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1.
Far-zone effects for different topographic-compensation models based on a spherical harmonic expansion of the topography 总被引:1,自引:1,他引:0
The determination of the gravimetric geoid is based on the magnitude of gravity observed at the surface of the Earth or at
airborne altitude. To apply the Stokes’s or Hotine’s formulae at the geoid, the potential outside the geoid must be harmonic
and the observed gravity must be reduced to the geoid. For this reason, the topographic (and atmospheric) masses outside the
geoid must be “condensed” or “shifted” inside the geoid so that the disturbing gravity potential T fulfills Laplace’s equation everywhere outside the geoid. The gravitational effects of the topographic-compensation masses
can also be used to subtract these high-frequent gravity signals from the airborne observations and to simplify the downward
continuation procedures. The effects of the topographic-compensation masses can be calculated by numerical integration based
on a digital terrain model or by representing the topographic masses by a spherical harmonic expansion. To reduce the computation
time in the former case, the integration over the Earth can be divided into two parts: a spherical cap around the computation
point, called the near zone, and the rest of the world, called the far zone. The latter one can be also represented by a global
spherical harmonic expansion. This can be performed by a Molodenskii-type spectral approach. This article extends the original
approach derived in Novák et al. (J Geod 75(9–10):491–504, 2001), which is restricted to determine the far-zone effects for
Helmert’s second method of condensation for ground gravimetry. Here formulae for the far-zone effects of the global topography
on gravity and geoidal heights for Helmert’s first method of condensation as well as for the Airy-Heiskanen model are presented
and some improvements given. Furthermore, this approach is generalized for determining the far-zone effects at aeroplane altitudes.
Numerical results for a part of the Canadian Rocky Mountains are presented to illustrate the size and distributions of these
effects. 相似文献
2.
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
3.
Lars E. Sjöberg 《Journal of Geodesy》2006,79(12):675-681
The application of Stokes’s formula to determine the geoid height requires that topographic and atmospheric masses be mathematically removed prior to Stokes integration. This corresponds to the applications of the direct topographic and atmospheric effects. For a proper geoid determination, the external masses must then be restored, yielding the indirect effects. Assuming an ellipsoidal layering of the atmosphere with 15% increase in its density towards the poles, the direct atmospheric effect on the geoid height is estimated to be −5.51 m plus a second-degree zonal harmonic term with an amplitude of 1.1 cm. The indirect effect is +5.50 m and the total geoid correction thus varies between −1.2 cm at the equator to 1.9 cm at the poles. Finally, the correction needed to the atmospheric effect if Stokes’s formula is used in a spherical approximation, rather than an ellipsoidal approximation, of the Earth varies between 0.3 cm and 4.0 cm at the equator and pole, respectively. 相似文献
4.
On the adjustment of combined GPS/levelling/geoid networks 总被引:12,自引:7,他引:5
A detailed treatment of adjustment problems in combined global positioning system (GPS)/levelling/geoid networks is given.
The two main types of `unknowns' in this kind of multi-data 1D networks are usually the gravimetric geoid accuracy and a 2D
spatial field that describes all the datum/systematic distortions among the available height data sets. An accurate knowledge
of the latter becomes especially important when we consider employing GPS techniques for levelling purposes with respect to
a local vertical datum. Two modelling alternatives for the correction field are presented, namely a pure deterministic parametric
model, and a hybrid deterministic and stochastic model. The concept of variance component estimation is also proposed as an
important statistical tool for assessing the actual gravimetric geoid noise level and/or testing a priori determined geoid
error models. Finally, conclusions are drawn and recommendations for further study are suggested.
Received: 9 September 1998 / Accepted: 8 June 1999 相似文献
5.
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
6.
In precise geoid determination by Stokes formula, direct and primary and secondary indirect terrain effects are applied for
removing and restoring the terrain masses. We use Helmert's second condensation method to derive the sum of these effects,
together called the total terrain effect for geoid. We develop the total terrain effect to third power of elevation H in the original Stokes formula, Earth gravity model and modified Stokes formula. It is shown that the original Stokes formula,
Earth gravity model and modified Stokes formula all theoretically experience different total terrain effects. Numerical results
indicate that the total terrain effect is very significant for moderate topographies and mountainous regions. Absolute global
mean values of 5–10 cm can be reached for harmonic expansions of the terrain to degree and order 360. In another experiment,
we conclude that the most important part of the total terrain effect is the contribution from the second power of H, while the contribution from the third power term is within 9 cm.
Received: 2 September 1996 / Accepted: 4 August 1997 相似文献
7.
Geoid determination using adapted reference field, seismic Moho depths and variable density contrast 总被引:4,自引:0,他引:4
The traditional remove-restore technique for geoid computation suffers from two main drawbacks. The first is the assumption
of an isostatic hypothesis to compute the compensation masses. The second is the double consideration of the effect of the
topographic–isostatic masses within the data window through removing the reference field and the terrain reduction process.
To overcome the first disadvantage, the seismic Moho depths, representing, more or less, the actual compensating masses, have
been used with variable density anomalies computed by employing the topographic–isostatic mass balance principle. In order
to avoid the double consideration of the effect of the topographic–isostatic masses within the data window, the effect of
these masses for the used fixed data window, in terms of potential coefficients, has been subtracted from the reference field,
yielding an adapted reference field. This adapted reference field has been used for the remove–restore technique. The necessary
harmonic analysis of the topographic–isostatic potential using seismic Moho depths with variable density anomalies is given.
A wide comparison among geoids computed by the adapted reference field with both the Airy–Heiskanen isostatic model and seismic
Moho depths with variable density anomaly and a geoid computed by the traditional remove–restore technique is made. The results
show that using seismic Moho depths with variable density anomaly along with the adapted reference field gives the best relative
geoid accuracy compared to the GPS/levelling geoid.
Received: 3 October 2001 / Accepted: 20 September 2002
Correspondence to: H.A. Abd-Elmotaal 相似文献
8.
Four different implementations of Stokes' formula are employed for the estimation of geoid heights over Sweden: the Vincent
and Marsh (1974) model with the high-degree reference gravity field but no kernel modifications; modified Wong and Gore (1969)
and Molodenskii et al. (1962) models, which use a high-degree reference gravity field and modification of Stokes' kernel;
and a least-squares (LS) spectral weighting proposed by Sj?berg (1991). Classical topographic correction formulae are improved
to consider long-wavelength contributions. The effect of a Bouguer shell is also included in the formulae, which is neglected
in classical formulae due to planar approximation. The gravimetric geoid is compared with global positioning system (GPS)-levelling-derived
geoid heights at 23 Swedish Permanent GPS Network SWEPOS stations distributed over Sweden. The LS method is in best agreement,
with a 10.1-cm mean and ±5.5-cm standard deviation in the differences between gravimetric and GPS geoid heights. The gravimetric
geoid was also fitted to the GPS-levelling-derived geoid using a four-parameter transformation model. The results after fitting
also show the best consistency for the LS method, with the standard deviation of differences reduced to ±1.1 cm. For comparison,
the NKG96 geoid yields a 17-cm mean and ±8-cm standard deviation of agreement with the same SWEPOS stations. After four-parameter
fitting to the GPS stations, the standard deviation reduces to ±6.1 cm for the NKG96 geoid. It is concluded that the new corrections
in this study improve the accuracy of the geoid. The final geoid heights range from 17.22 to 43.62 m with a mean value of
29.01 m. The standard errors of the computed geoid heights, through a simple error propagation of standard errors of mean
anomalies, are also computed. They range from ±7.02 to ±13.05 cm. The global root-mean-square error of the LS model is the
other estimation of the accuracy of the final geoid, and is computed to be ±28.6 cm.
Received: 15 September 1999 / Accepted: 6 November 2000 相似文献
9.
Any errors in digital elevation models (DEMs) will introduce errors directly in gravity anomalies and geoid models when used
in interpolating Bouguer gravity anomalies. Errors are also propagated into the geoid model by the topographic and downward
continuation (DWC) corrections in the application of Stokes’s formula. The effects of these errors are assessed by the evaluation
of the absolute accuracy of nine independent DEMs for the Iran region. It is shown that the improvement in using the high-resolution
Shuttle Radar Topography Mission (SRTM) data versus previously available DEMs in gridding of gravity anomalies, terrain corrections
and DWC effects for the geoid model are significant. Based on the Iranian GPS/levelling network data, we estimate the absolute
vertical accuracy of the SRTM in Iran to be 6.5 m, which is much better than the estimated global accuracy of the SRTM (say
16 m). Hence, this DEM has a comparable accuracy to a current photogrammetric high-resolution DEM of Iran under development.
We also found very large differences between the GLOBE and SRTM models on the range of −750 to 550 m. This difference causes
an error in the range of −160 to 140 mGal in interpolating surface gravity anomalies and −60 to 60 mGal in simple Bouguer
anomaly correction terms. In the view of geoid heights, we found large differences between the use of GLOBE and SRTM DEMs,
in the range of −1.1 to 1 m for the study area. The terrain correction of the geoid model at selected GPS/levelling points
only differs by 3 cm for these two DEMs. 相似文献
10.
M. E. Ayhan 《Journal of Geodesy》1997,71(6):362-369
In the analyses of 2D real arrays, fast Hartley (FHT), fast T (FTT) and real-valued fast Fourier transforms are generally
preferred in lieu of a complex fast Fourier transform due to the advantages of the former with respect to disk storage and
computation time. Although the FHT and the FTT in one dimension are identical, they are different in two or more dimensions.
Therefore, first, definitions and some properties of both transforms and the related 2D FHT and FTT algorithms are stated.
After reviewing the 2D FHT and FTT solutions of Stokes' formula in planar approximation, 2D FHT and FTT methods are developed
for geoid updating to incorporate additional gravity anomalies. The methods are applied for a test area which includes a 64×64
grid of 3′×3′ point gravity anomalies and geoid heights calculated from point masses. The geoids computed by 2D FHT and FTT are found to
be identical. However, the RMS value of the differences between the computed and test geoid is ±15 mm. The numerical simulations
indicate that the new methods of geoid updating are practical and accurate with considerable savings on storage requirements.
Received: 15 February 1996; Accepted: 22 January 1997 相似文献
11.
This paper is devoted to the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses in the most rigorous way. Such an expansion can be used to compute gravimetric topographic effects for geodetic and geophysical applications. It can also be used to augment a global gravity model to a much higher resolution of the gravitational potential of the topography. A formulation for a spherical harmonic expansion is developed without the spherical approximation. Then, formulas for the spheroidal harmonic expansion are derived. For the latter, Legendre’s functions of the first and second kinds with imaginary variable are expanded in Laurent series. They are then scaled into two real power series of the second eccentricity of the reference ellipsoid. Using these series, formulas for computing the spheroidal harmonic coefficients are reduced to surface harmonic analysis. Two numerical examples are presented. The first is a spherical harmonic expansion to degree and order 2700 by taking advantage of existing software. It demonstrates that rigorous spherical harmonic expansion is possible, but the computed potential on the geoid shows noticeable error pattern at Polar Regions due to the downward continuation from the bounding sphere to the geoid. The second numerical example is the spheroidal expansion to degree and order 180 for the exterior space. The power series of the second eccentricity of the reference ellipsoid is truncated at the eighth order leading to omission errors of 25 nm (RMS) for land areas, with extreme values around 0.5 mm to geoid height. The results show that the ellipsoidal correction is 1.65 m (RMS) over land areas, with maximum value of 13.19 m in the Andes. It shows also that the correction resembles the topography closely, implying that the ellipsoidal correction is rich in all frequencies of the gravity field and not only long wavelength as it is commonly assumed. 相似文献
12.
Essam Ghanem 《地球空间信息科学学报》2001,4(1):19-23
1 IntroductionDifferentgeoidsolutionswerecarriedoutforE gyptusingheterogeneousdataanddifferentmethodologies (El_Tokhey ,1 993) .ThemaingoalofthispaperistodetermineamostaccuratenewgeoidforEgypttakingadvantageofanewupdatedgravitydatabase,theinformationgivenby… 相似文献
13.
A 2×2 arc-minute resolution geoid model, CARIB97, has been computed covering the Caribbean Sea. The geoid undulations refer
to the GRS-80 ellipsoid, centered at the ITRF94 (1996.0) origin. The geoid level is defined by adopting the gravity potential
on the geoid as W
0=62 636 856.88 m2/s2 and a gravity-mass constant of GM=3.986 004 418×1014 m3/s2. The geoid model was computed by applying high-frequency corrections to the Earth Gravity Model 1996 global geopotential
model in a remove-compute-restore procedure. The permanent tide system of CARIB97 is non-tidal. Comparison of CARIB97 geoid
heights to 31 GPS/tidal (ITRF94/local) benchmarks shows an average offset (h–H–N) of 51 cm, with an Root Mean Square (RMS) of 62 cm about the average. This represents an improvement over the use of a global
geoid model for the region. However, because the measured orthometric heights (H) refer to many differing tidal datums, these comparisons are biased by localized permanent ocean dynamic topography (PODT).
Therefore, we interpret the 51 cm as partially an estimate of the average PODT in the vicinity of the 31 island benchmarks.
On an island-by-island basis, CARIB97 now offers the ability to analyze local datum problems which were previously unrecognized
due to a lack of high-resolution geoid information in the area.
Received: 2 January 1998 / Accepted: 18 August 1998 相似文献
14.
A synthetic Earth Gravity Model Designed Specifically for Testing Regional Gravimetric Geoid Determination Algorithms 总被引:1,自引:0,他引:1
I. Baran M. Kuhn S. J. Claessens W. E. Featherstone S. A. Holmes P. Vaníček 《Journal of Geodesy》2006,80(1):1-16
A synthetic [simulated] Earth gravity model (SEGM) of the geoid, gravity and topography has been constructed over Australia specifically for validating regional gravimetric geoid determination theories, techniques and computer software. This regional high-resolution (1-arc-min by 1-arc-min) Australian SEGM (AusSEGM) is a combined source and effect model. The long-wavelength effect part (up to and including spherical harmonic degree and order 360) is taken from an assumed errorless EGM96 global geopotential model. Using forward modelling via numerical Newtonian integration, the short-wavelength source part is computed from a high-resolution (3-arc-sec by 3-arc-sec) synthetic digital elevation model (SDEM), which is a fractal surface based on the GLOBE v1 DEM. All topographic masses are modelled with a constant mass-density of 2,670 kg/m3. Based on these input data, gravity values on the synthetic topography (on a grid and at arbitrarily distributed discrete points) and consistent geoidal heights at regular 1-arc-min geographical grid nodes have been computed. The precision of the synthetic gravity and geoid data (after a first iteration) is estimated to be better than 30 μ Gal and 3 mm, respectively, which reduces to 1 μ Gal and 1 mm after a second iteration. The second iteration accounts for the changes in the geoid due to the superposed synthetic topographic mass distribution. The first iteration of AusSEGM is compared with Australian gravity and GPS-levelling data to verify that it gives a realistic representation of the Earth’s gravity field. As a by-product of this comparison, AusSEGM gives further evidence of the north–south-trending error in the Australian Height Datum. The freely available AusSEGM-derived gravity and SDEM data, included as Electronic Supplementary Material (ESM) with this paper, can be used to compute a geoid model that, if correct, will agree to in 3 mm with the AusSEGM geoidal heights, thus offering independent verification of theories and numerical techniques used for regional geoid modelling.Electronic Supplementary Material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00190-005-0002-z 相似文献
15.
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
16.
Arne Bjerhammar 《Journal of Geodesy》1962,36(3):215-220
Summary The principal formulae of the geophysical geodesy are based on the famous explicit expression of Stokes (1849). Up to now,
there has been no method for a computation of the corresponding explicit expression of a (non-spherical) surface with masses
outside the geoid. In this paper there is a solution of this problem. Another paper on this subject was presented to “Nordiska
geodetm?tet” in Copenhagen May 1959 (in Swedish). 相似文献
17.
A. Ellmann 《Journal of Geodesy》2005,79(1-3):11-23
In regional gravimetric geoid determination, it is customary to use the modified Stokes formula that combines local terrestrial data with a global geopotential model. This study compares two deterministic and three stochastic modification methods for computing a regional geoid over the Baltic countries. The final selection of the best modification method is made by means of two accuracy estimates: the expected global mean square error of the geoid estimator, and the statistics of the post-fit residuals between the computed geoid models and precise GPS-levelling data. Numerical results show that the modification methods tested do not provide substantially different results, although the stochastic approaches appear formally better in the selected study area. The 2.8–5.3 cm (RMS) post-fit residuals to the GPS-levelling points indicate the suitability of the new geoid model for many practical applications. Moreover, the numerical comparisons reveal a one-dimensional offset between the regional vertical datum and the geoid models based upon the new GRACE-only geopotential model GGM01s. This gives an impression of a greater reliability of the new model compared to the earlier, EGM96-based and somewhat tilted regional geoid models for the same study area. 相似文献
18.
A detailed gravimetric geoid has been computed for the Nortwest Atlantic Ocean and Caribbean Sea area in support of the calibration
and evaluation of the GEOS-3 altimeter. This geoid, computed on a 15’ x 15’ grid was based upon a combination of surface gravity
data and the GSFC GEM-8 gravitational field model. This gravimetric geoid has been compared with passes of SKYLAB altimeter
data recorded in the Atlantic Ocean, and three typical passes are presented. The relative agreement of the two data types
is quite good with differences generally less than 2 meters for these passes. Sea surface manifestations of numerous short
wavelength (≈ 100 km) oceanographic features indicated in the altimeter data are also confirmed by the gravimetric geoid. 相似文献
19.
H. Fan 《Journal of Geodesy》1998,72(9):511-515
Historically, the mean Earth ellipsoid is obtained by fitting an ellipsoid of revolution to the geoid. Such an ellipsoid,
however, does not necessarily best fit the physical surface of the Earth due to the existence of topography outside the geoid.
In this paper, we present a purely geometrically defined Earth ellipsoid that best fits the physical surface of the Earth
so that the resulting ellipsoidal height attains minimum global mean square value. Using a global digital terrain model and
a global geopotential model, the size, shape and position of such an Earth ellipsoid have been numerically estimated.
Received: 6 September 1996 / Accepted: 1 April 1998 相似文献
20.
P. Nsombo 《Journal of Geodesy》1998,72(3):144-153
In the determination of the preliminary geoid over Zambia, we compare three methods of the modified Stokes formula: that
of Vincent and Marsh, a modified Wong and Gore method, and a modified spectral weighting method, with the final solution being
estimated by the modified Wong and Gore procedure. The geoid over Zambia (based on GRS80) is rising from north-east to south-west.
It coincides with the reference ellipsoid in the north-western and southern regions of Zambia. The preliminary estimate indicates
maximum and minimum values of about 13.7 and −16.8m, respectively. The mean geoid over the area is −2.8m. Formal analysis
of global root mean square errors for the three models leads us to conclude that for an integration cap radius of about 3
or less, the modified formula using optimal spectral weighting is superior to the Vincent and Marsh method, and to the modified
and unmodified Wong and Gore.
Received: 8 October 1996 / Accepted: 25 June 1997 相似文献