A quadrature formula over the sphere with application to high resolution spherical harmonic analysis

Journal of Geodesy -     

Approximation of harmonic covariance functions on the sphere by non-harmonic locally supported functions

Three methods to construct positive definite functions with compact support for the approximation of general geophysical harmonic covariance functions are presented. The theoretical background is given and simulations carried out, for three types of covariance functions associated with the determination of the anomalous gravity potential from gravity anomalies. The results are compared with those of the finite covariance function of Sansò and Schuh (1987). Received: 24 September 1998 / Accepted: 25 June 1999     

Global optimization of the Gauss conformal mappings of an ellipsoid to a sphere

The Gauss conformal mappings (GCMs) of an oblate ellipsoid of revolution to a sphere are those that transform the meridians into meridians, and the parallels into parallels of the sphere. The infinitesimal-scale function associated with these mappings depends on the geodetic latitude and contains three parameters, including the radius of the sphere. Gauss derived these constants by imposing local optimum conditions on certain parallel. We deal with the problem of finding the constants to minimize the Chebyshev or maximum norm of the logarithm of the infinitesimal-scale function on a given ellipsoidal segment (the region contained between two parallels). We show how to solve this minimax problem using the intrinsic function fminsearch of Matlab. For a particular ellipsoidal segment, we get the solution and show the alternation property characteristic of best Chebyshev approximations. For a pair of points relatively close in the ellipsoid at different latitudes, the best minimax GCM on the segment defined by these points is used to approximate the geodesic distance between them by the spherical distance between their projections on the corresponding sphere. This approach, combined with the best locally GCM if the points are on the same parallel, is illustrated by applying it to some case studies but specially to a 10° × 10° region contained between portions of two parallels and two meridians. In this case, the maximum absolute error of this spherical approximation is equal to 2.9 mm occurring at a distance about 1,360 km. This error decreases up to 0.94 mm on an 8° × 8° region of this type. So, the spherical approximation to the solution of the inverse geodesic problem by best GCM can be acceptable in many practical geodetic activities.     

Predictor stabilisation of reduction formulae for observations of heavily-damped simple harmonic motions

The paper summarises attempts which have been made to derive general, simple and stable formulae of adequate precision for the reduction of observations to the extremal positions of oscillations which are simple harmonic or near-simple harmonic. A fallacy in the thirty-five year old Schuler Mean is pointed out and general formulae are offered for the reduction of observations for any constant damping, within the range0<f≤1.     

New method for reduction of astrolabe observations using rectangular coordinates on the celestial sphere

A new reduction method usingx, y, z-coordinates is derived for astrolabe observations. By this method, the latitude and the longitude of the station are computed without the need of a priori knowledge of the station position. This method is a significant development in data reduction of astrolabe and other almucantar observations due to its mathematical exactness, simplicity, and the ease of handling the associated statistics.     

Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere

In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the \(4 \pi \) fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as \(2^{30}\,{\approx }\,10^9\). The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.     

A spatial error model with continuous random effects and an application to growth convergence

We propose a spatial error model with continuous random effects based on Matérn covariance functions and apply this model for the analysis of income convergence processes (\(\beta \)-convergence). The use of a model with continuous random effects permits a clearer visualization and interpretation of the spatial dependency patterns, avoids the problems of defining neighborhoods in spatial econometrics models, and allows projecting the spatial effects for every possible location in the continuous space, circumventing the existing aggregations in discrete lattice representations. We apply this model approach to analyze the economic growth of Brazilian municipalities between 1991 and 2010 using unconditional and conditional formulations and a spatiotemporal model of convergence. The results indicate that the estimated spatial random effects are consistent with the existence of income convergence clubs for Brazilian municipalities in this period.     

The harmonic representation of the disturbing potential

Summary The alternative harmonic representations of the disturbing potential, correct to the order of the flattening, are examined and an example is given where the incorrect use of a spherical harmonic expansion can give rise to fallacious results. The correct usage of the spherical harmonic expansion for the disturbing potential is given in the solution of the general surface integral to define the indirect effect in the case of the non-regularised geoid.     

Instability in spatial error models: an application to the hypothesis of convergence in the European case

This paper focuses on the hypothesis of stability in the mechanisms of spatial dependence that are usually employed in spatial econometric models. We propose a specification strategy for which the first step is to solve a local estimation algorithm, called the Zoom estimation. The aim of this stage is to detect problems of heterogeneity in the parameters and to identify the regimes. Then we resort to a battery of formal Lagrange Multipliers to test the assumption of stability in the processes of spatial dependence. The alternative hypothesis consists of the existence of several regimes in these parameters. A small Monte Carlo serves to confirm the behaviour of this strategy in a context of finite size samples. As an illustration, we solve an application to the case of the hypothesis of convergence for the per capita income in the European regions. Our results reveal the existence of a strong Centre-Periphery dichotomy in which instability extends to all the elements (coefficients of regression as well as parameters of spatial dependence) that intervene in a classical conditional β-convergence model.     

点云数据直接缩减方法及缩减效果研究

分析了国外点云数据处理中数据缩减方法的研究现状,针对实际工程中三维激光扫描数据采集的过密情况,提出利用扫描间隔进行点云数据直接缩减的方法,并编程实现所提出的算法。通过对真实的三维激光扫描点云数据的缩减处理,建立点云数据直接缩减的预测模型。     

On a self-consistent representation of earth models,with an application to the computing of internal flattening

Most realistic Earth models published as yet have been given in tabulated form, with the noticeable exception of three simple parametric Earth models derived by Dziewonski et al. (1975). Simple interpolation in these tables may lead to inconsistencies, when we consider certain effects which depend crucially on detailed density structure. We establish algorithmic formulae, which may be used to compute all the mechanical properties of a model in an entirely consistent way, once the density as well asP— andS— wave velocities are known. We then use this formulation to integrate Clairaut’s equation in a very efficient way, and thus obtain the hydrostatic flattening to the first order in smallness at any point inside the model. For most geodynamic purposes, we may suffice with this approximation. Finally, we show the results of some calculations of hydrostatic flattening to the first and second order, using an iterative technique of solving the integral figure equations, for an Earth model consistent with all geophysical data available at present. We find that the hydrostatic flattening at the surface should be about 1/298.8, instead of 1/296.961 as quoted by Nakiboglu (1979) for essentially the same model. Moreover, from our results, we estimate the actual flattening of the coremantle boundary to be about 1/390.3.     

Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies

The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30?km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength).     

An application of a summation formula to numerical computation of integrals over the sphere

Summary This paper discusses a method to approximate an integral over the (unit) sphere by a linear combination of the values of the integrand at given points. The main concept of this method is the observation, that on the sphere for each sufficiently smooth function the integral can be expressed by a summation formula. A method is given for optimizing the accuracy of the computation for a given set of sample points for the function being integrated.     

The reduction correction in North America

An inverse Poisson integral technique has been used to determine a gravity field on the geoid which, when continued by analytic free space methods to the topographic surface, agrees with the observed field. The computation is performed in three stages, each stage refining the previous solution using data at progressively increasing resolution (1^o×1^o, 5′×5′, 5/8′×5/8′) from a decreasing area of integration. Reduction corrections are computed at 5/8′×5/8′ granularity by differencing the geoidal and surface values, smoothed by low-pass filtering and sub-sampled at 5′ intervals. This paper discusses 1^o×1^o averages of the reduction corrections thus obtained for 172 1^o×1^o squares in western North America. The 1^o×1^o mean reduction corrections are predominantly positive, varying from −3 to +15mgal, with values in excess of 5mgal for 26 squares. Their mean andrms values are +2.4 and 3.6mgal respectively and they correlate well with the mean terrain corrections as predicted byPellinen in 1962. The mean andrms contributions from the three stages of computation are: 1^o×1^o stage +0.15 and 0.7mgal; 5′×5′ stage +1.0 and 1.6mgal; and 5/8′×5/8′ stage +1.3 and 1.8mgal. These results reflect a tendency for the contributions to become larger and more systematically positive as the wavelengths involved become shorter. The results are discussed in terms of two mechanisms; the first is a tendency for the absolute values of both positive and negative anomalies to become larger when continued downwards and, the second, a non-linear rectification, due to the correlation between gravity anomaly and topographic height, which results in the values continued to a level surface being systematically more positive than those on the topography.     

The convergence problem of collocation solutions in the framework of the stochastic interpretation

The problem of the convergence of the collocation solution to the true gravity field was defined long ago (Tscherning in Boll Geod Sci Affini 39:221–252, 1978) and some results were derived, in particular by Krarup (Boll Geod Sci Affini 40:225–240, 1981). The problem is taken up again in the context of the stochastic interpretation of collocation theory and some new results are derived, showing that, when the potential T can be really continued down to a Bjerhammar sphere, we have a quite general convergence property in the noiseless case. When noise is present in data, still reasonable convergence results hold true.

“Democrito che ’l mondo a caso pone” “Democritus who made the world stochastic” Dante Alighieri, La Divina Commedia, Inferno, IV – 136     

Formation of surface spherical harmonic normal matrices and application to high-degree geopotential modeling

 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     

基于球谐函数区域电离层模型建立

利用GPS双频观测数据建立高精度、准实时的区域电离层总电子含量(TEC)模型是电离层研究的一个重要手段。文中探讨IGS观测站数据结合4阶球谐函数建立区域电离层格网模型的方法,并对硬件延迟(DCB)和TEC建模结果的可靠性进行分析,结果表明,DCB解算精度在0.4ns以内,TEC内外精度优于1.4TECU(1TECU=1016电子数/m2)和1.5TECU,满足导航定位中电离层改正的需要。