The Applicatio of Coordiate Similarity Trasformatio Model for Stability Aalysis i High-precisio GS Deformatio Moitorig Network Jimig Guo, Migdua Zhou, Chao Wag, Liahui Mei School of Geodesy ad Geomatics, Wuha Uiversity, 9 Luoyu Road, Wuha, Chia - jmguo@sgg.whu.edu.c KEY WORDS: GS Deformatio Moitorig, Stability Aalysis, Coordiate Similarity Trasformatio Model, Hydropower Statio ABSTRACT: This paper firstly aalyzes the theory of coordiate similarity trasformatio, ad a ovel method of stability aalysis for datum poits applied i high-precisio GS deformatio moitorig etwork is the put forward. The coordiate similarity trasformatio model is adopted to calculate trasformatio parameters for statio coordiates of two adjacet periods. By comparig the trasformed results ad the etwork adjustmet solutios, the statio stability is verified. I order to judge the stability of statios, the threshold of statio stability ad statistical test based o variace ratio are developed. It is applied to stability aalysis of three periods datum poits i a A-order GS deformatio moitorig etwork for hydropower statio to verify the feasibility ad effectiveess of this method.. INTRODUCTION Stability aalysis of GS deformatio moitorig etwork is oe of the most importat part of deformatio observatio data processig. For the periodical measuremets of high-precisio GS deformatio moitorig etwork, the best situatio for deformatio aalysis is that a fixed datum is set up for all the time periods (Zhou Migdua et al, 0). The prerequisite of the above method is that adjustmet datum remais stable amog all the periods of observatio. But the stability of datum poits i high-precisio GS deformatio moitorig etwork eeds to be proved. We ca ot completely esure that the adjustmet datum poit still remais stable because of the impact of surroudig eviromet. If the datum is ot stable, ad we still adoptig the datum as fixed, the results of deformatio aalysis would be distorted (Gao Yapig, 00). I order to seek for stability aalysis methods of deformatio moitorig etwork, may scholars ad experts have doe a lot of research work. For example, Mea spacig method, data soopig method, full permutatio ad combiatio method, bayes discrimiatio approach ad robust iterative weights method ad so o are typical methods (Tao Bezao, 00; Hao Chuacai, 003; Huag Shegxiag et al, 00; Che Chao et al, 00; Huag Bigjie et al,0). O the basis of aalyzig the theory of coordiate similarity trasformatio ad cosiderig the characteristics that the trasformatio coordiates of commo poits are ot the same as the kow coordiates values because of the impact of errors of coordiates of commo poits, coordiate similarity trasformatio model is applied to stability aalysis of high-precisio GS deformatio moitorig etwork, ad a ovel stability aalysis method of datum poits for which the characteristics of similarity trasformatio have bee oly cosidered is put forward. This method has the advatages of simple calculatio, without cosiderig how to select adjustmet datum poit, ad without uifyig referece framework ad epoch. Thus, it is better practicability ad higher referece value.. STABILITY ANALYSIS MODEL BASED ON SIMILARITY TRANSFORMATION. Seve arameters Similarity Trasformatio Method Seve parameters similarity trasformatio method is oe of the most essetial ad commo models of three-dimesioal Cartesia coordiate trasformatio. It is show as Figure (Guo Jimig et al, 0). Figure. Seve parameters similarity trasformatio For the periodical measuremets of high-precisio GS deformatio moitorig etwork, if we deote the coordiates of oe datum poit for two periods are (, Y, Z) I ad (, Y, Z) II respectively, i the case of small rotatio agles, the similarity trasformatio equatio ca be expressed as: Z Y 0 Y ( k) Z Y Y0 Z Y Z Z II I 0 () where 0, Y 0, Z0 are three traslatio parameters;
, Y, Z are three rotatio parameters; k is a scale parameter. Geerally, the displacemet of datum poits is ot very large i high-precisio GS deformatio moitorig etwork, therefore, Eq. () is a liear model. Whe the umber of commo poits is ot less tha three poits, the optimal probability values of seve parameters ca be obtaied by least squares adjustmet. Ad the, the process of coordiate similarity trasformatio ca be realized ad stability aalysis model based o seve parameters similarity trasformatio ca be costructed by usig obtaied seve trasformatio parameters.. Evaluatio Method of Statio Stability I the process of similarity trasformatio, three-dimesioal Cartesia coordiates of both pre-periodical ad post-periodical have measuremet errors or eve may be the displacemet of datum poits, thus the calculated trasformatio parameters would be impacted if still usig Eq.(). Uder the assumptio coditio that the impact of measuremet errors of coordiates for trasformatio parameters is determied, the process of coordiate similarity trasformatio could be realized accordig to the calculated trasformatio parameters. For the impact of the poit positio displacemet of three-dimesioal Cartesia coordiates of both pre-periodical ad post-periodical, the differece of betwee the trasformed three-dimesioal Cartesia coordiates ad the kow coordiate values would become larger which ca be used as a evaluatio criterio. Ad the stability aalysis of poit positio i high-precisio GS deformatio moitorig etwork ca be carried out ad comprehesive evaluated by usig the threshold of differece method ad statistical test method based o variace ratio. (Guo Jimig et al, 0)... The Threshold for Statio Stability I high-precisio GS deformatio moitorig etwork, if three-dimesioal Cartesia coordiates of both pre-periodical ad post-periodical oly have measuremet errors without the displacemet of poit positio, the differeces of poit positio betwee the trasformed three-dimesioal Cartesia coordiates ad the kow coordiate values would be uder a limited value. I this way, the calculatio formula of the differeces of coordiate compoets (, Y ad Z respectively) betwee the trasformed three-dimesioal Cartesia coordiate ad the kow coordiate values ca be expressed as: V () Tras II II II Applyig the covariace propagatio law to Eq. (): (3) V Tras II II II ( k) () VII II I If, cosiderig the scale parameter k is I II too little geerally, so yieldig to: () VII II I For the periodical ad repetitious measuremet of high-precisio GS deformatio moitorig etwork, if assumig the omial precisio of GS receivers used i the field observatios of both pre-periodical ad post-periodical is = a ( b D ), where a is solid error, b is ratio error, D is real average side legth. For the specific datum poit, there is Y Z ; if lettig Y Z, there is: Substitutig Eq. (7) ito Eq. (), there is: (7) 3 VII (8) 3 Assumig that there are datum poits i the high-precisio GS deformatio moitorig etwork, sice measuremet errors of three-dimesioal Cartesia coordiates i both pre-periodical ad post-periodical observatios, the impact of differeces of coordiate compoets (, Y ad Z respectively) betwee the trasformed three-dimesioal Cartesia coordiates which are used by trasformatio parameters ad the kow coordiate values ca be expressed as: VII (9) 3 If twice accuracy of field measuremet (mea square error) is used as the allowable value of the limited error of differeces betwee the trasformed three-dimesioal Cartesia coordiates ad the kow coordiate values, the allowable value of the limited error at the directio of space poit positio should satisfy Eq. (0), or the datum poit is judged as chage. Also, applyig the covariace propagatio law to Eq. (): V (0) ( k) (4) Tras II I Substitutig Eq. (4) ito Eq. (3), there is:
.. Statistical Test Based o Variace Ratio: I high-precisio GS deformatio moitorig etwork, the similarity trasformatio method is carried out by usig three-dimesioal Cartesia coordiates of both pre-periodical ad post-periodical observatios. Accordig to the differece values betwee the trasformed coordiates ad the kow coordiate values, the estimatio of uit weight root mea square error ca be calculated by: deformatio moitorig etwork for a hydropower statio to be tested ad aalyzed. There are seve datum poits i the etwork which are observed by usig forced ceterig piers. The GS etwork figure is show as Figure. T ( ) V V 0 = () f where V is the differece value betwee the trasformed coordiates ad the kow coordiate value. f is the degree of freedom; is weight matrix of observatio value (i this paper, the uit weight E). After the coordiate similarity trasformatio, the poit with the largest differece value of space poit positio is marked as M. Assumig M poit may be the displacemet of poit positio ad so deletes it. The we do the process of similarity trasformatio agai by usig three-dimesioal Cartesia coordiates of the remaiig poits i pre-periodical ad post-periodical. Similarly, ( ) 0 ca be calculated. The the poit with the largest space poit positio is foud ad marked as M. Fially, we used F-test method ad let the statistic as: F - 0 ( ) 0 ( ) () Thus, it determies the acceptace regio of hypothesis test: where m= 3. -, F F f f m (3) If the sigificat level =0.0 is give, ad Eq.(3) is established, M poit with the largest differece value of space poit positio is cosidered relatively stable ad the all of the poits i the etwork are cosidered relatively stable; coversely, if Eq.(3) is ot established, M poit maybe is cosidered with the displacemet of poit positio. Repeatig the above process of similarity trasformatio ad applyig statistical test method based o uit weight variace ratio to judged stability of remai poits with the largest differece value of space poit positio util all of poits are seek for which they maybe chage. 3. EAMLE I this paper, it selects three periodical ad cotiuous GS observatio data which comes from a A-class GS Figure. GS deformatio moitorig etwork for the hydropower statio The etwork was firstly costructed ad observed from, Nov 008 to 8, Nov 008. Accordig to the distributio of IGS statios ad survey area situatio of GS statios, cosiderig the precisio of IGS statio coordiates ad data quality of IGS statio, fially, URUM statio ad GAUO statio (both of the global satellite trackig statio) were selected as datum statios. The coordiates of the above two statios were obtaied from IERS official website, ad the precisio of their geocetric coordiates was ±(3~8)mm. URUM statio, GAUO statio ad SK0 datum poit were used for costructig datum coordiate trasmissio etwork i order to obtai the adjustmet iitial coordiates for high-precisio GS deformatio moitorig etwork. The first repetitious measuremet of the etwork was observed from, Nov 009 to 9, Nov 009. The observatio data of URUM statio ad GAUO statio ca ot be obtaied because of power failure. Fially, KIT3 statio, LHAZ statio, ULAB statio (all of the global satellite trackig statio) ad SK0 datum poit were used for costructig datum coordiate trasmissio etwork i order to obtai adjustmet iitial coordiate for high-precisio GS deformatio moitorig etwork. The secod repetitious measuremet of the etwork was observed from, Sep 0 to Sep 0. To keep cosistet with the firstly coducted etwork, URUM statio, GAUO statio ad SK0 datum poit still were used for costructig datum coordiate trasmissio etwork i order to sequetially obtai the adjustmet iitial coordiate for high-precisio GS deformatio moitorig etwork i repetitious observatios. I the process of data processig, the same GS data aalysis software was used for calculatig ad aalyzig three periodical GS observatio data respectively. The baselie solutio strategies were completely idetical i data processig. Baselie vector, variace matrix ad other iformatio with basis equal precisio ad without gross error were obtaied, ad the, it made free et adjustmet processig ad stability aalysis of A-order datum poits for GS etwork respectively. 3. The Traditioal Method by Fixig Datum oits Uder the premise of esurig adjustmet datum is that the uity of referece framework ad epoch, we foud that SK0 datum poit located at a rocky which was cosidered as stable by meas of experiece, therefore, it could be used as the fixed datum poit for stability aalysis of poit positio. The
literature (Zhou Migdua et al, 0) described stability aalysis of A-class datum poits betwee the firstly costructed observatio ad the firstly repetitious measuremet, the results of aalysis show: withi the limits of measuremet error allowable value, SK0 was judged as chage ad other datum poits were stable relatively. After the field exploratio, we foud that there was a crack (width of about 8 cm, legth of about 00 m) alog the moutai body ad the distace from the crack to SK0 poit was about 00 m, as Figure 3 shows. Figure 3. The crack of ear SK0 poit Similarly, stability aalysis of A-order datum poit of twice repetitious measuremets was made by usig the method which referred the literature (Zhou Migdua et al, 0). Accordig to the differece values betwee three-dimesioal Cartesia coordiates of seve datum poits after later data processig ad of which previous correspodig coordiate results, ad comparig ad aalyzig the differece values betwee idepedet egieerig plae coordiate ad geodetic height after fixig some datum poit ad oe directio adjustmet, ad of which previous correspodig coordiate results, the differece values (the pre-periodical subtracts the post-periodical) were give i Table. Datum poit Differeces of three-dimesioal Cartesia coordiate Differeces of plae coordiate Differeces of geodetic height Y Z displacemet N E plae H Judgmet results SK0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Stable SK0 -.9. -.8 3.4-0..4.4 0. Stable relatively SK03 -.7-0. -..0 0. 3.4 3. -. Stable relatively SK04-0.9 -.3-4. 4.9-0. -0. 0. -4. Stable relatively SK0-4.0 48.3 88. 3.8-30.7 3.9 47. 308.0 Chage SK0 -.8 -. -3.4 4.0..9 3. -3.3 Stable relatively SK07 -.4 3. -.7 4. -3.3 3.0 4.. Stable relatively Table. The differece values of poit positio coordiates betwee two repetitious measuremets of GS deformatio moitorig etwork (uit: mm) As is show i Table, the differece values of poit positio coordiates betwee twice periodical observatio results show that the -directio, Y-directio ad Z-directio of datum poits icludig SK0, SK03, SK04, SK0 ad SK07 i WGS-84 coordiate system are withi 4.mm, ad the displacemets are amog.0~4.9mm, ad the geodetic height are withi 4.mm. I idepedet egieerig coordiate system by fixed oe poit ad oe directio adjustmet, N-directio ad E-directio are withi 3.4mm, ad the differece values of plae coordiate are amog.4~4.mm. If twice precisio of field measuremet (mea square error) is used as the allowable value of the limited error of differece betwee twice periodical observatio results, the five datum poits are judged as remaiig stable relatively i the rage of allowable value of measuremet error. But for the differece of three-dimesioal Cartesia coordiates of SK0 datum poit, -directio is 4.0mm, Y-directio is 48.3mm, Z-directio is 88.mm, the displacemet of poit positio is 3.8mm ad H-directio is 308.0mm. I idepedet egieerig coordiate system, the differece value of N-directio is 30.7mm, E-directio is 3.9mm ad the displacemet of poit positio is 47.mm. If twice mea square error of poit positio is used as the allowable value of the limited error of differece betwee twice periodical observatio results, datum poit SK0 is judged as chage i the rage of allowable value of measuremet error, ad the reasos eeded to be aalyzed. 3. The Threshold of Stability Accordig to the above metioed method, the correspodig program is developed usig C++ laguage. Stability aalysis model based o similarity trasformatio is costructed ad the chagig of poit positio betwee three-dimesioal Cartesia coordiate after similarity trasformatio ad kow coordiate is used as the evaluatio criterio. The results of umerical aalysis are give i Table.
The first similarity trasformatio The secod similarity trasformatio oit ame Differeces of space Threshold of Judgmet Differeces of Threshold of Judgmet poit positio differece results space poit positio differeces results SK0 0.0 0.0374 Stable relatively 0.000 0.034 Stable relatively SK0 0.0038 0.0374 Stable relatively 0.0034 0.034 Stable relatively SK03 0.0079 0.0374 Stable relatively 0.00 0.034 Stable relatively SK04 0.0 0.0374 Stable relatively 0.003 0.034 Stable relatively SK0 0.040 0.0374 Maybe chage ---- ---- ---- SK0 0.043 0.0374 Stable relatively 0.00 0.034 Stable relatively SK07 0.0 0.0374 Stable relatively 0.007 0.034 Stable relatively Table. Stability aalysis of the first measuremet ad the first repetitious measuremet of the GS deformatio moitorig etwork (uit: m) As is show i Table, i the process of the first similarity trasformatio, SK0 poit might have bee displaced ad it is judged as maybe chage. After SK0 poit is deleted, the secod similarity trasformatio is doe agai, the remaiig six poits are judged as stable relatively. Therefore, the above results show that all of the poits are stable relatively except SK0 poit. Similarly, stability aalysis of A-order datum poits of twice repetitious measuremets is doe; the results of umerical aalysis are give i Table 3. The first similarity trasformatio The secod similarity trasformatio oit ame Differeces of space Threshold of Differeces of Threshold of Judgmet results Judgmet results poit positio differece space poit positio differeces SK0 0.09 0.0374 Maybe chage 0.00 0.034 Stable relatively SK0 0.08 0.0374 Stable relatively 0.009 0.034 Stable relatively SK03 0.03 0.0374 Stable relatively 0.004 0.034 Stable relatively SK04 0.08 0.0374 Maybe chage 0.00 0.034 Stable relatively SK0 0. 0.0374 Maybe chage ---- ---- ---- SK0 0.04 0.0374 Maybe chage 0.000 0.034 Stable relatively SK07 0.03 0.0374 Stable relatively 0.008 0.034 Stable relatively Table 3. Stability aalysis of twice repetitious measuremets of GS deformatio moitorig etwork (uit: m) As is show i Table 3, i the process of the first similarity trasformatio, SK0, SK04, SK0, SK0 poits might have bee displaced ad it is judged as maybe chage. However, the displacemet of SK0 poit is the largest, thus, if SK0 poit is deleted, the secod similarity trasformatio is doe agai, the remaiig six poits are judged as stable relatively. Therefore, the above results show that all of the poits are stable relatively except SK0 poit. 3.3 Statistical Test Based o Variace Ratio Accordig to the suggested method, the statistical test method based o uit weight variace ratio is used for determiig stability of the datum poits. Firstly, two three-dimesioal Cartesia coordiates of seve datum poits of the first costructed observatio ad the first repetitious measuremet are used for doig coordiate similarity trasformatio, ad the (7) 0 =.9cm ca be obtaied. If SK0 poit is deleted because of the largest space differece value, the remaiig six poits whose three-dimesioal space Cartesia coordiates are used for doig the secod coordiate similarity trasformatio, also, () 0 =0.03cm ca be obtaied. Thus, the statistic ca be expressed as 7 F 89.7. However, F 0.0 4, =.74,ad obviously 7 F > F 4, 0.0.That is to say, SK0 poit maybe has bee displaced. If SK0 is deleted, the same approach is used for the remaiig six datum poits to do the similarity trasformatio agai. SK0 poit is got with the largest space differece value, therefore, if SK0 poit is deleted, () 0 =0.0cm ca be obtaied agai. Thus, the statistic ca be expressed as 0.0 8 F, =3.3, ad obviously F =.0. However, F < F, 0.0 8. That is to say, SK0 poit is a stable poit. Therefore, the above results show that all of the poits are stable except SK0 poit. Similarly, the datum poits of twice repetitious measuremets are used for doig stability aalysis by usig the same method. Firstly, two three-dimesioal Cartesia coordiates of seve datum poits are used for doig coordiate similarity trasformatio, ad the (7) 0 =3.88cm ca be obtaied. If SK0 poit is deleted because of the largest space differece value, the remaiig six poits whose three-dimesioal space Cartesia coordiates are used for doig the secod coordiate similarity trasformatio, also, () 0 =0.0cm ca be obtaied. Thus, the statistic ca be
expressed as obviously 7 F 794.00. However, 0.0 4, F > F 4, 7 0.0 F =.74,ad,That is to say, SK0 poit maybe has bee displaced. If SK0 is deleted, the same approach is used for the remaiig six datum poits to do the similarity trasformatio agai. SK0 poit is got with the largest space differece value, therefore, if SK0 poit is deleted, () 0 =0.0cm ca be obtaied agai. Thus, the statistic ca be expressed as obviously F =.00. However, F 0.0, 8 F < F, 0.0 8 =3.3,ad, That is to say, SK0 poit is a stable poit. Therefore, the above results show that all of the poits are stable except SK0 poit. Accordig to the above discussio, it is a effectual method to sythetically evaluate the stability of datum poits i high-precisio GS deformatio moitorig etwork by usig the threshold of differece method ad statistical test method based o variace ratio. Compared with the traditioal method, this method has the advatages of simple calculatio ad without cosiderig how to select adjustmet datum, ad the uity of referece framework ad epoch. 4. CONCLUSIONS O the basis of aalyzig the theory of coordiate similarity trasformatio, a ovel method of stability aalysis for datum poits applied i high-precisio GS deformatio moitorig etwork is put forward. Compared with the traditioal method, this method has the advatages of simple calculatio, without cosiderig how to select adjustmet datum poit, ad without uifyig referece framework ad epoch. The method applied to stability aalysis of a A-order GS deformatio moitorig etwork for a hydropower statio with three periodical GS observatio data, the aalysis results show that the suggested method i this paper ca be applied to stability aalysis of datum poits i high-precisio GS deformatio moitorig etwork, ad the judgmet results ca be i agreemet with the actual situatio. Therefore, the feasibility ad effectiveess of this method are verified. It provides a ew solvig thought ad approach for stability aalysis of datum poits. REFERENCES Che Chao, Zhag iazhou, 00. Applicatios of bayes criterio i aalysis o stability of deformatio moitorig poits based o false rate. Railway Ivestigatio ad Surveyig, (). pp.0-. Gao Yapig, 00.The base stability aalysis of GS deformatio moitorig ad suppressig the surveyig data oise by adaptive kalma filterig. Chag a Uiverstiy, i a. Guo Jimig, Wag Jiaguo, 0. Foudatio of Geodesy. Surveyig ad Mappig ress, Beijig. Guo Jimig, Zhou Migdua, 0.Statistical testig i the gross error aalysis of iitial datum for GNSS etwork costrait adjustmet, The st Iteratioal Workshop o Surveyig ad Geospatial Iformatio Systems, Fushu, Chia,pp.-70. Hao Chuacai, 003. Studyig several problems of the stability test of poits i deformatio moitorig etwork. Southwest Jiaotog Uiversity, Chegdu. Huag Shegxiag, Yi Hui, Jiag Zheg, 00.The data processig of deformatio moitorig (the secod editio). Wuha Uiversity ress, Wuha. Huag Bigjie, iao Jiahua, Ya iaopig et al, 0. Calculatio of robust iterative weights ad stability aalysis i deformatio moitorig etworks. Geotechical Ivestigatio & Surveyig, (3), pp.7-7. Tao Bezao,00.Free etwork adjustmet ad deformatio aalysis. Wuha Techical Uiversity of Surveyig ad Mappig ress, Wuha. Zhou Migdua, Guo Jimig, u Yigli et al,0.data processig ad stability aalysis of GS deformatio moitorig etwork for hydropower statio. Bulleti of Surveyig ad Mappig, (7), pp.30-33. BIBLIOGRAHY Jimig Guo is a professor i school of Geodesy ad Geomatics at Wuha Uiversity. He has B.Sc. ad M.Sc. degrees i Surveyig Egieerig from Wuha Techical Uiversity of Surveyig ad Mappig ad a h.d. i Satellite Geodesy from Wuha Uiversity. He has bee lecturig, researchig ad developig software i geodesy ad geomatics at Wuha sice 990. E-mail: jmguo@sgg.whu.edu.c Migdua Zhou is a h.d. cadidate i School of Geodesy ad Geomatics at Wuha Uiversity. He has a B.Sc. degree i Surveyig ad Mappig Egieerig from Guili Uiversity of Techology ad a M.Sc. degree i Geodesy ad Surveyig Egieerig form Wuha Uiversity. He is egaged i data aalysis theory for high precisio GS positioig ad algorithm research of etwork RTK positioig techology. E-mail: zmd_zry@3.com Chao Wag is a M.Sc. cadidate i School of Geodesy ad Geomatics at Wuha Uiversity. He has a B.Sc. degree i Surveyig ad Mappig Egieerig from Southwest Jiaotog Uiversity. He is egaged i high-precisio GS data processig. E-mail: wagchao08@.com Liahui Mei is a M.Sc. cadidate i School of Geodesy ad Geomatics at Wuha Uiversity. He has a B.Sc. degree i Surveyig ad Mappig Egieerig from Wuha Uiversity. He is egaged i high-precisio GS data processig ad deformatio aalysis. E-mail: yefeg34@.com ACKNOWLEDGEMENTS It ackowledges that the experimet data i this paper was provided by Istitute of Surveyig ad Mappig, Ivestigatio ad Desig for Water Resource ad Hydropower Research Istitute of ijiag. It ackowledges that GAMIT software was authorized by MIT. This work was supported by the Fudametal Research Fuds for the Cetral Uiversities (No. 0040000000).