants as well. This improved
boundary condition can be used for upward and downward continuations of the gravity field. Simple expressions are found representingC
nm
andS
nm
as explicit functions of the surface anomalies and its derivatives. The formula for the disturbing potential on the surface
is derived in terms of the surface anomalies. All the formulas do not involve the earth’s surface in clinations. 相似文献
13.
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
14.
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
15.
A technique for the analysis of low–low intersatellite range-rate data in a gravity mapping mission is explored. The technique
is based on standard tracking data analysis for orbit determination but uses a spherical coordinate representation of the
12 epoch state parameters describing the baseline between the two satellites. This representation of the state parameters
is exploited to allow the intersatellite range-rate analysis to benefit from information provided by other tracking data types
without large simultaneous multiple-data-type solutions. The technique appears especially valuable for estimating gravity
from short arcs (e.g. less than 15 minutes) of data. Gravity recovery simulations which use short arcs are compared with those
using arcs a day in length. For a high-inclination orbit, the short-arc analysis recovers low-order gravity coefficients remarkably
well, although higher-order terms, especially sectorial terms, are less accurate. Simulations suggest that either long or
short arcs of the Gravity Recovery and Climate Experiment (GRACE) data are likely to improve parts of the geopotential spectrum
by orders of magnitude.
Received: 26 June 2001 / Accepted: 21 January 2002 相似文献
16.
Variations in the accuracy of gravity recovery due to ground track variability: GRACE,CHAMP, and GOCE 总被引:4,自引:2,他引:2
J. Klokočník C. A. Wagner J. Kostelecký A. Bezděk P. Novák D. McAdoo 《Journal of Geodesy》2008,82(12):917-927
Following an earlier recognition of degraded monthly geopotential recovery from GRACE (Gravity Recovery And Climate Experiment)
due to prolonged passage through a short repeat (low order resonant) orbit, we extend these insights also to CHAMP (CHAllenging
Minisatellite Payload) and GOCE (Gravity field and steady state Ocean Circulation Explorer). We show wide track-density variations
over time for these orbits in both latitude and longitude, and estimate that geopotential recovery will be as widely affected
as well within all these regimes, with lesser track density leading to poorer recoveries. We then use recent models of atmospheric
density to estimate the future orbit of GRACE and warn of degraded performance as other low order resonances are encountered
in GRACE’s free fall. Finally implications for the GOCE orbit are discussed. 相似文献
17.
Throughout 2004 the GRACE (Gravity Recovery And Climate Experiment) orbit contracted slowly to yield a sparse repeat track of 61 revolutions every 4 days on 19 September 2004. As a result, we show from linear perturbation theory that geopotential information previously available to fully resolve a gravity field every month of 120× 120 (degree by order) in spherical harmonics was compressed then into about one-fourth of the necessary observation space. We estimate from this theory that the ideal gravity field resolution in September 2004 was only about 30 × 30. More generally, we show that any repeat-cycle mission for geopotential recovery with full resolution L × L requires the number of orbit-revolutions-to-repeat to be greater than 2L. 相似文献
18.
In satellite data analysis, one big advantage of analytical orbit integration, which cannot be overestimated, is missed in
the numerical integration approach: spectral analysis or the lumped coefficient concept may be used not only to design efficient
algorithms but overall for much better insight into the force-field determination problem. The lumped coefficient concept,
considered from a practical point of view, consists of the separation of the observation equation matrix A=BT into the product of two matrices. The matrix T is a very sparse matrix separating into small block-diagonal matrices connecting the harmonic coefficients with the lumped
coefficients. The lumped coefficients are nothing other than the amplitudes of trigonometric functions depending on three
angular orbital variables; therefore, the matrix N=B
T
B will become for a sufficient length of a data set a diagonal dominant matrix, in the case of an unlimited data string length
a strictly diagonal one. Using an analytical solution of high order, the non-linear observation equations for low–low SST range data can be transformed into a form to allow the application of the lumped concept.
They are presented here for a second-order solution together with an outline of how to proceed with data analysis in the spectral
domain in such a case. The dynamic model presented here provides not only a practical algorithm for the parameter determination
but also a simple method for an investigation of some fundamental questions, such as the determination of the range of the
subset of geopotential coefficients which can be properly determined by means of SST techniques or the definition of an optimal
orbital configuration for particular SST missions. Numerical results have already been obtained and will be published elsewhere.
Received: 15 January 1999 / Accepted: 30 November 1999 相似文献
19.
Least-squares collocation may be used for the estimation of spherical harmonic coefficients and their error and error correlations
from GOCE data. Due to the extremely large number of data, this requires the use of the so-called method of Fast Spherical
Collocation (FSC) which requires that data is gridded equidistantly on each parallel and have the same uncorrelated noise
on the parallel. A consequence of this is that error-covariances will be zero except between coefficients of the same signed
order (i.e., the same order and the same coefficient type C–C or S–S). If the data distribution and the characteristics of the data noise are symmetric with respect to the equator, then, within
a given order and coefficient type, the error-covariances amongst coefficients whose degrees are of different parity also
vanish. The deviation from this “ideal” pattern has been studied using data-sets of second order radial derivatives of the
anomalous potential. A total number of points below 17,000 were used having an equi-angular or an equal area distribution
or being associated with points on a realistic GOCE orbit but close to the nodes of a grid. Also the data were considered
having a correlated or an uncorrelated noise and three different signal covariance functions. Grids including data or not
including data in the polar areas were used. Using the functionals associated with the data, error estimates of coefficients
and error-correlations between coefficients were calculated up to a maximal degree and order equal to 90. As expected, for
the data-distributions with no data in the polar areas the error-estimates were found to be larger than when the polar areas
contained data. In all cases it was found that only the error-correlations between coefficients of the same order were significantly
different from zero (up to 88%). Error-correlations were significantly larger when data had been regarded as having non-zero
error-correlations. Also the error-correlations were largest when the covariance function with the largest signal covariance
distance was used. The main finding of this study was that the correlated noise has more pronounced impact on gridded data
than on data distributed on a realistic GOCE orbit. This is useful information for methods using gridded data, such as FSC. 相似文献
20.
Latitude-lumped coefficients (LLC) are defined, representing geopotential-orbit variations for dual-satellite crossovers (DSC). Formulae are derived for their standard errors from the covariances of geopotential field models. Numerical examples are
presented for pairs of the altimeter-bearing satellites TOPEX/Poseidon, ERS 1, and Geosat, using the error matrices of recent
gravity models. The DSC, connecting separate missions, will play an increasingly important role in oceanography spanning decades
only when its nonoceanographic signals are thoroughly understood. In general, the content of even the long-term averaged DSC
is more complex then their single satellite crossover (SSC) counterpart. The LLC, as the spatial spectra for the geopotential-caused
crossover effects, discriminate these source-differences sharply. Thus, the zero-order LLC in DSC data contains zonal gravity
information not present in SSC data. In addition, zero- and first-order LLC of DSC data can reveal a geocenter discrepancy
between the orbit tracking of the separate satellite missions. For example, DSC analysis from orbits computed with JGM 2 show
that the y-axis of the geocenter for Geosat in 1986–1988 is shifted with respect to T/P by 6–9 cm towards the eastern Pacific. Also,
where the time-gap is necessarily large (as between, say, Geosat and T/P missions) oceanographic (sea-level) differences in
DSC may corrupt the geopotential interpretation of the data. Most importantly, as we illustrate, media delays for the altimeter
(from the ionosphere, wet troposphere and sea-state bias) are more likely sources of contamination across two missions than
in SSC analyses. Again, the LLC of zero order best shows this contrast. Using the higher-order LLC of DSC for both Geosat-T/P
and ERS 1-T/P as likely representation of geopotential-only error, we show by comparison with the predicted standard errors
of JGM 2 that the latter's previously calibrated covariance matrix is generally valid.
Received: 14 February 1996 / Accepted: 27 March 1997 相似文献