Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt

 The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used, as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads. Received: 14 August 2000 / Accepted: 28 February 2001     

Time transfer using GPS carrier phase: error propagation and results

 A joint time-transfer project between the Astronomical Institute of the University of Berne (AIUB) and the Swiss Federal Office of Metrology and Accreditation (METAS) was initiated to investigate the power of the time transfer using GPS carrier phase observations. Studies carried out in the context of this project are presented. The error propagation for the time-transfer solution using GPS carrier phase observations was investigated. To this purpose a simulation study was performed. Special interest was focussed on errors in the vertical component of the station position, antenna phase-center variations and orbit errors. A constant error in the vertical component introduces a drift in the time-transfer results for long baselines in east–west directions. The simulation study was completed by investigating the profit for time transfer when introducing the integer carrier phase ambiguities from a double-difference solution. This may reduce the drift in the time-transfer results caused by constant vertical error sources. The results from the present time-transfer solution are shown in comparison to results obtained with independent time-transfer techniques. The interpretation of the comparison benefits from the investigations of the error propagation study. Two types of solutions are produced on a regular basis at AIUB: one based on the rapid orbits from CODE, the other on the CODE final orbits. The rapid solution is available the day after the observations and has nearly the same quality as the final solution, which has a latency of about one week. The differences between these two solutions are below the nanosecond level. The differences from independent time-transfer techniques such as TWSTFT (two-way satellite time and frequency transfer) are a few nanoseconds for both products. Received: 15 November 2001 / Accepted: 6 September 2002 Correspondence to:R. Dach     

Rapid GPS ambiguity resolution for short and long baselines

 A method of quick initial carrier cycle ambiguity resolution is described. The method applies to high-quality dual-band global positioning system observations. Code measurements on both frequencies must be available. The rapidity of the method is achieved through smoothing pseudoranges by phase observables and forming linear combinations between the phase observables. Two cases are investigated. Case 1: ionospheric bias is neglected (short distances); and case 2: the bias is taken into account (longer distances, more than, say, 10 km). The method was tested on six baselines, from 1 to 31 km long. In most cases, single-epoch ambiguity resolution was achieved. Received: 6 October 1999 / Accepted: 4 March 2002     

Topographic and atmospheric corrections of gravimetric geoid determination with special emphasis on the effects of harmonics of degrees zero and one

 The topographic and atmospheric effects of gravimetric geoid determination by the modified Stokes formula, which combines terrestrial gravity and a global geopotential model, are presented. Special emphasis is given to the zero- and first-degree effects. The normal potential is defined in the traditional way, such that the disturbing potential in the exterior of the masses contains no zero- and first-degree harmonics. In contrast, it is shown that, as a result of the topographic masses, the gravimetric geoid includes such harmonics of the order of several centimetres. In addition, the atmosphere contributes with a zero-degree harmonic of magnitude within 1 cm. Received: 5 November 1999 / Accepted: 22 January 2001     

A formula for gravimetric terrain corrections using powers of topographic height

Journal of Geodesy - ?The application of Stokes' formula to create geoid undulations requires no masses outside the geoid. However, due to the existence of the topography, terrain...     

Auto-covariance estimation of variable samples (ACEVS) and its application for monitoring random ionospheric disturbances using GPS

 Ionospheric variation may be considered as a stationary time series under quiet conditions. However, the disturbance of a stationary random process from stationarity results in the bias of corresponding samples from the stationary observations, and in the change of statistical model parameters of the process. From a general mathematical aspect, a new method is presented for monitoring ionospheric variations, based on the characteristic of time-series observation of GPS, and an investigation of the statistical properties of the estimated auto-covariance of the random ionospheric delay when changing the number of samples in the time series is carried out. A preliminary scheme for monitoring ionospheric delays is proposed. Received: 18 August 2000 / Accepted: 12 April 2001     

Geoid determination using adapted reference field, seismic Moho depths and variable density contrast

 The traditional remove-restore technique for geoid computation suffers from two main drawbacks. The first is the assumption of an isostatic hypothesis to compute the compensation masses. The second is the double consideration of the effect of the topographic–isostatic masses within the data window through removing the reference field and the terrain reduction process. To overcome the first disadvantage, the seismic Moho depths, representing, more or less, the actual compensating masses, have been used with variable density anomalies computed by employing the topographic–isostatic mass balance principle. In order to avoid the double consideration of the effect of the topographic–isostatic masses within the data window, the effect of these masses for the used fixed data window, in terms of potential coefficients, has been subtracted from the reference field, yielding an adapted reference field. This adapted reference field has been used for the remove–restore technique. The necessary harmonic analysis of the topographic–isostatic potential using seismic Moho depths with variable density anomalies is given. A wide comparison among geoids computed by the adapted reference field with both the Airy–Heiskanen isostatic model and seismic Moho depths with variable density anomaly and a geoid computed by the traditional remove–restore technique is made. The results show that using seismic Moho depths with variable density anomaly along with the adapted reference field gives the best relative geoid accuracy compared to the GPS/levelling geoid. Received: 3 October 2001 / Accepted: 20 September 2002 Correspondence to: H.A. Abd-Elmotaal     

Precise estimation of residual tropospheric delays using a regional GPS network for real-time kinematic applications

 A new method called Trop_NetAdjust is described to predict in real time the residual tropospheric delays on the GPS carrier phase observables using the redundant measurements from a network of GPS reference stations. This method can not only enhance the effectiveness and reliability of real-time kinematic users within the network, but also provide a valid approach to tropospheric parameter variation forecasting. Trop_NetAdjust is theoretically based upon LS prediction criteria and enables the prediction of residual tropospheric delays remaining after a standard model has been applied to the raw GPS measurements. Two cases are analyzed, namely a first case when the delay is required for an existing satellite at a new point within the network and a second case when the delay is required for a new satellite. Field tests were conducted using data collected in a network of 11 reference stations covering a 400×600 km region in southern Norway. The results were analyzed in the measurement domain (ionospheric-free double-difference residuals) and showed improvements of 20 to 65% RMS errors using Trop_NetAdjust. The estimates of the Trop_NetAdjust prediction accuracy were also obtained using the covariance analysis method. The agreement was consistently better than 30% when compared with data from a real network. Received: 28 February 2000 / Accepted: 9 January 2001     

GPS-assisted GLONASS orbit determination

 Using 1 week of data from a network of GPS/GLONASS dual-tracking receivers, 15-cm accurate GLONASS orbit determination is demonstrated with an approach that combines GPS and GLONASS data. GPS data are used to define the reference frame, synchronize receiver clocks and determine troposphere delay for the GLONASS tracking network. GLONASS tracking data are then processed separately, with the GPS-defined parameters held fixed, to determine the GLONASS orbit. The quality of the GLONASS orbit determination is currently limited by the size and distribution of the tracking network, and by the unavailability of a sufficiently refined solar pressure model. Temporal variations in the differential clock bias of the dual-tracking receivers are found to have secondary impact on the orbit determination accuracy. Received: 5 January 2000 / Accepted: 15 February 2001     

Random simulation and GPS decorrelation

 (i) A random simulation approach is proposed, which is at the centre of a numerical comparison of the performances of different GPS decorrelation methods. The most significant advantage of the approach is that it does not depend on nor favour any particular satellite–receiver geometry and weighting system. (ii) An inverse integer Cholesky decorrelation method is proposed, which will be shown to out-perform the integer Gaussian decorrelation and the Lenstra, Lenstra and Lovász (LLL) algorithm, and thus indicates that the integer Gaussian decorrelation is not the best decorrelation technique and that further improvement is possible. (iii) The performance study of the LLL algorithm is the first of its kind and the results have shown that the algorithm can indeed be used for decorrelation, but that it performs worse than the integer Gaussian decorrelation and the inverse integer Cholesky decorrelation. (iv) Simulations have also shown that no decorrelation techniques available to date can guarantee a smaller condition number, especially in the case of high dimension, although reducing the condition number is the goal of decorrelation. Received: 26 April 2000 / Accepted: 5 March 2001     

Improved convergence rates for the truncation error in gravimetric geoid determination

 When Stokes's integral is used over a spherical cap to compute a gravimetric estimate of the geoid, a truncation error results due to the neglect of gravity data over the remainder of the Earth. Associated with the truncation error is an error kernel defined over these two complementary regions. An important observation is that the rate of decay of the coefficients of the series expansion for the truncation error in terms of Legendre polynomials is determined by the smoothness properties of the error kernel. Previously published deterministic modifications of Stokes's integration kernel involve either a discontinuity in the error kernel or its first derivative at the spherical cap radius. These kernels are generalised and extended by constructing error kernels whose derivatives at the spherical cap radius are continuous up to an arbitrary order. This construction is achieved by smoothly continuing the error kernel function into the spherical cap using a suitable degree polynomial. Accordingly, an improved rate of convergence of the spectral series representation of the truncation error is obtained. Received: 21 April 1998 / Accepted: 4 October 1999     

Downward continuation and geoid determination based on band-limited airborne gravity data

 The downward continuation of the harmonic disturbing gravity potential, derived at flight level from discrete observations of airborne gravity by the spherical Hotine integral, to the geoid is discussed. The initial-boundary-value approach, based on both the direct and inverse solution to Dirichlet's problem of potential theory, is used. Evaluation of the discretized Fredholm integral equation of the first kind and its inverse is numerically tested using synthetic airborne gravity data. Characteristics of the synthetic gravity data correspond to typical airborne data used for geoid determination today and in the foreseeable future: discrete gravity observations at a mean flight height of 2 to 6 km above mean sea level with minimum spatial resolution of 2.5 arcmin and a noise level of 1.5 mGal. Numerical results for both approaches are presented and discussed. The direct approach can successfully be used for the downward continuation of airborne potential without any numerical instabilities associated with the inverse approach. In addition to these two-step approaches, a one-step procedure is also discussed. This procedure is based on a direct relationship between gravity disturbances at flight level and the disturbing gravity potential at sea level. This procedure provided the best results in terms of accuracy, stability and numerical efficiency. As a general result, numerically stable downward continuation of airborne gravity data can be seen as another advantage of airborne gravimetry in the field of geoid determination. Received: 6 June 2001 / Accepted: 3 January 2002     

The success rate and precision of GPS ambiguities

 An application of a theorem on the optimality of integer least-squares (LS) is described. This theorem states that the integer LS estimator maximizes the ambiguity success rate within the class of admissible integer estimators. This theorem is used to show how the probability of correct integer estimation depends on changes in the second moment of the ambiguity `float' solution. The distribution of the `float' solution is considered to be a member of the broad family of elliptically contoured distributions. Eigenvalue-based bounds for the ambiguity success rate are obtained. Received: 11 January 1999 / Accepted: 2 November 1999     

Geoid determination with density hypotheses from isostatic models and geological information

 Geoid determination by Stokes's formula requires a complete knowledge of the topographical mass density distribution in order to perform gravity reductions to the geoid boundary. However, deeper masses are also of interest, in order to produce a smooth field of gravity anomalies which will improve results from interpolation procedures. Until now, in most cases a constant mass density has been considered, which is a very rough approximation of reality. The influence on the geoid height coming from different mass density hypotheses given by the isostatic models of Pratt/Hayford, Airy/Heiskanen and Vening Meinesz is studied. Apart from a constant mass density value, additional density information deduced from geological maps and thick sedimentary layers is considered. An overview of how mass density distributions act within Stokes's theory is given. The isostatic models are considered in spherical and planar approximation, as well as with constant and lateral variable mass density of the topographical and deeper masses. Numerical results in a test area in south-west Germany show that the differences in the geoid height due to different density hypotheses can reach a magnitude of more than 1 decimetre, which is not negligible in a precise geoid determination with centimetre accuracy. Received: 7 January 2002 / Accepted: 20 September 2002 M. Kuhn now at: Western Australian Centre for Geodesy, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia Acknowledgements. The author would gratefully thank Prof. Dr.-Ing. B. Heck, who was the supervisor of my PhD thesis, and the second examiner Prof. Dr.-Ing. K.H. Ilk, as well as all other colleagues for their support of this work. Particular thanks go to the Landesvermessungsamt Baden–Württemberg (Survey Department of Baden–Württemberg), Bureau Gravimetrique International (BGI, France) for providing the gravity data and the Geologisches Landesamt Baden–Württemberg (Geological Department of Baden–Württemberg) for providing data and maps of the sediment layers within the Rhine Valley. Grateful thanks goes to Prof. W.E. Featherstone and the reviewers Prof. S.D. Pagiatakis, Dr. U. Marti as well as an unknown reviewer for their helpful comments on this paper.     

GPS network monitors the Western Alps' deformation over a five-year period: 1993–1998

  The Western Alps are among the best studied collisional belts with both detailed structural mapping and also crustal geophysical investigations such as the ECORS and EGT seismic profile. By contrast, the present-day kinematics of the belt is still largely unknown due to small relative motions and the insufficient accuracy of the triangulation data. As a consequence, several tectonic problems still remain to be solved, such as the amount of N–S convergence in the Occidental Alps, the repartition of the deformation between the Alpine tectonic units, and the relation between deformation and rotation across the Alpine arc. In order to address these problems, the GPS ALPES group, made up of French, Swiss and Italian research organizations, has achieved the first large-scale GPS surveys of the Western Alps. More than 60 sites were surveyed in 1993 and 1998 with a minimum observation of 3 days at each site. GPS data processing has been done by three independent teams using different software. The different solutions have horizontal repeatabilities (N–E) of 4–7 mm in 1993 and 2–3 mm in 1998 and compare at the 3–5-mm level in position and 2-mm/yr level in velocity. A comparison of 1993 and 1998 coordinates shows that residual velocities of the GPS marks are generally smaller than 2 mm/yr, precluding a detailed tectonic interpretation of the differential motions. However, these data seem to suggest that the N–S compression of the Western Alps is quite mild (less than 2 mm/yr) compared to the global convergence between the African and Eurasian plate (6 mm/yr). This implies that the shortening must be accomodated elsewhere by the deformation of the Maghrebids and/or by rotations of Mediterranean microplates. Also, E–W velocity components analysis supports the idea that E–W extension exists, as already suggested by recent structural and seismotectonic data interpretation. Received: 27 November 2000 / Accepted: 17 September 2001     

How to handle topography in practical geoid determination: three examples

 Three different methods of handling topography in geoid determination were investigated. The first two methods employ the residual terrain model (RTM) remove–restore technique, yielding the quasigeoid, whereas the third method uses the classical Helmert condensation method, yielding the geoid. All three methods were used with the geopotential model Earth Gravity Model (1996) (EGM96) as a reference, and the results were compared to precise global positioning system (GPS) levelling networks in Scandinavia. An investigation of the Helmert method, focusing on the different types of indirect effects and their effects on the geoid, was also carried out. The three different methods used produce almost identical results at the 5-cm level, when compared to the GPS levelling networks. However, small systematic differences existed. Received: 18 March 1999 / Accepted: 21 March 2000     

Confidence regions for GPS baselines by Bayesian statistics

 The global positioning system (GPS) model is distinctive in the way that the unknown parameters are not only real-valued, the baseline coordinates, but also integers, the phase ambiguities. The GPS model therefore leads to a mixed integer–real-valued estimation problem. Common solutions are the float solution, which ignores the ambiguities being integers, or the fixed solution, where the ambiguities are estimated as integers and then are fixed. Confidence regions, so-called HPD (highest posterior density) regions, for the GPS baselines are derived by Bayesian statistics. They take care of the integer character of the phase ambiguities but still consider them as unknown parameters. Estimating these confidence regions leads to a numerical integration problem which is solved by Monte Carlo methods. This is computationally expensive so that approximations of the confidence regions are also developed. In an example it is shown that for a high confidence level the confidence region consists of more than one region. Received: 1 February 2001 / Accepted: 18 July 2001     

The parameter distributions of the integer GPS model

 A parameter estimation theory is incomplete if no rigorous measures are available for describing the uncertainty of the parameter estimators. Since the classical theory of linear estimation does not apply to the integer GPS model, rigorous probabilistic statements cannot be made with reference to the classical results. The fact that integer parameters are involved in the estimation process forces a reappraisal of the propagation of uncertainty. It is with this purpose in mind that the joint and marginal distributional properties of both the integer and non-integer parameters of the GPS model are determined. These joint distributions can also be used to determine the distribution of functions of the parameters. As an important example, the distribution of the vector of ambiguity residuals is determined. Received: 30 January 2001 / Accepted: 31 July 2001