VERTICAL MOVEMENTS FROM LEVELLING, GRAVITY AND GPS MEASUREMENTS

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1 rd IAG / 2th FIG Symposium, Bade, May 22-24, 26 VERTICAL MOVEMENTS FROM LEVELLING, GRAVITY AND GPS MEASUREMENTS N. Hatjidakis, D. Rossikopoulos Departmet of Geodesy ad Surveyig, Faculty of Egieerig Aristotle Uiversity of Thessaloiki, hatji@topo.auth.gr, rossi@topo.auth.gr Abstract Height differeces from geometric levelig, gravity ad GPS measuremets carried out from 979 util ow i the etwork of Volvi area are systematically aalyzed. Various methods ad algorithmic techiques based o itegrated approach are rigorously applied i order to extract a more reliable estimatio of the vertical compoets of the deformatio field. Sigals, i.e. the gravity field parameters, their variatios i time ad the vertical displacemets are treated i a combied aalytical-stochastic approach. Fially the iterpretatio of the results at the iterestig area of the Volvi Lake is attempted.. Itroductio The estimatio of displacemet ad especially the height differeces at Volvi area is very importat i Greece due to its seismic activity. Volvi area is located at orther Greece (Cetral Macedoia) ad 4m from the city of Thessaloiki. Volvi take up a area of 4X45km icludig two lakes (Lagada ad Volvi) ad is cosidered as a geodyamically iterestig area. The largest earthquake took place i May 2, 978 (Ms= 5.8), Jue 9, 978 (Ms= 5.2) ad Jue, (Ms= 6.2) followed by a series of postseismic activity affected socially ad fiacially the city of Thessaloiki. I order to study the geodyamical behaviour ad ivestigate the crustal movemets a geodetic ad gravimetric etwork was established with 6 ad 26 pillars respectively. Classical measuremets are available for the epoch 979, gravity measuremets took place i 979 ad 999 while GPS measuremets performed i 994, 995, 996 ad 2. The object of this presetatio is the simultaeously aalysis of GPS baselies with classical geodetic observatios (height differeces from levellig) ad physical data related to the gravity field (absolute gravity values) with models of itegrated geodesy (e.g.[4]), havig as mai purpose the estimatio of orthometric heights i differet epochs. 2. The itegrated approach Itegrated geodesy i time deals with the aalysis of the above observatios for the study of etwork geometry ad its variatio with time, whe these observatios deped o the gravity field of the earth ad its temporal variatio. Itegrated adjustmet has bee itroduced as a rigorous adjustmet of observatios with both geometric ad gravimetric iformatio usig precise mathematical goals. Furthermore, itegrated geodesy is a method for the adjustmet of observatios depedig ot oly o discrete parameters but also o ukow fuctios. Parameters appearig i the mathematical model ca be treated as: a) determiistic b) stochastic c) aalytical stochastic way (e.g. [9], []) ad these leads to a adjustmet with

2 rd IAG / 2th FIG Symposium, Bade, May 22-24, 26 sigals. Applicatios of relevat cocepts i the estimatio of orthometric heights from GPS baselies have bee preseted by [6], [7]. Geodetic-gravimetric etwork deals with GPS baselies, levelig ad gravity observatios i differet epochs so orthometric heights are much easier to calculate. The difficult part of aalyzig simultaeously physical ad geometrical observatios i time is to fid out the gravity field behavior. This specifies the usefuless of time-space covariace model which help us to uderstad the previous thought. The local gravity behaviour cosists a covariace gravity model (e.g. [8]). The most iterestig field is the choice of local covariace model which icludes cotrasts betwee correlatio legths. As we ca see from the matrix below Variace σ g 2 (mgal 2 ) Correlatio legth d(m) Expoetial Reilly Moritz Poisso Table. Parameters of the covariace model the most proper choice is the expoetial model sice correlatio legth is more suitable for a small area. 2. Gravity ad geometry deformatio The behaviour of the gravity field is the iterestig part of this study. Calculatig the chages betwee the correctios i the geoid udulatios from 994 util 2 ad gravity aomaly perturbatios betwee 979 ad 999 epochs usig a time-space covariace (expoetial) model proved that o sigal chages are calculated from 979 util 2. The adjustmet with the itegrated model ca be preseted usig the followig geeral formula b b M b = A M A M O x x M M A x G + M G M O G M s s s M v + v M v The adjustmet of the observatios is carried out by applyig the least squares priciples (v P v + s s ) mi (2) α= T T α α α α α α = where x i cotais the determiistic parameters (orthometric heights), s i cotais all the stochastic parameters (gravity disturbaces, geoid udulatios) ad v i are the observatio errors for each epoch. Furthermore, the idicator ) refers to t epoch, P α = Q α is the weight observatio matrix at each epoch ad α ( α α () is the covariace matrix developed from sigals at each epoch. The results of the above algorith is preseted to the followig table δg 979 δg 999 δ t δg (mgal) δ t h(cm) δg 979 δg 999 δ t δg (mgal) δ t h(cm)

3 rd IAG / 2th FIG Symposium, Bade, May 22-24, Table 2. Gravity chages perturbatios betwee time period The aalytical formula betwee height ad gravity disturbaces is (e.g.[2]) δ t δg[mgal] =.86δ t h[cm] () max mv Table. Chages of geoid udulatios δν betwee 994 ad 2 time period

4 rd IAG / 2th FIG Symposium, Bade, May 22-24, δ t h (cm) / epoch pt (lat) / epoch Figure. Absolute gravity values - without space-time covariace model 4 variace depreciatio of gravity field geoid udulatios (mm) variace of gravity field pt (lat) / epoch -7 Figure 2. GPS baselies- sigal coectio with space-time covariace model (9 yr (2-994)) I figure 2 variace depreciatio of gravity field took place every ie (9) years. So the pheomeo of seismicity repeated every ie years, which is cofirmed by historical seismological data []. Because of earthquakes took place i 969,i 978,i 995 ad i 2 (every ie years) time space covariace model become a useful predictio tool i Seismology.

5 rd IAG / 2th FIG Symposium, Bade, May 22-24, 26 t y b Observatios LEV GPS GPS GPS GRAV GPS T T v Pv + s s = mi degrees of freedom f variace 2 ˆσ stadard deviatio σ ˆ Table 4. Aposteriori adjustmet parameters for 979, 994, 995, 996, 999 ad 2 epochs By calculatig gravity disturbaces or correctios of the geoid udulatios the differeces are too small, so the chages of absolute gravity observatios are due to height differeces. This simplifies the algorithmic problem ad makes calculatios be accessible. 2.2 Geometry deformatio at fixed gravity All observatios (GPS baselies, gravity field data ad levellig) ca be aalysed simultaeously eve if they measured i differet epochs. Sigals s are uchageably from 979 util 2 (Figure ) but gravity measuremets which took place i 979 ad 999 are chaged sice poits are moved. The adjustmet with the itegrated model ca be preseted b A x G v b A x + G = [] + v s (4) M M M O M M M M b A x G v where x i cotais the determiistic parameters (orthometric heights), s cotais all the stochastic parameters (gravity disturbaces, geoid udulatios commo for all epochs) ad v i are the observatio errors for each epoch. The adjustmet of the observatios is carried out by applyig the least squares priciples T T α α α s = v P v + s s mi (5) α= which leads to the best liear ubiased estimatio for the determiistic parameters x ad the best liear predictios for the stochastic oes s, v. The results of the above algorith is preseted to the followig table Height differeces (mm/yr)

6 rd IAG / 2th FIG Symposium, Bade, May 22-24, Table 5. Orthometric Heights ad stadard deviatio for each poit per epoch The mea height differece is 6.8mm/yr. Mea height differeces calculated ear to 5.7mm/yr from other scietific jobs developed i Volvi area. t y b T T v Pv + s s = mi f ˆσ σˆ Table 6. Aposteriori adjustmet parameters for 979, 994, 995, 996, 999 ad 2 epochs I figure vertical movemets are preseted.

7 rd IAG / 2th FIG Symposium, Bade, May 22-24, dth (cm) (5 years) pt (lat) / epoch Figure. Height differeces i cm (979-2) / pt (lat) / epoch 4. Coclusios We will defie itegrated geodesy as a powerful mathematical tool which maipulates physical ad geometrical observatios measured i differet epochs makig sigal coectio i space-time by the covariace gravity model. I the preset study space-time covariace model help us to uderstad gravity field behaviour i Volvi area. Height differeces calculated with high accuracy. Also, the biggest height differeces occurred ear to the poits 54, 64, 46, 77 which idicates the epicetre of the earthquakes. We would like to poit out that i the ear future a covariace model will be attempt to replace the classical expoetial model i volvi area which has the formula 2kM 2 si ϕs 2 (R + z) ge (S, z) = σ (6) 2kM 2 where b = si ϕ =. 8[],[5], []is a mea costat value i the whole area. (R + z) Refereces: [] Biro, P.:Time variatio of height ad gravity, Herbert wichma verlag, 98 [2] Gouaris, A., Arabelos D.N ad D.Rossikopoulos: Local vertical crustal movemets i the Mygdoia basi North Greece, resultig from gravity measuremets, Natural Hazards ad Earth System Scieces :-6, 24

8 rd IAG / 2th FIG Symposium, Bade, May 22-24, 26 [] Dermais, A. ad Rossikopoulos, D.:Modelig Alteratives i Four Dimesioal Geodesy, Ιteratioal Symposium for Itegrated Geodesy, Hugary Proceedigs, Volume 2,5-45, 99 [4] Hatjidakis ad Rossikopoulos: Orthometric Heights from GPS: The itegrated approach, Gravity ad Geoid 22, rd meetig of the iteratioal gravity ad geoid commissio, 22 [5] Heck, B. ad H. Mazler: Determiatio of vertical recet crustal movemets by levelig ad gravity data, 8 Cotributios to the workshop o precise levelig, held i Haover, 98 [6] Hei, G.: Orthometric height determiatio usig GPS observatios ad the itegrated geodesy adjustmet model, NOAA Techical reports NOS NGS 2, Rockville, MD, 985 [7] Hei, G., A.Leick ad S.Lambert: Orthometric height determiatio usig GPS ad gravity data, GPS 88Coferece o Egieerig. Applicatios of GPS Satellite Surveyig Techology, May -4, 988, Nashville, TN, 988 [8] Heiskae ad Moritz: Physical Geodesy,967 [9]Rossikopoulos, D.: Itegrated cotrol etworks, PhD dissertatio, Aristotle Uiversity of Thessaloiki, Departmet of Geodesy ad Surveyig egieerig, Greece, 986 [] Spiropoulos, I: Seismic Memorial i Greece, 997 [] Torge: Geodesy ( rd editio), 2

Confidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation

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