##
Karışık nivelman ağlarında stokastik model araştırması

Karışık nivelman ağlarında stokastik model araştırması

##### Dosyalar

##### Tarih

1996

##### Yazarlar

Coşkun, M. Zeki

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu çalışmanın amacı, farklı yöntemlerle ölçülmüş ve "Karışık Nivelman Ağları (Complex Levelling Nets)" olarak adlandırılan nivelman ağlarının dengelenmesinde, ölçülerin doğruluğunu etkileyen önemli hata kaynaklarını ve farklı ölçme yöntemlerini dikkate alan stokastik modellerin oluşturulması ve mevcut fonksiyonel modelleri genişleterek daha uygun modellerin geliştirilmesidir. Tezin ikinci Bölümü'nde nokta yükseklikleri hakkında kısa bilgi verildikten sonra Üçüncü Bölüm'de tigonometrik nivelman, Dördüncü Bölüm'de presizyonlu nivelman, Beşinci Bölüm'de ise vadi geçiş nivelmanı ölçme ve hesap modellerinden söz edilmiş ve yükseklik farklarına ait standart sapmaların hesabı açıklanmıştır. Altıncı Bölüm'de dengelemenin genel prensibleri özetlenerek bilinen fonksiyonel model dışında ölçek faktörünü, çekül sapması ve refraksiyon etkisini içeren genişletilmiş fonksiyonel modeller açıklanmış, karışık nivelman ağlarının dengelenmesinde kullanılabilecek bilinen ve tarafımızdan önerilen stokastik modeller hakkında bilgi verilmiştir. Yedinci Bölüm'de, iki farklı ağda yapılan uygulama sonuçları verilmiş, Sekizinci Bölüm'de ise uygulama sonuçları değerlendirilerek yorumlanmıştır.

The precise levelling is the main tool for the measurement of the national levelling nets. The levelling nets have been applied to many aspects in engineering surveying. For instance;. To determine recent crustal vertical movements. To determine possible vertical deformations in engineering structures (e.g. dams, bridges, tunnels, etc.). To establish high accuracy levelling points that are used for building of engineering structures (e.g. highways, railways, pipelines, etc.). To select the field for atomic electric stations and their control and maintenance. To appreciate the structures and machines which require high accuracy. To determine the geoid. To measure the national levelling nets In those types of applications, the required accuracy is about +0.5mmNkm. Precise levelling may not be used in some cases due to the topography of the land. Instead, trigonometric and valley cross levelling methods are used. In the end, a levelling net measured by different instruments and methods may be obtained. This type of the net is called "complex net". In the adjustment of the levelling nets, the weights are, in general, the inverse of the levelling line and the inverse of the square of levelling line for the precise and trigonometric levelling respectively. The weights derived from the above approaches may not represent reliable and proper stochastic models (Vanicek and Grafarend 1980). In these types of levelling nets, the accuracy of the observations is different from each other due to the use of different instruments and different observation and computational methods. Therefore, it is very important to define a proper and reliable a-priori stochastic model. Two types of unknowns can be used in the least-square adjustment. One of them is the scale factor and the other is the parameter of refraction and deviation of plumb line. The scale factor can be occurred between precise and the other levelling methods due to the different instruments and different observation and computational methods. The equation of the heigh difference between two bencmarks is as follows: Where k is the scale factor. If k is taken as unknown into the adjustment, the equation of residuals becomes the form of 1 l ff?-#,° Hl-H? Where 8k,: Unknown scale factor. If k0 = 1, the equation becomes vv = WJ-Wl-W1!-H?)Skl-H,l-Hf-Ah, The parameter of the refraction and deviation of plumb line are quite effected for the precise, trigonometric and valley cross levelling, especialy in the large area. The scale factor, parameter of the refraction and deviation of plumb line can be taken as unknown in the adjustment. The parameters of the refraction and deviation of plumb line are accepted to be equal in the two corrensponding points. This effect can be taken as a height difference and the equation of residuals then becomes vİJ=SHi-MJ+öh£ı-H°-H';-Ahij Where £hBi is the unknown of the height differences due to the sum of the effect of refraction and deviation of plumb line. Adjustment of levelling nets is performed in two ways: conditional and indirect approaches. In practice, the indirect approach is the most commonly used method. Because, the approach is more suitable to the programming and error estimate. The basic mathematical model is E(l}= Ax Following stochastic models are used in this research. MODEL 1) The basic equation of Pt=C/o* which shows relative accuracy between the weights, can be used for the adjustment of precise, trigonometric and valley cross levelling observations together. MODEL 2) The most commonly used models are XVI P=-L- p=± 'S ' n for precise levelling and s- for trigonometric and valley cross levelling. Where, S is the length of levelling line and n is the number of set-ups of the instrument. MODEL 3) Another stochastic model that is expanded by length of levelling line is p. - - e 2 MODEL 4) If the weights generally depend on the levelling lines. P = ± Sx can be written. In this case the weights are for precise levelling for valley cross levelling for trigonometric levelling XT " s-t The problem is to determine Xj. There are several ways for determining xj. For instance, the values, xj of the weights can be obtained from the iteration during the adjustment process. Here, the Xj values for a particular method are assumed to be 1 and the values for the other methods are then computed with an iteration process (e.g. xp = 1 for precise levelling). MODEL 5) Using the residuals related to each levelling line derived from the adjustment, the new weights can be computed from the following equation. The adjustment continues until the a-posteriori variance from adjustment equal to the 1. P =± M v? MODEL 6) Using residuals from adjustment, the standardised residuals, xvii 0 are performed. The standard deviations for each group o^=- (n-u); <&= - (»"«) '04 * can be written. Where nk is the number of observation for each group. The weights are calculated by multiple of variance of each group as a scale factor. OS Oo*0? The observations adjusted with the new weights and a20 are calculated once more. The weights are formed using the above equation. If a20 = 1 the procedure stops. The initial weights can be chosen a unit matrix or the weights mentioned above. MODEL 7) Another approach to the stochastic model determination is to use the Variance Component Estimation suggested by Helmert. This method can be summarised as follows. Step 1. The weight matrix, for each group of observations, W^,W2,- Wp is estimated before the least squares solution. The initial weights can be chosen to be unity. I.e.: P-j = P2 =...= Pp = I (identity matrix). Step 2. Using the initial weights, the normal matrices for each group, Nf,N2,-,Np and the global normal matrix N are formed. (Notice that N=N1 + N2+...+ NP) Step 3. From the least square's solution, the unknown parameters and the observation residuals are computed as following: x = N'1d (where d = ATPb) and Vf = A'fX - b-f xviii vp = ApX- bp Step 4. Then the following Helmert equation is formed rl\\ tin it\p i*2\ »»22 '" »»2p _fip\ flp2 '" flppA-b P-l YSsp. s2 C2 Where ci=viTpivj hjj = n; - 2Tr(N'1Nj) + Tr(N~1 NjN'1 Nj) hjj = Tr(N-1NjN-lNj) (for i * j) Si = S/2 Step 5. Having solved for s-|,S2,...,Sp. the new weights are computed as following: Pi = Pi.1/si.1 Step 6. If sj is not equal to 1 for all i=1,2,...,p the procedure returns to Step 2. When Sj=1 for all p groups the iteration is stopped. The initial weight matrix is sometimes very important for the last two models. It is suggested to start with the best estimate initial weight matrix. The value of the test of stochastic model is T = o? (f2) or T = ^ < (/,) o-o (f2) Where f is degrees of freedom and the variance must divide small variance. The Homogeneity criteria is the standard deviation of the standard deviation of the heights. Since the standard deviations related to u times heights (Oj,j= l,..u) are computed from any stochastic model, the standard deviation of the mean stansdard deviation XIX u(u - 1) Where a0 = - - '(6.94) After the values of \i and a° are computed for k times stochastic models, the homogeneity test is applied by comparing the values of ju for each two pairs. The values of juf and ns related to two stochastic models, the zero hypotesis is and the alternative hypotesis is HA=E(tf-tf)*0 The test magnitude of 2 rr> Mi 2 2 Tu = ~J » A ^ > Mj is compared with the F value derived from the Fischer table (F Table) for the degrees of freedom (/ = f.=u-\) and the confidence limit (1 - a ). When T < F li,j - rl-a,f"fj the zero hypotesis is valid. When T - T7 2 2 ILj " 2 > M-a,f"f} ?> A > Mj the stochastic model having ju;- is more superior than the other on the base of homogeneity. xx In order to classify the stochastic models whose superiorities cannot be defined (i.e. zero hypotesis), the mean accuracy criteria is used. In this case, the zero hypotesis is H0 = E(o°i) = E{o°j) and the alternative hypotesis is and the test magnitude is (7 - <7 For the degrees of freedom f=2(u-1) and the confidence limit (1 -a ) if the value computed from 6.102 (i.e. ty) is compared with the table value (ti-ocf) anc' 'f ti.^W the zero hypotesis is valid. In the other word the two stochastic models have the equal priority, if the model which has smaller mean standard deviation is more appropriate. In this research the first data is collected in so called the "Islands Levelling Nets" located in and around Istanbul. The data were collected by staff members of the Department of Geodesy and Photogrammetry in 1985. The Net has 71 benchmarks and 86 levelling lines. 5 of the levelling lines are measured by the valley cross levelling method, and other 5 of the levelling lines were measured in such a way that two theodolites were set up at two stations and observations were performed simultaneously. The rest of (76 lines) of the levelling lines were measured by the precise levelling method. The second data is collected in so called the "The Test Levelling Net of University of Selçuk" located in and around Konya. The net has 8 xxi benchmarks and 15 levelling lines. Levelling lines are measured by the precise levelling in such a way that two theodolites were set up at two stations and observations were performed simultaneously in several lengths (50 m, 100 m, 150 m, 200 m, 400 m). In this research, the problems of the adjustment of complex levelling nets and their solutions have been investigated. The functional zero hypotheses in the adjustment performed in the form of the expanded functional model were not fully accepted by all stochastic models. This result can be explained in two ways: (1) The systematic errors which affect observations, are filtered by the stochastic models that accept functional zero hypothesis. In other words, these stochastic models are called as best stochastic models. (2) The basic rule of adjustment is that the observations are in a form of normal distribution. If (1) is correct, the systematic errors are assumed as random errors, thus the achieved results will be wrong. The task of the stochastic models is to find out these types of systematic errors. In that case, final results by the stochastic models that do not except functional zero hypothesis, are more suitable and reliable. After that, in the examination of the results, the following criterion have been taken into account: Criteria 1, giving a result: the stochastic models which do not give a result, were not examined. Criteria 2, functional hypothesis test: in the expanded functional models, the stochastic models which accept functional zero hypothesis, were not examined. Criteria 3, stochastic model test: the test is applied after first two criterion. If the stochastic model excepts the stochastic zero hypothesis test, the model will be examined. Criteria 4, homogeneity and mean standard deviation: homogeneity and mean standard deviation tests are applied to the stochastic models which pass the first three models. 4.1 stochastic model is the best model at the end of all examination.

The precise levelling is the main tool for the measurement of the national levelling nets. The levelling nets have been applied to many aspects in engineering surveying. For instance;. To determine recent crustal vertical movements. To determine possible vertical deformations in engineering structures (e.g. dams, bridges, tunnels, etc.). To establish high accuracy levelling points that are used for building of engineering structures (e.g. highways, railways, pipelines, etc.). To select the field for atomic electric stations and their control and maintenance. To appreciate the structures and machines which require high accuracy. To determine the geoid. To measure the national levelling nets In those types of applications, the required accuracy is about +0.5mmNkm. Precise levelling may not be used in some cases due to the topography of the land. Instead, trigonometric and valley cross levelling methods are used. In the end, a levelling net measured by different instruments and methods may be obtained. This type of the net is called "complex net". In the adjustment of the levelling nets, the weights are, in general, the inverse of the levelling line and the inverse of the square of levelling line for the precise and trigonometric levelling respectively. The weights derived from the above approaches may not represent reliable and proper stochastic models (Vanicek and Grafarend 1980). In these types of levelling nets, the accuracy of the observations is different from each other due to the use of different instruments and different observation and computational methods. Therefore, it is very important to define a proper and reliable a-priori stochastic model. Two types of unknowns can be used in the least-square adjustment. One of them is the scale factor and the other is the parameter of refraction and deviation of plumb line. The scale factor can be occurred between precise and the other levelling methods due to the different instruments and different observation and computational methods. The equation of the heigh difference between two bencmarks is as follows: Where k is the scale factor. If k is taken as unknown into the adjustment, the equation of residuals becomes the form of 1 l ff?-#,° Hl-H? Where 8k,: Unknown scale factor. If k0 = 1, the equation becomes vv = WJ-Wl-W1!-H?)Skl-H,l-Hf-Ah, The parameter of the refraction and deviation of plumb line are quite effected for the precise, trigonometric and valley cross levelling, especialy in the large area. The scale factor, parameter of the refraction and deviation of plumb line can be taken as unknown in the adjustment. The parameters of the refraction and deviation of plumb line are accepted to be equal in the two corrensponding points. This effect can be taken as a height difference and the equation of residuals then becomes vİJ=SHi-MJ+öh£ı-H°-H';-Ahij Where £hBi is the unknown of the height differences due to the sum of the effect of refraction and deviation of plumb line. Adjustment of levelling nets is performed in two ways: conditional and indirect approaches. In practice, the indirect approach is the most commonly used method. Because, the approach is more suitable to the programming and error estimate. The basic mathematical model is E(l}= Ax Following stochastic models are used in this research. MODEL 1) The basic equation of Pt=C/o* which shows relative accuracy between the weights, can be used for the adjustment of precise, trigonometric and valley cross levelling observations together. MODEL 2) The most commonly used models are XVI P=-L- p=± 'S ' n for precise levelling and s- for trigonometric and valley cross levelling. Where, S is the length of levelling line and n is the number of set-ups of the instrument. MODEL 3) Another stochastic model that is expanded by length of levelling line is p. - - e 2 MODEL 4) If the weights generally depend on the levelling lines. P = ± Sx can be written. In this case the weights are for precise levelling for valley cross levelling for trigonometric levelling XT " s-t The problem is to determine Xj. There are several ways for determining xj. For instance, the values, xj of the weights can be obtained from the iteration during the adjustment process. Here, the Xj values for a particular method are assumed to be 1 and the values for the other methods are then computed with an iteration process (e.g. xp = 1 for precise levelling). MODEL 5) Using the residuals related to each levelling line derived from the adjustment, the new weights can be computed from the following equation. The adjustment continues until the a-posteriori variance from adjustment equal to the 1. P =± M v? MODEL 6) Using residuals from adjustment, the standardised residuals, xvii 0 are performed. The standard deviations for each group o^=- (n-u); <&= - (»"«) '04 * can be written. Where nk is the number of observation for each group. The weights are calculated by multiple of variance of each group as a scale factor. OS Oo*0? The observations adjusted with the new weights and a20 are calculated once more. The weights are formed using the above equation. If a20 = 1 the procedure stops. The initial weights can be chosen a unit matrix or the weights mentioned above. MODEL 7) Another approach to the stochastic model determination is to use the Variance Component Estimation suggested by Helmert. This method can be summarised as follows. Step 1. The weight matrix, for each group of observations, W^,W2,- Wp is estimated before the least squares solution. The initial weights can be chosen to be unity. I.e.: P-j = P2 =...= Pp = I (identity matrix). Step 2. Using the initial weights, the normal matrices for each group, Nf,N2,-,Np and the global normal matrix N are formed. (Notice that N=N1 + N2+...+ NP) Step 3. From the least square's solution, the unknown parameters and the observation residuals are computed as following: x = N'1d (where d = ATPb) and Vf = A'fX - b-f xviii vp = ApX- bp Step 4. Then the following Helmert equation is formed rl\\ tin it\p i*2\ »»22 '" »»2p _fip\ flp2 '" flppA-b P-l YSsp. s2 C2 Where ci=viTpivj hjj = n; - 2Tr(N'1Nj) + Tr(N~1 NjN'1 Nj) hjj = Tr(N-1NjN-lNj) (for i * j) Si = S/2 Step 5. Having solved for s-|,S2,...,Sp. the new weights are computed as following: Pi = Pi.1/si.1 Step 6. If sj is not equal to 1 for all i=1,2,...,p the procedure returns to Step 2. When Sj=1 for all p groups the iteration is stopped. The initial weight matrix is sometimes very important for the last two models. It is suggested to start with the best estimate initial weight matrix. The value of the test of stochastic model is T = o? (f2) or T = ^ < (/,) o-o (f2) Where f is degrees of freedom and the variance must divide small variance. The Homogeneity criteria is the standard deviation of the standard deviation of the heights. Since the standard deviations related to u times heights (Oj,j= l,..u) are computed from any stochastic model, the standard deviation of the mean stansdard deviation XIX u(u - 1) Where a0 = - - '(6.94) After the values of \i and a° are computed for k times stochastic models, the homogeneity test is applied by comparing the values of ju for each two pairs. The values of juf and ns related to two stochastic models, the zero hypotesis is and the alternative hypotesis is HA=E(tf-tf)*0 The test magnitude of 2 rr> Mi 2 2 Tu = ~J » A ^ > Mj is compared with the F value derived from the Fischer table (F Table) for the degrees of freedom (/ = f.=u-\) and the confidence limit (1 - a ). When T < F li,j - rl-a,f"fj the zero hypotesis is valid. When T - T7 2 2 ILj " 2 > M-a,f"f} ?> A > Mj the stochastic model having ju;- is more superior than the other on the base of homogeneity. xx In order to classify the stochastic models whose superiorities cannot be defined (i.e. zero hypotesis), the mean accuracy criteria is used. In this case, the zero hypotesis is H0 = E(o°i) = E{o°j) and the alternative hypotesis is and the test magnitude is (7 - <7 For the degrees of freedom f=2(u-1) and the confidence limit (1 -a ) if the value computed from 6.102 (i.e. ty) is compared with the table value (ti-ocf) anc' 'f ti.^W the zero hypotesis is valid. In the other word the two stochastic models have the equal priority, if the model which has smaller mean standard deviation is more appropriate. In this research the first data is collected in so called the "Islands Levelling Nets" located in and around Istanbul. The data were collected by staff members of the Department of Geodesy and Photogrammetry in 1985. The Net has 71 benchmarks and 86 levelling lines. 5 of the levelling lines are measured by the valley cross levelling method, and other 5 of the levelling lines were measured in such a way that two theodolites were set up at two stations and observations were performed simultaneously. The rest of (76 lines) of the levelling lines were measured by the precise levelling method. The second data is collected in so called the "The Test Levelling Net of University of Selçuk" located in and around Konya. The net has 8 xxi benchmarks and 15 levelling lines. Levelling lines are measured by the precise levelling in such a way that two theodolites were set up at two stations and observations were performed simultaneously in several lengths (50 m, 100 m, 150 m, 200 m, 400 m). In this research, the problems of the adjustment of complex levelling nets and their solutions have been investigated. The functional zero hypotheses in the adjustment performed in the form of the expanded functional model were not fully accepted by all stochastic models. This result can be explained in two ways: (1) The systematic errors which affect observations, are filtered by the stochastic models that accept functional zero hypothesis. In other words, these stochastic models are called as best stochastic models. (2) The basic rule of adjustment is that the observations are in a form of normal distribution. If (1) is correct, the systematic errors are assumed as random errors, thus the achieved results will be wrong. The task of the stochastic models is to find out these types of systematic errors. In that case, final results by the stochastic models that do not except functional zero hypothesis, are more suitable and reliable. After that, in the examination of the results, the following criterion have been taken into account: Criteria 1, giving a result: the stochastic models which do not give a result, were not examined. Criteria 2, functional hypothesis test: in the expanded functional models, the stochastic models which accept functional zero hypothesis, were not examined. Criteria 3, stochastic model test: the test is applied after first two criterion. If the stochastic model excepts the stochastic zero hypothesis test, the model will be examined. Criteria 4, homogeneity and mean standard deviation: homogeneity and mean standard deviation tests are applied to the stochastic models which pass the first three models. 4.1 stochastic model is the best model at the end of all examination.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1996

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1996

##### Anahtar kelimeler

Nivelman ağları,
Stokastik modeller,
Nivelman networks,
Stochastic models