Presizyonlu nivelman ağlarında dengeleme öncesi stokastik model için bir yaklaşım

Abstract

ÖZET Presizyonlu nivelman ölçmeleri uzun yıllardan beri uygulanan bir jeodezik ölçme tekniğidir. Günümüzde de gerek bilimsel amaçlı, gerekse mühendislik hizmetlerine yönelik ol arak yaygın bir biçimde kullanılmaktadır, özellikle, düşey yer kabuğu hareketlerinin izlen mesi, yüksek doğruluk gerektiren yapı ve makinaların aplikasyonu ve atom santralları gibi kritik yapıların bakım ve kontrolü, bu kullanım alanlarından birkaçıdır. Yüksek doğruluk istenen bu işlerin bazısında +0.1 mm lik yükseklik presizyonuna ulaşılması öngörülür. Presizyonlu nivelman tekniğinden talep edilen bu yüksek doğru luğun sağlanabilmesi, ölçmeleri etkileyen düzenli hata kaynaklarının yakından tanınması ve bunların etkisiz kılınması ile mümkündür. Presizyonlu nivelman ölçmelerini etkileyen sistematik hatalar, çeşitli yollarla azaltılabilirse de tümüyle giderilemez. Geriye artık sistematik hataların kalması kaçınılmazdır. Sözü edilen artık sistematik hatalar ölçüler arasında korelas yona sebep olurlar. Bu hataların davranışını ölçüler veya onlardan türetilen büyüklükler arasındaki istatistik bağımlılığa bakarak anlamak mümkündür. Yine bu hatalar, söz konusu büyüklükler üzerinde etkili olurlar. Bu etkinin gözönüne alınması, gözlemelerin dengelen mesi sırasında uygun bir stokastik modelin seçimi ile mümkündür. Oluşturulacak iyi bir stokastik model, dengelemeden amaçlanan büyük lüklerin, ve bunlara ait doğruluk kriterlerinin daha güvenilir olmasını sağlayacaktır. Bu çalışmada, özetle - İstasyon noktalan arasındaki korelasyonun da gözönüne alın masıyla bir stokastik model türetilmiş - H. Lucht'un kenar kapanmalarını kullanarak hatalar için ağırlık tersi hesaplamakta kullandığı stokastik model, burada istasyon kapanmalarından kenarlar için ağırlık hesaplamakta kullanılmış - Bu iki model ile birlikte, yaygın olarak kullanılan diğer iki model, Büyükada Presizyonlu nivelman ağına uygulanarak çeşitli dengelemeler yapılmış ve sonuçlar karşılaştırılmıştır. vı

A STUDY FOR STOCASTIC MODEL BEFORE ADJUSTMENT IN PRECISE LEVELLING SUMMARY Precise levelling measurements are a geodetic surveying technic which has been applied for many years. This technic has been applied too much both the science surveying and engineering surveying. Forex ample; - The determination of recent crustal vertical movements - To determine of possible vertical deformations in engineering structures like dams, bridges, tunnels etc. - To establish the high accuracy levelling points which are used in building of the engineering structures like highways, railways and pipelines etc. - To select the field for atomic electric stations and, their control and maintenance. - To applicate of the structures and machines which are required high accuracy - To determine the geoids shape - To measure the national levelling networks For this works, in general are required ±0.5 mm Am precision, however, *0.1 mm height precision is wanted for control and maintenance of the atomic reactors. This high accuracy in the precise levelling can only be obtained if all sources of error are studied closely and avoided strictly. The precise levelling errors can be seperated in two parts at the first time. 1- instrumental errors a) Errors due to the level - Not being horizontal of the sight axis of the level - To be not adjustment of the circular bubble axis VII- Compensation errors in the automatic levels - The other errors of the levels b) Errors of the rods (staff). - The graduation errors of the rods - The zero position error between two rods - Not being plane of the bottom of the rod - Not being perpendicular of the bottom of the rod to the face of the invar tape 2- External sources of the errors - Vertical movement error - Temperature error - Vertical refraction error - Influence of the magnetic field of the Earth - The effect of the Earths gravite field - Tidal effects The vertical movement error is an important error that can not be determine it's effect on the measurements exactly. Influence of the magnetic field on the compensator can be eliminated by design changes,. the others are required a higher expenditure of measuring techniques and calibration or field works. The effect of the temperature error and vertical refraction error can be reduced by additional data(like measuring of temperature, presure and wind speed etc.) and making corrections. The influence of gravity must be reduced into a well-defined height system. In order to do this gravity observations along the levelling lines should be made by and making corrections. Tidal effects lead to the error of deviating plump line. Maximum value of this error has been given as 0.1 mm 1 241. Whether these effects are modelled or not, a certain systematic distortion of the measurements will be remained. The existing of the systematic errors in precise levelling measurements and their significance can be detected by the statistical tests. In general, a precision of ±0.5 mm Am are required for these works and 0.1 mm a height precision has been expected for control and maintenance of atomic reactors. vmTherefore, following tests can be applicated to station and line discrepancies - The tests which can be applied to the station discrepancies are, 1- The testing of the mean station discrepancy belonging to the lines 2- The comparison of station discrepancies variances 3- The comparison of double run levelling measurements. 4- The tests for asymetri and exes 5- The comparison of stations discrepancies in all lines of the net 6- Normal distrubution tests, - The tests, which can be applied to the line discrepancies are, 1- The estimates the systematic and the random error components. These error components are calculated in 1 km by those equations. yk- 0.5öHk/2 &~ ; pk=pk-2.D.yk Here, pk; line discrepancey: D: distance of levelling line; y. : systematic error component; p`k: random error component; ph. : total line discrepancy 2- To test of line discrepancies whether significance or not 3- In line discrepancies asymetry and excess. 4- Normal distrubution. The rest systematic errors in precise levelling measurements are due to correlation between measures. The correlation between station points is given by following equation. n-0' rj J`max if1 (6r<)(6nj-*) 'i= 1-1 i/`^. ? n'J` n 1-1 1 ıVl 1+J Here, 6: station discrepancy, 6: mean station discrepancy (for a line), n: the number of stations in a line and j= 1,2,...n-2 or 3max = n/10- ixA criterion of relationship between the core! ation parameters can be detected by covariance, Therefore? correlation coefficient is C<5.,6.. j'r mi«m^j where C5. 5 is covariance, m- sample standart deviation for point 1,` H n Whether correlation coefficients are significant or not, can be detected by comparision to following test V fj'1=^Tf ; f``J`2 A suitable stochastic model in the levelling net works would provide reliability of the adjustment parameters and related accuracy criteria. The precision of the stochastic model which is required for the adjustment done by the line measurements, is related to m^ (standart deviation of the levelling lines) and consequently Çn^ (precision of variance-covariance matrix) found formed lines, In this study, a variance-covariance matrix (Q§5) for each line is derived by starting from the station point first, and by the help pf this matrix, standart deviations for the lines are calculated. Assuming that the measurements were correlated, the correlations between the stations are taken into consideration to some extend. For this, the station discrepancy âs divided into two parts which are systematic and random components, and related with these variance are written as follows: 6» V6r ; ml = ms+tnr where, s : systematic component of the station discrepancy 6r: random Then a vector which gives the covariances of station points is derived as follows; 2 2 Ç<$. = mg (s.., r12> r13,.,.,, 0,0) where; m : sample standart deviation of the systematic error part for a station point, Seperately, it is considered to be equal in every station for a levelling linec2 J/J s. = mi/ms formed written as Following this,-variance-covariance matrix for lines are and assuming s,= s£e, = sfi = sz and r-j= rn_-j n it is %6-- 2 s r^ r2-*.r-...0 s r1.,, r.,.,0 s2 r. Then, sample standart deviation for the height differences an a levelling line is calculated as: mr- (iT.Çhh.e)1/2 considering the relation between Ç and Qhh Wub -n (e..C `HB7 -68.e) 1/2 substituting C.., it is obtained as: 00 1 j 1/2 ``to* + 7 `V `s +2.z (n-i)ri * i»l 2 2 2 Assuming m /m = 3 and doing the concellation 1 1/2 ify= *-zms.{n(l+e£>2 X (n-i).^} is obtained. Using this last equation it is possible to find the sample standart deviation of the height differences of the line, giving different chosen values for 3 In an other method, the equations used by H, Lucht is obtaining the sample standart deviation from the lines discrepancy. In this method, station discrepancies are used instead of line discrepancies and again a correlation between the stations is accepted, n shows the total station discrepancy.in a line, n the station number and again r^ correlation coefficient a sample variance of a line is given by the following equation: XIml = mh (n+^rik) measurements are accepted to be done without correlation, mu is 111 aposteriori standard deviation of one station point on each line and also the correlation valu calculated by using to sample Using the theoretic method: and also the correlation value (Er^) in the equation, can be f using to sample and the theoretic methods. zr+ 2r, n-1 t, n n t=l a -1 Using the sample method: ir. n-1 m? ? ED* n t-1 2m£ t i k ut nQ The weight inverse of the line height difference (q`) is calculated by: mo 2lrt o.d n in this equation D: levelling l^ne length, d: average observation length in a station and o«= *10~V 2d, This last equation can be separated into different parts as follows: a) qH = D (km) b) qH * D (1* Jjll) mo c) qH = D. -T, mo 2lTt d) qH = d ° (H -İ) o -d The weight reciprocal corresponds to in a) the conventional assumption in b) additional consideration of the computed correlations in c) ` ` ` the varying accuracy associated with the height, in d) additional consideration of b) and c) xnAccording to this, in adjustment the precise levelling nets those models are tried as the stochastic models (weight models) la- qj = pj 2b- q. = p` 3c- qi r pT 4a- qi = pT 4b- qi = P: 4c- qi, pT The `m,` va = Di (km) = m?/c 2Er. ;Ü.Jo/oZ - n ° a. (1 + 2zr4 `c`..,_..._Jue in model `b` is the sample line standart deviation calculated using the line discrepancies, `m^.` value in is sample line standart deviation obtained using the correlation observations. These studies are applied on the Büyükada precise levelling net of the `Adalar Triangulation and Levelling Nets Sample Project` The measurements of the net are done by using three different instruments (Zeiss Nil, Wild N3, C.Zeiss Nİ004) and equipments by three different measurement teams (posts). All the calculations are done in the İTO Computer Center using the IBM 4341 computer. The drawings are plcted in the Epson PC microcomputer. All the rod readings are loaded in the computer and the measurements are analysed by using mathematical statistic tests, Then, with the weights explained above, a total of seventeen adjustment processes are done. The comparison of the adjustment results are shown both in numerical and graphical form. The following conclusions are obtained from these studies: 1°- In spite of all the preventions, it is understood from the statistical tests, that the systematic redundant errors have continued to effect the measurements. 2°- The systematic error components in one station for all three instruments and equipments used in the measurements are as fol 1 ows : rtig. * 0.018 mm ms =0.037 mm m^T1?0.012 mm ; I : Zeiss Nil (Oberk.) ; II : Wild N3 ; III : Zeiss Nİ004 (Jena) xi nThe value obtained for Zeiss Nİ004 is not very reliable beacause of the limited number of line measurements, 3°- For a 1 km. levelling way, a difference of up to 0.5 mm is obtained between the first line discrepancy (p -j ) and the total line discrepancy (p= p i«p 2 *P$)< In general, these discrepancies are sensitive for different errors. So, using total discrepancy value `p` instead of p-j more favorable, 4°- The weight models given above are used in adjustment the sample net. The following results are obtained: - A level difference of max. 0.8 mm and a standart deviation difference of max, 0.22mm are observed between the first and second models. - A level difference of max. 1.9 mm and a standart deviation difference of max. 0.53 mm are observed between the first and third models. This result shows the effects of the correlation and the importance of grouping the measurements. - A level difference of max. 1.5 mm and a standart deviation difference of max. 0.5 mm are observed between the first and the fourth models. Using these results, it is said that the correlation term in the weight models is quite effective on the adjustment results. 5°- In these studies, the correlation term used in the weight models is obtained from the station discrepancies instead of the line discrepancies. Thus, in long measurements, the change of the systematic error characters into errors of chance random characters is prevented, and more realistic and accurate criteria are obtained. xiv