NORMAL (GRAVIMETRIC) HEIGHTS VERSUS ORTHOMETRIC HEIGHTS

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1 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: ORL (GRVIETRI) EIGTS VERSUS ORTOETRI EIGTS Dr. bdlrhim Elizouli ohmd hmd* bstrct: Sinc th xistin odtic nd lvllin ntworks don t comply with th rquird ccurcy for th ntionl ntwork covrin th r of Sudn, prcis nd consistnt uniform odtic nd lvlin control ntwork ws stblishd binnin from lf town to Kjbr t th northrn bordr of Sudn towrds Shrik, Sblok nd Uppr tbr t th Southrn -Est of Sudn. This lvlin lin is stblishd for dms construction (Kjbr-Dl, Shrik, Sblo nd Stit-tbr Dms) nd irritions projct. sd on th omtric hiht diffrncs nd rvimtric obsrvtions procssin of th control points, norml hihts wr computd. onsistncy of th computd norml hihts with th xistin ons for th documntd xistnt lvllin bnchmrks which hv bn tid ws chckd. Du to som diffrncs btwn th computd norml hihts nd th xistin lvtions, nw hiht rfrnc systm ws dfind nd rlizd. This nw systm is norml hiht systm, with GRS8 s th rfrnc llipsoid. Tbls prsntd dscrib th rsults obtind nd th chivd ccurcy. omprison btwn norml hihts nd orthomtric hihts wr shown s th min purpos of this ppr. Kywords: Orthomtric hihts, norml hihts, GRS8, omtric hiht, opotntil. *ssocit Prof., Dp. of ivil En., Krry Univrsity, Sudn Vol. o. ovmbr 3 IJRES 68

2 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: ITRODUTIO Th intrst of rvimtric obsrvtions on lvllin ntwork is tht thy nbl to comput opotntil diffrncs btwn th rvimtric points. nd whrs th omtric hiht diffrnc btwn two points my dpnd on th lvllin wy, th opotntil diffrnc btwn ths points is uniqu. So it is prfrbl to mk th djustmnt of lvllin ntwork with opotntil diffrncs rthr thn with th rw omtric hiht diffrncs. Th common concpt of ltituds corrspond to wll dfind physicl quntity. Th Erth's rvity potntil V. indd, physiclly horizontl surfcs r surfc wr th rvity potntil is constnt. nd if n objct ( or wtr) is loctd t point with opotntil numbr V, it will im t fllin (or flowin) towrds plcs opotntil is lss thn V. Th opotntil diffrnc btwn two points nd is thorticlly dfind by V V. ds, Whr ds is th lmntry displcmnt lon th wy from to nd is th rvittionl cclrtion lon this wy. In prctic, th opotntil diffrnc btwn two rvimtric nd lvld points cn obtind by: V V h () Whr nd r th rvity vlus on ths points nd h is th omtricl hiht diffrnc msurd by lvlin [3]. On ch lvld zon in Sudn (Uppr tbr, Shrik, Kjbr nd Sblok), opotntil diffrncs wr thus computd btwn djcnt rvimtric point usin qution (). s th thr lvlin ntworks includ loops, ths rw opotntil diffrncs hd to b djustd. W thn obtind djust opotntil diffrncs btwn djcnt rvimtric points of thr ntworks.ths djustd opotntil diffrncs r th most suitbl quntitis to dscrib th "physicl rlity". owvr, thy cnnot b usd s thy r for civil ninrin or for nti onl hiht rfrnc systm. Indd, thy must b rltd to rfrnc point so tht hihts cn b computd (nd not hiht diffrncs). orovr, thir SI unit is m / s whrs hihts r xpctd to b xprssd in mtrs. For ths two rsons, hiht rfrnc systm hd to b dfind. Th intrst of rvimtric obsrvtions on lvllin ntwork is tht Vol. o. ovmbr 3 IJRES 69

3 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: thy nbl to comput opotntil diffrncs btwn th rvimtric points. nd whrs th omtric hiht diffrnc btwn two points my dpnd on th lvllin wy, th opotntil diffrnc btwn ths points is uniqu. So it is prfrbl to mk th djustmnt of lvllin ntwork with opotntil diffrncs rthr thn with th rw omtric hiht diffrncs [4].. ORTOETRI EIGTS Lt b th projction of point on th oid lon th rvity fild lin which crosss. In th cs of orthomtric hihts, th thorticl vlu for * is th mn vlu of lon th fild lin ~ m. ds. Fi. (): Difintion of orthomtric hihts Th orthomtric hiht of point is thus O m ~ O () It is th lnth hiht of th lin of forc which links to th oid. This dfinition shows tht orthomtric hihts lso hv physicl nd omtricl mnin, vn if thy r not quivlnt to th rvittionl potntil. owvr, thr is no wy to comput n xct orthomtric hiht. Indd, to dtrmin th mn rvity vlu ~ lon th fild lin O, on should know th rvity vlu vrywhr on this lin, which is impossibl. In prctic, is supposd to vry linrly lon th fild lin, so tht it cn b xprssd s: Vol. o. ovmbr 3 IJRES 7

4 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: O ~, whr is th mn rvity rdint lon th fild lin OY O. Th orthomtric hiht of point cn now b computd by: O OY OY (3) ut onc in, thr is no wy to comput th xct mn rvity rdint OY unlss w dispos of DT nd of th dnsity of th trrin. Usully, this rvity rdint is thus st to constnt: OY = -, s -, th so clld ( Poincr Pry rdint) [5]. So vn if orthomtric hihts thorticlly hv physicl mnin, thr is no wy to comput thm xctly. pproximtions hv to b don so tht computd orthomtric hihts do not rflct ny physicl rlity nymor. 3. ORL EGTS In th cs of norml hihts, * is not rfrrd to th rl rvity fild (lik for orthomtric hihts), but to thorticl rvity fild, clld norml rvity fild nd dfind s follows. ) Th norml rvity fild Th norml rvity fild is modl of th Erth s rvity fild such s: i) On of this quipotntil surfcs is odtic llipsoid (for xmpl GRS8). ii) Th norml potntil on this llipsoid quls th rl potntils on th oid. iii) This llipsoid rotts t th sm rt s th rth. iv) This llipsoid hs th sm mss s th rth + th tmosphr. Th rfrnc llipsoid GRS8 cn b dfind by four prmtrs: i) Its hlf mjor xis = m. ii) Its dynmic form fctor J = iii) Its rottionl rt ω = rd / s. iv) Th rvittionl constnt G = m 3 / s. From ths four fundmntl constnts, othr prmtrs cn b drivd: Th first xcntricity, th scond xcntricity nd th prmtr q which cn b obtind by pplyin th followin formuls: Vol. o. ovmbr 3 IJRES 7

5 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: o q 3 rctn 3 o 3J 5 G 3 q 3 o th omtricl flttnin f = th hlf minor xis b = (-f). m b G q o 3 rctn th norml rvity t th qutor E G b m m q 6q o th norml rvity t th pols P G m q 3q o t point t ltitud rvity thnks to Somilin, s formul: on th rfrnc llipsoid E, w cn now comput th norml E cos cos b b P sin sin (4) b) Dfinision of norml hiht Lt us dfin sphropotntil surfc s n quipotntil surfc of th norml rvity fild. ow, lt Q b th projction of on th sphropotntil surfc with norml potntil V, nd lt Q b th projction of on th) Dfinition of norml hihts rfrnc llipspod. Vol. o. ovmbr 3 IJRES 7

6 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: Fi. (): Dfinition of norml hihts In th cs of norml hihts, * is th mn norml rvity vlu ~ lon th fild lin Q Q, so tht th norml hiht of point is th lnth of this fild lin: ~ QQ c) omputtion of norml hihts In opposition to ~, th mn norml rvity vlu ~ is thorticl quntity nd cn thus b xctly computd by th followin formul: ~ f m f sin, whr is th ltitud of point. y rplcin ~ by its xprssion in th dfinition of norml hiht, on cn obtin n xct formul for th computtion of norml hihts: f m f sin (5) So, norml hiht do not rfr to physicl rlity sinc thy rprsnt th lnth of thorticl (norml) lin of forc. ut thir first dvnt ovr orthomtric hihts is tht thy cn b computd xctly. 4. OPUTTIO OF ORL D ORTOETRI EIGTS Th opotntil diffrncs hd bn computd nd djustd prviously btwn djcnt rvimtric points of Kjbr, Shrik, Sblok nd Uppr tbr lvllin ntworks. Th nxt stp ws to dfin hiht rfrnc systm for ch of ths four zons nd to Vol. o. ovmbr 3 IJRES 73

7 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: comput th rvimtric points hihts. In fct to stisfy Sudn ntionl dmnd, two hiht rfrnc systms wr dfind for ch zon, on with norml hihts nd th othr with orthomtric hihts. ow lt us xmin how ths systms wr dfind nd how th rvimtric points hihts wr computd. 4. ORL EIGTS SYSTES First of ll, in ch of th four zons, point (for which n old ( mn s lvl hiht from lxndri) hiht ws vilbl) ws chosn s th rfrnc point for th nw norml hiht systm. Its norml hiht in th nw systm ws st qul to its hiht in th old systm: LEX This rfrnc point ws KD in Kjbr (with.8596m of th sm norml nd orthomtric hiht), S in Shrik (with m of th sm norml nd orthomtric hiht), R6 in Sblok (with 4.69m of th sm norml nd orthomtric hiht)nd R7 in Uppr tbr (with m of th sm norml nd orthomtric hiht). Th opotntil numbr of this point in th nw systm ws computd usin th followin formul: f m f sin invrsof Eqution 5 For th rvimtric points, odtic coordints in th ITRF5 rfrnc frm wr usd. Thn, usin th djustd opotntil diffrncs, opotntil numbr Ws ssind to ch rvimtric point of th ntworks. Finlly, ths opotntil numbrs wr trnsformd into norml hihts usin qution (5). 4. ORTOETRI EIGTS SYSTES For th orthomtric hihts systms, th sm rfrnc points wr usd. This tim, thir orthomtric hihts in th nw systm wr st qul to thir hihts in th old systm: O LEX Th opotntil numbrs of ths points in th nw systm wr computd usin th followin formul: Vol. o. ovmbr 3 IJRES 74

8 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: O O OY (invrs of qution (3) nd th Poincr Pry rdint). Thn, usin th djustd opotntil diffrncs, opotntil numbr ws ssind to ch rvimtric point of th ntworks. Finlly, ths opotntil numbrs wr trnsformd into orthomtric hihts usin qution (3). 5. RESULTS D LYSIS Tbls, 3, 4 nd 5 bllow show th points odtic coordints with diffrnt kinds of hihts (llipsoid, norml nd orthomtric hihts with th diffrnc btwn norml nd orthomtric hiht of ch point). Ech tbl rprsnt on of th four zons (Kjbr-Dl, Shrik, Sblok, nd Uppr tbr) ovr th r of Sudn. Tbl (), sums up diffrncs btwn th computd norml nd orthomtric hihts. It contins th bist norml orthomtric diffrncs for ch lvlin zon. Ths diffrncs r xprssd in millimtrs. Tbl (): x. norml hiht orthomtric hiht diffrncs for ch lvlin zon Lvllin zon Kjbr-Dl Shrik Sblok Uppr tbr ximum diffrnc 3.6 mm 5.4 mm 7.5 mm.4 mm ommnts Thr is no mnin to th positiv or ntiv sin hr (it is th mttr of diffrnc only). Th ovrll mximl diffrnc is.4 mm, showin tht thr is no such n importnt diffrnc btwn th two kinds of hihts. hoosin on or nothr should not ffct ninrin works. ut it will hv crtin influnc t th scl of ntionl lvlin ntwork. 6. OLUSIOS Thorticlly, orthomtric hihts hv physicl mnin (lnth of lin of forc of th rl rvity fild ). ut in prctic, thy cn b computd only with pproximt formuls, so tht thy do not rflct ny physicl mnin ny mor. (In rlity, dos not vry linrly lon th lin of forc nd th rvity rdint is nithr constnt nor qul to th Poincr-Pry rdint). orml hihts hv no physicl mnin sinc thy rprsnt th lnth of lin of forc of th norml rvity fild. ut thir first dvnt ovr Vol. o. ovmbr 3 IJRES 75

9 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: orthomtric hihts is tht thy cn b computd xctly. Evntully, it my b prfrbl to us norml hihts xctly thn orthomtric hihts computd with pproximtions. 7. KOWLEGET ost of th dt incorportd in this ppr wr compild from Sudn Dms Implmnttion Unit (DIU). ccordinly, I wish to thnk mmbrs of Survy Dprtmnt of DIU. 8. REFEREES [] Duqunn,. ltituds, lvllin nd ltimtric rfrnc systms ours book for str PPD Jnury 5. [] Sudn Dms Implmnttion Unit rfrnc systm : dfinition nd rliztion V.4 uust 7. [3] Sudn Survy ontrol ntwork nd diitl ril photorphy nd orthophoto mppin projct -bsolut rvimtry v. Sptmbr 7. [4] Sudn Survy ontrol ntwork nd diitl ril photorphy nd orthophoto mppin projct -Rltiv rvimtry Rhd nd Knn rs - v. pril 8. [5] Wikko. isknn nd lmut oritz (98). Physicl Godsy. Institut of Physicl Godsy, Tchnicl Univrsity, Grs ustri. Tbl (): Orthomtric nd rvimtric norml hiht of Kjbr-Dl Ltitud orth Lonitud Est Ellipsoid hiht (m) Elvtion (m) Point o. D. in. Sc. D. in. Sc. orml hiht Orthomtric hiht Diffrncs (m) (orml Orthomtric) K K K K K K K K K K Th mximum (norml hiht orthomtric hiht) diffrnc 3.6 millimtrs Vol. o. ovmbr 3 IJRES 76

10 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: Tbl (3): Orthomtric nd rvimtric norml hiht of Shrik Ltitud orth Lonitud Est Ellipsoid hiht (m) Elvtion (m) Diffrncs (m) Point o. D. in. Sc. D. in. Sc. orml hiht Orthomtric (orml hiht Orthomtric) S S S S S S S S S S Th mximum (norml hiht orthomtric hiht) diffrnc 5.4 millimtrs Tbl (4): Orthomtric nd rvimtric norml hiht of Sblok Ltitud orth Lonitud Est Ellipsoid hiht (m) Elvtion (m) Diffrncs (m) D. in. Sc. D. in. Sc. orml hiht Orthomtric (orml hiht Orthomtric) Th mximum (norml hiht orthomtric hiht) diffrnc 7.5 millimtrs Vol. o. ovmbr 3 IJRES 77

11 Intrntionl Journl of dvncd Rsrch in Eninrin nd pplid Scincs ISS: Tbl(5): Orthomtric nd rvimtric norml hiht of Uppr-tbr Ltitud orth Lonitud Est Ellipsoid hiht (m) Elvtion (m) Diffrncs (m) Point o. D. in. Sc. D. in. Sc. orml hiht Orthomtric (orml hiht Orthomtric) Th mximum (norml hiht orthomtric hiht) diffrnc.4 millimtrs Vol. o. ovmbr 3 IJRES 78