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The regional gravimetric geoid solved using boundary-value problems of the potential theory is usually determined in two computational steps: (1) downward continuing ground gravity data onto the geoid using inverse Poisson's integral equation in a mass-free space and (2) evaluating geoidal heights by applying Stokes integral to downward continued gravity. In this contribution, the two integration steps are combined in one step and the so-called one-step integration method in spherical approximation is implemented to compute the regional gravimetric geoid model. Advantages of using the one-step integration method instead of the two integration steps include less computational cost, more stable numerical computation and better utilization of input ground gravity data (reduced in each integration step to avoid edge effects). A discrete form of the one-step integral equation is used to convert mean values of ground gravity anomalies into mean values of geoidal heights. To evaluate mean values of the integral kernel in the vicinity of the computation point, a fast and numerically accurate analytical formula is proposed using planar approximation. The proposed formula is tested to determine the regional gravimetric geoid of the Auvergne test area, France. Results show a good agreement of the estimated geoid with geoidal heights estimated at GNSS-levelling reference points, with the standard deviation for the difference of 3.3 cm. Considering the uncertainty of geoidal heights derived at the GNSS/levelling reference points, one can conclude the geoid models computed by the one-step and two-step integration methods have negligible differences. Thus, the one-step method can be recommended for regional geoid modelling with its methodological and numerical advantages. [ABSTRACT FROM AUTHOR]

This paper represents a milestone in the UNB effort to formulate an accurate and self-consistent theory for regional geoid determination. To get the geoid to a sub-centimetre accuracy, we had to formulate the theory in a spherical rather than linear approximation, advance the modelling of the effect of topographic mass density, formulate the solid spherical Bouguer anomaly, develop the probabilistic downward continuation approach, incorporate improved satellite determined global gravitational models and introduce a whole host of smaller improvements. Having adopted Auvergne, an area in France as our testing ground, where the mean standard deviation of observed gravity values is 0.5 mGal, according to the Institute Geographique Nationale (Duquenne in Proceedings of the 1st international symposium of the international gravity field service "gravity field of the earth", International gravity field service meeting, Istanbul, Turkey, 2006), we obtained the standard deviation of the gravity anomalies continued downward to the geoid, as estimated by minimizing the L2 norm of their residuals, to be in average 3 times larger than those on the surface with large spikes underneath the highest topographic points. The standard deviations of resulting geoidal heights range from a few millimetres to just over 6 cm for the highest topographic points in the Alpine region (just short of 2000 m). The mean standard deviations of the geoidal heights for the whole region are only 0.6 cm, which should be considered quite reasonable even if one acknowledges that the area of Auvergne is mostly flat. As one should expect, the main contributing factors to these uncertainties are the Poisson probabilistic downward continuation process, with the maximum standard deviation just short of 6 cm (the average value of 2.5 mm) and the topographic density uncertainties, with the maximum value of 5.6 cm (the average value of 3.0 mm). The comparison of our geoidal heights with the testing geoidal heights, obtained for a set of 75 control points (regularly spaced throughout the region), shows the mean shift of 13 cm which is believed to reflect the displacement of the French vertical datum from the geoid due to sea surface topography. The mean root square error of the misfit is 3.3 cm. This misfit, when we consider the estimated accuracy of our geoid, indicates that the mean standard deviation of the "test geoid" is about 3 cm, which makes it about 5 times less accurate than the Stokes–Helmert computed geoid. [ABSTRACT FROM AUTHOR]

In many European countries, normal heights referred to the quasi-geoid as introduced by Molodenskij in the mid-20th century are preferred to the classical height system that consists of orthometric heights and the geoid as a reference surface for these heights. The rationale for this choice is supposed to be that in the classical height system, neither the geoid, nor the orthometric height can be ever known with centimetre level accuracy because one would need to know the topographical mass density to a level that can never be achieved. The aim of this paper is to question the validity of this rationale. The common way of assessing the congruency of a local geoid model and the orthometric heights is to compare the geoid heights with the difference between orthometric heights provided by leveling and geodetic heights provided by GNSS. On the other hand, testing the congruency of a quasi-geoidal model with normal height a similar procedure is used, except that instead of orthometric heights, normal heights are employed. For the area of Auvergne, France, which is now a more or less standard choice for precise geoid or quasi-geoid testing, only the normal heights are supplied by the Institute Geographic National, the provider of the data. This is clearly the consequence of the European preference for the Molodenskij system. The quality of the height system is to be judged by the congruency of the difference of the geoid/quasi-geoid heights subtracted from the geodetic heights and orthometric/normal heights. To assess the congruency of the classical height system, the Helmert approximation of orthometric heights is typically used as the transformation between normal and Helmert's heights is easily done. However, the evaluation of the differences between Helmert's and the rigorous orthometric heights is somewhat more involved as will be seen from the review in this paper. For the area of interest, the differences between normal and Helmert's heights at the control leveling points range between -9.5 and 0 cm, differences between Helmert's and the rigorous orthometric heights vary between -3.6 and 1.1 cm. The local gravimetric geoid model of Auvergne, computed by the Stokes-Helmert technique, is used here to illustrate the accuracy of the classical height system. Results show a very reasonable standard deviation (STD) of 3.2 cm of the differences between geoid values, derived from control leveling points, and gravimetric geoid heights when Helmert's heights are employed and even a smaller STD of 2.9 cm when rigorous orthometric heights are used. A corresponding comparison of a quasi-geoid model, computed by Least-Squares Modification of Stokes method, with normal heights show an STD of 3.4 cm. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show a present state-of-the-art precise gravimetric geoid determination using the UNB Stokes-Helmert's technique in a simple schematic way. A detailed description of a practical application of this technique in the Auvergne test area is also provided. In this paper, we discuss the most problematic parts of the solution: correct application of topographic and atmospheric effects including the lateral topographical density variations, downward continuation of gravity anomalies from the Earth surface to the geoid, and the optimal incorporation of the global gravity field into the final geoid model. The final model is tested on 75 GNSS/levelling points supplied with normal Molodenskij heights, which for this investigation are transformed to rigorous orthometric heights. The standard deviation of the computed geoid model is 3.3 cm without applying any artificial improvement which is the same as that of the most accurate quasigeoid. [ABSTRACT FROM AUTHOR]

Spherical Radial Basis Functions (SRBFs) can express the local gravity field model of the Earth if they are parameterized optimally on or below the Bjerhammar sphere. This parameterization is generally defined as the shape of the base functions, their number, center locations, bandwidths, and scale coefficients. The number/location and bandwidths of the base functions are the most important parameters for accurately representing the gravity field; once they are determined, the scale coefficients can then be computed accordingly. In this study, the point-mass kernel, as the simplest shape of SRBFs, is chosen to evaluate the synthesized free-air gravity anomalies over the rough area in Auvergne and GNSS/Leveling points (synthetic height anomalies) are used to validate the results. A two-step automatic approach is proposed to determine the optimum distribution of the base functions. First, the location of the base functions and their bandwidths are found using the genetic algorithm; second, the conjugate gradient least squares method is employed to estimate the scale coefficients. The proposed methodology shows promising results. On the one hand, when using the genetic algorithm, the base functions do not need to be set to a regular grid and they can move according to the roughness of topography. In this way, the models meet the desired accuracy with a low number of base functions. On the other hand, the conjugate gradient method removes the bias between derived quasigeoid heights from the model and from the GNSS/leveling points; this means there is no need for a corrector surface. The numerical test on the area of interest revealed an RMS of 0.48 mGal for the differences between predicted and observed gravity anomalies, and a corresponding 9 cm for the differences in GNSS/leveling points. [ABSTRACT FROM AUTHOR]