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INVESTIGATION THE ACCURACY OF THE TRANSPETRO
GEODETIC NETWORK AND ITS TRANSFORMATION TO
SIRGAS2000
João Francisco G. Monico
1, Paulo de O. Camargo
1, Maurício Galo
1,
Leonardo C. de Oliveira
2, João P. Magna Junior
3, Lis M. L. Rabaco
4,
Fábio O. Fagundes
5, João R. P. Obregon
6Resumo
O propósito deste trabalho é avaliar a acurácia de um conjunto de estações, denominada de rede geodésica da Transpetro, que é referenciada ao SAD69 (Datum Sul Americano de 1969) e fornecer alternativas para a transformação de coordenadas de SAD69 para SIRGAS2000 (Sistema de Referência Geocêntrico para as Américas), juntamente com a estimativa da acurácia das coordenadas após a aplicação da transformação. A comparação foi realizada considerando a solução oficial fornecida pelo IBGE, ou seja, a partir do software ProGriD, que é baseado no NTv2, e o TPS – Thin-Plate Spline, que é um modelo que pode ser usado para interpolar dados irregularmente distribuídos. A transformação com base no TPS foi implementada no espaço tridimensional e os resultados foram avaliados em termos de discrepâncias usando o mesmo conjunto de dados utilizados no ProGriD. Embora estes resultados sejam preliminares, foi possível observar que a transformação com base no TPS proporcionou melhores resultados, quando comparados com a solução do ProGriD, uma vez que a primeira solução forneceu discrepâncias com menores RMSE e valores médios. Estes indicadores permitem concluir que o TPS é uma alternativa viável para transformar as coordenadas obtidas a partir de diferentes realizações dos sistemas de referência geodésicos.
Abstract
This investigation aims to assessing the accuracy of the so called Transpetro Geodetic Network which is referenced to SAD69 (South American Datum of 1969) and providing methodologies for coordinate transformation from SAD69 to SIRGAS2000 (Geocentric Reference System for the Americas in Portuguese) together with the estimation of the accuracy of the transformed coordinates. The comparison was performed considering the official solution provided by IBGE, i.e. ProGriD software, that is based in the NTv2 solution and the TPS – Thin-Plate Spline that is a model that can be used to interpolate irregularly distributed data. The transformation based on TPS was implemented in 3D space and the results were evaluated in terms of discrepancies using the same data set. Although these results are preliminary it was possible to observe that the transformation based on TPS provides better results when compared to the ProGriD solution since the former solution provided lower discrepancies in RMSE and average values, allowing to conclude that TPS is an potential alternative for transforming coordinates between geodetic reference frames.
1. Introduction
The evolution of geodetic positioning techniques for the coordinate’s determination suffered great development, mainly after that the NAVSTAR-GPS (NAVSTAR - Global Positioning System) was declared operational in 1994. To keep in pace with the evolution, several countries updated their geodetic reference system or turned to adopting new ones, usually geocentric, more directly compatible with the modern satellite based positioning techniques. Brazil adopted since the resolution R. PR. nº 1/2005 (IBGE, 2005), the SIRGAS2000 (Geocentric Reference System for the Americas) as the present official planimetric referential.
The process of referential changing is complex and studies about the impact of the adoption of the new reference system to the users are relevant. One of the most important factors is related to the procedure of referential conversion. Such procedure consists in estimating the coordinates from one point or from a set of points, in the referential of destiny by using the coordinates from the referential of origin and applying a mathematical model. In Brazil, during the year 2008 was made available by IBGE (Instituto Brasileiro de Geografia e Estatística) one system of transformation based in the ProGriD software (IBGE, 2009). This software performs the referential conversion by applying transformation grids in the NTv2 (National Transformation Version 2) format. The NTv2 was developed and used in Canada and is applied also in other countries as Australia and United States of America.
Other methods of referential conversion have been studied, as the ones presented by Oliveira et al. (2008) and Magna Júnior (2012). By the study of new methods, it is intended to adjust the peculiarities of each Geodetic Network involved in the conversion process, in terms of pattern and magnitude of the distortions, quantity and homogeneity of the points, etc.
Petrobras Transporte S.A. – Transpetro, an integrated subsidiary of Petrobras, has a set of geodetic stations, here stated as a Geodetic Network, composed of nine sub networks that guide all the geodetic surveys conducted by the company nationwide. All these networks were deployed after the year 2000, and the GNSS (Global Navigation Satellite System) technology was largely employed. To perform the conversion and update the existing geodetic network to SIRGAS2000 a process of coordinates’ transformation is needed. Aiming to a better quality transformation, some experiments were performed with stations from the Transpetro Network, applying the official Brazilian model for referential conversion (i.e., ProGriD) and also the Thin-Plate Splines 3D solution (TPS3D) developed by Magna Júnior (2012) and Magna Júnior et al (2014). In these experiments homologous points from the Transpetro Network were used, with coordinates in two reference systems: SAD69 (South American Datum of 1969), the previous planimetric referential, and in SIRGAS2000.
2. The Transpetro Geodetic Network
The Transpetro Geodetic Network is materialized by a set of 1740 geodetic stations and is divided into nine sub networks: Maceio–AL; Southeast pipelines (RJ, SP, MG); West pipelines (SP, MG, DF); Southern pipelines (PR, SC, RS) pipelines; Northeast terminals (BA, AL, SE, PE, RN, CE, MA); Cacimbas-Catu pipelines; Maranhão-Fortaleza pipelines; and Amazon Basin (Urucu-Manaus) pipelines. The geodetic stations from the network are used mainly for pipeline projects and according to Petrobras (2005) are classified in the following types: a) stations type “B” in number of 60 – to be used, in principle, in the implantation stage of the definitive project areas or to be located in the main terminals and edge points of right-of-way, pigging launch and gas treating stations; and b) stations type “C” – which should be pre-molded and as one option to the “B” type, of lighter weight, for sites of difficult access or in the middle of right-of-way pipelines. Figure 1 shows the distribution of a sample of stations linked to the Transpetro Geodetic Network project, extending across the country. This sample involves 831 stations of an estimated total of more than 1700 stations. As stated before, most these sub networks were deployed after the year 2000, and the GNSS technology has been largely employed. Therefore, at least in principle, the official transformation used in Brazil should be suitable to meet the Transpetro requirement in terms of accuracy. We have to call attention that such networks are a set of stations and not a network in the real meaning, because they are not connected among them.
Figure 1. Representation of sample of stations of the Transpetro Geodetic Network.
3. Coordinate transformation between Geodetic Reference Systems
The coexistence of several geodetic reference systems and their realizations requires mathematical models for conversion of coordinates among them. The conversion process consists in finding the position of known stations in a reference system (or realization) in another system (or realization) and vice versa. Several models for reference system conversion are presented in the literature as, for example: Junkins and Erickson (1996), Collier et al (1996), Oliveira (1998) and Collier (2002).
3.1 Coordinate transformation in the Brazilian Geodetic Network
The Brazilian Geodetic System has been using several methodologies for transformation of coordinates between referentials. In IBGE (1983) the simplified differential Molodenski equations were suggested. With the adoption of the SIRGAS2000, the IBGE (2005) published the parameters for transformation from SAD69 to SIRGAS2000.
In December 2008, IBGE released the ProGriD software (IBGE, 2009) that uses grids in NTv2 format for conversion of coordinates. The transformation grids are computed by using the set of homologous stations from Brazilian Geodetic System (SGB). The distortion model used by NTv2 is composed by one global model, which uses a complex polynomial for modelling the distortions in latitude and longitude, and a local model, that deals individually with the modelling in each component, latitude and longitude, in function of the global residual distortions (JUNKINS, ERICKSON, 1996). The modelled distortions are available in a NTv2 pattern, compatible to many GIS (Geographical Information System) softwares.
3.2 Coordinate Transformation using Thin-Plate Splines
making some modifications to the radial base function U and to the equation system. The function U(r) depends on the space of the real numbers to be used. According to Bookstein (1989), in the modelling of coordinates in the 2D space (R2), the function U is given by the Equation 1; in the tridimensional space (R3), U is given by the Equation 2.
2 2r
log
r
)
r
(
U
)
Y
,
X
(
Z
(1)r
)
r
(
U
)
Z
,
Y
,
X
(
Z
(2)where r is the Euclidian distance from the origin of the Cartesian system located at each control point.
From the radial base function it is possible to define the TPS mapping functions in X, Y and Z, by the following equations (MAGNA JÚNIOR, 2012; MAGNA JÚNIOR et al, 2014):
n i uiU ri Z a Y a X a a X2 0 1 1 2 1 3 1 1 ( )
n i viU ri Z a Y a X a a Y2 4 5 1 6 1 7 1 1 ( ) (3)
n i wiU ri Z a Y a X a a Z2 8 9 1 10 1 11 1 1 ( ) where: X2,Y2,Z2: Cartesian geodetic coordinates at the destination referential; X1,Y1,Z1: Cartesian geodetic coordinates at the referential of origin;
a0,a1,...,a11,u1,...,un,v1,...,vn,w1,...,wn: coefficients of the TPS function in X, Y e Z;
U(ri): the application of the Equation 2 having the Euclidian distance at the tridimensional space (ri) from a control point to the others.
To determine the TPS coefficients it is necessary the resolution of an equation system composed by the Equations 3 and the following constraints:
n i1ui 0
n i1uiXi 0
n i1viXi 0
n i1wiXi 0
n i1vi 0
n i1uiYi 0
n i1viYi 0
n i1wiYi 0
n i1wi 0
n i1uiZi 0
n i1viZi 0
n i1wiZi 0 (4)The set of Equations 3 warrants the linear growing of the TPS as the points are progressively far from the control points and the constrainsts (Equations 4) assures that the linear system to be solved have the same number of equations and unknowns. The system to be solved individually for each coordinate (X, Y and Z) is composed by n+12 equations and n+12 unknowns, where n is the number of control points. This formulation is called TPS3D and some details in the solution can be found in Magna Júnior (2012).
Once estimated the unknowns, the coordinates at the referential of destiny are computed using the Equations 3.
4. Preliminary results of the referential transformation on the Transpetro Geodetic Network
In order to conduct the experiments, a sample of geodetic stations of types C and B from the Transpetro Network had its coordinates determined with high accuracy using GNSS, following a very well established specification. Each station was independently occupied twice for a period of four hours. Also, some original files of GNSS observations were recovered from one previous campaign and the data processing was carried out. The methods for data processing involved were PPP (Precise Point Positioning) and relative positioning (MONICO, 2008). When both results agreed quite well (at centimeter level) the station was selected for use. Stations from the Amazon Basin were not considered and, from the 8 sub networks, a total of 294 stations were made available. The coordinates are given in SIRGAS2000 with high accuracy (cm).Using a set of 294 homologous stations known in both referentials, SAD69 and in SIRGAS2000, some experiments were performed to verify the network quality and the most adequate model to the coordinate transformation in the Transpetro network.
4.1 Coordinate transformation using the ProGriD
To verify the transformation quality applied to the Transpetro Network, the first step was to perform the transformation of the 294 stations given in SAD69 to SIRGAS2000 using the ProGriD. The transformation was performed using the option “SAD69 Doppler or GPS Technique”, option that performs the transformation applying only parameters of translations. The transformed coordinates were compared to the coordinates obtained by positioning and these discrepancies were used to compute the RMSE (Root Mean Square Error), the mean values and the extreme absolute errors. Table 1 shows a summary of the obtained results.
Table 1. Statistics of the coordinate’s discrepancies after SIRGAS2000 to SAD69 Transformation.
Statistics Discrepancies ProGriD
Latitude (m) Longitude (m)
RMSE 1,948 1,438
Abs maximum 9,215 7,944
Abs minimum 0,003 0,002
Mean 1,375 0,491
From Table 1 one can observe that the discrepancies can reach more than 9 m in Latitude and a little less in Longitude. These values are quite larger than what is expected by IBGE for those techniques based on spatial technology (maximum value of 46 cm) and classical network for the region of Transpetro network (up to 1m) (IBGE, 2006). Therefore, the results have shown that is quite important for Transpetro to apply a specific transformation for reaching their aim, once the ProGriD results did not meet the Transpetro accuracy requirements (20 cm).
4.2 Quality analysis of the transformations with the ProGriD and the TPS3D in points of check
The alternative solution investigated in this work was based on the TPS3D model, which is used for deformation interpolation in different applications, as mentioned. A so called TPS3D program was developed to apply the method in the Transpetro database. From the 294 stations available, 266 were used to determine the coefficients of the TPS3D model and 28 stations were used as reference or check points (Figure 2). From Figure 2 it is possible to see how disperse is the distribution of the stations in the network. However, this is a characteristic of networks built to support pipeline projects.
The Table 2 shows the statistic of the results based on the ProGriD and TPS3D solutions, allowing the comparison of the discrepancies based on the computed and the adjusted coordinates in the 28 check points
Table 2. Statistics of the coordinate’s discrepancies after SIRGAS2000 to SAD69 Transformation based on ProGriD and TPS3D for the 28 stations of checking
Statistics Discrepancies ProGriD Discrepancies TPS3D
Latitude (m) Longitude (m) Latitude (m) Longitude (m)
RMSE 2,018 1,380 0,067 0,147
Abs maximum 3,303 1,687 0,344 0,440
Abs minimum 0,003 0,003 0,000 0,000
Mean 1,516 0,521 0,016 0,020
From Table 2 it is quite clear show how better results are rendered when using TPS3D for transformations in the Transpetro database from SAD69 coordinates to SIRGAS2000. As can be observed, the Transpetro requirements for 20 cm are reached for this sample of stations. It may not be the case for the complete data set, but it is a quite enthusiastic preliminary result.
5. Final considerations
The preliminary experiments showed that the transformation using the TPS model has some advantages compared to the ProGriD solution when applied to the Transpetro Geodetic Network of points. The ProGriD solution transformation provides discrepancies of up to 10 m in the planimetric resultant, with RMSE of 2,4 m. In the quality test of the transformations based on checks points, the TPS3D model gives a reduction of 85 % in the value of discrepancies when compared with the ProGriD for the planimetric resultant. The values of RMSE reduced from 2,4 m to 0,16 m for the planimetric resultant.
It is important to point out that further investigation is still needed. In the next steps a boundary analysis will be carried out to answer questions related to the extrapolation of the transformation. Due the complexity of Transpetro Geodetic Network, other types of transformation must also be tested trying a comparison between local transformation and TPS3D.
6. Acknowledgments
The authors thank the Technological Transportation Program (PROTRAN) from Petrobras for supporting this research and Eng. Leopoldo Machado Paganelli for the suggestions presented in this article.
7. References
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IBGE – INSTITUTO BRASILEIRO DE GEOGRAFIA E ESTATÍSTICA ProGriD – guia do usuário. 2009. ftp://geoftp.ibge.gov.br/documentos/geodesia/projeto_mudanca_referencial_geodesico/progrid_guia_do_usuario.pdf IBGE – INSTITUTO BRASILEIRO DE GEOGRAFIA E ESTATÍSTICA. Ajustamento da Rede Planimétrica
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JUNKINS, D.; ERICKSON, C. Version 2 of the National Transformation between NAD27 and NAD83 and its importance for GPS positioning in Canada. Draft Report, Geodetic Survey Division, Geomatics Canada, 1996. MAGNA JÚNIOR, J. P. O uso de Thin-Plate Splines na transformação de coordenadas com modelagem de distorções
entre realizações de referenciais geodésicos. 2012. 117 p. Tese de Doutorado. Programa de Pós-Graduação em Ciências Cartográficas. FCT/UNESP. Presidente Prudente, SP.
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OLIVEIRA, L. C. Realizações do Sistema Geodésico Brasileiro associadas ao SAD69 – Um proposta metodológica de transformação. 1998. 197 p. Tese (Doutorado em Engenharia) – Escola Politécnica, Universidade de São Paulo, São Paulo.
OLIVEIRA, L. C.; SANTOS, M. C.; NIEVINSKI, F. G.; LEANDRO, R. F.; COSTA, S. M. A.; SANTOS, M. F.; MAGNA JÚNIOR, J. P.; GALO, M.; CAMARGO, P. O.; MONICO, J. F. G.; SILVA, C. U.; MAIA, T. B. Searching for the Optimal Relationships Between SIRGAS2000, South American Datum of 1969 and Córrego Alegre in Brazil. In: SIDERIS, M. G. Observing our Changing Earth. Itália: Springer Berlin Heidelberg, 2008. v. 133, 71-79.
