the spatial age and the new paradigms in geodesy: …cartografica.ufpr.br/docs/silvio/2013/spatial...

Revista Brasileira de Cartografia (2012) N0 64/6: 845-861Sociedade Brasileira de Cartografia, Geodésia, Fotogrametria e Sensoriamento RemotoISSN: 1808-0936

S B

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THE SPATIAL AGE AND THE NEW PARADIGMS IN GEODESY:

IMPLICATIONS FOR SURVEYING AND MAPPING IN BRAZIL

A Era Espacial e os Novos Paradigmas em Geodésia: Implicações nos

Levantamentos Geodésicos e Cartografia no Brasil

Sílvio Rogério Correia de Freitas1; Regiane Dalazoana1

& Vagner Gonçalves Ferreira2

1Universidade Federal do Paraná – UFPRSetor de Ciências da Terra / Departamento de Geomática

Caixa Postal: 19011, CEP: 81531-990, Curitiba – Paraná - [email protected]; [email protected]

2Hohai UniversitySchool of Earth Sciences and Engineering

Xikang Road 1, CEP: 210098, Nanjing - Jiangsu - [email protected]

Recebido em 28 de julho, 2012/ Aceito em 11 de agosto, 2012

Received on july 28, 2012/ Accepted on august 11, 2012

ABSTRACT

The satellite platforms allowed a very quick transition from local or regional Geodetic Reference Systems (GRSs) to the

global ones. The evolution of methods also brought changes in the standards of precision and accuracy. Networks

based on local Datums which had relative precision in the order of 1:100,000 to 1:150,000, like the classical networks in

the Brazilian Geodetic System (BGS), could be easily replaced by networks constrained in geocentric Datums which

reach relative precision of 1:10,000,000 or better and placed inside a concept of a Global Hierarchy. Besides the facilities

for positioning, the modern satellite platforms of remote sensors are also applied for satellite altimetry, multi spectral

imagery and on-board satellite gravimetry. This allows relevant geodetic information linked directly in global geocentric

GRSs. Then, the modern orbital platforms changed completely the ability to capture, model and manage spatial data with

strong implications in Geodesy and Cartography. The associated new data bases for Cartography have several new

implications coming from the applied surveying methods and present paradigms of precision. Most of the implications

come from the need of relating geometric and physical aspects. This happen due the necessary association of local

measurements affected by the local gravity field to global oriented geocentric GRS. The modern Global Geopotential

Models (GGMs) have a strong expectation for uses in the referred integration. In this paper are discussed the impacts of

“spatial age” on the geodetic methods for surveying and mapping. The possible applications of GGMs for realizing

fundamental geodetic networks, surveying and mapping are also discussed.

Keywords: Local and Global Geodetic Reference Systems; Gravity Field; Geodetic Surveying; Global Geopotential

Models.

846 Revista Brasileira de Cartografia, N0 64/6, p. 845-861, 2012

de Freitas S.R.C. et al.

RESUMO

As plataformas orbitais permitiram uma rápida transição de Sistemas Geodésicos de Referência (SGRs) com caráter local

ou regional para os globais. A evolução dos métodos geodésicos trouxe mudanças com grande incremento em precisão.

Redes clássicas baseadas em orientação topocêntrica em Data locais, com precisões relativas na ordem de 1:100.000 a

1:150.000, tais como as redes clássicas do Sistema Geodésico Brasileiro (SGB), podem agora ser facilmente substituídas

por redes globais com precisões relativas na ordem de 1:10.000.000 ou melhor, dentro de uma hierarquia global em vista

do moderno conceito de Datum geocêntrico. Juntamente com o posicionamento geodésico, as modernas plataformas de

sensores orbitais, tais como satélites altímetros, imageadores multiespectrais e sensores gravimétricos, propiciam

diretamente informações em SGRs geocêntricos. Desta forma, mudaram totalmente a capacidade de aquisição, modelagem

e gerenciamento de dados espaciais, com grandes implicações em Geodésia e Cartografia. As novas bases de dados

espaciais em Cartografia sofreram grandes melhorias decorrentes dos novos produtos e dos paradigmas atuais de

precisão decorrentes dos novos métodos geodésicos. Na abordagem clássica, as implicações de ordem física nas redes

entendidas como geométricas não eram, em geral, consideradas. No entanto, os levantamentos modernos têm exigências

da integração de aspectos correlatos ao campo da gravidade visando ao melhor aproveitamento da potencialidade

oferecida pela integração de levantamentos locais com as bases de dados globais. Sem dúvidas, os modernos Modelos

Globais do Geopotencial (MGGs) têm boas perspectivas de aplicação para a referida integração. Neste trabalho são

discutidas as mudanças que a “Era Espacial” produziu nos métodos de levantamentos geodésicos e que trazem implicações

para a Cartografia. As possíveis aplicações dos MGGs para levantamentos geodésicos e Cartografia são também

discutidas no trabalho.

Palavras chaves: Sistemas Geodésicos de Referência Locais e Globais; Campo da Gravidade; Levantamentos Geodésicos;

Modelos Globais do Geopotencial.

1. INTRODUCTION

In the last four decades Geodesy continuallydeveloped in a quick form because the modernmeasurement techniques mainly those based onEarth satellites as well as computational methods.These conditions generated new ideas and allowedto put in practice global positioning methods. Inconsequence it is natural a strong transition inconcepts mainly that related to Geodetic ReferenceSystems and Frames (GRSs and GRFs).

Nowadays, fundamental geodetic networkswith relative precision in the order of 1:100,000 to1:150,000 like classical triangulation and trilaterationare considered old-fashioned. Local or regionalclassical geodetic networks must be replaced bydensifications of global geodetic frames because thefacilities given by the Global Navigation SatelliteSystem (GNSS) technologies. Inside thisconsideration, it must be considered the presentneeds of accuracy related to data acquisition evenfor local geodetic and cartographic data basis. Theseneeds happened mainly because the globalreferencing for geodetic and imagery data comingfrom orbital platforms. To take the best profit fromthese data it is necessary to keep their registercharacteristics in a global GRS, avoidingtransformation for local classical GRF usually with

worst relative precision and accuracy than theacquired data.

For keeping the characteristics from globalGRF for complementary local surveys it is necessaryto take into account physical variables linked to theEarth Gravity Field and the planetary dynamic. Thishappens, e.g., for surveying based on modern highperformance Local Positioning System (LPS).

The mentioned aspects are in the presentefforts of International Association of Geodesy(IAG) and Fédération Internationale des

Géomètres (FIG). The IAG and FIG reorganizedtheir commissions and services because the evolutionfor global approaches. They point out new needsfor global GRFs and spatial data acquisition. Then,in the present there are considered three pillars ofGeodesy (Figure 1): geometry and kinematics(geokinematics); Earth’s gravity field; and Earth’srotation and orientation. Each one of the pillars isrelevant for definition and realization of a modernGRS as well as for related data acquisition.

The concept of modern global GRS considertwo components: the three-dimensional one that canbe associated with an ellipsoidal reference surface;and the vertical associated with a global equipotentialreference surface. Their realizations take intoaccount each one of the pillars. According to FIG,the correct use of modern GRFs in surveying implies

847Revista Brasileira de Cartografia, N0 64/6, p. 845-861, 2012

The spatial age and the new paradigms in geodesy: implications for surveying...

in to consider these characteristics. In consequence,we must classify the national, regional or local GRFsas densifications of a global GRF inside a hierarchy.In this sense the same definition of the global GRFmust be kept for regional and local ones.

Considering the Brazilian Geodetic System -BGS (Sistema Geodésico Brasileiro - SGB)evolution to the context of SIRGAS (SIstema de

Referência Geocêntrico para as AméricaS), twomain problems related to the modern global GRFsin Brazil are discussed in this paper:

· What are the steps to be accomplished forthe three-dimensional and vertical GRFs of BGS?

· How to integrate local surveys like levelingand Local Positioning Systems (LPS) based into themodern global concept?

The central role of the gravity field isconsidered as part of the answer to the questions.Because their impact in Geodesy, the modern GlobalGeopotential Models (GGMs) are considered.

2. GEODETIC SURVEY AND THE

EVOLUTION OF THE BRAZILIANGEODETIC SYSTEM (BGS)

Due to different techniques and employedequipments, control geodetic networks arecommonly divided into: horizontal; vertical; andthree-dimensional. In a classical approach, horizontalnetworks (often mistakenly called planimetricnetworks, because, in fact, they are related to acurved reference surface) were established by

triangulation, trilateration and traverse surveying. Theevolution of these networks is in direction of a uniqueglobal network. In this sense each local, regional,national and continental network is considered as adensification of determined realization of the IERSTerrestrial Reference System (ITRS) being underthe responsibility of IAG by its International EarthRotation and Reference Systems Service (IERS).Nowadays, the vertical networks realizations followthe usual approach of spirit leveling around the world.However, there are several propositions related toovercome the expensive and tedious process ofspirit leveling and for establishing a World HeightSystem (WHS) or its realization World HeightFrame (WHF). This process is coordinated by theIntercomission Project 1.2 (ICP1.2) of the IAG.This is pointed as an urgent task because the needof a unified global reference for heights to take profitfrom the registered data coming from spatialplatforms.

2.1 Evolution of Brazilian Horizontal Network

Towards a Densification of the ITRF/

SIRGAS

Since the 1940’s, the establishment of thehorizontal network in Brazil was carried out mainlyby triangulation technique, with the proposal ofhaving a net of quadrilaterals at each 2°. Thefundamental stations were established on ridges ofhills to enable the visibility between them. Followinginternational standards at the time, angularmeasurements were made with theodolites of 0.3"precision, the control baselines in the range of1:500,000 and the relative ellipsoidal heights weredetermined by trigonometric leveling with precisionin the order of the meter. The densification of thenetwork was usually made by traverse surveying.

The used topocentric orientation at the Datumin classical networks were usually determinedthrough six parameters: the coordinates of an originpoint (latitude and longitude); orientation (azimuth);the geoid height N = h - H, where h is the ellipsoidalheight and H is the orthometric height, and thecomponents of deflection of the vertical (meridionalcomponent x and prime vertical component h). Inpractice, there were determined the astronomiccoordinates at the origin point (Datum), the azimuthof a basis with origin in this point, and the values forthe geoidal height and vertical deviation componentswere arbitrated or estimated. Thus, the

Fig. 1: The pillars of modern Geodesy. Source:Adapted from Plag et. al. (2009).



transformation between astronomical and geodeticcoordinates could be processed by the followingequations (GEMAEL, 1999):

ϕξ −Φ= (1)

η

(2)

η

(3)

where:

Φ, Λ, AA: are respectively the astronomic

latitude, the astronomic longitude and the astronomicazimuth;

ϕ, λ, A: are respectively the geodeticlatitude, the geodetic longitude and the geodeticazimuth.

The equation (2) and (3) yields to theexpression:

(4)

The (4) is known as Laplace’s equation withwhich is possible to convert an astronomical azimuthinto a geodetic azimuth.

In the 1970’s, fundamental stations in thehorizontal network of the BGS have been establishedwith TRANSIT system, especially in the Amazonregion. In the 1990’s, Brazilian Institute ofGeography and Statistics (Instituto Brasileiro de

Geografia e Estatística – IBGE) began theextensive use of Global Positioning System (GPS)in the determination of the coordinates forfundamental horizontal network (COSTA, 1999);and from 1994 began to be deployed GPS highprecision regional networks in several states of theBrazilian federation. Brazilian fundamental horizontalnetwork underwent several adjustments over time,in view of developments in positioning techniquesand different horizontal geodetic reference systemsthat were adopted for the BGS.

The Córrego Alegre system was officiallyadopted in the country from 1950 to 1970. A setbased on the Hayford international ellipsoid of 1924and with topocentric orientation in Córrego Alegre

station, situated in Minas Gerais State (IBGE,1996). Astro Datum Chuá is considered aprovisional reference system between Córrego

Alegre and SAD 69, where some maps of the

systematic cartography were edited. The Astro

Datum Chuá system was based on the Hayfordellipsoid with topocentric orientation in the Chuá

station, situated in Minas Gerais State. It wasestablished with the purpose to be an essay forsupporting the definition of SAD 69 (IBGE, 2001).

The SAD 69 system was adopted for theBGS in the early of 1970’s. It was based on theGRS1967 and with topocentric orientation in Chuá

station (IBGE, 1998a). It is necessary to considerthat the IAG and the International Astronomic Union(IAU) immediately after the GRS1967 adoptionconsidered it not completely adequate because theevidences for different values for the best fittedreference ellipsoid parameters (MORITZ, 1988),mainly the equatorial radius a and the angular velocityw with implications in the flattening f.

For Córrego Alegre System the ellipsoidorientation was totally arbitrary, i.e. settling null valuesto the geoidal height and to the components of thevertical deviation at the Datum. This approach wasthe possible realization in practice at that time. Inthis way, the geodetic coordinates were equal tothe astronomical ones at Datum. Due to the arbitraryorientation, there was an acceptable adaptationellipsoid-geoid only in the region of Minas Gerais

and São Paulo, but at North or South regions, faraway from the Datum, the discrepancies were quiteevident. In the SAD 69 the ellipsoid orientation waspartly arbitrary, by estimating the values of thecomponents of vertical deviation through gravimetricsurveys around the Datum and looking for a betteragreement between ellipsoidal and orthometricheights in geodetic stations at continental edges. Itwas considered the null value for the geoidal heightin the Datum. By astronomical determinations inChuá and by estimating the values of the componentsξ and η it was possible to compute the geodeticcoordinates in the Datum by means of (1) and (2).

The first adjustment of the Brazilian HorizontalNetwork performed in computational environment,aiming the establishment of the SAD 69, wasconducted by the Defense Mapping Agency(DMA) of the United States (IBGE, 1996). In thisadjustment, the Brazilian Horizontal Network wasdivided into 10 areas that were processedseparately due to computational limitations (IBGE,1996). The measurements of new surveys, madefor the densification of the network, were adjustedconsidering fixed the coordinates of the existing



stations. This procedure entered distortions in thecoordinates of the stations, once systematic errorswere propagated through the different adjustments(COSTA, 1999). In general, there are three factorsthat cause distortions in classical networks (IBGE,1996):

· Poor geometry;· Lack of an adequate geoidal model for the

reduction of geodetic observations to ellipsoid, and;· The used and misused strategies for

adjustment.Still related with the distortions in the classical

network, according to Vanièek and Steeves (1996),the relative horizontal positioning is, as anymeasurement procedure, subject to random andsystematic errors. Random errors are estimatedduring the adjustment process and these estimationsmust be made available to the user. The systematicerrors in horizontal positioning are more difficult tobe controlled. They are originated from systematiceffects on measurement systems (e.g.: scaling errorthat results from the no calibration of the distancemeasurement system) and due to mismodeling (e.g.:neglecting the geoidal heights for reduction ofobservations; not take into account the deflectionof the vertical in the propagation of coordinates; theadjustment “by parts” of the network). Both errorsources cause distortions on the horizontal networks.These distortions can easily reach tens of meters inclassical networks of continental dimensions like inthe Brazilian case. For example, an error of scalecommonly found of 10 ppm (parts per million) inclassical networks, distorts horizontal positionsdistant of 1000 km from the Datum in 10 m(VANIÈEK and STEEVES, 1996).

In 1996, the horizontal network based onSAD 69 was submitted to a readjustment with aglobal character, i.e., all the network observationswere adjusted simultaneously (horizontal directions,baselines, astronomical azimuths, etc.) and GPS longbaselines observations were used as constraintsaccording to their accuracies. This adjustmentresulted in a new realization for SAD 69 system,with new coordinates for the horizontal networkstations. According to IBGE (1996), with theobtained results it was drawn for the first time aconsistent picture about the quality of the network.The quality of the network was improved due to theglobal treatment provided by the readjustment basedon the long baselines. Since 1997, IBGE began to

disclose only the coordinates and their precisions inthe new realization of SAD 69, which gave to theuser the knowledge about the quality of the stationscoordinates. This approach was not possiblepreviously. The mean value of the standard deviationafter the adjustment was of 10 cm for the GPSstations and 50 cm for the classical network stations.According to IBGE (1996), the horizontal offsetbetween the new and the old coordinates increasesalmost proportionally with the distance from theorigin (Datum Chuá), reaching approximately 15 mat South Brazil and Northeast Brazil, and in somecases reaching differences of around 50 m on stationslocated in Amapá State, Northern Brazil.

The satellite positioning methods naturallyprovide the acquisition of three-dimensional (3D)coordinates related to a reference system with originin the geocenter and with new paradigms ofprecision. Then, the classical systems which do nothave compatible precision related to the satellitepositioning techniques and each one with differenttopocentric orientation became old-fashioned. Foreach one it is necessary a set of transformationparameters and propagation of errors by differentcriteria. It is necessary to consider that in the processof transformation between GRSs the mandatory ruleis that the worst precision is propagated forobtaining the related transformation parameters.

For South America and within a historicalcontext, it must be mentioned the following facts:

I. In the 1980’s geodynamics controlnetworks using GPS were established in somecountries such as Venezuela, Ecuador, Peruand Chile;II. In June 1993 German Geodetic ResearchInstitute (Deutsches Forschungsinstitut

Geodätisches – DGFI), was responsible fora consult to the South-American countriesabout the interest in unifying their GRSs;III. In October 1993 was created theSIRGAS project with the initial purposes of:setting a geocentric GRS for South America(coordinate axes and Datum based on ITRSand its realizations and GRS80 ellipsoidparameters); and establish and maintain areference network (ITRF densification inSouth America). These activities related to thedefinition and realization of SIRGAS arecoordinated by the “Reference System”



Working Group I and II (WG-I and WG-II)respectively;IV. In May 1995 was held the first GPScampaign with the aim of realize SIRGASnetwork. This is considered the first realizationof SIRGAS (SIRGAS95), and it is composedof 58 stations distributed in South Americaand linked to ITRF94, at epoch 1995.4;V. From 1995 to 1997 data processing wasperformed by DGFI and by National Imageryand Mapping Agency (NIMA - USA);VI. In September 1997 was created the“Vertical Datum” Working Group (WG-III);VII. In May 2000 was held the second GPScampaign. This is considered the secondrealization of SIRGAS (SIRGAS2000),composed of 184 stations distributed in SouthAmerica, Central America and NorthAmerica, linked to ITRF2000 at epoch2000.4;VIII. In January 2001 CartographicConference of the United Nationsrecommended the adoption of SIRGAS;IX. In February 2001 the title SIstema de

Referência Geocêntrico para a América

do Sul was changed to SIstema de

Referência Geocêntrico para as

AméricaS;X. In January 2005 SIRGAS was adoptedas the new GRS for geodetic andcartographic works in Brazilian territory.Each country in South America is going to

integrate its individual geodetic network to SIRGASreference network. This allows setting each nationalnetwork as a densification of SIRGAS inside onehierarchy. In this sense, the responsible agencies ofmany countries have provided tools to the users forthe conversion between the classical references andSIRGAS, like the ProGriD platform of IBGE (seeIBGE home page). Today the emphasis is beinggiven to the deployment of continuous monitoringnetworks. In this sense, the third realization ofSIRGAS (SIRGAS-CON) is composed of morethan 230 continuous monitoring stations, 47 of themare part of the global IGS network (InternationalGNSS Service). The SIRGAS-CON coordinatesare processed weekly. These coordinates and thevelocities of the stations are made available to usersby the IGS Regional Network Associate AnalysisCentre for SIRGAS (IGS-RNAAC-SIR) under

responsibility of DGFI in Munich, Germany. Theseweekly coordinates refer to the same epoch ofobservation and reference system used by the IGSfor the computation of final GNSS satellites orbits,which currently is the IGS08 which involves onlyGNSS stations of ITRF2008 network. In additionto the weekly solutions the users can access multi-year solutions referred to the current ITRF and witha particular reference epoch. The integrationbetween different realizations of SIRGAS Networkis possible through the transformation parametersbetween correspondents ITRF, associated with thecoordinates reduction to the same reference epoch.

All the referred aspects related to the presentstructure of SIRGAS2000 in BGS must beconsidered for taking profit of its consistency withmodern techniques for spatial data acquisition. It isthe case for the optimization in integrating datacoming from remote sensors, LPS, etc., for obtainedspatial referencing of several kinds of relevantinformation. These aspects will be taken up in section3.

Even considering local or regional data bases,nowadays their development are strongly based inthe possibilities for capturing, modeling/processingand manage data, mainly those registered in globalGRS, without lose of their precision and accuracyin positioning. The use of an old-fashioned GRS bytransforming data coming from modern globalacquisition system usually implies in unnecessarylosing of precision and invested financial resources.

2.2 Evolution of Brazilian Fundamental

Altimetric Network Towards a WHS

The Brazilian Fundamental Altimetric Network- BFAN (RAFB – Rede Altimétrica Fundamental

do Brasil) is part of the BGS. Its realization beganin 1945 in Santa Catarina State by spirit leveling.At that time the country had not yet an official verticalDatum. The connection between the verticalnetwork with the tide gauge of Torres, Rio Grande

do Sul, in 1946, enabled the computation ofprovisory heights to the bench marks. The VerticalDatum of Torres had a provisory character since itwas defined with only one year of sea levelobservations (1919-1920). It was replaced by theBrazilian Vertical Datum of Imbituba (BVD-I) inImbituba harbor, Santa Catarina State, southernBrazil. The BVD-I was established in 1958 withnine years of sea level observations (1949-1957)



(ALENCAR, 1990). It is appropriate to mentionthe existence of a part of BFAN located at north ofthe Amazon River, in the Amapá State which is notreferred to BVD-I. The width of Amazon River doesnot allow the connection between the two segmentsof the BFAN by conventional spirit leveling surveys.The heights in this network are linked to the BrazilianVertical Datum of Santana (BVD-S), situated in theharbor of Santana, Amapá state. The currentconfiguration of the BFAN is shown in Figure 2,along with the Tide Gauge Network for Geodesy -TGNG (RMPG - Rede Maregráfica Permanente

para Geodésia) with their new control tide gauges.Today, the BFAN has approximately 69,000

stations and about 180,000 kilometers of leveledlines. The heights were obtained by spirit leveling,without the use of real gravity information and,therefore, it can not be displayed as a network withorthometric heights. The impacts of gravityinformation lack were partially reduced by theapplication of the correction related to the normalgravity derived from a Normal Earth Gravity FieldModel (DE FREITAS and BLITZKOW, 1999).This kind of model is based on the referenceellipsoid, and assigns the same mass and the sameangular velocity of the real Earth and its referencesurface is considered equipotential. With thiscorrection it is possible to consider only the effectof non-parallelism of the theoretical level surfaces,and thus the heights in this system are called normal-orthometric (LUZ, 2008).

Non-consideration of gravity informationgenerates distortions in leveling networks due toanomalous effects of topographic masses andvariations in the crust composition. The spirit levelingis a geodetic methodology which results are pathdependent. These aspects are not predicted in thetheoretical modeling. The only way for obtainingresults in a holonomic system is its association withgravimetry (VANIÈEK, 1982).

Both BVD, as already referred, wereestablished in association with the mean sea level(MSL). However, effects linked to the shift betweeneach local MSL from an equipotential surface W

0

best fitted to the global MSL were not taken intoaccount. The relationship between each BVD andW

0 can be determined by each local gravity potential

Wi.

The potential difference ∆W = W0 – W

i is

related to the topography of the MSL related to theglobal geoid, the so called Mean ocean DynamicTopography (MDT). In each Datum region the MDT

is given by:

where γ is the normal gravity in the consideredpoint. However, the computation of the potentialdifference ∆W is strongly affected by the localdisturbing potential T not predicted in GGMs. Thedetermination of these local components is part ofthe Geodetic Boundary Value Problem – GBVP. Inthe classical approach, like that used for obtainingthe MAPGEO2010, the solution of GBVP is basedon the use of gravity anomalies in the computationsby Stokes’ integral formula. This classical approach

Fig. 2: Brazilian Fundamental Altimetric Network(BFAN) (red lines) and the present Tide GaugeNetwork for Geodesy (TGNG) (black circles).(LUZ, 2008).

Fig. 3: Sequential historic adjustment and thediscrepancies in meter (green lines) related to theGPAA of the BFAN. (LUZ, 2008).



depends on the local vertical reference frame in avicious process (HECK, 1990).

The sequential historic adjustments of theBFAN are represented in Figure 3; they have beenevaluated by IBGE in 1993 with the conclusion ofthe first Global Preliminary Altimetric Adjustment(GPAA). Significant discrepancies were identifiedin this process of comparison. The showndiscrepancies are still present in the new globalnetwork readjustment concluded in 6/20/2011 asan update of BFAN (IBGE, 2011).

In view of the considered aspects and alsothe errors intrinsic to the leveling process, it can bementioned that BFAN is not consistent with thoseof other networks in South America(DALAZOANA, 2006). The same can be inferredbetween the part of BFAN linked to Imbituba andthe part linked to Santana. These aspects will betaken up in section 3.

Nowadays, a modern vertical referencesystem is defined by a type of physical height basedon the computation of geopotential numbers that inturn requires the integration of gravity observationsto spirit leveling data (TORGE, 2001). It is possibleto avoid the path dependency in the spirit levelingby using geopotential numbers. They allowrecovering the information about leveled heights andrealizing a height system with physical characteristics.These aspects are not present in the current BFAN.

The geopotential number in a point P (CP)

represents the potential difference between the geoid(W

0) and the equipotential surface passing through

the point P (WP). This geopotential number is equal

to the geopotential variation and therefore expressesthe work per unit of mass done by gravity in thepath O-P. In practice it can be determined withsufficient accuracy based on leveling and gravimetry(DE FREITAS; BLITZKOW, 1999; GEMAEL,1999):

∑∫ ∆≅=−= obs

m

P

PP ZggdZWWC0

0 (6)

In (6), gm is the mean gravity observed in the

extremes of a leveling section and ∆Zobs is theobserved level difference. If the geopotential numberis divided by a specific value of gravity, it results in adistance. The geopotential number of a point P onthe Earth surface is also equal to the work of gravityper unit of mass on an offset H along the plumb line

between the geoid and P. This work is equals to themean value of gravity g

G/P between the geoid and P

multiplied by the distance H. This implies in thedefinition of orthometric height of P:

PG

PP

g

CH

/

= (7)

The value gG/P

can be only estimated, suchthat the orthometric height can only be determinedbased on hypothesis about the crust structurebecause the gravity value in the geoid is not known.Because these referred aspects the so calledscientific heights are defined for keeping the physicalmeaning for heights at all, which are based on the(7), where the mean gravity value is estimated byany hypothesis and the path dependency is solvedby the geopotential number C

P. Depending on the

model or convention used to obtain in a consistentway the denominator of (7), it is possible to generateseveral kinds of Physical Height Systems. The kindof height or Height System depends only on the usedgravity value which acts as a scale factor. If g in thegeoid is determined by the Bouguer reduction theheight is in the so-called Helmert Height System,which is adopted in North America. Other relevantheight systems are: Dynamic Heights System,where the denominator of (7) is equal to a meanvalue of gravity, generally given by the value ofnormal gravity to latitude 45°. Once the gravity valueis constant, this is the only height system where allpoints on a same equipotential surface will have thesame height. This fact does not occur withorthometric height or with Helmert height. However,the use of an arbitrary constant divider introduces ascale distortion in the system. In the Molodenskii’sNormal Height System the denominator isdetermined by the mean theoretical gravity betweenthe ellipsoid and the point where the spheropotentialhas the same value than the geopotential in P. Thiskind of height preserves the physical meaning ofleveling survey and does not make use of reductionhypothesis. This is particularly useful for newconcepts of altimetric surveying such as theassociation of GNSS with gravimetry (FERREIRAand DE FREITAS, 2010) because the possibilitiesfor obtaining the disturbing potential at the Earth’ssurface.

Note that each of the reported physical heightsystems have as common characteristic the



preservation of gravimetric information associatedwith the leveling operation. It must be emphasizedthat this information is contained in geopotentialnumber C

P. This aspect does not exist in the present

BFAN where the heights are in the so-called normal-orthometric system and the empirical relationship putequivalent information to a theoreticalspheropotential number.

In the context of SIRGAS Project, theproblem concerning the unification of verticalsystems began to be discussed in 1997, with thecreation of WG-III. The first recommendations ofSIRGAS WG-III (IBGE, 1998b) indicated that:

i) SIRGAS Vertical System would be definedthrough two sets of heights, a geometrical and aphysical one, and their respective velocities;

ii) SIRGAS Vertical System would be realizedthrough a network of geodetic stations with GPSsurvey, spirit leveling and gravimetry;

iii) The network cited previously would beestablished on the basis of SIRGAS95 networkstations, with the addition of stations situated in theborder of South American countries and in its majortide gauges – this network was effectively establishedby SIRGAS 2000 GPS campaign,

iv) The geodetic authorities of the involvedcountries should organize the data necessary for thecomputation of geopotential numbers, with thepurpose of obtaining physical heights.

Subsequently, GT-III recommended theadoption of normal heights as the physical componentof the system (DREWES et al., 2002).

With respect to the recommendationspresented earlier, it must be remarked that severalactivities are in course nowadays in South America.The main activities towards a modern height systemaccording the most recent needs for users are:

a) The geometric component of SIRGASVertical System (ellipsoidal heights) is well definedfrom the adoption of SIRGAS by South Americancountries, even including digital maps for velocitiesrelated to this component;

b) Luz (2008) realized an extended analysisrelated to the BFAN including several propositionsaiming its modernization and for giving to users betterfoundations for its needs;

c) The computation of geopotential numbershas led countries to revise their first-order levelingnetworks, and verify the existing gravity information.In the Brazilian case a remarkable goal was a

complete revision of survey manual registers and thebuilding of a modern digital data base for the BFAN;

d) The determination of a reference surfacefor normal heights (quasigeoid) has led to studiesbased on GGMs derived from satellite missions asCHAMP, GRACE and GOCE, associated withterrestrial gravity data;

e) The BFAN is in a transitory process. Inline with the present activities of readjustment realizedby IBGE recently, several activities are in coursefor its modernization and for its link with a WHS.Some of these activities are presented in the nextsection.

3. PRESENT TOOLS OF GEODESY AND

THEIR IMPLICATION ON GEODETICSURVEY AND MAPPING

Geodesy is the science of measuring andmapping the Earth’s geometry, orientation and theexternal gravity field in a three-dimensional time-varying space. In the scope of Geodesy, it is worthyof notice the gravity-dedicated satellite missionsbased on LEOs (Low Earth Orbit satellites) suchas: CHAMP (CHAllenging Mini-satellite Payload),GRACE (Gravity Recovery And ClimateExperiment) and GOCE (Gravity field and steady-state Ocean Circulation Explorer). The main goalof these missions is to provide comprehensive globalinformation of the Earth’s gravity field associated tothe mass distribution and its change in time. Theglobal models and its derived functionals (e.g., geoidmodel, gravity anomalies and disturbances,components of deflection of the vertical - meridionalcomponent ξ and prime vertical component η, etc.)have an unprecedented spatial resolution in thehistory of Geodesy.

3.1 A New Generation of Global Geopotential

Models

The gravity-dedicated satellite missionsprovide accurate data that can be converted intogeopotential harmonic coefficients. Thesecoefficients can be obtained from several sourcesof gravity information and are the bases of GlobalGeopotential Model (GGM, as plural GGMs).There are essentially three classes of GGMs:

· Satellite-only GGMs: are derivedsolely from satellite tracking data (analysis ofsatellites’ orbits and gravity missions).

· Combined GGMs: are derived froma combination of a satellite tracking data, gravity



data, satellite altimeter-derived gravity data in marineareas, and gravity information extracted from DigitalElevation Models (DEMs).

· Tailored GGMs: are derived byrefining of existing (satellite-only or combined)GGMs using higher resolution regional gravity datawhich may have not been used previously.

Since the publication of the Earth GravityModel 1996 (EGM96), cf. Lemoine et al. (1997),considerable improvements on observationtechniques resulted in better quality GGMs. Thesehigh quality new GGMs, cf. Arabelos and Tscherning(2010), are consequence of the improvements inthe measuring techniques and systems which allowedobtaining new and more accurate data related tothe gravity ûeld even in areas with difficulties forconventional surveying techniques. Such data couldbe also obtained by improvements and reanalysisof old satellite altimetry data in association with newdata as well as from the gravity-dedicated satellitemissions.

Recently, improved models for the high degreecombined solutions were made possible by theintroduction of ellipsoidal harmonics. In addition,Jekeli (1988) proposed the conversion equationsto transform ellipsoidal harmonics coefficients tospherical harmonic coefficients. Pavlis et al. (2006)have compiled a very high-resolution globaltopographic database and applied it in thecomputation of all terrain-related quantities. This kindof information is necessary for the pre-processingof gravity data and for the development andsubsequent use of the Earth Gravitational Model2008 (EGM2008) released by the NationalGeospatial-Intelligence Agency (NGA), cf. Pavliset al., (2008). Such quantities include ResidualTerrain Model (RTM) effects, analytical continuationterms (G

1), topographic/isostatic gravitational

models, and the necessary models to convert heightanomalies into geoid height.

The EGM2008 is considered the mostconsistent GGM produced until now. The EGM2008is complete to spherical harmonic degree and order2,159, and contains additional coefficients extendingto degree 2,190 and order 2,159. EGM2008includes improved 5’×5’ gravity anomalies and hasbeen enhanced from the latest GRACE basedsatellite solutions. EGM2008 also includes improvedaltimetry-derived gravity anomalies estimated byusing PGM2007B and its implicit MDT model as

reference. For the Collocation prediction of the final5’×5’ mean gravity anomalies, PGM2007B is usedas reference model. It was also employed aformulation that predicts area-mean gravityanomalies which are effectively band limited.

3.2 Three-dimensional Geodetic Networks

and LPS/GNSS integration

In the past, how mentioned in the previouslysections, the three-dimensional (3D) geodeticnetworks were established as combinations ofhorizontal (2D) and vertical (1D) networks. Inaccordance with Hofmann-Wellenhof and Moritz(2006), the idea of a computation of a triangulationnetwork in space dates back to H. Bruns in 1878.Hotine (1969), based in Bruns’ ideas, presented theconcept of a classical geodetic network in a rigorousthree-dimensional way. Therefore, with the adventof GNSS technologies, especially the GPS, globalscale networks have gained more and moreimportance. It is the case of the IERS TerrestrialReference Frames realized in determined epoch yyyy(ITRFyyyy) which is the realization of ITRS whereseveral geodetic techniques are used. TheITRF2008 is the most recent realization of the ITRS.The IGS08 is a part of ITRF2008 composed onlyby GNSS stations.

Thus, today GPS is the best way to determine

global rectangular coordinates { }X ,Y ,Z . However,

in many practical surveys it is necessary to transforma global coordinate system •F into a local coordinatesystem ∗E , i.e., combine LPS and GPS. In doingso, it is necessary to know the Earth’s gravity fieldfor the association of GPS observations and TotalStation observations because the vertical axes ofthe instrument coincides with the direction of the local

gravity vector { }Γ = Γ . Deflection of the vertical

components and height anomaly { }, ,ξ η ζ at a point

P on the Earth’s surface provide the relationshipbetween a curvilinear astronomical coordinatesystem {Φ, Λ, H}

Γ and a curvilinear ellipsoidal

coordinate system { }, ,hϕ λ at the same point P .

Grafarend and Awange (2005) determineddeflection of the vertical by GPS and theodolitecombinations thereof (Total Station), the referredLPS. The Procrustes solution emerges as analternative to the solution of nonlinear equations.According to Awange (1999a), the Partial



Procrustes Problem, in adjustment, is formulatedas a problem of least squares. Given the coordinatesof a Global Reference System (e.g. coordinatesobtained by GPS) and coordinates in a LocalReference System (e.g. coordinates obtained byLPS) the Partial Procrustes Problem method maygive a solution for the deflection of the vertical, ascan be seen in (AWANGE, 1999a, 1999b).Considering •F the Cartesian coordinates of thedatum point in a global coordinate system and *ECartesian coordinates of that point on the localsystem, we have:

2 2 2

3 3 3

4 4 4

*

P P P

P P P

P P P

n P n P n P

x x y y z z

x x y y z z

x x y y z z

x x y y z z

− − − − − − = − − − − − −

A

M M M

E

(8)

and

2 2 2

3 3 3

4 4 4

P P P

P P P

P P P

n P n P n P

X X Y Y Z Z

X X Y Y Z Z

X X Y Y Z Z

X X Y Y Z Z •

− − −

− − − = − − − − − −

B

M M M

F

(9)

The partial Procrustes search exactly such asolution that:

= ⋅B A R (10)

A textbook treatment, which

A

refers to thelocal system and

B

to the global system to solve

R

is described in Schönemann (1996):

T= ⋅S A B

(11)T T

S=⋅ ⋅ ⋅S S V D V (12)

T T

S=⋅ ⋅ ⋅S S W D W (13)

V and

W

are obtained, for example, bySchur decomposition. The rotation matrix is

obtained by:

T= ⋅R W V

(14)This is the rotation matrix that relates A and

B

.Other solution for Procrustes problem is given

in Awange (1999a), with the peculiarity

T= ⋅A B R

instead of (10):T T⋅ = ⋅ ⋅A B U Σ V (15)

T= ⋅T V U (16)

The expression on the right-hand in equation(15) can be solved by decomposition in singularvalues. The matrix R is obtained from:

T=R T

(17)Finally, it is calculated {Λ

P, Φ

P, Σ}

Γ through

the extraction of elements of the matrix R :

=

31

32

r

rP atanΛ

(18)

+=

232

231

33

rr

rP atanΦ (19)

−=

13

23

r

ratanΣ (20)

With the (18) and (19) it is possible to establisha Local Astronomical System for combine LPSwith a GPS instead of a “Local Topographic System”(LTS). However, another system is necessary, in thiscase Local Geodetic System (LGS). This methodis physically and geometrically more consistent withthe systems related to LPS and GPS.

Another possibility with a good approximationfor local surveys with short extension is given byusing the moderns GGMs. From these models it ispossible to derive the local/regional values of thecomponents of deflection of the vertical (meridionalcomponent ξ and prime vertical component η).These values can be used for obtaining theastronomical coordinates Φ and Λ from geodeticcoordinates as well as local bases improvedorientation by using the GPS positioning and theequations (1) to (4). The relationship between theLPS local astronomical system and the GPS globalGRS is shown in Figure 4.

The coordinates of a generic point Pi {x

i, y

i,

zi} obtained in a local Cartesian astronomical system

with origin in a point P0 can be converted to a

Cartesian geocentric GRS, with some approximationby using the following equation:

Φ−°Λ−°+

=

i

i

i

i

i

i

z

y

x

SRR

Z

Y

X

Z

Y

X

223

0

0

0

)90()180(

(21)

Where {X0, Y

0, Z

0} are the geodetic

coordinates (e.g. that obtained with GPS) of theorigin of the local astronomical system. R

2 and R

3

are the rotation matrix around the y-axis and z-axis,



respectively, and S2 is the reflection matrix related

the y-axis. It must be remarked that thetransformation is not rigorous because theastronomical coordinates are obtained consideringconstant values for the deflection of the verticalcomponents which can vary slowly depending onthe dimension of survey area and disregarding effectsof the Earth’s curvature.3.3 Fundamental Altimetric Networks, Geoid/Quasigeoid and a World Height System

As mentioned in early section, the currentextent of BFAN has about four times the distancearound the world and it was dissociated of gravityobservations. In this context the heights associatedwith the BFAN are the normal-orthometric heightsbecause it was considered the non-parallelism ofthe normal equipotential surfaces. However, theadjustment of the leveling networks must be realizedin a holonomic (path independent) height system. Inthis context, the natural holonomic vertical coordinatesystem is that one based on the geopotentialnumbers. Taking into account these above mentioneddifficulties we can ask ourselves:

· Is it necessary re-leveling throughoutthe country by using spirit leveling associated withgravimetric observations?

· Is it possible to calculate definitely ageoid or quasigeoid model that achieves the precisionrequired for such work?

In regarding to the first question we canmention that the spirit leveling is a very time-consuming, tedious and an expensive operation.Taking into account the second question, the geoid(quasigeoid) is still a topic of fundamental importance

for emerging nations such as Brazil. Many regionsaround the world require improved gravimetric databases to support a very accurate geoid (quasigeoid)modeling. This is essential for the modernization ofheight systems by employing the emerging GNSStechnologies, in particular, the GPS. Geoid andquasigeoid computation have been of interest tomany Geodesists such as Moritz (2011), Ardalan etal., (2010), Zhang et al., (2009), Flury and Rummel(2009), among others.

Most of the South American Datums arebased on a determination of MSL at different tidegauges over a varying range of time intervals and atdifferent epochs. However, each vertical datum isreferred to a particular equipotential surface,associated with the MSL at tide gauge and reducedfor a specific epoch. In general, these surfaces arenot coincident with the global geoid. The oceansurface does not coincide with a level surface (e.g.the geoid) of Earth’s gravity field because geostrophiceffects which produce the Mean ocean DynamicTopography – MDT already mentioned in section2.2. Figure 5 displays the MDT around the BVD-I.

According Andersen and Knudsen (2009) aMDT model gives a global mean sea level biasrelated to a global geoid. However, Ferreira et al.(2010) found that the MDT at the BVD-I is -

Fig. 5: Mean Dynamic Topography at a regionaround of the Imbituba Brazilian Vertical Datum.

Fig. 4: The local Cartesian astronomical {x, y, z}system and its spatial relationship with the Cartesian{X, Y, Z} geocentric GRS.



0.31±0.01 m. We must emphasize that the usedmethodology for determining the offset of the BVD-I related to a World Height System (WHS) (or itsrealization World Height Frame, WHF) did not usethe computation of local geoids approach and, inconsequence, not contains error propagation relatedto those models (FREITAS et al., 2010).

De Freitas et al. (2011) showed that the MDTreflected in the Paraná state portion of the BFAN,or the Brazilian zero-reference is about -0.35±0.01m under a global zero reference. They used GPSpositioning integrated with disturbing potentialinformation refined by Residual Terrain Model(RTM), in the context of the Fixed GeodeticBoundary Value Problem (FGBV). The usedapproach was asked to be introduced in theSIRGAS project, WGIII, for linking verticalnetworks in South America.

Another consistent value for the shift of thezero height of BFAN related with a WHS wasobtained by Melo (2011). In his study they wereused several GGMs available in the InternationalCenter for Global Earth Models (ICGEM) of IAGand the MAPGEO2010 produced by IBGE. Theconsidered parameter for analysis was the geoidheight in reference stations obtained from the modelsminus the difference h - H obtained from the BGS,where h is the ellipsoidal height determined withGPS on bench marks and H is the normal-orthometric height from BFAN. It is possible to seein Table 1 that the mean discrepancy obtained in 21stations from all analyzed GGMs is 0.34 m abovethe BGS. The MAPGEO2010 has a discrepancyin opposite sense. It must be remarked that in thestudy area the MAPGEO2010 presents the largestdiscrepancies in Brazil.

Table 1: Discrepancies of geoid heightobtained for several GGMs and MAPGEO2010and the value obtained from GPS/leveling in the BGS

Recently, IAG by its Inter-CommissionProject 1.2: Vertical Reference Frames (Idhe, 2009)proposed a program of activities related to the studyof regional vertical systems and their relations to aWHS/WHF. The practical applications are amongthe main objectives. This goal is proposed to beachieved until 2012. This proposal is in accordingof SIRGAS WGIII statements for linking SouthAmerican Vertical Networks. There is a tendencyamong South American countries in to monitor andintegrate the vertical networks according to SIRGAS

project statements for taking profit of modern spacegeodetic observations in a continental basis. A unifiedvertical datum in South American is important formonitoring common problems related toenvironment, hydrodynamics and energy oftransporting mass, engineering, natural resources,land management, and so on.

Montecino et al. (2011) proposed threeapproaches for connecting the two portions ofBFAN linked respectively to the Datums of Imbitubaand Santana. These approaches are based indetermining the shift between the two Datums withbasis in a regional quasigeoid model. Then, the heightanomaly z in each Datum is determined accordingthe following models:

a) Satellite only model derived from theGOCE mission givem by theGO_CONS_GCF_2_TIM_R2 GGM (ICGEM,2011):

240( , ) ( , )N

GOCEζ ϕ λ ζ ϕ λ= 240( , ) ( , )N

GOCEζ ϕ λ ζ ϕ λ= (22)

b) Same model used in the precedentapproach but complemented by RTM:

240 max( , ) ( , ) ( , )N N

GOCE RTMζ ϕ λ ζ ϕ λ ζ ϕ λ>= +

240 max( , ) ( , ) ( , )N N

GOCE RTMζ ϕ λ ζ ϕ λ ζ ϕ λ>= + (23)

c) In addition to the precedents approachesby introducing the mean wave lengths in thegeopotential harmonic spectrum information givenby the EGM2008:

240 240 2190 max08( , ) ( , ) ( , ) ( , )N N N

GOCE EGM RTMζ ϕ λ ζ ϕ λ ζ ϕ λ ζ ϕ λ− >= + +

240 240 2190 max08( , ) ( , ) ( , ) ( , )N N N

GOCE EGM RTMζ ϕ λ ζ ϕ λ ζ ϕ λ ζ ϕ λ− >= + + (24)

Geopotential numbers, cf. Eq. (6), are still veryimportant in the calculation of physical heights forpractical use such as dynamic, Helmert/orthometric,and normal heights. Hofmann-Wellenhof and Moritz(2006, p. 318), Idhe (2009), and others havesuggested to apply the disturbing potential on theEarth’s surface associated with the normal potentialat the same point to calculate the geopotentialnumbers. Technical details on how to developgeopotential numbers from such a model must beworked out, so as to develop other types of heights(Roman et al, 2010). In this modern concept, tocarry out time-consuming leveling such as spirit andtrigonometric is not required any more. Nonetheless,from Molodenskii’s theory, which provides the



disturbing potential T at point P on the Earth’ssurface, we have:

( , , ) ( , , ) ( , , )W h U h T hϕ λ ϕ λ ϕ λ= + (25)

where ( , , )W hϕ λ is the geopotential requiredby:

0( , , ) ( , , )C h W W hϕ λ ϕ λ= − (26)

The geopotential numbers ( , , )C hϕ λ arecomputed in a direct way from gravity data. SeeEq. (6) for comparison with (23). It is more generalthan the geometric determination of the normal heightfrom ellipsoidal height associated with the quasigeoidmodel.

All countries in South America can benefitfrom the methodology discussed here, mainlyParaguay and Bolivia. Due to geographical limitation(landlocked countries) their vertical networks haveno direct connection with the MSL. The velocitycomponent related to temporal variation of the geoidis one aspect to be considered in the BFAN, aswell as in all leveling network in South America. TheBFAN should be planned as kinematic heightnetwork. This goal could be achieved by usingcombination of the SIRGAS-CON permanentstation network, the repeated leveling surveys,repeated gravity measurements, tide gaugemeasurements along the Brazilian coast lines as wellas the gravimetric quasigeoid model and GRACEdata.

3.4 Tide System and its Influence in Geodetic

Observations

Nowadays, it is necessary to consider thepermanent earth tide phenomena in several problemsrelated to Geodesy because the present standardsof precision. It is the case, e.g., for long baselinesdetermination with GNSS, leveling associated withgeopotential numbers, local and regional verticaldatums connections. Considering in special thefundamental altimetric networks connectionproblem, it is important that all measurements arereduced to the same tide system. Then, GPSpositioning, gravity measurements and local heightsystems must be reduced to the same permanenttide system (POUTANEN et al., 1996). There arethree approaches for dealing with the permanent tidaleffects on the geopotential and the Earth’s shape(MÄKINEN and IHDE, 2008):

· In the mean-tidal system: the meanpermanent effect of tides is not removed from theshape of the Earth.

· In the tide-free or non-tide system: thepermanent deformation is eliminated from the shapeof the Earth as well as the direct and indirect effectsof the potential associated with the tides.

· In the zero-tidal system only the directeffects of the potential associated with tides areremoved. Only the indirect effect is keep.

Conforming to the IAG Resolutions No 16adopted in Hamburg in 1983, gravity relatedcomponents are given in a zero tidal system as wellas the handling of the gravity data. However, theITRFyy coordinates are given in the tide-free system,the same holds for the EGM2008. In doing so,SIRGAS coordinates systems follows the same tidesystem related to the ITRF. Contrary to this, theBFAN heights (normal-orthometric heights) aregiven in the mean-tidal system since the tidalcorrections were not applied to the levelingobservations.

In Ekman (1989) are presented equations totransform heights and gravity values among tidalsystems. The transformations between a heightdifference

ZH∆ above the zero geoid, a height

difference mH∆ above the mean geoid, and a height

difference nH∆ above the non-tidal geoid, betweena northern and southern station, are given by(EKMAN, 1989):

( )2 229.6 sin sinm z N S

H H ϕ ϕ∆ − ∆ = − (27)

( )( )2 229.6 1 sin sinz n N S

H H γ ϕ ϕ∆ − ∆ = − − (28)

( )2 229.6 sin sinm n N S

H H γ ϕ ϕ∆ − ∆ = − (29)

Where N

ϕ and S

ϕ are the geodetic latitudesof the northern and southern points respectively andγ is a constant

γ

(EKMAN, 1989). Andfor gravity values, the differences between zerogravity , mean gravity mg and non-tidal gravity

ng are given by (EKMAN, 1989):

230.4 91.2sinm z

g g ϕ− = − + (30)

( )( )21 30.4 91.2sinz m

g g δ ϕ− = − − + (31)

( )230.4 91.2sinm n

g g δ ϕ− = − + (32)



Where δ is an arbitrary constant

1.53≈

(EKMAN, 1989).

4. SUMMARY

The Spatial Age brought new possibilities forGeodesy and Cartography. The modern orbitalplatforms changed completely the ability to capture,model and manage spatial data. Associated to thesefacts, new paradigms of precision must beconsidered. Then the classical geodetic networksbased in terrestrial methods became old-fashioned.Now the global GRS must be adopted in all stagesof the georeferencing.

The transition from local geodetic systems isa real need today for taking profit from the moderntechnologies. In the Brazilian case it is possible toobtain 3D positioning with accuracy on thecentimeter level all over the country based in theSIRGAS-CON network. However, the surveyneeds the adequate integration of LPS and GPSreference systems. In this case aspects linked to thegravity field must be considered.

The BFAN still remain with its classicalconception. The heights are in a non holonomicsystem. Its link with a WHS it is possible only byintroduction of gravity information along with levelinglines.

The modern GGMs allowed severalpossibilities for facing the absence of gravityinformation in practical applications, butsophisticated effects like permanent tides must benow considered.

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