Stability Determination of the Surface Area of the Prague Castle by the Periodically Measured Levelling Network and Robust Analysis

Transcription

1 243 Stablty Determato of the Surface Area of the Prague Castle by the Perodcally Measured Levellg Network ad Robust Aalyss Štroer, M., Urba, R. ad Kubí,. Czech echcal Uversty Prague, Faculty of Cvl Egeerg, hákurova 7, Prague, Czech Republc, Web ste: E-mal: Abstract I the area of Prague Castle there s already about 10 years of perodcal measuremet of the heght chages the buldgs ad structures beg performed. he measuremet method s precse levellg predomatly. Utl ow, these measuremets have bee evaluated oly locally wth respect to each buldg ad ts stablty wthout a overall vew of the stuato of possble movemets of dvdual parts of the surface Prague Castle. Whereas there are heght shft of some pots betwee epochs udoubtedly, a ew ad complete adustmet of each measured epoch ad mutual assessmet of chages betwee epochs usg robust aalyss was coducted. hs comparso shows the relatve movemet of certa parts agast aother. he results are cosstet wth curret kowledge of the geology the area of the Prague Castle. Key words: Robust Aalyss, Deformato Aalyss, Precse Levellg, Prague Castle 1 INRODUCION Prague Castle s oe of the most mportat hstorcal, poltcal ad tourst areas of the Czech Republc, sce 1918 also the seat of the Presdet of the Czech Republc. Accordg to (Prague Castle, 2014), the Prague Castle complex was created by sequetal addtos ad reovatos of the settlemet fouded the 9th cetury. Wth ts dmesos of 570 m legth ad 128 m wdth t s oe of the largest castle the world. It s cosdered to be ot oly symbol of the cty, but also the Czech statehood. Hstorc buldgs located the area are however affected by the agg process ad the effects of chages the surroudgs. I order to predct further developmets ths area, the log term perodc measuremets for determg the stablty of hstorc buldgs the area of the Prague Castle are carred out. Geology the area plays maor role, accordg to (Zálesky, 2001) t was orgally ot complcated, but athropogec actvtes related to structural modfcatos of Hradčay hll durg the last cetures made t cosderably more complcated. he bedrock of the area has bee reworked ad expaded by the flls of dfferet org. he frst measurg was coducted by the Departmet of Specal Geodesy 1999, sce the t s stll ogog ad S 7 Motorg of structures INGEO th Iteratoal Coferece o Egeerg Surveyg Prague, Czech republc, Aprl 3-4, 2014

2 244 INGEO 2014 have bee supported by several grats. he fdgs ad coclusos of the measuremets were summarzed (Procházka, 2011). hese measuremets were tally cocetrated o the fault motorg of dvdual buldgs, ad later coected va a etwork of referece pots for both heght measuremets (precse levellg) ad the posto measuremet. But because of ths o-systematc evoluto of the measuremet there are dffereces betwee epochs cofgurato of the etwork ad of motored pots, accordg to actual demad. So far these measuremets were evaluated oly locally wth respect to each buldg ad t s stablty wthout a overall vew of the stuato of possble movemets of dvdual parts of the surface of the Prague Castle area. A comprehesve evaluato of the heght shfts s further dscussed ths artcle. 2 GEODEIC MEASUREMENS A PRAGUE CASLE Geodetc measuremets were the area of Prague Castle carred out varous rage sce ts costructo, but the perodc motorg of selected hstorc buldgs ad slope stablty s a matter of the last 10 years. here are chages motored tlt ad relatve heght dvdual buldgs ad areas. For aalyss oly heght measuremet was chose. he reaso for ths decso s a very hgh ad log-term acheved precso of 0.7 mm / km, also the hgh relablty ad resstace of the method to systematc errors. Heght measuremets are almost etrely a matter of method levellg from the cetre wth the addto of precse trgoometrc method used to brdge the Jeleí příkop (Deer Moat). Scheme of the performed measuremets s o Fg. 1. Measuremets were coducted wth use of the Zess Ko 007 strumet, ad the last two epochs the dgtal levellg strumet rmble D 12 was used. Fgure 1 Scheme of the heght geodetc etwork the Prague Castle area

3 Štroer, M. et al.: Stablty Determato of the surface area of the Prague castle NEW CALCULAION OF EPOCH LEVELLING MEASUREMENS Overall, there were 18 epochs re-processed ad re-adusted, all of them measured betwee years 2004 ad Orgal teto of the measuremets was ot to carry out assessmets, but motorg of dvdual obects. Motored obects chaged durg the years ad dfferet epochs uequal sets of pots were measured. Because of the relatve soluto of all motorg, the measuremet oto the stable pots outsde the Prague Castle area was ot performed, ad therefore oe of the pots ca be cosdered to be stable. Processg was made wth regard to these facts epochs by the least squares adustmet, the GNU Gama software (gama-local ver. 1.13, more (Čepek, 2005)). As the measured values served averaged heght dffereces measured back ad forth. he a pror stadard devato was chose to be σ 0 = 0.7 mm, measuremets were coducted predomatly by the Zess KoN 007 strumet. he results of the ew adustmet are relatve heghts of pots at each epoch. For further aalyss, a stadard devato of heght of oe pot s assumed to be σ p = 0.36 mm as the average of stadard devatos of all pots all epochs. o detfy the shfts betwee the epochs, the heght dfferece betwee the epochs should exceed H = u p 2 σ p = 1.0 mm for the 95% (u p = 2) or 1.3 mm for the probablty of 99% (u p = 2.5). 4 CALCULAION OF ROBUS ANALYSIS he results of the adustmet are relatve heghts of pots at each epoch. It s ot possble to cosder ay of the pots to be stable, therefore the trasformato wth redudat measuremets was chose for the aalyss ad calculated wth use of the robust estmato, whch s hghly resstat agast the outlayg (here shfted) values. 4.1 HE BASIC PRINCIPLE OF A ROBUS CALCULAION Robust adustmet methods are mostly based o the prcple of maxmum lkelhood method ad ther basc property s (compared to geodesy wdely used the least squares method) hgh resstace agast the fluece (agast) of outlyg measuremets. he prcples ad dervato of least squares ad robust methods ca be foud (Štroer, 2011). Most practcally usable robust methods are based o adustg of the weghts the calculato method of least squares (reweghtg), such a calculato s the relatvely easy. Methods are preseted (Štroer, 2011) too. he calculato procedure of teratve adustmet s based o the assembly of ormal equatos the form: A PAdx = A Pl', (1) 1 ( A PA) A Pl' dx =, (2) where A s Jacob matrx, P dagoal matrx of weghts (o the dagoal are measuremets weghts P = K/σ 2 ), dx cremet vector of ukows, l vector of reduced measuremets. Robust weght chage: 1 ( A PWA) A PW l' dx =, (3) where robust weght chage s determed by the equato: W = dag ( w w,..., ); w f (, σ ) 1, 2 w =, (4) v

4 246 INGEO 2014 where correctos are determed by the equato v = Adx l'. (5) Robust chages are derved from the stadard devatos of measuremet ad correctos obtaed adustmet. Varous methods of calculatg of the chages weghts ca be used, derved o the bass of expected probablty dstrbuto of devatos from the ormal dstrbuto. Here t s worth metog Huber method (descrbed (Štroer, 2011)). Whe creatg a robust estmator Huber came out from the ormal radom varable dstrbuto. Hs soluto s based o the replacemet of the edge parts of the ormal probablty dstrbuto by the Laplace dstrbuto (a specal form of the expoetal dstrbuto), whch leads to greater probablty of outlayg measuremet o the dstrbuto s edges. For purposes of the aalyss, a L 1 orm was used, whch, as a fucto of the probablty dstrbuto, uses drectly Laplace dstrbuto, whch has comparso to a ormal dstrbuto a greater probablty of outlayg measuremets occurrece. For homogeeous measuremet (measuremets wth the same stadard devato) s a robust weght chage gve by the fucto w = 1/ v (6) ad there s o eed to kow the stadard devato. he calculato s doe teratvely, correctos used to calculate robust weghts chages are always used from a prevous calculato. More to the calculato procedure s (Štroer, 2011). 4.2 HE PROCEDURE OF CALCULAION USING HE L1 NORM he dvdual epochs were adusted ad for determg the pots, where betwee two epochs, were shfts, t s ecessary to trasform the matchg pots of the epoch to epoch. he equatos of lear trasformato: X = MRX, (7) + where X, X are vector of the coordates, M matrx of the scale coeffcets, R rotato matrx, traslato vector. here s oly a oe-dmesoal trasformato (oly heghts) eeded, scale betwee the epochs does ot chage ad therefore the trasformato equato for heghts H betwee epochs ad degrades as follows: H = H +. (8), Whe calculatg the relatoshp betwee the two epochs, t s determed oly by heght shft,, ad there s the average heght dfferece betwee the epochs: H k, H k, k= = 1,. (9) Ideally, ths shft wll exactly sut to all pots, though practcally t does ot, ad therefore for every pot = 1.. k ca be calculated correctos: ( H ) v H,, +, =. (10) hese correctos cota a compoet of measuremet accuracy, ad f there was a heght chage, ths fluece too. Mea as a method correspods wth the least squares method, ad the case of outlayg measuremets, here shfted pots, fals to gve proper results. For these reasos, t s advsable to use a robust method, whch does ot have such a

5 Štroer, M. et al.: Stablty Determato of the surface area of the Prague castle property. he heght dfferece betwee the two epochs s determed by a teratve calculato of the weghted average, where the weghts are calculated o the bass of correctos from prevous calculato (m-th terato). ( m+ 1) ( H k, H k, ) k = 1, = k= 1 ( m) w k ( m) w k. (11) he dvdual epochs were ot measured at regular tme tervals ad also measured pots chaged, so the procedure has bee used where at the selected epoch (amely 10) were gradually trasformed all the others. As the referece epoch was chose epoch 10, because most pots were measured ths epoch, both tal ad especally the termal epochs. he calculated shft, s ot mportat, sgfcat are dvdual correctos sgallg shfts of the pot betwee the epochs. 4.3 CALCULAION RESULS he calculato results are determed correctos after the trasformato, whch ca be terpreted as devatos of dvdual pots from the commo state from a commo level. Fgure 2 Example of the relatve pots shfts Whe plotted o a graph, these correctos ca gve a dea of the movemet of dvdual pot betwee epochs. Because of the large umber of pots t s ot possble to show all of t, a example s o the Fg. 2. Zero shft meas, that pot was ot measured the epoch. O Fg. 3 there are characterstc pots marked by the arrow characterzg ts relatve shft, grey dots marks pots cosdered to be stable.

6 248 INGEO CONCLUSIONS Fgure 3 Relatve pots shfts As a result of the ew evaluato of the deformato measuremets at the Prague Castle, the scheme of relatve shft pots was created, whch s show Fgure 2. A ew methodology was used for processg the results, whch volves the use of robust estmato, amely the L 1 orm. he results are cosstet wth observed pheomea the feld ad so the preseted methodology ca be cosdered to be approprate. ACKNOWLEDGEMENS he artcle was wrtte wth support of the teral grat of Czech echcal Uversty Prague SGS14 Optmzato of acqusto ad processg of 3D data for purpose of egeerg surveyg. REFERENCES Prague Castle, ct ZÁLESKÝ, J., CHAMRA, S.: Optmalzáca geotechckých koštrukcí: Proekt sledováí techckého stavu hstorckých budov. I: Optmalzáca geotechckých koštrukcí, září 2001, SvF SU Bratslava, Sloveská Republka, ISBN X. s PROCHÁZKA, J., JIŘIKOVSKÝ,., ZÁLESKÝ, J. et al.: Stablta hstorckých obektů. 1. vyd. Praha: České vysoké učeí techcké v Praze, s. ISBN ČEPEK, A.: GNU Gama Adustmet geodetc etworks. Edto 0.19, 2005 ŠRONER, M., HAMPACHER, M.: Zpracováí a aalýza měřeí v žeýrské geodéz. 1. vyd. Praha: CU Publshg House, s. ISBN