Odvodenie niektorých geometrických veličín z GPS meraní
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- Mabel Gallagher
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1 Acta Montanstca Slovaca Ročník 10 (2005), číslo 3, Odvodene nektorých geometrckých velčín z GPS meraní Adel Alfrehat 1, Janka Sabová a Marcel Mozeš 2 Dervaton of some geometrc parameters from GPS measurements Combnng GPS and terrestral data requres a common coordnate system. When the orgnal GPS vectors do not form a network, the 3D network adustment can not be performed. In ths case, n order to ntegrate the GPS measurements wth the terrestral observatons and to perform a combned network adustment, the GPS measurements should be transformed to ths common system. he GPS measurements whch are the usual output of the GPS post processng softwares are based on the WGS84 ellpsod and the S-JSK local datum s based on the Bessel ellpsod. hus, the reducton of measurements to the S-JSK mappng plane can not be started from the measurements resultng from GPS post processng softwares because GPS and S-JSK don t have the same ellpsod. Another vew of ths reducton wll be descrbed n ths paper. Key word: Datum transformaton, reducton of GPS measurements to S-JSK, stochastc models of the derved measurements. Výpočet škmých dĺžok z GPS meraní s ohľadom na Besselov elpsod Prvým krokom pre určene dĺžok z GPS meraní v S-JSK rovne e vykonane GPS meraní na plánovaných bodoch, kde sú potrebné dĺžky. Ďalším krokom e spracovane GPS dát vhodným softvérom. V prípade, že GPS vektory tvora seť, nasledue vyrovnane a na základe dentckých bodov zvolených v oblast GPS merana určene transformačných parametrov medz GPS a S-JSK dátumam. Kartezánsky súradncový systém všetkých bodov GPS sete [1] e určovaný Molodenského Badekasovým transformačným modelom. 1 ω -ω Z Y Y = Y + Y + (1 + δs) - ω 1 ω Y Y Z Z Z Z ω -ω 1 Z Z S JSK WGS 84 WGS 84 WGS 84 0 Y 0 (1) ypcké transformačné parametre sú tr posunuta (, Y, Z ), tr rotáce ( ω, ω, ω ) a eden Y Z merkový faktor ( δ S ). Súradnce ťažskového bodu medzí dentckým bodm sú, Y, Z. Škmá dĺžka S (1): medz dvoma GPS zameraným bodm a sa dá vypočítať s použtím daných súradníc bodov z rovnce ( - 2 ) ( - 2 ) ( - ) S = + Y Y + Z Z 2 (2) Redukca škmých dĺžok na dĺžky geodetckých čar na Besselovom elpsode Dĺžku tetvy kruhového oblúka na zvolene sfércke ploche e daná podľa [2] vzřahom t = S - h h h Rm Rm (3) 1 Ing. Adel Alfrehat,PhD, Doc. Dr. Ing. Janka Sabová,Dept. of Geodesy and GIS, echncal Unversty.of Košce, Letná 9, Slovaka. randogps84@hotmal.com, Janka.Sabova@tuke.sk 2 Doc. Ing. Marcel Mozeš, PhD, Dept. of theortcal Geodesy, Unv. Of Bratslava, Radlnskeho Bratslava, Slovaka marcel.mozes@stuba.sk. (Recenzovaná a revdovaná verza dodaná ) 310
2 Acta Montanstca Slovaca Ročník 10 (2005), číslo 3, prčom t e dĺžka tetvy kruhového oblúka, S e škmá dĺžka vypočítaná z rovnce (2), h e elpsodcké prevýšene dvoch koncových bodov, transformácou a R m h a h sú elpsodcké výšky koncových bodov získaných e stredný polomer krvost aproxmuúce guľove plochy, ktorého hodnota pre strednú o geodetckú šírku φ = územa SR e R = 6380,076 m m [2]. Dĺžka kruhového oblúka na Besselovom elpsode sa dá vypočítať podľa [2] z dĺžky tetvy t t = 2R arcsn m 2 Rm (4) Elpsodckú dĺžku merkového koefcentu podľa [2, 3] Redukca elpsodcke dĺžky do S-JSK rovny t e ďale potrebné redukovať do S-JSK rovny na t s použtím premerného k + 4k + k m t t 6 (5) kde k a k sú merkové koefcenty koncových bodov dĺžky a k e merkový koefcent stredu dĺžky. m Hodnoty merkových koefcentov pre Křovákovo zobrazene podľa [2, 3, 6] udáva vzťah = k = 0, , R 3, R + 1, R 1, R pre bod bod a dĺžku s použtím rovnce (6) platí: (6) = + R Y R = + R Y R Y + Y R = + R0 2 2 (9) kde R = e polomer Gaussove gule,,, Y a Y sú S-JSK rovnné súradnce koncových 0 bodov získané transformácou. Z nch e možné pramo vypočítať dĺžku v S-JSK rovne podľa ( - ) ( - ) t = + Y Y (10) akto určené dľžky z GPS meraní v S-JSK rovne e možné kombnovať s meraným velčnam získaným z terestrckého merana a prevedeným do kartografcke rovny. Presnosť S-JSK dĺžok určených z GPS meraní môže byť vypočítaná na základe zákona šírena chýb z kovarančných matíc každého súradncového typu ( ),, ( YZ,, ) ( YZ,, ) S YZ S σ = (11) S (,, Y, Y ) ( YZ,, ) ( YZ,, ) ( YZ,,) (,, Y, Y ) Kovarančná matca transformovaných elpsodckých súradníc e daná podľa [5] ( ϕ, λ, h) ( YZ,, ) ( YZ,, ) ( YZ,, ) ( ϕ, λ, h) = ( ϕλ,, h) (, Y, Z) ( YZ,, ) ( YZ,, ) ( YZ,, ) (, Y, Z) Kovarančná matca S-JSK rovnných súradníc (7) (8) 311
3 Adel Alfrehat,Janka Sabová a Marcel Mozeš: Odvodene nektorých geometrckých velčín z GPS meraní = ( Y, ) ( ϕλ, ) (13) kde podľa [7] e vyadrená vzťahom (, Y) ( R, D ) ( D, Š) ( U, V) = ( RD, ) ( DŠ, ) ( UV, ) ( ϕλ, ) (14) Presnosť S-JSK dĺžok bude nakonec vypočítaná podľa ( ) ( Y, ) ( Y, ) ( Y, ) ( ) t ( Y, ) ( Y, ) ( Y, ) t σ t =,, Y, Y,, Y, Y (15) Určene 2D súradncových rozdelov v S-JSK rovne z GPS meraní Súradncové rozdely a Y dĺžky v S-JSK sa daú vypočítať z S-JSK rovnných súradníc koncových bodov získaných transformácou za účelom ch ntegráce so získaným terestrckým meranam a spoločného vyrovnana sete v rovne S-JSK. eto súradncové rozdely sú = Y = Y Y kde,, Y, Y sú S-JSK rovnné súradnce koncových bodov dĺžky IJ získané transformácou. Kovarančnú matcu 2D súradncových rozdelov e možné vypočítať na základe zákona šírena varancí podľa [5] ( ) Y, ( Y, ) ( Y, ) (, ) = Y (, ) (, ) (, ) Y Y Y (17) Určene elpsodckých azmutov, zentových uhlov a 3D súradncových rozdelov v S-JSK z GPS meraní Elpsodcký azmut α podľa [8] - snλ + Y cosλ α = arctg sn ϕ cos λ Y sn ϕ sn λ + Z cosϕ kde,, Y, a Z sú rozdely súradníc koncových bodov dĺžky a ϕ, λ sú elpsodcké súradnce bodu získané z transformáce. Zentový uhol môže byť vypočítaný zo súradncových rozdelov zskaných transformácou a z elpsodcke šírky a dĺžky bodu [8, 9] (16) (18) cosϕ cos λ + Y cosϕsn λ + Z sn ϕ Ζ = arccos 2 + Y + Z (19) 3D súradncové rozdely, Y, a Z vektora sa daú vypočítať z kartezánskych súradníc koncových bodov získaných transformácou. eto súradncové rozdely sú vyadrené vzťahm = Y = Y Y Z = Z Z Kovarančnú matcu 3D súradncových rozdelov môžeme vypočítať na základe zákona šírena chýb [5] (20) 312
4 Acta Montanstca Slovaca Ročník 10 (2005), číslo 3, (, Y, Z ) = prčom ( YZ,, ) ( YZ,, ) ( YZ,, ) ( YZ,, ) ( YZ,, ) ( YZ,, ) = Škmá dĺžka vektora sa dá tež vypočítať z kartezánskych súradncových rozdelov získaných transformácou podľa (21) S Y = + + Z 2 Pr zvážení rovníc (18), (19) a (20), spoločná kovarančná matca pre (, α, a vypočítaná na základe zákona šírena chýb [5] ( ) = S,, (,, ) α Ζ Y Z kde S S S Y Z α α α = Y Z Ζ Ζ Ζ Y Z S Ζ (22) ) može byť (23) Výpočet 3D súradncových rozdelov v lokálnom geodetckom súradncovom systéme Kartezánske súradncové rozdely získané transformácou sa daú redukovať do lokálneho súradncového systému (obr.1) pomocou rovníc [10] n = sn ϕ cos λ Y sn ϕ sn λ + Z cosϕ e = sn λ + Y cos λ (25) u = cosϕ cos λ + Y cos sn sn ϕ λ + Z ϕ (26) kde, Y, a Z sú súradncové rozdely vypočítané z rovnce (20). Alternatívny výpočet e možný použtím, α a S Ζ n = S sn Ζ e = S sn Ζ u = S cos Ζ cosα sn α Kovarančná matca lokálnych súradncových rozdelov sa uskutoční podľa ( ) = n, e, u (,, ) Y Z (24) (27) (28) 313
5 Adel Alfrehat,Janka Sabová a Marcel Mozeš: Odvodene nektorých geometrckých velčín z GPS meraní kde sn ϕ cos λ sn ϕ sn λ cosϕ = sn λ cos λ 0 cosϕ cos λ cosϕ sn λ sn ϕ Výhoda lokálneho súradncového systému e v tom, že všetky 3D rozmery môžu byť vypočítané podobne ako v rovnc (27), (28) a (29) z meraných velčín totálnym stancam. eto lokálne systémy môžu byť kombnované s meranam GPS technológou za účelom spoločného vyrovnana sete metódou namenších štvorcov. e u n Obr. 1. Lokálny topocentrcký systém g. 1. Local topocentrc system Kartezánske súradncové rozdely môžu byť odvodené z lokálnych súradncových rozdelov získaných totálnym stancam a spoločné vyrovnane vykonané v kartezánskom rámc. eto súradncové rozdely sa daú vypočítať nasledovne: sn ϕ cos λ sn λ cosϕ cos λ n Y = sn ϕ sn λ cos λ cosϕ sn λ e Z cosϕ 0 sn ϕ u (29) Demonštračný príklad Ako e znázornené na obr. 2. bolo zvolené 4 body (A, B, C a D) zo ŠPS sete ako dentcké body, ktorých súradnce sú dané v ERS-89 a v S-JSK. ab. 1. Kartezánské súradnce bodov sete ab. 1. Cartesan coordnates of network s ponts Bod [m] σ [m] Y [m] σ [m] Y Z [m] σ [m] Z C ,9186 0, ,9094 0, ,3791 0,014 D ,4356 0, ,4463 0, ,0568 0,013 B ,3124 0, ,4424 0, ,8246 0,013 A ,5905 0, ,0165 0, ,7567 0,016 H ,0966 0, ,7319 0, ,8011 0, ,9997 0, ,8242 0, ,1878 0,014 G ,0091 0, ,0241 0, ,9754 0,
6 Acta Montanstca Slovaca Ročník 10 (2005), číslo 3, S použtím postupov transformáce bolo vypočítaných 7 lokálnych transformačných parametrov a podľa nch pretransformované súradnce troch nových bodov,g,h do S-JSK rovny. ransformáca súradníc všetkých bodov sete na základe Molodenského Badekasovho modelu podľa rovnce (1) a presnosť týchto súradníc e znázornená v tab. 1. Obr. 2 Konfguráca GPS sete g. 2. Confguraton of GPS network ab. 2. znázorňue určené škmé dĺžky vypočítané z transformovaných kartezánskych súradníc a redukcu týchto dĺžok do S-JSK rovny spolu s ch presnosťam podľa rovníc (2), (11), (3), (4), (5), (10) a (15). V sedmom stĺpc tab. 2 sú S-JSK dĺžky vypočítané z S-JSK transformovaných rovnných súradníc. Dĺžka ab. 2. určené škmé dĺžky vypočítané z transformovaných kartezánskych súradníc a ch redukce do S-JSK rovny ab. 2. he slope dstances determned from transformed Cartesan coordnates and ther reductons to the S-JSK plane S [m] σ [m] S t [m] t [m] t [m] t [m] σ t [m] A 11498,9835 0, , , , ,3674 0,006 BG 7766,2114 0, , , , ,9267 0,005 DH 7627,8611 0, , , , ,9200 0,005 C 9226,2679 0, , , , ,4883 0,006 CH 6904,3884 0, , , , ,3329 0,005 G 2941,7463 0, , , , ,2523 0,004 ab. 3. obsahue 2D súradncové rozdely nektorých dĺžok vypočítaných z S-JSK rovnných súradníc získaných transformácou a ch kovarančné matce, v zmysle rovníce (16) a (17). ab. 3. 2D súradncové rozdely určené z GPS meraní v S-JSK ab. 3. 2D coordnate dfferences determned from GPSmeasurements n S-JSK Dĺžka, Y [m] σ, σ Y (, Y) A E E E E-05 BG E E E E-05 V tabuľke 4 sú určené 3D kartezánske súradncové rozdely z GPS meraní na Besselovom elpsode a ch kovarančné matce, vď rovnce (20) a (21) ab. 4. 3D súradncové rozdely určené z GPS meraní v S-JSK ab. 4. 3D coordnate dfferences determned from GPS measurements n S-JSK Vektor, Y, Z [m] ( ) [m], Y, Z A 6442,409,8922E-04 9,7798E-05 2,9698E ,1923 9,7798E-05 7,8074E-05 1,1641E ,5689 2,9698E-04 1,1641E-04 4,1082E-04 BG -1761,3033 2,3430E-04 8,2457E-05 2,5769E ,4183 8,2457E-05 5,5351E-05 1,0063E ,1509 2,5769E-04 1,0063E-04 3,3811E-04 V tab. 5. sú určené škmé dĺžky, elpsodcké azmuty, zentové uhly z GPS meraní v S-JSK a ch kovarančné matce, podľa (20), (18), (19) a (23). 315
7 Adel Alfrehat,Janka Sabová a Marcel Mozeš: Odvodene nektorých geometrckých velčín z GPS meraní Vektor S, α, Ζ A BG ab. 5. určené škmé dĺžky, elpsodcké azmuty, zentové uhly z GPS meraní v S-JS ab. 5. Slope dstances, ellpsodal azmuths, zenth angles determned from GPS measurement n S-JSK σ, σ, σ S α Ζ [m] ( S, α, Ζ ) [m] 11498,9835 m 0,0064 4,05E-05-1,97E-10 1,69E ,1227-1,97E-10 3,54E-13 2,67E ,4715 1,69E-10 2,67E-15 5,22E m 0,0048 2,33E-05 1,33E-11-5,28E ,1281 1,33E-11 3,86E-13-1,58E ,6403-5,28E-10-1,58E-16 9,64E-12 Zmeny v lokálnom súradncovom systéme bol vypočítané podľa rovníc (24) až (26) spolu s ch kovarančným matcam vypočítaným podľa (28) a sú uvedené v tab. 6. ab. 6. 3D lokálne topocentrcké súradncové rozdely určené z GPS meraní v S-JSK ab. 6. 3D local topocentrc coordnate dfferences determned from GPS measurement n S-JSK Vektor e, n, u [m] σ e, σ n, σ u [m] ( e, n, u) A BG E E E E E E E E E E E E E E E E E E-04 Záver Určované velčny z GPS meraní vstupuú do spoločného vyrovnana spolu s meraným velčnam terestrckého merana., potom bude časť sete meraná GPS technológou a ďalša terestrckou metódou, a všetky merané velčny budú súčasne vyrovnané (2 D vyrovnane) v spoločnom súradncovom systéme, ktorý e v naše stuác S-JSK, kde musa byť terestrcké merané velčny redukované do tohto spoločného súradncového systému. Určované kvázmerané velčny môžu byť kombnované z velčín získaných terestrckým meranam a 3D vyrovnane metódou namenších štvorcov sa dá vykonať na Besselovom elpsode a výsledok tohto vyrovnana potom redukueme do S-JSK rovny. Lteratúra - References [1] Harvey, B., R.:ransformaton of 3D coordnates. he Australan Surveyor, vol. 33, no. 2, [2] Wess, G. a Šutt, J.: Geodetcké lokálne sete 1. U BERG, Košce, [3] Abelovč, J., et al.: Merane v geodetckých seťach. Alfa, Bratslava [4] Vykutl, J.: K převodu měřených délek do zobrazovací rovny souřadncového systému S-JSK. Geodetcký a kartografcký obzor, 20/62, 1974, 10, [5] Alfrehat, A.: Spracovane GPS meraní v S-JSK a novom zobrazení Slovenska, PhD thess, echncal Unversty of Kosce, Insttute of geodesy and GIS, June [6] Mervart, L. a Cmbálník, M. : Vyšší geodéze 1. ČVU, Praha, [7] Šutt, J.: Accuracy and relablty of plane networks transformed from WGS84 nto S-JSK, Acta Montanstca Slovaca, Ročnk 3 (1998), 4, [8] Hofmann- Wellenhof, B., Lchtenneger, H., a Collns, J.: Global Postonng System, heory and Practce. Wen, New York; Sprnger-Verlag, [9] Gunter, S. : Satellte Geodesy. Walter de Gruyter Inc; 2nd Rev edton. August 1, [10] Leck, A.: GPS Satellte Surveyng. New York, Chcester, oronto, Brsbane, Sngapore: John Wley and Sons Inc,
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