The Stability of Nodal Points in the Leveling Network in the Southwest Part of Israel

Transcription

1 he Stablty of Nodal Ponts n the Levelng Network n the Southwest Part of Israel Glad EVEN-ZUR, Israel Key words: levelng, deformaton, GPS, datum SUMMARY Repeated precse levelng surveys carred durng the past decade pont to an nstablty of the nodal ponts n the levelng network n the southwest part of Israel. hs stuaton prevents the adjustment of the levelng network properly whch would ensure the provdng of correct heghts for benchmarks. Modelng the regonal and the local vertcal movement of ponts wll enable adjustment of the levelng network, whch contans lnes that were measured n dfferent tmes. For the purpose of montorng the stablty of nodal ponts nne GPS campagns were carred n the regon over a perod of one calendar year. In ths study the large number of montorng sessons enabled the nvestgaton of the fluctuatons of the nodal ponts, based on physcal processes such as the cyclc effects of swellng and shrnkng of the ground. he GPS measurements were analyzed by a wo-step analyss. In the frst step the geodetc measurements were processed sequentally wthout modelng the varatons n the heght of the networks ponts. In the second step the varatons n the network geometry were modeled by means of a physcal model. 1/13

2 he Stablty of Nodal Ponts n the Levelng Network n the Southwest Part of Israel Glad EVEN-ZUR, Israel 1. INRODUCION Nodal ponts of a levelng network are used as a bass to determne the heghts of benchmarks n the natonal vertcal control network. Repeated precse levelng surveys durng the past decade ponted to an nstablty (of up to several cm) of the nodal ponts n the levelng network n the southwest part of Israel. hs renders any attempts for a network adjustment to falure. In order to eamne the behavor of the nodal benchmarks ntensve montorng measurements were carred usng GPS. he goal of the montorng network s two-fold, to estmate the model of any vertcal moton of the benchmarks and to defne areas n the network whch behave n a smlar way. Multple montorng sessons over a perod of a year provded the ablty to montor global and seasonal behavor of the nodal benchmarks. he vertcal accuracy of GPS measurements s not n the same qualty of the levelng measurements, but they are relatvely nepensve and fast. hese advantages enable the measurng of montorng network several tmes a year. Snce the prevous levelng campagns n the nvestgated area ponted to vertcal movements of several centmeters, the GPS measurements enabled the montorng of these movements. hs paper descrbes the desgn of the montorng network and the GPS measurement process. It presents the analyss of the montorng network usng a wo-steps analyss, based on two deformaton models whch descrbe the vertcal poston of a pont relatve to tme, the lnear moton model and the swellng and shrnkng dynamc model.. HE MEHOD OF WO-SEPS ANALYSIS wo types of models are pertnent n deformaton analyss, the mathematcal model that represents the geodetc measurements and the deformaton model. he mathematcal model s usually conceved as beng absolutely correct, whle the measurements are regarded as quanttes corrupted by measurement nose. he deformaton model should descrbe the physcal realty, but the valdty of the physcal model and ts system nose s frequently lmted. Estmaton of the deformaton parameters drectly from the geodetc measurements may lead to undesrable results. System nose due to nadequaces of the physcal model may cause sever dstortons of the parameter estmates. he relatonshp between a vector of measurements l and a vector of parameters can be epressed by a set of observaton equatons, gven as: A denotes the Jacoban matr and v s the measurement nose vector. It may be assumed that A s a column rank defcent due to the need for datum defnton. he parameters are /13 l = A + v (1)

3 calculated under the mnmum condton v Pv, where P s the weght matr of the observatons. Snce the vector of parameters can be epressed as a lnear functon of a vector of parameters s, another set of equatons may be created: = Bs + w () where B denotes the Jacoban matr and s usually a full column rank matr, and w s the model nose vector. he parameters s are calculated under the mnmum condton wpw, where P s the weght matr of the observatons n the second model. hs approach for solvng s ndrectly from the measurements vector l whle usng a vector of pseudo-measurements s referred to as a wo-step analyss (Papo and Perelmuter, 1993), where n the frst step s solved usng the measurement vector l, and n the second step s used as pseudo-measurements for solvng the parameter vector s. Substtutng equaton () nto equaton (1) yelds the One-Step analyss: l = A ( Bs + w) + v = ABs + Aw + v (3) where the measurements vector l conssts of a systematc part ABs, and two random parts Aw and v. hs paper employs the wo-steps analyss method for montorng the stablty of nodal ponts n the levelng network n the southwest part of Israel. For k sets of measurements l, each vector of parameters can be estmated ndependently as: ˆ 1 = (A P A ) A P l (4) wth the attached cofactor matr Σ ˆ = mˆ Q, where ˆ mˆ 1 ˆ = ( A P A ) = vˆ ˆ P v, and r Q and the varance-covarance matr r s the adjustment redundancy. he estmated soluton vector ˆ = [ ˆ ] 1 ˆ.. ˆ k resultng from the frst step wll be used as pseudomeasurements n the second step. he set of deformaton model parameters ŝ s solved by equaton () (Papo and Perelmuter, 1993): -1 sˆ = ( B P B) B P ˆ, (5) he weght matr P s produced by usng the varance-covarance matr Σ ˆ nstead of the cofactor matr Q ˆ (Even-zur, 3). he cofactor matr Q ŝ and the varance-covarance matr Σ ŝ are: 1 ˆ Q = ( B P B) ; Σ = m Q (6) sˆ sˆ sˆ 3/13

4 Whle mˆ s equal to (Even-zur, 3): ˆm k r = 1 = k -1 + wqw. (7) r + r 3. NEWORK DESIGN AND FIELD DAA COLLECION = 1 II he montorng network contaned 1 ponts. Preference was gven to nodal benchmarks estng n the levelng network, but some nodal ponts were destroyed over the years and nearby benchmarks were selected nstead. Some of the nodal ponts n the nvestgated area were not sutable for GPS measurements but the chosen benchmarks were ones establshed wth good geotechncal standards. Benchmarks wthout sutable stablty, for eample ones that were located on electrc poles or brdges, were dsqualfed. hree ponts from the Israel Geodetc-Geodynamc network were also chosen. hese ponts were bult accordng to hgh techncal specfcatons to ensure ther geotechncal stablty (Karcz, 1994) and t s mportant to montor ther stablty. After consderatons of the requrements, condtons and constrants, and after a tour n the feld ten benchmarks were chosen, (the locatons can be seen n fgure and ther constructon methods are detaled n table 1. An addtonal permanent GPS staton located n the nvestgated area was added to the network. For vertcal movement montorng purposes the GPS antenna set-up s of great mportance. o ncrease the relablty of the network a GPS antenna was set up twce, ndependently, at each pont n order to detect and dentfy gross errors resultng from the postonng of the GPS antenna at the marker ponts, dentfy algnment errors, and measure the antenna heght. Four rmble dual frequency GPS recevers (3 4SSE and 1 4SSI) wth geodetc antennas were used for the network measurements. he antennas were set up on trpods ecept at the G1 ponts, where a specal devce that was screwed to the pont and enabled a stable set up of the GPS antenna for long perods of tme was used. Fve sessons enabled measurng the network wth two ndependent antenna set ups n each pont. he duraton of each sesson was planned for 4.5 hours. In each campagn the same sessons were measured. Name Locaton Constructon 718/A Port of Ashdod Bolt on concrete banster on platform 4U Ashqelon Marna Bolt on concrete banster on platform 65/F Southern et of Yavne Bolt on sold rock 4/F Pelugot junc. 8 m drllng ppe 36/A Urm junc. 4 m drllng ppe 436/A Northern to Omer 4 m drllng ppe 477 Qama junc. Wall nal on concrete banster EZRA Ezrcam junc. 4.5 m drllng ppe (G1) ASHK Southeast to Ashkelon 4.5 m drllng ppe (G1) 4/13

5 Name Locaton Constructon OFKM Park of Ofaqm 4.5 m drllng ppe (G1) LHAV Kbbutz Lahav 5 drllng ppes to depth of 4 m. (permanent GPS ste) able 1: he network BM locaton and method of constructon. 4. FIRS SEP: EPOCH BY EPOCH DAA PROCESSING Each sesson was processed wth the permanent GPS ste (LHAV). Every processng contaned fve stes, whch enabled the producton of 1 baselnes. he coordnates of the permanent GPS staton were set as fed n each sesson processng. Solutons were determned usng the lnear combnaton method and a value of 15 degrees was used as the default cutoff elevaton angle for all the processng. he sessons were processed usng precse orbts, as dssemnated by the Internatonal GPS Servce (IGS). he average vector length was 7 km, whle the longest vector was 55 km long. he baselnes were adjusted nto a network, whle the 3 3varance-covarance matr assocated wth each baselne was used for weghng the observatons. o ensure that the error model would match the GPS observatons better, the varance-covarance matr was multpled by a constant scale factor. Detecton of gross errors was performed usng the w-test. Gross errors n the adjusted model wll result n the reducton of the quadratc form of the resduals (vˆ Pv) ˆ by (Chen, 1987 and Etrog, 1991): 1 R = vˆ PE(E PQ ˆ vpe) E Pv, (8) whle Q v s the Least Squares estmaton of the resduals cofactor matr and E s a matr wth the same number of columns as the number of suspected outlers, n, wth a unt value n the row whch correlates to the suspcous observaton and zeros n all other rows. he test statstc s Rn σ ~F α,n,, where α s the sgnfcance level and σ s the a-pror varance factor. When the observatons n the Least Squares adjustment are GPS vectors the E matr contans 3 tmes the number of columns, snce each vector contans three components. For j recevers there are j( j 1)/ sngle vectors, whle only j-1 of these vectors are ndependent. We therefore multply the normal matr by factor / j to compensate for the artfcal ncrease of redundancy resultng from the use of all possble vectors (Han and Rzos, 1995). he network contaned 11 ponts. Fve sessons were measured n each montorng epoch whch ncluded 5 stes (ecept the last montorng epoch, where one sesson was measured twce). en vectors were processed from each sesson, n total 5 vectors per montorng epoch. he number of unknown parameters to be solved was 3 ( ), and the number of observaton equatons was 15, so for the sum of 1 degrees of freedom could be epected n the adjustment process. he degrees of freedom are reduced by 3 for each vector that s removed from the system. he results of the data processng for each sesson can be seen n table. 5/13

6 Estmatons of the accuracy of the network ponts n a local horzon system n the free net soluton for all 9 montorng epochs can be seen n table 3. hese values present the accuracy of the measurements wthout any nfluence of outsde factors such as datum defnton. he accuracy of LHAV s sgnfcantly hgher than that of the other ponts. Includng LHAV n every sesson processng dramatcally ncreased the number of local degrees of freedom, ths whch resulted n a hgher accuracy. In the G1 ponts a specal devce that was screwed to the pont and enabled a stable set-up of the GPS antenna for long perods of tme was used, whle all the other ponts were measured usng a trpod. However, t can not be clamed that usng the specal devse for the GPS antenna set-up mproved the accuracy compared to ponts that were measured by trpod. Montorng number Average GPS day Number of days between sequental montorng Factor Degrees of freedom able : Summery of the nne montorng campagns. Pont name Accuracy (1σ) average [m] mnmum [m] mamum [m] LHAV A U OFKM U ASHK U F F U EZRA able 3: he accuracy of the network ponts n a local horzon system n the free net soluton. 5. SECOND SEP: DEFORMAION ANALYSIS he mnmum constrants soluton of the network ponts heght for each montorng epoch and ther varance-covarance matr are used as pseudo-measurements for the soluton of the Second Step, where n each soluton dfferent models can be tested to descrbe the vertcal poston of the network ponts. mˆ 6/13

7 5.1 he Deformaton Model wo deformaton models were tested to descrbe the vertcal poston of a pont relatve to tme, the lnear moton model and the swellng and shrnkng dynamc model. he heght of a pont h n tme t s descrbed n the lnear model as equal to h = h + h & t (9) where h s the vertcal locaton of the pont n standard tme t, h & s the lnear velocty of the pont and t = t t. In the dynamc model a cyclc factor s added for the descrpton of the vertcal poston. he model represent the vertcal locaton relatve to tme s receved from the ntegraton of the dynamc model descrbng the acceleraton && h of a pont: π c π && h = cos( t) (1) It s a smple snusodal model whch s dependant on the coeffcent c representng the swellng/shrnkng. he cycle perod s, and the acceleraton s calculate for tme t relatve to the reference tme t, where t = t t. he vertcal heght of a pont as derved from the ntegraton of equaton 1 s c π & h = h + h t cos( t) (11) π where the three parameters, h, h & and c are unknowns and should be solved. 5. he Statstcal ests Statstcal tests are appled for estmatng the correspondence of the moton model and ts sgnfcance. Parameter that s derved from the measurements s tested to see f t s sgnfcantly dfferent from parameter wth a certan confdence level (1 α ). he null hypothess (H ) s tested aganst any alternatve hypothess (H 1) : H : = H: 1 (1) When only one parameter s tested the Ch Square ( χ ) dstrbuton s used, and when a vector of parameters s tested the Fsher (F) dstrbuton (Cooper, 1987) s used. When Q s the cofactor matr of, the null hypothess s tested aganst the alternatve hypothess usng the value w obtaned from Equaton (13) (Hamlton, 1964, Koch, 1999): 7/13

8 w = [( ) Q ( )]/ d. (13) 1 he calculated value w s tested aganst F( α,h,r), whch s determned based on the Fsher dstrbuton wth the chosen sgnfcance level α, the degrees of freedom r,and d the rank of Q. he null hypothess s rejected f w > F( α,r,d). 5.3 Results he lnear moton model he free net soluton of the lnear moton model showed an a-posteror varance factor of unt weght equalng ˆm = 1.66, whle the model nose was wpw =15.. he rato between the a-posteror varance factor and the a-prory varance factor (m = 1.) s tested. he test statstc s (Cooper, 1987) ˆm Ω= F( α, r, ). (14) m he 9 sessons provded 118 degrees of freedom, a rather hgh value n the assessment of s, meanng that r can be defned as nfnty, r =. A falure of the test mght be caused by an ncompatble model, gross errors or a weghtng problem. he test s accepted f Ω< F. Settng the sgnfcance level at α= 5% for the lnear model resulted n F(5,118, ) = 1.7, and the test was accepted, margnally. At the second step, the velocty (h) & of the network ponts was tested. he null hypothess & & H :h= was tested aganst the alternatve hypothess H 1 :h. As the dmenson of P equaled and ts defect was 1, the rank of the matr was d = 1. hus F(5,1, ) = 1.83 was obtaned for an α= 5% level of accuracy. Accordng to Equaton (13) w = 1.4. Snce w < F the null hypothess was accepted, at a sgnfcance level of α= 5%, sgnfyng that there was no lnear movements durng the measurement campagns he swellng and shrnkng model he cycle perod = 1 was set for testng a yearly cycle. When settng t =.1, a mnmum model nose equalng wpw =11.4 was receved for the zero epoch. he a-prory varance factor was ˆm = It s obvously clear that ths dynamc model better fts the observatons then the lnear moton model. he h & and c were tested for the P dmenson of and ts defect, and the rank of the matr was d =. When settng an α= 5% level of sgnfcance, F(5,, ) = 1.57 was obtaned. w = 3.15 was calculated accordng to Equaton (13). Snce w > F the null hypothess was rejected at the sgnfcance level α= 5%, ndcatng that swellng and shrnkng movements occurred durng the measurement campagns. 8/13

9 o clarfy whch ponts moved a datum had to be defned. An S-transformaton was appled to transform vector s and ts covarance matr to a weght constrants soluton. Congruency testng was performed to determne the stable datum ponts (as presented n Equaton (13)) but the test statstc was appled only to the datum ponts. Fve ponts (36A, OFKM, 477, 4U, 4F) n the center of the network were selected as datum (see Fg. ). For an α= 5% level of sgnfcance, F(5,8, ) = 1.94 was obtaned. Accordng to Equaton (13) w =.63 was obtaned. Snce w < F the null hypothess was accepted verfyng that the 5 datum ponts were stable. he weght constrants soluton based on the datum defnton s presented n able 4. Pont name h [m] σ h h & [m/year] σ h & [m/year] c [m/year] σ c [m/year] [m] LHAV A A OFKM ASHK U F F A EZRA able 4: he weght constrants soluton of the swellng and shrnkng model, where ponts 36A, OFKM, 477, 4U and 4F (yellow background) defned the network datum. After settng a stable datum, a sngle pont test was carred for all the other ponts n the network. If the null hypothess was accepted for a sngle pont then the pont was consdered stable at a sgnfcance level α. Otherwse a sgnfcant movement was defned for the pont. Four ponts were found to have sgnfcantly moved (see Fg 1.); ponts (LHAV and 436A) n the southeast part of the network and ponts (4U and EZRA) n the center of the network near the coastlne. he results of the sngle pont testng are presents n able 5. Pont name h r F(α,h,r) k Sgnfcant LHAV yes 436A yes 4U yes 65F no 718A 3..7 no EZRA yes able 5: Sngle pont testng results for a sgnfcance level α= 5%. Fgure 1 depcts the behavor of the tested ponts for the shrnkng and swellng model, when compared wth the datum ponts. he blue ponts represent the measured heght n the 9 campagns. he black lne represents the lnear trend moton and the pnk lne represents the swellng and shrnkng model. 9/13

10 LHAV m year 436A m year 4U m year EZRA m year Fgure 1: he swellng and shrnkng model n the ponts wth sgnfcant movement relatve to the datum ponts (36A, OFKM, 477, 4U, 4F). 1/13

11 Fgure : Map of the montorng area. Green trangles represent montorng network pont. Black lnes defne the datum zone and the red lne defne the deformed zone 6. DISCUSSION AND CONCLUSIONS A number of dfferent sources may contrbute to the apparent varatons n the observed vertcal ste postons. hese sources nclude global and local phenomenon, where the local phenomenon s dvded to systematcal and ncdental dependant phenomenon. herefore, the vertcal poston of a pont may change n a constant manner, n a systematcal cyclc manner, n an ncdental cyclc manner or n an arbtrary manner. Constant movement may be caused by tectonc plate movements and can be dvded nto constant velocty and acceleraton. Cyclc movement, systematc or ncdental, can be caused by physcal processes of regular or rregular character lke sold earth, ocean and atmospherc 11/13

12 tdes, ground water, hydrodynamc and thermal epanson of bedrock etc. GPS measurements are characterzed by some modelng errors as well, such as errors n the modelng of satellte orbt; atmospherc and water vapor modelng and antenna phase center whch may have seasonal components as well. Based on the eamnaton of the contnuous GPS array daly solutons, Dong et al. () compared the relatve contrbutons of the dfferent sources and found that surface mass redstrbuton, ncludng ocean and earth tdes, as well as atmospherc and ground water loadng, are the prmary causes for the observed annual vertcal varatons of ste postons. he contrbutons of geophyscal sources and model errors to the observed annual vertcal varatons n ste postons are n the magntude of several mllmeters. It s logcal to assume that most of the global and regonal physcal processes whch cause nstablty wll have the same effects n a small area. Lookng at the 5 pont datum defned for ths research, sgnfcant shrnkng and swellng movements are evdent n two local areas, each defned by a par of ponts. he annual ampltude of the ponts LHAV and 436A relatve to the reference system s appromately 1mm, and for the ponts 4U and EZRA t s appromately 1mm. he lnear moton measured for ponts LHAV and 436A s smaller than the moton measured for ponts 4U and EZRA. he larger lnear moton of ponts 4U and EZRA may be caused by the ground water level and lnear changes n the Medterranean Sea level. hese two ponts are close to the coastlne and a Maro-graph located near to pont 4U showed postve lnear rsng of the sea level by 13cm over the montorng year. Attemptng to eplan the varatons n the poston of the ponts by sol epanson schemes accordng to the ranfall cycle s complcated. he swellng and shrnkng model s best fttng when defnng the zero epoch t as.1. In ths model the etreme ponts are receved n the begnnng of January and August (whch n Israel mark the mddle of wnter and summer, respectvely). he amount of precptaton over the nvestgated area s not constant, the ran fall vares between 5mm and 65mm. he ran fall season s lmted to a perod of four months (December to March). herefore, we could epect a mamum swellng at the end of the wnter when the sol s watered and mamum shrnkng at the end of the summer when the sol s completely dry. Any correlaton between the ponts constructon and ther movements s obvous apparent. hus, the tested swellng and shrnkng model may not be the best model to descrbe the local phenomenon n the nvestgated area. Further epermentaton wth dfferent models may be carred wth relatve ease at the second step of the analyss, untl a more sutable model s found to eplan the phenomena detected n the measurements. ACKNOWLEDGMENS he study reported n the above paper was supported by Survey of Israel. 1/13

13 REFERENCES Chen, Y.Q., Kavouuras M., Chrzanowsk A., 1987, A Strategy for Detecton of Outlyng Observatons n Measurements of Hgh Precson. he Canadan Surveyor. 41: Cooper, M. A. R., 1987, Control Surveys n Cvl Engneerng. Collns, London. Dong, D., Fang, P., Bock, Y., Cheng, M.K., and Myazak, S.,, Anatomy of Apparent Seasonal Varatons from GPS-derved Ste Poston me Seres. Journal of Geophyscal Research. 17(B4), 75, do:1.19/1jb573 Ethrog, U., 1991, Statstcal est of Sgnfcance for estng Outlyng Observatons. Survey Revew. 31:6-7. Even-zur, G., 3, Varance Factor Estmaton for wo-step Analyss of Deformaton Networks. Journal of surveyng engneerng. In prntng. Hamlton, W.C., 1964, Statstcs n Physcal Scence. he Ronald Press Company, New- York. Han, S. and Rzos, C., 1995, Selecton and Scalng of Smultaneous Baselnes for GPS Network Adjustment, or Correct Procedures for Processng rval Baselnes. Geomatcs Research Australasa. 63: Karcz, I., 1994, Geologcal Consderatons n Desgn of the Semnal Dead Sea Rft Network. Perelmuter Workshop on Dynamc Deformaton Models, Hafa, Israel. Koch, K.R., 1999, Parameter Estmaton and Hypothess estng n Lnear Models. Second Edton, Sprnger-Verlag, Berln Hedelberg New-York. Papo, HB, and Perelmuter, A, 1993, wo-step Analyss of Dynamcal Networks. Manuscrpta Geodetca, 18: BIOGRAPHICAL NOES Dr. s a senor lecturer at the echnon - Israel Insttute of echnology at the Faculty of Cvl and Envronmental Engneerng. Hs research nterests nclude GPS, Geodetc control networks, optmzaton of geodetc networks and Geodynamcs. CONACS Department of Cvl and Envronmental Engneerng echnon Israel Insttute of echnology Hafa 3 ISRAEL el Fa Emal: eventzur@t.technon.ac.l 13/13