GEOID IN THE WEST UKRAINE AREA DERIVED BY MEANS OF NON-CENTRAL MULTIPOLE ANALYSIS TECHNIQUE

Transcription

1 GEOID IN THE WEST UKRAINE AREA DERIVED BY MEANS OF NON-CENTRAL MULTIPOLE ANALYSIS TECHNIQUE INTRODUCTION Alexader Marcheko ad Oleg Abrkosov Faculty of Geodesy State Uversty "Lvv Polytechc" SBadera St Lvv Ukrae Amog the ma tasks of the West Ukrae Geod Project (WUG s the costructo of the aalytcal models of regoal ad local gravty feld for fast ad precse geod computatos As t follows from the prelmary studes (Marcheko 987a 987b; Marcheko et al 995) such models may be costructed effcetly by meas of gravty feld approxmato by a sets of o-cetral radal multpoles the frames of ether the prelmary multpole aalyss (PMA) techque (Marcheko 987b) or the sequetal multpole aalyss (SMA) techque (Marcheko et al 995) I the preset study whch reflects the part of vestgatos cocerg wth the WUGP such approach was developed for the gravmetrc geod costructo ad verfed o the gravty data processg the West Ukrae ad border areas THEORY SKETCH At a exteral pot P the aomalous gravty potetal T may be represeted as K m T( v( () r ( ) P where v ( s the dmesoless potetal of the -th degree radal multpole located at the pot the Bjerhammar sphere teror; m s the momet of the multpole; r P s the geocetrc dstace of the pot P I ths expresso we assume that each multpole has ts ow degree = O the groud of the cetral represetato () we ca derve represetatos for geod udulatos K ( ) m N( v ( () r ad for gravty aomales P P K m g( g( (3) r ( ) P as well as for ay geodetc fuctoal (Mortz 980) For the base fuctos seres (3) we get v ( g( s ( ) v( s where s s the relatve geocetrc dstace of the multpole (4) Please cte ths chapter as: Marcheko AN Abrkosov OA (995) Geod the West Ukrae Area Derved by Meas of No-Cetral Multpole Aalyss Techque I: H Sükel I Marso (eds) Gravty ad Geod: Jot Symposum of the Iteratoal Gravty Commsso ad the Iteratoal Geod Commsso Symposum No 3 pp Graz Austra September 994 Sprger ( DOI: 0007/ _66

2 Values of v ( ad thers dervatves regardg s may be foud from the followg recursve formulae (where P s the geocetrc sphercal dstace betwee pots ad : q v v 0 ( ) (cos q P s ) v ( ) v (5) q v s v s 0 ( ) (cos (cos cos q P 3 s P s ) v P v s ) s ( ) v v ( ) s (6) q s s cos (7) P Both of the PMA ad SMA techques use the sequetal approxmato of gravty data Correspodg computatoal procedure (SMA algorthm) cossts of the followg steps: Put =; Fd major absolute value of approxmatg gravty feld; ths value s chose further as the epceter of -th multpole; 3 Determe multpole's parameters s ad m ; 4 Remove the cotrbuto of -th multpole from the gravty feld; 5 If desrable accuracy of approxmato s acheved the computatos are stopped; else put =+ ad repeat all tems from the pot Relatve dstace s degree ad the momet m of -th multpole may be obtaed from the aalyss of the emprcal sotropc fucto features (EIF) (Marcheko 987b Marcheko et al 995) EIF s the dscrete fucto costructed by meas of data averagg (after ormalzato by the extreme value) wth the reasoable small sphercal cup aroud the extremum Local parameters of such fucto were defed by Marcheko 987b by aalogy to the essetal parameters of covarace fuctos (Mortz 980) I the cases of above cosdered base fuctos v ( ad g ( we get the followg expressos for metoed local parameters Magtude at the epceter v ( P 0) (8) s Please cte ths chapter as: Marcheko AN Abrkosov OA (995) Geod the West Ukrae Area Derved by Meas of No-Cetral Multpole Aalyss Techque I: H Sükel I Marso (eds) Gravty ad Geod: Jot Symposum of the Iteratoal Gravty Commsso ad the Iteratoal Geod Commsso Symposum No 3 pp Graz Austra September 994 Sprger ( DOI: 0007/ _66 s g ( P 0) (9) ( s )

3 3 Legth of decreasg e such value of sphercal dstace P o whch the magtude becomes prmarly equal to half of magtude at the epceter ad whch thus satsfes to codto f ( P ) f ( P 0) (0) where f s a term of a geodetc fuctoal multpole expaso (for stace () or () or (3)) The relatoshp (0) s o-lear equato wth respect to s It s evdetly o the groud of () - (3) ad (9) - (0) that ths equato does't cotas multpole's momet m ad thus allows determe s depedetly from m Curvature at the pot P =0 v P 0 P ( ) ( s ) 3 ( s ) () g P P 0 ( s ) 3 3 s ( s ) s () The procedure for determato of the cosderg multpole parameters cossts of the followg steps: Put =0; For fxed degree we determe value s o the bass of the legth of decreasg of EIF; 3 For fxed degree ad obtaed value s we determe multpole's momet m ether o the bass of magtude at the epceter (or curvature f t s eeded) of EIF or by meas of the least-squares approxmato of the gravty feld; 4 Put =+ ad repeat above metoed steps from the tem ; as a result we ca establsh the optmal degree of the multpole as such degree for whch best fttg ether of EIF or of gravty feld approxmato s reached Ths procedure s the most effcet realzato of the pot 3 of above metoed SMA algorthm It should be oted that for the better optmzato of costructg gravty model we ca apply fally total least-squares adjustmet to the whole set { m } Also we ca use such adjustmet perodcally durg SMA approxmato for some subsets of { m } RESULTS Developed theory was examed by costructo of gravmetrc geod for the West Ukrae as well as for the border areas Gravty data were aalyzed the frame of the CERGOP area whch cotas the areas of Hugary Polad (cotetal part together wth the small part of Baltc Sea) Czech ad Slovaka (merged data set) West Ukrae ad remaed part of Ukrae Most of these data are averaged by reasoable small Please cte ths chapter as: Marcheko AN Abrkosov OA (995) Geod the West Ukrae Area Derved by Meas of No-Cetral Multpole Aalyss Techque I: H Sükel I Marso (eds) Gravty ad Geod: Jot Symposum of the Iteratoal Gravty Commsso ad the Iteratoal Geod Commsso Symposum No 3 pp Graz Austra September 994 Sprger ( DOI: 0007/ _66

4 4 blocks (5'75' or 5'5') except the data Cetral ad Easter parts of Ukrae (see Table ) For geod determato the "remove-restore" procedure was realzed the followg way: The cotrbuto of the global gravty model wth approprate resoluto was removed from the tal gravty aomales For ths reaso GEM-T (3636) OSU8 (8080) ad OSU9A (360360) Earth gravty models were used The resdual gravty aomales were approxmated by the set of radal multpoles by meas of SMA techque descrbed the prevous secto We used sequetal approxmato oly wthout ay jot adjustmet of multpole momets Moreover we have used multpoles wth degrees >0 oly (e startg from radal dpoles) due to the well-kow approxmato's requremet of the physcal geodesy (Mortz 980) Thus all obtaed sets of radal multpoles do ot cota zero degree multpoles (pot masses) The rms of ft value mgal was the crtero of approxmato process completo because the tal gravty data accuracy was ear mgal Geodal udulatos were restored o the bass of combed gravty model cossts of both the global model harmocal coeffcets ad the set of o-cetral radal multpoles Computatos of the geod were made up for whole metoed CERGOP area as well as separately for each coutry For each terrtory the results obtaed from total data set ad from the dvdual data set were good agreemet Gravty feld approxmato results obtaed wth usg varous global gravty models were practcally detcal Besdes geod results deped o the global gravty model resoluto sgfcatly wth respect to ther accuracy although the cotrbutos of the above metoed gravty models to tred removal were essetally dfferet For llustrato see Table cossts of correspodg results for whole aalyzed data set) Thus the SMA techque allows use such (satellte derved) Earth gravty models as GEM-T to tred removal the frames of regoal gravmetrc geod costructo Table Characterstcs of the aalyzed data sets Coutry Number of data Sze of blocks Statstcs (mgal) m max mea Hugary 47 50'75' Czech ad Slovaka 40 50'75' Polad (terrestral area) '50' Polad (mare area) 45 50'50' West Ukrae '75' Ukrae 980 0'30' Please cte ths chapter as: Marcheko AN Abrkosov OA (995) Geod the West Ukrae Area Derved by Meas of No-Cetral Multpole Aalyss Techque I: H Sükel I Marso (eds) Gravty ad Geod: Jot Symposum of the Iteratoal Gravty Commsso ad the Iteratoal Geod Commsso Symposum No 3 pp Graz Austra September 994 Sprger ( DOI: 0007/ _66

5 5 Table Rms of dffereces betwee geods based o varous global gravty models Global gravty models Before SMA After SMA (OSU9A) - (OSU8) 06 m 09 m (OSU8) - (GEMT) 09 m 06 m (OSU9A) - (GEMT) 3 m 05 m Fg Geod soluto for the West Ukrae area refered to the GRS80 ellpsod (cotour terval: 0 m) I our opo obtaed fal geod soluto s more precse the West Ukrae lads because ths area s surrouded by the gravmetrc data ear the cetral part of the aalyzed Europea terrtory Geod soluto for the West Ukrae s show the Fg Please cte ths chapter as: Marcheko AN Abrkosov OA (995) Geod the West Ukrae Area Derved by Meas of No-Cetral Multpole Aalyss Techque I: H Sükel I Marso (eds) Gravty ad Geod: Jot Symposum of the Iteratoal Gravty Commsso ad the Iteratoal Geod Commsso Symposum No 3 pp Graz Austra September 994 Sprger ( DOI: 0007/ _66

6 6 CONCLUSIONS Thus the sequetal multpole aalyss techque was developed for a gravmetrc geod costructo regoal ad local scales By meas of ths approach we ca costruct a aalytcal model for the regoal gravty feld as the combato of the global gravty model ad the set of radal multpoles The gravty aomales the Cetral ad Easter Europe were approxmated wth the declared accuracy mgal ad the gravmetrc geod was costructed the followg step The total accuracy of the obtaed results ad thers realstc depedece o the resoluto of the global gravty models appled to tred removg allows cocludg that the SMA techque should be recommeded for fast geod determatos Therefore we ca coclude that the costructo of the global gravty model the form of the set of radal multpoles s ecessary wth respect to the buldg of mathematcally homogeeous models of the regoal ad local gravty feld REFERENCES Marcheko AN (987a) Descrpto of the Earth's gravty feld by the system of potetals of o-cetral multpoles I Theoretcal backgrouds Kematcs ad Physcs of Celestal Bodes 3 No 54-6 Marcheko AN (987b) Descrpto of the Earth's gravty feld by the system of potetals of o-cetral multpoles II Prelmary multpole aalyss Kematcs ad Physcs of Celestal Bodes 3 No Marcheko AN Abrkosov OA Romash PO (995) Improvemet of the gravmetrc geod the Ukrae area usg absolute gravty data I: Vermeer M (ed) Proceedgs of the Sesso G4 Latest Developmets the Computato of Regoal Geods XX Geeral Assembly EGS Reports of the Fsh Geodetc Isttute 95:7 pp 9- MASALA 995 pp 9- Mortz H (980) Advaced Physcal Geodesy SLWchma Please cte ths chapter as: Marcheko AN Abrkosov OA (995) Geod the West Ukrae Area Derved by Meas of No-Cetral Multpole Aalyss Techque I: H Sükel I Marso (eds) Gravty ad Geod: Jot Symposum of the Iteratoal Gravty Commsso ad the Iteratoal Geod Commsso Symposum No 3 pp Graz Austra September 994 Sprger ( DOI: 0007/ _66