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: Hata düzeltme yöntemleri, Jeodezik ağlar, Uyuşmazlık oranı, Error correction methods, Geodetic networks, Disagreement ratio
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
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-<» they come closer to nor mal distribution. In addition, since a0 becomes so small, the test will be an insensitive testing. The variance of unit weight o2 which is necessary in the calculation of the test value of Baarda's method is a the oretical concept. Since this value is generally unknown, if a v&ry realistic value is not estimated, the global test can not be applied, and instead the one dimensional hypoth esis test“Tau Test”which was found by Poppe (1976), can be applied. This test does not require theoretical varian ce and it uses a-posteriori variance factor 82 and is af fected by gross errors. Test magnitude is given by the following equation: T = 1 Vi 1 ~x Tk is called“Studentized Correction”. But this test va lue, like Wj,, is not in normal distribution. According to Poppe, Tk is in x Tau distribution. The test value is compared with the value c^ which is either going to be calculated for a or is going to be taken from x distribu tion table. If the test values of a t distribution greater than the limit value cx for Tau distribution with the degrees of freedom 1 and (n-q-1) is. xk>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 <2w The magnitude Piqvv=r1 shows the contribution of the mea surement li to thei degree of freedom of the network and is called“Partial Redundancy or Degree of Freedom Componen ts”. At the same time, this magnitude is the measure of the possibility of controlling Ii by other measurements. Since partial redundancy is related with 0^,0. and the de sign matrix (A) » it will define the geometry o¥ the network and it will show the contribution of systematic or gross errors in the 1th measurement of the resudial v±. In a re liable network the t±'s should be as homogeneous and big as possible. The redundancy share for reliable networks is required not to be smaller than (0.25). When the redundan cy measurement value is smaller, the reliability interval will increase and therefore the global test will be insen sitive. When the redundancy is large, the global test will be very sensitive to the small deviations from the chosen model, but the one-dimensional tests will be less sensiti ve. This situation can be eliminated by dividing the net work into small divisions that can be analyzed further. The controllability of measurements will give information about the internal reliability of the network. The inter nal reliability of the network can be defined by a lower limit value A,,^ for the inconsistancy value Ll£. The lower limit value A0Ij, is the inconsistancy value that can be re vealed by a definite“Minimum Reliability”of the test. This inconsistancy value for the Baarda Method: Vri for the Heck Method: °* fcı-*ı U-g-D Ao h. ' JPlTi and similarly for the Poppe Method: _ vl _ ö Tl-«o?l, (a-q-1) A“I,_ - *oxi * r i fF: i -«-ı is given by the above equations. This inconcistancy value which is a lower limit for a gross error is dependent on:o6 the precision”of the measurement, X± the non-central pa rameter, a the significance level, y power of the test the XI iredundancy share (ri=0YV.P22) °^ the 1th measurement in the total redundancy fiCr^n-g), and the geometry of the network. In the analysis of the quality of the geodetic network, the redundancy shares ri are the“Measurement of the Geometric Internal Decisiveness of the Network Configuration”, and the limit values A02i are used as the“Internal Reliability Measurement”. The internal reliability of a geodetic net work means that the measurements can be controlled against the errors, and is defined by the infinitesimal limit value which can be proven to be significant by the test value for the model errors. The limit value of the influence of the gross error that can be proven to be significant by the test power y0 to the coordinates or to the functions that are derived from these, is given by the following equation. A02f = Q^ âi Pi &0lj. Where A0x magnitude is dependent on datum parameters. It is important to know the effect of undefined (unelimi- nated) model errors on a function of the coordinates in order to determine the quality of the network. The effect of a measurement error with a magnitude of a limit value Gn a function of the coordinates f=hTx is shown as. A0fi t * 80 of v- « i M~ğ» A4 <3v^ When the function is used as a estimated value of a measu rement, jf.[=âj2f results, and the effect (influence) of the inconsistency value in the measurement ls to the estimated value is stated as: - a °o,i where 8(oi) is called the“Influence Factor”and is used as a measure' for external reliability of the geodetic network. The influence factor is a reliability measure which is in dependent of datum and it shows how a function of unknows will be affected by the limit error A,,^ of the i tb mea surement. In a reliable geodetic network, this factor is desired to be as small as possible. In this study after the weights for direction and sides we re calculated, the network was adjusted in angle-side, di rection, and side option, and then outlier tests were app lied. The resudials were analyzed in the three statistical test methods that were used. This analysis is always xiiinecessary because it will give the user the reliability of the results before the adjustment. In the adjustment that was applied to the primary test net work anc the four secondary test networks that were derived from the primary one, and to the result of outlier test the analysis of outlier were done in order to verify that they are really outlier. In the determination of outlier measu rements with the Baarda, Poppe and Heck test methods, Out lier tests were applied to the primary and to the four se condary networks one by one. In order to make an outlier analysis, and to protect the sensitivity of the test the network should be subdivided, and should be adjusted by free adjustment, or by unforced adjustment with enough qu antity of arbitrarily chosen external parameters. After the existence of the outlier was approved by applying a global test in order to verify which measurement was out lier, one of the Data-Snooping, Tau and t methods should be tested on each division of the network. This is also \ery helpful in the data preparation. It is wrong to search for outlier by making adjustments with the usage of more than necessary external parameters (coordinates that are chosen constant) in the network because, the given external para meters will force the measurements in order to keep them selves stable, and will cause an artificial increase in the corrections of the measurements. As a result the measure ments will be outlier. The sensibility of the test is also dependent on the quantity of the measurements. If the num ber of measurements pass certain limits, the power of the test will be decreased. This situation is analyzed by sub dividing the primary network into smaller networks each containing 15-20 points and by applying angle side, direc tion, and side adjustments with the three test methods to each individual division. The three outlier test criteria that were used are differ ent from each other due to the accepted a-posteriori facts. In the Data-Snooping method, it is necessary to know the theoretical variance value Oq whereas the other two methods do not use this assumption. If o* is known beforehand, the most sensitive test is“Data-Snooping”. If a* is not known beforehand, the global test cannot be applied and later on by identifying this value, the Tau test, which is as sensi tive as Data-Snooping, can be applied. When a* « 8*, one-dimensional Data-Snooping and Tau Tests which result the same statistics, Tki and T^ are similar to each other. These tests can not define the outlier because of the dete rioration of the variance. In this study, experimental error limits are determined by giving artificial errors to the measurements of the fourth division which was selected as a test network. First, test network #4 in angle-side adjustment is tested by the appli cation of one of the outlier test methods. In the second step the adjustment and test. methods were applied in order xivto find out the experimental error limits for the observa tion or the limits for finding the outlier for different test methods. For this purpose, internal or external to the network, small or large, for one or two measurements were changed (for example: in {(-) mines} direction the di rection angles and the sides diminished, 5CC, 15cc andSem, 15cm respectevly.) in the adjustments. Therefore in all the methods, artificial outliers can be calculated with the internal network measurements (short long; one, two) previously to the side measurements (short, long; one, two) on the perimeter of the network. The Data- Snooping method can determine the artificial error measure ments sooner than the other methods, and sometimes it can determine them at the same time as the t-distribution met hod. The Tau-Distribution method is always the last one to define these limits. When the artificial outlier are cal culated as side, angle, or two sides, two angles, this will not make any difference in the determination of the experi mental limit errors. In this study, it is proved that after the adjustment, the statistical test methods (Data-Snooping, Tau and t-Test) used in the calculation of large-scale errors are also sen sitive and efficient in determining small-scale errors. The reliability of the results is dependent on the precisi on of the measurements and how they are used in the model. The results can be summarized as follows: 1:) The superiority of the statistical tests that are used in search for gross errors are dependent on the least (scarcity, insufficiency) of assumptions. 2:) The most important subject in the outlier test is the decision of whether or not it is necessary to renew the measurement that is found as outlier. The criterion for this is the investigation of the situation from the pers pective of the suggested error limits in the specificati ons. In every geodetic study, the methods, the spesifica- tions, or contracts are determined, and whether or not the outlier can be renewed should be decided by taking into consideration the error limits of the related regulations. For example: In this study the goal is third order densi- fication. The targeted reliability of a third order densi- facation is stated in“The Preparation of Large Scale Map Regulations”. According to this regulation, the relative precision of the sides that is going to be calculated from the absolute coordinates of the points should not be grea ter than 1/50000. Therefore, it is concluded that in the third order networks, if the effect of outlier on the co ordinates is greater then 2 cm, then the repetition of the measurement will be inevitable. 3:) It is not always easy to find the location of outlier xvin the adjustment of geodetic networks. Therefore, post adjustment measures of precision (mx, my, mp) will be insuf ficient to determine the correctness and the reliability of the results, or if used they will give wrong information. For these reasons the correctness of the functional and stochastic model should be analyzed by statistical tests and possible model errors should be determined. 4:) When the measurement value 22 increases, i.e. in very large degree of freedom, the test power will decrease and will cause insensitivity. Therefore the adjustment is applied to the subdivided networks and it is necessary to eliminate the outlier from the calculations. 5:) Although the introduced and applied methods are effici ent enough, the research still continues because it has be en faced with problems related to error localization and there is a need for future improvement. 6:) There is a great need for more information on the quan tity, magnitude, condensation, collection, and distribution of the errors. 7:) There is also a need for future development in the Tau- Poppe and t-Heck methods which take into consideration theJtA type errors. 8:) In“The Preparation Of Large Scale Map Regulations”it would be appropriate if the effects of outlier to the in ternal and external reliability of the network were explai ned by an example, and if the other methods, Data-Snooping (Baarda) and Heck with t-distribution which are used in the search for outlier were explained. 9:) In the outlier test, the main input is the resudials. One resudial is affected by all other measurements. There fore the study should start with the largest test magni- tute. The measurement that belongs to the largest test magnitude which exceeds the limit value, should be subtrac ted from the measurement plan and then the test method should be renewed with a repeated adjustment calculation. Note: For the test network which is used in the application section of this study, a section of the Istanbul Metropoli tan Network that was directed by Istanbul Municipal (1986) is chosen. All the adjustment calculations in the applica tion section were calculated by an IBM 4381 in the Istanbul Technical University's Computer Center. xv 1
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