PRACTICAL ADVANTAGES OF USING THE MECHANICS OF CONTINUUM TO ANALYSE DEFORMATIONS OBTAINED FROM GEODETIC SURVEY

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1 PRACTICAL ADVANTAGES OF USING THE MECHANICS OF CONTINUUM TO ANALYSE DEFORMATIONS OBTAINED FROM GEODETIC SURVEY Mlan TALICH Rsarch Insttut of Godsy, Topography and Cartography, Zdby 98, CZ-5 66, Czch Rpublc Abstract: Th papr dmonstrats why applyng of th thory of contnuum s usful to dformaton analyss obtand from rpatd postonal survy n godsy. Th mchancs of contnuum may b appld as bass of communcaton durng th multdscplnar approach to godtc and gotchnc montorng. It srvs also as a tchnologcal and scntfc communcaton bass btwn godssts and spcalsts of othr profssons as gotchncs, gophyscs, buldng ngnrs tc. Th ndpndncy of rsultng dformaton paramtrs to appld coordnat fram s shown, too. It s not ncssary to try to fnd any condtons of placng th survy ntwork n th coordnat systm but th calculaton should b adjustd as fr ntwork. Thn rrors orgnatd from rronous pr-rqusts about stablty of som slctd ponts that ar takn durng normal calculaton as stabl (n th stabl part of such locaton) wll b compltly lmnatd. Whn compard to th sol lstnng of dsplacmnts ths procdur nabls to prsnt dformaton paramtrs n much mor objctv way and srvs as a tool to dmonstrat th rlatvly godynamcal trnds of th trrtory n quston.. INTRODUCTION Mor and mor rsults of godtc actvty ar usd n othr dscplns. Such cass ar,.g., godynamc rsarch, whr godsy offrs objctv and rlatvly suffcntly xact nformaton about motons and shap and dmnson changs of th trrstral surfac or buldng constructons, gnrally spakng of som montord objct or localty. Such nformaton s usd to furthr studs, physcal ntrprtatons and dtrmnng causatv factors. That mans thr s a multdscplnary lad approach to solvng such problms. Such approach rqurs cratng of tchnologcal and scntfc bas for communcaton btwn spcalsts of dffrnt profssons that wll b sutabl to all of thm. In (Szostak- Chrzanovsk t al. 6) s proposd th mchanc of contnuum to ths purpos as support of dformaton montorng, thr analyss and ntrprtaton. Lt us show othr rasons and bnfts ladng to applyng ths thory n godtc praxs. Homognous trrtory or objct n quston s th rqurd condton, of cours. It mans that f such condton s not fulflld, thn only dvson of th whol trrtory to svral

2 homognous parts must b trd and calculatons has to b don for ach on of thm sparatly. Analyss of dformatons of th trrstral surfac or of buldng constructons dos not blong to problms that many godssts solv n thr vryday praxs. Th man goal of dformaton analyss s dtrmnaton of dformaton mchansm f w look at th objct as at th mchancal systm undr dformatons accordng th laws of mchancs of contnuum. Aftr such gomtrc analyss physcal ntrprtaton of rsults may follow. Gomtrcal analyss, dscrbs th chang n shap and dmnsons of th montord objct, as wll as ts rgd body movmnts (translatons and rotatons). Th goal of th gomtrcal analyss s to dtrmn n th whol dformabl objct th dsplacmnt and stran flds n th spac and tm domans. Physcal ntrprtaton s basd on th rlatonshp btwn th causatv factors (loads) and dformatons. Godtc montorng of dformatons should follow n two stps. Th frst on s dtrmnaton of dsplacmnts of slctd ponts on th objct n quston (standard task). Aftr t dtrmnaton of dsplacmnt fld n contnuous form by thr ntrpolaton (gnralzd task) can follow. Th scond stp s dtrmnaton of dformatons paramtrs by gomtrc analyss of contnuum mchancs (stran analyss). Dtrmnaton of dsplacmnts n th frst stp s usually don by rpatd obsrvaton of godtc ntwork. But ralsaton of th scond stp s not so much usual among godssts. Nvrthlss stran analyss offrs many convnncs.. GEOMETRIC ANALYSIS OF DEFORMATIONS BY CONTINUUM MECHANICS As t was told arlr, th prncpl of godtc mthods applcatons s basd on rpatd masurmnt and comparson of rsults of ndvdual stags of masurmnts. Obtand dffrncs n postons of ponts rprsnt thr dsplacmnts. Th vctor of pont dsplacmnt s d = (u, u, u 3 ) T = x o - x t Whr x o (rsp. x t ) s th vctor of P pont coordnats of fundamntal (rsp. actual n t-tm) stag. Ths vctor may b xprssd as a functon of coordnats: u = (u, u, u 3 ) T = u(x) = (u (x), u (x), u 3 (x)) T = d, x = (x, y, z) T Th stran tnsor n P s dfnd as a gradnt of th functon n ths pont:

3 E = ( ) = grad d u x u = x u3 x u y u y u 3 y u z u z u 3 z In th dsplacmnt fld s vald nxt rlaton, s (Wlsch 983): d = E x + t whr d E x t s th dsplacmnt vctor, s th dsplacmnt gradnt, s th coordnat vctor, s th vctor of translaton lmnts. Th stran tnsor may b dvdd nto two parts: E = + Ω = ( jl ) + ( ω jl ) j,l =,,3 whr: s th symmtrc tnsor of dformaton, Ω s th antsymmtrc tnsor of rotaton, jl = ( jl + lj ) /, ω jl = ( jl lj ) /. = = ( + ) ( + ) ( + ) ( + ) ( + ) ( + )

4 Ω = ω ω3 ω ω 3 ω ω 3 3 = ( ) ( ) ( ) ( ) ( ) ( ) It could b wrttn: d = ( + Ω ) x + t W could dtrmn th dformaton paramtrs from and Ω. Ths may b dom as a 3D soluton as wll as n plan. Such plan could b.g., paralll wth XY or XZ or YZ or n a mor gnral way any plan of th local coordnat systm to whch spac dsplacmnts wll b projctd. It holds.g., for dsplacmnts projctd to XY of th local coordnat systm: Δ = + - total dlataton γ = - shar strans γ = - shar strans γ = + - total shar γ γ = ( Δ + γ ) - axs of maxmum stran = ( Δ γ ) - axs of mnmum stran γ ϕ = arctg γ - drcton of axs of maxmum stran ψ = ϕ + π for ω 4 - drcton of shar stran ψ = ϕ π for ω 4 < - drcton of shar stran Thortcal soluton and drvaton of ths formulas n quston may b found n many publcatons.g., (Szostak-Chrzanovsk 6), (Altnr 999), (Talch 994) and (Talch, Kostlcký, Vyskočl 993). 4

5 3. DEFORMATION PARAMETERS CHARACTERS AND BENEFITS RESULTING FROM THEM It s worth notng that all dsplacmnts dpnd on slctd coordnat fram. On contrary, all dformaton paramtrs of last quatons xcpt th ψ and φ drctons ar on usd coordnat fram ndpndnt, and nsnstv to translaton and rotaton. And ths s th rason why dformaton paramtrs and thr applcatons ar mportant n practc. In th cas of pont dsplacmnts calculaton th rsultng charactr of dsplacmnts s qut unambguously gvn by th appld coordnat fram or by condtons of godtc ntwork placng n th coordnat fram. Th frst condton that has to b fulflld s nt adjustmnt as fr ntwork to prvnt ts scal changs or vn ts dformng. Usually slctd ponts that ar xpctd to b n th stabl part of locaton ar chosn lk fxd. Mor xactly sad: n th cas of fr ntworks, ths ponts ar chosn lk ponts ncludd at th condton of bng placd n th coordnat fram (.g. by slctng among dntcal ponts durng th Hlmrt transformaton). Slctng of such ponts s usually basd on xpctd physcal proprts of th localty. Nvrthlss w ar nvr qut sur that our xpctatons about ponts stablty ar good and wll fulflld. Th stuaton may b shown on modls. Lt us xpct a fault n th localty n quston and movng of two qut rgd plats agan ach othr (ths may happn.g. by plungng of on plat undr th scond plat). Th followng pcturs to 3 show such modl stuaton. Thr ar thr dffrnt ways of fxd pont chosn ladng to qut dffrnt charactr of calculatd dsplacmnts of dtrmnd ponts of th sam ntwork. Ponts that ar slctd as fxd ponts and that ar supposd to b n th stabl part of locaton, ar always markd by rd trangl. To accntuat charactr of dtrmnd dsplacmnts thr ar on followng pcturs 4 to 6 also ntrpolatd dsplacmnts flds rprsntd. As sn from pcturs, followng physcal ntrprtaton of such calculatd dsplacmnts could lad to qut rronous or contradctory rsults. And ths all s nothng ls than nflunc of calculaton and data slctng. Fgur : Fxd pont AP, AP Fgur : Fxd pont BP, BP 5

6 Fgur 3: Fxd pont AP, BP Fgur 4: Fxd pont AP, BP Fgur 5: Fxd pont AP, AP Fgur 6: Fxd pont BP, BP At ths momnt t wll b qut convnnt to go to th scond stp, dtrmnaton of dformaton paramtrs accordng th mchancs of contnuum. Wth rgard to thr ndpndncy to th slctd coordnat fram and nsnstvty to translaton or rotaton w wll obtan from all possbl varants of dsplacmnts calculaton always th sam dformaton valus. It mans that t s not ncssary to dal wth condtons of placng th godtc ntwork n th coordnat fram (fxd ponts dclaraton). In othr words, whch ponts should b dclard as fxd or not. Th only rmanng thng s calculaton of ntwork adjustmnt as fr ntwork,.. to choos only ncssary numbr of condtons to thr placmnt n th coordnat fram and thus not dform th adjustd ntwork. Such condtons ar n th cas of horzontal ntwork wth gvn dmnson (at last on masurd 6

7 lngth) thr and n th cas of horzontal ntwork wthout gvn dmnson four. Dffrnt condton slctons do not man anythng ls n th cas of fr ntwork than applcaton of rotaton and translaton and calculaton of dformatons dos not dpnd on t, as was alrady sad. Th abov mntond s gvn on fg. 7 whch s common to all varants of calculatd dsplacmnts. Fgur 7: Dformatons paramtrs ar th sam for all varants of fxd ponts Ths mans n practc that rrors from bad (rronous) xpctatons about stablty of som slctd ponts, that w consdr at common calculatons of dsplacmnts from rpatd masurmnt as stabl (n th stabl part of localty), ar totally lmnatd. As such a mstak w could consdr for nstanc GPS antnna xchang (chang of phas cntr of a nw antnna aganst th old on) of a prmannt staton at accss pont of th GPS nt. Evn ths mstak wll b durng calculaton of dformaton paramtrs totally lmnatd provdd that t lads to th shft of whol ntwork and ths pont s not ncludd nto calculaton of th fld of dsplacmnts and dformaton. Th othr advantag s th fact that t s not ncssary (rgardng calculaton of dformaton paramtrs) to dscuss th problm of transformng dsplacmnts gvn.g. n coordnats fram ITRF nto ETRF, or to rduc dsplacmnts n ITRF by movmnts of tctonc plat accordng to som of godynamc modls as.g. APKIM (Drws and Angrmann ) or NNR-NUVEL (Shuanggn and Wnyao 4). In our modl cas t s possbl to dduc th ral godynamc actvts basd on gvn dformaton paramtrs. Abov all, th ral stuaton s dsclosd,.. movmnt of two 7

8 rlatvly consstnt plats on aganst th othr wth unambguously dfnd procss of dsrupton n th trrtory, as dscrbd n fgur 7. That mans that, contrastng to dsplacmnts, dformatons rprsnt objctv tool for dsclosur of ral rlatv godynamc trnds n th rsarchd trrtory. 4. PRACTICAL EXAMPLE Dformaton ntwork GEOSUD n th ara of Polsh Sudts and th For-Sudtc Block dscussd n (Cacon t al. 5) s chosn as a practcal xampl. It s godynamc ntwork of rpatd GPS masurmnts. Dformaton paramtrs calculatons ar mad by support of wb applcaton (Talch and Havrlant 8) accssbl at In fgur 8, rsultng dsplacmnts ar statd n coordnat fram ITRF. In fgur 9 rsdual dsplacmnts aftr thr rducton usng modl APKIM ar dsplayd. All dsplacmnts n ITRF ar approxmatly of th sam sz, 4 to 7 mm /yar, and approxmatly of th sam drcton. On th othr hand, rsdual dsplacmnts aftr rducton to local systm ar of dffrnt charactr wth sz from,3 to 3,7 mm / yar and dffrnt drctons. Ths trnds ar furthr mphaszd by dsplacmnts fld, whr dsplacmnts on masurd ponts ar dsplayd n rd colour and dsplacmnts statd by thr ntrpolaton n grd n blu colour. Fgur 8: GEOSUD ntwork, dsplacmnt fld n ITRF Howvr, both dsplacmnts valus lad to th sam fld of dformaton dsplayd n pctur. Th ndpndnc of dformaton paramtrs on translatons and rotatons was thus provd by practcal calculaton, n ths cas rprsntd by rducton n accordanc wth modl APKIM. In othr words, dsplacmnts rducton from ITRF nto local systm for dsclosur of ral rlatv godynamc trnds was not ncssary. 8

9 Fgur 9: GEOSUD ntwork, dsplacmnt fld n local systm Fgur : GEOSUD ntwork, dformaton paramtrs 9

10 5. CONCLUSION Th am of ths artcl was to show th practcal advantag of th usag of thory of mchancs of contnuum for dsclosur of ral rlatv godynamc ffcts n rsarchd trrtors, vntually of dynamcs rsarchd ntts or constructon structurs. Mchancs of contnuum gv rsults, that ar bttr undrstandabl vn for xprts from dffrnt flds than godsy. Thrfor t can srv as tchnologcal and scntfc bass for communcaton among xprts from dffrnt profssonal flds. Bsds that, ts usag also lmnats som mpacts of masurmnt and calculatons durng godtc ntworks procssng, whch nabls accss to mor crdbl and objctv rsults, spcally n comparson wth pur dsplacmnts dtrmnaton of ponts n th montord objct. Rfrncs Altnr, Y. (999). Analytcal Surfac Dformaton Thory for Dtcton of th Earth's Crust Movmnts. Sprngr-Vrlag Brln, Hdlbrg, Nw York. 999, X, pp. fgs., ISBN: Cacon, S., Vyskočl, P., Talch, M., Kontny, B., Bosy, J. (5). Dformatons analyss of th Polsh Sudts and For-Sudtc block. Rports on godsy, Insttut of Godsy and Godtc Astronomy of th Warsaw Unvrsty of Tchnology, No. (73), 5, Warsaw, p. 9, 5 obr., tab. Drws H. and Angrmann D. (). Th actual plat knmatc and crustal dformaton modl (APKIM ) as a godtc rfrnc systm. In: IAG Scntfc Assmbly, Budapst, Hungary, 5-6 Sptmbr,, (Postr), Kostlcký, J., Talch, M. and Vyskočl, P. (994). Crustal Dformaton Analyss n th Intrnatonal Cntr on Rcnt Crustal Movmnts. Journal of th Godtc Socty of Japan - 4/4 (994), p Shuanggn J. and Wnyao Z. (4). A rvson of th paramtrs of th NNR-NUVEL-A plat vlocty modl. Journal of Godynamcs, 38,, 4, p //a74fcf685d8f4677ac7a785df Szostak Chrzanowsk, A., Prószyñsk, W. and Gambn, W. (6). Contnuum Mchancs as a Support for Dformaton Montorng, Analyss, and Intrprtaton, In: Kahmn, H. and Chrzanowsk, A. (Eds.), Procdngs 3rd IAG / th FIG Symposum, Badn, May -4, 6, Badn, ISBN: Talch, M., Kostlcký, J. and Vyskočl, P. (993). Modllng of Horzontal Movmnts and Computng Mthod of Dformaton Flds at th Intrnatonal Cntr On Rcnt Crustal Movmnts. In: Zpráva o řšní vědckého projktu 4--6 Godéz a dynamka Změ v roc 993 (Rport about soluton of rsarch projct 4--6 Godsy and dynamc of th Earth n 993), VÚGTK, 993, p.4-5. Talch, M. (994). Usng GPS Data for Dformatons Analyss. In: 994 IERS and IGS Workshops Pars and Sant Mandé, 994, March -5, p.6.

11 Talch, M. (7). Analýza dformací v poddolovaných lokaltách z opakovaných polohových měřní s využtím wbové aplkac a mapových služb WMS. Acta Montanstca Slovaca, (7), mmoradn číslo 3, p , ISSN Talch, M. and Havrlant, J. (8). Applcaton of dformaton analyss and ts nw possblts. In: 3th FIG Symposum on Dformaton Masurmnts and Analyss (WG 6.) wth th 4th IAG Symposum on Godsy for Gotchncal and Structural Engnrng (SC 4.) - Masurng th Changs, Lsbon, Portugal, -5 May, 8, (n prnt). Wlsch, W. M. (983). Fnt lmnts analyss of stran pattrns from godtc obsrvatons across a plat margn. Tctonophyscs, 97, pp Elsvr. Amstrdam 983 Corrspondng author contacts Mlan TALICH Mlan.Talch@vugtk.cz Rsarch Insttut of Godsy, Topography and Cartography Ustcka 98, CZ 5 66, Zdby Czch Rpublc