Chapter 6 The Effect of the GPS Sytematc Error on Deformaton Parameter 6.. General Beutler et al., (988) dd the frt comprehenve tudy on the GPS ytematc error. Baed on a geometrc approach and aumng a unform atellte ky dtrbuton, they analytcally analed the effect of relatve tropophere, neglected tropophere, onophere, wrong GM-value (gravtatonal contant tme the ma of the Earth), wrong fxed coordnate and along track orbtal error on the GPS reult. Later, Santerre (99) tuded the mpact of the GPS non-unform atellte ky dtrbuton on thee reult. Kanuth (997) howed, n h tudy on the reference frame realzaton error on relatve potonng n regonal GPS network, that: frtly; bae of the reference frame element (atellte orbt, earth orentaton parameter and fducal pont coordnate) wll not only affect the abolute but alo the relatve poton of the network taton. Secondly, even mproper electon of fducal pont ha a mlar mpact on the relatve poton a the bae of the fducal pont coordnate. He alo howed that, the fducal pont coordnate bae and ther mproper electon have the larget mpact on the relatve pont poton among the other tuded reference frame ytematc error. Derman and Grafarend (993) alo analytcally analed the mpact of reference frame varaton on deformaton parameter. In the poneerng work of Beutler et al. (988), ytematc error n GPS meaurement have been clafed to two group: ) The frt group nclude ytematc error that produce heght bae n the network poton. Thee ytematc error are termed a Cla bae. ) The econd group nclude error that reult n cale bae n the network oluton. Th group of error ntroduced a Cla ytematc error. In th chapter, the effect of thee ytematc error on the parameter of deformaton wll be analytcally worked out. Th wll be done for both nfntemal and fnte deformaton. Snce by defnton tran the change n length per unt of length, t havng a cale nature.
Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter 84 Therefore, Cla ytematc error are expected to have a greater mpact on the tran tenor component compared to the Cla bae. Th encourage one to frtly anale the effect of Cla bae. Ioparametrc repreentaton of deformaton provde an eay and effcent way for th tudy. Therefore, the correpondng obervaton euaton, that : e l + e m + ezz n + e mn + ezx nl + e lm (6.a) ( ) ( ) zz zx + E l + + E m + + E n + E mn+ E nl+ E lm (6.b) are bac euaton n th tudy. Here, mlar to the notaton of Chapter 3, and are the baelne length before and after deformaton, l, m and n are t drecton cone, e, etc are the nfntemal tran component and fnally E, etc are the fnte one. If the vector of the unbaed etmate of baelne length at epoch and t correpondng vector of baed etmate due to ome ytematc error, theoretcally; the baed and unbaed etmate of the baelne length gve the baed ( e, e etc) veru unbaed etmate ( e, e, etc) for tran parameter repectvely. Clearly, for Cla and Cla ytematc error we can wrte: Cla : Cla : ( ) ( ) + ( ) (6.a) ( ) ( z ) + dz (6.b) z ( ) + (6.c)
85 Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter where and,, are the element of correpondng vector, the ytematc heght ba dz the unbaed heght dfference between the end pont of a baelne. z 6.. Cla Sytematc Error 6... Infntemal Stran Let u aume that the element of the vector of baelne length dfferent calng error, whch contruct the coordnate of the error vector are contamnated by at two dfferent epoch,. Theoretcally, at each pont of the network and n each epoch one can etmate the vrtual tran e,, etc. by formng the followng ytem of lnear euaton: e, e e l + e m + e n + e m n + e n l + e l m,,..., N (6.3a) () zz zx (6.3b) Where N the number of contrbutng pont to be ued for etmatng the tran parameter at the poton of the network taton, and are the baed and unbaed etmate of the th baelne length n the th epoch repectvely and fnally, l, m and n are the drecton cone of the th baelne n both epoch and. Smlarly, the unbaed etmate of the baelne length theoretcally provde u the unbaed tran e, e etc. of the tran tenor through: e l + e m + e n + e m n + e n l + e l m,,..., N (6.4a) zz zx (6.4b)
Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter 86 To anale the effect of the calng error (6.c) on nfntemal deformaton, we hould try to etablh a functonal relaton between the unbaed deformaton of Euaton (6.4) and the baed one. The baed deformaton parameter e, e, e zz, etc are theoretcally derved from the baed etmate of the baelne length,.e., by olvng the over determned multaneou ytem of euaton: e l + e m + e n + e m n + e n l + e zz zx l m,,..., N (6.5a) (6.5b) If (, ) an calng error on a ngle baelne of length, we can wrte: ( + ) ( + ) (6.6) Applyng Euaton (6.6) to Euaton (6.5b) reult: ( + ) ( + ) ( + ) ( + ) ( + ) + (6.7a) Ung Euaton (6.4b) and (6.c) gve ( ) + + ( + ) ( + ) + + ( + ) ( + ) (6.7b) (6.7c)
87 Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter Applyng Euaton (6.3b) to the relatve length change of the frt epoch n Euaton (6.7c) gve re to the euaton: () + + ( + ) ( + ) (6.7d) Agan, by applyng Euaton (6.3b) to the relatve length change of the econd epoch n E- uaton (6.7d), th euaton can be further reduced to: () () () + + ( + ) ( + ) (6.7e) Fnally, the applcaton of euaton (6.6) to euaton (6.7e) reult: + + + ( + ) ( ) + + (6.7f) Expreng,, () and () n term of the correpondng tran parameter that are gven n Euaton (6.5), (6.4) and (6.3), one wll come up wth the followng et of multaneou euaton for each baelne: + ( ) + + + ( ) + + + zz zz ( ) + zz zz + + ( ) + + (6.8)
Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter 88 + xz xz ( ) + xz xz + + ( ) + + 6... Fnte tran Smlar to nfntemal deformaton, to anale the effect of calng error on fnte deformaton we hould try to etablh a functonal relaton between the unbaed tran derved from: + E l + + E m + + E n + E mn + E nl + E l m,,..., N (6.9a) ( ) ( ) ( zz ) zx (6.9b) and the baed one that are gven by: + E l + + E m + + E n + E mn+ E nl+ E lm,,..., N (6.0a) ( ) ( ) ( zz) zx (6.0b) Ung Euaton (6.6) and acceptng the approxmaton O ( ) uare of ytematc error, we can wrte:,.e. gnorng product and + + (6.a)
89 Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter ( ) O + + + + + + O ( ) [( ) ] ( ) [ ] ( [ ]) (6.b) + + O ( + ) + O ( ) (6.c) + + O And fnally: + (6.a) Subttutng Euaton (6.9a) and (6.0a) n Euaton (6.a) gve: E E + E E + Ezz E zz + E E + Exz E xz + E E + (6.b) 6.3. Cla Sytematc Error 6.3.. Infntemal tran Ung Euaton (6.5b) and (6.a) we can wrte: ( ) + ( ) ( ) + ( ) / / / ( ) + ( ) (6.3a)
Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter 90 Where and are defned n Euaton (6.b). Bnomal expanon of contrbutng term n Euaton (6.3a) gve: ( ) ( ) ( ) ( ) ( ) ( ) 4 ( ) ( ) + + O 3 ( ) 4 4 + + O + + O 3 3 (6.3b) Therefore: ( ) ( ) + + ( ) (6.3c) z Subttutng Euaton (6.b) n th euaton and ung the approxmaton O gve: + z dz z dz z dz + (6.4a) or approxmately: z dz z dz z dz z dz + +... ( ) ( ) ( ) (6.4b)
9 Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter 6.3.. Fnte tran Ung Euaton (6.0b) and (6.a) one can wrte: ( ) ( ) ( ) + ( ) + (6.5a) z Agan, ubttutng Euaton (6.b) n th euaton and ung the approxmaton O gve: + z + z dz ( ) dz ( ) (6.5b) + +... ( ) ( ) dz dz z z (6.5c) And fnally: dz dz + z z ( ) ( ) (6.6) Snce to a good approxmaton: dz dz (6.7a)
Chapter 6: The Effect of the GPS Sytematc Error on Deformaton Parameter 9 ( ) ( ) dz dz (6.7b) Euaton (6.6) can be further mplfed to + p ( ) ( p ) z z dz (6.8) where p or.