Quality-control issues relating to instantaneous ambiguity resolution for real-time GPS kinematic positioning

An integrated method for the instantaneous ambiguity resolution using dual-frequency precise pseudo-range and carrier-phase observations is suggested in this paper. The algorithm combines the search procedures in the coordinate domain, the observation domain and the estimated ambiguity domain (and therefore benefits from the integration of their most positive elements). A three-step procedure is then proposed to enhance the reliability of the ambiguity resolution by: (1) improving the stochastic model for the double-differenced functional model in real time; (2) refining the criteria which distinguish the integer ambiguity set that generates the minimum quadratic form of residuals from that corresponding to the second minimum one; and (3) developing a fault detection and adaptation procedure. Three test scenarios were considered, one static baseline (11.3 km) and two kinematic experiments (baseline lengths from 5.2 to 13.7 km). These showed that the mean computation time for one epoch is less than 0.1 s, and that the success rate reaches 98.4% (compared to just 68.4% using standard ratio tests). Received: 5 June 1996; Accepted: 16 January 1997     

Apropos laser tracking to GPS satellites

. Laser tracking to GPS satellites (PRN5 and 6) provides an opportunity to compare GPS and laser systems directly and to combine data of both in a single solution. A few examples of this are given in this study. The most important results of the analysis are that (1) daily SLR station coordinate solutions could be generated with a few cm accuracy; (2) coordinates of nine stations were determined in a 2.3-year-long arc solution; (3) the contribution of laser data on the `SLR-GPS' combined orbit, resulting from the simultaneous processing of SLR and GPS data, is significant and (4) laser-only orbits have an accuracy of 10–20 cm, 1-day predictions of SLR orbits differ from IGS orbits by about 20–40 cm, 2-day predictions by 50–60 cm. Received: 1 October 1996 / Accepted: 14 February 1997     

Evidence and cause of climate cycles in polar motion

 The long-term variation of polar motion contains a number of periods similar to climate cycles. Two possible causes for these long-term variations are mass redistributions produced by variations of atmospheric and oceanic circulation, and mass exchanges between the cryosphere and hydrosphere. Inner-core wobble, which can be inferred from the observed motion of the geomagnetic pole, is another phenomenon with periods similar to climate cycles. Only observations relating to mass redistributions caused by atmosphere dynamics and inner-core wobble are available for sufficiently long periods of time to investigate their influence on climate cycles in polar motion. Both processes contribute to climate cycles in polar motion, but they cannot completely explain these cycles. Possible sources of climate cycles are discussed. Received: 20 December 1999 / Accepted: 28 August 2000     

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     

Grid point search algorithm for fast integer ambiguity resolution

A Grid Point Search Algorithm (GRIPSA) for fast integer ambiguity resolution is presented. In the proposed algorithm, after the orthogonal transformation of the original ambiguity parameters, the confidence ellipsoid of the new parameters is represented by a rectangular polyhedron with its edges parallel to the corresponding axes. A cubic grid covering the whole polyhedron is then identified and transformed back to the original coordinate system. The integer values of the corresponding transformed grid points are obtained by rounding off the transformed values to their nearest integer values. These values are then tested as to whether they are located inside the polyhedron. Since the identification of the grid points in the transformed coordinate system greatly reduces the search region of the integer ambiguities, marked improvements are obtained in the computational effort. Received: 13 October 1997 / Accepted: 9 June 1998     

Recent Improvements to IERS Bulletin A Combination and Prediction

Driven by a need for increased accuracy in real-time Earth orientation parameters (EOPs), the Bulletin A (Rapid Servce and Predictions) of the International Earth Rotation Service (IERS) has recently made several major changes to its combination and prediction procedures. Changes to the process ob combining multi-technique results include creation of a daily Bulletin A updata, inclusion of several new data sets, and use of polar motion rantes for the latest epoch. Notably, the contributions from GPS observations have grown steadily in significance, both for polar motion and Universal Time (UT1). The prediction procedure has, in turn, benefited from these changes as well as improvements to the polar motion prediction model. As a result, demanding real-time applications, such as for satellite orbit extrapolations should observe a major improvement in the accuracy of our real-time EOP products. All results, together with supporting and diagnostic information, are available at the website http://maia.usno.navy.mil. The maximum EOP errors (root-mean-squared sense) that a real-time user would experience using the latest available update of Bulletin A are currently estimated to be ∼0.9 milliarcseconds (mas) for polar motion and ∼0.15 milliseconds (ms) for UT1-UTC. The data latency (the lag since the most recent observations) for EOP predictions need not exceed ∼41 hours for users who avail themselves of the daily updates. Over the past four years, the accuracy for real-time applications has improved by nearly a factor of 4 in polar motion and a factor of 10 in UT1. This is primarily due to the large reduction in data latency, which in turn is mostly possible due to the Rapid product delivery of the International GPS Service (IGS) (see Mireault et al, 1999). ? 2001 John Wiley & Sons, Inc.     

Linear drift and periodic variations observed in long time series of polar motion

 Two long time series were analysed: the C01 series of the International Earth Rotation Service and the pole series obtained by re-analysis of the classical astronomical observations using the HIPPARCOS reference frame. The linear drift of the pole was determined to be 3.31 ± 0.05 milliarcseconds/year towards 76.1 ± 0.80° west longitude. For the least-squares fit the a priori correlations between simultaneous pole coordinates x _p , y _p were taken into account, and the weighting function was calculated by estimating empirical variance components. The decadal variations of the pole path were investigated by Fourier and wavelet analysis. Using sliding windows, the periods and amplitudes of the Chandler wobble and annual wobble were determined. Typical periods in the variable Chandler wobble and annual wobble parameters were obtained from wavelet analyses. Received: 21 January 2000 / Accepted: 28 August 2000     

Analytical solar radiation pressure modelling for GLONASS using a pixel array

 A new method for calculating analytical solar radiation pressure models for GNSS spacecraft has been developed. The method simulates the flux of light from the Sun using a pixel array. The method can cope with a high level of complexity in the spacecraft structure and models effects due to reflected light. Models have been calculated and tested for the Russhar global navigation satellite system GLONASS IIv spacecraft. Results are presented using numerical integration of the force model and long-arc satellite laser ranging (SLR) analysis. The integrated trajectory differs from a precise orbit calculated using a network of global tracking stations by circa 2 m root mean square over a 160 000-km arc. The observed − computed residuals for the 400-day SLR arc are circa 28 mm. Received: 23 December 1999 / Accepted: 28 August 2000     

Prediction of UT1–UTC,LOD and AAM χ<Subscript>3</Subscript> by combination of least-squares and multivariate stochastic methods

This article presents the application of a multivariate prediction technique for predicting universal time (UT1–UTC), length of day (LOD) and the axial component of atmospheric angular momentum (AAM χ ₃). The multivariate predictions of LOD and UT1–UTC are generated by means of the combination of (1) least-squares (LS) extrapolation of models for annual, semiannual, 18.6-year, 9.3-year oscillations and for the linear trend, and (2) multivariate autoregressive (MAR) stochastic prediction of LS residuals (LS + MAR). The MAR technique enables the use of the AAM χ ₃ time-series as the explanatory variable for the computation of LOD or UT1–UTC predictions. In order to evaluate the performance of this approach, two other prediction schemes are also applied: (1) LS extrapolation, (2) combination of LS extrapolation and univariate autoregressive (AR) prediction of LS residuals (LS + AR). The multivariate predictions of AAM χ ₃ data, however, are computed as a combination of the extrapolation of the LS model for annual and semiannual oscillations and the LS + MAR. The AAM χ ₃ predictions are also compared with LS extrapolation and LS + AR prediction. It is shown that the predictions of LOD and UT1–UTC based on LS + MAR taking into account the axial component of AAM are more accurate than the predictions of LOD and UT1–UTC based on LS extrapolation or on LS + AR. In particular, the UT1–UTC predictions based on LS + MAR during El Niño/La Niña events exhibit considerably smaller prediction errors than those calculated by means of LS or LS + AR. The AAM χ ₃ time-series is predicted using LS + MAR with higher accuracy than applying LS extrapolation itself in the case of medium-term predictions (up to 100 days in the future). However, the predictions of AAM χ ₃ reveal the best accuracy for LS + AR.     

利用小波分解改进极移预报模型

为进一步提高极移预报精度，将小波分解引入极移预报中。首先利用小波分解对极移序列进行分解，分离低频分量与高频分量，然后对低频分量建立最小二乘外推模型，获得极移序列的趋势项外推值与残差序列，最后采用自回归（autoregressive，AR）模型对高频分量与残差序列之和进行预报，最终极移的预报值为最小二乘外推值与AR模型预报值之和。结果表明，小波分解可以明显改善最小二乘外推与AR组合模型的极移预报精度，尤其对于中长期预报改善更为明显。     

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     

Is there utility in rigorous combinations of VLBI and GPS Earth orientation parameters?

Combinations of station coordinates and velocities from independent space-geodetic techniques have long been the standard method to realize robust global terrestrial reference frames (TRFs). In principle, the particular strengths of one observing method can compensate for weaknesses in others if the combination is properly constructed, suitable weights are found, and accurate co-location ties are available. More recently, the methodology has been extended to combine time-series of results at the normal equation level. This allows Earth orientation parameters (EOPs) to be included and aligned in a fully consistent way with the TRF. While the utility of such multi-technique combinations is generally recognized for the reference frame, the benefits for the EOPs are yet to be quantitatively assessed. In this contribution, which is a sequel to a recent paper on co-location ties (Ray and Altamimi in J Geod 79(4–5): 189–195, 2005), we have studied test combinations of very long baseline interferometry (VLBI) and Global Positioning System (GPS) time-series solutions to evaluate the effects on combined EOP measurements compared with geophysical excitations. One expects any effect to be small, considering that GPS dominates the polar motion estimates due to its relatively dense and uniform global network coverage, high precision, continuous daily sampling, and homogeneity, while VLBI alone observes UT1-UTC. Presently, although clearly desirable, we see no practical method to rigorously include the GPS estimates of length-of-day variations due to significant time-varying biases. Nevertheless, our results, which are the first of this type, indicate that more accurate polar motion from GPS contributes to improved UT1-UTC results from VLBI. The situation with combined polar motion is more complex. The VLBI data contribute directly only very slightly, if at all, with an impact that is probably affected by the weakness of the current VLBI networks (small size and sparseness) and the quality of local ties relating the VLBI and GPS frames. Instead, the VLBI polar motion information is used primarily in rotationally aligning the VLBI and GPS frames, thereby reducing the dependence on co-location tie information. Further research is needed to determine an optimal VLBI-GPS combination strategy that yields the highest quality EOP estimates. Improved local ties (including internal systematic effects within the techniques) will be critically important in such an effort.     

The least-squares ambiguity decorrelation adjustment: its performance on short GPS baselines and short observation spans

The least-squares ambiguity decorrelation adjustment is a method for fast GPS double-difference (DD) integer ambiguity estimation. The performance of the method will be discussed, and although it is stressed that the method is generally applicable, attention is restricted to short-baseline applications in the present contribution. With reference to the size and shape of the ambiguity search space, the volume of the search space will be introduced as a measure for the number of candidate grid points, and the signature of the spectrum of conditional variances will be used to identify the difficulty one has in computing the integer DD ambiguities. It is shown that the search for the integer least-squares ambiguities performs poorly when it takes place in the space of original DD ambiguities. This poor performance is explained by means of the discontinuity in the spectrum of conditional variances. It is shown that through a decorrelation of the ambiguities, transformed ambiguities are obtained which generally have a flat and lower spectrum, thereby enabling a fast and efficient search. It is also shown how the high precision and low correlation of the transformed ambiguities can be used to scale the search space so as to avoid an abundance of unnecessary candidate grid points. Numerical results are presented on the spectra of conditional variances and on the statistics of both the original and transformed ambiguities. Apart from presenting numerical results which can typically be achieved, the contribution also emphasizes and explains the impact on the method's performance of different measurement scenarios, such as satellite redundancy, single vs dual-frequency data, the inclusion of code data and the length of the observation time span. Received: 31 October 1995 / Accepted: 21 March 1997     

Polar motion modeling,analysis, and prediction with time dependent harmonic coefficients

A time dependent amplitude model was proposed for the analysis and prediction of polar motion time series. The formulation was implemented to analyze part of the new combined solution, EOP (IERS) C 04, daily polar motion time series of 14 years length using a statistical model with first order autoregressive disturbances. A new solution approach, where the serial correlations of the disturbances are eliminated by sequentially differencing the measurements, was used to estimate the model parameters using weighted least squares. The new model parsimoniously represents the 14-year time series with 0.5 mas rms fit, close to the reported 0.1 mas observed pole position precisions for the x and y components. The model can also predict 6 months into the future with less than 4 mas rms prediction error for both polar motion components, and down to sub mas for one-step ahead prediction as validated using a set of daily time series data that are not used in the estimation. This study is dedicated to the memory of Prof. Urho Uotila (1923–2006) whose teaching of “Adjustment Computations” over the years influenced so much, so many of us who had the privilege of being his students.     

The impact of celestial pole offset modelling on VLBI UT1 intensive results

Very Long Baseline Interferometry (VLBI) Intensive sessions are scheduled to provide operational Universal Time (UT1) determinations with low latency. UT1 estimates obtained from these observations heavily depend on the model of the celestial pole motion used during data processing. However, even the most accurate precession- nutation model, IAU 2000/2006, is not accurate enough to realize the full potential of VLBI observations. To achieve the highest possible accuracy in UT1 estimates, a celestial pole offset (CPO), which is the difference between the actual and modelled precession-nutation angles, should be applied. Three CPO models are currently available for users. In this paper, these models have been tested and the differences between UT1 estimates obtained with those models are investigated. It has been shown that neglecting CPO modelling during VLBI UT1 Intensive processing causes systematic errors in UT1 series of up to 20 μas. It has been also found that using different CPO models causes the differences in UT1 estimates reaching 10 μas. Obtained results are applicable to the satellite data processing as well.     

Residual errors in altimetry data detected by combinations of single- and dual-satellite crossovers

 The single- and dual-satellite crossover (SSC and DSC) residuals between and among Geosat, TOPEX/Poseidon (T/P), and ERS 1 or 2 have been used for various purposes, applied in geodesy for gravity field accuracy assessments and determination as well as in oceanography. The theory is presented and various examples are given of certain combinations of SSC and DSC that test for residual altimetry data errors, mostly of non-gravitational origin, of the order of a few centimeters. There are four types of basic DSCs and 12 independent combinations of them in pairs which have been found useful in the present work. These are defined in terms of the `mean' and `variable' components of a satellite's geopotential orbit error from Rosborough's 1st-order analytical theory. The remaining small errors, after all altimeter data corrections are applied and the relative offset of coordinate frames between altimetry missions removed, are statistically evaluated by means of the Student distribution. The remaining signal of `non-gravitational' origin can in some cases be attributed to the main ocean currents which were not accounted for among the media or sea-surface corrections. In future, they may be resolved by a long-term global circulation model. Experience with two current models, neither of which are found either to cover the most critical missions (Geosat & TOPEX/Poseidon) or to have the accuracy and resolution necessary to account for the strongest anomalies found across them, is described. In other cases, the residual signal is due to errors in tides, altimeter delay corrections or El Ni?o. (Various examples of these are also presented.) Tests of the combinations of the JGM 3-based DSC residuals show that overall the long-term data now available are well suited for a gravity field inversion refining JGM 3 for low- and resonant-order geopotential harmonics whose signatures are clearly seen in the basic DSC and SSC sets. Received: 15 January 1999 / Accepted: 9 September 1999     

Assessment of periodic sub-diurnal Earth rotation variations at tidal frequencies through transformation of VLBI normal equation systems

We present an empirical model for periodic variations of diurnal and sub-diurnal Earth rotation parameters (ERPs) that was derived based on the transformation of normal equation (NEQ) systems of Very Long Baseline Interferometry (VLBI) observing sessions. NEQ systems that contain highly resolved polar motion and UT1-TAI with a temporal resolution of 15 min were generated and then transformed to the coefficients of the tidal ERP model to be solved for. To investigate the quality of this model, comparisons with empirical models from the Global Positioning System (GPS), another VLBI model and the model adopted by the conventions of the International Earth Rotation and Reference Systems Service (IERS) were performed. The absolute coefficients of these models agree almost completely within 7.5 μ as in polar motion and 0.5 μs in UT1-TAI. Several bigger differences exist, which are discussed in this paper. To be able to compare the model estimates with results of the continuous VLBI campaigns, where signals with periods of 8 and 6 h were detected, terms in the ter- and quarter-diurnal band were included in the tidal ERP model. Unfortunately, almost no common features with the results of continuous VLBI campaigns or ERP predictions in these tidal bands can be seen.     

Distribution of vertical crustal movement rates in the Tohoku district, Japan, predicted by least-squares collocation

Least-squares prediction using an empirically deduced local covariance function was performed to investigate the temporal change in the rates of vertical crustal movements in the Tohoku district, Japan. Levelling data covering an area of approximately 450 × 275 km² observed between 1966 and 1995 were used and the results shown in the form of contour maps. Firstly we derived a covariance function of the rates of vertical crustal movement with a Gaussian form function. We used this function to estimate the spatial distribution of the rates of vertical crustal movements. By the present method, a steady tilt of the Tohoku district to the east, toward the Japan Trench and an areal uplift in the southwestern part were well reproduced. Moreover, a significant temporal change in vertical movement rates is clearly seen. Received: 15 July 1996      

Results of three year observation with a superconducting gravimeter at the GeoForschungsZentrum Potsdam

The superconducting gravimeter (SG) TT70 has been continuously recording gravity data at the GeoForschungsZentrum (GFZ) Potsdam absolute gravity site since July 1992. The recorded data are edited and preprocessed by spike and step detection and elimination and gap filling. An atmospheric pressure correction is carried out on gravity data in the time domain with a complex admittance before tidal fitting. The atmospheric pressure admittance is calculated from tide free output of SG data and local atmospheric pressure using the cross spectral method. The ground water level admittance is determined by a single coefficient. Improvements with these corrections are shown in analysis results. New tidal parameters for Potsdam site are determined and compared with recordings of an Askania spring gravimeter at a nearby site. Deviations against the Wahr-Dehant-model are shown. Polar motion data of the IERS (International Earth Rotation Service, Paris) are used to calculate variations of centrifugal acceleration caused by polar motion (pole tide). These are compared with the corrected tide free output of SG series. For drift determination the polar motion correction is applied on SG data. The nutation period equivalent to the Earth's Nearly Diurnal Free Wobble is calculated from the SG data with a value of T_FCN = (437.4 ± 1.5) sidereal days. This result is compared with those obtained from other SG stations. Received 19 December 1995; Accepted 13 September 1996     

Combination of modeled short-term angular momentum function forecasts from atmosphere, ocean, and hydrology with 90-day EOP predictions

Angular momentum forecasts for up to 10 days into the future, modeled from predicted states of the atmosphere, ocean and continental hydrosphere, are combined with the operational IERS EOP prediction bulletin A to reduce the prediction error in the very first day and to improve the subsequent 90-day prediction by exploitation of the revised initial state and trend information. EAM functions derived from ECMWF short-range forecasts and corresponding LSDM and OMCT simulations can account for high-frequency mass variations within the geophysical fluids for up to 7 days into the future primarily limited by the accuracy of the forecasted atmospheric wind fields. Including these wide-band stochastic signals into the first days of the 90-day statistical IERS predictions reduces the mean absolute prediction error even for predictions beyond day 10, especially for polar motion, where the presently used prediction approach does not include geophysical fluids data directly. In a hindcast experiment using 1 year of daily predictions from May 2011 till July 2012, the mean prediction error in polar motion, compared to bulletin A, is reduced by 32, 12, and 3 % for prediction days 10, 30, and 90, respectively. In average, the prediction error for medium-range forecasts (30–90 days) is reduced by 1.3–1.7 mas. Even for UT1-UTC, where AAM forecasts are already included in IERS bulletin A, we obtain slight improvements of up to 5 % (up to 0.5 ms) after day 10 due to the additional consideration of oceanic angular momentum forecasts. The improved 90-day predictions can be generated operationally on a daily basis directly after the publication of the related IERS bulletin A product finals2000A.daily.