Abstact.Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-Level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analyzed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared to combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITGGrace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-months GOCE gravity field models of 10 cm in terms of geoid heights, and 3 mGal in terms of gravity anomalies, globally at a resolution of 100 km (degree/order 200).

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The EGM2008 model is nowadays one of the description of the global gravitational field at the highest resolution. It is delivered with two, not fully consistent, sources of information on its error: spherical harmonic coefficient variances and a geographical map of error variances, e.g. in terms of geoid undulation. In the present work, the gravity field information derived from a GOCE satellite-only global model is used to improve the accuracy of EGM2008 model in the low to medium frequencies, especially in areas where no data were available at the time of EGM2008 computation. The key issue is to set up the error covariance matrices of the two models for an optimal least-squares combination: the full error covariance matrix of GOCE spherical harmonic coefficients is approximated by an order-wise block-diagonal matrix, while for EGM2008, the pointwise error variances are taken from the provided geoid error map and the error spatial correlations from the coefficient variances. Due to computational reasons the combination is directly performed in terms of geoid values over a regular grid on local areas. Repeating the combination for overlapping areas all over the world and then performing a harmonic analysis, a new combined model is obtained. It is called GECO and extends up to the EGM2008 maximum degree. Comparisons with other recent combined models, such as EIGEN-6C4, and a local geoid based on new gravity datasets in Antarctica are performed to evaluate its quality. The main conclusion is that the proposed combination, weighting the different input contributions not only on a global basis but also according to some local error information, can perform even better than other more sophisticated combinations in areas where the input global error description is not reliable enough

goGPS is a positioning software application designed to process single-frequency code and phase observations for absolute or relative positioning. Published under a free and open-source license, goGPS can process data collected by any receiver, but focuses on the treatment of observations by low-cost receivers. goGPS algorithms can produce epoch-by-epoch solutions by least squares adjustment, or multi-epoch solutions by Kalman filtering, which can be applied to either positions or observations. It is possible to aid the positioning by introducing additional constraints, either on the 3D trajectory such as a railway, or on a surface, e.g., a digital terrain model. goGPS is being developed by a collaboration of different research groups, and it can be downloaded from http://​www.​gogps-project.​org. The version used in this manuscript can be also downloaded from the GPS Toolbox Web site http://​www.​ngs.​noaa.​gov/​gps-toolbox. This software is continues to evolve, improving its functionalities according to the updates introduced by the collaborators. We describe the main modules of goGPS along with some examples to show the user how the software works

goGPS is a free and open source satellite positioning software package aiming to provide a collaborative platform for research and teaching purposes. It was first published in 2009 and since then several related projects are on-going. Its objective is the investigation of strategies for enhancing the accuracy of low-cost single-frequency GPS receivers, mainly by relative positioning with respect to a base station and by a tailored extended Kalman filter working directly on code and phase observations. In this paper, the positioning algorithms implemented in goGPS are presented, emphasizing the modularity of the software design; two specific strategies to support the navigation with low-cost receivers are also proposed and discussed, namely an empirical observation weighting function calibrated on the receiver signal-to-noise ratio and the inclusion of height information from a digital terrain model as an additional observation in the Kalman filter. The former is crucial when working with high-sensitivity receivers, while the latter can significantly improve the positioning in the vertical direction. The overall goGPS positioning accuracy is assessed by comparison with a dual-frequency receiver and with the positioning computed by a standard low-cost receiver. The benefits of the calibrated weighting function and the digital terrain model are investigated by an experiment in a dense urban environment. It comes out that the use of goGPS and low-cost receivers leads to results comparable with those obtained by higher level receivers; goGPS has good performances also in a dense urban environment, where its additional features play an important role.

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The modernization of the global positioning system and the advent of the European project Galileo will lead to a multifrequency global navigation satellite system (GNSS). The presence of new frequencies introduces more degrees of freedom in the GNSS data combination. We define linear combinations of GNSS observations with the aim to detect and correct cycle slips in real time. In particular, the detection is based on five geometry-free linear combinations used in three cascading steps. Most of the jumps are detected in the first step using three minimum-noise combinations of phase and code observations. The remaining jumps with very small amplitude are detected in the other two steps by means of two-tailored linear combinations of phase observations. Once the epoch of the slip has been detected, its amplitude is estimated using other linear combinations of phase observations. These combinations are defined with the aim of discriminating between the possible combinations of jump amplitudes in the three carriers. The method has been tested on simulated data and 1-second triple-frequency undifferenced GPS data coming from a friendly multipath environment. Results show that the proposed method is able to detect and repair all combinations of cycle slips in the three carriers

Collocation is widely used in physical geodesy. Its application requires to solve systems with a dimension equal to the number of observations, causing numerical problems when many observations are available. To overcome this drawback, tailored step-wise techniques are usually applied. An example of these step-wise techniques is the space-wise approach to the GOCE mission data processing. The original idea of this approach was to implement a two-step procedure, which consists of first predicting gridded values at satellite altitude by collocation and then deriving the geo-potential spherical harmonic coefficients by numerical integration. The idea was generalized to a multi-step iterative procedure by introducing a time-wise Wiener filter to reduce the highly correlated observation noise. Recent studies have shown how to optimize the original two-step procedure, while the theoretical optimization of the full multi-step procedure is investigated in this work. An iterative operator is derived so that the final estimated spherical harmonic coefficients are optimal with respect to the Wiener–Kolmogorov principle, as if they were estimated by a direct collocation. The logical scheme used to derive this optimal operator can be applied not only in the case of the space-wise approach but, in general, for any case of step-wise collocation. Several numerical tests based on simulated realistic GOCE data are performed. The results show that adding a pre-processing time-wise filter to the two-step procedure of data gridding and spherical harmonic analysis is useful, in the sense that the accuracy of the estimated geo-potential coefficients is improved. This happens because, in its practical implementation, the gridding is made by collocation over local patches of data, while the observation noise has a time-correlation so long that it cannot be treated inside the patch size. Therefore, the multi-step operator, which is in theory equivalent to the two-step operator and to the direct collocation, is in practice superior thanks to the time-wise filter that reduces the noise correlation before the gridding. The criteria for the choice of this filter are investigated numerically