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1.
E. W. Grafarend 《Journal of Geodesy》2001,75(7-8):363-390
In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is
aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational
potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the
Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere
(IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical
gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry
space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of
horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections
given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions
specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the
vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators
for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space
are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity
to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward
continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced.
Received: 10 May 2000 / Accepted: 26 February 2001 相似文献
2.
Two numerical techniques are used in recent regional high-frequency geoid computations in Canada: discrete numerical integration
and fast Fourier transform. These two techniques have been tested for their numerical accuracy using a synthetic gravity field.
The synthetic field was generated by artificially extending the EGM96 spherical harmonic coefficients to degree 2160, which
is commensurate with the regular 5′ geographical grid used in Canada. This field was used to generate self-consistent sets of synthetic gravity anomalies and
synthetic geoid heights with different degree variance spectra, which were used as control on the numerical geoid computation
techniques. Both the discrete integration and the fast Fourier transform were applied within a 6∘ spherical cap centered at each computation point. The effect of the gravity data outside the spherical cap was computed using
the spheroidal Molodenskij approach. Comparisons of these geoid solutions with the synthetic geoid heights over western Canada
indicate that the high-frequency geoid can be computed with an accuracy of approximately 1 cm using the modified Stokes technique,
with discrete numerical integration giving a slightly, though not significantly, better result than fast Fourier transform.
Received: 2 November 1999 / Accepted: 11 July 2000 相似文献
3.
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
4.
H. Nahavandchi 《Journal of Geodesy》2002,76(6-7):345-352
It is suggested that a spherical harmonic representation of the geoidal heights using global Earth gravity models (EGM) might
be accurate enough for many applications, although we know that some short-wavelength signals are missing in a potential coefficient
model. A `direct' method of geoidal height determination from a global Earth gravity model coefficient alone and an `indirect'
approach of geoidal height determination through height anomaly computed from a global gravity model are investigated. In
both methods, suitable correction terms are applied. The results of computations in two test areas show that the direct and
indirect approaches of geoid height determination yield good agreement with the classical gravimetric geoidal heights which
are determined from Stokes' formula. Surprisingly, the results of the indirect method of geoidal height determination yield
better agreement with the global positioning system (GPS)-levelling derived geoid heights, which are used to demonstrate such
improvements, than the results of gravimetric geoid heights at to the same GPS stations. It has been demonstrated that the
application of correction terms in both methods improves the agreement of geoidal heights at GPS-levelling stations. It is
also found that the correction terms in the direct method of geoidal height determination are mostly similar to the correction
terms used for the indirect determination of geoidal heights from height anomalies.
Received: 26 July 2001 / Accepted: 21 February 2002 相似文献
5.
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
6.
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
7.
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
8.
Fast spherical collocation: theory and examples 总被引:2,自引:4,他引:2
It has long been known that a spherical harmonic analysis of gridded (and noisy) data on a sphere (with uniform error for
a fixed latitude) gives rise to simple systems of equations. This idea has been generalized for the method of least-squares
collocation, when using an isotropic covariance function or reproducing kernel. The data only need to be at the same altitude
and of the same kind for each latitude. This permits, for example, the combination of gravity data at the surface of the Earth
and data at satellite altitude, when the orbit is circular. Suppose that data are associated with the points of a grid with
N values in latitude and M values in longitude. The latitudes do not need to be spaced uniformly. Also suppose that it is required to determine the
spherical harmonic coefficients to a maximal degree and order K. Then the method will require that we solve K systems of equations each having a symmetric positive definite matrix of only N × N. Results of simulation studies using the method are described.
Received: 18 October 2001 / Accepted: 4 October 2002
Correspondence to: F. Sansò 相似文献
9.
Three different methods of handling topography in geoid determination were investigated. The first two methods employ the
residual terrain model (RTM) remove–restore technique, yielding the quasigeoid, whereas the third method uses the classical
Helmert condensation method, yielding the geoid. All three methods were used with the geopotential model Earth Gravity Model
(1996) (EGM96) as a reference, and the results were compared to precise global positioning system (GPS) levelling networks
in Scandinavia. An investigation of the Helmert method, focusing on the different types of indirect effects and their effects
on the geoid, was also carried out. The three different methods used produce almost identical results at the 5-cm level, when
compared to the GPS levelling networks. However, small systematic differences existed.
Received: 18 March 1999 / Accepted: 21 March 2000 相似文献
10.
G. Ramillien 《Journal of Geodesy》2002,76(3):139-149
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
11.
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
12.
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
13.
One of the aims of the Earth Explorer Gravity Field and Steady-State Ocean Circulation (GOCE) mission is to provide global
and regional models of the Earth's gravity field and of the geoid with high spatial resolution and accuracy. Using the GOCE
error model, simulation studies were performed in order to estimate the accuracy of datum transfer in different areas of the
Earth. The results showed that with the GOCE error model, the standard deviation of the height anomaly differences is about
one order of magnitude better than the corresponding value with the EGM96 error model. As an example, the accuracy of the
vertical datum transfer from the tide gauge of Amsterdam to New York was estimated equal to 57 cm when the EGM96 error model
was used, while in the case of GOCE error model this accuracy was increased to 6 cm. The geoid undulation difference between
the two places is about 76.5 m. Scaling the GOCE errors to the local gravity variance, the estimated accuracy varied between
3 and 7 cm, depending on the scaling model.
Received: 1 March 2000 / Accepted: 21 February 2001 相似文献
14.
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
15.
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
16.
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
17.
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
18.
L. E. Sjöberg 《Journal of Geodesy》2001,75(5-6):283-290
The topographic and atmospheric effects of gravimetric geoid determination by the modified Stokes formula, which combines
terrestrial gravity and a global geopotential model, are presented. Special emphasis is given to the zero- and first-degree
effects. The normal potential is defined in the traditional way, such that the disturbing potential in the exterior of the
masses contains no zero- and first-degree harmonics. In contrast, it is shown that, as a result of the topographic masses,
the gravimetric geoid includes such harmonics of the order of several centimetres. In addition, the atmosphere contributes
with a zero-degree harmonic of magnitude within 1 cm.
Received: 5 November 1999 / Accepted: 22 January 2001 相似文献
19.
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
20.
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