A new integral approach to geopotential coefficient determinations from terrestrial gravity or satellite altimetry data

The spherical harmonic coefficients of the Earth’s gravitational potential are conveniently determined by integration of gravity data or potential data (derived from satellite altimetry) over a sphere. The major problem of such a method is that the data, given on the non-spherical surface of the Earth, must be reduced to the sphere. A new integral formula over the surface of the Earth is derived. With this formula improved first order topographic corrections to the spherical formulas are obtained.     

由SLR观测的地心周期性运动(1993-2006年)

地球质心(CM)是整个地球的质量中心,地心运动是由地球系统的质量重新分布激励的,特别是流体圈层.利用SLR对LAGEOS 1/2卫星的距离观测,解算1993-2006年期间的地心运动时间序列,然后分别利用小波变换和最小二乘法分析该序列,发现地心运动存在长期和周期性变化.地心的长期运动表明地壳形状在变化.季节性变化是地心运动的主项,主要是由地球流体圈层的质量分布造成的,如海洋、大气和陆地水等.地心运动还存在其他周期性和准周期性变化.地心运动存在2～5年的长周期变化.许多周期都存在渐变,这表明整个地球系统质量和环境存在不规则的变化.

Abstract：

The center of mass of Earth (CM) is the center of total Earth's mass. The geocenter motion may be excit-ed by the mass redistribution of Earth system, especially the fluid layer. A time series of geocenter motion meas-ured with SLR on LAGEOS 1/2 is estimated and then analyzed with the wavelet transformation and the least squaresmethod, respectively. The secular and periodic variations of geocenter motion are detected. The long-term move-ment indicates that the crustal figure is changing, the north hemisphere and 180-degree hemisphere are shrinking, and the south hemisphere and O-degree hemisphere are swelling. The seasonal variations are the main componentswhich may be caused from the mass distribution of Earth fluid layer, e.g. ocean, atmosphere and continental wa-ter, There are many other periodic or quasi-periodic variations. There are long periodic variations through 2 to 5 years, Many periods gradually change, which indicates that there exist nonregular fluxes for the environment and mass of the whole Earth system.     

Estimating atmospheric pressure loading regression coefficients from GPS observations

The loading exerted by atmospheric pressure on the surface of the Earth causes deformations, mainly in vertical direction. Consequently, these deformations are also subject to pressure variations. At present this effect is only modeled by a few research groups in the post-processing of very long baseline interferometry (VLBI) and global positioning system (GPS) observations. As the displacements may clearly exceed the accuracy goals, we implement vertical pressure loading regression coefficients as a new estimable parameter type in the Bernese GPS software. This development is applied to a network of 60 European permanent GPS stations extending from 35 to 79° northern latitude. The analysis comprises 1,055 days of observations between January 2001 and February 2004. During that period pressure variations as large as 80 hPa occurred at high latitude sites. A least squares solution including all observations and all relevant parameters yields significant regression coefficients for all stations but reveals also some critical issues with regard to the capability of this geodetic approach to verify results based on the geophysical convolution method.An erratum to this article can be found at     

The development and analysis of quasi-linear map projections

Map projections are an essential component of coordinate systems used in applications such as surveying, topographic mapping, and engineering, and care needs to be taken to select ones that minimize distortion for each case. This article explores the selection process for near-linear features on the surface of the Earth and derives limits for the extent of a project that can be projected within specified distortion tolerances. It is then demonstrated that a multifaceted set of projections of the Earth may be used to extend this concept to the mapping of features such as highways and railways that are quasi-linear but do not exactly follow a standard geometrical line (a great circle or a small circle) on the surface of the Earth. A continuous, conformal coordinate system may be derived in such situations, extending to indefinite length and applicable over a swath of several kilometers width, but it cannot be extended to cover situations with extensive variations in height. Instead, the Snake Projection is analyzed, and it is shown that this can be used to develop continuous (non-zonal) projected coordinate systems for major engineering projects extending for hundreds of kilometers and having extensive height ranges. Examples are shown of the application to railway projects.     

DORIS and the Determination of the Earth’s Polar Motion

DORIS (Détermination d’Orbite et Radiopositionnement Intégrés par Satellite) is a system used for precise orbit determination (POD) and ground-station positioning. It has been implemented on-board various satellites: the SPOT (Système pour l’Observation de la Terre) remote sensing satellites SPOT-2, SPOT-3, SPOT-4, SPOT-5, TOPEX/Poseidon and more recently on its successors Jason-1 and ENVISAT. DORIS is also a terrestrial positioning system that has found many applications in geophysics and geodesy; in particular, it contributes to the realization of the International Terrestrial Reference Frame, ITRF2000 and the forthcoming ITRF2005. Although not its primary objective, DORIS can bring information on Earth orientation monitoring, mainly polar motion and length of day (LOD) variations that complement other astrogeodetic techniques. In this paper, we have analyzed various recent polar motion solutions derived from independent analysis centers using different software packages and applying various analysis strategies. Comparisons of these solutions to the International Earth Rotation and Reference Systems Service (IERS) C04 solution are performed. Depending on the solutions, the accuracy of DORIS polar components are in the range of 0.5–1 mas corresponding to a few centimeters on the Earth’s surface. This is approximately ten times larger than results derived from GPS, which are typically 0.06 mas in both components. This does not allow DORIS results to be taken into account in the IERS–EOP combinations. A gain in the precision could come from technical improvements to the DORIS system, in addition to improvement of the orbit, tropospheric, ionospheric and Earth gravity field modeling.     

Assessment of observing time-variable gravity from GOCE GPS and accelerometer observations

An assessment has been made of the possibility to estimate time-variable gravity from GPS-derived orbit perturbations and common-mode accelerometer observations of ESA’s GOCE Earth Explorer. A number of 20-day time series of Earth’s global long-wavelength gravity field have been derived for the period November 2009 to November 2012 using different parameter setups and estimation techniques. These techniques include a conventional approach where for each period, one set of gravity coefficients is estimated, either excluding or including empirical accelerations, and the so-called Wiese approach where higher frequency coefficients are estimated for the very long wavelengths. A principal component analysis of especially the time series of gravity field coefficients obtained by the Wiese approach and the conventional approach with empirical accelerations reveals an annual signal. When fitting this annual signal directly through the time series, the sine component (maximum in spring) displays features that are similar to well-known continental hydrological mass changes for the low latitude areas, such as mass variations in the Amazon basin, Africa and Australia for spatial scales down to 1,500 km. The cosine component (maximum in winter), however, displays large signals that can not be attributed to actual mass variations in the Earth system. The estimated gravity field changes from GOCE orbit perturbations are likely affected by missing GPS observations in case of high ionospheric perturbations during periods of increased solar activity, which is minimal in Summer and maximal towards the end of autumn.     

The gravity field variation caused by inner core super rotation

Due to the super rotation of the Earth’s inner core, the tilted figure axis of the inner core would progress with respect to the mantle and thus cause the variation of the Earth’s external gravity field. This paper improves the present model of the gravity field variation caused by the inner core super rotation. Under the assumption that the inner core is a stratifying ellipsoid whose density function is fitted out from PREM and the super rotation rate is 0.27～0.53°/yr, calculations show that the global temporal variations on the Earth’s surface have a maximum value of about 0.79～1.54×10?3 μGal and a global average intensity of about 0.45～0.89×10?3 μGal in the whole year of 2007, which is beyond the accuracy of the present gravimetry and even the super conducting gravimeter data. However, both the gravity variations at Beijing and Wuhan vary like sine variables with maximal variations around 0.33 μGal and 0.29 μGal, respectively, in one cycle. Thus, continuous gravity measurements for one or two decades might be able to detect the differential motion of the inner core.     

Moon-based Earth observation: scientific concept and potential applications

Although Earth’s surface parameters obtained from satellite data have become more and more precise, it is still difficult to guarantee temporal consistency and spatial continuity for large-scale geoscience phenomena. Developing new Earth observation platforms is a feasible way to improve the consistency and continuity of such data. As the planet’s only natural satellite, the Moon has special advantages as a platform for observing Earth, including long lifetime, whole disk view, tectonic stability and unique perspective. After presenting the observation geometry constructed by using the ephemeris, this paper mainly discusses the characteristics of a lunar platform and the proper Moon-based sensors, as well as the scientific objectives of Moon-based Earth observation. Solid Earth dynamics, the energy budget of Earth, Earth’s environmental elements and the Earth-space environment are four potential applications analysed in this paper.     

Using NHDPlus as the Land Base for the Noah-distributed Model

The National Elevation, Hydrography and Land Cover datasets of the United States have been synthesized into a geospatial dataset called NHDPlus which is referenced to a spheroidal Earth, provides geospatial data layers for topography on 30 m rasters, and has vector coverages for catchments and river reaches. In this article, we examine the integration of NHDPlus with the Noah-distributed model. In order to retain compatibility with atmospheric models, Noah-distributed utilizes surface domain fields referenced to a spherical rather than spheroidal Earth in its computation of vertical land surface/atmosphere water and energy budgets (at coarse resolution) as well as horizontal cell-to-cell water routing across the land surface and through the shallow subsurface (at fine resolution). Two data-centric issues affecting the linkage between Noah-distributed and NHDPlus are examined: (1) the shape of the Earth; and (2) the linking of gridded landscape with a vector representation of the stream and river network. At mid-latitudes the errors due to projections between spherical and spheroidal representations of the Earth are significant. A catchment-based "pour point" technique is developed to link the raster and vector data to provide lateral inflow from the landscape to a one-dimensional river model. We conclude that, when Noah-distributed is run uncoupled to an atmospheric model, it is advantageous to implement Noah-distributed at the native spatial scale of the digital elevation data and the spheroidal Earth of the NHDPlus dataset rather than transforming the NHDPlus dataset to fit the coarser resolution and spherical Earth shape of the Noah-distributed model.     

地球椭球向径和平均曲率半径的积分表达式

引入地球向径积分平均值和地球平均曲率半径积分平均值的概念,借助计算机代数系统推导出了两者的符号表达式,并将它们表示为偏心率e的幂级数形式。将地球向径积分平均值和地球平均曲率半径积分平均值分别与平均球半径、等面积球半径、等距离球半径、等体积球半径这4种常用球体半径进行比较,研究表明地球向径积分平均值与4种常用球体半径间的差异更小。由于地球是一个旋转椭球体,向径与曲率半径是背离的,向径最大时,曲率半径最小, 向径最小时,曲率半径最大,传统思维所认为的曲率半径并不能准确地代表地球半径平均值,因此在一定程度上,地球向径的积分平均值更能代表地球半径平均值。这些研究结果可为地球科学、空间科学、导航定位提供基础理论依据。     

Four dimensional studies in Earth space

Summary A system of reference which is directly related to observations, is proposed for four dimensional studies in Earth space. The requisite data is used to define both global control network and also polar wandering. The determination of variations of the Earth’s gravitational field with time also forms part of such a system. Techniques are outlined for the unique definition of the motion of the geocenter, and the changes in the location of the axis of rotation of an instantaneous Earth model, in relation to values at some epoch of reference. The instantaneous system referred to is directly related to a fundamental equation in geodynamics. The reference system defined would provide an unambiguous frame for long period studies in Earth space, provided the scale of the space were specified. Presented at the I.A.G./A.G.U. Symposium on Earth Gravity Models related problems, St Louis, Missouri, U.S.A., 16–18 August 1972.     

Atmospheric Angular Momentum Time-Series: Characterization of their Internal Noise and Creation of a Combined Series

The atmosphere induces variations in Earth rotation. These effects are classically computed using the “angular momentum approach”. In this method, the variations in Earth rotation are estimated from the variations in the atmospheric angular momentum (AAM). Several AAM time-series are available from different meteorological centers. However, the estimation of atmospheric effects on Earth rotation differs when using one atmospheric model or the other. The purpose of this work is to build an objective criterion that justifies the use of one series in particular. Because the atmosphere is not the only cause of Earth rotation variations, this criterion cannot rely only on a comparison of AAM series with geodetic data. Instead, we determine the quality of each series by making an estimation of their noise level, using a generalized formulation of the “three-cornered hat method”. We show the existence of a link between the noise of the AAM series and their correlation with geodetic data: a noisy series is usually less correlated with Earth orientation data. As the quality of the series varies in time, we construct a combined AAM series, using time-dependent weights chosen so that the noise level of the combined series is minimal. To determine the influence of a minimal noise level on the correlation with geodetic data, we compute the correlation between the combined series and Earth orientation data. We note that the combined series is always amongst the best correlated series, which confirms the link established before. The quality criterion, while totally independent of Earth orientation observations, appears to be physically convincing when atmospheric and geodetic data are compared     

Self-consistent treatment of tidal variations in the geocenter for precise orbit determination

We show that the current levels of accuracy being achieved for the precise orbit determination (POD) of low-Earth orbiters demonstrate the need for the self-consistent treatment of tidal variations in the geocenter. Our study uses as an example the POD of the OSTM/Jason-2 satellite altimeter mission based upon Global Positioning System (GPS) tracking data. Current GPS-based POD solutions are demonstrating root-mean-square (RMS) radial orbit accuracy and precision of \({<}1\)  cm and 1 mm, respectively. Meanwhile, we show that the RMS of three-dimensional tidal geocenter variations is \({<}6\)  mm, but can be as large as 15 mm, with the largest component along the Earth’s spin axis. Our results demonstrate that GPS-based POD of Earth orbiters is best performed using GPS satellite orbit positions that are defined in a reference frame whose origin is at the center of mass of the entire Earth system, including the ocean tides. Errors in the GPS-based POD solutions for OSTM/Jason-2 of \({<}4\)  mm (3D RMS) and \({<}2\)  mm (radial RMS) are introduced when tidal geocenter variations are not treated consistently. Nevertheless, inconsistent treatment is measurable in the OSTM/Jason-2 POD solutions and manifests through degraded post-fit tracking data residuals, orbit precision, and relative orbit accuracy. For the latter metric, sea surface height crossover variance is higher by \(6~\hbox {mm}^{2}\) when tidal geocenter variations are treated inconsistently.     

Measuring Earth orientation with the global positioning system

Summary A globally distributed network of high-precision receivers which obtain data from the full Global Positioning System (GPS) configuration of 18 or more satellites may soon become an efficient and economical method for the rapid determination of short-term variations in Earth orientation. A covariance analysis has been performed to evaluate the errors associated with GPS monitoring of Earth orientation. Earth orientation parameters were modeled either as constants over observing windows of various lengths, or as stochastic process-noise variables. The sensitivity of Earth orientation estimates to systematic errors in selected model parameters was also examined. GPS measurements appear to be highly competitive with those from other techniques, and have the potential to generate nearly continuous centimeter-level Earth orientation information to aid both spacecraft navigation and the study of high-frequency Earth orientation-related processes.     

椭球上的测地变换和Voronoi图的生成——地理空间度量

地理空间的度量不宜再采用借助地图投影转换到欧氏空间内的近似解决途径,在定义了大地线尺度作地理空间度量的基础上,采用地图代数完成了全形态（点、线、面）图形在地球尺度空间的精密距离变换,并生成了相应的Voronoi图,为地球上相应于距离的各实际问题的解决提供了严密、合理的解决途径。     

Stokes和Hotine积分离散求和的快速算法

提出了一种用于Stokes积分和Hotine积分直接离散求和的快速算法。该算法将积分核表达为计算点纬度、流动点纬度和两点间经度差的函数,充分利用核函数的对称性,相同纬度的所有计算点只需计算一组核函数,计算次数远少于普通离散求和。基于EGM2008地球重力位模型的模拟实验表明,快速算法的计算效率远高于普通算法,有效解决了离散求和计算速度太慢的数值问题,且保留了球面积分的特性,可取代一维FFT用于计算Stokes积分和Hotine积分。     

Integral formulas for geopotential coefficient determination from gravity anomalies versus gravity disturbances

Integral formulas are derived for the determination of geopotential coefficients from gravity anomalies and gravity disturbances over the surface of the Earth. First order topographic corrections to spherical formulas are presented. In addition new integral formulas are derived for the determination of the external gravity field from surface gravity. Taking advantage of modern satellite positioning techniques, it is suggested that, in general, the external gravity field as well as individual coefficients are better determined from gravity disturbances than from gravity anomalies.     

日长和地球引力场参数J_2的长期和长周期变化

用高精度的现代日长观测结果、近 2 0年的J2 变化的结果和古地球自转结果 ,讨论了日长和J2 变化的长期和长周期的主要特性 ,从理论和实测两方面分析了日长和J2 变化的关系及其异同 ,认为J2 变化的结果在地球动力学研究中的作用应予以重视 ,综合利用日长和J2 将使这两种资料的潜力得以进一步发挥。     

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.     

Numerical solution of the linearized fixed gravimetric boundary-value problem

The fixed gravimetric boundary-value problem (FGBVP) represents an exterior oblique derivative problem for the Laplace equation. Terrestrial gravimetric measurements located by precise satellite positioning yield oblique derivative boundary conditions in the form of surface gravity disturbances. In this paper, we discuss the boundary element method (BEM) applied to the linearized FGBVP. In spite of previous BEM approaches in geodesy, we use the so-called direct BEM formulation, where a weak formulation is derived through the method of weighted residuals. The collocation technique with linear basis functions is applied for deriving the linear system of equations from the arising boundary integral equations. The nonstationary iterative biconjugate gradient stabilized method is used to solve the large-scale linear system of equations. The standard MPI (message passing interface) subroutines are implemented in order to perform parallel computations. The proposed approach gives a numerical solution at collocation points directly on the Earth’s surface (on a fixed boundary). Numerical experiments deal with (i) global gravity field modelling using synthetic data (surface gravity disturbances generated from a global geopotential model (GGM)) (ii) local gravity field modelling in Slovakia using observed gravity data. In order to extend computations, the memory requirements are reduced using elimination of the far-zone effects by incorporating GGM or a coarse global numerical solution obtained by BEM. Statistical characteristics of residuals between numerical solutions and GGM confirm the reliability of the approach and indicate accuracy of numerical solutions for the global models. A local refinement in Slovakia results in a local (national) quasigeoid model, which when compared with GPS-levelling data, does not make a large improvement on existing remove-restore-based models.