Bir jeodezik ağın farklı bölgelerindeki uyuşumsuz ölçülerin değişik yöntemlerle saptanması ve dengeleme sonuçlarına etkilerinin araştırılması
On the detection of outliers in the different parts of a geodetic network by different methods and the analysis of their inflence on the adjustment results
- Tez No: 39742
- Danışmanlar: PROF.DR. AHMET AKSOY
- Tez Türü: Doktora
- Konular: Jeodezi ve Fotogrametri, Geodesy and Photogrammetry
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 132
Özet
where y is a variable in normal distribution and u is a square form with u~x2 (Jc) chi-square distribution with deg ree of freedom Jc which is independent of y, will be in“t (Student) Distribution”with degree of freedom Jc. The dist ribution parameters of measurements are usually unknown and therefore approximate values which are estimated from the measurements are used. One of the assumption methods is the“Least Squares Method”in the Gauss-Markoff Model. In the Gauss-Markoff Model in order to make an estimation about E(l) and £ parameters, with the help of a normal distribution measurement vectori. the following equation is derived: s(i) =ax £ = olzr1 Here, the parameters vector x has an unknown value and it has a functional relationship with the expected values E(l), A is a design matrix with the dimension {n, u) an ob servations value n and an unknown parameter value u, and İT1 1 is a weight coefficient matrices with the dimension (n, n). Therefore, in the Gauss-Markoff Model the first equation is called“Functional Model”, the second equation is called“Stochastic Model”and the two together are call ed“Mathematical Model”. For E(l) the estimated value is l=2.+y:, and for of the es timated value is following. A2 _ XT Ex n-u The y=Ax-1 resudials with x estimated values for x pa rameters should satisfy xr£z=nûn condition. The estima ted x values will be the solution for the equation AT£Ax-AT21=Q. and the solution results are as follows: Matrix of vari- ance-covariance of x estimated value is, matrix of variance-covariance of resudials is, £w = 01(^-2^^) = ^ viii
Özet (Çeviri)
where y is a variable in normal distribution and u is a square form with u~x2 (Jc) chi-square distribution with deg ree of freedom Jc which is independent of y, will be in“t (Student) Distribution”with degree of freedom Jc. The dist ribution parameters of measurements are usually unknown and therefore approximate values which are estimated from the measurements are used. One of the assumption methods is the“Least Squares Method”in the Gauss-Markoff Model. In the Gauss-Markoff Model in order to make an estimation about E(l) and £ parameters, with the help of a normal distribution measurement vectori. the following equation is derived: s(i) =ax £ = olzr1 Here, the parameters vector x has an unknown value and it has a functional relationship with the expected values E(l), A is a design matrix with the dimension {n, u) an ob servations value n and an unknown parameter value u, and İT1 1 is a weight coefficient matrices with the dimension (n, n). Therefore, in the Gauss-Markoff Model the first equation is called“Functional Model”, the second equation is called“Stochastic Model”and the two together are call ed“Mathematical Model”. For E(l) the estimated value is l=2.+y:, and for of the es timated value is following. A2 _ XT Ex n-u The y=Ax-1 resudials with x estimated values for x pa rameters should satisfy xr£z=nûn condition. The estima ted x values will be the solution for the equation AT£Ax-AT21=Q. and the solution results are as follows: Matrix of vari- ance-covariance of x estimated value is, matrix of variance-covariance of resudials is, £w = 01(^-2^^) = ^ viiiÖZET Günümüzün Jeodezik çalışmalarında, beklentileri karşılamak üzere yüksek presizyonlu sonuç gerektiren istekler artmış tır. Bu istekler; gelişen ölçme aletlerinde, fiziksel koşulların dikkate alınarak değişik etkilerin isleme katıl ması ve hesap yöntemlerinde de yeni teorik gelişmelerin uygulanmasını gerektirmektedir. Sonuçların yeterli doğruluk ta olabilmesi, Jeodezik Ağların (Nirengi, Nivelman ve Gra- vite) yeterli doğrulukta belirlenmesine dayanmaktadır. Bu nun sağlanabilmesi için, ölçmelerin uygun koşullarda ve ye terli doğrulukta ölçülmesi, stokastik varsayıma uygun dağı lımda olması, uyusumsuz ölçülerin araştırılması ve ayrıca Jeodezik Ağın uygun geometrik yapıda olmasını gerektirmektedir. Bu çalışmanın yapılmasındaki amaçlar: * Bir Nirengi Ağında Uyusumsuz ölçüleri araştırmak için ge liştirilmiş olan istatistik Test Yöntemlerinin seçilen mo del ag'a ve bu ağdan ayrılan küçük kısımlara uygulanarak Test Gücünün incelenmesi ve karşılaştırılması, olarak özetlenebilir. Araştırma çalışmalarında istanbul Metropolitan Nirengi Ağı nın bir bölümü Model Ag olarak seçilmiştir. Bu amaçla: * İki boyutlu Gauss-Krüger düzlemine indirgenmiş ölçülerle ağın Acı-Kenar, Kenar ve Doğrultu olması durumunda serbest ag olarak dengelemesinde varsa Uyusumsuz ölçülerin Data- Snooping (Baarda), Tau (Poppe) ve t (Heck) Test Yöntemleri ile ayrı ayrı araştırılması, * Model ag' dan ayrılan küçük ağların birinde doğrultu ve kenar ölçülerine birlikte veya ayrı ayrı olarak verilen yapay hatalarla kullanılan uyusumsuz ölçü testlerinde, de neysel olarak hata sınırlarının veya, testin hatalı ölçüyü bulabilme sınırlarının (test gücünün) saptanması, istatistik Test yöntemlerinin özelliklerinin kendi içle rinde ve seçilen model ağlardaki durumlarıyla karşılaştı rılması, amaçlanmış, bu konularla ilgili programlar bilgisayara uygulanmış, çeşitli dengeleme hesabı yapılarak sonuçlar karşılaştırılmıştır.SUMMARY ON THE DETECTION OF OUTLIERS IN THE DIFFERENT PARTS OF A GEOOETIC NETWORK BY DIFFERENT METHODS AND THE ANALYSIS OF THEIR INFLUENCE ON THE ADJUSTMENT RESULTS In the analysis of the geodetic values which are the re sults of measurements using mathematical statistical met hods, we have to know which distribution the sets that are made by these values represent. The sets that are created by geodetic measurements can be proved to be in“Normal Distribution”, and the distribution of linear and nonlinear functions that are dependent on the measurements can be calculated by these character of their values. For example: * The random variables y in the linear equation X=Ax+£, which is dependent on X~-N(\l, E) is in“Normal Distribu tion”with (A\k+£, A^A7) parameters. * If the random variables x are in normal distribution with (41, D parameters, and the multiplication of a less positive semidefinite A matrix with AH, is generating an idempotent matrix which has the value of (A £) (A L) =A Zc then the square form y=ktA X, will be in the“Non-central Chi-Square Distribution”which the degree of freedom will be equal to the A 's rank. Here the noncentral parameter is defined by the following equation X=p.TAiL. * The following equation, (v/n) which is formed by u and v square forms that are in u~xl2{m,\) and v~x2 (n) distribution with the condition of being independent of each other and with the noncentral parameter X and the degrees of freedom m and n, will be in the“Non-central F- (Fisher) Distribution”. As a special condition, if u and v are in central chi-square distribu tion with the degrees of freedom m and n, the same ratio will be in central F distribution with the degrees of freedom m and n. * The following equation, Vllwhere y is a variable in normal distribution and u is a square form with u~x2 (Jc) chi-square distribution with deg ree of freedom Jc which is independent of y, will be in“t (Student) Distribution”with degree of freedom Jc. The dist ribution parameters of measurements are usually unknown and therefore approximate values which are estimated from the measurements are used. One of the assumption methods is the“Least Squares Method”in the Gauss-Markoff Model. In the Gauss-Markoff Model in order to make an estimation about E(l) and £ parameters, with the help of a normal distribution measurement vectori. the following equation is derived: s(i) =ax £ = olzr1 Here, the parameters vector x has an unknown value and it has a functional relationship with the expected values E(l), A is a design matrix with the dimension {n, u) an ob servations value n and an unknown parameter value u, and İT1 1 is a weight coefficient matrices with the dimension (n, n). Therefore, in the Gauss-Markoff Model the first equation is called“Functional Model”, the second equation is called“Stochastic Model”and the two together are call ed“Mathematical Model”. For E(l) the estimated value is l=2.+y:, and for of the es timated value is following. A2 _ XT Ex n-u The y=Ax-1 resudials with x estimated values for x pa rameters should satisfy xr£z=nûn condition. The estima ted x values will be the solution for the equation AT£Ax-AT21=Q. and the solution results are as follows: Matrix of vari- ance-covariance of x estimated value is, matrix of variance-covariance of resudials is, £w = 01(^-2^^) = ^ viiivalue will be in the“Student Distribution with (n-q-1) Degree of Freedom”. If the below condition is satisfied then the measurement 1± is an outlier. Here t is a frac tional value chosen from the t Distribution table for con- fidance level i-(a0/2) and {n-q-1) is an degree of free dom. The test magnitudes with x and t distribution are functionally dependent on each other. In an outlier for the error possibility afc=aT, x and t tests will absolutely give the same results, and for n-Cx_at, then the mea surement 21 is in grossly error and is called“Outlier”. The significance level aT for testing is dependent ona which is the significance level of the total system and when at=a0, it can be calculated with the following equa tion. o0 = 1 - (l - a ) When the adjustment is calculated by using one measurementlj, with the inconcistancy value AIj, the Functional Model of the adjustment will not comply with the Stochastic Model which is dependent on random errors. If the inconcistancy value is desired to be calculated, the following equation XIis used. Pi
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