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1.
J. Li 《Journal of Geodesy》2005,79(1-3):64-70
Integral formulas are derived which can be used to convert the second-order radial gradient of the disturbing potential, as boundary values, into the disturbing potential, gravity anomaly and the deflection of the vertical. The derivations are based on the fundamental differential equation as the boundary condition in Stokes’s boundary-value problem and the modified Poisson integral formula in which the zero and first-degree spherical harmonics are excluded. The rigorous kernel functions, corresponding to the integral operators, are developed by the methods of integration. 相似文献
2.
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
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.
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
5.
A function having some properties of a wavelet and being harmonic around a given point in R
3 is defined, and three models showing the local relationships between the disturbing density, the disturbing potential and
the disturbing gravity are established by using the function as the kernel function of the integrals in the models. The local
relationship has two meanings. One is that we can evaluate with a high accuracy the integrals in the models by using mainly
high-accuracy and high-resolution data in a local area. The other is that we can obtain a stable solution with high resolution
when we invert the integrals in the models because of the rapid decrease of the kernel function of the integrals. As a result,
with these models we evaluate one quantity with high resolution, in a band limited by the maximum degree of a set of geopotential
coefficients or by the resolution (spacing) of the local data, from another quantity (or quantities) in a local area, and
the resulting solution is stable.
Received: 6 April 1998 / Accepted: 16 June 1999 相似文献
6.
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
7.
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
8.
GPS-based precise orbit determination of the very low Earth-orbiting gravity mission GOCE 总被引:5,自引:0,他引:5
A prerequisite for the success of future gravity missions like the European Gravity field and steady-state Ocean Circulation
Explorer (GOCE) is a precise orbit determination (POD). A detailed simulation study has been carried out to assess the achievable
orbit accuracy based on satellite-to-satellite tracking (SST) by the US global positioning system (GPS) and in conjunction
the implications for gravity field determination. An orbit accuracy at the few centimeter level seems possible, sufficient
to support the GOCE gravity mission and in particular its gravity gradiometer.
Received: 21 January 2000 / Accepted: 4 July 2000 相似文献
9.
P. Moore 《Journal of Geodesy》2001,75(5-6):241-254
Dual satellite crossovers (DXO) between the two European Remote Sensing satellites ERS-1 and ERS-2 and TOPEX/Poseidon are
used to (1) refine the Earth's gravity field and (2) extend the study of the ERS-2 altimetric range stability to cover the
first four years of its operation. The enhanced gravity field model, AGM-98, is validated by several methodologies and will
be shown to provide, in particular, low geographically correlated orbital error for ERS-2. For the ERS-2 altimetric range
study, TOPEX/Poseidon is first calibrated through comparison against in situ tide gauge data. A time series of the ERS-2 altimeter
bias has been recovered along with other geophysical correction terms using tables for bias jumps in the range measurements
at the single point target response (SPTR) events. On utilising the original version of the SPTR tables the overall bias drift
is seen to be 2.6±1.0 mm/yr with an RMS of fit of 12.2 mm but with discontinuities at the centimetre level at the SPTR events.
On utilising the recently released revised tables, SPTR2000, the drift is better defined at 2.4±0.6 mm/yr with the RMS of
fit reduced to 3.7 mm. Investigations identify the sea-state bias as a source of error with corrections affecting the overall
drift by close to 1.2 mm/yr.
Received: 25 May 2000 / Accepted: 24 January 2001 相似文献
10.
Stokes' formula from 1849 is still the basis for the gravimetric determination of the geoid. The modification of the formula,
originating with Molodensky, aims at reducing the truncation error outside a spherical cap of integration. This goal is still
prevalent among various modifications. In contrast to these approaches, some least-squares types of modification that aim
at reducing the truncation error, as well as the error stemming from the potential coefficients, are demonstrated. The least-squares
estimators are provided in the two cases that (1) Stokes' kernel is a priori modified (e.g. according to Molodensky's approach)
and (2) Stokes' kernel is optimally modified to minimize the global mean square error. Meissl-type modifications are also
studied. In addition, the use of a higher than second-degree reference field versus the original (Pizzetti-type) reference
field is discussed, and it is concluded that the former choice of reference field implies increased computer labour to achieve
the same result as with the original reference field.
Received: 14 December 1998 / Accepted: 4 October 1999 相似文献
11.
The Cartesian moments of the mass density of a gravitating body and the spherical harmonic coefficients of its gravitational
field are related in a peculiar way. In particular, the products of inertia can be expressed by the spherical harmonic coefficients
of the gravitational potential as was derived by MacCullagh for a rigid body. Here the MacCullagh formulae are extended to
a deformable body which is restricted to radial symmetry in order to apply the Love–Shida hypothesis. The mass conservation
law allows a representation of the incremental mass density by the respective excitation function. A representation of an
arbitrary Cartesian monome is always possible by sums of solid spherical harmonics multiplied by powers of the radius. Introducing
these representations into the definition of the Cartesian moments, an extension of the MacCullagh formulae is obtained. In
particular, for excitation functions with a vanishing harmonic coefficient of degree zero, the (diagonal) incremental moments
of inertia also can be represented by the excitation coefficients. Four types of excitation functions are considered, namely:
(1) tidal excitation; (2) loading potential; (3) centrifugal potential; and (4) transverse surface stress. One application
of the results could be model computation of the length-of-day variations and polar motion, which depend on the moments of
inertia.
Received: 27 July 1999 / Accepted: 24 May 2000 相似文献
12.
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
13.
Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt 总被引:1,自引:0,他引:1
C. C. Tscherning Awar Radwan A. A. Tealeb S. M. Mahmoud M. Abd El-Monum Ramdan Hassan I. El-Syaed K. Saker 《Journal of Geodesy》2001,75(7-8):343-348
The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and
the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity
data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not
using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with
gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general
gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan
Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using
only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was
in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used,
as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave
difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the
agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances
and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account
for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads.
Received: 14 August 2000 / Accepted: 28 February 2001 相似文献
14.
Iterative vector methods for computing geodetic latitude and height from rectangular coordinates 总被引:4,自引:4,他引:4
J. Pollard 《Journal of Geodesy》2002,76(1):36-40
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
15.
Journal of Geodesy - ?The application of Stokes' formula to create geoid undulations requires no masses outside the geoid. However, due to the existence of the topography, terrain... 相似文献
16.
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
17.
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
18.
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
19.
The gravitational potential and its derivatives for the prism 总被引:24,自引:12,他引:12
As a simple building block, the right rectangular parallelepiped (prism) has an important role mostly in local gravity field
modelling studies when the so called flat-Earth approximation is sufficient. Its primary (methodological) advantage follows
from the simplicity of the rigorous and consistent analytical forms describing the different gravitation-related quantities.
The analytical forms provide numerical values for these quantities which satisfy the functional connections existing between
these quantities at the level of numerical precision applied. Closed expressions for the gravitational potential of the prism
and its derivatives (up to the third order) are listed for easy reference.
Received: 18 August 1999 / Accepted: 15 June 2000 相似文献
20.
Global mean sea surface heights (SSHs) and gravity anomalies on a 2′×2′ grid were determined from Seasat, Geosat (Exact Repeat Mission and Geodetic Mission), ERS-1 (1.5-year mean of 35-day, and
GM), TOPEX/POSEIDON (T/P) (5.6-year mean) and ERS-2 (2-year mean) altimeter data over the region 0∘–360∘ longitude and –80∘–80∘ latitude. To reduce ocean variabilities and data noises, SSHs from non-repeat missions were filtered by Gaussian filters
of various wavelengths. A Levitus oceanic dynamic topography was subtracted from the altimeter-derived SSHs, and the resulting
heights were used to compute along-track deflection of the vertical (DOV). Geoidal heights and gravity anomalies were then
computed from DOV using the deflection-geoid and inverse Vening Meinesz formulae. The Levitus oceanic dynamic topography was
added back to the geoidal heights to obtain a preliminary sea surface grid. The difference between the T/P mean sea surface
and the preliminary sea surface was computed on a grid by a minimum curvature method and then was added to the preliminary
grid. The comparison of the NCTU01 mean sea surface height (MSSH) with the T/P and the ERS-1 MSSH result in overall root-mean-square
(RMS) differences of 5.0 and 3.1 cm in SSH, respectively, and 7.1 and 3.2 μrad in SSH gradient, respectively. The RMS differences
between the predicted and shipborne gravity anomalies range from 3.0 to 13.4 mGal in 12 areas of the world's oceans.
Received: 26 September 2001 / Accepted: 3 April 2002
Correspondence to: C. Hwang
Acknowledgements. This research is partly supported by the National Science Council of ROC, under grants NSC89-2611-M-009-003-OP2 and NSC89-2211-E-009-095.
This is a contribution to the IAG Special Study Group 3.186. The Geosat and ERS1/2 data are from NOAA and CERSAT/France, respectively.
The T/P data were provided by AVISO. The CLS and GSFC00 MSS models were kindly provided by NASA/GSFC and CLS, respectively.
Drs. Levitus, Monterey, and Boyer are thanked for providing the SST model. Dr. T. Gruber and two anonymous reviewers provided
very detailed reviews that improved the quality of this paper. 相似文献