Showing papers in "Journal of Geodesy in 2006"

Plate Motion of India and Interseismic Strain in the Nepal Himalaya from GPS and DORIS Measurements

Abstract: We analyse geodetically estimated deformation across the Nepal Himalaya in order to determine the geodetic rate of shortening between Southern Tibet and India, previously proposed to range from 12 to 21 mm yr^(−1). The dataset includes spirit-levelling data along a road going from the Indian to the Tibetan border across Central Nepal, data from the DORIS station on Everest, which has been analysed since 1993, GPS campaign measurements from surveys carried on between 1995 and 2001, as well as data from continuous GPS stations along a transect at the logitude of Kathmandu operated continuously since 1997. The GPS data were processed in International Terrestrial Reference Frame 2000 (ITRF2000), together with the data from 20 International GNSS Service (IGS) stations and then combined using quasi observation combination analysis (QOCA). Finally, spatially complementary velocities at stations in Southern Tibet, initially determined in ITRF97, were expressed in ITRF2000. After analysing previous studies by different authors, we determined the pole of rotation of the Indian tectonic plate to be located in ITRF2000 at 51.409±1.560°N and−10.915± 5.556°E, with an angular velocity of 0.483±0.015°. Myr^(−1). Internal deformation of India is found to be small, corresponding to less than about 2 mm yr^(−1) of baseline change between Southern India and the Himalayan piedmont. Based on an elastic dislocation model of interseismic strain and taking into account the uncertainty on India plate motion, the mean convergence rate across Central and Eastern Nepal is estimated to 19 ± 2.5 mm yr^(−1), (at the 67% confidence level). The main himalayan thrust (MHT) fault was found to be locked from the surface to a depth of about 20km over a width of about 115 km. In these regions, the model parameters are well constrained, thanks to the long and continuous time-series from the permanent GPS as well as DORIS data. Further west, a convergence rate of 13.4 ± 5 mm yr^(−1), as well as a fault zone, locked over 150 km, are proposed. The slight discrepancy between the geologically estimated deformation rate of 21 ± 1.5 mm yr^(−1) and the 19 ± 2.5 mm yr^(−1) geodetic rate in Central and Eastern Nepal, as well as the lower geodetic rate in Western Nepal compared to Eastern Nepal, places bounds on possible temporal variations of the pattern and rate of strain in the period between large earthquakes in this region.

An Optimal Adaptive Kalman Filter

Abstract: In a robustly adaptive Kalman filter, the key problem is to construct an adaptive factor to balance the contributions of the kinematic model information and the measurements on the state vector estimates, and the corresponding learning statistic for identifying the kinematic model biases. What we pursue in this paper are some optimal adaptive factors under the particular conditions that the state vector can or cannot be estimated by measurements. Two optimal adaptive factors are derived, one of which is deduced by requiring that the estimated covariance matrix of the predicted residual vector equals the corresponding theoretical one. The other is obtained by requiring that the estimated covariance matrix of the predicted state vector equals its theoretical one. The two related optimal adaptive factors are given. These are analyzed and compared in theory and in an actual example. This shows, through the actual computations, that the filtering results obtained by optimal adaptive factors are superior to those obtained by adaptive factors based on experience.

Precise orbit determination for the GRACE mission using only GPS data

Abstract: The GRACE (gravity recovery and climate experiment) satellites, launched in March 2002, are each equipped with a BlackJack GPS onboard receiver for precise orbit determination and gravity field recovery. Since launch, there have been significant improvements in the background force models used for satellite orbit determination, most notably the model for the geopotential. This has resulted in significant improvements to orbit accuracy for very low altitude satellites. The purpose of this paper is to investigate how well the orbits of the GRACE satellites (about 470 km in altitude) can currently be determined using only GPS data and based on the current models and methods. The orbit accuracy is assessed using a number of tests, which include analysis of orbit fits, orbit overlaps, orbit connecting points, satellite Laser ranging residuals and K-band ranging (KBR) residuals. We show that 1-cm radial orbit accuracy for the GRACE satellites has probably been achieved. These precise GRACE orbits can be used for such purposes as improving gravity recovery from the GRACE KBR data and for atmospheric profiling, and they demonstrate the quality of the background force models being used.

Coastal Gravity Anomalies from Retracked Geosat/GM Altimetry: Improvement, Limitation and the Role of Airborne Gravity Data

Abstract: We process geophysical and waveform data records of the Geosat/GM (geodetic mission) satellite altimeter mission for waveform retracking and applications. An improved threshold retracker is developed. The performances of the Beta-5, threshold and improved threshold retrackers are assessed over waters around Taiwan. The improved threshold retracker outperforms the other two. The improvement in the accuracy of sea surface height (SSH) is investigated according to marine zone and the distance of waters to the shore. The improvement rate increases closer to the land, with the largest improvement rate of about 20% in waters within 10 km of the shore. Over waters around islands and coasts, there are still retracked SSHs with large errors. Least-squares collocation is used to compute gravity anomalies from the Geosat/GM altimeter data. Use of retracked SSHs improves the accuracy of gravity anomalies by about 11%. Adding airborne gravity data further improves the accuracy, especially in the immediate vicinity of the coasts. Tide model errors over coastal waters remain a problem in altimetry applications, even if the waveforms are properly retracked.

Regional gravity modeling in terms of spherical base functions

Abstract: This article provides a survey on modern methods of regional gravity field modeling on the sphere. Starting with the classical theory of spherical harmonics, we outline the transition towards space-localizing methods such as spherical splines and wavelets. Special emphasis is given to the relations among these methods, which all involve radial base functions. Moreover, we provide extensive applications of these methods and numerical results from real space-borne data of recent satellite gravity missions, namely the Challenging Minisatellite Payload (CHAMP) and the Gravity Recovery and Climate Experiment (GRACE). We also derive high-resolution gravity field models by effectively combining space-borne and surface measurements using a new weighted level-combination concept. In addition, we outline and apply a strategy for constructing spatio-temporal fields from regional data sets spanning different observation periods.

Determination of Postglacial Land Uplift in Fennoscandia from Leveling, Tide-gauges and Continuous GPS Stations using Least Squares Collocation

Abstract: Precise leveling, tide-gauge recordings and time series from continuous GPS stations are all sources of information about the ongoing postglacial land uplift in Fennoscandia. This article describes how to gather these three sources of data together in a least-squares collocation adjustment in order to calculate uplift rates, as well as new heights for the benchmarks in the leveling network. The estimated reliability of the resulting uplift model is in general better than 0.4 mm/year, decreasing in areas of fewer data.

Variance Component Estimation in Linear Inverse Ill-posed Models

Abstract: Regularization has been applied by implicitly assuming that the weight matrix of measurements is known. If measurements are assumed to be heteroscedastic with different unknown variance components, all regularization techniques may not be proper to apply, unless techniques of variance component estimation are directly implemented. Although variance component estimation techniques have been proposed to simultaneously estimate the variance components and provide a means of regularization, the regularization parameter is treated as if it were also an extra variance component. In this paper, we assume no prior information on the model parameters and do not treat the regularization parameter as an extra variance component. Instead, we first analyze the biases of estimated variance components due to the regularization parameter and then propose bias-corrected variance component estimators. The results have shown that they work very well. Finally, we propose and investigate through simulations an iterative scheme to simultaneously estimate the variance components and the regularization parameter, in order to eliminate the effect of regularization parameter on variance components and the effect of incorrect prior weights or initial variance components on the regularization parameter.

Pseudo-Stochastic Orbit Modeling Techniques for Low-Earth Orbiters

Abstract: The Earth’s non-spherical mass distribution and atmospheric drag cause the strongest perturbations on very low-Earth orbiting satellites (LEOs). Models of gravitational and non-gravitational accelerations are utilized in dynamic precise orbit determination (POD) with GPS data, but it is also possible to derive LEO positions based on GPS precise point positioning without dynamical information. We use the reduced-dynamic technique for LEO POD, which combines the geometric strength of the GPS observations with the force models, and investigate the performance of different pseudo-stochastic orbit parametrizations, such as instantaneous velocity changes (pulses), piecewise constant accelerations, and continuous piecewise linear accelerations. The estimation of such empirical orbit parameters in a standard least-squares adjustment process of GPS observations, together with other relevant parameters, strives for the highest precision in the computation of LEO trajectories. We used the procedures for the CHAMP satellite and found that the orbits may be validated by means of independent SLR measurements at the level of 3.2 cm RMS. Validations with independent accelerometer data revealed correlations at the level of 95% in the along-track direction. As expected, the empirical parameters compensate to a certain extent for deficiencies in the dynamic models. We analyzed the capability of pseudo-stochastic parameters for deriving information about the mismodeled part of the force field and found evidence that the resulting orbits may be used to recover force field parameters, if the number of pseudo-stochastic parameters is large enough. Results based on simulations showed a significantly better performance of acceleration-based orbits for gravity field recovery than for pulse-based orbits, with a quality comparable to a direct estimation if unconstrained accelerations are set up every 30 s.

A New Data Processing Strategy for Huge GNSS Global Networks

Abstract: In Global Positioning System (GPS) data analyses, large networks are usually divided into sub-networks to solve the conflict between increasing amounts of data and limited computer resources, although an integrated analysis would provide better results. This conflict becomes even more critical with the increasing number of stations, and low-Earth-orbiting satellites and the Galileo system coming into operation. The major reason is that a huge number of ambiguity parameters are kept in the normal equation for sequential integer ambiguity fixing. In this paper, the problem is solved by a special procedure of parameter elimination for both real-valued and ambiguity-fixed solutions, based on an adapted ambiguity-fixing approach where the covariance-matrix of ambiguity parameters is not required anymore. It is demonstrated that, with the new strategy, the required memory can be reduced to one-tenth and the computation time to at least one-third compared to the existing methods, and huge GPS networks with several hundred stations can be processed efficiently on a personal computer.

An integrated GPS-accelerometer data processing technique for structural deformation monitoring

Abstract: Global Positioning System (GPS) is being actively applied to measure static and dynamic displacement responses of large civil engineering structures under winds. However, multipath effects and low sampling frequencies affect the accuracy of GPS for displacement measurement. On the other hand, accelerometers cannot reliably measure static and low-frequency structural responses, but can accurately measure high-frequency structural responses. Therefore, this paper explores the possibility of integrating GPS-measured signals with accelerometer-measured signals to enhance the measurement accuracy of total (static plus dynamic) displacement response of a structure. Integrated data processing techniques using both empirical mode decomposition (EMD) and an adaptive filter are presented. A series of motion simulation table tests are then performed at a site using three GPS receivers, one accelerometer, and one motion simulation table that can simulate various types of motion defined by input wave time histories around a pre-defined static position. The proposed data processing techniques are applied to the recorded GPS and accelerometer data to find both static and dynamic displacements. These results are compared with the actual displacement motions generated by the motion simulation table. The comparative results demonstrate that the proposed technique can significantly enhance the measurement accuracy of the total displacement of a structure.

A corrective model for Jason-1 DORIS doppler Data in relation to the South Atlantic Anomaly

Abstract: The DORIS Doppler measurements collected by Jason-1 are abnormally perturbed by the influence of the South Atlantic Anomaly (SAA). The DORIS ultra-stable oscillators on-board Jason-1 are not as stable as they should be; their frequency is sensitive both to the irradiation rate and to the total irradiation encountered in orbit. The consequence is that not only are the DORIS measurement residuals higher than they ought to be, but also large systematic positioning errors are introduced for stations located in the vicinity of the SAA. In this paper, we present a method that has been devised to obtain a continuous observation of Jason-1 frequency offsets. This method relies on the precise determination of the station frequency and troposphere parameters via the use of other DORIS satellites. More than 3 years of these observations have then been used to construct a model of response of the oscillators of Jason-1 to the SAA. The sensitivity of the Jason-1 oscillators to the SAA perturbations has evolved over time, multiplied by a factor of four between launch and mid-2004. The corrective performances of the model are discussed in terms of DORIS measurement residuals, precise orbit determination and station positioning. The average DORIS measurement residuals are decreased by more than 7 % using this model. In terms of precise orbit determination, the 3D DORIS-only orbit error decreases from 5 to 4.2 cm, but the DORIS+SLR orbit error is almost unaffected, due to the already good quality of this type of orbit. In terms of station positioning, the model brings down the average 3D mono-satellite monthly network solution discrepancy with the International Terrestrial Reference Frame ITRF2000 from 11.3 to 6.1 cm, and also decreases the scatter about that average from 11.3 to 3.7 cm. The conclusion is that, with this model, it is possible to re-incorporate Jason-1 in the multi-satellite geodetic solutions for the DORIS station network.

Multi-technique comparison of tropospheric zenith delays derived during the CONT02 campaign

Abstract: In October 2002, 15 continuous days of Very Long Baseline Interferometry (VLBI) data were observed in the Continuous VLBI 2002 (CONT02) campaign. All eight radio telescopes involved in CONT02 were co-located with at least one other space-geodetic technique, and three of them also with a Water Vapor Radiometer (WVR). The goal of this paper is to compare the tropospheric zenith delays observed during CONT02 by VLBI, Global Positioning System (GPS), Doppler Orbitography Radiopositioning Integrated by Satellite (DORIS) and WVR and to compare them also with operational pressure level data from the European Centre for Medium-Range Weather Forecasts (ECMWF). We show that the tropospheric zenith delays from VLBI and GPS are in good agreement at the 3–7 mm level. However, while only small biases can be found for most of the stations, at Kokee Park (Hawaii, USA) and Westford (Massachusetts, USA) the zenith delays derived by GPS are larger by more than 5 mm than those from VLBI. At three of the four DORIS stations, there is also a fairly good agreement with GPS and VLBI (about 10 mm), but at Kokee Park the agreement is only at about 30 mm standard deviation, probably due to the much older installation and type of DORIS equipment. This comparison also allows testing of different DORIS analysis strategies with respect to their real impact on the precision of the derived tropospheric parameters. Ground truth information about the zenith delays can also be obtained from the ECMWF numerical weather model and at three sites using WVR measurements, allowing for comparisons with results from the space-geodetic techniques. While there is a good agreement (with some problems mentioned above about DORIS) among the space-geodetic techniques, the comparison with WVR and ECMWF is at a lower accuracy level. The complete CONT02 data set is sufficient to derive a good estimate of the actual precision and accuracy of each geodetic technique for applications in meteorology.

Twenty years of evolution for the DORIS permanent network: from its initial deployment to its renovation

Abstract: The ground network is one of the major components of the DORIS system. Its deployment, managed by the French national mapping agency [Institut Geographique National, (IGN)], started in 1986 at a sustained pace that allowed it to reach 32 stations upon the launch of the first DORIS-equipped satellite (SPOT-2) in 1990. For the first generation of transmitting antennas, the installation procedures were adapted to the decimetre performance objective for the DORIS system. During the second era of the deployment of an even denser network, the antenna support layouts gradually evolved towards a better quality, thus improving the long-term stability of the antenna reference point, and a new antenna model allowed a more accurate survey. As the positioning accuracy of the DORIS system improved, it was necessary to review the antenna stability for the whole network. A first stability estimation, using criteria like antenna model and support design, was followed by a major renovation effort which started in 2000 and is now almost complete. In 6 years, through the renovation or installation of 43 stations and the implementation of new installation procedures to meet more stringent stability requirements, significant improvement in network quality was achieved. Later a more analytical approach, taking into account the characteristics of each element that support the antenna, has been taken to assess the potential stability of all DORIS occupations. IGN is also in charge of its operational maintenance, an intensive activity on account of the significant failure rate of the successive generations of equipment. Nevertheless, thanks to its unique density and homogeneity, DORIS has maintained a very good coverage rate of the satellite orbits. Through 38 well-distributed current co-locations with the Global Positioning System, Satellite Laser Ranging and Very Long Baseline Interferometry techniques in its current 56-station network, DORIS contributes significantly to the realisation of the International Terrestrial Reference System. DORIS stations in areas where no other space geodesy technique is available provide a significant contribution to the study of plate tectonics. Many stations co-located with tide gauges contribute to the monitoring of sea level changes. Although it has several advantages over similar techniques, there is still room for improvement in the DORIS network.

The International DORIS Service: genesis and early achievements

Abstract: All space-geodetic techniques are now organized as separate services of the International Association of Geodesy (IAG), supporting the first pilot project “Global Geodetic Observing System (GGOS)”. The International DORIS (Determination d’Orbite et Radiopositionnement Integres par Satellite) Service (IDS) was created in mid-2003 to organize a DORIS contribution to this project and to foster a larger international cooperation on this topic. The goal of this paper is to summarize the key steps that were taken to create this structure and to present its current organization and recent results. At present, more than 50 groups from 35 different countries participate in the IDS at various levels, including 43 groups hosting DORIS stations in 32 countries all around the globe. Four Analysis Centres (ACs) provide results, such as estimates of weekly or monthly station coordinates, geocentre variations or Earth polar motion, that will soon be used to generate IDS-combined products for geodesy and geodynamics. As a first test, a preliminary combination was performed for all the 2004 data from these four ACs. Three of them show RMS of weighted station residuals with respect to this combination solution between 1 and 2 cm. The main topic under investigation is a discrepancy in the scale factor of the terrestrial reference frame (TRF) to map the individual solutions into the combination solution, which reaches 6 cm (multiplying the unit-less scale factor by the Earth radius to get convert scale to millimetre in vertical at the Earth’s surface). Finally, foreseen improvements of the DORIS technology are discussed as well as future improvements concerning the service organization itself and the accuracy and reliability of its scientific products.

A Quaternion-based Geodetic Datum Transformation Algorithm

Abstract: This paper briefly introduces quaternions to represent rotation parameters and then derives the formulae to compute quaternion, translation and scale parameters in the Bursa–Wolf geodetic datum transformation model from two sets of co-located 3D coordinates. The main advantage of this representation is that linearization and iteration are not needed for the computation of the datum transformation parameters. We further extend the formulae to compute quaternion-based datum transformation parameters under constraints such as the distance between two fixed stations, and develop the corresponding iteration algorithm. Finally, two numerical case studies are presented to demonstrate the applications of the derived formulae.

Error Analysis of Weekly Station Coordinates in the DORIS Network

Abstract: Twelve years of DORIS data from 31 selected sites of the IGN/JPL (Institut Geographique National/Jet Propulsion Laboratory) solution IGNWD05 have been analysed using maximum likelihood estimation (MLE) in an attempt to understand the nature of the noise in the weekly station coordinate time-series. Six alternative noise models in a total of 12 different combinations were used as possible descriptions of the noise. The six noise models can be divided into two natural groups, temporally uncorrelated (white) noise and temporally correlated (coloured) noise. The noise can be described as a combination of variable white noise and one of flicker, first-order Gauss–Markov or power-law noise. The data set as a whole is best described as a combination of variable white noise plus flicker noise. The variable white noise, which is white noise with variable amplitude that is a function of the weekly formal errors multiplied by an estimated scale factor, shows a dependence on site latitude and the number of DORIS-equipped satellites used in the solution. The latitude dependence is largest in the east component due to the near polar orbit of the SPOT satellites. The amplitude of the flicker noise is similar in all three components and equal to about 20 mm/year1/4. There appears to be no latitude dependence of the flicker noise amplitude. The uncertainty in rates (site velocities) after 12 years is just under 1 mm/year. These uncertainties are around 3–4 times larger than if only variable white noise had been assumed, i.e., no temporally correlated noise. A rate uncertainty of 1 mm/year after 12 years in the vertical is similar to that achieved using Global Positioning System (GPS) data but it takes DORIS twice as long to reach 1 mm/year than GPS in the horizontal. The analysis has also helped to identify sites with either anomalous noise characteristics or large noise amplitudes, and tested the validity of previously proposed discontinuities. In addition, several new offsets were found in the time-series that should be used or at least flagged in future work.

Degradation of geopotential recovery from short repeat-cycle orbits : application to GRACE monthly fields

Abstract: Throughout 2004 the GRACE (Gravity Recovery And Climate Experiment) orbit contracted slowly to yield a sparse repeat track of 61 revolutions every 4 days on 19 September 2004. As a result, we show from linear perturbation theory that geopotential information previously available to fully resolve a gravity field every month of 120× 120 (degree by order) in spherical harmonics was compressed then into about one-fourth of the necessary observation space. We estimate from this theory that the ideal gravity field resolution in September 2004 was only about 30 × 30. More generally, we show that any repeat-cycle mission for geopotential recovery with full resolution L × L requires the number of orbit-revolutions-to-repeat to be greater than 2L.

Combination of temporal gravity variations resulting from superconducting gravimeter (SG) recordings, GRACE satellite observations and global hydrology models

Abstract: Gravity recovery and climate experiment (GRACE)-derived temporal gravity variations can be resolved within the μgal (10−8 m/s2) range, if we restrict the spatial resolution to a half-wavelength of about 1,500 km and the temporal resolution to 1 month. For independent validations, a comparison with ground gravity measurements is of fundamental interest. For this purpose, data from selected superconducting gravimeter (SG) stations forming the Global Geodynamics Project (GGP) network are used. For comparison, GRACE and SG data sets are reduced for the same known gravity effects due to Earth and ocean tides, pole tide and atmosphere. In contrast to GRACE, the SG also measures gravity changes due to load-induced height variations, whereas the satellite-derived models do not contain this effect. For a solid spherical harmonic decomposition of the gravity field, this load effect can be modelled using degree-dependent load Love numbers, and this effect is added to the satellite-derived models. After reduction of the known gravity effects from both data sets, the remaining part can mainly be assumed to represent mass changes in terrestrial water storage. Therefore, gravity variations derived from global hydrological models are applied to verify the SG and GRACE results. Conversely, the hydrology models can be checked by gravity variations determined from GRACE and SG observations. Such a comparison shows quite a good agreement between gravity variation derived from SG, GRACE and hydrology models, which lie within their estimated error limits for most of the studied SG locations. It is shown that the SG gravity variations (point measurements) are representative for a large area within the accuracy, if local gravity effects are removed. The individual discrepancies between SG, GRACE and hydrology models may give hints for further investigations of each data series.

Surface Ice Flow Velocity and Tide Retrieval of the Amery Ice Shelf using Precise Point Positioning

Abstract: Five days of continuous GPS observation data were collected in the frontal zone of the Amery ice shelf and subsequently post-processed using precise point position (PPP) technology based on precise orbit and clock products from the International GNSS service. The surface ice flow velocity of the observed point was derived from PPP to be 2.25 m/day toward the northeast with an azimuth of 41°. Major semi-diurnal and diurnal oceanic tide constituents could be recovered from the 5 days of PPP-derived height variations and compared well with a hydrodynamic ocean tide model. The PPP technique can replace double-difference GPS positioning in remote or hostile environments, and be used to retrieve the surface ice flow velocity without any reference station. Furthermore, the solution can be derived epoch-by-epoch with accuracy in the centimeters to decimeter range.

Multiple Parameter Regularization: Numerical Solutions and Applications to the Determination of Geopotential from Precise Satellite Orbits

Abstract: Kaula’s rule of thumb has been used in producing geopotential models from space geodetic measurements, including the most recent models from satellite gravity missions CHAMP. Although Xu and Rummel (Manuscr Geod 20 8–20, 1994b) suggested an alternative regularization method by introducing a number of regularization parameters, no numerical tests have ever been conducted. We have compared four methods of regularization for the determination of geopotential from precise orbits of COSMIC satellites through simulations, which include Kaula’s rule of thumb, one parameter regularization and its iterative version, and multiple parameter regularization. The simulation results show that the four methods can indeed produce good gravitational models from the precise orbits of centimetre level. The three regularization methods perform much better than Kaula’s rule of thumb by a factor of 6.4 on average beyond spherical harmonic degree 5 and by a factor of 10.2 for the spherical harmonic degrees from 8 to 14 in terms of degree variations of root mean squared errors. The maximum componentwise improvement in the root mean squared error can be up to a factor of 60. The simplest version of regularization by multiplying a positive scalar with a unit matrix is sufficient to better determine the geopotential model. Although multiple parameter regularization is theoretically attractive and can indeed eliminate unnecessary regularization for some of the harmonic coefficients, we found that it only improved its one parameter version marginally in this COSMIC example in terms of the mean squared error.

Non-Singular Expressions for the Gravity Gradients in the Local North-Oriented and Orbital Reference Frames

Abstract: The conventional expansions of the gravity gradients in the local north-oriented reference frame have a complicated form, depending on the first- and second-order derivatives of the associated Legendre functions of the colatitude and containing factors which tend to infinity when approaching the poles. In the present paper, the general term of each of these series is transformed to a product of a geopotential coefficient $$\overline{C}_{n,m} $$ and a sum of several adjacent Legendre functions of the colatitude multiplied by a function of the longitude. These transformations are performed on the basis of relations between the Legendre functions and their derivatives published by Ilk (1983). The second-order geopotential derivatives corresponding to the local orbital reference frame are presented as linear functions of the north-oriented gravity gradients. The new expansions for the latter are substituted into these functions. As a result, the orbital derivatives are also presented as series depending on the geopotential coefficients $$\overline{C}_{n,m} $$ multiplied by sums of the Legendre functions whose coefficients depend on the longitude and the satellite track azimuth at an observation point. The derived expansions of the observables can be applied for constructing a geopotential model from the GOCE mission data by the time-wise and space-wise approaches. The numerical experiments demonstrate the correctness of the analytical formulas.

Short Note: On the Relativistic Doppler Effect for Precise Velocity Determination using GPS

Abstract: The Doppler effect is the apparent shift in frequency of an electromagnetic signal that is received by an observer moving relative to the source of the signal. The Doppler frequency shift relates directly to the relative speed between the receiver and the transmitter, and has thus been widely used in velocity determination. A GPS receiver-satellite pair is in the Earth's gravity field and GPS signals travel at the speed of light, hence both Einstein's special and general relativity theories apply. This paper establishes the relationship between a Doppler shift and a user's ground velocity by taking both the special and general relativistic effects into consideration. A unified Doppler shift model is developed, which accommodates both the classical Doppler effect and the relativistic Doppler effect under special and general relativities. By identifying the relativistic correction terms in the model, a highly accurate GPS Doppler shift observation equation is presented. It is demonstrated that in the GPS "frequency" or "velocity" domain, the relativistic effect from satellite motion changes the receiver-satellite line-of-sight direction, and the measured Doppler shift has correction terms due to the relativistic effects of the receiver potential difference from the geoid, the orbit eccentricity, and the rotation of the Earth.

Plate kinematics of Nubia-Somalia using a combined DORIS and GPS solution

Abstract: We have used up to 12 years of data to assess DORIS performance for geodynamics applications. We first examine the noise characteristics of the DORIS time-series of weekly station coordinates to derive realistic estimates of velocity uncertainties. We find that a combination of white and flicker noise best explains the DORIS time-series noise characteristics. Second, weekly solutions produced by the Institut Geographique National/Jet Propulsion Laboratory (IGN/JPL) DORIS Analysis Centre are combined to derive a global velocity field. This solution is combined with two independent GPS solutions, including 11 sites on Nubia and 5 on the Somalia plate. The combination indicates that DORIS horizontal velocities have an average accuracy of 3 mm/year, with best-determined sites having velocity accuracy better than 1 mm/year (one-sigma levels). Using our combined velocity field, we derive an updated plate kinematics model with a focus on the Nubia–Somalia area. Including DORIS data improves the precision of the angular velocity vector for Nubia by 15%. Our proposed model provides robust bounds on the maximum opening rates along the East African Rift (4.7–6.7 mm/year). It indicates opening rates 15 and 7% slower than values predicted by NUVEL-1A for the southern Atlantic Ocean and Indian Ocean, respectively. These differences are likely to arise from the fact that NUVEL-1A considered Africa as a single non-deforming plate, while here we use a more refined approach.

Wavelet Modeling of Regional and Temporal Variations of the Earth’s Gravitational Potential Observed by GRACE

Abstract: This work is dedicated to the wavelet modeling of regional and temporal variations of the Earth’s gravitational potential observed by the GRACE (gravity recovery and climate experiment) satellite mission. In the first part, all required mathematical tools and methods involving spherical wavelets are provided. Then, we apply our method to monthly GRACE gravity fields. A strong seasonal signal can be identified which is restricted to areas where large-scale redistributions of continental water mass are expected. This assumption is analyzed and verified by comparing the time-series of regionally obtained wavelet coefficients of the gravitational signal originating from hydrology models and the gravitational potential observed by GRACE. The results are in good agreement with previous studies and illustrate that wavelets are an appropriate tool to investigate regional effects in the Earth’s gravitational field.

‘DEOS_CHAMP-01C_70’: a model of the Earth’s gravity field computed from accelerations of the CHAMP satellite

Abstract: Performance of a recently proposed technique for gravity field modeling has been assessed with data from the CHAMP satellite. The modeling technique is a variant of the acceleration approach. It makes use of the satellite accelerations that are derived from the kinematic orbit with the 3-point numerical differentiation scheme. A 322-day data set with 30-s sampling has been used. Based on this, a new gravity field model – DEOS CHAMP-01C 70 - is derived. The model is complete up to degree and order 70. The geoid height difference between the DEOS CHAMP-01C 70 and EIGEN-GRACE01S models is 14 cm. This is less than for two other recently published models: EIGEN-CHAMP03Sp and ITG-CHAMP01E. Furthermore, we analyze the sensitivity of the model to some empirically determined parameters (regularization parameter and the parameter that controls the frequency-dependent data weighting).We also show that inaccuracies related to non-gravitational accelerations, which are measured by the on-board accelerometer, have a minor influence on the computed gravity field model.

Transformation from Cartesian to Geodetic Coordinates Accelerated by Halley’s Method

Abstract: By using Halley’s third-order formula to find the root of a non-linear equation, we develop a new iterative procedure to solve an irrational form of the “latitude equation”, the equation to determine the geodetic latitude for given Cartesian coordinates. With a limit to one iteration, starting from zero height, and minimizing the number of divisions by means of the rational form representation of Halley’s formula, we obtain a new non-iterative method to transform Cartesian coordinates to geodetic ones. The new method is sufficiently precise in the sense that the maximum error of the latitude and the relative height is less than 6 micro-arcseconds for the range of height, −10 km ≤ h ≤ 30,000 km. The new method is around 50% faster than our previous method, roughly twice as fast as the well-known Bowring’s method, and much faster than the recently developed methods of Borkowski, Laskowski, Lin and Wang, Jones, Pollard, and Vermeille.

Frequency-dependent data weighting in global gravity field modeling from satellite data contaminated by non-stationary noise

Abstract: Satellite data that are used to model the global gravity field of the Earth are typically corrupted by correlated noise, which can be related to a frequency dependence of the data accuracy. We show an opportunity to take such noise into account by using a proper noise covariance matrix in the estimation procedure. If the dependence of noise on frequency is not known a priori, it can be estimated on the basis of a posteriori residuals. The methodology can be applied to data with gaps. Non-stationarity of noise can also be dealt with, provided that the necessary a priori information exists. The proposed methodology is illustrated with CHAllenging Mini-satellite Payload (CHAMP) data processing. It is shown, in particular, that the usage of a proper noise model can make the measurements of non-gravitational satellite accelerations unnecessarily. This opens the door for high-quality modeling of the Earth’s gravity field on the basis of observed orbits of non-dedicated satellites (i.e., satellites without an on-board accelerometer). Furthermore, the processing of data from dedicated satellite missions – GRACE (Gravity Recovery and Climate Experiment) and GOCE (Gravity field and steady-state Ocean Circulation Explorer) – may also benefit from the proposed methodology.

Estimating the noise in space-geodetic positioning: the case of DORIS

Abstract: The noise spectrum in DORIS ground- station motion is investigated by means of the Allan variance method applied to the decomposition of the 3D signal into its principal components in the time domain. Sets of weekly position time-series from 1994 to 2005 derived by three IDS Analysis Centres (IGN-JPL, INASAN, and LEGOS-CLS) for 119 stations at 69 sites are considered. The observing satellites are SPOT-2, SPOT-3, SPOT-4, and SPOT-5, TOPEX/Poseidon, and ENVISAT. Annual and semi-annual perturbations, as well as the 117.3-day term associated with the TOPEX/Poseidon orbit, are found at most stations. Their amplitudes reach up to 19.3, 23.7, and 13.3 mm, respectively, for the three analysis centres (ACs). When corrected for these components and a linear drift, the time-series dominantly show white noise (WN) at the 10–45mm level the noise level is the highest in the East direction, probably in connection with the high orbit inclinations. The noise level is minimum for the high latitude stations, mostly and intensively observed by the SPOT satellites, and the determination of the noise type is unclear; longer observation spans would be needed to decide between interannual variations and flicker noise. The improvement in positioning due to the DORIS constellation extension from three to five satellites in 2002, and the network rejuvenation program initiated in 2000, results in a decrease of the noise level by a factor of 1.7 in a WN context, both before and after the changes. One example of the benefit of studying the signal in the time eigenspace domain is the detection of anomalously large WN in the East direction for station HBKB (Hartebeesthoek, Africa) that masks the above-mentioned improvement. Studying the projection on the local frame of the second and third time-eigenspace components, a noise excess is detected in the North direction for some of the ACs. Station stability derived from our time-series analysis confirms, in general, the expected performance based on the careful technical review of the station components (antenna, pillar, etc.). The respective merits of our noise qualification method, based on direct time-series analysis in the time-eigenspace domain without any a priori statistical model, in comparison with other methods, such as the selection of a mixed-noise model by maximum likelihood estimation, are discussed.

The relation between rigorous and Helmert’s definitions of orthometric heights

Abstract: Following our earlier definition of the rigorous orthometric height [J Geod 79(1-3):82–92 (2005)] we present the derivation and calculation of the differences between this and the Helmert orthometric height, which is embedded in the vertical datums used in numerous countries. By way of comparison, we also consider Mader and Niethammer’s refinements to the Helmert orthometric height. For a profile across the Canadian Rocky Mountains (maximum height of ~2,800 m), the rigorous correction to Helmert’s height reaches ~13 cm, whereas the Mader and Niethammer corrections only reach ~3 cm. The discrepancy is due mostly to the rigorous correction’s consideration of the geoid-generated gravity disturbance. We also point out that several of the terms derived here are the same as those used in regional gravimetric geoid models, thus simplifying their implementation. This will enable those who currently use Helmert orthometric heights to upgrade them to a more rigorous height system based on the Earth’s gravity field and one that is more compatible with a regional geoid model.

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

Abstract: 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