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1.
Spherical cap harmonic analysis: a comment on its proper use for local gravity field representation 总被引:1,自引:0,他引:1
Spherical cap harmonic analysis is the appropriate analytical technique for modelling Laplacian potential and the corresponding
field components over a spherical cap. This paper describes the use of this method by means of a least-squares approach for
local gravity field representation. Formulations for the geoid undulation and the components ξ, η of the deflection of the
vertical are derived, together with some warnings in the application of the technique. Although most of the formulations have
been given by another paper, these were confusing or even incorrect, mainly because of an improper application of the spherical
cap harmonic analysis.
Received: 16 January 1996 / Accepted: 17 March 1997 相似文献
2.
An analysis of vertical deflections derived from high-degree spherical harmonic models 总被引:5,自引:4,他引:1
C. Jekeli 《Journal of Geodesy》1999,73(1):10-22
The theoretical differences between the Helmert deflection of the vertical and that computed from a truncated spherical harmonic
series of the gravity field, aside from the limited spectral content in the latter, include the curvature of the normal plumb
line, the permanent tidal effect, and datum origin and orientation offsets. A numerical comparison between deflections derived
from spherical harmonic model EGM96 and astronomic deflections in the conterminous United States (CONUS) shows that correcting
these systematic effects reduces the mean differences in some areas. Overall, the mean difference in CONUS is reduced from
−0.219 arcsec to −0.058 arcsec for the south–north deflection, and from +0.016 arcsec to +0.004 arcsec for the west–east deflection.
Further analysis of the root-mean-square differences indicates that the high-degree spectrum of the EGM96 model has significantly
less power than implied by the deflection data.
Received: 9 December 1997 / Accepted: 21 August 1998 相似文献
3.
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
4.
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
5.
Prediction of vertical deflections from high-degree spherical harmonic synthesis and residual terrain model data 总被引:6,自引:4,他引:2
Christian Hirt 《Journal of Geodesy》2010,84(3):179-190
This study demonstrates that in mountainous areas the use of residual terrain model (RTM) data significantly improves the
accuracy of vertical deflections obtained from high-degree spherical harmonic synthesis. The new Earth gravitational model
EGM2008 is used to compute vertical deflections up to a spherical harmonic degree of 2,160. RTM data can be constructed as
difference between high-resolution Shuttle Radar Topography Mission (SRTM) elevation data and the terrain model DTM2006.0
(a spherical harmonic terrain model that complements EGM2008) providing the long-wavelength reference surface. Because these
RTM elevations imply most of the gravity field signal beyond spherical harmonic degree of 2,160, they can be used to augment
EGM2008 vertical deflection predictions in the very high spherical harmonic degrees. In two mountainous test areas—the German
and the Swiss Alps—the combined use of EGM2008 and RTM data was successfully tested at 223 stations with high-precision astrogeodetic
vertical deflections from recent zenith camera observations (accuracy of about 0.1 arc seconds) available. The comparison
of EGM2008 vertical deflections with the ground-truth astrogeodetic observations shows root mean square (RMS) values (from
differences) of 3.5 arc seconds for ξ and 3.2 arc seconds for η, respectively. Using a combination of EGM2008 and RTM data for the prediction of vertical deflections considerably reduces
the RMS values to the level of 0.8 arc seconds for both vertical deflection components, which is a significant improvement
of about 75%. Density anomalies of the real topography with respect to the residual model topography are one factor limiting
the accuracy of the approach. The proposed technique for vertical deflection predictions is based on three publicly available
data sets: (1) EGM2008, (2) DTM2006.0 and (3) SRTM elevation data. This allows replication of the approach for improving the
accuracy of EGM2008 vertical deflection predictions in regions with a rough topography or for improved validation of EGM2008
and future high-degree spherical harmonic models by means of independent ground truth data. 相似文献
6.
D. Arabelos 《Journal of Geodesy》1985,59(2):109-123
The evaluation of deflections of the vertical for the area of Greece is attempted using a combination of topographic and astrogeodetic
data. Tests carried out in the area bounded by 35°≤ϕ≤42°, 19°≤λ≤27° indicate that an accuracy of ±3″.3 can be obtained in
this area for the meridian and prime vertical deflection components when high resolution topographic data in the immediate
vicinity of computation points are used, combined with high degree spherical harmonic expansions of the geopotential and isostatic
reduction potential. This accuracy is about 25% better than the corresponding topographic-Moho deflection components which
are evaluated using topographic and Moho data up to 120 km around each station, without any combination with the spherical
harmonic expansion of the geopotential or isostatic reduction potential. The accuracy in both cases is increased to about
2″.6 when the astrogeodetic data available in the area mentioned above are used for the prediction of remaining values. Furthermore
the estimation of datum-shift parameters is attempted using least squares collocation. 相似文献
7.
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
8.
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
9.
Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high
degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of
orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from
a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics.
This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals
of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and
obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests
with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes
and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series
also offers a computational savings of at least one third. 相似文献
10.
The classical integral formula for determining the indirect effect in connection with the Stokes–Helmert method is related
to a planar approximation of the sea level. A strict integral formula, as well as some approximations to it, are derived.
It is concluded that the cap- size truncated integral formulas will suffer from the omission of some long-wavelength contributions,
of the order of 50 cm in high mountains for the classical formula. This long-wavelength information can be represented by
a set of spherical harmonic coefficients of the topography to, say, degree and order 360. Hence, for practical use, a combination
of the classical formula and a set of spherical harmonics is recommended.
Received: 10 March 1998 / Accepted: 16 November 1998 相似文献
11.
As shown in previous work, dynamical effects of a realistic model of a heterogeneous, compressible, stably stratified liquid
core may be obtained by means of a simple analysis of the generalized two-dimensional Laplace tidal equation which describes
tidal flows of an incompressible and non-gravitating fluid in a thin spherical layer with mobile boundaries. The solution
was presented in the form of expansions in powers of a small parameter κ being the ratio of nutational motion frequency in
space to the frequency of the Earth's diurnal rotation. Whereas in an earlier paper only first-order terms were taken into
account, our present approach includes not only main second-order terms in the spherical harmonic expansions of the solutions,
but also the terms of higher orders. These effects are calculated numerically for realistic models of the Earth's outer liquid
core, solid inner core and anelastic mantle (PREM model). All tables are found in electronic version at http://www.tu-darmstadt.de/fb/vw/ipg/Welcome2.html
Received: 12 June 1997 / Accepted: 11 December 1997 相似文献
12.
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
13.
We present an alternate mathematical technique than contemporary spherical harmonics to approximate the geopotential based
on triangulated spherical spline functions, which are smooth piecewise spherical harmonic polynomials over spherical triangulations.
The new method is capable of multi-spatial resolution modeling and could thus enhance spatial resolutions for regional gravity
field inversion using data from space gravimetry missions such as CHAMP, GRACE or GOCE. First, we propose to use the minimal
energy spherical spline interpolation to find a good approximation of the geopotential at the orbital altitude of the satellite.
Then we explain how to solve Laplace’s equation on the Earth’s exterior to compute a spherical spline to approximate the geopotential
at the Earth’s surface. We propose a domain decomposition technique, which can compute an approximation of the minimal energy
spherical spline interpolation on the orbital altitude and a multiple star technique to compute the spherical spline approximation
by the collocation method. We prove that the spherical spline constructed by means of the domain decomposition technique converges
to the minimal energy spline interpolation. We also prove that the modeled spline geopotential is continuous from the satellite
altitude down to the Earth’s surface. We have implemented the two computational algorithms and applied them in a numerical
experiment using simulated CHAMP geopotential observations computed at satellite altitude (450 km) assuming EGM96 (n
max = 90) is the truth model. We then validate our approach by comparing the computed geopotential values using the resulting
spherical spline model down to the Earth’s surface, with the truth EGM96 values over several study regions. Our numerical
evidence demonstrates that the algorithms produce a viable alternative of regional gravity field solution potentially exploiting
the full accuracy of data from space gravimetry missions. The major advantage of our method is that it allows us to compute
the geopotential over the regions of interest as well as enhancing the spatial resolution commensurable with the characteristics
of satellite coverage, which could not be done using a global spherical harmonic representation.
The results in this paper are based on the research supported by the National Science Foundation under the grant no. 0327577. 相似文献
14.
Based upon a data set of 25 points of the Baltic Sea Level Project, second campaign 1993.4, which are close to mareographic
stations, described by (1) GPS derived Cartesian coordinates in the World Geodetic Reference System 1984 and (2) orthometric
heights in the Finnish Height Datum N60, epoch 1993.4, we have computed the primary geodetic parameter W
0(1993.4) for the epoch 1993.4 according to the following model. The Cartesian coordinates of the GPS stations have been converted
into spheroidal coordinates. The gravity potential as the additive decomposition of the gravitational potential and the centrifugal
potential has been computed for any GPS station in spheroidal coordinates, namely for a global spheroidal model of the gravitational
potential field. For a global set of spheroidal harmonic coefficients a transformation of spherical harmonic coefficients
into spheroidal harmonic coefficients has been implemented and applied to the global spherical model OSU 91A up to degree/order
360/360. The gravity potential with respect to a global spheroidal model of degree/order 360/360 has been finally transformed
by means of the orthometric heights of the GPS stations with respect to the Finnish Height Datum N60, epoch 1993.4, in terms
of the spheroidal “free-air” potential reduction in order to produce the spheroidal W
0(1993.4) value. As a mean of those 25 W
0(1993.4) data as well as a root mean square error estimation we computed W
0(1993.4)=(6 263 685.58 ± 0.36) kgal × m. Finally a comparison of different W
0 data with respect to a spherical harmonic global model and spheroidal harmonic global model of Somigliana-Pizetti type (level
ellipsoid as a reference, degree/order 2/0) according to The Geodesist's Handbook 1992 has been made.
Received: 7 November 1996 / Accepted: 27 March 1997 相似文献
15.
Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic 总被引:3,自引:2,他引:1
We present software for spherical harmonic analysis (SHA) and spherical harmonic synthesis (SHS), which can be used for essentially
arbitrary degrees and all co-latitudes in the interval (0°, 180°). The routines use extended-range floating-point arithmetic,
in particular for the computation of the associated Legendre functions. The price to be paid is an increased computation time;
for degree 3,000, the extended-range arithmetic SHS program takes 49 times longer than its standard arithmetic counterpart.
The extended-range SHS and SHA routines allow us to test existing routines for SHA and SHS. A comparison with the publicly
available SHS routine GEOGFG18 by Wenzel and HARMONIC SYNTH by Holmes and Pavlis confirms what is known about the stability of these programs. GEOGFG18 gives errors <1 mm for latitudes [-89°57.5′, 89°57.5′] and maximum degree 1,800. Higher degrees significantly limit the range
of acceptable latitudes for a given accuracy. HARMONIC SYNTH gives good results up to degree 2,700 for almost the whole latitude range. The errors increase towards the North pole and
exceed 1 mm at latitude 82° for degree 2,700. For a maximum degree 3,000, HARMONIC SYNTH produces errors exceeding 1 mm at latitudes of about 60°, whereas GEOGFG18 is limited to latitudes below 45°. Further extending the latitudinal band towards the poles may produce errors of several
metres for both programs. A SHA of a uniform random signal on the sphere shows significant errors beyond degree 1,700 for
the SHA program SHA by Heck and Seitz. 相似文献
16.
Simplified techniques for high-degree spherical harmonic synthesis are extended to include gravitational potential second
derivatives with respect to latitude.
Received: 23 July 2001 / Accepted: 12 April 2002
Acknowledgement. The authors would like to thank Christian Tscherning for recommending Laplace's equation as an accuracy test. Our use of
Legendre's differential equation, as the most direct means for extending our simplified synthesis methods to second-order
derivatives, was a direct result of this suggestion.
Correspondence to: S. A. Holmes 相似文献
17.
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
18.
An algorithm (differential mode) is presented for the improvement of harmonic tidal analysis along T/P tracks, in which the differences between the observed sea surface heights at adjacent points are taken as observations. Also, the observation equations are constrained with the results of the crossover analysis; the parameter estimations are performed at 0.1° latitude intervals by the least squares. Cycle 10 to 330 T/P altimeter data covering the China Sea and the Northwest Pacific Ocean (2°-50° N,105°-150° E) are adopted for a refined along-track harmonic tidal analysis, and harmonic constants of 12 constituents in 8 474 points are obtained, which indicates that the algorithm can efficiently remove non-tidal effects in the altimeter observations, and improve the precision of tide parameters. Moreover, parameters along altimetry tracks represent a smoother distribution than those obtained by traditional algorithms. The root mean squares of the fitting errors between the tidal height model and the observations reduce from 11 cm to 1.3 cm. 相似文献
19.
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
20.
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