Reliability of the Conventional Deformation Analysis Methods for Vertical Networks

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1 Reliability of the Conventional Deformation Analysis Methods for Vertical Networs Serif HEKIMOGLU, Hüseyin DEMIREL and Cüneyt AYDIN, urey Key words: Deformation analysis, vertical deformation, reliability, homoscedasticity heteroscedasticity. ABSRAC Reliability of the conventional deformation analysis methods is defined with mean success rate. he mean success rate is given as the number of successes divided by the number of experiments. Four different vertical networs are generated by simulation. he observations for two epochs and deformations are also generated. he mean success rates of the methods are computed for certain number of deformed points, for a given interval and for inds of deformation. Consequently, the reliability of the methods changes depending on number of points, magnitude of deformations, degrees of freedom and number of deformed points. he reliability of the methods increases when the degrees of freedom and the magnitudes of deformations increase. It decreases when the number of points and the number of deformed points increase. As nown, the random errors variances are changing depending on the distance of the levelling lines. his Gauss-Marov model is called as heteroscedastic. If the distances are approximately equal to each other, the random errors have a common variance. So, the model is called as homoscedastic. If the model is homoscedastic, the reliability of the analysis methods increases very rapidly with respect to the ones of the heteroscedastic model. CONAC Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Yildiz echnical University Department of Geodesy and Photogrammetry Engineering Istanbul URKEY el Fax heim@yildiz.edu.tr, hdemirel@yildiz.edu.tr and caydin@yildiz.edu.tr Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 1/13

2 Reliability of the Conventional Deformation Analysis Methods for Vertical Networs Serif HEKIMOGLU, Hüseyin DEMIREL and Cüneyt AYDIN, urey 1. INRODUCION One of the main aims of geodesy is detection of the deformations imposed on an object or an area which is characterized with points of a geodetic networ. Since it is essential to detect deformations for many purposes (monitoring plate tectonics, determination of global datum, taing precautions for a construction which may be under damage, etc.), many considerable efforts and investigations have been performed on deformation analysis (Chen 1983; Chrzanowsi et al. 198; Liu and Chen 1998; Niemeier 1985; Pelzer 1971; Welsch et al. 000). Conventionally, for all types of networs (vertical, horizontal or 3D networs), for detecting deformations, same points estimated coordinates obtained from least-squares adjustment of observations made at different epochs are compared with each other by using the statistical tests. herefore, this procedure is called as conventional or geometrical analysis which comprises global congruency test and localization steps (Welsch and Heunece 001). Although it is also used widely to verify stabilities of some points (to define reference points) or instabilities of others, it is not nown exactly that if it gives correct results in all probable cases or which situations may increase its reliability according to imposed deformations. o get information about reliability of the method, true deformations must be nown in advance. However, in real cases, it is not entirely possible to estimate what deformation magnitudes are imposed on which points. For that reason, some authors use simulation that creates probable cases in model to find some methods are effectual (Betti et al. 1999; Liu and Chen 1998). In this study, we investigated how reliability of the conventional deformation analysis methods can be measured and how it changes for the vertical networs. herefore, we adapted the reliability concept introduced by Heimoglu and Koch (1999; 000) to the conventional deformation analysis. he reliability is defined as a mean success rate obtained from number of success divided by the number of experiments to identify outliers. In this concept, we put the deformations instead of outliers. For this purpose, we simulated four vertical networs. In the vertical networs, the random errors variances are changing depending on the distance of levelling lines. So, the Gauss-Marov model is called as heteroscedastic in robust statistics (Carroll and Ruppert 198; Heimoglu and Berber 001). We now that if the heteroscedasticity is strong, it affects on the reliability of robust estimators badly (Heimoglu and Berber 001). herefore, for the vertical networs observations, we considered two random error types: heteroscedastic and homoscedastic. hen, simulated deformations for the vertical networs are added to certain number of points heights at present epoch. hey are defined for different inds and a given interval of deformation. After the deformed points detected separately with using two localization methods (Gauss Elimination and Implicit Hypothesis) are compared with nown deformed Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs /13

3 points, the mean success rates of these methods are obtained for the vertical networs.. CONVENIONAL DEFORMAION ANALYSIS In the conventional deformation analysis (CDA), the deformations are accepted as significant geometrical differences verified with statistical test, i.e., F-test. So, detection of the deformations basically depends on comparison of geometry of the geodetic networs observed at different epochs, e.g., initial and present epochs (Welsch and Heunece 001)..1 Free Adjustment of the Vertical Deformation Networs In order to remove geodetic datum deficiency, heights and their cofactor matrix of each epoch of the vertical networ are computed individually by free adjusment method. All of the heights of the networ are incorporated to the model as unnowns. So, the Gauss-Marov model can be expressed as (Koch 1999) v = A x l, (1) 1 l P = Q, () where is the number of epochs, v is the n 1 residual vector, A is the n u design matrix, x is the u 1 unnown vector, l is the n 1 observation vector, P is the n n diagonal weight matrix of observations, Q l is the n n weight coefficient matrix of the observations, u is the number of unnowns (equal to number of points in vertical networs), n is the number of observations. When the conditions v P v = min, x x = min are satisfied, estimated unnown vector can be formulated as x ˆ = Q A P l, (3) where xˆ xˆ Q xˆ xˆ is the cofactor matrix of estimated unnowns, + Q xˆ xˆ = ( A P A ), (4) + where ( A P A ) is the pseudo inverse of the normal equation (Koch 1999; eunissen 1985). he estimated variance factor as fallows: s 0 v P v =, (5) f where f is the degrees of freedom. If the two epochs estimated variance factors are equal to each other statistically, the pooled variance factor is written by s 0 v 1 P1 v1 + v P v =, f = f 1 + f. (6) f Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 3/13

4 . Global Congruency est Whether the difference between the estimated unnown vectors for two epochs is the result of the deformations or random errors of the observations is investigated with global congruency test. For this aim, expected values of the estimated unnown vectors are assumed equal to each other by the following null-hypothesis H { ˆ } E{ x } 0 : E 1 ˆ x =. (7) he influence of the hypothesis can be expressed as + R = d Q d, (8) dd where d is the difference vector of the estimated unnown vectors d = xˆ x, (9) ˆ1 Q dd is the its cofactor matrix, Q = Q + Q. (10) dd xˆ 1xˆ 1 xˆ xˆ If the null-hypothesis is true, the test quantity G follows the central F-distribution, R G = h, f h s 0 F, (11) where h is the ran of Q dd. If G F h, f, 1 α ( 1 α is confidence level), the null hypothesis is rejected. In other words, the difference is accepted as a result of deformation. herefore, the next step is localization of deformations..3 Localization of Deformations Among the many localization methods presented in Welsch et al. (000), Gauss elimination and implicit hypothesis methods are reviewed below..3.1 Localization with Gauss elimination + he difference vector d and Q dd are divided into sub-vectors and sub-matrices as follows (Niemeier, 1985; Welsch et al. 000) df + PFF PFB d =, Q dd = d, (1) B PBF PBB where index B describes assumed deformed point P B, index F is for the other points. o distinguish independently the effect of the point P B on R in (8), the following equation is obtained by using Gauss elimination method d 1 B db + PBB PBF df =. (13) Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 4/13

5 So, the partial effect of P B on quadratic form R is given by R B B = d P d. (14) BB B Sequentially, each point is considered as deformed point. he point which gives maximum R B is accepted as deformed point and it defines the current datum. o test whether remaining points are deformed, d and Q dd is transformed into the current datum with using S transformation (eunissen 1985) d i = Si d, Q d d = Si Qdd Si, ( i = 1,, u ). (15) i i Using elements related to the remaining points in (15), the new global congruency test is applied. he same localization procedure is performed until the test can not verify existence of deformation any more. As a result of the localization, deformed points are detected..3. Localization with implicit hypothesis Some points estimated unnowns are considered stable for two epochs by the nullhypothesis (Welsch et al. 000) H { x ˆ1F} E{ xˆ F} = xf ˆ 0 : E =. (16) he hypothesis is included into the following observation equation implicitly xˆ F A1F A1B o l1 v H = xˆ 1B, (17) AF o AB l x ˆ B where ˆx 1B and ˆx B are unnown vectors related to assumed as the deformed point. he influence of the hypothesis can be expressed as = v P v ( v P v + v P v ), (18) R I H H where P = diag ( P 1, P ). he point i that gives minimum R I is defined as the deformed point. For the new global test, the minimum value of R I and degrees of freedom (h min = h 1) are used. If the test quantity is greater than F h min, f,1 α, the observation equation (17) is augmented as fallows: xˆ F ˆ x1b A1F A1B o A1i o l1 v ˆ H = x B, (19) A F o A B o A i ˆ l x1i ˆ x i where ˆx 1i and ˆx i are unnown vectors of point i obtained as deformed from previous localization. Its location in the model is not changed until the end of the procedure. Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 5/13

6 3. HE RELIALIBIY CONCEP: HE MEAN SUCCESS RAE he reliability concept is defined as the mean success rate which is obtained by number of success divided by the number of experiments to identify outliers (Heimoglu and Koch 1999). We adapt this concept to the CDA with the aim of obtaining its reliability. Vertical deformations occurs as uplift or subsidence. In other words, the signs of the deformations are plus or minus. herefore, random and influential deformations are used and called as ind of deformations. Random deformations (RD) have signs selected randomly, while influential deformations (ID) have the same signs, i.e., only all plus or all minus. If a point is translated as ~ d ( u i i = 1,, ) in vertical direction between initial and present epochs, u 1 vertical deformation vector d ~ and the set of deformed points M can be expressed as d ~ ~ = [ 0 0 di 0 0 ], (0) { 0,0, i0,0 } M =. (1) ~ It is considered that the magnitute of di lies in the interval (int) as fallows: int = aσ< ~ di < bσ, a 3, a b and b > a, () where σ is the variance of unit weight. So the initial and present epoch s observation vectors are written as l 1 h + e1 A1 H 0 l =, (3) ~ = h + e + A d A H, (4) 0 where h is the true height differences vectors, e 1 and e are n 1 normal distributed random error vectors and H 0 is the vector of approximate heights of points. Both observation vectors l 1 and l constitute a woring sample. If the set M is equal to the set of detected points M L from one of the localization methods that use the estimated unnown vectors and cofactor matrices from l 1 and l, the method was considered as successful. According to the generating different random error vectors ( e 1, e ) and different deformation vector d ~, we can produce many woring samples. When the deformation vector d ~ contains n d (1 n d < u) number of deformation of any inds with any magnitudes in the given interval, we can define the mean success rate as fallows (Heimoglu and Koch 000) γ r mean N q ( L, l 1, l, int, n d, n, u) =, (5) N s= 1 Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 6/13

7 where r is the deformation ind, L is the localization method, q is the number of success, s denotes a certain woring sample, N is the number of woring samples. hus, the different mean success rates for the different deformation inds can be computed. he smallest one of them is accepted here as the reliability of the CDA for a certain number of deformation and for a given interval (int): Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs r { γ (L, l, l, int, n,n,u), r 1, } γ ( L, l 1, l, int, nd,n,u) = minimum 1 d =. (6) 4. SIMULAION In order to obtain the reliability of the CDA for vertical application areas, we simulated four vertical networs as shown in Figure 1a-d. Number of observations, number of unnown parameters and degrees of fredoom for free networs are given in able 1. able 1. Number of observations, number of unnown parameters, degrees of freedom for vertical networs Networ I II III IV Number of observations (n) Number of unnown parameters (u) Degrees of freedom (f) Generating of Simulated Deformations ~ a) Random deformations (RD): he magnitude of d i of one deformation is generated by the uniform distribution for a given interval as fallows: ~ int = aσ < di < bσ, (7) ~ d =, ( i = 1,, u ), (8) i sign(t1i ) ~ d 0i sign(t ~ d 1i + t1i > 0.5 ) =, 0 < t1i 1, (9) t1i 0.5 = aσ + t1i (bσ a ), i = u t i, 0 < t i < 1, a 3, b > a, (30) 0i σ where t 1i and t i are distributed uniformly (Heimoglu and Koch 1999; 000). his algorithm has been computed 500 times for each woring sample in (3) and (4). ~ ~ he magnitudes d i, d j of two deformations, and etc. are generated correspondingly by the uniform distribution. We used the three intervals for the magnitutes of deformations as follows: 3σ 6σ, 3σ 10σ, 10σ 50σ. b) Influential deformations (ID): ID s magnitudes are generated also by the uniformed distributions for a given interval as done for the RD. Only they all have the same sign, i.e., all 7/13

8 plus or all minus a b c d Fig. 1a-d. Configurations of simulated networs: a Networ I, b Networ II, c Networ III, d Networ IV 4. Observations If random errors are independent and identically distributed (iid), the Gauss-Marov model is called as homoscedastic in robust statistics (Carroll and Ruppert 198). However, the random errors variances in vertical networs are changing depending on distance levelling lines. he model is called as heteroscedastic (Carroll and Ruppert 198; Heimoglu and Berber 001). o investigate the effect of these concepts on the CDA, we produced two types of random errors of observations for the networs as follows. a) Random errors are heteroscedastic (ype 1) We produced random error vectors with normal random error generator in MALAB v.5.1 software for one initial and present epochs as fallows: e = [ e1 e e n ], = 1,, (31) where e j ( j = 1,,, n) comes from the different normal distributions N( µ j = 0, σ j = σ L j ), σ =1mm / 1m is standart deviation and L j is the distance of levelling line. hen, the observation vectors l 1 and l are obtained by using (3) and (4). For stochastic model, we selected the distance of levelling lines in the interval of m. wo epoch s observations distances are the same, so the weight matrices of observations are equal to each other as follows: P = P = diag ( L / L, L / L,, L / L ), (3) where L 0 is the unit distance (1m) of levelling line. 0 n Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 8/13

9 b) Random errors are homoscedastic (ype ) In this type, the distances of levelling lines are chosen the same for all the networs, i.e., L j = 1 m. So, random errors of the observations come from the same distribution and are independent, i.e., the model is homoscedastic. 4.3 Analysis o obtain a mean success rate of the CDA for a given magnitude interval, a deformation ind and n d number of deformed point in a vertical networ, we produced randomly 500 different woring samples by applying mentioned above procedures. hen, the observation vectors of two epochs were free adjusted individually using the same design matrices and the same weight matrices for two epochs in the vertical networ. All the estimated variance factors for the woring samples are approximately equal to 1 mm as expected, because there are not any ouliers in the observations. he model test which checs the null hypothesis H 0 : E{ s 01} = E { s 0 } was applied to the samples to verify whether they were used for the deformation analysis (%95 confidence level is here used for all statistical tests). We used two localization methods for checing the results obtained from them. Applying both methods, we computed the mean success rates according to (5). 5. RESULS he same results were obtained from two localization methods performed to the same woring samples. herefore, the results from one of two methods are presented only as the mean succes rate of the CDA. First, to obtain whether the CDA produces deformations for the networs, we used 500 samples which do not have any deformations. As shown in able, the analysis may produce deformations with the rate of %6.3 on the average. able. he mean succes rates in the case of no added deformations for the vertical networs Networ type 1 Networ number I II III IV I II III IV he mean succes rates he Vertical Networs of the ype 1 he results of the mean success rates for ype 1 are shown in able 3 and able 4. he mean success rates under the 51% are shaded in tables to compare them with the others easily. Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 9/13

10 5.1.1 For random deformations In the networs I, II, III and IV, the mean success rates are decreased while the number of deformed point n d is increased for all of the intervals. For the interval of 10σ 50σ, the mean success rates are quite bigger than the ones for the intervals of 3σ 6σ and 3σ 10σ. he networs I and II have the same number of points, but different degrees of freedom. For the intervals of 3σ 6σ and 3σ 10σ, the mean success rates of the networ I are bigger than the ones for the networ II since the degrees of fredoom are increased. he networs III and IV have the same number of points, but different degrees of freedom. For the intervals of 3σ 6σ and 3σ 10σ, the mean success rates of the networ III are bigger than the ones which are obtained for networ IV because of the increasing of the degrees of freedom. Moreover, the networs II and III have different number of points, but the same degrees of freedom. Although the number of observation of the networ III is smaller than the ones of the networ II, the mean success rates of the networ III are larger than the ones of the networ II for the interval of 3σ 6σ and 3σ 10σ for nd For influential deformations he mean success rates were obtained for only all plus (uplift) and only all minus (subsidence) deformations. he features of the mean success rates gained for RD are the same for ID. However, the mean success rates are decreased related to the ones that are computed for the RD when n d >. he results of ID were not given in able 3 and 4. Magnitude of deformation able 3. he mean success rates for RD in the networs I and II (ype 1) Number of Deformed Points NEWORK I (n=36, u=16, f=1) 3σ-6σ σ-10σ σ-50σ NEWORK II (n=7, u=16, f=1) 3σ-6σ σ-10σ σ-50σ Magnitude of deformation able 4. he mean success rates for RD in the networs III and IV (ype 1) Number of Deformed Points NEWORK III (n=19, u=8, f=1) NEWORK IV (n=15, u=8, f=8) 3σ-6σ σ-10σ σ-50σ Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 10/13

11 5. he Networs of the ype he mean success rates for ype are shown in able 5 and able 6. he mean success rates under the 51% are shaded in tables to compare them with the others easily For random deformations he mean success rates for 3σ 6σ and 3σ 10σ are quite bigger than the ones that were obtained from the ype 1. As seen the results from able 5 and able 6, the features of the mean success rates obtained for the networs of ype 1 are the same for ype. 5.. For influential deformations he mean success rates for 3σ 6σ and 3σ 10σ are bigger than the ones that were achieved from the ype 1. he results of ID are not presented in able 5 and 6. Magnitude of Deformation able 5. he mean success rates for RD in the networs I and II (ype ) Number of Deformed Points NEWORK I (n=36, u=16, f=1) 3σ-6σ σ-10σ σ-50σ NEWORK II (n=7, u=16, f=1) 3σ-6σ σ-10σ σ-50σ able 6. he mean success rates for RD in the networs III and IV (ype ) Number of Deformed Points Magnitude of Deformation NEWORK III (n=19, u=8, f=1) NEWORK IV ( n=15, u=8, f=8) 3σ-6σ σ-10σ σ-50σ CONCLUSION In this paper, we pointed out that the reliability of the conventional deformation analysis for the vertical networs can be measured by using the mean success rates. he reliability of the CDA changes depending on the many factors, such as, the number of unnowns, the degrees of freedom, the number of deformed points, the magnitudes of deformation, the deformation inds (RD and ID), especially types (heteroscedasticity or homoscedasticity) of the random errors. According to the results of the numerical simulations, if the degrees of freedom and the magnitudes of the deformations increase, the reliability of the CDA increases. On the contrary, when the number of unnowns, number of deformed points increase, the reliability of the CDA decreases. Furthermore, if the random errors come from the same normal Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 11/13

12 distribution, i.e., the model is homoscedastic, much more reliable results are obtained from CDA. When the points have vertical deformations whose magnitudes are bigger than approximately 10σ, CDA s reliability is high in all vertical networs used in this paper. If the magnitudes are smaller than 10σ, the reliability of the CDA s starts decreasing rapidly. However, this decreasing rate in the mean success rates of ype 1 is bigger than the ones that in ype. Although vertical networ II and networ III have the same degrees of freedom, the number of unnowns of networ III are smaller than networ II. When the mean success rates of these networs are compared with each other, the ones of networ III are bigger than the ones for networ II. It can be interpreted that the number of points must be small to get more reliable results. Consequently, the reliability of the CDA may increase very much if the points of a vertical networ are established, so that the distances of levelling lines are approximately equal to each other. Moreover, in this networ, when the assumed number of deformed points and especially the number of points are taing small, the reliability of the CDA increases. So, even if the deformation magnitutes of the points are smaller than approximately 10σ, more reliable results may be obtained since the Gauss-Marov model is homoscedastic. REFERENCES Betti B., Biagi, L., Crespi, M., Riguzzi, F., 1999, GPS sensitivity analysis applied to nonpermanent deformation control networs, Journal of Geodesy, vol. 73, pp Carroll, R. J. and Ruppert, D.,198, Robust estimation in heteroscedastic linear models, he Annals of Statistics, vol. 10, no., pp Chen, Y., Q., 1983, Analysis of deformation surveys-a generalized method, Department of Surveying Engineering, University of New Brunswic, echnical Report No. 94. Chrzanowsi, A, Chen, Y. Q., Secord J., M, 198, On the analysis of deformation surveys, 4 th Canadian Symposium on Mining Surveying and Deformation Measurements, he Canadian Institute of Surveying, Banff, , Proceedings. Heimoglu, S., and Koch, K-R., 1999, How can reliability of robust methods be measured?, hird urish-german Joint Geodetic Days, Istanbul, vol.1, pp Heimoglu, S., and Koch, K-R., 000, How can reliability of the test for outliers be measured?, AVN 107, pp Heimoglu, S., and Berber, M., Reliability of robust methods in heteroscedastic linear models, IAG 001 Scientific Assembly, September , Budapest, Hungary. Koch, K., R., 1999, Parameter estimation and hypothesis testing in linear models, Springer- Verlag, New Yor, nd Edition. Liu, Q., W. and Chen Y., Q., 1998, Combining the geodetic models of vertical crustal deformation, Journal of Geodesy, vol. 7, pp Niemeier, W., 1985, Deformationanalyse, Geodaetische Netze in Landes-und Ingeniuervermessung II, Herausg. von H. Pelzer, Konrad Wittwer, Stuttgart, pp Pelzer, H., 1971, Zur Analyse Geodaetischer Deformationmessungen, Deutsche Geodaetiche Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 1/13

13 Kommission, Reihe C, No. 164, München. eunissen, P. J., 1985, Zero order design: Generalized inverses, adjustments, the datum problem and S-transformations. In: Grafarend E. W., Sanso F (eds), Optimization and design of geodetic networs. Springer, Berlin Heidelberg New Yor, pp Welsch, W., Heunece, O., Kuhlmann, H., 000, Auswertung Geodaetischer Überwachungsmessungen, Herbert Wichman Verlag, Heidelberg. Welsch, W., M. and Heunece, O., (001), Models and terminology for the analysis of geodetic monitoring observations, he 10th International Symposium on Deformation Measurements, Orange, California, USA, pp Serif Heimoglu, Hüseyin Demirel and Cüneyt Aydin Reliability of the Conventional Deformation Analysis Methods for Vertical Networs 13/13