A Critical Analysis of a Recent Test of the Lense–Thirring Effect with the LAGEOS Satellites

In this paper, we quantitatively discuss the impact of the current uncertainties in the even zonal harmonic coefficients J _l of the Newtonian part of the terrestrial gravitational potential on the measurement of the general relativistic Lense–Thirring effect. We use a suitable linear combination of the nodes Ω of the laser-ranged LAGEOS and LAGEOS-II satellites. The one-sigma systematic error due to mismodelling of the J _l coefficients ranges from ~ 4% for the EIGEN−GRACE02S gravity field model to ~ 9% for the GGM02S model. Another important source of systematic error of gravitational origin is represented by the secular variations j _l of the even zonal harmonics. While the relativistic and J _l signals are linear in time, the shift due to j _l is quadratic. We quantitatively assess their impact on the measurement of the Lense–Thirring effect with numerical simulations obtaining a 10−20% one-sigma total error over 11 years for EIGEN-GRACE02S. Ciufolini and Pavlis (Nature 431:958–960, 2004) claim a total error of 5% at the one-sigma level.     

地球引力位球谐级数展开的收敛性研究(英文)

Given a continuous boundary value on the boundary of a "simply closed surface"S that encloses the whole Earth, a regular harmonic fictitious field V*(P) in the domain outside an inner sphere K i that lies inside the Earth could be determined, and it is proved that V*(P) coincides with the Earth’s real field V(P) in the whole domain outside the Earth. Since in the domain outside the inner sphere Ki and the fictitious regular harmonic function V*(P) could be expressed as a uniformly convergent spherical harm...     

On Definition of Geograghic Variable Scaling

IntroductionThe map is a basic form of geographic informationvisualization[1]. To provide space attributes or geo-graphic orders is the basic function of a map. Incartography, according to the different measure ofphenomenal quantitative attribute, four fo…     

Global Ionosphere Maps of VTEC from GNSS,satellite altimetry,and formosat-3/COSMIC data

For space geodetic techniques, operating in microwave band, ionosphere is a dispersive medium; thus signals traveling through this medium are in the first approximation affected proportional to inverse of the square of their frequencies. This effect allows gaining information about the parameters of the ionosphere in terms of Total Electron Content (TEC) or the electron density (N _e ). TEC or electron density can then be expressed by means of spherical harmonic base functions to provide a Global Ionosphere Map (GIM). The classical input data for development of GIMs are obtained from dual-frequency observations carried out at Global Navigation Satellite Systems (GNSS) stations. However, GNSS stations are in-homogeneously distributed around the world, with large gaps particularly over the oceans; this fact reduces the precision of the GIM over these areas. On the other hand, dual-frequency satellite altimetry missions such as Jason-1 provide information about the ionosphere precisely above the oceans; and furthermore Low Earth Orbiting (LEO) satellites, such as Formosat-3/COSMIC (F/C) provide well-distributed information of ionosphere globally. This study investigates on global modeling of TEC through combining GNSS and satellite altimetry data with global TEC data derived from the occultation measurements of the F/C mission. The combined GIMs of vertical TEC (VTEC) show a maximum difference of 1.3–1.7 TEC units (TECU) with respect to the GNSS-only GIMs in the whole day. The root mean square error (RMS) maps of combined solution show a reduction of about 0.1 TECU in the whole day. This decrease of RMS can reach up to 0.5 TECU in areas where no or few GNSS observations are available, but high number of F/C measurement is carried out. This proves that the combined GIMs provide a more homogeneous global coverage and higher reliability than results of each single method. All comparisons and validations made within this study provide vital information regarding combination and integration of various observation techniques in the Global Geodetic Observing System of the International Association of Geodesy.     

On the approximation of external gravitational potential with closed systems of (trial) functions

Summary Let S be the (regular) boundary-surface of an exterior regionE _e in Euclidean space ℜ³ (for instance: sphere, ellipsoid, geoid, earth's surface). Denote by {φ_n} a countable, linearly independent system of trial functions (e.g., solid spherical harmonics or certain singularity functions) which are harmonic in some domain containingE _e ∪ S. It is the purpose of this paper to show that the restrictions {ϕ_n} of the functions {φ_n} onS form a closed system in the spaceC (S), i.e. any functionf, defined and continuous onS, can be approximated uniformly by a linear combination of the functions ϕ_n. Consequences of this result are versions of Runge and Keldysh-Lavrentiev theorems adapted to the chosen system {φ_n} and the mathematical justification of the use of trial functions in numerical (especially: collocational) procedures.     

Evolution of the Earth's principal axes and moments of inertia: the canonical form of solution

 Harmonic coefficients of the 2nd degree are separated into the invariant quantitative (the 2nd-degree variance) and the qualitative (the standardized harmonic coefficients) characteristics of the behavior of the potential V ₂(t). On this basis the evolution of the Earth's dynamical figure is described as a solution of the time-dependent eigenvalues–eigenvectors problem in the canonical form. Such a canonical quadratic form is defined only by temporal variations of the harmonic coefficients and always remains finite, even within an infinite time interval. An additional condition for the correction or the determination of temporal variations of the 2nd degree is obtained. Temporal variations of the fully normalized sectorial harmonic coefficients are estimated in addition to ˙Cˉ ₂₀, ˙Cˉ ₂₁, and ˙Sˉ ₂₁ of the EGM96 gravity model. In addition, a non-linear hyperbolic model for Cˉ _2m (t), Sˉ _2m (t) is constructed. The trigonometric form of the hyperbolic model leads to the consideration of the potential V ₂(ψ) instead of V ₂(t) within the closed interval −π/2≤ψ≤+π/2. Thus, it is possible to evaluate the global trend of V ₂(t), the Earth's principal axes and the differences of the moments of inertia within the whole infinite time interval. Received: 25 September 1998 / Accepted: 28 June 2000     

A new global TEC model for estimating transionospheric radio wave propagation errors

Space-based navigation and radar systems operating at single frequencies of <10 GHz require ionospheric corrections of the signal delay or range error. Because this ionospheric propagation error is proportional to the total electron content of the ionosphere along the ray path, a user friendly TEC model covering global scale and all levels of solar activity should be helpful in various applications. Since such a model is not available yet, we present an empirical model approach that allows determining global TEC very easily. Although the number of model coefficients and parameters is rather small, the model describes main ionospheric features with good quality. Presented is the empirical approach describing dependencies on local time, geographic/geomagnetic location and solar irradiance and activity. The non-linear approach needs only 12 coefficients and a few empirically fixed parameters for describing the broad spectrum of TEC variation at all levels of solar activity. The model approach is applied on high-quality global TEC data derived by the Center for Orbit Determination in Europe (CODE) at the University of Berne over more than half a solar cycle (1998–2007). The model fits to these input data with a negative bias of 0.3 TECU and a RMS deviation of 7.5 TECU. As other empirical models too, the proposed Global Neustrelitz TEC Model NTCM-GLis climatological, i.e. the model describes the average behaviour under quiet geomagnetic conditions. During severe space weather events the actual TEC data may deviate from the model values considerably by more than 100%. A preliminary comparison with independent data sets as TOPEX/Poseidon altimeter data reveals similar results for NeQuick and NTCM-GL with RMS deviations in the order of 5 and 11 TECU (1 TECU = 10¹⁶ electrons/m²) for low and high-solar activity conditions, respectively. The more extended data base of ionosphere information that accumulates in the coming years will help in further improving the set of coefficients of the model.     

Spherical cap harmonic model for mapping and predicting regional TEC

An approach to modeling the regional ionospheric total electron content (TEC) based on spherical cap harmonic analysis is presented. This approach not only provides a better regional TEC mapping accuracy, but also the capability for ionospheric model prediction based on spectrum analysis and least squares collocation. Unlike conventional approaches, which predict the immediate TEC with models using current observations, the spherical cap harmonic approach utilizes models using past observations to predict a model which will provide future TEC values. A significant advantage in comparison with conventional approaches is that the spherical cap harmonic approach can be used to predict the long-term TEC with reasonable accuracy. This study processes a set of GPS data with an observation time span of 1 year from two GPS networks in China. The TEC mapping accuracy of the spherical cap harmonic model is compared with the polynomial model and the global ionosphere model from IGS. The results show that the spherical cap harmonic model has a better TEC mapping accuracy with smoother residual distributions in both temporal and spatial domains. The TEC prediction with the spherical cap harmonic model has been investigated for both short- and long-term intervals. For the short-term interval, the prediction accuracies for the latencies of 1-day, 2-days, and 3-days are 2.5 TECU, 3.5 TECU, and 4.5 TECU, respectively. For the long-term interval, the prediction accuracy is 4.5 TECU for a 2-month latency.     

Le projet gradio et la determination a haute resolution du geopotentiel

Résumé Une des techniques de détermination fine et globale du champ de gravitation terrestre U est la gradiométrie spatiale, dans laquelle on mesure à bord d'un satellite sur orbite basse certaines combinations linéaires des composantes du tenseur ∂² U/∂x_i ∂x_j dans des axes {x _i } liés au satellite. Un tel projet, appelé GRADIO, est actuellement à l'étude en France et pourrait aboutir à partir de 1990. Après avoir rappelé les objectifs scientifiques d'une telle mission, nous en donnons les spécifications—étayées par une série d'études analytiques; nous définissons ensuite le satellite porteur et ses caractères techniques, en insistant sur les points délicats de la faisabilité (facteurs d'échelle des micro-accéléromètres constituant l'appareil, connaissance de l'attitude...) et en présentant des idées de solution en cours d'approfondissement.

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Summary Satellite gradiometry arises as one of the methods for improving our knowledge of the global Earth gravity field at high resolution: by means of micro-accelerometers on board a low orbiting spacecraft, linear combination of the gravity tensor components ∂² U/∂x_i ∂x_j are measured in a satellite-fixed reference frame {x _i }. Based on this technique, a project named GRADIO is presently under study in France and could fly in 1990 at the earliest. After the scientific objectives of that experiment have been reviewed, the measurement specifications are given as coming from various analytical studies. The platform and its characteristics are then defined: the critical realization problems (scale factors of the micro-accelerometers, spacecraft attitude control and restitution) are pointed out together with some ideas for their solution which are under analysis and require further study.

     

One centimeter-level observations of diurnal ocean tides from global monthly mean time-variable gravity fields

A method of analyzing GRACE satellite-to-satellite ranging data is presented which accentuates signals from diurnal ocean tides and dampens signals from long-period non-tidal phenomena. We form a time series of differences between two independent monthly mean gravity solutions, one set computed from range-rate data along strictly ascending arcs and the other set computed from data along descending arcs. The solar and lunisolar diurnal tides having alias periods longer than a few months, such as K ₁, P ₁, and S ₁, present noticeable variations in the monthly ascending and descending ‘difference’ solutions, while the climate-related signals are largely cancelled. By computing tidal arguments evaluated along the actual GRACE orbits, we decompose and estimate residual tidal signals with respect to our adopted prior model GOT4.7. The adjustment in the tidal height is small yet significant, yielding maximum amplitudes of 4 cm mostly under the Antarctic ice shelves and ~1 cm in general at spatial scales of several hundred kilometer. Moreover, the results suggest there are possible 1-cm errors in the tide model even over oceans well-covered by decades of radar altimetry missions. Independent validation of such small adjustments covering wide areas, however, is difficult, particularly with limited point measurements such as tide gauge.     

Low-degree earth deformation from reprocessed GPS observations

Surface mass variations of low spherical harmonic degree are derived from residual displacements of continuously tracking global positioning system (GPS) sites. Reprocessed GPS observations of 14 years are adjusted to obtain surface load coefficients up to degree n _max = 6 together with station positions and velocities from a rigorous parameter combination. Amplitude and phase estimates of the degree-1 annual variations are partly in good agreement with previously published results, but also show interannual differences of up to 2 mm and about 30 days, respectively. The results of this paper reveal significant impacts from different GPS observation modeling approaches on estimated degree-1 coefficients. We obtain displacements of the center of figure (CF) relative to the center of mass (CM), Δr _CF–CM, that differ by about 10 mm in maximum when compared to those of the commonly used coordinate residual approach. Neglected higher-order ionospheric terms are found to induce artificial seasonal and long-term variations especially for the z-component of Δr _CF–CM. Daily degree-1 estimates are examined in the frequency domain to assess alias contributions from model deficiencies with regard to satellite orbits. Finally, we directly compare our estimated low-degree surface load coefficients with recent results that involve data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission.     

Mapping beech (Fagus sylvatica L.) forest structure with airborne hyperspectral imagery

Estimating forest structural attributes using multispectral remote sensing is challenging because of the saturation of multispectral indices at high canopy cover. The objective of this study was to assess the utility of hyperspectral data in estimating and mapping forest structural parameters including mean diameter-at-breast height (DBH), mean tree height and tree density of a closed canopy beech forest (Fagus sylvatica L.). Airborne HyMap images and data on forest structural attributes were collected from the Majella National Park, Italy in July 2004. The predictive performances of vegetation indices (VI) derived from all possible two-band combinations (VI_(i,j) = (R_i − R_j)/(R_i + R_j), where R_i and R_j = reflectance in any two bands) were evaluated using calibration (n = 33) and test (n = 20) data sets. The potential of partial least squares (PLS) regression, a multivariate technique involving several bands was also assessed. New VIs based on the contrast between reflectance in the red-edge shoulder (756–820 nm) and the water absorption feature centred at 1200 nm (1172–1320 nm) were found to show higher correlations with the forest structural parameters than standard VIs derived from NIR and visible reflectance (i.e. the normalised difference vegetation index, NDVI). PLS regression showed a slight improvement in estimating the beech forest structural attributes (prediction errors of 27.6%, 32.6% and 46.4% for mean DBH, height and tree density, respectively) compared to VIs using linear regression models (prediction errors of 27.8%, 35.8% and 48.3% for mean DBH, height and tree density, respectively). Mean DBH was the best predicted variable among the stand parameters (calibration R² = 0.62 for an exponential model fit and standard error of prediction = 5.12 cm, i.e. 25% of the mean). The predicted map of mean DBH revealed high heterogeneity in the beech forest structure in the study area. The spatial variability of mean DBH occurs at less than 450 m. The DBH map could be useful to forest management in many ways, e.g. thinning of coppice to promote diameter growth, to assess the effects of management on forest structure or to detect changes in the forest structure caused by anthropogenic and natural factors.     

Measurement and Scaling of Carbon Dioxide (CO<Subscript>2</Subscript>) Exchanges in Wheat Using Flux-Tower and Remote Sensing

The present study investigates the characteristics of CO₂ exchange (photosynthesis and respiration) over agricultural site dominated by wheat crop and their relationship with ecosystem parameters derived from MODIS. Eddy covariance measurement of CO₂ and H₂O exchanges was carried out at 10 Hz interval and fluxes of CO₂ were computed at half-hourly time steps. The net ecosystem exchange (NEE) was partitioned into gross primary productivity (GPP) and ecosystem respiration (R _e) by taking difference between day-time NEE and respiration. Time-series of daily reflectance and surface temperature products at varying resolution (250–1000 m) were used to derive ecosystem variables (EVI, NDVI, LST). Diurnal pattern in Net ecosystem exchange reveals negative NEE during day-time representing CO₂ uptake and positive during night as release of CO₂. The amplitude of the diurnal variation in NEE increased as LAI crop growth advances and reached its peak around the anthesis stage. The mid-day uptake during this stage was around 1.15 mg CO₂ m⁻² s⁻¹ and night-time release was around 0.15 mg CO₂ m⁻² s⁻¹. Linear and non-linear least square regression procedures were employed to develop phenomenological models and empirical fits between flux tower based GPP and NEE with satellite derived variables and environmental parameters. Enhanced vegetation index was found significantly related to both GPP and NEE. However, NDVI showed little less significant relationship with both GPP and NEE. Furthemore, temperature-greenness (TG) model combining scaled EVI and LST was parameterized to estimate daily GPP over dominantly wheat crop site. (R ² = 0.77). Multi-variate analysis shows that inclusion of LST or air temperature with EVI marginally improves variance explained in daily NEE and GPP.     

Study on Recovering the Earth＇s Potential Field Based on GOCE Gradiometry

Given the second radial derivative Vrr（P） ｜δs of the Earth＇s gravitational potential V（P） on the surface δS corresponding to the satellite altitude, by using the fictitious compress recovery method, a fictitious regular harmonic field rrVrr（P）^＊ and a fictitious second radial gradient field V：（P） in the domain outside an inner sphere Ki can be determined, which coincides with the real field V（P） in the domain outside the Earth. Vrr^＊（P）could be further expressed as a uniformly convergent expansion series in the domain outside the inner sphere, because rrV（P）^＊ could be expressed as a uniformly convergent spherical harmonic expansion series due to its regularity and harmony in that domain. In another aspect, the fictitious field V^＊（P） defined in the domain outside the inner sphere, which coincides with the real field V（P） in the domain outside the Earth, could be also expressed as a spherical harmonic expansion series. Then, the harmonic coefficients contained in the series expressing V^＊（P） can be determined, and consequently the real field V（P） is recovered. Preliminary simulation calculations show that the second radial gradient field Vrr（P） could be recovered based only on the second radial derivative V（P）｜δs given on the satellite boundary. Concerning the final recovery of the potential field V（P） based only on the boundary value Vrr （P）｜δs, the simulation tests are still in process.     

Downward continuation of the free-air gravity anomalies to the ellipsoid using the gradient solution and terrain correction—An attempt of global numerical computations

The formulas for the determination of the coefficients of the spherical harmonic expansion of the disturbing potential of the earth are defined for data given on a sphere. In order to determine the spherical harmonic coefficients, the gravity anomalies have to be analytically downward continued from the earth's surface to a sphere—at least to the ellipsoid. The goal of this paper is to continue the gravity anomalies from the earth's surface downward to the ellipsoid using recent elevation models. The basic method for the downward continuation is the gradient solution (theg ₁ term). The terrain correction has also been computed because of the role it can play as a correction term when calculating harmonic coefficients from surface gravity data. Theg ₁ term and the terrain correction were expanded into the spherical harmonics up to180 ^th order. The corrections (theg ₁ term and the terrain correction) have the order of about 2% of theRMS value of degree variance of the disturbing potential per degree. The influences of theg ₁ term and the terrain correction on the geoid take the order of 1 meter (RMS value of corrections of the geoid undulation) and on the deflections of the vertical is of the order 0.1″ (RMS value of correction of the deflections of the vertical).     

Comparison between the harmonic and response methods of tidal analysis using TOPEX/POSEIDON altimetry

Three years of TOPEX/POSEIDON altimeter data have been processed at Delft Institute for Earth-Oriented Space Research (DEOS) to solve the major diurnal and semi-diurnal constituents of the global ocean tide using the two classical methods of tidal analysis, i.e. the harmonic and response analyses. Some experiments with the parameters in the response formalism show that the tidal admittance in both the diurnal and semi-diurnal band can be adequately described with a lag interval of 2 days and a number of lags of three. Results of both methods are evaluated from the differences with the most recent Grenoble hydrodynamic model (FES95.2) and from the fit with the harmonic constants of a globally distributed set of tide gauges. It was found that the solutions of the two methods differ at the millimeter level and are thus fully equivalent, which is confirmed by the tide gauges and the differences with FES95.2. From the comparisons with the Grenoble model it was found that the M ₂ and S ₂ solutions of that model likely contain bathymetric errors which are of the order of 1–2 cm for M ₂ and 0.5 cm for S ₂. Received: 18 December 1996 / Accepted: 12 May 1997     

Green's function solution to spherical gradiometric boundary-value problems

 Three independent gradiometric boundary-value problems (BVPs) with three types of gradiometric data, {Γ _rr }, {Γ _{r θ},Γ _{r λ}} and {Γ_θθ−Γ_λλ,Γ_θλ}, prescribed on a sphere are solved to determine the gravitational potential on and outside the sphere. The existence and uniqueness conditions on the solutions are formulated showing that the zero- and the first-degree spherical harmonics are to be removed from {Γ _{r θ},Γ _{r λ}} and {Γ_θθ−Γ_λλ,Γ_θλ}, respectively. The solutions to the gradiometric BVPs are presented in terms of Green's functions, which are expressed in both spectral and closed spatial forms. The logarithmic singularity of the Green's function at the point ψ=0 is investigated for the component Γ _rr . The other two Green's functions are finite at this point. Comparisons to the paper by van Gelderen and Rummel [Journal of Geodesy (2001) 75: 1–11] show that the presented solution refines the former solution. Received: 3 October 2001 / Accepted: 4 October 2002     

Triangulated spherical splines for geopotential reconstruction

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.     

National height datum,the Gauss–Listing geoid level value w0 and its time variation 0 (Baltic Sea Level Project: epochs 1990.8, 1993.8, 1997.4)

 A methodology for precise determination of the fundamental geodetic parameter w ₀, the potential value of the Gauss–Listing geoid, as well as its time derivative ₀, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w ₀ and ₀ values (w ₀=62 636 855.75 ± 0.21 m²/s², ₀=−0.0099±0.00079 m²/s² per year, w ₀/&γmacr;=6 379 781.502 m,₀/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m²/s²) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1) the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums of countries around the Baltic Sea. Received: 14 August 2000 / Accepted: 19 June 2001     

Earth＇s Temporal Principal Moments of Inertia and Variable Rotation

Based on the gravity field models EGM96 and EIGEN-GL04C, the Earth＇s time-dependent principal moments of inertia A, B, C are obtained, and the variable rotation of the Earth is determined. Numerical results show that A, B, and C have increasing tendencies; the tilt of the rotation axis increases 2.1×10^ 8 mas/yr; the third component of the rotational angular velocity, ω3 , has a decrease of 1.0×10^ 22 rad/s^2, which is around 23% of the present observed value. Studies show in detail that both 0 and ω3 experience complex fluctuations at various time scales due to the variations of A, B and C.