DEFORMATION ANALYSIS OF A GEODETIC MONITORING NETWORK

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1 Gematca. Vl. 55, N. 3, 00. DEFORMAION ANALYSIS OF A GEODEIC MONIORING NEWORK Halm Setan an Ranjt Sngh Cente f Inustal Measuement an Engneeng Suveyng Faculty f Genfmatn Scence an Engneeng, Unvest eknlg Malaysa, Jh Bahu, Malaysa Results f efmatn analyss ae ectly elevant t the safety f human lfe. heefe ne has t be vey caeful n assessng the ata f a mntng netwk t av wng ntepetatn f the splacements. hs pape pesents a efmatn analyss pceue that cnssts f netwk ajustment f nvual epchs, ten analyss f the splacement fel an mellng f the efmatn. Dung ten analyss, tw bust meths an a nn-bust meth have been apte an apple n etemnng the ten f mvements f all the cmmn pnts n a mntng netwk. he ten f mvements then fm a bass f pelmnay entfcatn f the efmatn mels. he evelpe pceue has been mplemente n a pgam package knwn as NEDEFAN (NEwk an DEFmatn ANalyss). A numecal example s als gven by usng knwn ata f a am mntng netwk. Intuctn he uppe layes f the eath s cust ae n cnstant mtn bth hzntally an vetcally ue t facts such as change f gun wate level, tectnc phenmena, lan sles, etc. heefe any lage man-mae stuctues such as bges, hgh se bulngs, ams, etc., whch ae bult n the suface f the eath ae subject t efmatn. hs efmatn nees t be mnte cntnuusly f safety assessment pupse. Geneally, the efmatn measuement technques can be ve nt getechncal, stuctual an geetc meths. Getechncal an stuctual meths ae ect measuement meths, whch use specal equpment t measue changes n length, nclnatn, elatve heght, stan, etc. [eskey an Pte 988; Chzanwsk 986]. On the the han, n the geetc meth thee ae tw basc types f geetc mntng netwks; namely the efeence (abslute) an elatve netwks [Chzanwsk et al. 986]. In a efeence netwk, sme f the pnts statns ae assume t be lcate utse f the efmable by bject, thus sevng as efeence pnts f the etemnatn f the abslute splacements f the bject pnts. Hweve, n a elatve netwk, all suveye pnts ae assume t be lcate n the efmable by. hs pape wll fcus nly n the geetc meth usng a efeence netwk. In a geetc mntng netwk, the bject aea une nvestgatn s usually epesente by a numbe f pnts whch ae pemanently mnumente make. All the pnts ae then bseve n tw me epchs f tme. he geetc mntng netwk can be ethe a cnventnal (teestal) netwk, a phtgammety (.e., aeal clse-ange) netwk, Glbal Pstnng System (GPS) netwk a cmbnatn f these netwk types. Defmatn analyss usng the geetc meth manly cnssts f a tw-step analyss va nepenent ajustment f the netwk f each epch, fllwe by efmatn etectn between the tw epchs. Dung efmatn analyss t s mptant t etemne the ten f mvements (splacements) f all the cmmn pnts n a mntng netwk. he ten f mvements then fm a bass f pelmnay entfcatn f the efmatn mels ung the mellng f efmatn. Althugh efmatn analyss can be apple n ne-mensnal (-D), twmensnal (-D) an thee-mensnal (3-D) mntng netwks, ths pape wll fcus nly n the -D netwk f the pupse f smplcty an ease unestanng. Netwk Ajustment

2 Gematca. Vl. 55, N. 3, 00. Defmatn measuements may cnsst f a cmbnatn f bsevables such as stances, azmuths, cnates, ectns, cnate ffeences, etc. he numbe f bsevatns usually excees the mnmum numbe eque t etemne the unknwn quanttes paametes. In engneeng suveyng, these unknwn quanttes ae the cnates f the pnts. he eunant measuements ae useful f checkng gss es utles n the measuements, pecsn f the unknwn quanttes an qualty f the netwk. he meth f least squaes estmatn (LSE) s an mptant tl n estmatng the unknwn paametes fm eunant ata. he esult btane by LSE s knwn as the best lnea unbase estmate (BLUE). he functnal mel elatng the measuements an paametes t be estmate can be expesse as l = f(x) () whee l s the vect f bsevatns an x s the vect f paametes t be estmate. In geneal, equatn () s nn-lnea, an t nees t be lneaze by usng ayl s theem. Afte lneazatn the bsevatn equatn s wtten as vˆ = A xˆ + b () whee, vˆ s the vect f esuals, A s the esgn matx, xˆ s the vect f cectns t the appxmate values ( x ) an b s the msclsue vect. In ths stuy, the atum efect pblem s ealt wth by fxng a mnmum numbe f paametes cnates (.e., a mnmum cnstants atum) n e t efne the geetc atum. F a -D netwk thee s a maxmum f fu atum paametes,.e., tw tanslatns, ne tatn an ne scale. heefe t s necessay t ntuce fu nepenent paametes (f example the cnates f tw pnts) n e t efne the geetc atum, hence leang t a full ank. Hweve, sme f the atum paametes can als be efne by cetan bsevables, f example, the stance efnes the scale an the azmuth efnes the tatn f the netwk. he nmal equatn wth a full ank an the a p vaance fact ( σ ) s assume t be knwn (.e., σ = ), can then be wtten as whee; N xˆ + U = 0 (3) N = A WA, ceffcent matx U = A Wb b = l - l l = vect f actual bsevatns l = vect f cmpute bsevatns xˆ = -N - U = -( A WA) - A Wb xˆ a = xˆ + x, the upate paametes Q = ( A WA) -, cfact matx f xˆa W = Q l, the weght matx xˆ a vˆwvˆ ˆσ =, a pste vaance fact n u n = numbe f bsevatns u = numbe f paametes Q vˆ = W - AN - A, cfact matx f the esuals lˆ a = l + vˆ, ajuste bsevatns Q = AN - A, cfact matx f lˆa ajuste bsevatns Althugh the estmate paametes, xˆ a (.e., cnates f the pnts) an the cfact matx, Q xˆ a, ae atum epenent base n the chce f ze-vaance cmputatnal base, thee exst functns such as vˆ, Q vˆ, lˆ a, ˆσ an, whch ae atum nvaant. Q lˆa Dung LSE, the mptant aspects that nee t be cnsee ae the glbal test (Chsquae), lcal test (au test Baaa Meth), pecsn, accuacy an elablty (ntenal an extenal) analyss. F futhe etals n LSE the nteeste eae ae efee t Wlf an Ghlan [997], Sngh [997], Caspay [987], Cpe [987], Sngh [999], Setan an Sngh [999], an Setan [995], t name a few. Befe efmatn analyss can be cae ut t s mptant t pefm ntal checkng n the nput ata an test n the a pste vaance facts f bth epchs. Intal Checkng f Data an est n Vaance Rat

3 Gematca. Vl. 55, N. 3, 00. Intal checkng f ata s mptant t ensue that cmmn pnts, same appxmate cnates an same pnts names ae use n the tw epchs. If the numbe f pnts n the fst epch s nt the same as n the secn epch, then efmatn analyss s pefme nly n the exstng cmmn pnts n bth epchs. hs can be easly cae ut by extactng the cmmn pnts n bth epchs. he numbe f atum efects epens n the unn f the atum efects n bth epchs [Chen 983; Chen et al. 990]. F example, f the fst epch s a tlateatn netwk whch has thee atum efects (.e., tw tanslatns an ne tatn), whle the secn epch s a tangulatn netwk wth a atum efect f fu (.e., tw tanslatns, ne tatn an ne scale), then the unn f atum efects s equal t the atum efects n the secn epch (.e., fu). he a pste vaance facts f bth epchs ae then teste f the cmpatblty. he null an altenatve hypthess f ths test ae [Setan 995; Caspay 987; Chen et al. 990; Cpe 987; Sngh 999] an H : H a : ˆ ˆ σ = ˆ σ > ˆ σ (4) ˆ σ ˆ ˆ σ > ˆ σ (5) wth σ an σ beng the pste vaance facts f the fst an secn epchs espectvely. he test statstc s ˆ σ j = ~ F(α,f j,f ) (6) ˆ σ wth j an epesent the lage an smalle vaance facts, F s the Fshe s stbutn, α s the chsen sgnfcance level (typcally α = 0.05) an f an f j ae the egees f feem f epchs an j espectvely. he abve test s accepte f < F(α,f j,f ) at a sgnfcance level α. he falue f the abve test may be cause by ncmpatble weghtng between the tw epchs ncect weghtng scheme, an any futhe analyss shul be stppe at ths stage. en Analyss Afte the test n the vaance at (equatn 6) s accepte, the splacement vect (cnates ffeences) an ts cfact matx can be cmpute as = xˆ - xˆ (7) Q = Q xˆ + Q xˆ (8) whee, xˆ an xˆ ae the estmate cnates f all the cmmn pnts n the fst an secn epchs espectvely (wth same atum efntn), Q xˆ an Q xˆ ae the cfact matx f the estmate cnates xˆ an xˆ, s the splacement vect an Q s the cfact matx f. In ths stuy, tw well-establshe gus meths knwn as the bust an cnguency testng meths have been apple f estmatng the ten f mvements f all the cmmn pnts n a mntng netwk. Rbust Meths Caspay an Butta [987] have gven a etal explanatn abut thee bust meths namely Dansh, M-estmatn (Hube) an Least Abslute Sum (LAS) meth, f the estmatn f the ten f mvements. Accng t Caspay an Butta [987], the mst bust s the Dansh meth, fllwe by LAS an M- estmatn (Hube) meth. Chen [983] has ppse a bust meth knwn as an teatve weghte smlaty tansfmatn (IWS). hs bust meth was evelpe at the Unvesty f New Bunswck, Canaa. hs pape wll fcus n tw bust meths va the LAS an IWS f estmatng the ten f mvements f a mntng netwk. Such meths wee chsen ue t the we applcatn n efmatn etectn. he LAS an IWS meths ae base n S-tansfmatn (smlaty Helmet tansfmatn) as belw [Chen 983; Chen et al. 990; Setan an Sngh 998b, c; Sngh an Setan 999a, b, c; Sngh 999; Setan an Sngh 999] (k+) =[I-G(G W (k) G) - G W (k) ] (k) =S (k) (k) (9) whee

4 Gematca. Vl. 55, N. 3, 00. I = entty matx k = numbe f teatns = splacement vect (equatn 7) S = S-tansfmatns matx W = weght matx G = 0 y x 0 x y 0 y x 0 x y y x 0 m m 0 x y m m (0) whee x, y ae the cnates f pnt P whch ae euce t the cent cente f gavty f the netwk,.e., x = x - y = y - m = m m = m x y () () wth x, y the appxmate cnates f pnt P an m s the numbe f cmmn pnts n the netwk. he fst tw ws f the nne cnstant matx (G ) take cae f the tanslatns n the x an y ectns, whle the th w efnes the tatn abut the vetcal (z) axs an the last w efnes the scale f the netwk. F a tlateatn netwk, the last w f G s mtte [Caspay 987; Cpe an Css 99; Setan 997; Chen et al. 990; Sngh 999]. he man ffeence between the LAS an IWS meths s n fmng the weght matx, W. In the fst tansfmatn (k = ) the weght matx s taken as entty (W (k) = I) f all the cmmn pnts, then n the (k+) tansfmatn the weght matx s efne as F the IWS meth W (k) = ag ( k ) F the LAS meth (3) W (k) = ag ( k ) + ( k ) ( x ) ( y ) (4) In equatn (3), s the splacement vect as n equatns (7) an (9). Hweve, n equatn (4), x an y efe t the splacement cmpnents n x an y axes espectvely. It s mptant t mentn that the abve weghtng schemes (equatns 3 an 4) ae nly apple n the cmmn atum pnts (.e., ethe the efeence pnts f a efeence netwk a gup f pnts n a stable blck f a elatve netwk), whlst f the bject pnts the weght s set as ze (.e., W (k) = 0). he teatve pceue cntnues untl the abslute ffeences between the successve tansfme splacements f all the cmmn pnts (.e., (k+) - (k) ) ae smalle than a tleance value δ (say [m]). It s pssble that ung the teatns sme ( k ) ( x, y ) may appach ze, causng numecal nstabltes, because W (k) (equatns 3 an 4) becmes vey lage. hee ae tw ways t slve ths pblem, ethe () Settng a lwe bun value (e.g., ( k ) [m]). If s smalle than the lwe bun value, ts weght s set t ze, () Replacng equatns 3 an 4 as W (k) = ag W (k) =ag ( ) k + δ (5) ( k ) ( k + + ) ( x δ ) ( y + δ ) (6) In equatn (6), the tleance value s nly ae t thse splacement cmpnents (ethe x y ) whch ae smalle than ts lwe bun value. In ths stuy, slutn () was apple t the IWS meth, whle slutn () was apple t the LAS meth. hs technque was fun t be useful n avng numecal nstabltes. In ths pceue, the LAS meth mnmzes the sum f the lengths f the ( ) splacements (.e., x ) + ( y

5 Gematca. Vl. 55, N. 3, 00. mnmum), whle the IWS meth mnmzes the ttal sum f abslute values f the splacement cmpnents (.e., mnmum). In the fnal teatn the cfact matx f the splacement vect s cmpute as Q (k+) = S (k) Q (S (k) ) (7) he stablty nfmatn f each cmmn pnt j can then be etemne thugh a sngle pnt test as belw [Setan 995; Setan an Sngh 998c) ( j = whee; ) ( k + ) j ( Q σ ( k + ) ( k + ) j ) j ~F(α,,f) (8) ˆ j, Q j = splacement vect an ts cfact matx espectvely f each cmmn pnt j [ f ( ˆ ) f ( ˆ )] ˆσ = σ + σ, cmmn f ple vaance fact ˆσ, ˆσ = a pste vaance facts f epchs an espectvely f, f = egees f feem f epchs an f = f + f, sum f egees f feem f epchs an α = sgnfcance level (usually chsen as 0.05) If the abve test passes (.e., j < F(α,,f)) then the pnt s assume t be stable at a sgnfcance level α. Othewse, f the test fals (.e., j F(α,,f)) then the pnt s assume t be efme (mve). F futhe eang n bust meths, Caspay [987], Caspay an Butta [987], Chen [983], Chen et al. [990], Sngh [999], an Setan an Sngh [998c] ae ecmmene. Cnguency estng Meth Anthe gus meth knwn as cnguency testng has been apple n estmatng the splacements f all cmmn pnts n a mntng netwk. As ppse t the bust meths, cnguency testng wll teatvely emve ne atum pnt at a tme untl the cnguency test passes. Bascally the apte pceue f cnguency testng cnssts f the fllwng steps [Setan 997] () ansfmatn f splacement vect () an ts cfact matx (Q ) espectvely, f bth epchs nt a cmmn atum. () Detemnatn f stable atum pnts by cnguency testng [Fase an Gueng 985]. (3) Lcalzatn f efmatn thugh sngle pnt test, S-tansfmatn an cnguency testng. (4) Fnal testng f efmatn by a sngle pnt test. ansfmatn f Bth Epchs nt a Cmmn Datum Dung efmatn analyss by the cnguency testng (an bust) meths, t s mptant that the splacement vect, (equatn 7) an ts cfact matx, Q (equatn 8) ae efee t the same atum cmputatnal base. he ntal atum cul be ethe the efeence pnts n a efeence mntng netwk, a gup f selecte pnts base n a p nfmatn n a elatve mntng netwk. In ths stuy, the S-tansfmatn has been apple t tansfm matx an Q nt a cmmn atum efntn (ethe mnmum tace patal mnmum tace slutns) [Caspay 987; Cpe 987; Fase an Gueng 985; Bacs an eskey 990] = S (9) Q = SQ S (0) S = I - G( G WG) - G W () whee an Q ae the splacement vect an ts cfact matx espectvely base n the new atum cmputatnal base, G s the nne cnstants matx cnstucte epenng n the unn f the atum efects n the tw epchs an n the numbe f cmmn pnts, an W s the weght matx wth agnal value f ne f atum pnts an ze elsewhee. Matx S s symmetc nly f the mnmum tace slutns

6 Gematca. Vl. 55, N. 3, 00. (.e., all pnts n the netwk wee efne as atum). he gup f selecte atum pnts s then teste f ts stablty by usng cnguency testng. Cnguency estng n Intal Selecte Datum Pnts Cnguency testng s pefme t etemne whethe a gup f selecte atum pnts have sgnfcantly mve between the tw epchs. he null an altenatve hypthess f cnguency testng ae [Setan 995; Sngh 997; Cpe 987; Fase an Gueng 985; Bacs an eskey 990; Sngh 999] an whee + H : E( ) = 0 () (n sgnfcant efmatn f a gup f atum pnts) H a : E( ) 0 (3) (exstence f efmatn f a gup f atum pnts) he test statstc s atum nepenent,.e., + Q ω = h ˆ σ Q ~ F(α,h,f) (4), = splacement vect an ts cfact matx f the atum pnts cmmn n bth epchs h = ank ( Q ) = (n-) f a -D netwk wth n numbe f cmmn atum pnts an numbe f atum efects ˆσ = cmmn vaance fact Q =( Q + G G ) - - G ( G G G G ) - G, pseu nvese α = sgnfcance level, typcally α = 0.05 he nne cnstants matx G s cnstucte epenng n the numbe f cmmn atum pnts an the unn f the atum efects n the tw epchs. Geneally, matx s sngula, thus a pseu nvese s Q use f the cmputatn f ω. he null hypthess s accepte at the α level f sgnfcance f the test statstc (equatn 4) es nt excees the ctcal value f the F- stbutn (.e., ω < F(α,h,f)). he ejectn f the null hypthess ncates the exstence f efmatn n the gup f selecte atum pnts (.e., acceptng the altenatve hypthess). he falue f the test leas t lcalzatn f the efmatn n lcatng the atum pnt whse splacements cause the netwk t change n shape. he netwk s then tansfme nt a new cmputatnal base. he lcalzatn pceue cntnues untl all the emanng gup f atum pnts wee beng vefe as stable by the cnguency test. Lcalzatn f Defmatn If the cnguency test (equatn 4) s ejecte, lcalzatn f efmatn s then pefme. In ths stuy, the evelpe pceue f lcalzatn f efmatn cnssts f the fllwng steps,.e., the sngle pnt test n the atum pnts, S-tansfmatn nt a new cmputatnal base an the cnguency testng n the emanng cmmn atum pnts [Setan 995]. () Sngle Pnt est n the Datum Pnts he pupse f the sngle pnt test s t lcate the atum pnt wth the lagest statstcal value ( unstable atum pnt), an these pnt wll be elmnate fm the cmputatnal base atum. he sngle pnt test f a -D netwk,.e., neglectng the celatn between the atum pnts, s [Sngh 997; Setan 995] jq j j j = (5) ˆ σ whee j an Q j ae the splacement vect an ts cfact matx f each atum pnt j, an ˆσ s the ple vaance fact (equatn 8). Instea f the sngle pnt test (equatn 5), ne can als apply the ecmpstn f the

7 Gematca. Vl. 55, N. 3, 00. quaatc fm n e t etemne the sgnfcantly unstable atum pnt. Me etals ae gven n Caspay [987], Setan [995], Cpe [987], an Fase an Gueng [985]. he atum pnt wth the lagest j s assume t have cause the change f shape n the netwk, an t wll be elmnate fm the cmputatnal base. he netwk s then tansfme nt a new cmputatnal base whch s efne by the emanng atum pnts. () ansfmatn f the Netwk nt a New Cmputatnal Base An S-tansfmatn s then cae ut t tansfm an Q nt a new cmputatnal base, hence equatns 9 t can be wtten as = S (6) Q = S Q S (7) S = I - G( G WG) - G W (8) whee an Q ae the splacement vect an ts cfact matx efeng t the new atum efntn an W s the weght matx (wth agnal value f ne f the emanng atum pnts an ze elsewhee). he etane atum pnts ae then teste f stablty by cnguency testng. () Cnguency estng n a Gup f Remanng Datum Pnts he statstcal test s the same as n equatn 4, except that ths test s pefme nly n the emanng atum pnts. he hyptheses f ths test ae an H : E( ) = 0 (9) (n sgnfcant efmatn f a gup f emanng atum pnts), H a : E( ) 0 (30) (exstence f efmatn f a gup f emanng atum pnts) he test statstc then becmes (efe equatn 4) whee ω = Q + ˆ ( h k) σ an ~ F(α,h-k,f) (3) Q ae the splacement vect an ts cfact matx f the emanng atum pnts, an k s the numbe f pnts emve fm the atum efntn. If the null hypthess s ejecte (.e., ω F(α,h-k,f)), then the lcalzatn pcess wll be epeate untl the null hypthess s accepte. Othewse, f the null hypthess s accepte (.e., ω < F(α,h-k,f)) then the stablty nfmatn f all the cmmn pnts n the netwk s etemne thugh the fnal testng f efmatn. Fnal estng f Defmatn by Sngle Pnt est Afte the cnguency test (equatn 3) passes, the sngle pnt test s then cae ut n all the cmmn pnts n the netwk at a sgnfcance level α. he null an altenatve hypthess f ths test ae an s H : j = [x j y j ] = 0 (3) (n efmatn f each cmmn pnt n the netwk), H a : j = [x j y j ] 0 (33) (exstence f efmatn f each cmmn pnt n the netwk) he test statstc (efe equatns 8 an 5) j = j Q j ˆ σ j ~ F(α,,f) (34) If the abve test passes (.e., j < F(α,,f)) at a sgnfcance level α (typcally α = 0.05), then the pnt j s cnsee as stable (.e., the vect f splacements les wthn ts cnfence egn). Othewse, the ejectn f the abve test ncates that the pnt j s sgnfcantly unstable (.e., the vect f splacements les utse ts cnfence egn).

8 Gematca. Vl. 55, N. 3, 00. All the fnal atum pnts shul be vefe as stable by the sngle pnt test. If thee stll exst any unstable atum pnts, then the atum pnt wth the lagest statstcal value (equatn 34) shul be elmnate fm the atum, an the lcalzatn pcess has t be epeate. F a bette ntepetatn f the splacements f the pnts, a gaphc pesentatn n the fm f splacement vect an e ellpses s use. hs gaphc pesentatn s useful n vefyng the esults f the sngle pnt test. Geneally, bth the gaphc pesentatn an the sngle pnt test shul pve entcal esults. One can see that the fmulatns an pceues nvlve f the cnguency testng meth ae much me cmplex cmpae t the bust meths. he bust an cnguency testng meths ae mptant tls f etemnng the ten f mvements f all the cmmn pnts n the netwk, whch fm a bass f pelmnay entfcatn f the efmatn mels ung efmatn mellng. Mellng f Defmatn In ths stuy the pceue f mellng f efmatn s base n the well knwn genealze meth evelpe at the Unvesty f New Bunswck, Canaa [Chen 983]. Defmatn mellng can be cae ut by usng ethe the bsevatn ffeences appach splacement (cnate ffeences) appach. hs pape wll fcus nly n the splacement appach. Hweve, the nteeste eae can efe t Chen [983] f a etal explanatn f the tw appaches. he genealze meth cnssts f thee basc pcesses () Pelmnay entfcatn f the efmatn mel () Estmatn f the efmatn paametes (3) Dagnstc checkng f the efmatn mels an the fnal selectn f the best mel Pelmnay Ientfcatn f the Defmatn Mel he entfcatn f the efmatn mels s base n a p nfmatn n ten analyss f the splacements. In a -D analyss the fllwng efmatn paametes must be cnsee () w cmpnents f the g by splacement (a an b ) () A tatn paamete n xy axes (ω(x,y)) () w nmal stan cmpnents n x an y axes (ε x (x,y) an ε y (x,y)) an sheang stan n xy axes (ε xy (x,y)) he efmatn f a blck s fully escbe f a splacement functn (x,y) s gven f the whle blck. In a geneal case the splacement functn s etemne thugh a plynmal appxmatn f the splacement fel as x = a + a x + a y + a 3 xy + a 4 x + (35) y = b + b x + b y + b 3 xy + b 4 x + whee x an y ae the splacements n the x an y axes, an x an y ae the cnates f the cmmn pnts (.e., ethe appxmate estmate cnates). Hweve, n ths stuy nly the lnea mel has been taken nt cnseatn an equatn (35) can be futhe smplfe as x = a + a x + a y (36) y = b + b x + b y x = a + ε x x + ε xy y - ωy (37) y = b + ε xy x + ε y y + ωx Equatn (37) can be wtten n matx fm as (gnng tatn paamete ω) = B ĉ (38) x y 0 x 0 y = 0 0 y x a b ε x (39) ε y ε xy whee s the vect f splacements (equatn 7), B s the esgn matx an ĉ s the vect f

9 Gematca. Vl. 55, N. 3, 00. unknwn efmatn paametes ( ceffcents f the plynmals). Equatn (38) s well knwn as the efmatn mel. Sme examples f the typcal efmatn mels can be fun n Chzanwsk et al. [983], Chen [983], an Chzanwsk et al. [986]. In ths stuy, sx typcal efmatn mels have been apte an apple.e., () N glbal efmatn (x = 0, y = 0) () Stable pnts (x = 0, y = 0) (3) Sngle pnt mvement (x = a, y = b ) (4) Rg by mvement (x = a, y = b ) (5) Hmgeneus stan (x = ε x x + ε xy y, y = ε xy x + ε y y) (6) Rg by plus hmgeneus stan (x = a + ε x x + ε xy y, y = b + ε xy x + ε y y). Estmatn f the Defmatn Paametes he efmatn mel equatn (38) s base n the Gauss-Makv mel whch can be wtten as + vˆ = B ĉ (40) whee vˆ s vect f esuals. In e t etemne the vect ĉ, the numbe f knwn splacements f the cmmn pnts must be at least equal t the numbe f unknwn efmatn paametes n the efmatn mel. If the numbe f knwn splacements s lage than that f the unknwn efmatn paametes, then the vect ĉ s etemne thugh LSE. he slutn s then btane fm ĉ = ( B P B) - B P (4) whee P s the weght matx an s calculate as [Chen et al. 990; Chen 983; Sngh an Setan 999b, c; Sngh an Setan 998; Sngh 999] () f all the cmmn pnts n the netwk: P =[S Q S +G( G G) - G ] - -G( G G) - G (4) wth P = N (N + N + G G ) - N (43) S = I - G( G G) - G (44) whee, Q s gven n equatn (8) an N an N ae the ceffcent matx (.e., full matx) f epchs an espectvely. Matx N, N, G an Q ae fme wth espect t all the cmmn pnts n the netwk. () f a ptn f cmmn pnts n the netwk: P =[S G wth Q S +G ( G G ) - G ] - -G ( G G ) - (45) P = N ( N + N + G G ) - N (46) S = I - G ( G G ) - G (47) N N N = (48) N N N N N = N N (49) N = N - N N N (50) N = " N - whee matx S, " N N Q, G, " " N (5) N an fme nly f a ptn f cmmn pnts. he cfact matx f ĉ s ĉ " N ae Q = ( B P B) - (5) he pceue f calculatn as shwn abve shul be fllwe n e t av any numecal pblems. Accng t Chen [983] the tatn paamete ω can be mtte (cnsee as beng ze) fm the efmatn mel (equatn 40) f the weght matx P s calculate as n equatns (4) (43) an (45) (46). Only n ths case the mssn f ω s

10 Gematca. Vl. 55, N. 3, 00. justfe when n extenal entatn f the netwk s nclue n the bsevables. he tatn paamete ω nly plays the le f a atnal nusance paamete wthut any pactcal meanng. Me etals ae gven n Chen [983]. Dagnstc Checkng f the Defmatn Mel an Selectn f the Best Mel he glbal apppateness f the efmatn mel can be teste usng the quaatc fm f the esuals vˆ. he glbal test s base n the fllwng hyptheses an H : H a : ˆσ ĉ = ˆσ ĉ ˆσ (53) ˆσ (54) whee ˆσ s the cmmn vaance fact an ˆσ ĉ s calculate as [Chen 983; Chzanwsk et al. 983; Kuang 996; Sec 985; Sngh an Setan 998; Sngh an Setan 999b, c; Sngh 999] whee ˆσ ĉ = vˆ P vˆ f c (55) vˆ = B ĉ -, vect f esuals, equatn (40) P = weght matx (equatn 7) f c = n - - m c, wth n numbe f pnts n fmng matx P, s numbe f atum efects an m c s numbe f paametes t be estmate, ĉ. he statstc then becmes ˆ σ ĉ = ˆ σ = vˆ P vˆ c ˆ σ f ~ F(α,f c,f) (56) he test passes the null hypthess s accepte f the statstc es nt excee the ctcal value (.e., F(α,f c,f)) at a sgnfcance level α (typcally α = 0.05). Othewse the null hypthess s ejecte f the statstc excees the ctcal value (.e., > F(α,f c,f)). he efmatn mel s acceptable f the null hypthess s nt ejecte at a sgnfcance level α. he sgnfcance f the nvual paamete ĉ a gup f paametes ĉ whch s a subset f ĉ (equatn 4), s gven by the fllwng hyptheses [Chen 983; Kuang 996; Sngh 999] H : ĉ = 0 an H a : ĉ 0 (nvual paamete) (57) H : ĉ = 0 an H a : ĉ 0 (a gup f paametes) (58) he lcal test becmes [Chen 983; Kuang 996; Chzanwsk et al. 983; Chzanwsk et al. 986; Sec 985; Sngh an Setan 999b; Sngh 999] whee = cˆ ~ F(α,,f) (nvual ˆ σ qcˆ paamete) (59) cˆ Q cˆ cˆ g = ˆ σ u ~ F(α,u,f) (a gup f paametes) (60) q ĉ = the th agnal element f matx Q ĉ ĉ = nvual paamete ˆσ = cmmn vaance fact f = sum f egees f feem f epch an epch ĉ = subset f ĉ u = numbe f paametes n ĉ Q = submatx f Q ĉ ĉ α = sgnfcance level, usually α = he null hypthess (equatns 57 58) wll be ejecte at a sgnfcance level α f > F(α,,f) g > F(α,u,f) (6)

11 Gematca. Vl. 55, N. 3, 00. he teste paametes ae expecte t be statstcally sgnfcant. If the paametes ae nt sgnfcant, then a new efmatn mel nees t be selecte an e-evaluate [Kuang 996]. hs pcess s epeate untl the glbal test passes an all the paametes ae statstcally sgnfcant. Snce the behav f the efmable by s usually nt cmpletely knwn, thee s ften me than ne pssble mel that may be apppate. he best mel s selecte base n the fllwng ctea () he mel passes the glbal test an all the paametes nvlve ae statstcally sgnfcant () If me than ne mel satsfes ctea (), the mel wth the fewest paametes s then selecte () On the bass f the estmate ten f mvements a p the nfmatn (makes mechancal sense) Pgam Package NEDEFAN he apte pceue (.e., netwk ajustment, ten analyss an mellng f efmatn) has been mplemente n a pgam package knwn as NEDEFAN (NEwk an DEFmatn ANalyss) evelpe at the Cente f Inustal Measuement an Engneeng Suveyng, Unvest eknlg Malaysa [Sngh 999]. NEDEFAN s evelp usng FORRAN77, an can hanle nly -D mntng netwks. he pgam package cnssts f thee mules knwn as COMPU, DEFORM an SRANS [Setan an Sngh 997; Setan an Sngh 998a; Sngh 997]. Pgam COMPU s esgne f the netwk ajustment f each epch, wth thee man ptns,.e., () LSE f each epch () Pe-analyss () Vaance cmpnent estmatn Pgam DEFORM s esgne f efmatn analyss, an cnssts f seven man ptns,.e., () en analyss by cnguency testng meth () en analyss by IWS meth () en analyss by LAS meth (v) en analyss by ect cnate ffeences [Sngh an Setan 999a] (v) Senstvty analyss f estmatng the mnmum etectable efmatn f a mntng netwk (v) Mellng f efmatn (v) Stan analyss Only ptn () f pgam COMPU an ptns (), (), () an (v) f pgam DEFORM ae mentne n ths pape. Pgam SRANS s an atnal pgam f S-tansfmatns f LSE esults f a sngle epch fm ne atum efntn t anthe. he nput f pgam COMPU ae the bsevatns an the appxmate cnates fles f each epch. Pgam COMPU puces LSE summay fles, efmatn fles an plt fles f epchs an. he efmatn fle f each epch cntans nfmatn f a pste vaance fact, egees f feem, numbe f atum efects, types f atum efects, appxmate cnates, estmate cnates an the lwe tangle f the cfact matx f the estmate cnates. he efmatn fles f epch an ae then use as nput f pgam DEFORM f efmatn analyss. he efmatn fle f each epch s als use as nput f pgam SRANS f tansfmatn f LSE esults nt a new atum efntn. All the abve pgams pve plt fles whch ae use as nput f a specal pgam calle Gaphcal Pesentatn f Defmatn Suvey Vesn (GPDSV), f gaphcal splay f the splacement vects an the stana ellpses espectvely [Sngh 999]. Defmatn Analyss f a Dam Mntng Netwk A knwn ata f a am mntng netwk (efe Fgue ), taken fm Caspay [987], have been use t test the evelpe pceue an the pgam package.

12 Gematca. Vl. 55, N. 3, 00. y x 6 Refeence pnts (,,3,4,6,7,9) Object Pnts (0,,,3,4) Fgue : A am mntng netwk [Caspay 987]. he mntng netwk cnssts f twepchs f bsevatns wth pnts.e., 7 efeence pnts (,, 3, 4, 6, 7 an 9) an 5 bject pnts (0,,, 3 an 4). Pnts 0 thugh 4 ae lcate n the cest f the am thus epesent the bject une nvestgatn. Each epch cnssts f 49 ectns (stana e f secn) an 6 hzntal stances (stana e f 0.3 [mm]). Smulate efmatn wee gven t pnts 3,, an 3 as shwn n able. able : Smulate efmatn [Caspay 987]. Pnt x (mm) y (mm) Ajustment f the Netwk Netwk ajustment f each epch s cae ut by usng pgam COMPU by fxng cnates x, y an x (mnmum cnstants slutn) an leang t 9 egees f feem an 6 paametes (.e., cnates an 5 entatn paametes). he sgnfcance levels f the Ch-squae an au tests wee chsen as he ctea f cnvegence was set t [m], an bth epchs cnvege at the secn teatn. he estmate vaance facts wee 0.89 an f the fst an secn epch espectvely. he cmbne vaance fact s he LSE esults f each epch passe bth the glbal an lcal tests. he aveage eunancy numbe s 0.53 f bth epchs, thus ncatng that the netwk pssesses a hgh egee f elablty. en Analyss f the Dsplacement Fel he ten f mvements f the mntng netwk ae etemne by usng pgam DEFORM wth thee slutns as belw () en analyss by the cnguency testng meth wth pnts,, 3, 4, 6, 7 an 9 efne as a atum () en analyss by IWS meth wth pnts,, 3, 4, 6, 7 an 9 efne as a atum () en analyss by LAS meth wth pnts,, 3, 4, 6, 7 an 9 efne as a atum. he sgnfcance level f efmatn etectn s specfe as he tleance value an the lwe bun value wee taken as [m] an [m] espectvely. he test n the vaance at passes at 0.05 sgnfcance level (.e.,.30 <.86), thus ncatng the cmpatablty between the tw epchs. he cnguency test (equatn 4) fale f slutn (), thus cnfmng the exstence f efmatn f a gup f selecte atum pnts. he lcalzatn pcess then emves unstable atum pnts fm the cmputatnal base ne at a tme untl the cnguency test passes. hs pceue emve pnt 3 an 6, esulte n 5 atum pnts (.e.,,, 4, 7 an 9) f fnal cmputatn f the splacements. All atum pnts passe the sngle pnt test (equatn 34) an wee cnfme as stable. Slutn () vefe pnts 3, 6,, an 3 as sgnfcantly efme an the est f the pnts as stable. Slutns () an () cnvege at the secn teatn. Slutn () vefe pnts 3, an 3 as mve an the est f the pnts as stable. Whle slutn () vefe pnts 3, 6, an 3 as mve an the est f the pnts as stable. he cnate ffeences btane f the thee slutns ae shwn n able.

13 Gematca. Vl. 55, N. 3, 00. able : Estmate splacements f slutns () t () (* Pnt vefe as mve). pnt x (mm) slutn () slutn () slutn () y sp. x y sp. x y (mm) vect. (mm) (mm) Vect. (mm) (mm) sp. vect. (mm) (mm) (mm) * * * * * * * * * * * * mel n. estmate efmatn paametes an the statstcal testng f a gup f paametes x 3 = -0.8 y 3 = > 3.3 (sgnfcance) x 6 = -0.5 y 6 = > 3.3 (sgnfcance) x = -0.8 y = > 3.3 (sgnfcance) x = 0.4 y = > 3.3 (sgnfcance) x 3 = 0. y 3 =..55 > 3.3 (sgnfcance) x 3 = -0.7 y 3 = > 3.3 (sgnfcance) x = 0.5 y = > 3.3 (sgnfcance) x 3 = 0. y 3 = > 3.3 (sgnfcance) x 3 = -0.9 y 3 = > 3.3 (sgnfcance) x 6 = -0.5 y 6 = > 3.3 (sgnfcance) x = 0.5 y = > 3.3 (sgnfcance) x 3 = 0. y 3 = > 3.3 (sgnfcance) glbal test passe 0.55 <.96 fale.64 >.84 fale.3 >.89 Selectn f the Best Dsplacements 3 he stablty nfmatn (.e., stable an mve pnts) vaes between the thee slutns (.e., cnguency testng, IWS an LAS). he best splacements s then selecte base n the esults f mellng f sngle pnt mvement. hee mels wee selecte as belw () Pnts 3, 6,, an 3 ae expeencng sngle pnt mvement n sepaate blcks, whle the est f the pnts as a stable blck () Pnts 3, an 3 ae expeencng sngle pnt mvement n sepaate blcks, whle the est f the pnts as a stable blck () Pnts 3, 6, an 3 ae expeencng sngle pnt mvement n sepaate blcks, whle the est f the pnts as a stable blck. Y X 7 Fgue : Gaphcal splay f the best splacements tgethe wth the 95% cnfence ellpses f mel (nt t scale) Mellng f Defmatn 6 he glbal test (equatn 3) fale f mels () an (). Mel () was theefe selecte as the best mel because the glbal test passe an all the gup f paametes ae statstcally sgnfcant. he esults f sngle pnt mvement ae shwn n able 3. Fgue epct the best splacements va the cnguency testng meth (mel ). able 3: Results f mellng f sngle pnt mvement f mel () t (). A ten analyss f the splacement fel (efe Fgue ) leas t a chce f seveal pssble efmatn mels as belw () N glbal efmatn mel n all the pnts n the netwk () Pnts 0,,, 3 an 4 ae expeencng hmgeneus stan () Pnts 3, 6 an ae expeencng sngle pnt mvement n sepaate blcks, pnts an 3 ae expeencng g by mvement an the est f the pnts as a stable blck (v) Pnts 3 an 6 ae expeencng sngle pnt mvement n sepaate blcks, pnts 0,,, 3 an 4 ae expeencng g by mvement plus hmgeneus stan an the est f the pnts as a stable blck.

14 Gematca. Vl. 55, N. 3, 00. he esults f the selecte efmatn mels ae shwn n able 4. All mels (except mel ) fale the glbal test. Mel () was then selecte as the best mel because t passes the glbal test an all the gup f paametes ae statstcally sgnfcant. able 4: he estmate efmatn mels. 3 3 mel n. Estmate efmatn paametes an the statstcal testng f a gup f paametes glbal test fale.85 >.49 ε x = 0.98 μstan ε y = μstan ε xy = 7.37 μstan 6.70 >.70 (sgnfcance) fale.65 >.5 6 x 3 = -0.8 mm y 3 = -0.5 mm 35.9 > 3.3 (sgnfcance) x 6 = -0.5 mm y 6 = 0. mm 4. > 3.3 (sgnfcance) x = -0.8 mm y = 0.5 mm.46 > 3.3 (sgnfcance) a = 0.3 mm b =.4 mm 6.60 > 3.3 (sgnfcance) v x 3 = -. mm y 3 = -0.5 mm 7.5 > 3.3 (sgnfcance) x 6 = -0.6 mm y 6 = 0. mm 6. > 3.3 (sgnfcance) a = -0.9 mm b = -.8 mm 0.75 > 3.3 (sgnfcance) passe 0.60 <.89 fale 4.36 >.9 unefme efme at epch (Nte:- nt t scale) Fgue 3: Gaphcal pesentatn f the best efmatn mel. ε x = 8. μstan ε y = 9.7 μstan ε xy = -.4 μstan 7.50 >.75 (sgnfcance) heefe t can be cnclue that the by was expeencng g by mvement f pnts an 3, whle pnts 3, 6 an ae expeencng sngle pnt mvement n sepaate blcks. he esults f mel () (efe able 4) ae smla t the smulate efmatn (.e., able ). he best mel s gaphcally pesente n Fgue 3. Caspay [987] nly estmate the ten f mvements f ths mntng netwk (.e., usng cnguency testng an bust (LAS) meth) an mellng f efmatn was nt pefme. heefe n futhe cmpasn can be mae wth Caspay [987] egang the esults f mellng (efe able 4). Cnclusns hs pape has pesente a pceue f efmatn analyss, whch cmpses f the netwk ajustment f each epch, ten analyss an the mellng f efmatn. he evelpe pceue pves a systematc stepby-step analyss an t can be apple t ethe efeence elatve mntng netwk. he sx typcal efmatn mels apple n ths stuy wee fun t be ptmal enugh n etemnng the behav f the by bject une nvestgatn. Hweve, the secn an th e mels (equatn 35) shul als be taken nt cnseatn n e t btan a eepe nsght f the behav f the by. Dung ten analyss f the splacement fel, seveal meths have been cmpae n e t av wng ntepetatn f the ten f mvements f the bject une nvestgatn. he best splacements ae then selecte base n the esults f mellng f sngle pnt mvement befe efmatn mellng s pefme. he esults btane f the am mntng netwk shw that the evelpe pceue an the pgam package ae applcable f the gemetcal analyss f efmatn.

15 Gematca. Vl. 55, N. 3, 00. Acknwlegement hs eseach s pat f the eseach pject Vt 7070 spnse by IRPA. Refeences Bacs, Z.F., an W.F. eskey Defmatn analyss f suvey netwks wth nteactve hypthess testng an cmpute gaphcs, CISM JOURNAL ACSGC, Vl. 4, N. 4, pp Caspay, W.F Cncepts f Netwk an Defmatn Analyss, st. e., Schl f Suveyng, he Unvesty f New Suth Wales, Mngaph, Kensngtn, N.S.W. Caspay, W.F., an H. Butta Rbust estmatn n efmatn mels, Suvey Revew, Vl. 9, N. 3, pp Chen, Y.Q Analyss f Defmatn Suveys - A Genealze Meth, echncal Rept N. 94, Depatment f Suveyng Engneeng, Unvesty f New Bunswck, Feectn, N.B. Chen, Y.Q., A. Chzanwsk, an J.M. Sec A stategy f the analyss f the stablty f efeence pnts n efmatn suveys, CISM JOURNAL ACSGC, Vl. 44, N., pp Chzanwsk A Getechncal an the nn-geetc meths n efmatn measuements, Pceengs f the Defmatn Measuements Wkshp, Bstn, Massachusetts, 3 Octbe- Nvembe, Massachusetts Insttute f echnlgy, Cambge, M.A., pp Chzanwsk A., Y.Q. Chen, an J.M. Sec On the stan analyss f tectnc mvements usng fault cssng geetc suveys, ectnphyscs, 97, pp Chzanwsk A., Y.Q. Chen, an J.M. Sec Gemetcal analyss f efmatn suveys, Pceengs f the Defmatn Measuements Wkshp, Bstn, Massachusetts, 3 Octbe- Nvembe, Massachusetts Insttute f echnlgy, Cambge, M.A., pp Cpe, M.A.R Cntl Suveys n Cvl Engneeng, Wllam Cllns Sns & C. Lt., Lnn. Cpe, M.A.R., an P.A. Css. 99. Statstcal cncepts an the applcatn n phtgammety an suveyng (cntnue), Phtgammetc Rec, Vl. 3, N. 77, pp Fase, C.S. an L. Gueng he analyss f phtgametc efmatn measuements n tutle muntan, Phtgammetc Engneeng an Remte Sensng, Vl. 5, N., pp Kuang, S.L Geetc Netwk Analyss an Optmum Desgn: Cncepts an Applcatns, Ann Ab Pess, Inc., Chelsea, Mchgan. Sec, J.M Implementatn f a Genealze Meth f the Analyss f Defmatn Suveys, echncal Rept N. 7, Depatment f Suveyng Engneeng. Unvesty f New Bunswck, Feectn, N.B. Setan, H Functnal an Stchastc Mels f Gemetcal Detectn f Spatal Defmatn n Engneeng: A Pactcal Appach, Ph.D. hess, Depatment f Cvl Engneeng, Cty Unvesty, Lnn. Setan, H A flexble analyss pceue f gemetcal etectn f spatal efmatn, Phtgammetc Rec, Vl. 5, N. 90, pp Setan, H. an R. Sngh Pengesanan efmas -D secaa gemetkal engan kaeah ujan knguens, Buletn Genfmas, Vl., N., pp Setan, H. an R. Sngh. 998a. -D gemetcal analyss f efmatn, Semna Penyelkan alam Bang Elektk, Elektnk, Aeangkasa, eknlg Maklumat an elekmunkas, 8 Mach, Unvest eknlg Malaysa, Jh Bahu, pp Setan, H. an R. Sngh. 998b. Pengesanan efmas secaa gemet menggunakan kaeah bust, Pesente at 998 Annual Semna f Genfmatn Engneeng (Genfmatn 98), 7-8 Septembe, Unvest eknlg Malaysa, Kuala Lumpu. Setan, H. an R. Sngh. 998c. Cmpasn f bust an nn-bust meths n efmatn suveys. he 7th. JSPS-VCC Semna n Integate Engneeng, 7-8 Decembe, Unvest Malaya, Kuala Lumpu. Setan, H. an R. Sngh Cmpasn f Dffeent Stateges f en Analyss f the Dsplacement Fel n Defmatn Suveys, Unpublshe. Sngh, R Analss Defmas -D engan kaeah Ujan Knguens, Bachel Degee Pject, Faculty f Genfmatn Scence an Engneeng, Unvest eknlg Malaysa, Jh Bahu. Sngh, R Pelaasan an Analss Jangan Pengawasan Untuk Pengesanan Defmas Secaa Gemet, M.Sc. hess, Unvest eknlg Malaysa, Jh Bahu. Sngh, R. an H. Setan Cmpute pgam f stan analyss f geetc mntng netwk, Pesente at Malaysan Scence an echnlgy Cngess 98, 0- Nvembe, Unvest Sans Malaysa, Pulau Pnang. Sngh, R. an H. Setan. 999a. Cmpasn f ffeent atum efntns n etectn f efmatn f a geetc mntng netwk, Pesente at Reseach Semna n Cnstuctn, Mateals an Envnmental echnlgy, 3-4 Febuay, Unvest eknlg Malaysa, Jh Bahu. Sngh, R. an H. Setan. 999b. Gemetcal analyss f efmatn - Ajustment, ten analyss an mellng, Pesente at Wl Engneeng Cngess an Exhbtn (WEC99), 9- July, Kuala Lumpu. Sngh, R. an H. Setan. 999c. Stan Analyss f Geetc Mntng Netwk n Defmatn Suveys, Unpublshe. eskey, W.F. an.r. Pte An ntegate meth f mntng the efmatn behav f engneeng stuctues, Pceengs f 5th. Intenatnal (FIG) Sympsum n Defmatn Measuements an 5th. Canaan Sympsum n Mnng Suveyng an Rck Defmatn Measuements, E. by Chzanwsk, A. an W. Wells, Depatment f Suveyng Engneeng, Unvesty f New Bunswck, Feectn, N.B., pp Wlf, P.R. an C.D. Ghlan Ajustment Cmputatns: Statstcs an Least Squaes n Suveyng an GIS, Jhn Wley & Sns, Inc., New Yk. Auths

16 Gematca. Vl. 55, N. 3, 00. D. Halm Setan s an asscate pfess at the Faculty f Genfmatn Scence an Engneeng, Unvest eknlg Malaysa (UM). He hls a B.Sc. (Hns.) n Suveyng an Mappng Scences fm Nth East Lnn Plytechnc (984), a M.Sc. n Geetc Scence fm Oh State Unvesty, USA (988) an a Ph.D fm Cty Unvesty, Lnn (995). Hs cuent eseach nteest s n efmatn mntng, least squaes estmatn an nustal metlgy. M. Ranjt Sngh hls Cetfcate n lan suveyng fm the Sultan Haj Ahma Shah Plytechnc (POLISAS) n Pahang, Malaysa, n 99, a Dplma n lan suveyng fm the Ungku Oma Plytechnc (PUO) n Iph, Malaysa, n 994, an a BSc egee n lan suveyng, wth a fst class hns, fm the Unvest eknlg Malaysa (UM) n Jh Bahu, Malaysa, n 997. He then cntnue hs MSc n the fel f engneeng suveyng at the Cente f Inustal Measuement an Engneeng Suveyng (CIMES), UM, Jh Bahu, Malaysa. He s the c-auth f abut 0 techncal papes n geetc an engneeng suveys,.e., netwk ajustment an efmatn analyss.