ALBERT

Description:    A new global navigation satellite system (GNSS) carrier-phase attitude model and its solution are introduced in this contribution. This affine-constrained GNSS attitude model has the advantage that it avoids the computational complexity of the orthonormality-constrained GNSS attitude model, while it still has a significantly improved ambiguity resolution performance over its unconstrained counterpart. The functional and stochastic model is formulated in multivariate form, for one-, two- and three-dimensional antenna arrays, tracking GNSS signals on an arbitrary number of frequencies with two or more antennas. The stochastic model includes the capability to capture variations in the antenna-quality within the array. The multivariate integer least-squares solution of the model parameters is given and the model’s ambiguity success-rate is analysed by means of the ambiguity dilution of precision (ADOP). A general closed-form expression for the affine-constrained ADOP is derived, thus providing an easy-to-use and insightful rule-of-thumb for the ambiguity resolution capabilities of the affine constrained GNSS attitude model. Content Type Journal Article Category Original Article Pages 1-17 DOI 10.1007/s00190-011-0538-z Authors P. J. G. Teunissen, Department of Spatial Sciences, GNSS Research Centre, Curtin University of Technology, Bentley, Australia Journal Journal of Geodesy Online ISSN 1432-1394 Print ISSN 0949-7714

Description:    In some geodetic and geoinformatic parametric modeling, the objectives to be minimized are often expressed in different forms, resulting in different parametric values for the estimated parameters at non-zero residuals. Sometimes, these objectives may compete in a Pareto sense, namely a small change in the parameters results in the increase of one objective and a decrease of the other, as frequently occurs in multiobjective problems. Such is the case with errors-in-all-variables (EIV) models, e.g., in the geodetic and photogrammetric coordinate transformation problems often solved using total least squares solution (TLS) as opposed to ordinary least squares solution (OLS). In this contribution, the application of Pareto optimality to solving parameter estimation for linear models with EIV is presented. The method is tested to solve two well-known geodetic problems of linear regression and linear conformal coordinate transformation. The results are compared with those from OLS, Reduced Major Axis Regression (TLS solution), and the least geometric mean deviation (GMD) approach. It is shown that the TLS and GMD solutions applied to the EIV models are just special cases of the Pareto optimal solution, since both of them belong to the Pareto-set of the problems. The Pareto balanced optimum (PBO) solution as a member of this Pareto optimal solution set has special features and is numerically equal to the GMD solution. Content Type Journal Article Category Original Article Pages 1-15 DOI 10.1007/s00190-011-0536-1 Authors B. Paláncz, Department of Photogrammetry and Geoinformatics, Budapest University of Technology and Economy, 1521 Budapest, Hungary J. L. Awange, Western Australian Centre for Geodesy and The Institute for Geoscience Research, Curtin University, Perth, Australia Journal Journal of Geodesy Online ISSN 1432-1394 Print ISSN 0949-7714

Description:    The issue of combining high-resolution gravity models, based on observations taken on the Earth surface, with those derived from satellite-only observations is of increasing importance, due to the new data provided by gravity satellite missions, CHAMP, GRACE and GOCE. The paper addresses this issue with a twofold purpose. On the one hand, it is an attempt to discuss and assess general concepts, well known in literature, such as achievable resolution, regularization in the least-squares sense or in an infinite dimensional setup, combination criteria, symmetry and block diagonal structures. In particular, as for the symmetry question, a well-defined result, generalizing known facts, is derived. On the other hand, the outcomes of the general discussion are specifically applied to the combination of a high-resolution model (e.g. EGM08) with a GOCE gravity model estimated by the so-called space-wise approach. Small numerical examples are developed to clarify the property of the proposed solution. Content Type Journal Article Category Original Article Pages 1-16 DOI 10.1007/s00190-011-0526-3 Authors Mirko Reguzzoni, Politecnico di Milano, DIIAR, P.za Leonardo da Vinci 32, 20133 Milano, Italy Fernando Sansò, Politecnico di Milano, Polo Territoriale di Como, DIIAR, Via Valleggio 11, 22100 Como, Italy Journal Journal of Geodesy Online ISSN 1432-1394 Print ISSN 0949-7714

Description:    Ambiguity resolution dedicated to a single global positioning system (GPS) station can improve the accuracy of precise point positioning. In this process, the estimation accuracy of the narrow-lane fractional-cycle biases (FCBs), which destroy the integer nature of undifferenced ambiguities, is crucial to the ambiguity-fixed positioning accuracy. In this study, we hence propose the improved narrow-lane FCBs derived from an ambiguity-fixed GPS network solution, rather than the original (i.e. previously proposed) FCBs derived from an ambiguity-float network solution. The improved FCBs outperform the original FCBs by ensuring that the resulting ambiguity-fixed daily positions coincide in nature with the state-of-the-art positions generated by the International GNSS Service (IGS). To verify this improvement, 1 year of GPS measurements from about 350 globally distributed stations were processed. We find that the original FCBs differ more from the improved FCBs when fewer stations are involved in the FCB estimation, especially when the number of stations is less than 20. Moreover, when comparing the ambiguity-fixed daily positions with the IGS weekly positions for 248 stations through a Helmert transformation, for the East component, we find that on 359 days of the year the daily RMS of the transformed residuals based on the improved FCBs is smaller by up to 0.8 mm than those based on the original FCBs, and the mean RMS over the year falls evidently from 2.6 to 2.2 mm. Meanwhile, when using the improved rather than the original FCBs, the RMS of the transformed residuals for the East component of 239 stations (i.e. 96.4% of all 248 stations) is clearly reduced by up to 1.6 mm, especially for stations located within a sparse GPS network. Therefore, we suggest that narrow-lane FCBs should be determined with ambiguity-fixed, rather than ambiguity-float, GPS network solutions. Content Type Journal Article Category Original Article Pages 1-11 DOI 10.1007/s00190-011-0537-0 Authors Jianghui Geng, Nottingham Geospatial Institute, University of Nottingham, Nottingham, NG7 2TU UK Chuang Shi, GNSS Center, Wuhan University, Wuhan, 430079 China Maorong Ge, GeoForschungsZentrum Helmholtz Center, 14473 Potsdam, Germany Alan H. Dodson, Nottingham Geospatial Institute, University of Nottingham, Nottingham, NG7 2TU UK Yidong Lou, GNSS Center, Wuhan University, Wuhan, 430079 China Qile Zhao, GNSS Center, Wuhan University, Wuhan, 430079 China Jingnan Liu, GNSS Center, Wuhan University, Wuhan, 430079 China Journal Journal of Geodesy Online ISSN 1432-1394 Print ISSN 0949-7714

Description:    The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30 km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength). Content Type Journal Article Category Original Article Pages 1-22 DOI 10.1007/s00190-011-0533-4 Authors G. Balmino, CNES, Groupe de Recherches de Geodesie Spatiale, Geosciences Environnement Toulouse, Observatoire Midi-Pyrenees, 14, Avenue Edouard Belin, 31400 Toulouse, France N. Vales, CNES, Groupe de Recherches de Geodesie Spatiale, Geosciences Environnement Toulouse, Observatoire Midi-Pyrenees, 14, Avenue Edouard Belin, 31400 Toulouse, France S. Bonvalot, IRD, Bureau Gravimétrique International, Geosciences Environnement Toulouse, Observatoire Midi-Pyrenees, 14, Avenue Edouard Belin, 31400 Toulouse, France A. Briais, CNRS, Bureau Gravimétrique International, Geosciences Environnement Toulouse, Observatoire Midi-Pyrenees, 14, Avenue Edouard Belin, 31400 Toulouse, France Journal Journal of Geodesy Online ISSN 1432-1394 Print ISSN 0949-7714