Robust estimator for correlated observations based on bifactor equivalent weights

 A new robust parameter estimator for the adjustment of correlated observations is developed based on a `bifactor reduction' model of weight elements. A shrinking factor for weight elements is proposed. The new equivalent weight matrix composed by the bifactor weight elements preserves the symmetry and keeps the original correlation coefficients unchanged. The new parameter estimator with its error influence function is derived. The robustness and efficiency of the new robust estimator is demonstrated with a simulated example and some conclusions are drawn. Received: 5 March 2001 / Accepted: 17 January 2002     

Simplified formulae for the BIQUE estimation of variance components in disjunctive observation groups

 General rigorous and simplified formulae are reported for the best invariant quadratic unbiased estimates of the variance–covariance components, which can be applied to all least-squares adjustments with the general linear stochastic model. Simplified procedures are given for two cases frequently recurring in geodetic applications: uncorrelated groups of correlated or uncorrelated observations, with more than one variance component in each group. Received: 19 November 1998 / Accepted: 21 March 2000     

Regularization of gravity field estimation from satellite gravity gradients

 The performance of the L-curve criterion and of the generalized cross-validation (GCV) method for the Tikhonov regularization of the ill-conditioned normal equations associated with the determination of the gravity field from satellite gravity gradiometry is investigated. Special attention is devoted to the computation of the corner point of the L-curve, to the numerically efficient computation of the trace term in the GCV target function, and to the choice of the norm of the residuals, which is important for the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) in the presence of colored observation noise. The trace term in the GCV target function is estimated using an unbiased minimum-variance stochastic estimator. The performance analysis is based on a simulation of gravity gradients along a 60-day repeat circular orbit and a gravity field recovery complete up to degree and order 300. Randomized GCV yields the optimal regularization parameter in all the simulations if the colored noise is properly taken into account. Moreover, it seems to be quite robust against the choice of the norm of the residuals. It performs much better than the L-curve criterion, which always yields over-smooth solutions. The numerical costs for randomized GCV are limited provided that a reasonable first guess of the regularization parameter can be found. Received: 17 May 2001 / Accepted: 17 January 2002     

The Bayesian approach applied to GPS ambiguity resolution. A mixture model for the discrete–real ambiguities alternative

 The problem of phase ambiguity resolution in global positioning system (GPS) theory is considered. The Bayesian approach is applied to this problem and, using Monte Carlo simulation to search over the integer candidates, a practical expression for the Bayesian estimator is obtained. The analysis of the integer grid points inside the search ellipsoid and their evolution with time, while measurements are accumulated, leads to the development of a Bayesian theory based on a mathematical mixture model for the ambiguity. Received: 29 March 2001 / Accepted: 3 September 2001     

Regularization of geopotential determination from satellite data by variance components

 Different types of present or future satellite data have to be combined by applying appropriate weighting for the determination of the gravity field of the Earth, for instance GPS observations for CHAMP with satellite to satellite tracking for the coming mission GRACE as well as gradiometer measurements for GOCE. In addition, the estimate of the geopotential has to be smoothed or regularized because of the inversion problem. It is proposed to solve these two tasks by Bayesian inference on variance components. The estimates of the variance components are computed by a stochastic estimator of the traces of matrices connected with the inverse of the matrix of normal equations, thus leading to a new method for determining variance components for large linear systems. The posterior density function for the variance components, weighting factors and regularization parameters are given in order to compute the confidence intervals for these quantities. Test computations with simulated gradiometer observations for GOCE and satellite to satellite tracking for GRACE show the validity of the approach. Received: 5 June 2001 / Accepted: 28 November 2001     

Gravitational perturbation theory for intersatellite tracking

 An analytical gravitational perturbation theory for the intersatellite tracking range and range-rate measurement between two satellites is developed. The satellite-to-satellite tracking (SST) range data measure the difference between the position perturbations of two satellites along the direction of the intersatellite range. The SST range-rate data measure the difference between the velocity perturbations along the direction of the intersatellite range, and the difference of the position perturbation along the direction perpendicular to the intersatellite range (cross-range). The SST range and range rate depend on different orbital excitations for mapping the gravity field. For the Gravity Recovery and Climate Experiment (GRACE), approximately 97% of the geopotential coefficient pairs produce perturbations with a root-mean-square larger than 1 m on the range and 0.1 m/sec on the range rate based on the EGM96 gravity field truncated at degree and order 140. Results in this study showed that ocean tides produce significant perturbations in the range and range-rate measurements. An ocean tide field with a higher degree and order (>70) is required to model the ocean tide perturbations on the intersatellite range and range-rate measurement. Received: 17 May 2000 / Accepted: 3 September 2001     

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     

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     

Improvements in height datum transfer expected from the GOCE mission

 One of the aims of the Earth Explorer Gravity Field and Steady-State Ocean Circulation (GOCE) mission is to provide global and regional models of the Earth's gravity field and of the geoid with high spatial resolution and accuracy. Using the GOCE error model, simulation studies were performed in order to estimate the accuracy of datum transfer in different areas of the Earth. The results showed that with the GOCE error model, the standard deviation of the height anomaly differences is about one order of magnitude better than the corresponding value with the EGM96 error model. As an example, the accuracy of the vertical datum transfer from the tide gauge of Amsterdam to New York was estimated equal to 57 cm when the EGM96 error model was used, while in the case of GOCE error model this accuracy was increased to 6 cm. The geoid undulation difference between the two places is about 76.5 m. Scaling the GOCE errors to the local gravity variance, the estimated accuracy varied between 3 and 7 cm, depending on the scaling model. Received: 1 March 2000 / Accepted: 21 February 2001     

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     

An optimality property of the integer least-squares estimator

A probabilistic justification is given for using the integer least-squares (LS) estimator. The class of admissible integer estimators is introduced and classical adjustment theory is extended by proving that the integer LS estimator is best in the sense of maximizing the probability of correct integer estimation. For global positioning system ambiguity resolution, this implies that the success rate of any other integer estimator of the carrier phase ambiguities will be smaller than or at the most equal to the ambiguity success rate of the integer LS estimator. The success rates of any one of these estimators may therefore be used to provide lower bounds for the LS success rate. This is particularly useful in case of the bootstrapped estimator. Received: 11 January 1999 / Accepted: 9 July 1999     

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     

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     

Seasonal length-of-day changes and atmospheric angular momentum oscillations in their temporal variability

Recently, effective atmospheric-angular-momentum (AAM) functions as calculated from National Centers for Environmental Prediction (NCEP) (formerly National Meteorological Center, NMC) and National Center for Atmospheric Research (NCAR) Reanalyses have become available for the years 1958 to 1998. Concerning the wind terms, the top level in the atmosphere used here is 10 hPa. Compared with earlier NMC model versions, which incorporate wind fields up to 100 hPa since 1976 and up to 50 hPa since 1981, the reanalyses have produced improved data series over a longer period than before. The axial AAM component χ₃ is associated with changes in length of day (LOD). Motivated by better quality and continuity of the series AAM (NCEP) Reanalysis, the problem of the seasonal imbalances in the solid Earth–atmosphere axial angular momentum budget is re-examined. To assess better the estimates of the annual and semiannual oscillations in LOD and AAM and of the residual oscillations derived as difference series between LOD and AAM, the series of LOD data from three analysis centers [International Earth Rotation Service (IERS), GeoForschungZentrum Potsdam (GFZ) and Jet Propulsion Laboratory Pasadena (JPL)] and of AAM data in terms of χ₃(W), χ₃(P) and χ₃(P+IB) from four meteorological centers [NCEP, Japan Meteorological Agency (JMA), European Centre for Medium-Range Weather Forecasts (ECMWF) and the UK Meteorological Office (UKMO)] are used in this study. The main analysis steps were removing gaps, filtering out the seasonal oscillations, calculating optimal estimates of the parameters of the oscillations and calculating the difference series between the LOD and AAM systems as well as the residuals in the axial angular momentum budget in the LOD–AAM systems. The results derived as difference series between the different LOD, AAM and LOD–AAM systems show to what extent the variations reflect systematic differences and significant signals, respectively, which is important for future activities in this field. Received: 2 February 1999 / Accepted: 30 November 1999     

The probability distribution of the ambiguity bootstrapped GNSS baseline

 The purpose of carrier phase ambiguity resolution is to improve upon the quality of the estimated global navigation satellite system baseline by means of the integer ambiguity constraints. However, in order to evaluate the quality of the ambiguity resolved baseline rigorously, its probability distribution is required. This baseline distribution depends on the random characteristics of the estimated integer ambiguities, which in turn depend on the chosen integer estimator. In this contribution is presented an exact and closed-form expression for the baseline distribution in the case that use is made of integer bootstrapping. Also presented are the bootstrapped probability mass function and easy-to-compute measures for the bootstrapped baseline's probability of concentration. Received: 28 September 2000 / Accepted: 11 January 2001     

A modified Wiener-type filter for geodetic estimation problems with non-stationary noise

 One of the most basic and important tools in optimal spectral gravity field modelling is the method of Wiener filtering. Originally developed for applications in analogue signal analysis and communication engineering, Wiener filtering has become a standard linear estimation technique of modern operational geodesy, either as an independent practical tool for data de-noising in the frequency domain or as an integral component of a more general signal estimation methodology (input–output systems theory). Its theoretical framework is based on the Wiener–Kolmogorov linear prediction theory for stationary random fields in the presence of additive external noise, and thus it is closely related to the (more familiar to geodesists) method of least-squares collocation with random observation errors. The main drawback of Wiener filtering that makes its use in many geodetic applications problematic stems from the stationarity assumption for both the signal and the noise involved in the approximation problem. A modified Wiener-type linear estimation filter is introduced that can be used with noisy data obtained from an arbitrary deterministic field under the masking of non-stationary random observation errors. In addition, the sampling resolution of the input data is explicitly taken into account within the estimation algorithm, resulting in a resolution-dependent optimal noise filter. This provides a more insightful approach to spectral filtering techniques for noise reduction, since the data resolution parameter has not been directly incorporated in previous formulations of frequency-domain estimation problems for gravity field signals with discrete noisy data. Received: 1 November 2000 / Accepted: 19 June 2001     

On long-period polar motion

 Five separate polar motion series are examined in order to understand what portion of their variations at periods exceeding several years represents true polar motion. The data since the development of space-geodetic techniques (by themselves insufficient for study of long-period motion), and a variety of historical astrometric data sets, allow the following tentative conclusions: retrograde long-period polar motion below about −0.2 cpy (cycles per year) in pre-space-geodetic data (pre-1976) is dominantly noise. For 1976–1992, there is poor agreement between space-geodetic and astrometric series over the range −0.2 to +0.2 cpy, demonstrating that classical astrometry lacked the precision to monitor polar motion in this frequency range. It is concluded that all the pre-1976 astrometric polar motion data are likely to be dominated by noise at periods exceeding about 10 years. The exception to this is possibly a linear trend found in some astrometric and space geodetic series. At frequencies above prograde +0.2 cpy (periods shorter than about 5 years), historical astrometric data may be of sufficient quality for comparisons with geophysical excitation time series. Even in the era of space geodesy, significant differences are found in long-period variations in published polar motion time series. Received: 27 March 2001 / Accepted: 15 October 2001     

Improvement of the separability of a survey scheme for monitoring crustal deformations in the area of an active fault

 Monitoring of the crustal movements along a tectonic fault is of particular importance in the study of the mechanism of an earthquake. There are several techniques to gauge crustal deformations, including terrestrial survey methods, space-positioning techniques and permanently installed geotechnical instruments. Each technique or method has its own advantages and limitations. Integration of the various techniques into a monitoring scheme is recommended. It is discussed how a proper integrated system can significantly improve the separability of a monitoring scheme at little additional expense. Separability is the ability of a monitoring scheme to distinguish among potential deformation models, and can be used for the optimum design of monitoring schemes. Discussion concentrates on the separability between a dislocation model and a rigid movement model in the area of an active fault. The addition of a few strain observations to a conventional terrestrial survey scheme can better distinguish between the above-mentioned models. A simulated example is presented to demonstrate the idea. Received: 4 November 1997 / Accepted: 9 July 2001     

Global mean sea surface and marine gravity anomaly from multi-satellite altimetry: applications of deflection-geoid and inverse Vening Meinesz formulae

 Global mean sea surface heights (SSHs) and gravity anomalies on a 2^′×2^′ grid were determined from Seasat, Geosat (Exact Repeat Mission and Geodetic Mission), ERS-1 (1.5-year mean of 35-day, and GM), TOPEX/POSEIDON (T/P) (5.6-year mean) and ERS-2 (2-year mean) altimeter data over the region 0^∘–360^∘ longitude and –80^∘–80^∘ latitude. To reduce ocean variabilities and data noises, SSHs from non-repeat missions were filtered by Gaussian filters of various wavelengths. A Levitus oceanic dynamic topography was subtracted from the altimeter-derived SSHs, and the resulting heights were used to compute along-track deflection of the vertical (DOV). Geoidal heights and gravity anomalies were then computed from DOV using the deflection-geoid and inverse Vening Meinesz formulae. The Levitus oceanic dynamic topography was added back to the geoidal heights to obtain a preliminary sea surface grid. The difference between the T/P mean sea surface and the preliminary sea surface was computed on a grid by a minimum curvature method and then was added to the preliminary grid. The comparison of the NCTU01 mean sea surface height (MSSH) with the T/P and the ERS-1 MSSH result in overall root-mean-square (RMS) differences of 5.0 and 3.1 cm in SSH, respectively, and 7.1 and 3.2 μrad in SSH gradient, respectively. The RMS differences between the predicted and shipborne gravity anomalies range from 3.0 to 13.4 mGal in 12 areas of the world's oceans. Received: 26 September 2001 / Accepted: 3 April 2002 Correspondence to: C. Hwang Acknowledgements. This research is partly supported by the National Science Council of ROC, under grants NSC89-2611-M-009-003-OP2 and NSC89-2211-E-009-095. This is a contribution to the IAG Special Study Group 3.186. The Geosat and ERS1/2 data are from NOAA and CERSAT/France, respectively. The T/P data were provided by AVISO. The CLS and GSFC00 MSS models were kindly provided by NASA/GSFC and CLS, respectively. Drs. Levitus, Monterey, and Boyer are thanked for providing the SST model. Dr. T. Gruber and two anonymous reviewers provided very detailed reviews that improved the quality of this paper.     

The ERS-2 altimetric bias and gravity field enhancement using dual crossovers between ERS and TOPEX/Poseidon

 Dual satellite crossovers (DXO) between the two European Remote Sensing satellites ERS-1 and ERS-2 and TOPEX/Poseidon are used to (1) refine the Earth's gravity field and (2) extend the study of the ERS-2 altimetric range stability to cover the first four years of its operation. The enhanced gravity field model, AGM-98, is validated by several methodologies and will be shown to provide, in particular, low geographically correlated orbital error for ERS-2. For the ERS-2 altimetric range study, TOPEX/Poseidon is first calibrated through comparison against in situ tide gauge data. A time series of the ERS-2 altimeter bias has been recovered along with other geophysical correction terms using tables for bias jumps in the range measurements at the single point target response (SPTR) events. On utilising the original version of the SPTR tables the overall bias drift is seen to be 2.6±1.0 mm/yr with an RMS of fit of 12.2 mm but with discontinuities at the centimetre level at the SPTR events. On utilising the recently released revised tables, SPTR2000, the drift is better defined at 2.4±0.6 mm/yr with the RMS of fit reduced to 3.7 mm. Investigations identify the sea-state bias as a source of error with corrections affecting the overall drift by close to 1.2 mm/yr. Received: 25 May 2000 / Accepted: 24 January 2001