Determination of a geoid model for Ghana using the Stokes-Helmert method. Michael Adjei Klu. BSc Geodetic Engineering

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1 Deeminaion of a geoid model fo Ghana using he Sokes-Helme mehod by Michael Adjei Klu BSc Geodeic Engineeing A Thesis Submied in Paial Fulfillmen of he Requiemens fo he Degee of Mase of Science in Engineeing in he Gaduae Academic Uni of Geodesy and Geomaics Engineeing Supeviso: Pee Dae PhD. Geodesy and Geomaics Eng. Examining Boad: Macelo Sanos PhD Geodesy and Geomaics Eng. Juak Janak PhD Geodesy and Geomaics Eng This hesis is acceped by he Dean of Gaduae Sudies THE UNIVERSITY OF NEW BRUNSWICK Augus 2015 Michael Adjei Klu 2015

2 ABSTRACT One of he geaes achievemens of humankind wih egad o posiioning is Global Navigaion Saellie Sysem GNSS. Use of GNSS fo suveying has made i possible o obain accuacies of he ode of 1 ppm o less in elaive posiioning mode depending on he sofwae used fo pocessing he daa. Howeve he elevaion obained fom GNSS measuemen is elaive o an ellipsoid fo example WGS84 and his endes he heighs fom GNSS vey lile pacical value o hose equiing ohomeic heighs. Convesion of geodeic heigh fom GNSS measuemens o ohomeic heigh which is moe useful will equie a geoid model. As a esul he aim of geodesis in he developed counies is o compue a geoid model o cenimee accuacy. Fo developing counies which include Ghana hei siuaion will no even allow a geoid model o decimee accuacy. In spie of he spase eesial gaviy daa of vaiable densiy disibuion and qualiy his hesis se ou o model he geoid as accuaely as achievable. Compuing an accuae geoid model is vey impoan o Ghana given he wide spead of Global Posiioning Sysem GPS in he fields of suveying and mapping navigaion and Geogaphic Infomaion Sysem GIS. The gavimeic geoid model fo Ghana developed in his hesis was compued using he Soke-Helme appoach which was developed a he Univesiy of New Bunswick UNB [Ellmann and Vaníček 2007]. This mehod uilizes a wo space appoach in solving he associaed bounday value poblems including he eal and Helme s spaces. The UNB appoach combines obseved eesial gaviy daa wih long-wavelengh gaviy infomaion fom an Eah Gaviy Model EGM. All he eesial gaviy daa used in his compuaion was obained fom he Geological Suvey Depamen of Ghana due o difficulies in obaining daa fom BGI and GETECH. Since ii

3 some pas of Ghana lack eesial gaviy daa coveage EGM was used o pad hose aeas lacking in eesial gaviy daa. Fo he compuaion of opogaphic effecs on he geoid he Shule Radio Topogaphy Mission SRTM a Digial Elevaion Model DTM geneaed by NASA and he Naional Geospaial Inelligence Agency NGA was used. Since he eain in Ghana is elaively fla he opogaphic effec ofen a majo poblem in geoid compuaion is unlikely o be significan. This fis gavimeic geoid model fo Ghana was compued on a 1' 1' gid ove he compuaion aea bounded by laiudes 4ºN and 12ºN and longiudes 4ºW and 2ºE. GPS/ igonomeic levelling heighs wee used o validae he esuls of he compuaion. Keywods: Gavimeic geoid Sokes s fomula Eah Gaviy Model Topogaphic effec Digial Teain Model Bounday value poblem GPS/igonomeic levelling. iii

4 DEDICATION To my lae mohe fahe bohe sise and all who have suppoed and susained me duing my eseach and wiing a he Univesiy of New Bunswick. iv

5 ACKNOWLEDGEMENTS The fea achieved by his eseach could no have been possible wihou he help and suppo of he following: I am highly indebed o Pof Vaníček fo his unflinching suppo houghou my sudies a he Univesiy of New Bunswick. His invaluable guidance advice paience ciique and ques o go he exa mile o mainained eseach sandads ae gealy appeciaed. I feel pivileged and honoed o wok unde his guidance. My special hanks go o D. Sanos fo his invaluable inellecual guidance pofessional discussion and advice duing my sudies a Univesiy of New Bunswick. My pofound gaiude goes o D. Dae my supeviso fo his consucive feedback on his eseach and wie-up unflinching suppo pofessional discussions flexibiliy and availabiliy houghou he sudy peiod. Am gaeful o D. Kingdon fo his wondeful assisance and availabiliy fo discussions on many eseach issues which canno be lised hee. Many hanks go o M. Michael Sheng fo his wondeful assisance wih egad o giving me a fim gounding on he use of he SHGeo sofwae. I am vey hankful o M. Faoughi who povided me wih all he Malab pogams fo he daa pocessing I am gaeful o Tius Tienaa fo his advice and encouagemen houghou my sudies. His suppo and kindness ae examples of a dea fiend. I acknowledge he vaious oganizaions and hei heads who have funded my sudies especially D. Kaikai of he Land Adminisaion Pojec D. Odame of he Lands Commission D. Ben Quaye of Lands Commission M. Abeka of Land Adminisaion Pojec and M. Odameey of Suvey and Mapping Division of he Lands Commission. v

6 I am vey hankful o M. Ben Okang and M. Opoku of he Geodeic Secion of he Suvey and Mapping Division of he Lands Commission fo he moal suppo. Las bu no he leas am gaeful o all my family fiends back in Ghana and Canada who suppoed me and made my say in Canada a memoable one. Above all I will like o hank Jehovah he God I seve and woship fo binging me his fa and helping me houghou my sudies. May you name Jehovah be paised foeve. All I can say is hank you vey much. vi

7 Table of Conens ABSTRACT... ii DEDICATION... iv ACKNOWLEDGEMENTS...v Table of Conens... vii Lis of Tables...x Lis of Figues... xi Lis of Symbols Nomenclaue o Abbeviaions... xii Chape 1. Inoducion Descipion of chapes Backgound The geoid Uses of a geoid model Reseach objecive and conibuion... 9 Chape 2. Lieaue eview Inoducion The goal fo cenimee geoid Techniques fo geoid compuaion Remove-compue-esoe using Sokes-Helme mehod of geoid compuaion Leas Squaes Modificaion Mehod LSMS also called KTH mehod Chape 3. Theoeical backgound of Sokes-Helme s appoach o geoid deeminaion Inoducion Fomulaion of geodeic bounday-value poblem in eal space Gaviy anomaly Tansfomaion of gaviy anomalies fom he eal space o Helme space Effec of opogaphical masses on gaviaional aacion Diec Topogaphic Effec DTE Diec Amospheic Effec DAE Seconday Indiec opogaphical Effec SITE Downwad coninuaion of gaviy anomalies vii

8 3.10 Refeence Field Soluion of Sokes s bounday value poblem Tansfomaion fom Helme s space o eal space Pimay Indiec Topogaphic Effec PITE Pimay Indiec Amospheic Effec PIAE Chape 4. Daa acquisiion Inoducion Teesial gaviy daa in Ghana Compuing he gaviy anomalies and esidual gaviy anomalies Saisical es fo oulies Tansfomaion fom eal space o Helme s space Gidding of he esidual gaviy anomalies Digial Elevaion Model DEM Compuaion of Diec Topogaphic Effec DTE Diec Amospheic Effec DAE Seconday Indiec Topogaphic Effec SITE Helme anomalies on he opogaphy Downwad coninuaion Refeence Field Ellipsoidal coecions Compuaion of he esidual co-geoidal heighs Refeence Spheoid Tansfomaion fom Helme s space o eal space Pimay Indiec Topogaphic Effec PITE Pimay Indiec Amospheic Effec PIAE Geoid-Ellipsoid sepaaion GNSS/igonomeic-levelling daa Chape 5. Conclusions and ecommendaion Conclusion Limiaions of his eseach Recommendaion viii

9 Chape 6. Refeences Cuiculum Viae ix

10 Lis of Tables Table 1 Ageemen of geoid model heighs and GNSS/Tigonomeic-levelling Table 2 Ageemen of geoid model heighs using ±10 mgal cieion and GNSS/Tigonomeiclevelling x

11 Lis of Figues Figue 1-1 The suface of he Eah geoid and ellipsoid Souce: [Bukhad Figue 4-1 Gaviy obsevaions in Ghana Souce: [Davis Figue 4-2 Disibuion of obseved eesial gaviy coveage Souce of map: Google Figue 4-3 Hisogam of esidual gaviy anomaly daa Figue 4-4 Fee-ai anomalies mgal Figue 4-5 Diec Topogaphic Effec DTE in mgal Figue 4-6 Diec Amospheic Effec DTE in mgal Figue 4-7 Seconday Indiec Topogaphic Effec SITE in mgal Figue 4-8 Helme anomalies in mgal Figue 4-9 Diffeences beween Helme anomalies on opogaphy and downwad coninuaion in mgal Figue 4-10 Refeence field in mgal Figue 4-11 Ellipsoidal coecion fo gaviy in mgal Figue 4-12 Ellipsoidal coecion fo sphee in mgal Figue 4-13 Helme co-geoidal heighs in mees Figue 4-14 Refeence spheoid in mees Figue 4-15 Pimay Indiec Topogaphic Effec PITE in mees Figue 4-16 Seconday Indiec Amospheic Effec SIAE in mees Figue 4-17 Geoid-ellipsoid sepaaion in mees Figue 4-18 Conou plo of he geoid-ellipsoid sepaaion in mees Figue 4-19 Locaion of igonomeic levelling saions Souce of map: Google Figue 4-20 Enlagemen of he locaion of he igonomeic saions Souce of map: Google.. 81 xi

12 Lis of Symbols Nomenclaue o Abbeviaions AC BVP BGI DAE DDM DTE EGM FBM GIS GNSS GPS GSM LSMS NASA NGA ORSTOM PIAE PITE SRTM Addiive Coecions Bounday Value Poblem Bueau Gavimeique Inenaional Diec Amospheic Effec Digial Densiy Model Diec Topogaphic Effec Eah Gaviy Models Fundamenal Bench Mak Geogaphic Infomaion Sysem Global Navigaion Saellie Sysem Global Posiioning Sysem Geological Suvey Museum Leas Squaes Modificaion of Sokes Naional Aeonauics and Space Adminisaion Naional Geospaial Inelligence Agency Office de la Recheche Scienific e Technique Oue-Me Pimay Indiec Amospheic Effec Pimay Indiec Topogaphic Effec Shule Rada Topogaphic Mission xii

13 Wha we call in he geomeic sense he suface of he Eah is nohing else bu ha suface which inesecs he diecion of gaviy a igh angles and fom which he suface of he wold ocean is pa. C.F. Gauss 1828 Chape 1. Inoducion 1.1 Descipion of chapes Befoe dealing wih he main subjec of his hesis he compuaion of he gavimeic geoid model of Ghana and he conclusion which can be dawn fom i Chape 1 gives a bief inoducion o geodesy wha he geoid is and he impoance of a geoid model. This chape coninues wih he eseach objecives and backgound o geoid compuaion in Ghana. Chape 2 povides a lieaue eview of he vaious mehods of geoid compuaion and gives a bief descipion of hee majo mehods of compuing a gavimeic geoid. All hese hee mehods use a combinaion of eesial gaviy daa and Eah Gaviy Models EGM. This chape places emphasis on he vaious seps one has o go hough in ode o compue a geoid model using any such echnique. The hee mehods ae: he Remove-Compue-Resoe -c- appoach wih he Helme condensaion mehod fo handling he opogaphy [Sansò and Rummel 1997] he second mehod of compuing he geoid is he Leas Squaes Modificaion of Sokes LSMS wih Addiive Coecions AC [Ågen e al. 2009] he hid is he Sokes-Helme appoach o geoid compuaion he mehod of compuing he geoid as devised a Univesiy of New Bunswick. 1

14 This is followed by Chape 3 which is dedicaed o he UNB Sokes-Helme s mehod seleced fo he compuaion of he gavimeic geoid model of Ghana. This Chape sas wih a wo-space se-up used fo fomulaing he bounday value poblem and defining gaviy quaniies which would be appopiae fo downwad coninuaion fom he Eah s suface o he geoid level. This is followed by he descipion of he efeence gaviy field and he spheoid and he efomulaion of he Sokes bounday value poblem fo he highe-degee efeence spheoid. This includes he vaious seps necessay in he UNB s applicaion of he Sokes-Helme mehod in solving he geodeic bounday value poblem of geodesy. Chape 4 is dedicaed o he descipion of gaviy daa used in he compuaion of he geoid. This Chape sas wih he backgound of gaviy daa acquisiion in Ghana. This includes a bief hisoy of insumen used in he daa acquisiion pecauionay measues aken duing he daa acquisiion pocess educion of he aw gaviy daa compuaion using leas squaes adjusmen and he sandad deviaion of he juncion poins. I is explained ha in ode o efe he compued gaviy values o he Posdam sysem a link was esablished beween he gaviy suvey newok in Ghana wih ha of he Unied Kingdom. Compuaions such as Fee-ai gaviy anomalies equied fo he compuaion of he geoid hen follows. This compuaion pocess also includes he descipion of sofwae used in gidding he gaviy daa he Eah Gaviy Models EGMs and SRTM gidded daa a diffeen densiies needed in he compuaion by he SHGeo sofwae. Chape 5 follows wih a discussion and assessmen of he esuls of he compued geoid model. This assessmen also includes a ecommendaion abou he need o impove his compued geoid model in he nea fuue by including he aibone gaviy daa in any 2

15 new compuaion of a geoid model fo Ghana. I also conains he essenial poblems encouneed when assessing he accuacy of his geoid model as hee is a lack of GPS/levelling daa ha would coves he enie couny. This Chape concludes wih ecommendaion fo fuhe wok which have o done o impove he geoid model. 1.2 Backgound Geodesy as defined by Fiedich Robe Helme 1880 is he science ha deals wih he measuemen and epesenaion of he eah s suface. Toge [2001] exends his definiion o include he deeminaion of he gaviy field of he eah in a hee dimensional ime vaying space. This exension of he definiion of geodesy by Toge follows up on a definiion of geodesy inoduced by Vaníček and Kakiwsky [1986]. Howeve geodesy has been aound fo cenuies. Humankind has been concened abou he eah on which hey live and cay ou viually all hei aciviies. Duing vey ealy imes mos of hese human aciviies wee limied o his o he immediae viciniy. The developmen of means of anspoaion enabled human o avel o disan lands and as a esuls humans became ineesed in he size and shape whole wold [Bukhad 1985]. Accoding o Home B.C. c he Eah was a convex dish suounded by an infinie ocean which he called Oceanus [Bullen 1975]. Thales c.625-c.547 B.C of Mileus povides he fis documen abou Home s ideas egading he shape of he eah. To Thales he eah is a disc-like body floaing on an infinie ocean [Vaníček and Kakiwsky 1986]. To Anaximande B.C a conempoay of Thales he eah was cylindical wih he axis oiened in an eas-wes diecion [Vaníček and Kakiwsky 3

16 1986]. Anaximande was he fis o advocae he noion of a celesial sphee - an idea ha pemeaed asonomical hinking in his ea [Bullen 1975]. Anaximenes a pupil of Anaximande believed songly ha he eah was ecangula and he eah is floaing on an infinie cicumfeenial ocean held in space by compessed ai [Vaníček and Kakiwsky 1986]. Pyhagoas a mahemaician believed ha he eah shape is spheical. Accoding o Bukhad [1985] Pyhagoas easoned ha he gods would ceae he mos pefec figue and his pefec figue o him was a sphee. Aisole suppoed his idea of a spheical eah by Pyhagoas abou a hunded yeas lae. Afe acceping he heoy of a spheical eah came he effos o deemine he lengh of is cicumfeence. Eaoshenes lef an accoun of a mehod of esimaing he cicumfeence of he eah. Plao Achimedes and Posidonius also conibued in deemining he cicumfeence of he eah. I is now acceped ha he eah is flaened a he poles and bulges aound he equao. The geomeical figue used in geodesy o nealy appoximae he shape of he eah is an ellipsoid of evoluion [Vaníček and Kakiwsky 1986]. Because of he above definiion geodesy is consideed o have wo pas: The fis pa which concens measuemen and epesenaion aspec is ofen efeed o as geomeical geodesy. Geomeical geodesy deals wih he deeminaion of he size and shape of he eah ineconinenal ies among land masses of he eah and he deeminaion of posiions lenghs of lines and azimuhs [Ewing and Michell 1970]. The second pa called physical geodesy is he sudy of he shape of he eah and is gaviy field. In sudying he gaviy field of he eah use is made of gaviy anomalies and ohe daa. Physical geodesy is pimaily concened wih he use of gaviy measuemens and dynamic saellie geodesy o deemine he shape of he geoid and 4

17 deflecion of he veical [Coss 1985]. The aims of physical geodesy accoding o [Bomfod1975] include: 1. Sudies of he vaiaions in he inensiy and diecion of gaviy a he eah s suface and he deeminaion of he iegulaiies in he fom of he geoid and he exenal equipoenial sufaces. 2. The vaiaions in he inensiy and diecion of gaviy wih ime which conibues diecly o conol of geodeic famewok hough Sokes s and elaed inegal. 3. Some consideaion of he vaiaion of he densiy in he cus which cause he iegulaiies found. This sudy is augmened by ohe geophysical daa. 4. Measuemens of hoizonal and veical movemens a he Eah s suface including ides. 5. The use of aificial saellies o sudy he vaiaion in he inensiy and diecion of gaviy a he Eah s suface and he iegulaiies in he geoid. In sudying he gaviy field of he eah use is made of gaviy poenial ahe han gaviy which is a veco quaniy. The use of poenial of gaviy is o enable easie handling of he poenial mahemaically because he diffeence beween he acual and nomal poenial called he disubing poenial is quie small [Vaníček and Kakiwsky 1986]. Thus he gaviy poenial of he eah can be sepaaed ino wo pas: one due o nomal poenial and he ohe aising fom he mass disibuion wihin he Eah [Bomfod 1975]. Poenials of opogaphic masses fom he heoeical basis fo educing gaviy measued on he suface of he eah o he geoid. The gadien of he poenials gives gaviy and his gaviy is needed fo he 5

18 compuaion of he geoid. Addiionally he gadien of he gaviy gives he diffeenial equaion whose soluion is equied in fee space in ode o compue he geoid. 1.3 The geoid Even hough he sudy of gaviy is useful in deemining he shape of he eah bu no he size he deeminaions of shape and he size of he eah ae no independen objecs of sudy. Geodesiss ae ineesed in gaviy because he oucome of geodeic measuemens ses of coodinaes he lengh and azimuh of a line ae caied ou on he Eah s physical suface in he domain of acion of eesial gaviy. While i is necessay o make obsevaions and measuemen on o nea he physical suface of he eah i is impossible o pefom deail mahemaical compuaion on he Eah s physical suface. This is because he suface of he eah is exemely uneven and no definable mahemaically [Coss 1985]. A possible suface fo compuaion is mean sea level o he geoid. The geoid is defined as he equipoenial suface of he Eah s aacion and oaion which on he aveage coincides wih Mean Sea Level of he Eah in he absence of exenal influences such as wind and ocean cuen [Vaníček and Kakiwsky 1986]. Accoding o Gauss he geoid is he mahemaical figue of he eah [Bukhad 1985]. Geodesy places a significan emphasis on he geoid because of is ole as he efeence suface fo heigh sysems. Ohomeic heigh is defined a disance beween he geoid and he poin of inees measued along he plumb line [Vaníček and Kakiwsky 1986]. 6

19 Mos counies in he wold use ohomeic heigh fo hei naional heigh sysem. The geoid is physically meaningful because i is ied o he Eah s gaviy field hough he plumblines and also epesens he level a which seawaes would sabilize if hey wee homogenously a es [Kingdon 2012]. A level suface is eveywhee hoizonal i.e. pependicula o he diecion of he plumb line. Level sufaces ae sufaces of consan poenial and he geoid is one of hem [Moiz 1990]. The geoid suface is much smoohe han he naual eah suface. Howeve he shape of he geoid is iegula and unsuiable as mahemaical suface fo pefoming geomeic compuaion. A suiable suface fo compuaion which appoximaes he geoid in shape is he ellipsoid of evoluion geneally efeed o as biaxial ellipsoid [Vaníček and Kakiwsky 1986]. Figue 1 shows he hee basic geodeic sufaces which seves as possible definiions of he figue of he eah. The biaxial ellipsoid which is a mahemaical suface has is dimensions and shape specified in such a way ha is depaue fom he geoid is small. This depaue of he geoid fom he ellipsoid is called geoid undulaion o geoid ellipsoid sepaaion. Such an ellipsoid is consideed a nomal fom o figue of he eah and is coesponding poenial and gaviy ae called nomal poenial and nomal gaviy. Deemining he geoid equies exensive gaviy measuemens and compuaions [Inebayeva 2010]. The poblem of compuing he geoid would be much simplified if hee wee no masses ouside he geoid. Absence of opogaphic masses masses ouside he geoid would make he poenial of he eah ouside he geoid a hamonic funcion. Accoding o Sokes he poblem of deemining he shape of he geoid can be solved if he disibuion of gaviy ove he geoid suface wee known [Bomfod 1975]. 7

20 Figue 1-1 The suface of he Eah geoid and ellipsoid Souce: [Bukhad Uses of a geoid model An essenial poblem of physical geodesy is he deeminaion of he gaviy field of he eah fom vaious ypes of measuemens by solving a bounday value poblem. The aim in solving his bounday value poblem is o ge a geoid model accuae o one cenimee level. The wide use of GPS fo geodeic heighing is consideed o be he main eason fo a cenimee geoid model. Convesion of geodeic heighs he heigh sysem in which GPS deemined heighs ae given o ohomeic heighs equies an accuae geoid model. The impoance of deemining a geoid model can be explained as follows: 8

21 1. The use of GPS has eplaced he ime consuming adiional mehods of suveying which has ansfomed he fields of suvey and mapping navigaion and Geogaphic Infomaion Sysems GIS. Having a geoid model enables effecive use of GPS deemined heighs in some applicaions whee is possible o use GNSS fo suveying such as when is possible o eceive he signals fom he saellies and avoids he use of adiional mehods of levelling in some cases which ae expensive as compaed o he use of GPS [Abdalla and Faihead 2011]. 2. Gaviy field infomaion is necessay fo pedicing he posiions of saellies in hei obi. Since he geoid eflecs he vaious vaiaions in he gaviy field of he Eah a good undesanding of he geoid enables a bee pedicion of he saellies in he obis [Inebayeva 2010]. 3. Knowledge of he geoid is essenial fo modelling hydogaphic suveys and maine navigaion. 4. The geoid seves as he efeence suface of ohomeic heighs. 5. Vaiaions in he gaviy field ae due o changes in densiy of mae wihin he Eah. This seves as valuable infomaion fo locaing naual esouces such as oe deposis as well as oil and gas [Inebayeva 2010]. 1.5 Reseach objecive and conibuion In Ghana almos all suvey woks especially hoizonal conol posiioning ae caied ou using GPS. Ye he main mehod of deemining heighs fo benchmaks fo all 9

22 geodeic and engineeing suvey woks ae sill based on spii levelling pocedue. This appoach of poviding benchmaks is vey expensive edious and inefficien. The objecive of his eseach is o povide a geoid model fo Ghana. Since Ghana is a developing couny a educion in cos of suvey woks will have an impac on he economy. Moeove availabiliy of a geoid model fo Ghana will eplace he adiional mehod of spii levelling wih is aendan disadvanages. An advanage of spii levelling echnique is is accuacy i.e. he abiliy o esimae o millimee level in odinay spii levelling and sub-millimee level wih pecise levelling. The eseach quesion is; wha is he level of geoid model accuacy ha can be achieved by a developing couny such as Ghana wih spase gaviy daa coveage? The UNB appoach o geoid compuaion has been used o model he geoid fo counies such as Canada Ausalia and he Unied Saes. These counies ae developed and have huge gaviy daa coveage even hough no evenly disibued. This daa coveage makes he geoid model fo hese counies elaively quie accuae. A new insigh will be gained by applying he UNB mehod o a spase gaviy daa se wih vaiable densiy fom he developing wold. Fuhe hee has no been any aemp o compue he geoid model fo Ghana since 1924 when he Biish esablished he iangulaion and avese newok which sill seves as a basis fo hoizonal conol posiioning and mapping. 10

23 Accoding o ou opinion we have o deemine numeically in he fuue he deivaions of he plumbline as long as hey have visible oigin namely by a opogaphic suface of he coninenal elief by a geological deeminaion of he mass densiy of is consiuens and by a sysemaic suvey of he oceans accoding o well-esablished mehod.we shall call he peviously defined mahemaical suface of he eah of which he ocean suface is a pa geoidal suface of he Eah o he geoid. J.B. Lising 1873 Chape 2. Lieaue eview 2.1 Inoducion Ideas abou he geoid as he mahemaical suface of he eah as disinguished fom he ellipsoid had been developed and expounded by enowned mahemaicians of he eigheenh cenuy such as Gauss bon 1777 and Besselbon 1784 [Moiz 1990]. The quesion hen is how is he sepaaion beween he geoid and he efeenced ellipsoid deemined? One appoach owads a soluion is aso-geodeic mehod. Howeve such a soluion would only be possible on he coninens. A heoy by G. G. Sokes [1849] made i possible o compue he geoid-ellipsoid sepaaion houghou he wold. His deeminaion of he geoid is based on gaviy obsevaions [Vaníček and Chisou 1993].Since ha ime he mahemaical mehods obsevaion echniques and modelling mehods have made a gea pogess. The 11

24 availabiliy of saellies ajecoy daa since he 1960s has led o an inceased knowledge of he long wavelengh shape of he geoid and also made he compuaion of he geoid possible since Sokes appoach o geoid deeminaion will equie gaviy daa houghou he Eah [Amalvic and Boavida 1993]. This chape sas wih a discussion on he need fo a cenimee geoid and he difficulies in achieving such accuacy in compuing he geoid compuaion [Sansò and Rummel 1997]. Molodensky mainained ha an accuae geoid compuaion has o ake ino accoun opogaphical densiy vaiaions which would always be unknown. To wha level of accuacy hen could he geoid be modeled? Reseach a UNB has shown ha Helme s second condensaion mehod when used in combinaion wih he heoy by Sokes woks easonably well fo educing he effec of limied knowledge of he opogaphic densiy on he accuacy of geoid deeminaion [Vaníček e al. 2013]. This chape concludes wih a eview of ohe echniques of geoid compuaions such as emove-compue-esoe appoach and he Leas Squaes Modificaion mehods ogehe wih a bief commen on he limiaions of he emove-compue esoe echnique as well as quesions egading he validiy of he heoy behind he Leas Squaes Modificaion echnique. 2.2 The goal fo cenimee geoid Cuenly one of he bigges asks facing he geodeic communiy is he deeminaion of a cenimee geoid model. A deficiency in he accuacy of geoid models limis he use of Global Navigaion Saellie Sysem GNSS echnology i.e. he echnique o do leveling 12

25 using GNSS echnology and he gavimeic geoid model [Sjöbeg 2013]. Advances in GNSS echnology made i possible o deemine geodeic heigh wih an accuacy of a few cenimees depending on he obsevaion and pocessing echniques. Impovemen in echnology may lead o inceased accuacy in GNSS posiioning in he fuue. This impovemen in accuacy will also equie moe accuae geoid models. The soluion o his heigh poblem will enable he geodeic communiy o benefi fully fom GNSS echnology. Howeve his ques fo he cenimee geoid deeminaion is no easy paiculaly in mounainous egions. Knowledge of he mass densiy disibuion wihin he Eah s opogaphy is equied in ode o compue such an accuae geoid. Unfounaely he densiy disibuion wihin he Eah is no known o sufficien accuacy [Kingdon 2012]. As he deail densiy disibuion of he opogaphy is unknown Molodensky developed his famous echnique by inoducing he quasi-geoid. Howeve accoding o Vaníček e al.[2012] deemining he quasi-geoid using he Molodensky mehod has a faal poblem wih he geomey of he Eah s suface. Inegaing gaviy ove he suface of he Eah which is much oughe hen he geoid is no possible in ceain aeas and in ohe aeas will esul in unpedicable eos. They ague ha veical ock sufaces epesen locaions of disconinuiy and hee ae ohe aeas whee he suface of he Eah canno be descibed as a mahemaical funcion of hoizonal posiions. In hese locaions he Molodensky echnique fails. Hence in hei opinion he geoid which is a faily smooh and convex suface wihou any kinks edges o ohe iegulaiies is a bee suface fo inegaion [Vaníček e al. 2012]. Thee have been seveal aemps o addess his densiy disibuion wihin he opogaphy issue. 13

26 Mainec [1993] made his fis aemp o model he opo-densiy effecs of laeal hoizonal anomalies. Mainec s appoach modeled he opogaphy as discee mass columns. Huang e al.[2001] caied ou a pacical applicaion of Mainec s ideas in wesen Canada. Ohe advanced echnique of opogaphical densiy modelling includes ecangula paallelepiped pism by Nagy e al.[2000] pisms wih inclined sufaces e.g [Smih 2000] and bilinea sufaces e.g [Tsoulis e al. 2003]. All hese echniques enable compuaion of any hee-dimensional densiy disibuion of opogaphic masses o he desied accuacy using close fomula [Nagy e al. 2000]. Addiionally densiies could be assigned o each pism independen of any neighboing densiies. Since no close-fom soluion fo pism whose ops ae in shape of a cuve has been found all he above menioned echniques use plane sufaces in modelling he densiy effecs which is an appoximaion fo he ue opogaphy [Smih 2000]. Reseach a UNB owads he use of laeally-vaying Digial Densiy Model DDM of he opogaphy fo geoid modelling ceaed by digiizing geological maps and assigning appopiae ock densiies wee caied ou by Fase e al.[1998] Tenze e al.[2005] Sanos e al.[2006]. Thei mehods ae compaable o ha due o Mainec [1993]. They epoed ha he effec of laeal densiy vaiaion on he geoid compuaion is a mos a few decimees wih a sandad deviaion of less han 2 cenimees [Kingdon 2012]. Addiionally Mainec e al.[1995] invesigaed he influence of he adial changes of densiies of opogaphic masses beween he geoid and he Eah s suface veical densiy vaiaion. They epoed a vaiaion of less han 5 cm even unde vey exeme condiions and unde ealisic condiions ae no likely o exceed 2-3 cm [Kingdon 2012]. Again eseach by Vaníček e al.[2013] using a synheic gaviy field shows ha he geoid could be 14

27 modeled o a sandad deviaion of abou 25 mm and a maximum ange of abou 200 mm. I is impoan o menion ha hese eseaches did no use any coecive measues such as suface fiing o biases and ils so he esuling eos eflec only he eos in modelling he geoid. Thus he ques fo cenimee geoid is possible if opo-densiy infomaion is incopoaed in he geoid compuaion. As shown by he eseaches a UNB he opogaphic densiy issue can be esolve o a few cenimees if he densiy wihin he cus is easonably well known [Kingdon 2012]. 2.3 Techniques fo geoid compuaion Mehods of compuing he geoid depends on he daa used fo he compuaion pocess. This includes eesial gaviy daa aibone and maine gaviy daa deflecions of he veical GNSS/levelling daa and saellie daa. The daa souces can be combined in one fom o anohe o deemine he geoid. Machenko e al.[2002] combined aibone gaviy daa wih gaviy daa obained by diffeen echniques o compue he geoid. Sjöbeg and Eshagh [2009] also invesigaed compuaion of a geoid model fom aibone gaviy daa. Thei echnique combined aibone gaviy daa wih saellie posiioning daa poins. Hi e al.[2009] validaed a geoid model fo mounainous egion of he Geman Alps using asogeodeic mehod of geoid compuaion. They deemined veical deflecion a 100 saions wih a spacing of abou 230m aanged in a pofile of 23 km lengh. Repeaed obsevaion a 38 saions in diffeen nighs evealed an obsevaional accuacy of abou Compaison of he compued asogeodeic pofile 15

28 wih GPS/levelling daa yielded diffeences of 10 mm. Abd-Elmoaal and Küheibe [2014] used Aiy isosaic hypohesis o opogaphically-isosaically educe he deflecion of he veical obsevaions. The educed deflecions wee used o inepolae deflecion of he veical o fom a dense gid. The gidded educed deflecions wee used o compue asogeodeic geoid. They epo a good fi wih GPS/levelling daa. Teesial gaviy daa se have been used o compue he geoid fo seveal counies aound he wold. Remove-compue-esoe and leas squaes modificaion mehods ae wo main appoaches used when compuing he geoid using eesial gaviy daa and a eview of hese mehods ae shown in secion Remove-compue-esoe using Sokes-Helme mehod of geoid compuaion The Naional Suvey and Cadase of Denmak KMS and he Geophysics Depamen of he Neils Boh Insiue of Univesiy of Copenhagen developed he pue emovecompue esoe -c- mehod of geoid compuaion [Fosbeg 1985]. Thee ae seveal appoaches o geoid model compuaion using he -c- echnique. Each of hese echniques handles he opogaphy in a diffeen way pio o gaviy anomaly daa being used as inpu ino he Sokes fomula. The basic seps of geoid compuaion using emove-compue-esoe and Helme condensaion echnique fo handling he opogaphy ae as follows: 1. Remove he effec of opogaphy fom he gaviy daa on he suface of he eah Calculae fee-ai gaviy anomalies fom he gaviy obsevaion daa. 16

29 Conve he fee-ai gaviy anomalies o Bougue gaviy anomalies using plana appoximaion. Plana Bougue anomaly is smoohe han fee-ai anomaly and hus easie o gid. Gid he plana Bougue gaviy anomaly Conve he gidded plana Bougue anomaly back o fee-ai gaviy anomaly Apply Helme second condensaion mehod o emove he effec of he opogaphy above geoid o saisfy he bounday condiions of Sokes inegaion 2. Compue he downwad coninuaion of he gidded gaviy anomalies 3. Remove he long-wavelengh pa of he gaviy signal fom he eain educed gaviy daa The long-wavelengh pa of gaviy signal pediced fom he geopoenial model up o a chosen spheical hamonic degee and ode is emoved fom he eain educed gaviy anomalies. The esul is efeed o as esidual gaviy anomaly. 4. Compue he esidual co-geoid by applying Sokes inegal. The esidual co-geoid undulaions ae calculaed fom he esidual gaviy anomalies using a modified Sokes funcion. 17

30 5. Resoe he long-wavelengh pa of he gaviy signal subaced ealie using he same EGM o he same degee and ode by compuing he efeence spheoid. 6. Compue he pimay indiec opogaphic effec on he geoid. 7. Obain he gavimeic geoid by adding he esidual co-geoid he efeence spheoid and he pimay indiec opogaphic effec on he geoid. The scheme of emove-compue-esoe using Helme condensaion descibed above suffes fom uncaion eos because he pocess neglecs he fa-zone conibuion in he compuaion of he esidual co-geoid. A moe igoous appoach which accouns fo he fa-zone conibuion is he Sokes-Helme mehod of geoid compuaion developed a UNB. 2.5 Leas Squaes Modificaion Mehod LSMS also called KTH mehod The Royal Insiue of Technology also known as KTH in Sweden developed he KTH mehod of geoid compuaion. This KTH mehod is also called Leas Squaes Modificaion wih addiive coecions. This echnique of geoid compuaion is based on gaviy anomaly daa. Howeve he echnique does no equie gaviy educion bu ahe includes addiive coecions fo he opogaphic effec downwad coninuaion amospheic and ellipsoidal coecions fo he shape of he eah [Yildiz e al. 2012]. A eview of he compuaion seps fo geoid modelling using he KTH mehod accoding o Ågen e al. [2009] is as follows 18

31 1. Compue gaviy anomalies fom he gaviy obsevaion daa 2. Use a gidding algoihm o gid he gaviy anomalies daa 3. Compue an appoximae geoid-ellipsoid sepaaion using he eesial gaviy anomalies and he Eah Gaviy Model gaviy anomalies up o degee and ode M. In compuing he appoximae geoid sepaaion a modified Sokes kenel is used. 4. Compue he combined coecion due o he opogaphic effec using DEMs. 5. Compue he downwad coninuaion effec 6. Compue he amospheic effec 7. Compue he ellipsoidal coecion o he modified Sokes fomula 8. Compue he geoid by adding all he coecions o he appoximae geoid compued in sep 3 As indicaed ealie he above-menioned seps of compuing he geoid he Leas Squaes Mehod applies coecions o he appoximae geoid compued fom suface gaviy anomalies diecly wihou applying any of he adiional gaviy educions pio o Sokes inegaion such as downwad coninuaion of he educed gaviy anomalies. Fuhemoe he mehod does no apply indiec effecs of he opogaphy and 19

32 he amosphee [Inebayeva 2010]. Accoding o Sjöbeg [2003] he aim of his Leas Squaes Modificaion appoach o geoid compuaion is o modify find a new appoach o he adiional pocedue used in geoid compuaion in view of a cenimee geoid model. A big quesion is whehe his pocedue has a sound heoeical foundaion. As poined ou in he efeence made in secion 4.12 downwad coninuaion of gaviy anomalies is only possible if hee ae no masses wihin he ange of he coninuaion pocess. Ove he pas wo decades eseaches a Univesiy of New Bunswick have used he Sokes-Helme appoach o geoid compuaion. Thei appoach uses a wo-space se-up; eal space used fo defining gavimeic quaniies appopiae fo downwad coninuaion fom he Eah s suface o he geoid level i.e. solid gaviy anomalies and Helme s space see below fo fomulaing and solving he Sokes bounday value poblem [Ellmann and Vaníček 2007]. In addiion he opogaphic effecs ae fomulaed in hei spheical fom. Thei soluion o he Sokes bounday value poblem employs a modified bu diffeenly fom Sjobeg s modificaion Sokes s fomula in conjuncion wih he low-degee conibuion fom an Eah Gaviy Model EGM. They epoed ha hei appoach hough esing and he echnique in he mounainous egions of Canada and on he Ausalian synheic gaviy field is suiable fo deemining geoid model wih a sandad deviaion of cenimee accuacy geoid depending on he availabiliy of eesial gaviy daa coveage and qualiy of he gaviy daa [Vaníček e al. 2013]. The nex chape gives a deailed heoeical basis fo he UNB Sokes-Helme appoach o geoid compuaion. 20

33 Fo a long ime mahemaicians fel ha ill-posed poblems canno descibe eal phenomena and objecs. A.N. Tikhonov and V.Ya. Asenin 1977 Chape 3. Theoeical backgound of Sokes-Helme s appoach o geoid deeminaion 3.1 Inoducion The heoeical foundaion which foms he basis fo he compuaion of he geoid was deived by Sokes in 1849 [Vaníček and Chisou 1993]. Accoding o his heoy he geoid can be obained fom gaviy obsevaion on he geoid and hee should be no mass ouside he geoid [Tenze e al. 2003]. This condiion is difficul o aain since gaviy measuemens ae caied ou on he suface of he Eah. To saisfy his bounday condiion specified by he heoem gaviy anomalies need o be downwad coninued fom he suface of he Eah o he geoid. Hamonic quaniies ae needed fo he downwad coninuaion and hus a numbe of diffeen coecions elaed o he exisence of opogaphy and he amosphee needed o be accouned fo caefully [Ellmann and Vaníček 2007]. This educion pocess equies knowledge of opogaphical mass densiy which can be assumed fom geological maps and hey assue quie a high accuacy of he geoid compuaion [Huang e al. 2001]. Helme [1884] in his fis aemp o saisfy his condiion suggesed ha he Eah s opogaphical masses and amosphee can be

34 eplaced by a condensaion laye of an infiniesimal hickness locaed inside he geoid. This condense mass laye has aeal densiy which equals he poduc of he mean densiy of he opogaphical column and he heigh ohomeic of opogaphical column above he poin [Vaníček and Mainec 1994]. In his second condensaion model Helme placed he condensaion laye on he geoid [Heck 1993]. Mainec and Vaníček [1994a] used Newonian aacion o fomulae he effec of opogaphy on he gaviaional poenial fo laeally vaying opogaphical densiy disibuion fo he spheical appoximaion of he geoid. Vaníček e al.[1987] used an idea inoduced by Molodensky which seeks o modify he Sokes funcion by inoducing highe degee gaviy field as a efeence field. The heoy of he efeence gaviy field he efeence spheoid and he efomulaion of he Sokes s bounday value poblem fo he highe-degee efeence spheoid have been descibed by Vaníček and Sjöbeg [1991] and Vaníček and Feahesone [1998]. The bounday-values Helme s gaviy anomalies on he geoid ae compued by he use of he Poisson inegal equaion fo he downwad coninuaion. The pinciple of he Sokes-Helme s scheme fo geoid compuaion accoding o Tenze e al. [2003] can be summaized as follows: 1. Fomulaion of he bounday-value poblem in eal space. 2. Tansfomaion of he bounday-value poblem fom he eal ino a hamonic space i.e. ansfomaion of gaviy anomalies fom he eal o Helme space accoding o he second Helme s condensaion echnique whee he opogaphical and amospheic masses ae condensed diecly ono he geoid which consis of adding he Diec Topogaphic Effec Diec Amospheic Effec

35 and ohe smalle effecs o he fee-ai anomalies on he Eah suface. Helme s anomalies ae no vey diffeen in chaace fom he eal fee-ai anomalies. 3. Soluion of Diichle s bounday-value poblem i.e. he downwad coninuaion of Helme s gaviy anomalies fom he suface of he Eah o he geoid by applying he Poisson inegal equaion. This is he mos difficul of he whole pocess. 4. Refomulaion of he geodeic bounday-value poblem by decomposiion of Helme s gaviy field ino low fequency and high fequency pas. I should be noed ha he esidual anomalies ae somewha smalle han he oiginal anomalies. 5. Soluion of he Sokes bounday-value poblem fo he esidual high-fequency Helme gaviy field by using he modified spheoidal Sokes kenel and wheeby he esidual Helme geoid called Helme esidual co-geoid is hus efeed o he efeence spheoid obained fom saellie geopoenial model of a degee L. The complee Helme co-geoid is obained by adding ogehe he Helme efeence spheoid and he Helme esidual co-geoid. 6. Tansfomaion of he complee Helme co-geoid fom Helme space ino he eal space. This is done by adding he Pimay Indiec Topogaphic Effec PITE o complee Helme s co-geoid. 3.2 Fomulaion of geodeic bounday-value poblem in eal space

36 This poblem is fomulaed as deemining he geoid by ansfoming he Sokes bounday value poblem ino Helme space whee he Sokes bounday value poblem epesens he sandad fee bounday value poblem of geodesy. The fomulaion closely follows ha of Vaníček and Mainec [1994].Thee ae an infinie numbe of equipoenial sufaces of he of he eah s gaviy field on which of couse he poenial is consan. Among hem hee is only one suface which appoximaes he mean sea level mos closely and i is given a special significance. This suface is denoed by W W cons. 3.1 o and is called he geoid. Similaly hee ae an infinie numbe of equipoenial sufaces of he nomal gaviy field. Among hese hee is only one such suface which coincides wih he efeence ellipsoid and is denoed by U W g 3.2 This nomal gaviy field is seleced such ha i saisfies he Poisson equaion ouside he geneaing ellipsoid: 2 2 U whee is he gadien opeao. The nomal gaviy denoed by is he gadien of U. The diffeence beween he Eah s gaviy poenial W and nomal gaviy poenials U is called he disubing poenial denoed by T and eckoned anywhee is wien as T W U 3.4 whee is he geocenic disance adius is he geocenic angle denoing he pai he spheical co-laiude and longiude. In his sequel he agumens is

37 he posiion in hee dimensions. In solving he geoid and elaed coecions he mahemaical opeaions ofen needs o be aken ove a oal solid angle [ ]. 0 The disubing poenial saisfies he Laplace equaion ouside he eah and is amosphee. In he absence of opogaphic masses and he eah s amosphee he disubing poenial will be hamonic above he geoid so ha 2 T whee denoes he gadien opeao. If he values of he disubing poenials ae known of he geoid he geoid ellipsoid sepaaion can be compued using Buns s fomula T N g whee is he nomal gaviy on he efeence ellipsoid. Since he disubing 0 poenial canno be measued diecly hen a bounday value poblem of he hid kind has o be fomulaed and solved. In geoid deeminaion some ype of gaviy anomalies efeed o he geoid level seve as he bounday values of his poblem [Mainec e al. 1993]. To find he elaion beween he disubing poenial and he gaviy anomalies he adial deivaive of he disubing poenial is inoduced: T W U 3.7 Equaion 3.7 evaluaed a he suface of he Eah can be appoximaed accoding o [Vaníček and Novák 1999] by T g g 3.8 g g

38 whee he diffeence beween he acual gaviy g and nomal gaviy is he gaviy disubance g and is he ellipsoidal coecion due o he g eplacing of he deivaive wih espec o ellipsoidal nomal n by a moe convenien deivaive wih espec o o he gaviy disubance. The geocenic adius is obained by adding he ohomeic heigh o o H o he i.e. H. The appoximaion is due o H o being measued along he plumb-line beween he geoid and he suface of he Eah which is somewha cuved. g g 3.3 Gaviy anomaly In he pe-gnss ea geodeic heigh was no available and gaviy anomalies ahe han gaviy disubances had o be used. The wold daa bases ae full of gaviy anomalies and hee ae few gaviy disubances. The eason is ha nomal gaviy canno be evaluaed on he suface of he eah as his equies he knowledge of he geodeic heigh h of he poin. Hence he gaviy anomalies have o be ansfomed ino gaviy disubance [Vaníček and Novák 1999]. In geneal gaviy is measued on he suface of he eah. Befoe hese measuemens can be used fo geodeic o geophysical puposes hey mus be conveed ino gaviy anomalies. Geophysics use gaviy anomalies o deduce vaiaions in he mass wihin he eah. This helps in inepeaion of he undelining subsuface sucue. Fo geodesis gaviy anomalies ae used o define he figue of he Eah he geoid [Hackney and Feahesone 2003]. Fuhemoe Gaviy anomalies ae diffeeniaed accoding o he way in which he obseved o nomalous gaviy was deduced. Suface gaviy anomaly

39 does no equie he knowledge of he veical gadien of he acual gaviy wihin he eah. The exac value of he nomal gaviy on he elluoid needed fo suface gaviy anomaly is obained fom nomal heigh N H. Nomal heigh is compued by upwad coninuaion of nomal gaviy fom he geocenic efeence ellipsoid [Vaníček and Kakiwsky 1986]. Gaviy anomalies ae calculaed fom gaviy measuemens on he suface of he eah as g g 3.9 T whee g is he obseved gaviy on he suface of he eah and is he T nomal gaviy evaluaed on he elluoid. Nomal gaviy is a heoeical value epesening he acceleaion of gaviy ha is geneaed by he efeence ellipsoid accoding o Somigliana-Pizzei heoy. Nomal gaviy on he ellipsoid is evaluaed by he use of Somigliana fomula due o Somigliana. The fomula fom Vaníček and Kakiwsky [1986] eads 2 2 a a cos b b sin a cos b sin whee a b ae he majo and mino semis-axes of he ellipsoid and ae nomal a b gaviy a he equao and he pole of he ellipsoid especively. Since he Somigliana s fomula deemines he nomal gaviy on he efeence ellipsoid a heigh coecion is needed o accoun fo a change in heoeical gaviy due o he locaion of he elluoid above o below he ellipsoid. Hisoically his heigh coecion has been incoecly associaed wih he ohomeic heigh H no he geodeic heigh h [Li and Göze 2001]. As a second appoximaion using a Taylo seies expansion fo he

40 heoeical gaviy above he ellipsoid wih a posiive diecion downwad along he geodeic nomal o he efeence ellipsoid accoding o [Heiskanen and Moiz 1967] is given by h 1 1 f m 2 f sin h h 2 a a 3.11 The diffeence h is he coecion fo heigh above he efeence ellipsoid. Accoding o [Moiz 1980] his heigh coecion is given by h 2 1 a f 5 m 3 f sin e h 2 a h Subsiuing he values fo he GRS80 ellipsoid ino Eq gives h sin h h 3.14 The ansfomaion of gaviy disubance o gaviy anomaly is achieved by adding a em o he gaviy disubance ha accouns fo he change in nomal gaviy due o he diffeence beween he geodeic heigh h and he ohomeic heigh o H [Vaníček and Novák 1999]. The gaviy anomaly is hus elaed o he gaviy disubance g by he following fomula: N g g [ 0 H ] 3.15 whee N H is nomal heigh and 0 is he geocenic adius a funcion of laiude of he efeence ellipsoid. Using Molodensky s appoach he diffeence of he nomal gaviy on he o N Eah s suface: g H o h and nomal gaviy H on he elluoid in Eq can be evaluaed as:

41 O H R N n gad H 3.16 whee n is he deivaive of nomal gaviy aken wih espec o he nomal n o he efeence ellipsoid and is he Molodenskij heigh anomaly. Using Buns s spheical fomula he expession on he igh-hand side of Eq accoding o [Vaníček and Novák 1999] can be ewien as N H R H R H T n n O O 3.17 Subsiuing Eq ino Eq leads o N H R H T n g g O 3.18 Applying spheical appoximaion 2 1 n H R N T T n H O 3.19 whee n he ellipsoidal coecion fo he spheical appoximaion allows o eplace he ellipsoidal em by a moe simple em: 2 T. Subsiuing Eq. 3.8 and Eq ino Eq he fundamenal fomula of physical geodesy akes he following fom [Ellmann and Vaníček 2007] 2 n g T T g 3.20 The above equaion fomulaed in eal space can be applied in Helme space fo he pupose of he compuaion of Helme s and any ohe gaviy anomaly: h g

42 3.4 Tansfomaion of gaviy anomalies fom he eal space o Helme space On he coninens he geoid is mosly locaed inside he opogaphical masses. To compue he geoid by Sokes fomula equies educion of he disubing poenial o he gaviy anomalies o he geoid. This educion pocess is called downwad coninuaion [Sun and Vaníček 1996]. An impoan equiemen fo he exisence of he downwad coninuaion is ha he funcion has be hamonic. The pesence of opogaphic masses and he amosphee violae his hamoniciy condiion. To esablish he hamoniciy of he disubing poenial eveywhee above he geoid implies ha he amosphee and he opogaphic masses have o be eliminaed [Ellmann and Vaníček 2007]. One way of eliminaing opogaphy and amosphee is by esimaing he effec of he opogaphic masses and he amospheic effec by means of he Helme s second condensaion echnique which condenses he opogaphical masses ono he geoid. This echnique causes elaively small indiec opogaphic effec see lae. Fo his eason UNB favos his appoach in compuaion of he geoid [Ellmann and Vaníček 2007]. The gaviy field of he Helme Eah is diffeen fom he eal Eah and is given by W h a W V V 3.21 whee he supescip h denoes he quaniies efeing o Helme s Eah o given in Helme s space V is he esidual opogaphical poenial i.e. he diffeence beween he poenial of he opogaphical masses and he poenial of he condensaion laye as shown below: c V V V 3.22

43 Similaly he esidual amospheic poenial V a is he diffeence beween he poenial of he amospheic masses and he poenial of he amospheic condensaion laye on he geoid as shown below a a ca V V V 3.23 Using he analogy wih he eal Eah Helme disubing poenial is defined as T h h a W U T V V 3.24 Helme disubing poenial is hamonic ouside he geoid i.e. 2 T h Helme gaviy g h is he negaive adial deivaive of he Helme gaviy poenial which is paallel o he definiion of gaviy in he eal space [Ellmann and Vaníček 2007]. Helme gaviy on he suface of he eah is obained fom obseved gaviy on he suface of he eah by adding o he obseved gaviy g he diec opogaphical effec A and diec amospheic effec A a : h h W a g g A A 3.26 Applicaion of he elaion g W uilized in Eq is only an appoximaion since he adial deivaive is aken ove he complee poenial ahe han ove he disubing quaniy. This is he eason fo using he symbol in Eq The impoance of his elaion is ha i makes he link beween acual gaviy and coesponding gaviy anomaly and Helme s gaviy anomaly moe anspaen. Fuhe he deivaive in Eq should be aken along he plumbline insead of. As

44 a esul hee is he need o inoduce an ellipsoidal coecion due o his appoximaion [Ellmann and Vaníček 2007]. Helme s gaviy disubance g h is defined as he negaive gadien of he Helme disubing poenial and can be descibed as a sum of he negaive adial deivaive of he Helme disubing gaviy poenial H T and he ellipsoidal coecion g o he gaviy disubance: a g g h h A A g T g 3.28 The elaion beween he gaviy disubance g h and he gaviy anomaly h g in Helme s space can be obained fom he fundamenal gavimeic equaion of physical geodesy Eq as [Tenze e al. 2003] ] [ 0 h N h g h h H T T g n h g h T T 3.30 Helme gaviy anomalies can be expessed via fee-ai anomalies g as [Vaníček and Novák 1999] 2 a h A V A g g 2 n g a V 3.31 The second and hid ems on he igh-hand side of Eq ae diec and seconday indiec opogaphic effecs on he gaviaional aacion. The fouh and five ems of

45 Eq ae he diec and seconday indiec amospheic effecs on he gaviaional aacion. The non-opogaphical coecions o he Helme gaviy anomaly ae hen as follows: diec amospheic effec seconday indiec amospheic effec ellipsoidal coecion fo he gaviy disubance and ellipsoidal coecion fo he spheical appoximaion. Impoanly he poduc of coesponding Helme anomaly and geocenic adius g h is a hamonic funcion above he geoid and heefoe such a field can be coninued downwads o he geoid level [Vaníček and Novák 1999]. Nomal heighs ae used fo esimaing he nomal gaviy in Eq and Eq If ohomeic heighs ae used ahe han nomal heighs hen a geoid-quasigeoid coecion has o be applied o Eq [Ellmann and Vaníček 2007]. 3.5 Effec of opogaphical masses on gaviaional aacion Evaluaion of Helme s gaviy anomaly on he suface of he eah accoding o Eq will equie a compuaion of he diec effec of aacion of he opogaphy. The opogaphical effec on gaviaional aacion eckoned on he suface of he eah is epesened by diec and seconday indiec opogaphical effec DTE and SITE especively [Tenze e al. 2003]. The effecs of opogaphical masses ae fomulaed in spheical fom insead of plana appoximaion. Based on a conclusion by Vaníček e al. [2001] he spheical model should be used wheneve highe accuacy is equied. The spheical model is close o ealiy as he plana model does no allow he fomulaion of any physically meaningful condensaion models [Ellmann and Vaníček 2007].

46 The gaviaional aacion of he eal opogaphy is evaluaed using he Newon inegal ove he volume of opogaphy. Addiionally he Newon inegal can be spli ino he Bougue shell aacion and spheical oughness em fo he compuaion. This spliing has been inoduced o eliminae he singulaiy of Newon s inegal which occus a he a he compuaion poins. Accoding o Mainec [1993] he poenial of opogaphic masses he condensaion of which obeys he law of consevaion of masses can be esimaed as V 4 G o 2 R H H 1 H 1 R 3 R 2 G G o O O 0 RH 1 l [ ] 0 RH 0 RH 1 R 2 dd 2 l [ ] dd 3.32 whee G is he gaviaional consan [ ] and ae he spaial l disance and geocenic angle beween he compuaional and inegaion poins d is he aea of he inegaion elemen. The fis em on he igh-hand side of Eq is he gaviaional poenial of he spheical Bougue shell of mean densiy o and he hickness equal o he ohomeic heigh H 0 of he compuaion poin. The second em sands fo he gaviaional poenial of he spheical opogaphical oughness em and he hid em epesens he effec of he anomalous opogaphical densiy disibuion on he gaviaional poenial. Similaly Mainec [1993] saes ha he poenial of he condensaion masses can be wien as

47 c V R H R H H R G o O d R l G o ] [ O d R l R G ] [ whee he fis em on he igh-hand side of Eq is he gaviaional poenial of he condensed spheical maeial of single laye spheical Bougue shell he second em sands fo he gaviaional poenial of he spheical oughness em of he condensed opogaphical masses and he hid em epesens he effec of he anomalous condensed opogaphical densiy disibuion on he gaviaional poenial. The seconday indiec opogaphic effec on gaviaional aacion hid em on he igh-hand side of Eq which efes o he eah s suface is compued by subacing Eq fom Eq i.e. ] [ 2 2 c V V V Diec Topogaphic Effec DTE The diec opogaphic effec is obained by aking he diffeence of he adial deivaives of Eq and Eq The aacion of he opogaphical masses accoding o [Mainec 1993] eads V R H R H H R G o

48 O H R H R o d d l G ] [ O H R R d d l G ] [ 3.35 whee he fis em on he igh-hand side is he negaive gaviaional aacion of he spheical Bougue shell. The second em is he gaviaional aacion of he spheical oughness em i.e. he spheical eain coecion and he hid em epesens he effecs of he anomalous opogaphical densiy on he gaviaional aacion. The aacion of he condensed masses can be expessed as c V R H R H H R G o O d R l G o ] [ 3 O d R l R G ] [ whee he fis em on he igh-hand side is he gaviaional aacion of he condensed spheical Bougue shell he second em is he gaviaional aacion of he spheical oughness em of he condensed opogaphical masses and he hid em epesens he effec of he anomalous condensed opogaphical densiy disibuion on he gaviaional aacion. I should be noed ha he fis ems of Eq and Eq ae equal. Theefoe he Bougue shell conibuions cancel each ohe ou when Eq is subaced fom Eq The final expession fo he diec opogaphic effec in Helme s space hen becomes

49 V V A c } { 2 1 [ H R H R o d d l G O d R l G o ] [ 3 O H R R d d l G 2 1 ] [ O d R l R G ] [ The fis em on he igh-hand side of Eq is he spheical eain coecion he second em is he spheical condensed eain coecion and he hid and fouh ems epesen ogehe he conibuion of he laeal vaying densiy o he diec opogaphic effec. 3.7 Diec Amospheic Effec DAE The equiemen hee should no masses ouside he geoid suface indicaes ha he amosphee mus also be condensed ono he geoid suface. This condensing of he amosphee on he suface of he geoid is caied ou as Diec Amospheic Effec DAE on he fee-ai gaviy anomalies and is compued as p p a g g a c g g a A A A 3.38 whee g g a c A is he gaviaional aacion of he condense amosphee and p p a A is he gaviaional aacion of he eal amosphee. A digial elevaion model povides opogaphical heighs infomaion fo he compuaion poins in compuing he DAE.

50 3.8 Seconday Indiec opogaphical Effec SITE The effec of condensaion of opogaphy on nomal gaviy is called seconday indiec opogaphical effec SITE and ha of he amosphee is called seconday indiec amospheic effec SIAE. SIAE is so small ha i is aely calculaed whiles SITE is calculaed accoding o [Vaníček and Novák 1999] by 2 p p Vc p p V p p 3.39 g p 3.9 Downwad coninuaion of gaviy anomalies Gaviy anomalies on he geoid ae needed as inpu fo he compuaion of he geoid. Afe ansfomaion of he gaviy anomalies a he opogaphic suface fom eal space o Helme space hey mus be coninued down o he geoid in Helme s space. The upwad coninuaion is descibed by Poisson inegal [Heiskanen and Moiz 1967]. This is a fomula fo upwad coninuaion of hamonic funcions fom a sphee of adius R. The Poisson inegal fo gaviy anomalies is given by [Kellogg 1929]: h R h g K[ R] g g d whee [ R] is he spheical Poisson inegal kenel. K When a downwad coninuaion is sough Eq is inveed yielding an inegal of Fedholm of he fis kind. I can be e-wien as an inegal equaion whose soluion is sough hough disceizaion ha leads o a sysem of linea equaions. In he maixveco fom i can be pesened as follows:

51 h h g K[ R] g 3.41 g whee g h is he veco of he gaviy anomalies efeed o he suface of he eah g h he veco of he gaviy anomalies efeed o he geoid suface and g K [ R] is he maix of he values of he Poisson inegal kenel muliplied by he emaining elemens on he igh-hand side of Eq Equaion 3.40 yields he gaviy anomalies on he suface of he eah given he gaviy anomalies on he geoid. Thus downwad coninuaion an invese of Eq i.e. povides values of he gaviy anomalies on he geoid given he gaviy anomalies on he opogaphy. As a esul of he invese opeaion he Poisson downwad coninuaion is known o be numeically unsable. Due o he insabiliy exising eos in g h may appea magnified in he soluion. Howeve when mean Helme s gaviy anomaly values ae used insead of poin values his poblem is somewha alleviaed [Ellmann and Vaníček 2007] Refeence Field The Eah Gaviy Model EGM is a global epesenaion of he Eah gaviaional field poenial geoid heighs gaviy anomalies gaviy disubances and deflecion of he veical in ems of spheical hamonic expansion [Hackney and Feahesone 2003]. These models ae classified ino wo ypes: saellie-only EGMs deived fom acking aificial Eah saellies fo example CHAMP and GRACE and combined EGMs. These lae models ae deived fom combinaion of eesial gaviy saellie alimey gaviy daa in maine aeas and saellie-only model [Rodíguez-Cadeo e al. 2006].

52 The EGMs povide long wavelengh infomaion of he eah s gaviy field and conibue o he UNB Sokes-Helme geoid deeminaion as a Refeence field of a seleced degee and ode nowadays ypically 90 by 90 aken only fom pue saellie soluion. The use of a combined EGM would esul in he eesial infomaion being used wice in he final esul. The Refeence Field in he fom of low fequency 90 by 90 o so Helme s anomalies is subaced fom he Helme anomalies on he geoid befoe he Sokes inegal is applied [Vaníček e al. 1995]. The Sokes inegaion of hese esidual gaviy anomalies esuls in he esidual cogeoid. The esidual co-geoid can be viewed as a co-geoid efeed o a Refeence Spheoid of degee and ode 90 by 90 so ha he final co-geoid would consis of he sum of hese wo. Thus in compuing he co-geoid his way global o lage-scale feaues of he geoid ae expessed by spheical hamonic expansion of he gaviaional poenial while highe ems ae conibued by eesial gaviy daa [Vaníček e al. 1995] The EGM poenial coefficiens ae conveed o disubing poenial coefficiens and hen o EGM gaviy anomaly in Helme space. The efeence field plays he ole of educing he magniude of he compued quaniies and also seves as a lineaizaion ool [Vaníček e al. 1995] Soluion of Sokes s bounday value poblem Afe ansfoming he Helme gaviy anomalies fom he suface of he eah o he geoid he gaviy anomalies ae conveed ino he Helme disubing poenial T. Helme disubing poenial is he diffeence beween he Helme gaviy poenial on he co-geoid and he nomal poenial on he Refeence Ellipsoid. Gaviy anomalies ae

53 ansfomed ino he disubing poenials hough he fundamenal equaion of gavimey as shown by Eq This disubing poenial is ansfomed ino geoidal heigh via Buns s fomula shown in Eq. 3.6.This appoximaion by Buns s fomula indicaes ha N can be compued if an expession fo T can be found. An accuae spheical soluion is found by using Geen funcions fo a sphee which is subsiued ino Sokes s inegal. [Heiskanen and Moiz 1967] give ansfoming gaviy anomalies on he geoid ino disubing poenial using Sokes s inegal as R T gs d whee S is he esiced Geen funcion known as Sokes s funcion given by 1 2 S 6sin 1 5cos 3cos lnsin sin sin The use of he Sokes inegal wih Sokes funcion on a sphee esuls in some eos. These eos ae gealy diminished in he UNB Sokes-Helme s echnique by use of a highe degee efeence Field as descibed above. Thus he Sokes inegaion of esidual Helme s anomalies ahe han he complee anomalies complemened by he addiion of he Refeence Spheoid esuls in a much moe accuae co-geoid. Thee ye is anohe eason fo having inoduced he Refeence Field/Refeence Spheoid of degee and ode M in he UNB Sokes-Helme s echnique. The use of he oiginal Sokes s fomula equies gaviy anomalies o be known all ove he suface of he eah. In pacice he gaviy anomalies ae available only in and aound he aea of inees. When Vaníček and Kleusbeg [1987] inoduced he Sokes-Helme echnique hey ealized ha he Sokes funcion M S fo compuing he esidual co-geoid by means

54 of esidual gaviy anomalies esembles moe closely he Diac disibuion as is dies ou wih gowing angle much moe quickly han he oiginal Sokes s funcion S does. The low-fequency pa of he co- geoid is descibed by he EGM as a spheoid of degee M. The esidual co-geoid is compued as N h es R 4 o g h es g S M d N o h FZ 3.44 M whee he appopiae Sokes s funcion S can be compued accoding h o Vaníček and Kleusbeg [1987] g is he Helme esidual gaviy anomaly es g h on he geoid and N FZ is he fa zone conibuion i.e. he conibuion of he anomalies fom he es of he wold. Equaion 3.44 uses he high-degee esidual Helme gaviy anomalies obained by subacing he long-wavelengh RF conibuion fom he complee gaviy on he geoid. The fa zone conibuion o uncaion eo is evaluaed in a specal fom using an EGM o a highe degee and ode han M. o h N FZ 3.12 Tansfomaion fom Helme s space o eal space Evaluaion of he Sokes bounday value poblem in Helme space esuls in an equipoenial suface called Helme co-geoid. To find he geoid in eal space will equie an evaluaion of he pimay indiec opogaphic effec PITE and pimay indiec amospheic effec PIAE. The ansfomaion is achieve by adding PITE and PIAE o he Helme co-geoid as:

55 N Helme Co geoid V R N O a g 3.45 whee N is he geoid sepaaion V R is PITE and a N g is PIAE. O The applicaions of hese wo effecs ae shown in secions 3.13 and Pimay Indiec Topogaphic Effec PITE Even hough he diec and seconday indiec opogaphic effecs ae evaluaed a he suface of he eah hee is an impoan opogaphic effec ha needs o be accouned fo a he geoid level. As a esul of he condensaion of he opogaphic and amospheic masses he Helme poenial becomes slighly diffeen fom he acual poenial. Consequenly he Helme co-geoid does no coincide exacly wih he geoid in eal space. The effec causing his is known as Pimay Indiec Topogaphical Effec PITE. One of he impoan advanages of using Helme second condensaion is ha PITE is nowhee lage han 2 m woldwide. Similaly he Pimay Indiec Amospheic Effec is less han 1 cm globally. PITE is accouned fo sepaaely as a coecion o he Helme co-geoid. Moe explicily PITE is he ansfomaion em added o he co-geoid in Helme space o obain he geoid in he eal space. PITE is compued fom he gaviaional poenials of he opogaphical and condensed masses boh of which efe o he geoid level. The final expession fo he PITE accoding o [Mainec 1993] is as follows:

56 o O O o O G R H H G R V ] [ O H R H R d d l ] [ O d R R l G o O ] [ O H R R O d d R l G ] [ O d R R l R G o O ] [ Pimay Indiec Amospheic Effec PIAE Simila o PITE he pimay indiec amospheic PIAE effec is calculaed accoding o [Novák 2000] as 0 g p g p a g g g a c g a V V N 3.47 whee p g g a c V is he gaviaional poenial induced by he condensed amospheic masses a he geoid and p g p a V is he gaviaional poenial induced by he eal amospheic masses a he geoid. This value mus be subaced fom he compued co-geoid. This concludes he heoy behind he Sokes-Helme mehod of geoid compuaion developed a he Univesiy of New Bunswick.

57 Gaviy is a mysey of he body invened o conceal he defecs of he mind LaRochefoucauld Chape 4. Daa acquisiion 4.1 Inoducion All eesial gaviy measuemens consis of poin measuemens of he Eah s gaviy on he Eah s physical suface. Acquiing eesial gaviy daa is an expensive undeaking which akes ime effo and equies access o he place of obsevaion. As esuls mos of he gaviy obsevaions in Ghana wee caied ou along he majo and mino oads in he couny see Figue 4.1. The main agency esponsible fo acquiing and mainaining gaviy daa in Ghana is he Geological Suvey Depamen. Mos of he eesial gaviy daa gaheed by he Geological Suvey Depamen ae concenaed a he souh-wesen he middle and he noh-easen pas of Ghana. These aeas coincide wih he main mining aeas of Ghana. The Geological Suvey Depamen also has aibone gaviy daa se in hei eposioy. Howeve fo his geoid compuaion only eesial gaviy daa wee used. This is because he aibone gaviy daa equies special eamen befoe he daa can be used and ime consains did no allow me o do i popely. Effos wee made o ask he Bueau of Gavimeique Inenaional BGI and he Univesiy of Leads o povide addiional gaviy daa in hei possession fo he use in his geoid compuaion. Howeve all effo o secue he daa failed.

58 The eesial gaviy daa se of he Geological Suvey Depamen also includes hoizonal coodinaes i.e. laiude and longiude ohomeic heigh he fee-ai gaviy anomalies and he Bougue gaviy anomalies. The nomal gaviy used in compuing he gaviy anomalies in he daa se wee deemined fom he Inenaional Gaviy fomula of 1930 which is no longe valid. Hence hese anomalies wee no used. The nomal gaviy used in he compuaion of geoid models nowadays ae all based on Geodeic Refeence Sysem 1980 GRS80. Since hee ae aeas wihin he scaeed eesial gaviy daa ha lack any gaviy daa coveage gaviy anomalies compued using EGMs wee used in padding he aeas in Ghana ha lack gaviy daa. This chape descibes all he eesial gaviy daa ha wee used in compuing he geoid model fo Ghana. The souce maeial fo he hisoy of gaviy suvey in Ghana is ha due o Davis [1958].

59 Figue 4-1 Gaviy obsevaions in Ghana Souce: [Davis 1958

60 4.2 Teesial gaviy daa in Ghana The Biish Govenmen in collaboaion wih Ghana Govenmen caied ou he fis gaviy suvey in Ghana beween 1957 and This gaviy suvey wok was pa of Ghana s conibuion o he inenaional Geophysical Yea pogam in This pogam aimed o esablish a woldwide newok of known gaviy values. Gaviy obsevaions wee made a 128 sies. The newok of gaviy obsevaions was linked o he Office de la Recheche Scienific e Technique Oue-Me ORSTOM newok of he neighboing counies. Thee sies in he ORSTOM newok ouside Ghana wee used: Abidjan Ivoy Coas Lome Togo and Ouagadougou Bukina Faso. The ORSTOM newok includes seveal pendulum saions such as Pais Fance and esablishing a link wih hese base saions will ensue ha he Ghana newok would be compaible wih local daa. The value of gaviy fo each sie was deemined wih an unceainy of less han 0.2 mgal. The Ghana newok was also linked wih he well-esablished gaviy newok of he Unied Kingdom. The sandad deviaions of he connecion wee all below 0.27 mgal and 70% of hem wee below 0.05 mgal. The newok is made up of 20 inenal loops involving only he connecions and hee ohe loops a he edge of he newok involving he base saions as well. The smalles miss closue is 0.01 mgal and he lages 0.34 mgal. Half of he miss closue is less han 0.10 mgal. Ideally he loops should be iangles o quadilaeals bu owing o he lack of navigable oads half he loops ae many-sided polygons some wih ove 20 sides. The newok was adjused using leas squaes adjusmen.mehod.

61 Since i was necessay o link he Ghanaian gaviy newok o he gaviy newok of he neighboing counies he calibaion faco of he gavimee used fo he gaviy suvey wok was eaed as unknown in he adjusmen. A weigh was assigned o each leg as a measue of is eliabiliy. The nomal pacice is o egad each leg as having equal weigh and hus afe he usual pinciple of combining sandad deviaions each leg is assigned a weigh invesely popoional o he squae oo of he numbe of connecions consiuing i in manne simila o opogaphical suvey. Howeve due o he wide spead of he individual sandad deviaion fom nealy 0.00 o 0.27 mgal i was fel he leg conaining many connecions would be abnomally heavily weighed. The compomise adoped was all connecions having sandad deviaion fom nealy 0.00 mgal o 0.01mGal ae eaed as if hey had a sandad deviaion of 0.02 mgal. A Pegasus compue made by Feani Ld was used fo he compuaion. The ipod heigh allowance i.e. he heigh of he gavimee above gound was accouned fo. The ORSTOM newok is based on he Posdam sysem. Howeve a decision was made o esablish an independen link o he new newok in Ghana. The sies occupied duing he linking fom London Heahow Aipo wee Rome Aipo in Ialy Tipoli Aipo in Libya Kano Aipo in Nigeia and finally Acca Aipo. The Geological Suvey and Museum GSM London supplied gaviy values fo he London Aipo. Wollad supplied he gaviy values fo Rome Tipoli and Kano aipos. The connecions beween he saions ae as follows: 1. London Rome London 2. Rome Tipoli Rome 3. Tipoli Kano Tipoli

62 4. Kano Acca Kano To cay ou he dif-ae deeminaions befoe and afe each secion of he link i was necessay o avel by a seies of flighs on successive days. The measuemens wee made beween 12 h and 15 h Decembe Afe he gaviy daa collecion campaign hee wee fuhe gaviy suveys done in Ghana caied ou by he Ghana Geological Suvey Depamen. Mos of hose suveys wee caied ou in collaboaion wih he Developmen Panes mosly fom he Euopean Union. Figue 4-2 shows he map of he obseved eesial gaviy daa in Ghana used fo his compuaion of he geoid model. In oal hee ae ove10081 eesial gaviy daa coveing he enie couny. Howeve he gaviy daa ae no unifomly disibued wihin he couny. They ae concenaed a he souhen and easen pas of he couny wih he easen half coveed mainly by aibone gaviy suvey. The wesen pa of he couny ha bodes Coe D Ivoie is only spasely coveed wih gaviy daa. These aeas wee padded wih EGM daa. I is ou hope ha aibone gaviy suvey could be exended o cove hose aeas in he nea fuue. Such gaviy daa coveage will impove he geoid model fo ha pa of he couny. Howeve he spase gaviy coveage will no be much of a poblem since abou 80 pecen of Ghana is no hilly wih he highes mounain Afajao abou 800m above sea level.

63 Figue 4-2 Disibuion of obseved eesial gaviy coveage Souce of map: Google

64 4.3 Compuing he gaviy anomalies and esidual gaviy anomalies Equaion 3.9 was used o compue gaviy anomalies on he elluoid fo all he gaviy poins in Ghana. Because of he spase gaviy daa coveage of Ghana a decision was made o pad he aeas in Ghana ha lacked gaviy daa wih daa geneaed fom EGM08. The padding was caied ou as follows: 1. A file conaining lis of laiude longiude and ohomeic heigh of he scaeed gaviy daa poins fom Ghana eesial gaviy suvey daa wee used as inpu daa o compue he gaviy anomalies fom he EGM. GafLab a MATLAB pogam was used o compue of he gaviy anomalies [Bucha and Janák 2013]. 2. This was followed by a compuaion of esidual gaviy anomalies which ae he diffeence beween he eesial gaviy anomalies and gaviy anomalies fom he EGM08 i.e. g g g 4.1 RES obsev EGM 08 whee g RES is he esidual gaviy anomaly gobsev is he eesial suface gaviy anomaly and g EGM 08is he gaviy anomaly compued fom he EGM A plo of he esidual anomalies shown in Fig. 4.3 indicaes ha he esidual anomalies ae nomally disibued bu no andomly disibued aound he mean of zeo and hee ae oulies. Hence saisical es wee compued and oulies wee eliminaed 4. The esidual gaviy anomalies wee gidded. The SURFER Sofwae wih invese disance squaed inepolaion mehod was used fo was used fo gidding. The esidual anomalies wee gidded a 5'5' inevals.

65 5. Thee columns of he oupu file of he DTE compuaion i.e. he laiude longiude and heighs he DTE gid wee used as inpu daa ino he GafLab pogam o compue fee-ai gaviy anomalies fom EGM08. This DTE gid is bounded by laiude 2ºS and 18ºN and longiudes 10ºW and 8ºE 6. The fee-ai anomalies compued fom he EGM08 wee added o he esidual anomalies o obain fee-ai gaviy anomalies. 4.4 Saisical es fo oulies The accuacy of a compued geoid model depends on he qualiy of gaviy daa used in he compuaion pocess. To ensue ha he eesial gaviy daa used fo compuing he geoid model is fee of goss and sysemaic eos as much as possible saisical es was caied ou on he esidual gaviy anomalies compued in Secion 4.3. The aim of his saisical es is fo he deecion of oulies. The assumpion made was he esidual gaviy values should be zeo mean. The compuaion esuled in a mean of mgal and sandad deviaion of mgal. A plo of he hisogam of he esidual is shown in Fig 4.3. A 99.99% confidence ineval fo a wo ail es was seleced and a nomal es of a single obsevaion caied ou. This esuled in a ciical egion of mGal< < mGal. Thus esidual gaviy anomalies below mGal and hose above mGal wee ejeced as oulies. Of he poins eesial gaviy daa poins 117 eesial gaviy poins wee ejeced as oulies pobably due o blundes.

66 Figue 4-3 Hisogam of esidual gaviy anomaly daa 4.5 Tansfomaion fom eal space o Helme s space Fee-ai gaviy anomaly obseved on he Eah suface once ansfomed ino Helme s space is called Helme s anomaly on he Eah suface [Vaníček e al. 2012]. In ansfoming he gaviy anomalies ino Helme s space he cux of he opeaion is he assessmen of opogaphic and amospheic effec on he eal gaviy and hus on he anomalies. The emoval of opogaphy and he addiion of he condensed opogaphy laye on he geoid cause he opogaphic effec called diec opogaphic effec DTE and ha of he amosphee is called diec amospheic effec DAE. The compuaion of he diec opogaphic effec and diec amospheic effec was caied ou using downloaded

67 daa fom STRM and ACE2 websies. The 3 3 ac-secs ac-secs 5 5 acmin and global opogaphy daa ses wee used in compuing he DTE and DAE. Each daa se is equied in a paicula daa foma fo compuaion using SHGeo Sofwae. The final Helme gaviy anomalies a he Eah s suface wee compued by adding he DTE and SITE and subacing he DAE fom he gidded fee-ai anomalies. The inclusion of SITE is o ensue ha Helme anomalies saisfy he definiion of a gaviy anomaly. 4.6 Gidding of he esidual gaviy anomalies Sufe sofwae fom Golden Sofwae was used o gid he esidual gaviy anomalies. An Invese Disance Squae mehod was used o pedic he anomalies on a 5'5' gid. The gid nodes of he esidual gaviy anomalies wee made o coincide wih he gid nodes of he Diec Topogaphic Effec. The eason fo he coincidence of he gid nodes is o simplify he compuaion of Helme anomalies as shown in secion 4.9. Figue 4-4 shows he gaviy anomalies. The maximum value 300mGal is locaed on he islands of Sao Tome and Pincipe and Malabo nea Cameon in he Alanic Ocean. Fo mos pa of he compuaion bounday and in Ghana he fee-ai anomalies ae beween plus 50mGal and minus 100mGal.

68 Figue 4-4 Fee-ai anomalies mgal 4.7 Digial Elevaion Model DEM Vey accuae and deailed esoluion infomaion abou he opogaphy is equied fo he educion of gaviy measuemens fom he suface of he eah o he geoid in ode o saisfy he condiions of he bounday values poblem of geodesy. These heigh daa ae available as Digial Elevaion Model DEM. Addiionally he pesence of opogaphy indicaes ha he gaviy obsevaions ae no on a level suface and his conadics he basic equiemens fo he Sokes s heoy in geoid compuaions [Sansò and Sideis

69 2013]. The lages effec on he gaviy measuemen comes fom he mass anomalies in he manle and he coe and i is of long wavelengh chaace [Mainec e al Bajachaya 2003]. Even hough DEM s ae impoan inpu daa in calculaing he effec of he opogaphy on he gaviy signal anohe impoan pa especially fo pecise compuaion of he geoid model is addiional infomaion abou he opo-densiy o evaluae he opogaphical effecs [Vaníček 1990]. Fou Digial Elevaion Models DEM wee used fo he compuaion of he diec opogaphic effec. The SHGeo sofwae fo compuing DTE equies hese fou diffeen daa ses. This is because o he DTE_Helme_global.c module of he SHGeo sofwae used fo compuing he DTE equies DEM in hese fou diffeen esoluions. The fis is he gidded opogaphy wih block size of 3" 3" fom he Shule Rada Topogaphy Mission SRTM. The ohe wo ae he 30"30" and 5'5' blocks size gid fom his ACE2 websie and finally a global 1º1º daa. The SRTM was a collaboaive effo beween Naional Aeonauics and Space Adminisaion NASA he Naional Geospaial-Inelligence Agency NGA he Geman and Ialian space agencies o geneae a nea-global Digial Elevaion Model DEM. The SRTM daa as saed is sampled a 3 ac-seconds which is 1/1200h of a degee of laiude and longiude o abou 90 mees in laiude and 90 cosphi m in longiude. The 30 ac-second gid oughly anslaes o 1 kilomee in laiude. Howeve on Sepembe he Whie House announced ha he highes esoluion opogaphic daa geneaed fom NASA s SRTM in 2000 will be eleased globally ove he nex yea. The new daa have a 1 ac-second o abou 30 mee esoluion sampling ha eveals he full esoluion of he oiginal

70 measuemen [NASA 2015]. Howeve his 1 ac second daa was no used in his compuaion of he DTE. 4.8 Compuaion of Diec Topogaphic Effec DTE Compuaion of he Diec Topogaphical Effec was caied ou using he pogam DTE_Helme_global.c module of he SHGeo sofwae. Fig 4-5 shows he DTE compuaion esuls. Figue 4-5 Diec Topogaphic Effec DTE in mgal

71 4.9 Diec Amospheic Effec DAE The module DAE_H_and_NT.cc of he SHGeo sofwae was used fo he compuaion of he amospheic effec on gaviy. The effec of he amosphee on gaviy anomalies fo he compuaion aea is shown in Fig 4-6. Figue 4-6 Diec Amospheic Effec DTE in mgal

72 4.10 Seconday Indiec Topogaphic Effec SITE The SITE.cc module of SHGeo pogam was used fo he compuaion of he seconday indiec opogaphic effec efeed o he suface of he eah. Fig. 4-7 shows he effec of seconday indiec effec of opogaphy on he gaviy anomaly. Figue 4-7 Seconday Indiec Topogaphic Effec SITE in mgal 4.11 Helme anomalies on he opogaphy

73 Helme gaviy anomalies on he suface of he eah wee compued by combining he esuls of he fee-ai gaviy anomalies and DTE DAE and SITE compuaions. The compuaion of Helme gaviy anomalies ae as follows: Helme anomalies = fee-ai anomalies + DTE + SITE DAE 4.2 Fig. 4-8 shows he compuaion of Helme anomalies on he opogaphy. The esuls fom he above compuaion can now be downwad coninued o he geoid. Figue 4-8 Helme anomalies in mgal 4.12 Downwad coninuaion

74 Downwad coninuaion accoding o Poisson is a mahemaical pocess involving suface inegaion o calculae wha he gaviy anomalies measuemen would be if hey wee compued on he geoid [Heiskanen and Moiz 1967]. Gaviy anomalies can be downwad coninued o he geoid if hee ae no disubing masses wihin he ange of he coninuaion [Neleon 1976]. In compuing he downwad coninuaion of gaviy anomalies o he geoid suface he module Downwad_coninuaion.c [Kingdon and Vaníček 2011] of he SHGeo sofwae suie was used. The pogam equies an opion file wih he following: 1. An inpu file of Helme gaviy anomalies ha efe o he suface of he eah 2. Anohe inpu file conaining he coesponding heighs of hese Helme gaviy anomalies. 3. An oupu file of he compuaion pocess which is he gaviy anomalies ha efes o he geoid ogehe wih hei coesponding laiudes and longiudes. 4. The bodes in laiude and longiude of he downwad coninuaion aea. 5. The spacing of he gids wihin he compuaion bounday. Fo his eseach a compuaion gid spacing of 1 1 ac-minue was seleced. 6. An ieaion oleance of 0.05 i.e. he value below which he ieaion pocess should be eminaed. Fig. 4-9 shows he diffeence beween he Helme anomalies on he opogaphy and ha on he geoid.

75 Figue 4-9 Diffeences beween Helme anomalies on opogaphy and downwad coninuaion in mgal

76 4.13 Refeence Field The long-wavelengh pa of he gaviy signal pediced fom he EGM08 up o degee 90 is emoved fom he Helme gaviy anomalies on he geoid. This emoves he uneliable low-fequency pa of he eesial gaviy daa and indicaes he limis of he o inegaion i.e. he 0 2 of inegaion cap aound he compuaion poins. The degee and ode 90 of EGM08 was esed and found o give elaively eliable esuls. Compuaion of he efeence field was caied ou using he pogam Refeence_field_2001_HighDegee.f in he SHGeo package. The efeence field is shown Fig 4-10.

77 Figue 4-10 Refeence field in mgal 4.14 Ellipsoidal coecions As explained ealie he ellipsoidal coecions ae wo: gaviy disubance coecion and spheical appoximaion coecion. These coecions aise fom he fac ha he bounday fo which he BVP is fomulaed is he geoid. I is in pacice appoximaed by an ellipsoid which in un is fuhe appoximaed by he sphee. The pogam Ellips_coecions_masPeseve.fo pogam of SHGeo was used o calculae he gaviy disubance coecion and spheical appoximaion coecion. The esuls of he

78 ellipsoid coecions fo gaviy disubance and spheical appoximaion ae shown in Fig 4-11 and Fig Figue 4-11 Ellipsoidal coecion fo gaviy in mgal

79 Figue 4-12 Ellipsoidal coecion fo sphee in mgal 4.15 Compuaion of he esidual co-geoidal heighs The pogam Sokes_inegal was used o compue he Sokes inegaion using a modified Sokes s kenel [Vaníček and Sjöbeg 1990]. This modificaion was bough abou by Molodenskuj and fis used in Canada by Vaníček and Kluesbeg 1987 as indicaed ealie. Residual Helme gaviy anomalies on he geoid o which he ellipsoidal coecions have been applied wee used as inpu daa ino he Sokes fomula

80 fo he compuaion of he esidual Helme s co-geoidal heighs. A wo-degee spheical cap was used fo his compuaion in spaial fom and he fa zone conibuion i.e. he conibuion of he anomalies fom he es of he wold is compued in specal fom. By use of his echnique UNB s Sokes-Helme mehod of geoid compuaion avoids he uncaion eos which mos of he geoid compuaion sofwae ae unable o accoun fo. The heoeical basis fo his echnique was explained in secion Fig 4-13 shows he esul Sokes inegaion compuaion.

81 Figue 4-13 Helme co-geoidal heighs in mees 4.16 Refeence Spheoid Since he efeence field of degee and ode 90 of EGM 08 was subaced fom Helme s anomalies on he geoid pio o Sokes inegaion he efeence spheoid up

82 o degee and ode 90 of EGM08 was added back o he Helme esidual cogeoid afe he Sokes inegaion. The pogam Refeence_spheoid_2011_HighDegee.f of he SHGeo package was used fo he compuaion of he efeence spheoid. Fig shows he esuls of his compuaion. Figue 4-14 Refeence spheoid in mees

83 4.17 Tansfomaion fom Helme s space o eal space The addiion of pimay indiec opogaphic effec PITE and pimay indiec amospheic effec PIAE ansfoms he co-geoidal heighs in Helme space back o geoidal heigh eal space [Mainec and Vaníček 1994]. The deails of he PITE and PIAE compuaions ae shown below Pimay Indiec Topogaphic Effec PITE This indiec effec of opogaphy is compued using he pie_io.cc pogam module of he SHGeo sofwae. The esul of he PITE compuaion is shown in Fig 4-15.

84 Figue 4-15 Pimay Indiec Topogaphic Effec PITE in mees 4.19 Pimay Indiec Amospheic Effec PIAE Simila o he PITE he PIAE on he compued geoid has o be accouned fo. This compuaion was caied ou using piae_h.cc module of he SHGeo Sofwae. Figue 4-16 shows he compuaion of he effec of pimay indiec amospheic effec. The PIAE is

85 quie small as shown in he colo ba. UNB s SHGeo sofwae suie accouns fo his effec and his makes SHGeo geoid compuaion eally obus. Figue 4-16 Seconday Indiec Amospheic Effec SIAE in mees

86 4.20 Geoid-Ellipsoid sepaaion The esuling geoid fo Ghana is shown in Fig A conou plo of he geoid is also shown in Fig Figue 4-17 Geoid-ellipsoid sepaaion in mees

87 Figue 4-18 Conou plo of he geoid-ellipsoid sepaaion in mees

88 4.21 GNSS/igonomeic-levelling daa Measuemens fom GNSS eceives povide geodeic heighs which ae heighs efeed fom he suface of efeenced ellipsoid o he poins of inees and measued along he nomal of he ellipsoid. Ohomeic heigh is efeenced fom he geoid o he poin of inees and is measued along he plumbline. Ohomeic heighs ae obained mainly by spii levelling and o a less degee of accuacy by igonomeic levelling. Equaion 4.3 below shows he elaionship beween he geoidal heigh geodeic heigh and ohomeic as: N h H 4.3 whee N is he geoidal heigh h is he geodeic heigh and H is he ohomeic heigh. GNSS/igonomeic-leveling is a echnique based on he use of he above equaion i.e. N = h H. If he geodeic heigh h is obained fom GNSS measuemens and he ohomeic heigh H is available fom levelling hei diffeences can be compaed agains he compued N. GNSS/igonomeic-levelling echnique can be used o assess he accuacy of he compued gavimeic geoid model and vise-vesa. Unfounaely ohomeic heighs obained fom spii levelling ogehe wih is GNSS daa ae no eadily available in Ghana as a he ime of esing his geoid model. Howeve less accuae igonomeic heighs can be exaced fom he old iangulaion newok esablished by he Biish fom 1924 o 1926 [Clendinning 1926]. These igonomeic heighs wee povided by he examinaion secion of he Suvey and Mapping Division of he Lands Commission. These ohomeic heighs wee compued fom igonomeic levelling daa duing he

89 iangulaion of he hen Gold Coas and ae hough o have geneally a sandad deviaion of abou 25cm [Clak and Clendinning 1923]. The oal numbe of such poins ha can be used fo he assessmen of he Ghanaian gavimeic geoid model is 13. The 13 poins coves mainly he souhen poions of he couny usually efeed o as he golden iangle. Thee may be inconsisencies in he compuaion of he igonomeical heighs since he compuaions wee no adjused homogeneously. GNSS deemined geodeic heighs wee obseved on hese igonomeic poins duing he fis phase of he Land Adminisaion pojec in Each poin was obseved fo a minimum of 12 hous. As shown in he able below he diffeences beween GNSS/igonomeic-levelling deemined geoidal heighs and hose deemined in ou compuaion anges fom m o m. These elaively lage diffeences ae due o he eos in he elevaion values fom he igonomeic levelling newok since he newok has no been adjused homogenously and addiionally only has hieen poins. These poins ae locaed on high mounains see Figs and As a esul hese igonomeically deemine heighs ae no good enough o use in evaluaing geoid model accuacy. Heigh deemined fom pecise levelling will be a bee measue of assessing he accuacy of he geoid model. Again igonomeically deemined heighs suffe fom efacion coecion eos which ae vey significan see lae. Fom able 1 and able 2 i can be seen ha he poins wih highe elevaions have elaively highe diffeences. I is woh menioning ha he compuaion of he geoid model if caied ou using a cieion ha made esidual anomalies less han -10 mgal and ha geae han +10 mgal as

90 oulies poduces only slighly bee elaively o GNSS/igonomeic-levelling diffeences. This indicaes ha he lage diffeences in he assessmen of he accuacy of he compued geoid model ae due o he eos in he igonomeic levelling and no in he gaviy daa. Table 2 shows he esuls of his compuaion. Figue 4-19 shows he locaions of he igonomeic poins used fo assessing he geoid model. An enlaged fom of hese locaions is in figue One may also check he accuacy of he compued geoid model by using GOCE TIM5 EGM up o degee and ode d/o 200 combined by EGM2008 fom d/o 201 o d/o 2190.

91 Name Tigonomeic heigh m NGNSS / Tigonome ic Levelling m N Geoid m Diffeence m CFP 145R CFP 150R CFP CFP CFP CFP CFP CFP CFP CFP GCS GCS GCS s dev Table 1 Ageemen of geoid model heighs and GNSS/Tigonomeic-levelling

92 Name Tigonomeic heigh m NGNSS / Tigonome ic Levelling m N Geoid m Diffeence m CFP 145R CFP 150R CFP CFP CFP CFP CFP CFP CFP CFP GCS GCS GCS s dev Table 2 Ageemen of geoid model heighs using ±10 mgal cieion and GNSS/Tigonomeiclevelling

93 Figue 4-19 Locaion of igonomeic levelling saions Souce of map: Google Figue 4-20 Enlagemen of he locaion of he igonomeic saions Souce of map: Google