GRAVIMETRIC DENSIFICATION IN THE STATE OF SÃO PAULO AIMING A GEOID MODEL

ABSTRACT. Gravitydata coverage is quite complete in São Paulo state for a 5’ resolution. In addition, field works have been conducted to fill in gaps around the state. These efforts are the results of the Fundac¸˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP) Thematic Project that seeks to achieve the establishment of a geoid model, a height system, and the study of mean sea level. This paper presents a comparison between two geoidal height models; the first with information available up to 2008 (beginning of the project) and the second including all data collected. Both models have been derived using the modified Stokes integral through the FFT technique. The SHGEO package was used to derive the Helmert gravity anomalies, while the EGM2008 model was selected as a reference geopotential field restricted to degree and order 150, to obtain the long and medium wavelength components. The model was validated using the geoidal heights from 170 GPS observations on Bench Marks of the spirit leveling network. The corresponding height anomalies derived from recent geopotential models (GO CONS GCF 2 TIM R2, GOCO02S and EIGEN-6C) were also compared with the same quantities at the points of leveling, besides the official Brazilian model, MAPGEO2010. The findings showed that GEOIDSP2011 displayed better consistency with respect for the GPS/leveling than GEOIDSP2008 and the geopotential models.Keywords: geodesy, geoid, geopotential model. RESUMO. O estado de São Paulo possui uma cobertura bastante completa de dados gravimétricos para uma resolução de 5’. Além disso, trabalhos de campo estão sendo realizados com o objetivo de preencher os vazios no entorno do estado. Os recentes esforços nos levantamentos são devido ao Projeto Temático da FAPESP que tem, entre outras finalidades o estabelecimento de um modelo geoidal, além do estudo sobre sistemas de altitude e monitoramento do nível médio dos mares. Este artigo apresenta a comparação entre doismodelos de alturas geoidais; o primeiro cominformações disponíveis até 2008 (fase inicial do projeto) e o segundo incluindo os dados coletados até o momento. Ambos os modelos foram calculados utilizando a integral modificada de Stokes por meio da técnica FFT. O pacote computacional SHGEO foi empregado para determinar as anomalias de gravidade de Helmert, enquanto omodelo do geopotencial EGM2008 restrito até grau e ordem 150 foi selecionado para obter as componentes de longo e médio comprimento de onda. Osmodelos foramvalidados a partir da comparação das alturas geoidais em 170 marcos da rede de nivelamento onde observações GPS foram conduzidas. As anomalias de altura dos modelos mais recentes do geopotencial (GO CONS GCF 2 TIM R2, GOCO02S e EIGEN-6C) também foram comparadas com as mesmas grandezas nos pontos de nivelamento, além do modelo oficial brasileiro, o MAPGEO2010. Os resultados mostraram melhor consistência na comparação com os pontos GPS/nivelamento do modelo GEOIDSP2011 em relação ao GEOIDSP2008 e os modelos do geopotencial.Palavras-chave: geodésia, geoide, modelo do geopotencial.

Modeling topographical density for geoid determinationThis article is one of a series of papers published in this Special Issue on the theme GEODESY.

In geoid computation, effects of real three-dimensional topographic masses on the Earth’s gravity field must be accurately quantified and, in the Stokes–Helmert scheme, replaced with effects of those masses condensed on the geoid. The most comprehensive modern schemes for evaluation of topographical effects account for terrain effects, use a spherical model of topography, and incorporate two-dimensionally varying models of topographical mass density. In this contribution, we employ a three-dimensionally varying model of topographical density. We use Newton’s integration to determine the direct topographical effect (DTE) on gravity and primary indirect topographical effect (PITE) on gravity potential. Lastly, we apply Stokes’ integration to calculate the DTE, PITE, and secondary indirect topographical effect (SITE) on geoidal height. We focus here on validation of our results and demonstration of our software’s capabilities. We present results for the simple geometrical shape of a disc under various rotations and for the anomalous density of lake waters. Effects on geoidal height for these simulations reach centimetre level, up to 2.2 cm in magnitude. For a simulation of the effects of neglected mass anomalies of the lakes, we find results reaching 0.8 cm in magnitude. We examine the behavior of our results as calculated using various step sizes for numerical integration and by comparing numerical results with analytical results for the specific case of a disc. These results suggest that the maximum percent error of our results is about 23.5% for the DTE on gravity and 7.6% for the PITE on gravity potential.

Distortions of Canadian Geodetic Networks due to the Neglect of Deflections of the Vertical and Geoidal Heights

In North America, geodetic observations have not been, generally, rigorously reduced to the reference ellipsoid. To investigate the effects on the coordinates of the points in some networks, several adjustments have been performed. In each adjustment, the individual contributions, as well as the total effect, of the geoidal height and deflections of the vertical have been investigated. The distortions are shown to be systematic, and are of practical significance in a continental context.