An improvement to ionospheric delay correction for single-frequency GPS users – the APR-I scheme

 For the commonly used GPS wide-area augmentation systems (WAAS) with a grid ionospheric model, the efficient modelling of ionospheric delays in real time, for single-frequency GPS users, is still a crucial issue which needs further research. This is particularly necessary when differential ionospheric delay corrections cannot be broadcast, when users cannot receive them, or when there are ionospheric anomalies. Ionospheric delays have a severe effect on navigation performance of single-frequency receivers. A new scheme is proposed which can efficiently address the above problems. The robust recurrence technique is based on the efficient combination of single-frequency GPS observations by users and the high-precision differential ionospheric delay corrections from WAAS. Its effectiveness is verified with examples. Received: 24 December 1999 / Accepted 21 February 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     

GPS vector configuration design for monitoring deformation networks

 The performance of geodetic monitoring networks is heavily influenced by the configuration of the measured GPS vectors. As an effective design of the GPS measurements will decrease GPS campaign costs and increase the accuracy and reliability of the entire network, the identification of the preferred GPS vectors for measurement has been highlighted as a core problem in the process of deformation monitoring. An algorithm based on a sensitivity analysis of the network, as dependent upon a postulated velocity field, is suggested for the selection of the optimal GPS vectors. Relevant mathematical and statistical concepts are presented as the basis for an improved method of vector configuration design. A sensitivity analysis of the geodetic geodynamic network in the north of Israel is presented, where the method is examined against two deformation models, the Simple Transform Fault and the Locked Fault. The proposed method is suggested as a means for the improvement of the design of monitoring networks, a common practice worldwide. Received: 30 July 2001 / Accepted: 3 June 2002 Acknowledgments. It is my pleasant duty to thank the Survey of Israel and Dr. E. Ostrovsky for providing the variance–covariance matrix of the G1 network in northern Israel. I would like to thank the reviewers of this paper for their constructive and helpful remarks.     

Combination procedure for length-of-day time series according to the noise frequency behavior

 Length-of-day (LOD) estimates from seven Global Positioning System (GPS) and three satellite laser ranging (SLR) analysis centers were combined into an even-spaced time series for a 27-month period (1996–1998). This time series was compared to the multi-technique Earth-orientation-parameter (EOP) combined solution (C04) derived at the Central Bureau of the International Earth Rotation Service (IERS/CB). Due to inhomogeneities in the different series derived from the various techniques (time, length, quality, and spatial resolution), the concept of a combined solution is justified. The noise behavior in LOD for different techniques varies with frequency; the data series were divided into frequency windows after removing both biases and trends. Different weight factors were assigned in each window, discriminating by technique, and produced one-technique combined solutions. Finally, these one-technique combined solutions were combined to obtain the final multi-technique solution. The LOD combined time series obtained by the present method based on the frequency windows combined series (FWCS) is very close to the IERS C04 solution. It could be useful to generate a new LOD reference time series to be used in the study of high-frequency variations of Earth rotation. Received: 28 March 2000 / Accepted: 15 February 2001     

Improvement of edge effect of the wavelet time–frequency spectrum: application to the length-of-day series

 The edge effect in the wavelet time–frequency spectrum of a time series is treated. The time series is first extended on both ends by applying a non-linear model, namely the leap-step time series analysis (LSTSA) model, prior to the wavelet transform. The results of a series of simulation experiments and an application to the observed length-of-day (LOD) series demonstrate that the edge effect is effectively reduced this way. Thus, the application of the LSTSA model improves the wavelet time–frequency spectrum, especially enhancing the ability to detect the low-frequency signals. Received: 15 September 1998 / Accepted: 4 October 1999     

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     

Adaptive filtering of continuous GPS results

 An adaptive finite-duration impulse response filter, based on a least-mean-square algorithm, has been used to mitigate multipath effects, and to derive tectonic and fault movement signals from continuous global positioning system (CGPS) data. By applying the filter on both pseudo-range and carrier-phase multipath sequences from CGPS observations on consecutive days, multipath models have been reliably derived. The standard deviations of residual time series are reduced to about one-quarter on pseudo-range and to about one-half on carrier phase. The adaptive filter is then used to process baseline solutions from a five-station array. Tectonic and fault movements have been resolved, which are in good agreement with previous studies involving many more CGPS stations. Received: 11 February 2000 / Accepted: 28 June 2000     

Analysis of the first year of Earth rotation parameters with a sub-daily resolution gained at the CODE processing center of the IGS

 The solutions of the CODE Analysis Center submitted to the IGS, the International Global Position System (GPS) Service for Geodynamics, are based on three days of observation of about 80–100 stations of the IGS network. The Earth rotation parameters (ERPs) are assumed to vary linearly over the three days with respect to an a priori model. Continuity at the day boundaries as well as the continuity of the first derivatives are enforced by constraints. Since early April 1995 CODE has calculated a new ERP series with an increased time resolution of 2 hours. Again continuity is enforced at the 2-hours-interval boundaries. The analysis method is described, particularly how to deal with retrograde diurnal terms in the ERP series which may not be estimated with satellite geodetic methods. The results obtained from the first year of data covered by the time series (time interval from 4 April 1995 to 30 June 1996) are also discussed. The series is relatively homogeneous in the sense of the used orbit model and the a priori model for the ERPs. The largest source of excitation at daily and sub-daily periods is likely to be the effect of the ocean tides. There is good agreement between the present results and Topex/Poseidon ocean tide models, as well as with models based on Very Long Baseline Interferometry (VLBI) and Satellite Laser Ranging (SLR) data. Non-oceanic periodic variations are also observed in the series. Their origin is most probably a consequence of the GPS solution strategy; other possible sources are the atmospheric tides. Received: 13 July 1999 / Accepted: 21 March 2000     

GPS measurements of ocean loading and its impact on zenith tropospheric delay estimates: a case study in Brittany, France

 The results from a global positioning system (GPS) experiment carried out in Brittany, France, in October 1999, aimed at measuring crustal displacements caused by ocean loading and quantifying their effects on GPS-derived tropospheric delay estimates, are presented. The loading effect in the vertical and horizontal position time series is identified, however with significant disagreement in amplitude compared to ocean loading model predictions. It is shown that these amplitude misfits result from spatial tropospheric heterogeneities not accounted for in the data processing. The effect of ocean loading on GPS-derived zenith total delay (ZTD) estimates is investigated and a scaling factor of 4.4 between ZTD and station height for a 10° elevation cut-off angle is found (i.e. a 4.4-cm station height error would map into a 1-cm ZTD error). Consequently, unmodeled ocean loading effects map into significant errors in ZTD estimates and ocean loading modeling must be properly implemented when estimating ZTD parameters from GPS data for meteorological applications. Ocean loading effects must be known with an accuracy of better than 3 cm in order to meet the accuracy requirements of meteorological and climatological applications of GPS-derived precipitable water vapor. Received: 16 July 2001 / Accepted: 25 April 2002 Acknowledgments. The authors are grateful to H.G. Scherneck for fruitful discussions and for his help with the ocean loading calculations. They thank H. Vedel for making the HIRLAM data available; D. Jerett for helpful discussions; and the city of Rostrenen, the Laboratoire d'Océanographie of Concarneau, and the Institut de Protection et de S?reté Nucléaire (BERSSIN) for their support during the GPS measurement campaign. Reviews by C.K. Shum and two anonymous referees significantly improved this paper. This work was carried out in the framework of the MAGIC project (http://www.acri.fr/magic), funded by the European Commission, Environment and Climate Program (EC Contract ENV4-CT98–0745). Correspondence to: E. Calais, Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN 47907-1397, USA. e-mail: ecalais@purdue.edu Tel. : +1-765-496-2915; Fax:+1-765-496-1210     

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 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     

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     

Assessment of three numerical solution strategies for gravity field recovery from GOCE satellite gravity gradiometry implemented on a parallel platform

 The recovery of a full set of gravity field parameters from satellite gravity gradiometry (SGG) is a huge numerical and computational task. In practice, parallel computing has to be applied to estimate the more than 90 000 harmonic coefficients parameterizing the Earth's gravity field up to a maximum spherical harmonic degree of 300. Three independent solution strategies (preconditioned conjugate gradient method, semi-analytic approach, and distributed non-approximative adjustment), which are based on different concepts, are assessed and compared both theoretically and on the basis of a realistic-as-possible numerical simulation regarding the accuracy of the results, as well as the computational effort. Special concern is given to the correct treatment of the coloured noise characteristics of the gradiometer. The numerical simulations show that the three methods deliver nearly identical results—even in the case of large data gaps in the observation time series. The newly proposed distributed non-approximative adjustment approach, which is the only one of the three methods that solves the inverse problem in a strict sense, also turns out to be a feasible method for practical applications. Received: 17 December 2001 / Accepted: 17 July 2002 Acknowledgments. We would like to thank Prof. W.-D. Schuh, Institute of Theoretical Geodesy, University of Bonn, for providing us with the serial version of the PCGMA algorithm, which forms the basis for the parallel PCGMA package developed at our institute. This study was partially performed in the course of the GOCE project `From E?tv?s to mGal+', funded by the European Space Agency (ESA) under contract No. 14287/00/NL/DC. Correspondence to: R. Pail     

A solution to the downward continuation effect on the geoid determined by Stokes' formula

 The analytical continuation of the surface gravity anomaly to sea level is a necessary correction in the application of Stokes' formula for geoid estimation. This process is frequently performed by the inversion of Poisson's integral formula for a sphere. Unfortunately, this integral equation corresponds to an improperly posed problem, and the solution is both numerically unstable, unless it is well smoothed, and tedious to compute. A solution that avoids the intermediate step of downward continuation of the gravity anomaly is presented. Instead the effect on the geoid as provided by Stokes' formula is studied directly. The practical solution is partly presented in terms of a truncated Taylor series and partly as a truncated series of spherical harmonics. Some simple numerical estimates show that the solution mostly meets the requests of a 1-cm geoid model, but the truncation error of the far zone must be studied more precisely for high altitudes of the computation point. In addition, it should be emphasized that the derived solution is more computer efficient than the detour by Poisson's integral. Received: 6 February 2002 / Accepted: 18 November 2002 Acknowledgements. Jonas ?gren carried out the numerical calculations and gave some critical and constructive remarks on a draft version of the paper. This support is cordially acknowledged. Also, the thorough work performed by one unknown reviewer is very much appreciated.     

Multi-level arc combination with stochastic parameters

 The method of square root information filtering and smoothing (SRIF/S) is reviewed and has been implemented in the combined square root information filter and smoother (CSRIFS) program. CSRIFS is a part of the GEOSAT space geodesy software developed at Forsvarets forskningsinstitutt (FFI, The Norwegian Defence Research Establishment). The state vectors and complete variance–covariance matrices from the analyses of a number of independent arcs of space geodesy data can be combined using CSRIFS. Four parameter levels are available and any parameter can, at each level, be represented as either a constant or a stochastic parameter (white noise, colored noise, or random walk). The batch length (i.e. the time interval between the addition of noise to the SRIF array) can be made time and parameter dependent. CSRIFS was applied in the combination of 623 very long baseline interferometry (VLBI) observing sessions. The purpose of this test was to validate the computer implementation of the SRIF/S method and to give an example of how this method can be used in the analysis of a large number of space geodetic observations. The results show that the implementation is very satisfactory. Received: 28 May 1999 / Accepted: 15 June 2000     

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     

First use of IGEX precise ephemerides for intercontinental GLONASS P-code time transfer

 Until recently, the Global Positioning System (GPS) was the only operational means of distributing time to an arbitrary number of users and of synchronizing clocks over large distances with a high degree of precision and accuracy. Over the last few years it has been shown that similar performance can be achieved using the Russian Global Navigation Satellite System (GLONASS). GLONASS time transfer between continents was initially hampered by the lack of post-processed precise ephemerides. Results from the International GLONASS Experiment (IGEX) campaign are now available, however, and this paper reports on the first use of IGEX precise ephemerides for GLONASS P-code intercontinental time links. The results of GLONASS P-code and GPS C/A-code time transfer are compared under similar conditions. Received: 31 January 2000 / Accepted: 10 July 2000     

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     

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     

Prediction of Earth orientation parameters by artificial neural networks

 Earth orientation parameters (EOPs) [polar motion and length of day (LOD), or UT1–UTC] were predicted by artificial neural networks. EOP series from various sources, e.g. the C04 series from the International Earth Rotation Service and the re-analysis optical astrometry series based on the HIPPARCOS frame, served for training the neural network for both short-term and long-term predictions. At first, all effects which can be described by functional models, e.g. effects of the solid Earth tides and the ocean tides or seasonal atmospheric variations of the EOPs, were removed. Only the differences between the modeled and the observed EOPs, i.e. the quasi-periodic and irregular variations, were used for training and prediction. The Stuttgart neural network simulator, which is a very powerful software tool developed at the University of Stuttgart, was applied to construct and to validate different types of neural networks in order to find the optimal topology of the net, the most economical learning algorithm and the best procedure to feed the net with data patterns. The results of the prediction were analyzed and compared with those obtained by other methods. The accuracy of the prediction is equal to or even better than that by other prediction methods. Received: 6 February 2001 / Accepted: 23 October 2001