S-Transformasyonu ve yatay kontrol ağlarında deformasyon analizi

Abstract

ÖZET Bu çalışmada yatay kontrol ağlarıyla, geometrik deformasyon analizinin genel ilkeleri teorik esaslarıyla açıklanmış; ayrıca bu çalışmanın ağırlıklı konusunu oluşturan S-transformasyonu ve onun yardımıyla deformasyonların belirlenme ilkeleri ele alınmıştır. Deformasyon analizinde, varsayımlardan mümkün olduğunca sakınmak gerektiğinden ağın konumlandırılması, yönetilmesi ve ölçeklendirilmesi, yani ağın datumu üzerinde varsayımlara yer vermeyen ve ağın iç prezisyonunu gerçekçi bir biçimde yansıtan serbest ağ dengelemesi uygulanır. Bu konu üçüncü bölümde ana hatlarıyla işlenmiştir. Kaçınılmaz varsayımlar ise matematik istatistik yöntemlerle test edilmesi gere¬ kir. Bu nedenle ölçüler ve bilinmeyenler arasındaki matematiksel ve fiziksel gerçekliği temsil eden, ölçme ve hipotezlere dayalı bir fonksiyonel ve stokastik ilişkiler kümesi biçiminde tanımlanan `Gauss-Markoff` modelinde yapılan kabullerin sorgulanması ve modelin gerçeğe yakınlığının test edilmesi gerekir. Bu mutlak gereklilikleri yerine getirmek için ikinci bölümde jeodezik değerlerin özellikleri, istatistik testler ve hipotez testleri, dördüncü bölümde ise model testleri genel olarak ele alınmıştır. Uygulama bölümünde ise İstanbul'da bir yapı çukurundaki derin kazı destekleme sisteminin yatay hareketlerinin belirlenmesi ve yorumlanması için kurulan yatay kontrol ağının 19 noktadan oluşan bir bölümüne ait dört periyot ölçü değerlendirilmiştir. Xİ

SUMMARY S-TRANSFORMATION AND DEFORMATION ANALYSIS İN HORIZONT AL CONTROL NETWORKS in this study, basic principles of geometrical deformation analysis applied in ho- rizontal control networks and the strategy of analyzing deformation measurements by means of S-transformation are discussed. A rigorous analysis of the measureınents became feasible only recently with the ad- vent of modern computers. These permit data processing using sophisticated matiıe- matical models and are capable of dealing simultaneously with large numbers of da¬ ta. The development of new and improyed instruments of signifıcantly higher accu- racy has also opened new fıeld of application. The purpose of the monitöring of deformation usually is, *Establishment and/or verifıcation of hypothesis in geophysics, geology, glaciology and engineering sciences, *Assessment of safety and performance of engineering structures, *Protection of population from hazards caused by rock slides, slope creep and engi¬ neering structures, *Determination of the responsibility for damages caused by mining, tunneling and similar activities. The object ör area under investigation is usually represented by a number of points which are monumented ör marked durably. Geodetic observations transform the cluster of points into a geodetic network. The number of points depends on the sub- ject and on the deformations anticipated. To investigate the deformations of an object ör an area the geodetic observations are repeated at different epochs of time. The observations of each epoch are adjusted independently through free netvvork adjustment using the same approximate coor- dinates. From the coordinate differences bet/yeen the epochs the parameters of a deformation model are estimated and conclusions on the object deformations are drawn. The Gauss-Markoff model, which is a linear mathematical model consisting of func- tional and stochastic relations, is given as, xiiü-Aİ-i £, - ol ö, where Y is(nxl)vectorofresiduals, A (nxu)matrixofunknownparameters, î (uxl) vector of unknowns, l (nxl) vector of observations, Z; (nxn) variance- covariance matrâ, Q (nxn) cofactor matrix of observations and a% is the a- priori variance factor. in the procedure of free network adjustment, the coordinates of ali points of the netvvork are considered as unknowns. For a 2d-network with a rank deficiency of d = 4 four constraints are required. If the arrangement of the coordinates of the p point is, £r = (*ı> :vı> ** J^...> w/>) then, l O l O... O l O l... GT - - - - - -?1 *ı ^ X2... *i y ı *2 y~2... xt and y ı are the coordinates shifted to the center of gravity of the network. - / x MJ -, x MI **-W0- py*-W0- p where (x{) and (y,.) denote the coordinates in the original system. To achieve numerical stability for computer solutions, the matrk fi is to be normalized. ı Geometrically, the two fırst rows of Q7 takecareofthetranslations in x and y direction. The third row defınes tiıe rotation about the vertical axis and the fourth row adjust the scale of the observations to that given by the approximate coordinates. If some observations contain datum information, then the corresppnding rows of <2r are omitted. in the netvvorks consisting of directions and distances ör only distances the fourth row of G1 matrix is omitted. If certain points are to be xiii> excluded from contributing to the datum, the corresponding coeffîcients of G.T are simply replaced by zeros leading to the matrix R. With the coefficient matrix R of datum constraints, the solution equations of the Linear Gauss-Markoff Model are g iven below. qt-(& + RRTY1U(ll + RRTYl £-(&+ RRTYl ATPL Qv~£:1 -AQ£AT V--QVEI 2 y.TEY d°D-T` where £ is (nxn) weight matrix of observations, Q£ is (uxu) cofactor matrk of unknowns, a% is a posteriori variance factor and f is the degree of freedom. The Gauss-Markoff model is defined as a set of functional and stochastic relationship based on observations and hypothesis, which represent the physical reality. There- fore it is always essential to consider and question the model assumption and to check how the model conforms with reality. The fırst test which is applied after the adjustment is the global test on the a posteri¬ ori variance factor o0. The null hypothesis of the Global Test is `The model is correct and complete`. This can be expressed as: #0:*(*o) = ol and the alternative hypothesis, Ha:E(ö20)* 4 and leads to the last stochastic, r--î` - *TE* 4 f «l xivThe critical value, Fl_al2.j^ ör (x2ı-«/2;/)/ / ıs completed ör taken from the table of concerning distribution. v2 T<r pX(l-«/2j/) 1 ^ r(l-a/2y,~) `j, then the null hypothesis is accepted. If test fails, then H0 is rejected. There may be more than öne reason for rejection for example, *Incorrect estimate of vveights, *İncorrect mathematical model, *Blunders in the observation. We may not know which öne of the above reasons caused the failure of the test confining ourselves to the third possible cause for rejection, i.e. blımders in the observations, an alternative hypothesis Ha is introduced. The simplest Ha considers a possible shift VI of the probability distribution of the observations l. E(l/Ha)-E(l/H0) f VI Under the null hypothesis the expectations of residuals is zero, E l y. l HQ/°*Q but under the alternative hypothesis E l y. l Ha/°°Vl. The relationship between VI and Vv is known as VY - Q £ VI V in practice, a further simplifıed alternative hypothesis is used since it is not easy to know which observations are the erroneous ones. The new Ha assumes that only öne observation at the time is erroneous. The new null hypothesis, HQ:E(VL)~0 and alternative hypothesis, Ha:E(VL)*0 XVwhere VI is the gross error in the i th observation. in Baarda's Data-Snooping method, the test statistic is given as follows Iv.l Iv.l.OT 0«_ =' °v</ fa where v. is the residual of i th observation and qv is the elements of Q. If the maximum value Wwa. of W. is greater than the critical value Nl_al2 or 4/^ı-tt. ı ` then i. th observation contains gross error. ' *j in T Test, developed by Pope, the true variance factor o0 is unknown. His approach is based on the studentized residuals: Iv.i Iv.l Ttîî- Öv,*o ^ The test statistic Tt is compared with the value tl_(t;/. in Heck's t-Test, the observation under investigation is omitted and adjustment procedure is re- peated. The test statistic for this method is similar those of above, lv,l *i ` A /- > *l-«/2;/-l °ı V^< where âî is the calculated a posteriori variance factor which contribution of suspicious observation excluded from quadratic form û. The robust strategy of detecting of outliers is different from the methods outiined above. The basic idea is that large residuals indicate less accurate observations and vice versa. Therefore, after a conventional Least Square estimation of para- meters the Gauss-Markoff model, the a priori weights are replaced by new ones being functions of residuals. Then a new estimation is carried out and leads to new residual, from which again new weights are computed. This process of esti¬ mation and modifıcation of weights repeated until convergence is achieved. The equations of this iteration method are as follows. xviJ *t-(ArXtA)-lArKtl Wt-LSİ(¥^l)k- 1,2,3,... ff(ü)-' where JÜ7.K)-1/ is a weight function and k is the number of iterations. Coordinates and their variances and covariances are always related to datum they are based on. The coordinates and error ellipses computed by an adjustmerit are different whether the adjustment is performed as a free network adjustment ör as an adjustment with some coordinates fıxed. The transformation from öne result to another (each called S-system) can be done without performing a new adjustment. The algorithmic instrument for this is called an S-transformation and the formulae of the S-transformation from system i to system j are the follovving; for the trans¬ formation of the coordinates, *, - *j *, for the transformation of cofactor matrix, Q - S. Q S.T ^X) *.] ***, *j where S. is the transformation matrix, a-z-fîfjRffî)`1^ j/ j l j I is the unit matrix; Q is a matrix containing conditions for a free netvvork to allow for the computation of the coordinates; R, is the part of Q. containing only the rows of the datum points. in the application of deformation analysis, each epoch is adjusted separately as a free net. in the procedure of analysis by the strategy of S-transformation, ali epochs are compared in two groups. If the monitoring network comprises reference points and object points, the parameter vector are partitioned accordingly into the subvector xr referring to reference block and x° referring to the object point. The problem of investigating the stability of the reference block is solved by a test xvii jof the null hypotheses : H.:X(İ>)-E(Ç) prpvided that the single şpoch adjustments have been performed using the same a priori variance factpr OQ and have been based on the same geodetic datum, the pooled variance estimate is computed from :,2 ûı + ûa + - + û* Q o «= - /l +/2 + -+/*f f - /! - /2 + - + A ; o - Q! + o2 +... + ot ı The quadratic form is calculated by explicitly introducing the conditions associated HO ^M^-^)rK + ö;r(^-^r) where Qr and Qr are the cofactor matrices pertaining to J1` and £f res- pectively. The test statistic for the global test is read as follovvs T--*_ A öo where, ft-^(ö;+ö4) if, T>F(h,f-, l~a) then null hypothesis (the common points of both epochs are stable) is rejected. The point in the reference block whose contribution to the quadratic form R is maximum is considered responsible for the rejection of HO and tested. This procedure is xviiicarried out until the null hypothesis is satisfıed. After having defmed the cpmmon reference block to the epochs, the parameter vectors and their cofactor matrices are transformed into the new datum by the S-transformation. After this step, the difference vector, *-*>-*! and its weight matrix, *.-(V<U are partitioned accordingly, yielding: MI£ £n I <-U *<-*<-; P II » II~~0r ^00 where subscripts r and o denote the reference points and the object points respective- ly. The test quantity for a displacement vector d of point p is given as, < <C d» f _ p pp p 26* Where Q is the cofactor matrix referring to point p. If the test quantity T ex~ ceeds the critical value ^I1_tt;2,/ of F-distribution, then the displacement vector d has been proved statistically. in the application part of the study, the above mentioned two-epoch analysis in the static model has been carried out for every two-epoch combinated from four epochs in prder to detect horizontal movements in a control network consisting of nineteen points. As a result of the analysis, displacement up to 30.3 mm have been detected. xix