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1.
Srinivas V. Bettadpur 《Journal of Geodesy》1995,69(3):135-142
Hotine's (1969) partially nonsingular geopotential formulation is revisited to study its utility for the computation of geopotential acceleration and gradients from high degree and order expansions. This formulation results in the expansion of each Cartesian derivative of the potential in a spherical harmonic series of its own. The spherical harmonic coefficients of any Cartesian derivative of the potential are related in a simple manner to the coefficients of the geopotential. A brief overview of the derivation is provided, along with the fully normalized versions of Hotine's formulae, which is followed by a comparison with other algorithms of spherical harmonic synthesis on a CRAY Y-MP. The elegance and simplicity of Hotine's formulation is seen to lead to superior computational performance in a comparison against other algorithms for spherical harmonic synthesis. 相似文献
2.
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
3.
Adaptive filtering of GOCE-derived gravity gradients of the disturbing potential in the context of the space-wise approach 总被引：1，自引：0，他引：1
Filtering and signal processing techniques have been widely used in the processing of satellite gravity observations to reduce measurement noise and correlation errors. The parameters and types of filters used depend on the statistical and spectral properties of the signal under investigation. Filtering is usually applied in a non-real-time environment. The present work focuses on the implementation of an adaptive filtering technique to process satellite gravity gradiometry data for gravity field modeling. Adaptive filtering algorithms are commonly used in communication systems, noise and echo cancellation, and biomedical applications. Two independent studies have been performed to introduce adaptive signal processing techniques and test the performance of the least mean-squared (LMS) adaptive algorithm for filtering satellite measurements obtained by the gravity field and steady-state ocean circulation explorer (GOCE) mission. In the first study, a Monte Carlo simulation is performed in order to gain insights about the implementation of the LMS algorithm on data with spectral behavior close to that of real GOCE data. In the second study, the LMS algorithm is implemented on real GOCE data. Experiments are also performed to determine suitable filtering parameters. Only the four accurate components of the full GOCE gravity gradient tensor of the disturbing potential are used. The characteristics of the filtered gravity gradients are examined in the time and spectral domain. The obtained filtered GOCE gravity gradients show an agreement of 63–84 mEötvös (depending on the gravity gradient component), in terms of RMS error, when compared to the gravity gradients derived from the EGM2008 geopotential model. Spectral-domain analysis of the filtered gradients shows that the adaptive filters slightly suppress frequencies in the bandwidth of approximately 10–30 mHz. The limitations of the adaptive LMS algorithm are also discussed. The tested filtering algorithm can be connected to and employed in the first computational steps of the space-wise approach, where a time-wise Wiener filter is applied at the first stage of GOCE gravity gradient filtering. The results of this work can be extended to using other adaptive filtering algorithms, such as the recursive least-squares and recursive least-squares lattice filters. 相似文献
4.
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
5.
利用最小二乘直接法反演卫星重力场模型的MPI并行算法 总被引：2，自引：0，他引：2
针对海量卫星重力数据反演高阶次地球重力场模型的密集型计算任务与高内存耗用问题,基于MPI实现了最小二乘直接法恢复高阶次位系数的并行算法。引入并行读写、分块存储与分块计算等方式完成了设计矩阵的构建、法方程的形成与求解等密集型计算任务的并行算法,数值计算结果表明三者的并行相对效率峰值可分别达到95%、68%、63%。利用GOCE轨道跟踪和径向扰动重力梯度数据(共518 400个历元)分别反演了120、240阶次地球重力场模型,计算时间仅为40 min、7 h,内存耗用峰值仅为290 MB、1.57 GB;采用与GOCE同等噪声水平的观测数据恢复的重力场模型精度与GOCE已发布模型的解算精度相一致,联合GRACE和GOCE的解算模型能够实现二者独立信息的频谱互补,表明本文方法可高效稳定地恢复高阶次地球重力场模型。 相似文献
6.
Least-squares collocation may be used for the estimation of spherical harmonic coefficients and their error and error correlations
from GOCE data. Due to the extremely large number of data, this requires the use of the so-called method of Fast Spherical
Collocation (FSC) which requires that data is gridded equidistantly on each parallel and have the same uncorrelated noise
on the parallel. A consequence of this is that error-covariances will be zero except between coefficients of the same signed
order (i.e., the same order and the same coefficient type C–C or S–S). If the data distribution and the characteristics of the data noise are symmetric with respect to the equator, then, within
a given order and coefficient type, the error-covariances amongst coefficients whose degrees are of different parity also
vanish. The deviation from this “ideal” pattern has been studied using data-sets of second order radial derivatives of the
anomalous potential. A total number of points below 17,000 were used having an equi-angular or an equal area distribution
or being associated with points on a realistic GOCE orbit but close to the nodes of a grid. Also the data were considered
having a correlated or an uncorrelated noise and three different signal covariance functions. Grids including data or not
including data in the polar areas were used. Using the functionals associated with the data, error estimates of coefficients
and error-correlations between coefficients were calculated up to a maximal degree and order equal to 90. As expected, for
the data-distributions with no data in the polar areas the error-estimates were found to be larger than when the polar areas
contained data. In all cases it was found that only the error-correlations between coefficients of the same order were significantly
different from zero (up to 88%). Error-correlations were significantly larger when data had been regarded as having non-zero
error-correlations. Also the error-correlations were largest when the covariance function with the largest signal covariance
distance was used. The main finding of this study was that the correlated noise has more pronounced impact on gridded data
than on data distributed on a realistic GOCE orbit. This is useful information for methods using gridded data, such as FSC. 相似文献
7.
The computational requirements in the simulations of geopotential estimation from satellite gravity gradiometry are discussed. Fast algorithms for spherical harmonic synthesis and least squares accumulation on a vectorizing supercomputers are presented. Using these methods, in a test case estimation of 2595 coefficients of a degree and order 50 gravity field, sustained program execution speeds of 275 Mflops (87 % peak machine speed) on a single processor of a CRAY Y-MP were achieved, with spherical harmonics computation accounting for less than 1 % of total cost. From the results, it appears that brute-force estimation of a degree and order 180 field would require 537 Million Words of memory and 85 hours of CPU time, assuming mission duration of 1 month, and execution speed of 1 Gflops. Both memory size and execution speed requirements are within the capabilities of modern multi-processor supercomputers. 相似文献
8.
Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis 总被引：5，自引：3，他引：2
Collocation is widely used in physical geodesy. Its application requires to solve systems with a dimension equal to the number
of observations, causing numerical problems when many observations are available. To overcome this drawback, tailored step-wise
techniques are usually applied. An example of these step-wise techniques is the space-wise approach to the GOCE mission data
processing. The original idea of this approach was to implement a two-step procedure, which consists of first predicting gridded
values at satellite altitude by collocation and then deriving the geo-potential spherical harmonic coefficients by numerical
integration. The idea was generalized to a multi-step iterative procedure by introducing a time-wise Wiener filter to reduce
the highly correlated observation noise. Recent studies have shown how to optimize the original two-step procedure, while
the theoretical optimization of the full multi-step procedure is investigated in this work. An iterative operator is derived
so that the final estimated spherical harmonic coefficients are optimal with respect to the Wiener–Kolmogorov principle, as
if they were estimated by a direct collocation. The logical scheme used to derive this optimal operator can be applied not
only in the case of the space-wise approach but, in general, for any case of step-wise collocation. Several numerical tests
based on simulated realistic GOCE data are performed. The results show that adding a pre-processing time-wise filter to the
two-step procedure of data gridding and spherical harmonic analysis is useful, in the sense that the accuracy of the estimated
geo-potential coefficients is improved. This happens because, in its practical implementation, the gridding is made by collocation
over local patches of data, while the observation noise has a time-correlation so long that it cannot be treated inside the
patch size. Therefore, the multi-step operator, which is in theory equivalent to the two-step operator and to the direct collocation,
is in practice superior thanks to the time-wise filter that reduces the noise correlation before the gridding. The criteria
for the choice of this filter are investigated numerically. 相似文献
9.
Exploring gravity field determination from orbit perturbations of the European Gravity Mission GOCE 总被引：5，自引：0，他引：5
A comparison was made between two methods for gravity field recovery from orbit perturbations that can be derived from global
positioning system satellite-to-satellite tracking observations of the future European gravity field mission GOCE (Gravity
Field and Steady-State Ocean Circulation Explorer). The first method is based on the analytical linear orbit perturbation
theory that leads under certain conditions to a block-diagonal normal matrix for the gravity unknowns, significantly reducing
the required computation time. The second method makes use of numerical integration to derive the observation equations, leading
to a full set of normal equations requiring powerful computer facilities. Simulations were carried out for gravity field recovery
experiments up to spherical harmonic degree and order 80 from 10 days of observation. It was found that the first method leads
to large approximation errors as soon as the maximum degree surpasses the first resonance orders and great care has to be
taken with modeling resonance orbit perturbations, thereby loosing the block-diagonal structure. The second method proved
to be successful, provided a proper division of the data period into orbital arcs that are not too long.
Received: 28 April 2000 / Accepted: 6 November 2000 相似文献
10.
利用GOCE模拟观测反演重力场的Torus法 总被引：1，自引：1，他引：0
在介绍Torus方法反演地球重力场模型的基本原理和方法的基础上,基于圆环面上均匀分布的卫星引力梯度模拟观测值解算了200阶次的地球重力场模型,在无误差情况下,Torus方法解算模型的阶误差RMS小于10-16,验证了该方法的严密性。利用61dGOCE卫星轨道上无误差的模拟引力梯度观测值解算了200阶次的地球重力场模型,分析了格网化误差、极空白对解算精度的影响,迭代3次后,在不考虑低次系数情况下,模型的大地水准面阶误差和累积误差均较小,最大值仅为0.022mm和0.099mm。在沿轨卫星引力梯度模拟数据中加入5mE/Hz1/2的白噪声,基于Torus方法和空域最小二乘法解算了200阶次的地球重力场模型,Torus方法的精度略低于空域最小二乘法的精度,在不考虑低次项的情况下,两种方法解算模型的大地水准面阶误差最大值分别为1.58cm和1.45cm,累积误差最大值分别为6.37cm和5.55cm。但由于采用了二维快速傅里叶技术和块对角最小二乘法,极大地提高了计算效率。本文数值结果说明Torus方法是一种独立有效的方法,可用于GOCE任务海量卫星引力梯度观测值反演重力场的快速解算。 相似文献
11.
GOCE采用的高低卫-卫跟踪和卫星重力梯度测量技术在恢复重力场方面各有所长并互为补充,如何有效利用这两类观测数据最优确定地球重力场是GOCE重力场反演的关键问题。本文研究了联合高低卫-卫跟踪和卫星重力梯度数据恢复地球重力场的最小二乘谱组合法,基于球谐分析方法推导并建立了卫星轨道面扰动位T和径向重力梯度Tzz、以及扰动位T和重力梯度分量组合{Tzz-Txx-Tyy}的谱组合计算模型与误差估计公式。数值模拟结果表明,谱组合计算模型可以有效顾及各类数据的精度和频谱特性进行最优联合求解。采用61天GOCE实测数据反演的两个180阶次地球重力场模型WHU_GOCE_SC01S（扰动位和径向重力梯度数据求解）和WHU_GOCE_SC02S（扰动位和重力梯度分量组合数据求解）,结果显示后者精度优于前者,并且它们的整体精度优于GOCE时域解,而与GOCE空域解的精度接近,验证了谱组合法的可行性与有效性。 相似文献
12.
The Meissl scheme for the geodetic ellipsoid 总被引：2，自引：1，他引：1
We present a variant of the Meissl scheme to relate surface spherical harmonic coefficients of the disturbing potential of
the Earth’s gravity field on the surface of the geodetic ellipsoid to surface spherical harmonic coefficients of its first- and second-order normal derivatives on the same or any other ellipsoid.
It extends the original (spherical) Meissl scheme, which only holds for harmonic coefficients computed from geodetic data
on a sphere. In our scheme, a vector of solid spherical harmonic coefficients of one quantity is transformed into spherical
harmonic coefficients of another quantity by pre-multiplication with a transformation matrix. This matrix is diagonal for
transformations between spheres, but block-diagonal for transformations involving the ellipsoid. The computation of the transformation
matrix involves an inversion if the original coefficients are defined on the ellipsoid. This inversion can be performed accurately
and efficiently (i.e., without regularisation) for transformation among different gravity field quantities on the same ellipsoid,
due to diagonal dominance of the matrices. However, transformations from the ellipsoid to another surface can only be performed
accurately and efficiently for coefficients up to degree and order 520 due to numerical instabilities in the inversion. 相似文献
13.
The paper deals with data filtering on closed surfaces using linear and nonlinear diffusion equations. We define a surface finite-volume method to approximate numerically parabolic partial differential equations on closed surfaces, namely on a sphere, ellipsoid or the Earth’s surface. The closed surface as a computational domain is approximated by a polyhedral surface created by planar triangles and we construct a dual co-volume grid. On the co-volumes we define a weak formulation of the problem by applying Green’s theorem to the Laplace–Beltrami operator. Then the finite-volume method is applied to discretize the weak formulation. Weak forms of elliptic operators are expressed through surface gradients. In our numerical scheme we use a piece-wise linear approximation of a solution in space and the backward Euler time discretization. Furthermore, we extend a linear diffusion on surface to the regularized surface Perona–Malik model. It represents a nonlinear diffusion equation, which at the same time reduces noise and preserves main edges and other details important for a correct interpretation of the real data. We present four numerical experiments. The first one has an illustrative character showing how an additive noise is filtered out from an artificial function defined on a sphere. Other three examples deal with the real geodetic data on the Earth’s surface, namely (i) we reduce a stripping noise from the GOCE satellite only geopotential model up to degree 240, (ii) we filter noise from the real GOCE measurements (the component $T_{zz})$ , and (iii) we reduce a stripping noise from the satellite only mean dynamic topography at oceans. In all experiments we focus on a comparison of the results obtained by both the linear and nonlinear models presenting advantages of the nonlinear diffusion. 相似文献
14.
This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84(3):165–178, 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first- and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first- and second-order derivatives in terms of the geopotential coefficients. These equations are converted into recurrent relations from which the coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission. 相似文献
15.
M. Kern T. Preimesberger M. Allesch R. Pail J. Bouman R. Koop 《Journal of Geodesy》2005,78(9):509-519
The satellite missions CHAMP, GRACE, and GOCE mark the beginning of a new era in gravity field determination and modeling. They provide unique models of the global stationary gravity field and its variation in time. Due to inevitable measurement errors, sophisticated pre-processing steps have to be applied before further use of the satellite measurements. In the framework of the GOCE mission, this includes outlier detection, absolute calibration and validation of the SGG (satellite gravity gradiometry) measurements, and removal of temporal effects. In general, outliers are defined as observations that appear to be inconsistent with the remainder of the data set. One goal is to evaluate the effect of additive, innovative and bulk outliers on the estimates of the spherical harmonic coefficients. It can be shown that even a small number of undetected outliers (<0.2 of all data points) can have an adverse effect on the coefficient estimates. Consequently, concepts for the identification and removal of outliers have to be developed. Novel outlier detection algorithms are derived and statistical methods are presented that may be used for this purpose. The methods aim at high outlier identification rates as well as small failure rates. A combined algorithm, based on wavelets and a statistical method, shows best performance with an identification rate of about 99%. To further reduce the influence of undetected outliers, an outlier detection algorithm is implemented inside the gravity field solver (the Quick-Look Gravity Field Analysis tool was used). This results in spherical harmonic coefficient estimates that are of similar quality to those obtained without outliers in the input data. 相似文献
16.
First GOCE gravity field models derived by three different approaches 总被引：28，自引：10，他引：18
Roland Pail Sean Bruinsma Federica Migliaccio Christoph F?rste Helmut Goiginger Wolf-Dieter Schuh Eduard H?ck Mirko Reguzzoni Jan Martin Brockmann Oleg Abrikosov Martin Veicherts Thomas Fecher Reinhard Mayrhofer Ina Krasbutter Fernando Sans�� Carl Christian Tscherning 《Journal of Geodesy》2011,85(11):819-843
Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analysed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared with combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITG-Grace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-month GOCE gravity field models of 10?cm in terms of geoid heights, and 3?mGal in terms of gravity anomalies, globally at a resolution of 100?km (degree/order 200). 相似文献
17.
The static gravity field model DGM-1S from GRACE and GOCE data: computation, validation and an analysis of GOCE mission’s added value 总被引：4，自引：2，他引：2
We present a global static model of the Earth’s gravity field entitled DGM-1S based on GRACE and GOCE data. The collection of used data sets includes nearly 7 years of GRACE KBR data and 10 months of GOCE gravity gradient data. The KBR data are transformed with a 3-point differentiation into quantities that are approximately inter-satellite accelerations. Gravity gradients are processed in the instrumental frame. Noise is handled with a frequency-dependent data weighting. DGM-1S is complete to spherical harmonic degree 250 with a Kaula regularization being applied above degree 179. Its performance is compared with a number of other satellite-only GRACE/GOCE models by confronting them with (i) an independent model of the oceanic mean dynamic topography, and (ii) independent KBR and gravity gradient data. The tests reveal a competitive quality for DGM-1S. Importantly, we study added value of GOCE data by comparing the performance of satellite-only GRACE/GOCE models with models produced without GOCE data: either ITG-Grace2010s or EGM2008 depending on which of the two performs better in a given region. The test executed based on independent gravity gradients quantifies this added value as 25–38 % in the continental areas poorly covered with terrestrial gravimetry data (Equatorial Africa, Himalayas, and South America), 7–17 % in those with a good coverage with these data (Australia, North America, and North Eurasia), and 14 % in the oceans. This added value is shown to be almost entirely related to coefficients below degree 200. It is shown that this gain must be entirely attributed to gravity gradients acquired by the mission. The test executed based on an independent model of the mean dynamic topography suggests that problems still seem to exist in satellite-only GRACE/GOCE models over the Pacific ocean, where noticeable deviations between these models and EGM2008 are detected, too. 相似文献
18.
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
19.
A spatiospectral localization method is discussed for processing the global geopotential coefficients from satellite mission
data to investigate time-variable gravity. The time-variable mass variation signal usually appears associated with a particular
geographical area yielding inherently regional structure, while the dependence of the satellite gravity errors on a geographical
region is not so evident. The proposed localization amplifies the signal-to-noise ratio of the (non-stationary) time-variable
signals in the geopotential coefficient estimates by localizing the global coefficients to the area where the signal is expected
to be largest. The results based on localization of the global satellite gravity coefficients such as Gravity Recovery And
Climate Experiment (GRACE) and Gravity and Ocean Circulation Explorer (GOCE) indicate that the coseismic deformation caused
by great earthquakes such as the 2004 Sumatra–Andaman earthquake can be detected by the low-low tracking and the gradiometer
data within the bandwidths of spherical degrees 15–30 and 25–100, respectively. However, the detection of terrestrial water
storage variation by GOCE gradiometer is equivocal even after localization. 相似文献
20.
Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients 总被引：2，自引：2，他引：0
Four widely used algorithms for the computation of the Earth’s gravitational potential and its first-, second- and third-order
gradients are examined: the traditional increasing degree recursion in associated Legendre functions and its variant based
on the Clenshaw summation, plus the methods of Pines and Cunningham–Metris, which are free from the singularities that distinguish
the first two methods at the geographic poles. All four methods are reorganized with the lumped coefficients approach, which
in the cases of Pines and Cunningham–Metris requires a complete revision of the algorithms. The characteristics of the four
methods are studied and described, and numerical tests are performed to assess and compare their precision, accuracy, and
efficiency. In general the performance levels of all four codes exhibit large improvements over previously published versions.
From the point of view of numerical precision, away from the geographic poles Clenshaw and Legendre offer an overall better
quality. Furthermore, Pines and Cunningham–Metris are affected by an intrinsic loss of precision at the equator and suffer
from additional deterioration when the gravity gradients components are rotated into the East-North-Up topocentric reference
system.
Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. 相似文献