Düşey kontrol ağlarında deformasyon analizi
Deformation analysis in vertical control networks
- Tez No: 46535
- Danışmanlar: PROF.DR. AHMET AKSOY
- Tez Türü: Yüksek Lisans
- Konular: Jeodezi ve Fotogrametri, Geodesy and Photogrammetry
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 62
Özet
ÖZET Bu çalışma, yerkabuğundaki hareketlerin belirlenmesi ve izlenmesi amacıyla kurulan bir jeodezik kontrol ağının 1991 ve 1992 peryodlan arasındaki düşey hareketlerin saptanması ve analizi konusundadır. Jeodezik ölçülerin dengelemesi matematiksel modellere dayanır. Matematiksel model de, ölçülerin ümit değeri ile bilinmeyen parametreler vektörü arasında fonksiyonel ilişkiler ve bir fonksiyon ile kavranamayan fiziksel etkileri yani ölçüler arasındaki korelasyonu ve onların duyarlıklarını gösteren stokastik ilişkiler vardır. Bu modellerin temsil ettikleri fiziksel gerçeğe ne derece uydukları bilinmelidir. Çünkü, dengeleme sonuçlarına ve bunların duyarlıklarına ilişkin yargılar ancak matematiksel modelin gerçeğe uyması halinde doğrudur. Deformasyon analizine geçmeden önce matematiksel modelin gerçeğe yakınlığı test edilmelidir. Çünkü, model varsayımlarında yapılacak bir hata deformasyon analizi sırasında yanlış yorumların yapılmasına sebep olabilir. Ölçülerdeki kaba ve sistematik hatalar, bu ölçülere bağlı olarak tahmin edilecek parametreleri olumsuz olarak etkileyecekleri için, kaba hatalarla yüklü ölçüler hesaplara esas olacak ölçüler kümesinden ayrılmalıdır. Bu da ancak matematik istatistik testlerle mümkün olur. Çalışmada deformasyon kavramı hakkında genel bilgiler, matematik istatistik testler konusundaki teorik bilgiler verildikten sonra Gauss-Markoff modeli ve bu modelde lineer hipotezlerin test edilmesi, daha sonra ise hipotez testleriyle düşey deformasyon analizi konusu ele alınmıştır. Verilen kontrol ağının dengelenmesi ve deformasyon analizi ile ilgili olarak yapılan uygulamalar yedinci bölümde sunulmuştur. ix
Özet (Çeviri)
SUMMARY DEFORMATION ANALYSIS IN VERTICAL CONTROL NETWORKS Deformation measurements and analysis of movement are an essential task in the field of engineering surveys. During the past years, several strategies have been developed to predict the earthquakes by determining movements. Probability of succesfully predicting an earthquake increases with the variety and significance of observed precursors. Different precursors have been observed and described in the literature. Crustal movements along active fault zones are one of these. For the purpose of investigating the deformation or movements of an area or an object, a geodetic control network is established on the area. The aim of geodetic deformation analysis is to detect displacements of points which are expected to move because of the geophysical or geological structure of the ground on which they are located. Geodetic measurements of the network are repeated at different periods. Observations of each epoch are adjusted independently by the least squares method using the same approximate value of unkown parameters. In a region located at Taşkesti (Mudurnu Walley) where earthquakes have been occured and are suspected to occur, a local geodetic control network was established. The measurements have been made in many epochs by“ İ.T.Ü. Geodesy Working Group as well as Bonn Institute for Applied Geodesy ”. Aim of this study is to evaluate height measurements of the above mentioned network and to detect deformations and displacements based on repeatedly observed geodetic networks. To determine vertical movements, height measurements of epochs are evaluated with the least squares adjustment using the same approximate heights of network points. After that, with height differences between the epochs the deformation model is formed by using the evaluated parameters of this adjusting model. By solving the deformation model, active point or point groups are determined and tested if the differences between the epochs are significant or not. In geodetic control networks, established for the purpose of applying deformation analysis, free network adjustment is prefered because it is necessary to avoid any assumption in deformation analysis as much as possible. The fixed point assumption is not reasonable. So, the free network adjustment is applied which shows the interior precision of the network in the most reliable way and does not permit to assumptions on the datum of the network.The evaluation of the geodetic observations are based on mathematic models. The most common model used for adjusting is the Gauss-Markoff Model. This model consists of two components. One is functional and the other is stochastic model. Functional model defines the geometric relations between the expected value of the observations and the unknown parameters. The stochastic model gives the physical effects which can not be defined with a function, namely the correlation between the observations and their precision. The Gauss-Markoff model is given by the following equations, v= Ax- J where v is (nxl) vector of residuals, A is (nxu) matrix of coefficients having the geometric relations between the parameters and the observations, x is (uxl) vector of unknowns, J is (nxl) vector of observations, Zu is (nxn) variance-covariance matrix of observations, Q is (nxn) cofactor matrix of observations and c^ is a-priori variance. In the free network adjustment in vertical control networks, the heights of all points are considered as unknowns. In this case, a rank deficiency occurs in matrix A. So, it is required to deal with the rank deficiency which is also called as rank defect or datum defect in networks (d=l shift in vertical direction in vertical control networks). This problem is solved by the g eigen vector which responds to the eigen values of matrix N = ATP A and is given in the following form, - Vu where u is the number of unknown parameters. By using this vector the following equations are obtained, N = ATPA x = (N + ggT)_1ATPJ Q^N+ggV-gg7 v = -(P“1-AQ AT)P / - v- - -xx - ~ Q = P~1-AQ AT 2 VT£V mo -' 7 0 n-u+d where P is (nxn) weight matrix of observations, Q is (uxu) cofactor matrix of unknowns, m^ is a posteriori variance and (n-u) is degree of freedom of the network. XIIt is always necessary to take into consideration and question the model assumptions and check how the model reflects the reality of the model. According to the estimates of the Gauss-Markoff model, we can assume some values (hypothesis) for the theoritical values of the problem. We have to examine if this assumption can be accepted or not. This procedure is called ”hypothesis testing“. Hypothesis testing is always based on null hypothesis and alternative hypothesis. A null hypothesis, denoted by Ho is a statement about the value of a parameter that reflects the absence of an effect. An alternative hypothesis reflects the presence of an effect on the value of a parameter and is denoted by Ha. By applying the Ho and Ha hypothesis we can obtain some test statistic from the results of Gauss-Markoff model. For the hypothesis testing procedure a confidence interval is determined next. The selection of a risk level depends on the purpose of hypothesis testing. After the risk level a is selected, the critical value of the test statistic is determined according to this risk level. The critical values can be taken from the tables of disribution of the test statistic. There are some blunders (errors) in the geodetic obeservations. In the theory of errors, they are classified into three groups; random errors, systematic errors and gross errors. It is assumed that the random errors obey the normal distribution since they are unavoidable and are small differences between the observations and their expectations. Systematic errors occur because of physical causes and instrumental factors. They can be eleminated in the mathematic model theoretically. The gross errors is caused by the surveyor or instrument used for observations. Since the least squares estimation is sensitive to the blunders (gross errors, outliers), the blunders distort the solution which are determined by the conventional least squares procedure and reliable solutions can be obtained after eliminating the blunders. The above written equation shows that, the V; residual of one observation is effected by all the others. So, the observations are tested to find the blunders by means of the tests of outliers. In the test of outliers procedure it is assumed that the observations are not correlated to each other. The normalized residuals are used for the test of outliers and given as follows, Wj vi where v, is the residual of the i.th observation, av. is the variance of the residual and w; is the normalized residual. The testing outliers procedures are based on above given equation. The most used ones are Data Snooping, x test and t-test. xnIn Data Snooping, normalized residuals are used as test statistic and given as follows, Wi=-^ = - P=-~N{0,1> % °o^ and if the maximum value of the test statistic is bigger than the critical value of the normal distribution wmax. > Nl-ccQ/2 ~ -y/ Fl,«=,l-a0 the i.th observation is considered as containing gross errors. qvv. is the element of cofactor matrix of residuals for concerning Vi. The expected value of the unit observation's variance 0^ which is used for calculating the test statistic w;, is a theoritical concept. So, the estimated value of variance denoted by â* which is obtained after the adjustment procedure can be used. In this case the test statistic ( according to Poppe) is given by the following equation, Ti= - =. / -~T{f> If the maximum value of the test x ; test statistic denoted by x max. is bigger than c=x f,1-a critical value of x (tau) distribution, the concerning observation is considered as a blunder. There is a little absence theoritically in calculating the test statistic for tests of outliers by the above mentioned equations. It is also not reasonable to use the estimated variance value if there is a blunder in a /; observation because the model assumption will not be right. In this case, if the effect of investigated observations can be ommited from the fi= vTPv (quadratic summation of all residuals), the a-posteriori variance factor which is obtained without the effect of the blunders can be used in the test procedure. The estimated value for the variance factor can be obtained as follows, ~7 1 v? 'Xrl Then the test statistic ( t-test ) is given in the following form, v.- ' N i i f-1 where âj is a-posteriori variance after the renewed adjustment, f is the degree of freedom. xiuIf the t; test statistic satisfies the following equation, t; > t 2 then, the concerning observation is considered as having blunder and ommited from the adjustment procedure. The adjustment is continued until the t ; test statistic for all observations will be less than the critical value of the t-distribution. Since the later statistical testing is valid only under the assumption of normally distributed observations, gross errors have to be eliminated during the adjustment of each epoch. The geometrical or physical functional relationship between variables X;, of an object at certain epochs is described by observations /;. For an analysis of deformation there must be such observations for several epochs. The main task of the analysis is to test the general hypothesis that the variables of different epochs are related to one another. These relations can be formulated as conditions between the variables of the epochs as follows Hx = w where x is the vector of variables, H is the functional matrix and w is the vector of constant. In a procedure of geometrical deformation analysis, the null hypothesis Ho, which states that there are no deformations between the variables of two epoch, will either be accepted or rejected. The corresponding conditions might be ; X{ - Xj = 0 To prove the validity of this hypothesis, one has to consider two different steps of analysis. In the first step, the assumed conditions are not taken into account. Under the assumption that the observations of each epoch are independent from the other epoch, the observations of the each epoch is evaluated by the least square adjustment separately. If the quadratic forms of each epoch are denoted by O ; and Hj and the degrees of freedom of each epoch are denoted by f; and fj, in the case of adjustment of these two epochs together the same forms will be as follows Q = Qi + Qj and f=fi+fj By adding the conditions to the adjustment model, the quadratic form O H is obtained. The difference of two quadratic forms is a quadratic form again and is stated by R = nH-fi and h = fH-f If the null hypothesis is accepted, the division of two quadratic forms which is divided by their degrees of freedom, is distributed in the F (Fisher) distribution. xivR/h If the T test statistic is bigger than F i_ ”; h, f critical value of the F-distribution, the null hypothesis will not be valid with the risk level a. In this case, it is considered that there are movements between the epochs. The difference of heights between the investigated epochs can be given with the d = Hx matrix equation. Then, the displacement quantity for a point of k, is stated by d = Hkj -Hki This difference has to be tested if it is significant or not. This is done by the following null hypothesis, d = 0 The test statistic for this hypothesis is given as follows, 2 m0 When the T test statictic which is calculated with the above given equation satisfies the following equation, T k > F i_ a ; i ) f it is considered that the k point is active between the two epochs. In this study, the two epochs are adjusted with different stochastic models to see which stochastic model reflects the properties of the adjustment model best. The results obtained by using the 1 mAh; stochastic model were the best results for the two epochs. After the deformation analysis, displacement up to 16.4 mm. have been detected and three points (3,10,12) have been found as active. xv
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