A Window on the Future of Geodesy

In its function as ITRS Combination Center and IERS Combination Research Center, DGFI is involved in the combination of space geodetic observations. Based on the latest space geodetic solutions provided by individual analysis centers and/or technique centers, DGFI has computed a combined solution for station positions and velocities. Major goals are: (1) a validation of the various components of the ITRS Combination Center; (2) a verification and enhancement of the combination strategy; (3) an external comparison with ITRF2000; (4) the identification of still existing deficiencies; and (5) proposals for future ITRF realisations.

This paper investigates the time evolution of the terrestrial reference frame (TRF). We analysed time series of site positions and datum parameters obtained from VLBI, SLR, GPS and DORIS solutions with respect to non-linear motions (e.g. jumps, seasonal signals) and systematic differences. The time series of the translation and scale parameters were used to identify solution- and technique-related problems, and to investigate the contribution of the different techniques to realise the TRF origin and scale. The position time series reveal non-linear motions and jumps for many sites, which have to be considered for future TRF realisations.

We give an overview of time series analyses of permanent GPS stations using solutions of the IGS and FinnRef® networks. Lomb periodograms show in most cases a statistically significant annual period both in station coordinates and inter-station distances. In regional networks the scale of the whole network changes periodically, and in some cases there is also a secular trend. There are several possible causes of scale variations, which may not be separable in the data. These include computational artefacts, periodic systematic errors in satellite orbits, signal path delay variations, and geophysical causes like loading and postglacial rebound. We discuss possible reasons, their significance, and their consequences on high-precision GPS observations. Additional constraints, e.g. time series from the superconducting gravimeter, are also discussed.

The Global Positioning System (GPS) data observed in the Tsukuba 2000 and 2001 GPS/MET dense network campaigns were analyzed. In these campaigns about 75 GPS receivers were installed in a 20 × 20 km area with 1.5km – 3.0km intervals. The 2000 campaign was carried out late fall for 31 days and the 2001 campaign in the summer season for 53 days. GAMIT/GLOBK GPS analyzing software was applied to the analysis. In the analysis of the Tsukuba 2000 campaign data, the time variation of the vertical component changed simultaneously at all the sites, making the correlated motion among the campaign sites. It might be attributed to the fact that constraints on the vertical component of Tsukuba IGS site (TSKB), a fiducial site of the campaign, were too loose. The variation was in the range of 20 – 30 mm with a one-sigma repeatability of 4 – 7 mm. When we examine the vertical component of the baseline vectors of each site with reference to one site, TSKB, the time variation of the vertical components with reference to the TSKB site was much smaller than for single stations. The time variation was in the range of 10 – 20 mm with the one-sigma repeatability of 2 – 6 mm. in this case for the 2000 campaign. The repeatability of the baseline vector among the network sites had a weak dependency on the baseline length, with the zero length repeatability of 3.17 mm and a baseline length dependency of 0.11 ppm. Meanwhile, in the 2001 campaign, the time variation of the baseline height was around 10 mm and almost random. The relationship between the baseline length and vertical repeatability of the baseline vector among the network sites was strong, with the zero length repeatability of 3.36 mm and the baseline dependency of 0.17 ppm during the 2001 campaign.

This paper presents preliminary estimates of horizontal crustal motion inferred from a GPS network in the UK. The British Isles are on the western part of the Eurasian plate and the effects of active tectonism are expected to be small. However, the British Isles are believed to undergo non-rigid plate behaviour due to glacial isostatic adjustment (GIA) occurring in northern Britain. Non-rigid plate behaviour can generally be shown to cause velocity biases when compared to velocities derived from models such as NNR-NUVEL-1A, which are based upon assumed rigid plate behaviour. As a first assessment, in this paper, horizontal station velocities obtained from the GPS analysis are compared to the ITRF2000 angular velocity of rigid Eurasia.

The geocentric reference system for the Americas (SIRGAS) was initially realized for South America only. A GPS campaign in 1995 covered 58 sites. In 2000 these stations were re-observed and the network was extended to North and Central America as well as the Caribbean. The objectives of the project were completed by the establishment of a unified vertical reference system connecting the classical national height systems. The processing of the 2000 observation data was performed by three analysis centres at DGFI and BEK in Munich as well as IBGE in Rio de Janeiro. The Bernese and GIPSY/OASIS software packages were used. The processing and the comparisons with the 1995 coordinates are described.

New Zealand Geodetic Datum 2000 (NZGD2000) is a ‘semi-dynamic’ datum in that coordinates are published in terms of their values at the reference epoch of 1 January 2000 and it incorporates an as yet unpublished deformation model which allows positions at other times to be extrapolated. The proposed deformation model has two components, 1) a national deformation component using a latitude/longitude grid which is spatially and temporally continuous, and 2) a number of ‘patches’ used to model specific deformation events such as earthquakes. The national model will comprise a series deformation grids at fixed time intervals between which the deformation can be interpolated. The patches are localised triangulation based models of limited extent and time that can represent arbitrarily complex deformation. This is seen as meeting both the needs of the geodetic community, who require accurate coordinates at arbitrary times, and the mapping community, who prefer static coordinates, but expect them to reflect major deformation events.

It has been pointed out that GPS results in mountain areas are usually less accurate than those in flat areas, especially in the vertical component, when the height difference among observation stations is large. It is thought that the decrease of accuracy is caused by insufficient corrections for tropospheric delay of the GPS microwave.In order to investigate the difference of tropospheric delays at sites located on high mountains and those at valleys, and furthermore to improve the accuracy of GPS results in mountain areas, we have repeated GPS measurements since September 1997 at Yamainudan (about 1400 m above sea level) near the top of Mt. Sobatsubu in the Southern Alps of Japan. Their data were processed with the data at four stations of surrounding valleys (300 – 400 m) by using GAMIT software, to investigate the relation of the coordinates obtained and zenith tropospheric delays. We found a negative correlation between the difference of zenith tropospheric delay and height difference between Yamainudan and a site at valley. These results scarcely changed when a mapping function used was changed.We also processed some GPS data obtained by the Geographical Survey Institute (GEONET data) in the mountain areas. A positive correlation between the difference of tropospheric zenith delay and height difference was found in the Northern Alps, and no significant correlation was found in the Central Alps of Japan.We think that these different correlations were caused by the difference of the shape of the valleys.

Uncertainty of vector components estimation, obtained from processing GPS data using either commercial or scientific software, represents rather an internal consistency than the accuracy of positioning. Short period biases, including non-modelled effects, some with unstable amplitudes, that affect GPS solutions are as systematic-type terms not reflected in uncertainty estimation. Therefore, calculated uncertainty is usually much too optimistic as the accuracy estimate. In the routine GPS data processing one observing session provides a single solution. Data that form such observing session can be represented by a series of overlapped sessions. Comparison of solutions obtained from processing overlapped sessions leads to more reliable accuracy estimation than the one based on uncertainty estimation of individual solutions. The strategy of GPS solutions quality analysis based on the concept of overlapping sessions with optimum length and temporal resolution is presented. The strategy was verified with use of data from the EPN stations processed with both Bernese and Pinnacle software packages.

The classical concept in geodesy of an exclusively stochastic assessment of the total error budget of observation data is extended. Uncertainty due to remaining systematics (imprecision) is treated consistently by interval mathematics. The superposition of both random variability (stochasticity) and imprecision yields uncertainty measures such as extended point confidence regions which are more realistic. The new concept is applied to GPS phase measurements. It is exemplarily discussed for a synthetic GPS network.

Within the framework of EUREF, the Federal Agency for Cartography and Geodesy (BKG) has developed a new technique using the open Internet for the real-time collection and exchange of GNSS data, as well as for broadcasting derived products. A major purpose of these activities is the dissemination of Differential GPS corrections (DGPS) for precise positioning and navigation. This paper describes the http-based technique for streaming GNSS data to mobile users connected to the Internet via Mobile IP-Networks like GSM, GPRS, EDGE, or UMTS. The technique establishes a format called Networked Transport of RTCM via Internet Protocol (Ntrip).

When GPS satellite signals are transmitted through the atmosphere they are affected by the media. In the neutral atmosphere the refraction is a function of pressure, temperature, and humidity along the signal path, and in the GPS positioning process this effect is normally handled by utilising global tropospheric delay models. For high accuracy differential positioning over baselines lengths where the differential effect of the signal delay from the neutral atmosphere is significant, these global models of the signal delay are not sufficiently accurate, and this is especially the case during abnormal weather conditions.This paper describes a new approach where numerical weather predictions (NWPs) are introduced in the GPS data processing instead of global tropospheric delay models. NWPs are predictions of the meteorological conditions for a given area and epoch in time, and can as such be used for estimating the tropospheric delay for a satellite signal by numerical integration along the signal path through the NWP. For the tests described in this paper, the signal delays are determined as a zenith delay through the NWP combined with a mapping function. This approach is useful for kinematic and shorter static GPS applications. The paper describes the theory of the method, and the applicability of the method is evaluated by analysing position accuracies obtained by introducing NWP-derived signal delays in kinematic and static processing of GPS data. Improved position accuracies are obtained for most of the test scenarios, indicating that the method does have a potential.

In airborne applications (e.g. photogrammetry, gravimetric measurements and laser scanning) the positions of the sensors are determined by Global Navigation Satellite Systems (GNSS). Traditionally, this has been done with differential techniques using GNSS observations from one reference station only. The distance and height dependent errors may be significant in airborne differential positioning using one reference station only. In this work the alternative approach of Multi-Base-Station (MBS) differential processing is discussed and verified. In the MBS approach, data from a network of permanent reference stations are used to estimate the distance dependent errors. Improved reference data can be materialized as e.g. observations from a Virtual Reference Station (VRS). Results from a test flight over a dedicated test field in Fredrikstad, Norway, are presented. Reference positions are estimated using photogrammetric aerial triangulation and compared to the positions from various GNSS processing strategies. The results verify that the method of Multi-Base-Station processing gives significantly more precise and reliable results than the traditional approach using one base station only.

Autonomous point positioning was the original aim of GPS. However, due to errors caused by satellite ephemerides, satellite clock corrections, ionosphere, troposphere, multipath and noise, the accuracy of autonomous point positioning cannot be better than a couple of meters, even after Selective Availability (SA) was turned off.Nowadays, with the emergence of Point Real-Time Kinematic (P-RTK) positioning, a lot of attention is again focused on standalone positioning. This method arose from the availability of precise ephemeris and satellite clock corrections. The concept behind this method is to make use of un-differenced code and carrier phase observations along with precise corrections to get the best accuracy out of GPS. The concepts, challenges, processing steps and results of the P-RTK system are presented in this paper.Currently at the University of Calgary, a software system has been developed that achieves centimeter to decimeter level accuracy with a single GPS receiver. All components of P-RTK have been incorporated in this software, which is portable and user-friendly, containing many graphical interface tools for analyzing data.

One application in structural engineering is measuring the movement of suspension bridges. Under unfavourable wind conditions or heavy traffic loads cable suspension bridges may move up to a few metres. Therefore, a positioning system used to monitor bridge movements can provide extremely valuable information. For such monitoring applications it is desirable for the positioning system to deliver equal precision in all position components, all the time.GPS has the potential to deliver cm-level positioning accuracy, through use of the very precise carrier-phase measurement. However, the same level of accuracy cannot be guaranteed in all three-position components, twenty-four hours a day, in every situation.Additionally, GPS signals are very weak and easily obstructed, especially in built-up urban environments. One solution to improving the GPS satellite geometry and the availability of ranging signals is to use ground-based transmitters of GPS-like signals (called “pseudolites”). A major advantage of pseudolite devices is the ability to control where they are located, improving a geometrically poor GPS constellation.In this paper proof of concept for the use of pseudolites in the monitoring of suspension bridges is demonstrated through an experimental trial conducted on the Parsley Bay suspension footbridge in Sydney, Australia.

Analytical finite element (FE) modelling, modal testing and FE model updating are approaches which are employed to detect structural dynamic characteristics in the mechanical and aerospace engineering communities. Their application to civil engineering structures is more recent. To successfully apply these approaches, field data acquisition plays a very important role in validating and updating the analytical model.As one of the feasible data collection tools for modal testing, GPS has been employed with other sensors by the IESSG at The University of Nottingham to monitor large structural deformations. While demonstrating promising advantages in positioning precision and flexibility in instrumentation installation, the requirements from the structural engineering perspective need to be well defined before this technology can be fully implemented.In this paper prototypes of hybrid sensor systems coupled with a computational FE model are introduced. Data collected from a test bed bridge are processed and analysed to demonstrate the feasibility of proposed systems. The results reveal that it is possible to achieve 3D millimetre positioning precision for detecting high dynamic structural deformations, and obtaining a highly accurate FE model for the purpose of structural health monitoring (SHM).

In many cases landslide areas can be divided into several blocks, which are moving with different velocities in different directions. So, if we are able to detect block boundaries, landslide monitoring can be performed more efficiently. The information about the relative movement of these blocks is a very important indicator for future movement behavior, if monitored very precisely at the block boundaries with continuously measuring geotechnical sensors.To detect the boundaries of the blocks the following algorithm is used: the displacement vectors of the observed points (out of a geodetic deformation analysis) will be analysed by an affine coordinate transformation. The assignment of the observed points to the different blocks is done by an iterative algorithm; the thresholds for the several steps of the algorithm are calculated by a fuzzy system. The input parameters for this fuzzy system are e.g. the standard deviations of the transformations and strain parameters calculated from these transformation steps. Finally, an example for application of this fuzzy system will be given.

In the past there has been a wide range of research work on landslides. However, most of this research has been carried out by a single discipline such as geology or geodesy. Therefore an international and interdisciplinary project was started. The proponents of the project, sponsored by the European Union, believe that a multidisciplinary integration of different methods has the greatest potential for substantial progress in natural hazards management. The goal of the project is the development of methods that allow: The detection of potential landslides, an efficient and continuous observation of critical areas, the derivation of real time information about actual risks.This can be achieved by three steps: •Detecting of potential landslides by large scale monitoring.•Detecting of taking-off-domains by local scale monitoring.•Impact and risk assessment and development for strategies for alert systems.

The paper presents three examples of how geotechnical knowledge can be used for the analysis of geodetic deformation data. The knowledge is represented as facts and heuristic rules which are implemented in a knowledge-based system. The system is supposed to support the geotechnical expert on site by an automatic detection of specific deformation phenomenas. With this contribution the application field of NATM-tunnelling (New Austrian Tunnelling Method) is addressed.

We started the development of a quasi real time technique for spacecraft positioning by using the Internet-VLBI system in collaboration with the Institute of Space and Astronautical Science (ISAS). A series of VLBI observations receiving the signals from Japanese spacecraft GEOTAIL and NOZOMI have been carried out since June, 2002 to establish an observation method and to evaluate the measurement accuracy. At present it is possible to obtain the time delay which is a VLBI observable to use for satellite positioning within a few days after observations. Orbital determination software is under development in parallel to the improvement of observation data processing software. Phase delay measurement software is also under development to increase the measurement accuracy.

At present, promptness of the data processing of the international Very Long Baseline Interferometry observations is limited by the time required to ship observation tapes or disks from radio telescope sites to a correlation processing site. It can take more than two weeks to transport all the data tapes to the correlation processing site in typical observation sessions for Earth Orientation Parameter measurements. The e-VLBI technique uses high speed communication networks to transport observation data electrically and it has a possibility to dramatically improve the promptness of the data processing and hence the accuracy of the near real-time prediction of the Earth Orientation Parameters. To realize the e-VLBI observations by using a shared network environment, PC based data acquisition systems and data processing systems have been developed. By using the newly developed system, international e-VLBI sessions were performed twice and a rapid estimation of UTI-UTC in less than one day after the observations was demonstrated by the second session performed on June 27, 2003.

In April 2002 the International VLBI Service for Geodesy and Astrometry (IVS) set up the Pilot Project — Tropospheric Parameters, and the Institute of Geodesy and Geophysics (IGG), Vienna, was put in charge of coordinating the project. Seven IVS Analysis Centers have joined the project and regularly submitted their estimates of tropospheric parameters (wet and total zenith delays, horizontal gradients) for all IVS-R1 and IVS-R4 sessions since January 1st, 2002. The individual submissions are combined by a two-step procedure to obtain stable, robust and highly accurate tropospheric parameter time series with one hour resolution. The zenith delays derived by Very Long Baseline Interferometry (VLBI) are compared with those provided by the International GPS Service (IGS). At sites with co-located VLBI and GPS antennas, biases are found between the Global Positioning System (GPS) and VLBI derived zenith delays, although short-term variabilities generally show a good agreement. Possible reasons for these biases are discussed. Additionally, the time series submitted by the individual Analysis Centers are compared to the combined IGS time series for the CONT02 campaign of 15 days of continuous observations in the second half of October 2002.

In geodetic Very Long Baseline Interferometry (VLBI) the observations are performed at two distinct frequencies (2.3 and 8.4 GHz) in order to determine ionospheric delay corrections. This allows information to be obtained from the VLBI observables about the sum of electrons (total electron content — TEC) along the ray path through the ionosphere. Due to the fact that VLBI is a differential technique, only the differences in the behavior of the propagation media over the stations determine the values of the observed ionospheric delays. However, in a first simple approach, an instrumental delay offset per baseline shifts the TEC measurements by a constant value. This offset is independent of the azimuth and elevation of the observed radio source what allows separation of the ionospheric parameters for each station from the instrumental delay offsets per baseline in a least-squares adjustment. In first tests of this method Fourier coefficients up to the 4th order plus a constant value and a linear trend were estimated to represent the vertical TEC (VTEC). Slant TEC (STEC) values are converted into VTEC values by a mapping function. A disadvantage of this approach is the assumption that these values are assigned to the station coordinates but not to the geographical coordinates of the intersection point of the ray path and the infinitely thin ionospheric layer. The precision of the estimated values is about ±5 to ±7 TEC units (TECU). The results obtained from VLBI agree with a standard deviation of ±10 TECU with other techniques like GPS, and the differences rarely exceed 20 TECU. A second approach, developed at the TU Vienna, using piecewise linear functions (VTM — Vienna TEC model) is also tested.

Space Very Long Baseline Interferometry (SVLBI) is primarily a radio astronomical technique, which extends conventional VLBI baselines into the space. Its potential applications in Global Geodesy include interconnection of celestial and terrestrial reference frames, and orbit determination using the delay and delay rate observables. In the past two decades, there have been significant theoretical developments in this field. With the launch of the first SVLBI satellite of the VLBI Space Observatory Programme (VSOP), Japan, this technique has now been realized. The Geodesy Demonstration Experiment (GEDEX) is designed to explore the feasibility of the geodetic applications of SVLBI. In this paper, some results of the numerical simulations for geodetic parameter estimation from VSOP data are presented.With the availability of real data from VSOP, it is necessary to repeat the precision analysis carried out earlier, in order to re-estimate the precision under realistic conditions. Numerical simulations for this purpose have been initiated, and the first results are presented in this paper. Conclusions are drawn, based upon the results obtained, and the implications for possible dedicated geodetic SVLBI missions in the future are discussed.

When very long baseline interferometry (VLBI) is applied to observation of finite distance radio source from observer and the curvature of the the wave front cannot be approximated as plane wave. Thus current standard VLBI model (‘Consensus model’) does not have enough accuracy and an alternative precise VLBI delay model for finite distance radio source corresponding to the ‘Consensus model’ is required in observation of radio source in the solar system (e.g. planets, asteroids, and spacecraft). We derived a formular of relativistic VLBI delay model for finite distance radio source by taking into account coordinates transformation by using linearized PPN metric. This model is valid in order of several pico seconds accuracy when the radio source is at distance beyond 109 m. Our model include the ‘Consensus model’ as a special case that radio source is at infinite distance form the observer.

Although Very Long Baseline Interferometer (VLBI) operations require considerable resources compared to GPS, VLBI is an essential tool for geodesy in maintaining International Terrestrial Reference Frame and providing Earth Orientation Parameters for precise GPS surveying. This paper reviews GSI’s VLBI programs focusing on the synergy with GPS in Japan. One of the largest fruits is the realization of Geodetic Coordinates 2000 based on the inter-technique solution of VLBI and GPS for national geodetic application.

In Japan’s lunar mission called SELENE (SE-Lenological and ENgineering Explorer), detailed measurements of the lunar gravity field will be made by using two methods. The first is the differential VLBI measurement with two sub-satellites, and the second is the four-way Doppler measurement by using a sub-satellite which relays radio waves from the main lunar orbiter. The lunar far-side gravity field will be directly measured for the first time by using the second method, which will give us new information about the inner structure of the Moon. For the four-way Doppler measurement, the confirmation of the link establishment is an important issue because there is no telemetry relay from the main satellite to the sub-satellite in SELENE. The instruments are now under development, and a compatibility test of the instruments to the ground antenna station was carried out in March 2003. We have applied recently proposed algorithms for the confirmation of the link establishment and confirmed that they work successfully for data obtained during this test. The accuracy of the 2- and 4-way Doppler measurement are also measured and confirmed to be the expected level.

Four-way Doppler measurements and differential VLBI will be executed by SELENE to obtain highly accurate global mapping and improve the model of lunar gravity. Two small sub-satellites: Relay Satellite (Rstar) and VLBI Radio Satellite (Vstar), which are separated from SELENE Main Orbiter, perform these selenodetic observation using Relay Satellite Transponder (RAST) and VLBI Radio Sources (VRAD). These sub-satellites have no thrusters to control orbits and attitude to yield precise measurements of orbits perturbed by gravity anomaly. Lack of thrusters, however, causes the instability of attitude that interfere the selenodesy observation. The tip off at the separation and the solar radiation pressure torque dominantly affect the attitude, hence, we evaluated them in the critical design phase of the sub-satellite development. Properties of the newly developed release mechanism were obtained by ground tests, and the satellite attitude inclination by disturbances was analyzed. These results show that the design of the sub-satellites has enough properties to produce selenodesy data.

We are proposing a selenodetic mission, e.g. In situ Lunar Orientation Measurement (ILOM) to study lunar rotational dynamics by direct observations of the lunar physical liberation and the free librations from the lunar surface with an accuracy of 1 millisecond of arc in the post-SELENE project. Year-long trajectories of the stars provide information on various components of the physical librations and they can also be used to possibly detect the lunar free librations in order to investigate the lunar mantle and the liquid core. The PZT on the moon is similar to that used for latitude observations of the Earth except the half mirror above the objective, a CCD with high well capacity, and the heater beneath the mercury pool. Although a star position on the focal plane does not change even if the telescope inclines in principle, the tilt of the telescope affects the star position due to aberrations of the objective in the actual case. We obtained the relation between the deviation of the star position in CCD and the tilt of the telescope by ray tracing method and found that the effect of the tilt less than 100 arc seconds upon the star position does not exceed 1 mas. Thermal test of a tube made of CFRP showed that the tube did not incline by more than 100 seconds of arc even in the conditions of the lunar surface. We have a prospect to attain an accuracy of positioning of better than 1 mas from simulated experiments in laboratory using a CCD.

GPS plays an important role for the definition of the terrestrial reference frame and for the monitoring of Earth orientation parameters based on observations acquired by the global IGS (International GPS Service) tracking network. With an increasing number of low Earth orbiters (LEOs) carrying high-quality dual-frequency GPS receivers the question arises whether the inclusion of orbiter receiver data may help to further improve parameters estimated in a global GPS data analysis.In the following study the satellite JASON-1 was introduced a posteriori into the global analysis performed at the CODE Analysis Center of the IGS. A relatively simple strategy was used allowing for a double difference data analysis applying correct correlations and taking advantage of the fact that preprocessed files with fixed ambiguities were already available for the ground network.An improvement of the LEO orbit as a result from the combined processing could not be clearly shown. The impact on GPS orbits and geocenter coordinates caused by the inclusion of JASON is, on the other hand, significantly larger than expected from the addition of one receiver to a 120 station tracking network. This indicates a potential for improvements with respect to these parameters by means of a combined processing. Modeling insufficiencies and inconsistencies, however, need to be solved to reach this goal.

Reduced-dynamic orbit determination for spaceborne GPS receivers of low Earth orbiting satellites is a successful method promising highest precision. We review the principles of (reduced) dynamic orbit determination and develop the mathematical background for different efficient stochastic orbit parametrizations (e.g., piecewise constant accelerations which provide not only continuous but also differentiable orbits) using least squares methods. Simulated as well as real data from the CHAMP GPS receiver show, to some extent, the equivalence of the different parametrizations and reveal the impressive performance of stochastic orbit modeling techniques. Independent comparisons with orbits determined by other groups and validations with SLR measurements show that our orbits are of high quality.

In this paper we first present approaches and results in precise orbit determination (POD) for satellites in Low Earth Orbit (LEO) based on one or two frequency GPS measurements and, secondly, we focus on the relations between kinematic POD and gravity field determination. Using GPS measurements of the CHAMP satellite we show that it is possible to estimate kinematic positions of a LEO satellite with the same level of accuracy (≈ 1–3 cm w.r.t. SLR) as with the widely applied reduced-dynamic or dynamic approaches. Kinematic precise orbit determination (POD) as presented here is based on GPS phase measurements and is independent of satellite dynamics (e.g. gravity field, air-drag, etc.) and orbit characteristics (e.g. orbit height, eccentricity, etc.). We also looked at the LEO POD based on GPS measurements from only one frequency, where we make use of what we call the LP linear combination. We show that with this linear combination LEO POD can be performed with one frequency at the 10 cm level. In the second part of the paper the use of kinematic POD is discussed in the framework of the CHAMP, GRACE and GOCE gravity missions. With simulated GPS measurements we studied the impact of ambiguity resolution for the kinematic baseline between the two GRACE satellites. At the end of the paper we present an alternative approach in gravity field determination by measuring the gravitational frequency shift between (optical) atomic clocks in space. In this approach we require very accurate clocks positions, which may be obtained from kinematic POD.

A procedure for gravity field determination is presented based on the analysis of short arcs of low flying satellites as CHAMP and GRACE. The analysis technique can be applied to high-low as well as to low-low satellite-to-satellite tracking in a consistent way. The method allows to recover the global gravity field of the Earth, combined with a regional gravity field refinement in regions with rough gravity field features. To detect regional residual signals in the observations a preprocessing step is being applied based on the energy balances along the short arcs. This procedure will be applied as well as the validation of the regional gravity field refinement in a post-processing step. First results are presented by using Post-processed Science Orbits (PSO) of CHAMP for a region with rough gravity field features. To demonstrate the applicability to a tailored low-low satellite-to-satellite tracking experiment the procedure is applied to a GRACE simulation scenario.

Satellite altimetry and tide gauges provide observations of the instantaneous sea surface. As a further source an oceanographic model of the Baltic Sea also provides sea-level heights. The different sea-surface heights are of comparable precision and complement each other well in terms of their spatial and temporal resolution and their information content.When the different sea-level information are combined, variations in the time series can be reduced, the quality of the observations can be checked, and phenomena of special interest can be separated.One application for the combination of observed and modelled heights is the estimation of secular sea-level changes. By reducing a large part of the high frequency sea-level variations in the observations, the trend estimation from short time series can be improved. This is of interest especially for satellite altimeter observations, which, though having a high spatial resolution, cover a rather short time span.

Cooperative campaigns by space geodetic techniques have been actively carried out around the world to develop global GIS infrastructures. Asia Pacific Regional Geodetic Project (APRGP), as a regional campaign, has promoted annual observations mainly by GPS to densify the International Terrestrial Reference Frame (ITRF). To construct an ITRF based frame, not only positions but also velocities are necessary. Nevertheless, establishing a velocity field in the region has not been achieved yet. In this paper, we estimate a velocity field, using GPS data obtained from 1998 to 2001 campaings. The result shows an excellent agreement with the plate model.

In 1999, an International DORIS Experiment was initiated under the aegis of the International Association of Geodesy. Since then, 3 new DORIS satellites have been launched carrying improved Doppler receivers. In addition, the DORIS tracking network has also been steadily improved. In 2001, a DORIS analysis campaign was launched. Several groups participated using different software and providing weekly and monthly time series of results, making them available on Internet. The goal of this paper is to discuss the status of the DORIS Pilot experiment, to analyze the progress made during the past 2 years and to propose the official creation within the IAG of an International DORIS Service (IDS). Such a service would operate in conjunction with the already existing IGS, IVS and ILRS as part of the proposed International Global Geodetic Observing System (IGGOS).

One promising method for the external validation and calibration of the upcoming GOCE satellite mission data is the use of ground gravity field data continued upward to satellite altitude. There is a unique situation for Hungary in this respect since surface gravity gradients are available at 20143 points over an approximately 48700 km2 area, measured by the classical E6tv6s torsion balance. The concept of this contribution is to test the usability of these point gravity gradient observations for upward continuation to the GOCE satellite orbit in combination with different geopotential models and other gravity field information.The computations are based on the least squares collocation method and the direct numerical integration of the torsion balance data. For the latter method, the spectral combination technique and the classical integration kernels are considered. Furthermore, various other data sources, such as the T_zz gravity gradients based on the gravity and terrain data collected within the frame of the European Geoid Project, are utilized for comparisons.Besides the comparisons between the different satellite gravity gradient computations, an error analysis of the results is presented.

Absolute gravity measurements have been repeatedly made with the FG5 #210 at the superconducting gravimeter station in Matsushiro, Japan. Four out of the five experiments so far performed were successful. The data from the absolute gravimeter are used to calibrate the instrumental sensitivity and drift of the superconducting gravimeter TO 11 with good results. Drift of the absolute gravimeter is investigated at the same time. Long-term gravity changes are discussed by collocating the data from the absolute and superconducting gravimeters.

We started the development of gravimetry using helicopter in 1998 and attained a preliminary success in gravity measurement in 2000. For this purpose we have manufactured a new gravimeter (SEGAWA Model) with a servo accelerometer as a gravity sensor and an optical fibre gyro to keep vertical. Our success owes mainly to precise positioning of helicopter, and delicate correction for the effect of helicopter’s horizontal acceleration. We have so far conducted test and/or practical measurements by helicopter (mostly using Bell 412). The advantages of a helicopter over a fixed-wing aircraft in gravity measurement is the high resolution of gravity anomaly obtained because of low altitude flight and low and stable flight speed. The main objective of the measurements is to get continuous profiles of gravity anomalies in the land-to-sea boundary zones, so that gravity void or data discrepancy across coastal zones may be removed and that the active seismic faults running across coasts, which are reflected on gravity anomalies, may be retraced as far as the sea floor. In this paper two results have been selectively described: One is the result of measurement conducted in April 2000 along west-to-east tracks from Saitama to Kashima-Nada Sea, which has disclosed inconsistency between land and marine gravity. Another is the result of survey along the east-to-west tracks in the Enshu-Nada Sea (Tokai area), which shows the changes of free air anomaly and/or gravity disturbance from the coast to the sea floor associated with active seismic faults.

In order to investigate subsurface density structures with the highest precision, two major gravity databases and some unpublished data files, which cover Southwest Japan, are recompiled. This paper presents some examples of the detailed gravity anomaly maps in Kyushu District, southwest Japan. These maps indicate various fine subsurface structures, which may arose with tectonic movements and/or volcanic activities of this region. With an aid of geological and topographical information, we discussed relationships between characteristic patterns of the detailed gravity anomalies and subsurface structures in this region. The most conspicuous feature is an existence of a steep horizontal gradient zone of gravity anomalies running through the Aso Caldera in the ENE-WSW direction. Although the location of the westward extension of the Median Tectonic Line (MTL) in this region is still debatable, this zone seems to be the westernmost part of MTL.

This paper examines gravity structures of the Ishikari Lowland of Hokkaido, Japan, by focusing relief-shaded Bouguer gravity, specifically to relocate the faults (the Ishikari Teichi Touen Fault Zone, ITTFZ), and to present gravity-based fault mapping as another approach for estimate of fault segmentation. Gravity analyses are strictly based on dense gravity data measured by various institutes and organizations. Bouguer relief is produced by illuminating the light from eight directions to effectively display the detailed gravity features varying laterally along the azimuthal direction. A striking linearity of the relief-shaded Bouguer gravity along the ITTFZ is found on most of the relief maps, particularly on the maps for the azimuth of the due east and west directions. In the central part of the ITTFZ, however, the lineament in the relief-shaded Bouguer map does not bear a good correlation with the known fault distributions. In addition, the gravity relief for the azimuth of the south-west direction exhibits a remarkable lineament, extending south-ward from near Bibai, whose southward continuation can be traced to the south-east until 20∼25 km south of Atsuma along the westernmost boundaries of pre-Neogene volcanics. This implies that the geometry of the southern end of the ITTFZ provides distinct continuity along the relief-shaded Bouguer lineament roughly to the south-east.

Various institutions have collected shipborne gravimetric measurements during the last decades. Due to different standards used for the processing of the observations and the necessary corrections, significant inconsistencies exist between different cruises. This contribution aims at producing a consistent marine gravity data set surrounding Europe, which can then be used for high precision geoid modelling, dynamic sea surface topography estimation, and other applications.Besides our own marine gravity data holdings, data were collected from the Bureau Gravimétrique International (BGI), the National Imagery and Mapping Agency (NIMA, formerly DMA), and the National Geophysical Data Center (NGDC). The area of investigation is spanning the latitudes from 10 °N to 90 °N and the longitudes from 60 °W to 60 °E. The quality of the data varies between the individual cruises, as they originate from many projects at different epochs. Hence, systematic errors are likely to exist. Such errors can be significantly reduced by a crossover adjustment of the individual ship tracks. Because the track information was not available for all cruises, it had to be regenerated by different procedures. Furthermore, duplicate sources were removed before the crossover adjustment. The crossover adjustment is based on a bias per track error model. The adjustment of about 1.5 million observations in nearly 17,000 tracks led to a consistent high quality marine gravity data set. The RMS of the about 80,000 crossover differences is 15.5 mgal for the original data set, 8.4 mgal for an edited data set, and 4.7 mgal for the final crossover adjusted data set.The second part of this contribution describes the evaluation of the marine gravity data set by altimeter derived gravity anomalies from different sources. These comparisons also prove the effectiveness of the crossover adjustment.

The determination of the geoid represents a central task in geodesy. In this matter airborne gravimetry provides a fast and reliable method for geoid determination especially for those regions, which are difficult to survey. In addition with respect to the spectral representation of the earth’s gravity field in terms of spherical harmonics airborne gravimetry is capable to fill the gap with regard to the medium wavelength contribution. With current airborne gravity systems an accuracy of 2–3 mgal (rms), with a resolution of 6–15 km can be achieved. Data processing becomes more significant to meet the demands for a geoid with centimetre-accuracy. Besides the task of pre-processing which results in band-limited gravity disturbances at flight level, the choice of a proper downward continuation method gains in importance.Besides the problem of regularization most calculations are based on block mean values, which has disadvantages in the approximation accuracy. This paper will compare two different approaches of downward continuation of simulated airborne gravimetry data. In the first approach gravity disturbances derived at flight altitude are directly transformed to the disturbing potential at geoid level. This method derives the disturbing potential in terms of block mean values, which are not continuously differentiable. With respect to that problem an alternative approach is proposed. This method will make use of space localizing spline functions to represent the disturbing potential at geoid level. It will be shown that this will increase the accuracy regarding the resulting geoid noise.

This paper describes experiments conducted to examine the combination of gravity values predicted from bathymetric data with observed gravity data to increase the resolution of offshore gravity data. The density parameter in the prediction is optimised empirically.The gravity effect resulting from bathymetric features can provide short wavelength features of the gravity field, but at longer wavelengths geophysical features below the seafloor begin to influence the gravity signal. A model of the isostatic response might help to determine these longer wavelength features, but these models are inconclusive so in this case the calculations are restricted to the bathymetry.By using a combination of observed gravity and gravity predicted from bathymetry, the long wavelength features present in the direct observations can be preserved while using the bathymetric data to fill-in the short wavelength features.The computation scheme is described, and some numerical results are presented. The effect of increasing the distance between the observed gravity data is examined.For the calculation of the gravity signal arising from the bathymetry, a single density value is chosen for the entire area. Due to density variations this density may not be optimal. Different density values are tested to examine the effect this choice has on the resulting predicted gravity.

GOCE will be the first satellite ever to measure the second derivatives of the Earth’s gravitational potential in space. With these measurements it is possible to derive a high accuracy and resolution gravitational field if systematic errors have been removed to the extent possible from the data and the accuracy of the gravity gradients has been assessed. It is therefore necessary to understand the instrument characteristics and to setup a valid calibration model. The calibration parameters of this model could be determined by using GOCE data themselves or by using independent gravity field information. Also the accuracy or error assessment relies on either GOCE or independent data. We will demonstrate how state-of-the-art global gravity field models, terrestrial gravity data and observations at satellite track crossovers can be used for calibration/validation. In addition we will show how high quality terrestrial data could play a role in error assessment.

The primary instrument on board of the future ESA gravity field observing satellite GOCE is a gravity gradiometer. The observations from this instrument will be passed through a 7th-order Butterworth filter in order to reduce high-frequency noise and aliasing due to the output rate of 1 Hz. This filtering will distort the gravity gradients that are to be measured, which can however be largely compensated by a time tag correction. An assessment has been made of the remaining observation error after this correction and its impact on gravity field recovery performance. It has been found that the filtering leads to a very small increase of the gravity field recovery error budget at the medium to short wavelengths after application of the optimal time tag correction. Only at the very long wavelengths a slight degradation might take place. However, it is anticipated that the GOCE gravity field recovery will be improved/dominated by GPS satellite-to-satellite tracking observations at these long wavelengths.

Least-squares collocation (LSC) has been used to test frame transformation, filtering and gridding of GOCE gradiometer data. Initially the function of the GRAVSOFT program GEOCOL was checked by using noise-free data with 5 s sampling provided by IAG along a 1 month realistic orbit. Data in a 2° × 2° area was used. For these data it was verified that 0.1° frame transformation and gridding in an interior 1° × 1° area of one quantity from the same quantity could be done with an error below 5 % of the signal standard deviation. If several components of the gravity gradient were used, the error decreased slightly.Noise with standard deviations of 0.050 and 0.150 Eötvös Units (E=10−9/s2), respectively, and a 1 s correlation distance was generated and added to the data. For the 0.050 E noise the result of the gridding was as for the data without noise, when 1 s sampling was used. For the 0.150 E data, the use of totally 2 months of simulated data and all 3 diagonal components of the gradient matrix were necessary in order to obtain a result below 10 % of the signal standard deviation. Further improvements are possible if data from a larger area or from the planned second measurement phase of GOCE are used.

The high frequency accelerometer can provide non-gravitational perturbation force acceleration acting on CHAMP satellite. In CHAMP standard science product level-2 data, the acceleration, star camera, thruster firing and some corrections data are available. They play important roles in the determination of the earth gravity field. In this paper, the level-2 data (Day of year 1–100, 2002) are processed. Before applying the corrections, the curves of mean acceleration per day are very rough. After corrections, the curves show clear trend. On the other hand, the precise orbit of CHAMP is necessary for the determination of the earth gravity field. Now, the rapid science orbit data are provided by two organizations, GFZ and JPL. The data of a randomly chosen period (Day of year 267–269, 2002) from the two organizations are compared. The results of the comparison show that the difference between the two RSO data is notable, especially at the end part of orbit files. After getting rid of the end part of each file, there is no obvious system bias exists between the two RSO data and the magnitude of the difference keeps only about 0.1 m. Finally, some conclusions are drawn and some suggestions are given.

A full year of champ gravity field solutions has been calculated using the energy integral approach. The monthly solutions in the time frame 03.2002–02.2003 were based solely on kinematic orbits from champ gps orbit tracking and accelerometry. These kinematic orbits have not been contaminated by a priori gravity field information.Recovery of medium-wavelength time-variable gravity signal from champ is expected to be at the edge of feasibility. This expectation is validated by comparison to an SLR solution of seasonal gravity variations and by comparison to oceanographic and meteorological models. It is shown that the error level of monthly CHAMP solutions is insufficient for revealing these time variations. Orbit decay—and consequently ground track variation—is a main contributor to this effect.

In satellite gravity gradiometry, the gravitational signals from the Earth’s topography and its isostatic compensation still exist in the gravity gradients observed along the satellite orbit. Due to the high-frequency behaviour of the combined topographic-isostatic effect, downward continuation of the gravitational signal from satellite height to sea level is rather difficult, requiring some mathematical method of regularization. On the other hand, the complete calculation of topographic-isostatic effects according to, say, the Airy-Heiskanen isostatic model is too laborious for the practical evaluation of gravity gradiometry data. In this paper, another approach is proposed, which is based on a generalized condensation model corresponding to Helmert’s first condensation model; here the condensed masses are assumed to be situated on a surface at a constant depth D below the geoid. The respective formulae representing the effects of the topographic and condensation masses on the vertical gravity gradient (V_rr) at satellite level are derived. A simulation based on the JGP95E rock-equivalent terrain model proves that the order of magnitude of both topographic and condensation effects is about 10 E.U. The magnitude of the combined topographic-condensation effect is much smaller, amounting to 0.6 E.U. and 0.06 E.U. for Helmert’s first and second condensation model, respectively.

Long-wavelength differences (in the ∼1,100 km to ∼6,500 km waveband) are evident between Australian free-air gravity anomalies and those implied by EIGEN-2 to degree 32, which is derived purely from CHAMP data. They reach ∼11 mGal for the Australian land gravity anomalies and over ∼15 mGal for the [unadjusted] ship-track gravity anomalies. These long-wavelength differences are used to quantify the effect on the gravimetric geoid model. Using an unmodified Stokes kernel and no limit on the cap radius (i.e., all terrestrial gravity in the region), the resulting effect on the geoid is between −0.72m and 0.13m, though the large negative value is attributed to a lack of gravity data to the north of Australia. Using a modified Stokes kernel and a one-degree cap radius as a high-pass filter, the resulting effect on the geoid is between −0.28m and 0.10m. Applying the latter and EIGEN-2 to degree 32 as corrections to AUS-Geoid98 yields a small, but insignificant, improvement in the fit to nation-wide GPS-levelling data.

A set of new gravity data, not previously available in the Nordic NKG data base, was obtained from NIMA. Based on existing gravity data and these additional data several different new geoids were estimated. The inclusion of the NIMA data set give local geoid changes of up to a few dm.The new CHAMP based EIGEN-2S geopotential model was compared to the EGM96 model using a two step procedure. Results from the two steps show that the EIGEN-2S geopotential model give better fit to GPS and MSSH/MDT data than EGM96 up to degree 50, but applying the full EGM96 model still gives the best resulting geoid.

Using a realistic orbit for GOCE, IAG SC7 has created a one month gravity gradient dataset from EGM96 to degree 300, with gradients referring to an instrument frame aligned with the velocity vector and the z-axis in the plane formed by this vector and the position vector. From the second order derivative of the potential V _zz , we subtracted the contribution of EGM96 to degree 24. The resulted (noise free) data set was used to predict radial gravity gradient values in a 0.5° grid, covering the area of the Earth from −83°to + 83° latitude using local Least-Squares Collocation (LSC). The standard deviation of differences between predicted gridded values and values computed from EGM96 (degree 24 – 300) was between 1.0 and 0.5mE (Eötvös unit = E, 1 E=10−9s−2). Correlated noise with a 3 mEU standard deviation and a 35° correlation distance was added to the simulated data and the gridding was repeated. The formal LSC error-estimates were 2 mE. This was confirmed by comparing radial derivatives from EGM96 with the values predicted from the data with noise. The simulated data sets were used to generate spherical harmonic coefficients of the gravity potential to degree 300 using Fast Spherical Collocation (FSC), with a global covariance function. Both, a grid of noise-free data and a grid obtained from the data with noise were used. Both results agreed with EGM96 within the error-bounds of the FSC estimate.

The homogeneous and repeated data coverage over coastal regions from satellite radar altimetry is one important data source for oceanographic and geodetic applications. However, the sea surface heights (SSH) extracted from the altimeter data are often in error close to the coast, due in part to the complex nature of echoes returned from rapidly varying coastal topographic surfaces (both land and sea) and the generally rougher sea state. This paper presents improved SSH results derived from ERS-2 altimeter waveform data (two cycles of 35-day repeat orbit, March to May 1999) near the Australian coast using a coastal retracking system. This system was developed based upon a systematic and comprehensive analysis of return waveforms. Central to the system is the use of two retracking techniques: the iterative least squares fitting and the threshold retracking algorithms. Using the AUSGeoid98 geoid grid as a quasi-independent ground reference and comparing with a broad contaminated distance of ∼10 km obtained from a previous study before retracking, the use of the retracking system is able to reduce this contaminated distance to ∼5 km. However, improved SSH data cannot be recovered by waveform retracking less than ∼5 km from the coastline due to predominant land returns in the waveform range window.

The paper presents numerical results for the solution of the altimetry-gravimetry boundary value problem with compatibility conditions along the coastline. The solution is based on the application of spherical Shannon wavelets based on the Abel-Poisson kernel and pseudo-diffferential operators. The orthogonality of Shannon wavelets can be used for the numerical solution overcoming some technical problems with generally non-orthogonal spherical wavelets.A numerical experiment has been conducted in eastern Canada and the results have been compared with the geoid heights computed from gravity anomalies by applying ID FFT spherical Stokes convolution with 50 km integration radius. The effect of compatibility conditions along the coastline on the solution are discussed and the following conclusions are drawn: spherical Shannon wavelets can be very successfully applied in the numerical solution of the AGBVP II and the smoothing effect of spherical wavelets is much more effective than the classical planar type of wavelets.

A series of KMS high resolution near global gravity fields from altimetry have been released during the past 7 years. This presentation focuses on describing the improvements leading to the release of the new KMS2002 field.Significant improvements have been achieved through careful fine-tuning of the procedure and parameters used for the derivation of gravity from satellite altimetry.Particularly emphasis has been made in improving the gravity field determination at medium wavelengths ranging between 100–500 km, and at shorter wavelength in coastal regions. Emphasis was made in recovering the gravity field in lakes, which have normally been deleted from global marine gravity field maps. Extensive comparisons carried out at the National Imagery and mapping Agency is also presented to document the improvements made.

In the frame of the EU-sponsored GAVDOS project the need of a new high-resolution and high-accuracy geoid model for the calibration of altimeters onboard satellites like JASON-1, ENVISAT and EURO-GLOSS and for sea level monitoring purposes has become apparent. That was mainly due to the fact that the already available models have been estimated using outdated datasets and fail to meet the wanted, cm-level, accuracy requirements. To determine the new geoid models multi-satellite (ERS1 and GEOSAT) altimetry and land and marine gravity data have been used. The EGM96 global geopotential model has been employed, while the effect of the bathymetry has been taken into account using recently developed local Digital Depth Models (DDMs). Several solutions have been estimated based on the different datasets used and the two main methodologies followed, i.e., the Fast Fourier Transform (FFT) based Input Output System Theory (IOST) and Least Squares Collocation (LSC). The accuracy of the new models was assessed through comparisons with TOPEX/POSEIDON (T/P) data and the GEOMED geoid solution for the area under study. Finally, the consistency between the estimated solutions has been determined by comparing the geoid height value they provide at the Gavdos Tide Gauge (TG) station on the isle of Gavdos. From the results it was found that the precision of the new geoid models is between ±0.9 and ±3.3 cm, their accuracy ranges between ±5 and ±10 cm and their consistency is at the ±0.5 − 6 cm level.

New Zealand currently uses 13 disparate vertical datums, each connected to a separate tide gauge. In 1998, a new national datum, NZGD2000, was implemented based on GPS observations. This leads to a 3D geocentric datum. As further development of the spatial infrastructure in New Zealand, Land Information New Zealand approved the implementation of a new national vertical datum that is independent of local mean sea level. This new national vertical datum will be based on ellipsoidal heights, and the relationship between the separate existing vertical datums relative to the ellipsoid will be established using a high-precision regional gravimetric geoid model.Phase one of this programme is the development of a regional geoid model for New Zealand. This paper will present the current status of development of the regional geoid model. Two geoids have been computed to determine the effectiveness of a ‘gravity reconstruction technique’ in New Zealand. The models computed are based on a combination of the EIGEN-2 satellite-only global geopotential model, which uses CHAMP dedicated satellite gravity data, and EGM96. Residual geoid undulations were computed from 40,000 land gravity observations and satellite altimeter-derived marine gravity anomalies. GPS and first-order spirit levelling data was used in conjunction with the geoid model to estimate offsets among the 13 vertical datums.

The vertical gravity gradients play an important role in the reduction of absolute gravity measurements and in the geoid determination, too. In order to enhance the precision of the gravity reductions and the geoid computations, the difference between the vertical gravity gradient of the real and the normal gravity fields should be taken into account.In this paper the vertical gravity gradients are computed by prism integration using constant density for the crustal masses and a high-resolution (gridsize 10m) digital elevation model to represent the topography in the Sóskút test area. On the other hand, gravity gradients are also predicted from point gravity data using spectral techniques.The calculated gradient values are compared to each other, and are validated using terrestrial measurements at 6 points in the Sóskút test area. The measurements of the vertical gravity gradients were carried out by the Loránd Eötvös Geophysical Institute and have an accuracy better than 80E, and are within the interval of 3189–3805E. The vertical gradient values calculated from the DEM range from 2926 to 3330E, and the values fitted the measured values with the standard deviation of ±244E, and the maximal residual of 547E.The vertical gravity gradient has been investigated by Csapó — Papp (2000) in the study area, using the lithospheric model developed by Papp. Their results fitted to the observed values with the standard deviation of ±561E, with a maximal residual of −1298E.

A synthetic Earth gravity model (SEGM) generates exact gravity field quantities and therefore is well suited to validate methods used in gravity field modelling. This paper describes the construction of a SEGM by forward gravity modelling using available data describing the Earth’s mass distribution. In particular, the global 5-arc-min × 5-arc-min JGP95E digital elevation model (DEM), the most recent global compilation of crustal thickness and mass heterogeneities (CRUST2.0) and the S12WM13 mantle model of seismic velocity anomalies have been used. Results for the SEGM show in general a good agreement with the anomalous Earth’s gravity field (geoid height). However, in some areas the differences with respect to the observed gravity field almost reach the magnitude of the geoid height itself. Possible explanations can be un-modelled mass anomalies situated in the upper and middle mantle.

Different isostatically compensated gravity models are assessed in view of a synthetic reference Earth gravity model. Among the models considered are a model produced by the analysis of the density and stratification information of the CRUST 2.0 global crustal database, and an isostatically compensated gravity model based on a simple spherical layer compensation mechanism. The deduced coefficients are used for direct computations of geoidal heights in a global scale. Appropriate filtering of the models relates the computed fields to known structures of the Earth’s interior, like the topography of the core-mantle boundary (CMB).

Usually the gravity field of the Earth is modeled by means of a series expansion in terms of spherical harmonics. However, the computation of the series coefficients requires preferably homogeneous distributed global data sets. Since wavelet functions localize both in the spatial and in the frequency domain, regional and local structures may be modeled by means of a spherical wavelet expansion. Wavelet-based techniques allow the decomposition of a given data set into frequency-dependent detail signals. This paper deals with two methods to achieve a multi-resolution representation, namely a wavelet-only solution and a combined approach. The latter consists of a spherical harmonic expansion for the low-frequency part and a spherical wavelet expansion for the remaining medium- and high-frequency parts of the gravity field. Parameter estimation procedures are presented to determine the unknown series coefficients of both approaches.

For the computation of high resolution regional geoid models, gravity and terrain data in connection with a global geopotential model play a very important role. The data sets are usually combined in a remove-restore procedure. In many cases, the transformation from gravity anomalies to geoid undulations is done using Stokes’s integration kernel or a modified integration kernel, e.g., based on the spectral combination technique. Least squares collocation may be used for this task as well, but for continental-scale computations the integration techniques are often preferred due to their high computational efficiency.Besides the classical integration techniques, the wavelet technique is investigated in this contribution. The wavelet technique also uses residual gravity field quantities in a remove-restore procedure. However, the computations are carried out in two steps. The first step consists of a convolution of the residual gravity data with several wavelet functions, being contracted or dilated variants of one prototype (“mother”) wavelet function. This leads to a decomposition of the whole spectrum of the original data into a set of filtered detail signals with unique spatial resolution. This type of space and frequency analysis is called multi-scale analysis (MSA). The second step then convolves the residual gravity details with an integration kernel (e.g., Stokes) and leads to corresponding geoid undulations. The second step, applied to every decomposed detail (scale) of the original data, corresponds to the classical integration techniques.In this contribution, both the classical integration and spherical wavelet techniques are applied using Europe as a test area. The differences in methodology and numerical performance of both techniques are investigated. Finally, the results are evaluated by independent GPS and levelling control points.

Precise geoid models are essential for the conversion of GPS-derived heights to heights above sea level. Such a model is under development for the continent of Africa, as part of the African Geoid Project. A uniform 5′ grid of gravity anomalies has been derived from terrestrial gravity data and has been combined with a 5′ grid for the marine areas derived from satellite altimetry. The combined data set has been used with the EGM96 geopotential model in a remove-restore process to compute the geoid using two-dimensional convolution. The final result is a 5′ grid of geoidal heights covering the land mass of Africa. There are significant gaps in the available terrestrial gravity data — these gaps, coupled with the effects of errors in the DEM used for calculating the G₁ term and in interpolating gravity anomalies, mean that the accuracy of this geoid model will be variable and generally less than desirable. Nevertheless, comparison with GPS/levelling data covering a small part of South Africa shows an RMS agreement of better than 10cm. Over a larger region (all of Egypt) the agreement is less satisfactory.

The determination of an optimal, in terms of resolution and accuracy, marine geoid model for the Atlantic coastal region of Argentina is investigated using satellite altimetry and shipborne gravity data. The altimetric data used are those of the geodetic phase of the ERS 1 mission while marine gravity data have been employed as well to determine a gravimetric geoid solution. Furthermore, the effect of the Quasi-Stationary Sea Surface Topography (QSST) was taken into account in correcting the altimetric Sea Surface Heights (SSHs) to derive geoid undulations. Special emphasis was placed on reducing the effects of the Sea Surface Variability (SSV) on the densely spaced altimetric SSHs with low-pass filtering. The satellite and shipborne data were combined in the spectral domain to improve the accuracy of the altimetric solution close to the coastline and derive a more rigorous solution. The accuracy of the final geoid models is assessed through comparisons with stacked TOPEX/POSEIDON (T/P) SSHs, known for their high precision. From the results achieved it was concluded that an altimetric geoid accurate to about 5–8 cm (1σ) is feasible in some areas, while the gravimetric solution gives poorer results by about 5–6 cm. The combination of satellite and shipborne data with the proposed algorithm improves the accuracy of the gravimetric geoid model by about 2 cm.

Newton’s integral can be evaluated either in the space domain by a direct integration approach or in the frequency domain by a spherical harmonic approach. In the first approach Newton’s integral is evaluated by a discretised numerical integration using the gravitational potential (or its derivatives) caused by regularly shaped bodies such as prisms or tesseroids (i.e., spherical or elliptical volume elements, respectively). The second approach expresses Newton’s integral by spherical harmonic expansions of height or density functions, which can be interpreted as the gravitational effect of different mass layers. This paper studies the theoretical and practical differences between both approaches. The gravitational effect of the global topographic masses, described here by equivalent rock heights, is evaluated using both approaches and the results compared. Numerical results for the gravitational potential are given for evaluation points located directly on the Earth’s surface and at different heights (e.g., flight levels) above it.

The paper presents a method of evaluating some singular geodetic integrals using wavelets. The wavelet transform is a powerful tool in evaluating some singular geodetic integrals. Due to its localization properties in both of the time (space) and frequency (scale) domains, and because the kernels of some geodetic integrals have singular points and decay smoothly and quickly away from the singularities, many numbers of wavelet transform coefficients of the kernels become zeros or negligible, and only small number of wavelet transform coefficients are significant. It is thus possible to significantly compress the kernels of these integrals in a wavelet basis by neglecting the zero coefficients and the small coefficients below a certain threshold. Therefore, wavelets provide a convenient way for efficiently evaluating these integrals in terms of fast computation and savings of computer memory. In this contribution, a modified algorithm for the wavelet evaluation of Stokes integrals is presented. And the same modified algorithm is applied to the evaluation of Vening Meinesz and terrain corrections integral, whose kernels have stronger singularities than of Stokes’ kernel. Numerical examples illustrate the efficiency and accuracy of the wavelet methods.

As a preliminary analysis for the development and evaluation of a precise gravimetric geoid for Argentina, different gravimetric geoid solutions were computed using different gravity reduction techniques in the roughest area of the country.The rugged area bounded by latitude 200° S to 420° S and longitude 720° W to 67° W was chosen to compute the geoids using the following terrain reductions methods: Helmert’s second condensation method, Rudzki’s inversion method, the Airy-Heiskanen topographic-isostatic reduction, and the residual terrain model (RTM) method. The remove-restore method was employed using the most homogenised gravity database (with data blunders removed) referenced to a unified datum, the EGM96 global geopotential model and GTOPO30 digital elevation model, to take into account the short-wavelength component of the gravity signal. Stokes’s formula was evaluated using the approximated spherical kernel by the two-dimensional fast Fourier transform algorithm.The external accuracy of the gravimetric geoid models was evaluated by comparing them with undulations derived from GPS/levelling data. A four-parameter transformation was used to remove the systematic datum differences between the gravimetric geoid and the GPS/levelling undulations, and the possible long wavelength errors of the geoid

In regional gravimetric geoid determination, it has become customary to utilize the modified Stokes formula, which combines local terrestrial data with a global geopotential model (GGM). A modification method, proposed by L.E. Sjöberg in 1984 (with later developments), allows least squares minimization of the influence of any error source in geoid modelling. In this approach, depending on the local gravity data quality, the chosen radius of integration, and the characteristics of the used GGM, the modification parameters s _n , vary. New satellite gravity missions are expected to improve significantly the accuracy of geopotential models. Of particular interest of this study is to evaluate the impact of future (i.e. post-GOCE) geopotential coefficients. A set of least squares modification parameters is determined from the system of linear equations, aiming at minimizing the global mean square error of geoid estimator. Some difficulties may be encountered when practically computing the modification parameters. In particular, for certain parameters the design matrix suffers from numerical ill-conditioning. Importantly, Tikhonov regularization is satisfactory in providing a solution for the modification parameters. Numerical results are presented to illustrate the applicability of the obtained parameters in geoid modelling by comparing with GPS-levelling data.

This paper reviews the new method (recently derived by L.E. Sjöberg) of the ellipsoidal correction for Stokes’ formula. Importantly, the correction can also be expressed in a series of spherical harmonics of the disturbing potential, which is a considerable computational advantage. In order to assess the applicability of this approach, it is numerically compared with two other methods. The results reveal that the new method practically coincides with one of the earlier methods, although their derivational approaches and resulting expressions are considerably different. The new method is adapted for the modified Stokes formula, which combines regional terrestrial gravity data with a global geopotential model. The magnitude of the ellipsoidal correction in the modified Stokes formula does not exceed cm level, globally.

For calibration purposes, oftentimes various datasets are compared in such a way that observations enter the coefficient matrix of a Linear Model ("errors-in-variables"). In such a case, the Total Least-Squares approach would be appropriate that was pioneered by G. Golub and C. van Loan in the early eighties. In essence, rather than solving the usual normal equations system for the estimated parameters, the smallest singular values of a slightly extended system is set to be zero, and its eigenvector is re-scaled to provide the estimated parameter vector. The authors have recently presented their studies that show the potential of this technique to provide improved variograms for geostatistical Kriging applications.Sometimes, however, stability or slow convergence problems may occur with the algorithm as designed so far. In order to increase the stability, additional parameters could be introduced to represent the functional model under investigation, but with a number of constraints that keep the original redundancy unchanged. In the end, the same Total Least-Squares Fit is supposed to result after fewer iterations from the newly developed scheme that, for the first time, allows the integration of constraints between the parameters, thus solving a case that was long considered "untreatable" by the original TLS algorithm.

The present study is motivated by the problem of GNSS carrier phase ambiguity resolution. We know, when the ambiguity success rate of the usual GNSS model is large enough, that the ‘fixed’ baseline estimator outperforms its ‘float’ counterpart in the sense that its probability of being close to the true but unknown baseline is larger than that of the ‘float’ baseline, provided that the ambiguity success rate is sufficiently large. This is a strong result, but unfortunately only valid if the probability of correct integer estimation is sufficiently close to one. We therefore pose the question whether it is possible to devise a baseline estimator which always outperforms its ‘float’ counterpart. We will show that this is indeed possible, be it that the result comes at the price of having to use a weaker performance criterion. As our main result we present a Gauss-Markov-like theorem which introduces a new minimum variance unbiased estimator that is always superior to the well-known best linear unbiased estimator (BLU) of the Gauss-Markov theorem. This minimum variance unbiased estimator is referred to as the best integer equivariant (BIE) estimator.

The goal of ambiguity resolution is to make optimal use of the integerness of the ambiguities, and it is the key to high precision GNSS positioning and navigation. However, it should only be applied in case the probability of correct integer ambiguity resolution, i.e. the success rate, is very close to one. In that case, the probability that the fixed baseline will be closer to the true but unknown baseline is larger than that of the float baseline. Clearly, this condition will not be fulfilled for each measurement scenario, and this means that for low success rates a user will prefer the float solution.However, there exists a baseline estimator that will always be superior to its float and fixed counterparts, albeit that this superiority is measured using a weaker optimality condition. This baseline estimator is the Best Integer Equivariant (BIE) estimator, which is unbiased and of minimum variance within the class of integer equivariant estimators.In this contribution, the three different estimators are compared. For that purpose, we will focus on the geometry-free GNSS models, either single frequency or dual frequency. The performance of the estimators is compared based on their probability density functions, the variances of the different estimators, and the probabilities that the baseline estimators are within a certain convex region symmetric with respect to the true baseline. This will provide information on whether or not the BIE estimator could be useful in positioning applications.

Averaging is a simple, yet very effective, technique that can be used for the fusion of repeated data sets and/or multiple estimates of a single parameter vector. With the increasing number of geodetic data sensors and the availability of numerous geodetic signal realizations from various satellite missions, averaging offers an important tool which can significantly reduce the data load while improving the quality of the recovered signal information. A preliminary study of weighted averaging methods for multivariate data ensembles is given in this paper, along with a novel approach for optimal weight determination directly from the available data.

The method of GPS-levelling for obtaining orthometric heights is not a new concept. In fact, many studies have proven its usefulness and the question of whether GPS-levelling can provide a viable alternative to traditional techniques is no longer an issue. An important question, however, that has yet to be satisfactorily solved is, ‘What accuracy level can be achieved using this approach?’ Over the past decade, numerous advances have been made which have placed us in a position where we can begin to address the issue with more confidence, namely (i) improved mathematical models/techniques for dealing with GPS and geoid data, (ii) increased data availability for gravimetric geoid models, and (iii) improved data processing capabilities. In this paper a statistical approach for estimating the variance components of heterogeneous groups of observations is used in the combined adjustment of GPS, geoid and levelling data. Specifically, the iterative minimum norm quadratic unbiased estimation algorithm is employed to determine the individual variance components for each of the three height types. The challenges encountered when implementing this well-known algorithm in practice with real data are discussed. The analysis provides some indication into the practicality and effectiveness of estimating variance components in mixed vertical networks. Notably, the estimation of realistic variance components provides us with important insight regarding the GPS-levelling problem in addition to other uses of combined GPS, geoid and levelling data, such as assessing the accuracy of a gravimetric geoid model.

The problem of finding the density distribution of the Earth from gravity data is called the inverse gravimetric problem. It is well known that this problem has not a unique solution. A possible approach to force the uniqueness of the solution is to impose the solution to realize a minimum of energy, that is to be an equilibrium distribution.Several authors have considered this possibility by setting the energy to the sole energy of the potential field. In this paper, we show that such approaches correspond to the maximization of a quadratic form in the set of the density distributions considered as an Hilbert space. As a consequence, they cannot deliver realistic results without a constraint that reduces the set of possible density distributions to a bounded domain of the considered Hilbert space. Moreover, under such constraints, the result will necessarily be located at the boundary of the searched domain.In order to express a steady state condition leading to a freer solution, we propose to introduce a compression energy. We choose the state equation of Murnaghan that is known to apply reasonably in the upper mantle. We give an integral form of the compression energy and show that the resulting total energy admits a lower bound, which assess the existence of a solution. Finally, we derive a differential equation verified by that the absolute minima from the equilibrium equation under its local form.

For many purposes a sphere or an ellipsoid of revolution approximate the figure of the Earth fairly well. However, the Earth’s topography causes effects that continuously stimulate a considerable research effort. In contrast to the concept of the so-called shrinking parameter in the classical treatment of the effect, successive approximations are applied within a weak formulation of the respective boundary-value problem in this paper. In this connection functional-analytic estimates are investigated so as to clarify the convergence of the iteration process. In particular the ellipticity of a bilinear form associated with the problem under consideration is discussed for the solution domain of an ellipsoidal boundary. The apparatus of ellipsoidal harmonics is substantially used for this purpose. Quantitative estimates for the respective parameters were derived, when the potential is assumed to follow a finite degree model.

One of the most important stages in the computation of a global geopotential model is the computation of the spherical harmonic coefficients from gravity anomalies on the normal ellipsoid. None of the existing methods provides an exact solution, and all show severe shortcomings in the high degrees of the spectrum. In this paper, a new, theoretically exact method is proposed, which is moreover easily applicable up to very high degree and order (2160 and beyond). The solution of the geopotential coefficients is presented as a weighted sum over “spherically approximated” coefficients of equal order, where the gravity anomalies are presumed to reside on a sphere. The weights depend solely upon the degree and order of the coefficient and the definition of the normal ellipsoid and its gravity field. Numerical comparisons with existing methods show substantial differences, especially in the high degrees, which can be explained by the fact that all previous methods are of limited accuracy

A harmonic development of the Earth tide generating potential (TGP) by an improved spectral method over two thousands years, 1000–3000, is done. The expansion is directly made to Poisson series where both amplitudes and frequencies of the series’ terms are high-degree polynomials of time as opposed to classical Fourier analysis where the terms’ amplitudes and frequencies are constants.The final development of the TGP, named KSM03, includes some 27,000 terms of amplitudes down to the level of 1*10−8 m2/s2. The respective accuracy in calculation of gravity tides at a midlatitude station is 0.025/0.39 nGal (the r.m.s./maximum error) when comparing with a benchmark gravity tide series numerically computed with use of the most accurate ephemeris DE-405 at every hour within 1600–2200. It exceeds the accuracy of any previously made harmonic development of the TGP in time domain by a factor of at least 3.

Assume that the density anomaly is linearly related to the topography by a convolution of the topography and an isotropic kernel function. Hence, it can be shown that the attraction of the compensating masses is also a convolution of the topography and an isotropic isostatic response function, which can be determined by deconvolution. The paper gives the necessary derivation of such a deconvolution by means of global spherical harmonics. A practical determination of the isotropic isostatic response of the earth’s crust needs the harmonic analysis of both the topography and the attraction of the compensating masses. To avoid the assumption of an isostatic model, the principle of inverse isostasy has been employed. The harmonic analysis of the Bouguer anomalies is thus a combination of the harmonic analysis of the topographic potential and the already existed global (free-air) reference models. The needed harmonic analysis of the topography has been carried out using different global digital height models. The results show that the isostatic response of the earth’s crust derived by inverse isostasy behaves in the same sense as those given by the exact solution of the Vening Meinesz isostatic model.

This paper presents a contribution to the localization of the main basement structures of the Paraná Basin using a Geographic Information System (GIS). This Ordovician-Cretaceous sedimentary basin is intracratonic and located in the S-SE portion of Brazilian territory in South America. The main result of this study, based upon an isostatic modeling approach, was to compute the gravimetric anomaly caused by sediments and igneous rocks that fill the basin, and the “root” created by the isostatic adjustment. Approximately 8,000 gravity stations took part in this study, and through modeling results allowed to identify variations in the basement gravimetric signature, revealing different compositions. Over the years, a great number of gravimetric surveys have been done in the study area and countless data sets were produced. In attempt to summarize and synthesize some of these data sets, a GIS approach was used to create a data bank with topographic, gravimetric and stratigraphic information.

When gravity field modelling is carried out in larger regions a subdivision of the region into smaller areas may be carried out in order to handle the data volume. This may create problems during the estimation since the use of the optimal local empirical covariance function tends to give different results at the borders when the statistical characteristics change from one area to the next. In this paper an approximation to the covariance function that allows spatial variations in both magnitude and in spectral characteristics is presented. The different magnitudes are taken into account by a scaling function. Differences in the spectral characteristics are taken into account by changing the depth to the Bjerhammar sphere, so that it is located at a shallow depth in areas with a rough gravity field. In areas with a smooth gravity field the Bjerhammar sphere is located more deeply.

This study presents an asymptotic theory for calculating displacements excited by a point dislocation in a spherically symmetric earth model as an approximation of the dislocation theory for a spherical earth model (Sun et al., 1996), employing the reciprocity theorem (Okubo, 1993) and asymptotic solutions of an elastic earth to tidal, load and shear forces (Okubo, 1988). This theory is mathematically simple and physically reasonable since it reflects the earth’s sphericity and radial structure. Comparison of the asymptotic results with both the exact results and the corresponding flat-earth results shows that for any distances the exact results are approximated better by the asymptotic results than by the flat-earth results. Especially, the asymptotic theory is obviously superior to the flat-earth theory due to a big sphericity effect for the tensile opening on a vertical fault. Owing to its mathematical simplicity, this theory can be applied easily to calculate co-seismic displacements. The validity of the theory for a stratified earth needs to be further tested and discussed.

There is a need for a proper validation procedure of gravity field solutions, especially for those high precise ones which are derived from the dedicated gravity field missions as CHAMP, GRACE and in future GOCE. In this paper the balance equations for the energy and for the energy exchange of a satellite orbiting around the Earth are proposed as analysis tool to validate gravity field solutions together with orbits derived with precise reduced dynamic or kinematic orbit determination procedures. The theoretical foundation of the analysis tool is presented as well as simulation results and applications to real orbits of a GPS satellite and of CHAMP based on various gravity field solutions. It is shown that the validation procedure can be used to detect deficiencies in the orbit modelling and in gravity field recovery results.

In some respect, the continuous representation of surfaces such as the geoid, the topography, or other spatial phenomena, is superior to discrete forms like the TIN (Triangular Irregular Network) or a raster DEM (Digital Elevation Model) as long as these surfaces exhibit a certain degree of local smoothness. In this contribution, we shall concentrate on the special study of biharmonic spline wavelets and of generalized multi-quadrics, with the emphasis on increased efficiency while maintaining the local approximation quality up to the desired resolution.To support these interpolation techniques a newly developed global search algorithm will be adapted to this problem. It is based on heuristic methods that would allow the user to handle gridded as well as arbitrarily scattered data. The number of coefficients for any surface representation using the global optimization will be extremely small while maintaining a magnitude of deviations that is still acceptable. On the other hand, long computation times have to be taken into account. Concerning the functional model one may face fewer restrictions for its structure when using global methods because no derivatives have to be computed and no approximate values need to be provided.Some geodetic examples will show the potential of the new techniques, particularly in view of the more classical Fourier analysis.

The premise in using principles of vibration-based damage identification in structural health monitoring (SHM) is that when damage occurs, the mechanical properties of the structure will be altered and thus the vibration response of the structure to externally applied loads and environmental conditions will be changed. The low frequency of civil structures and the limited effect of local damage on vibration characteristics of these structures make the development of efficient SHM systems challengeable.This paper introduces the Wavelet multi-resolution analysis as a promising technique for damage identification. Analytical analysis of simulated sensor signals significantly corrupted by other noise signals showed that the wavelet multi-resolution analysis was able to separate the different frequency components and to identify the original simulated structural response. Combining wavelet multi-resolution analysis with artificial neural networks makes it possible to pattern structural dynamics during healthy performance and detect changes of the structural dynamics when damage is introduced to the system.

Two discretization schemes of the Poisson integral have been proposed to date. Although they are mathematically solvable, they produce different gravity on the geoid for the same input data. This discrepancy arises because of different discretization techniques of the Poisson kernel; still, this problem has not received adequate attention. The question is whether the system of the discretization is reasonable. Methods to discretize the Poisson integral are investigated in this study. For this purpose, a single mean scheme is presented to evaluate numerically the Poisson integral. A comparison between the point and mean schemes shows that, for a limit topographical grid size, the point discretization scheme results in a serious theoretical problem since it greatly underestimates gravity on the geoid, and even gives incorrect results for extreme cases. A careful construction of the coefficient matrix for the discrete system is much more important than using point gravity as input.

The hydrological cycle induces mass exchange between the oceans and continents, and redistribution of water within the continents. In turn, relative sea level passively responds to the change in shape of both the geoid and the ocean bottom. The time variation in these two surfaces must self-consistently relate to the total mass redistribution. Here we show formally how, if given a time series of station coordinates from a global network of geodetic stations, we can invert for the separate contributions to the total mass redistribution due to water on the continents and water in the oceans. If we characterize the deformation of the solid Earth in terms of a truncated vector spherical harmonic expansion, the solution to this problem can be formulated in terms of Clebsch-Gordan coefficients, which are familiar for the calculation of branching ratios in elementary particle physics. As part of a rigorous solution to this problem, we formally address the definition of reference frame by applying the concept of an “isomorphic frame transformation,” which conserves the formal theoretical relationship between surface deformation and load distribution by a simultaneous translation of origin and change of degree-one load Love numbers.

Algebra, in particular the "Abelian Group" and the "Semi Group" (also known as "Monoid") axioms, which form a “ring with identity”, are employed to define the "polynomial ring". Polynomial ring theory enables the solution of geodetic observations that can be converted into (algebraic) polynomials. The advantages of algebraic approaches are that they provide exact solutions to problems requiring closed form approaches (e.g. solving for geocentric coordinates from Helmerts projection through minimum distance mapping) and also act as tools to control iterative procedures. As a motivation, we present several examples of geodetic problems solved algebraically. These examples include; nonlinear analysis of bending angles in GPS-Meteorology, transformation of geocentric Cartesian coordinates into ellipsoidal, densification problems etc. The overriding advantage of the algebraic approach is the removal of the requirement of approximate starting values; they are non-iterative and enable detection of outliers.

During the past few years a methodology has been developed for processing data collected by GPS networks consisting of a mixed set of single-frequency and dual-frequency receivers. The strategy is to deploy a few permanent GPS stations with dual-frequency, geodetic-quality receivers surrounding an ‘inner’ deformation monitoring network of low-cost, single-frequency GPS receivers. The dual-frequency GPS network is used to generate a file of ‘corrections’, analogous to Wide Area DGPS correction models for the distance dependent biases. These ‘corrections’ are then applied to the double-differenced phase observations from the inner receivers to enhance baseline accuracy, primarily through empirical modelling of the residual atmospheric biases that otherwise would be neglected. Moreover, epoch-by-epoch baseline solutions are preferred in order to detect deformational signals in (near) real-time. Data from two continuous GPS networks have been used to investigate the performance of this configuration under severe ionospheric conditions and in different geographical regions. In the mid-latitude region the L1 baseline repeatability has clearly been improved by 40–50%, while an improvement of about 20% has been achieved in the equatorial region. The findings also indicate that the proposed procedure is sensitive to extreme ionospheric conditions, such as those experienced in close proximity to the geomagnetic equator during solar cycle maximum periods.

Many local geodynamic geodetic GPS networks are increasingly being exploited for monitoring crustal movements and deformations. It is thus very important that such monitoring networks should be developed and observed in the most economical way and at the same time satisfy some precision and reliability criteria set by the user. In this report we present our solution of the optimal design of two-dimensional geodynamic GPS networks which guarantees the required precision of parameters for the chosen deformation model with the least cost. A common cost criterion is used as a target function and the non-linear programming problem is solved for an optimal set of GPS vectors. The proposed methodology is illustrated using two optimization examples of working GPS networks using simple deformation models.

The present-day surface velocity field of the South American continent reflects the recent geodynamic features. It varies from quasi rigid body motion in the eastern part to mountain range deformation in the western Andes. Space geodetic observations provide velocities at discrete points only. To represent the continuous velocity field, an adequate deformation model has to be developed. Two different models are applied, the least squares collocation approach (LSC) and the finite element method (FEM). The input data are given by 329 velocities derived from continuously observing GPS stations and several GPS geodynamics projects. The different data sets are transformed to a common kinematic datum by deriving the rotation vector of the South American plate from station motions of the IGS Regional Network (RNAAC-SIR) in the rigid eastern part and reducing these plate motions from all the data sets. The resulting residual motions define the boundary conditions in the FEM and the input signals in the LSC. For the FEM a network of approximately 75000 linear elements is generated. The model theology is a homogeneous elastic material (Young’s modulus 70 GPa and Poisson ratio 0,3). For the LSC empirical covariance functions are derived from the observed velocity vectors. The comparison of both methods shows an agreement in the mm/a level. The result is a continuous surface velocity model for the South American continent which may be used for interpolation of point motions in geodetic networks and reference frames. A detailed geodynamic interpretation has not yet been done due to the simplicity of the model (homogeneous material, rough fault structures, no sophisticated dynamics).

China continent, constructed by many blocks, is a dynamical system with complicated geological structure, topography and physiognomy. Since Cenozoic, it takes on different tectonic deformation in different blocks and the relative movement between neighboring blocks does always exist, which may be the key inducement of the strong seismic activity at the inner of China continent (Zhang Peizhen, 1999). In terms of tectonic movement feature, the continent can be divided into several different scalar crustal and lithosphere blocks, which include 6 active block regions that are Qinghai-Tibet, Xingjiang, Northeast, North China, South China and South China Sea block region, two active plate boundary tectonic zones and 29 active blocks that are Lhasa, Tian Shan etc. (Deng Qidong, Zhang Peizhen et al., 2003). In this paper, based on the bi-cubic Bessel spline function method, we inverted the present-day movement field of active blocks in China continent by combining with GPS measurements, seismic moment tensors of earthquakes and fault slip rates in China and its neighboring areas, considering about the geological and geophysical parameters at the same time. Furthermore, we inverted the strain rate field, strain energy density rate field, then we calculated the shearing strain rate field, area strain rate field and analyzed the relationship between the strong seismic activity and strain rate fields. The results reveal that in the middle and west of China continent, tectonic activity is stronger, particularly along the boundary zone of Himalayan block, in ChuanDian area of Qinghai-Tibet block and Tian Shan area of Xingjiang block, shearing strain rate is larger, the shearing strain rate fields are in correspondence with the tendency of present-day tectonic movement; from the distribution of principal axes of tectonic stress field, we can infer that present-day movement of active blocks in China continent is mainly caused by posthumous movement. The continent of China deformation has not only the strike-slip faulting and block’s transverse slip feature but also the crustal shortening and thickening feature, it doesn’t support the continental eastward escape theory of high-rate of slip.

The National Survey and Cadastre (KMS) is responsible for the geodetic definition of the reference network in Denmark. Permanent GPS stations play an important role in the monitoring and maintenance of the geodetic network. During 1998 and 1999 KMS established three permanent GPS station in Denmark, SMID, SULD and BUDP. Using almost 4.5 years of continuous data from the Danish station and the Swedish station, ONSA, we analyse the daily GPS solution due to crustal deformation caused by glacial isostatic adjustment (GIA). Although, displacements due to GIA are only 1–3 mm/year at the Danish GPS sites, the current precision of positioning using GPS allows us to observe these effects. The modelled horizontal GIA velocities and the observed horizontal residuals obtained from GPS show almost the same direction for all station. However, the observed velocity residuals are larger than the modelled GIA velocities. Furthermore, the difference of the observed vertical velocity residual rate between ONSA and SMID, and between ONSA and SULD seems to agree with the expected rate according to the post-glacial model by Milne. However, the vertical velocity rate of BUDP seems to disagree with the model. Moreover, the analysis presented here uses a relative short period of data. The error of the vertical velocities (including reference frame drift) is order of ±1.1 mm/yr.

Jakarta is the capital city of Indonesia with a population of about 12 million people, inhabiting an area of about 25-km by 25-km. It has been reported for quite sometime that several locations in Jakarta are subsiding at different rates. Leveling surveys performed in 1982, 1991 and 1997 have detected the subsidence up to about 80 cm during the period of 1982–1991, and up to about 160 cm during the 1991–1997 period; while GPS surveys observed the subsidence up to about 50 cm during the period of 1997–2002. InSAR technique using JERS-1/SAR L-band data estimated a subsidence rate of about 5 to 10cm during the period of 1993 to 1995. Maximum subsidence were found in the northwestern and central eastern parts of Jakarta, while minimum subsidence were found in the southern part. InSAR results show good correspondence with the results from Leveling and GPS Surveys

The Atera fault is one of the most active both geomorphologically and geologically in Japan. In the southern part of the fault, the Sakashita body (Ueno volcano) erupted about 1.5 km southwest of the fault in a direction parallel to it (early Pleistocene). A local strike direction change, reversal of vertical displacement (SW uplift), and branching of the fault line occurred in the southern tip of the fault (in and around the Dendahara district). Our microgravity survey suggests that a high-density body (probably basaltic) prevents ordinary activity of the Atera fault (left-lateral and NE uplift).

A miniaturized absolute gravimeter, the FG5L was tested at Ogiri geothermal field. One day of measurements at a site provided a gravity value with an error of about 10 microGal. However, the fluctuations in the daily values greatly exceeded the daily estimated error. The FG5L is far from being useful for geothermal reservoir monitoring. The FG5, which was upgraded from the FG5L, was tested at Yanaizu-Nishiyama geothermal field. The FG5 requires only a few days to provide useful results at each survey location. Therefore one FG5 gravimeter can be used at all the geothermal fields in Japan that are being monitored. We have already used it at the 5 existing gravity networks.

The International VLBI Service for Geodesy and Astrometry (IVS) set up a 15-day campaign of continuous VLBI observations, named CONT02, that took place October 16–31, 2002. The goal of CONT02 was to acquire state of the art VLBI data over a continuous two-week period to demonstrate the highest accuracy of which VLBI is capable. Using the two weeks of CONT02 data from VLBI and the global GPS data for the same time span, daily unconstrained normal equations were generated taking much care to use identical models and the same parameterization of the common parameters for both space geodetic techniques. Based on these normal equations first steps towards a fully rigorous combination of all common parameters were performed by including station coordinates and Earth orientation parameters (EOP) into the solutions. In addition, the impact of the local tie information on the combined solutions was studied. To assess the quality of the individual and combined solutions, polar motion and UTI-UTC parameters were set up with a two-hour time resolution and the results were compared with sub-daily models for ocean tide variations derived from altimetry and other space geodetic techniques.

The paper presents short-term and long-term predictions of Earth rotation parameters (ERP) (Length-of-Day and polar motion) up to one year by means of ANFIS (Adaptive Network based Fuzzy Inference System). The ERP C04 time series of the International Earth Rotation and Reference Systems Service (IERS) with daily values are used as data basis. The impact of the tides of the solid Earth and the oceans and seasonal variations of the atmosphere are removed from the C04 series. The residual series are used both for training and validation of the ANFIS. For network optimization purposes, different architectures are considered. The results of the prediction are compared with those from other methods. The ERP values predicted by ANFIS have RMS errors, which are equal or even less than those from other methods. The most striking advantage of using ANFIS instead of other approaches is its significantly reduced complexity.

We study the oceanic excitation of polar motion by using six different time series of the non-tidal oceanic angular momentum which are either publicly available or more experimental in nature. The oceanic excitation series are compared to each other and to the difference between the excitation inferred from the polar motion data and the corresponding atmospheric contributions. Comparisons show good agreement between the observed residual excitation and most of the ocean series. By performing computations separately for the seasonal sinusoids and four non-overlapping spectral components decadall, interannual, seasonal and intraseasonal) we demonstrate that the quality of the ocean products depends on the considered frequency band. We also discuss the importance of using data-constrained ocean models in future efforts to improve ocean angular momentum estimates.

We study how atmospheric models can be best used to understand excitations of polar motion. The set of models consists of those contributed worldwide by meteorological centers participating in the Atmospheric Model Intercomparison Project, which are forced solely by a prescribed distribution of sea surface temperatures. We have calculated the monthly mean values of excitations from the resulting surface pressure fields over the 17-year study period. With such historical variability and with the spread inherent in the suite of models, a measure of the excitations of polar motion can be obtained, including variability at intraseasonal and seasonal scales; from these models we can also view the resulting interannual variability, including the impact of El Nino events. We compare the results with the excitations determined by atmospheric analyses, as well as with the geodetic excitation functions themselves, determined from analysis of observations. Based on the suite of atmospheric models we estimate the spread in the polar motion excitation from the ensemble of 19 different models.

The non-linear gyroscopic model DyMEG has been developed at DGFI in order to study the interactions between geophysically and gravitationally induced polar motion and the Earth’s free wobbles, in particular the Chandler oscillation. The model is based on a triaxial ellipsoid of inertia. It does not need any explicit information concerning amplitude, phase, and period of the Chandler oscillation. The characteristics of the Earth’s free polar motion are reproduced by the model from theological and geometrical parameters. Therefore, the traditional analytical solution is not applicable, and the Liouville equation is solved numerically as an initial value problem. The gyro is driven by consistent atmospheric and oceanic angular moment. Mass redistributions influence the free rotation by rotational deformations. In order to assess the dependence of the numerical results on the initial values and theological or geometrical input parameters like the Love numbers and the Earth’s principal mo- ments of inertia, a sensitivity analysis has been per- formed. The study reveals that the pole tide Love number k ₂ is the most critical model parameter. The dependence of the solution on the other mentioned parameters is marginal.

We examine the problem of obtaining average surface mass load changes in a local area (for example, water content in a river basin) from time-variable Stokes coefficients available from the GRACE mission. A basin function is unity inside a defined geographical region, and zero outside, and can be represented exactly only if Stokes Coefficients of all degrees and orders are available. GRACE Stokes coefficients will contain errors that generally increase with degree, and will be of limited degree range, perhaps to 100 or so. Load variations within a basin should be estimated by minimizing some quantity that accounts for both GRACE measurement error, and leakage error, associated with a finite degree range. To understand this problem, we use Fourier series, the 1-D equivalent to spherical harmonics. The solution of this problem is not unique, and we examine several different approaches. We use Monte Carlo experiments to test the performance of various methods derived. Time series of gridded soil moisture, snow, ocean bottom pressure, atmospheric surface pressure and Gaussian random noise are utilized in an experiment to recover load variations within the Nile basin.