Deformation vector differences between two dimensional (2D) and three dimensional (3D) deformation analysis

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1 Deformation vector differences between two dimensional (D) and three dimensional (3D) deformation analysis H. HAKAN DENLI Civil Engineering Faculty, Geomatics Division Istanbul Technical University Maslak, Istanbul TURKEY SEDA CETIN Civil Engineering Faculty, Geomatics Division Yildiz Technical University 340 Esenler, Istanbul TURKEY Abstract: Compact position results in three-dimension (3D) obtained from GPS (Global Positioning System) technology encouraged in the last two decade almost all geoscientists to use this technique for different purposes. Determination of the deformation of a geodetic network is an objective of the analysis of various geodetic deformation measurements. GPS for deformation investigations became also a systematic contribution towards global and regional geodynamics. The weaknesses of the vertical position component against the horizontal components on using GPS are well known. This article discusses the differences between a two-dimensional and a tree-dimensional deformation analysis using the same sample data. Scientific aspects are discussed and the results are presented. Keywords: GPS, deformation, vector, analysis, similarity, transformation 1. Introduction Monitoring of the motion of the lithospheric plates has grown in importance in the last two decades. Neotectonic and geomorphological studies may supply important information and estimates of the long- term deformation. Seismological and geodetic measurements supply precise information about the shortterm behavior of crustal deformation, limited by the duration of the observational period. Modeling studies, based on geodetic data, can be considered as a fundamental scientific process, because these studies attempt to understand the physical processes on the Earth s crust. The plates move continuously with different direction and magnitude depending on location. Until recently, geodesy used networks with geodetic coordinates that were invariant in time. With the advances in GPS measurement and data analysis, the increasing number and improving quality of GPS receivers deployed in active tectonic areas, have contributed to our understanding of earthquake sources and interseismic deformation, and time variation of coordinates became measurable. These variations are the result of plate motion and deformations at the Earth s surface. For detecting and determining significant displacement, deformation analysis using the S-transformation has been applied. This model has been used both for the twodimensional horizontal (D) and the threedimensional (3D) deformation analysis. The main objective of the present contribution is to show the differences between the D and 3D deformation analysis, carried out separately using the same sample test data with an appropriate deformation analysis model for datum independent GPS networks. ISBN:

2 . Data collection and processing The test data for this experiment is taken from the international geodynamic polyproject MARMARA, a collaboration of geodetic measurements in North West Anatolia between Istanbul Technical University (ITU, Turkey) and The Federal Institute of Technology, Zurich (ETHZ, Switzerland), lasted from 1990 to The network used in this experiment comprised of 36 stations. The GPS data processing was carried out using Bernese software [1], following the concepts as described in []. This is a suite of programs with each one performing a distinct task. All components of these 3D spatial vectors are referred to a local (topocentric) frame which is parallel to the terrestrial (Earth fixed) geocentric reference frame defined by the precise ephemeris (e.g., WGS84, ITRF97) [3]. The combination of daily solutions has been configured as a free network. 3. Deformation monitoring When monitoring deformations by means of precision surveys, several reference stations are typically set up, against which the displacements of object points are calculated. To ensure that sound conclusions are made based upon analysis of the displacements, it is necessary to ensure that the reference stations are, in fact, stable. To obtain the displacements of the object points, the stability of the reference points must be confirmed and any unstable points identified. The main issue of deformation analysis with GPS is the datum problem. Coordinates and their variances and covariances are always related to the datum they are based on. The transformation from one system to another (each called an S-system) can be done without performing a new adjustment. The algorithmic tool for this is called as S- transformation and the formulae of the S- transformation from system k to i are as follows. For the transformation of the coordinates x i = S i x k (1) For the transformation of the cofactor matrix Q i = S i Q k S i T where S i is the transformation matrix. () S i = I - G(G T E i G) -1 G T E i (3) I is the identity matrix; G is a matrix containing the conditions for a free network to allow for the computation of the coordinates; E i is the matrix for determining the datum, with 1 on the diagonal for the points determining the datum and 0 for the others [4]. Matrices are shown with underlined variables. For analyzing on local plane using 3D data, the data should be transformed to the local plane. Transformation of the variance-covariance matrix to the local system should also be performed in a similar way [5]. In the application of deformation analysis, each epoch is adjusted separately as a free network. If the monitoring network is comprised of reference points and object points, the parameter vector is partitioned accordingly into the subvector x r containing the reference block and x o, the object points. The analyses were carried out both in the local plane (D) and in (3D). The problem in investigating the stability of the reference block is solved by a test of the null hypotheses: H 0 : x 1 r = x r Against the alternative hypothesis: H A : x 1 r x r (4) (5) in an attempt to determine if the differences between the results are significant. Acceptance of the null hypothesis will reflect no difference between the results, thus the two sets of coordinates could be seen as identical. For ISBN:

3 the global test of common points the equations are; d f = x i r - x i r1 (6) (Q dd ) r = (Q i rr) 1 + (Q i rr) (7) R = d T r (Q dd ) r + d r (8) s 0 = T = f s f 1 01 R s 1 0 h r + f s + f 0 (9) (10) h r = r g (Q x1 r + Q x r ) (11) where d f is the difference vector of the coordinates, (Q dd ) + r is the Moore-Penrose inverse of the common cofactor matrix, subscripts 1 or indicates the epoch respectively, subscripts and superscripts r indicate the reference points which are assumed to be stable, f is the freedom of the network, s 01 and s 0 the variance factor, s 0 is the common variance, r g denotes the rank and F the Fisher distribution and 1-α is the confidence interval. If; T > F (h, f; 1-α) (1) then the null hypothesis (the common points of both epochs are stable) is rejected. The point in the reference block, whose contribution to the quadric form R is maximum is considered responsible for the rejection of H 0. This procedure is carried out until the null hypothesis is satisfied [6, 7, 8]. After having defined the common reference block to the epochs, the parameter vectors and their cofactor matrices have to be transformed into the new datum by the S-transformation. After this step, the differences of a single point should be tested by the test statistic T to investigate the compatibility of each point. With d = x x 1 (13) the length of the difference vector for all the object points; d = d T d (14) will be tested for the null hypothesis; H 0 : d = 0 (15) If the test quantity T exceeds the critical value F h, f; 1-α of F-distribution, then the displacement vector d has been proved statistically [6, 9, 10]. To the GPS results deformation analysis is performed in (3D). Next, the (3D) data was transformed to the local plane. With these local plane coordinates and their corresponding variance-covariance matrices, a secondary deformation analysis following the same procedure is performed in (D) to compare the differences between (D) and (3D) deformation analysis with the same GPS data. 4. Comparison of (D) and (3D) deformation analysis results One of the goals of this research is to call attention to the possible differences of deformation monitoring in (D) and (3D) so that others may consider both experimenting with and using it to detect deformations as illustrated below. In fact, this paper does not intend to focus so much on the results of the particular application as it does to stimulate the reader s interest in. In order to define the stable points within the network the procedure following equations (4)-(1) and the explanations given in section 3 Deformation monitoring is used for both the (D) and (3D) data set. The null hypothesis in (3D) analysis is satisfied with 4 geodetic points. The same analysis procedure has been applied to the (D) data, given the same stable points as obtained for the (3D) data in an addition of more points. Stable points obtained from both applications ISBN:

4 referring the Eurasian plate are in harmony with past studies in the region [11, 1]. After having defined the common reference block to the epochs with regard to the points obtained from the global test, the parameter vectors and their cofactor matrices are transformed into the new datum by the S-transformation. This procedure has been applied separately both for (D) and (3D) approaches. To compare the (D) and (3D) results, the (3D) deformation vectors are transformed to the local plane as described in section 3 Deformation monitoring. The differences of the deformation vectors in direction and length between the (D) and (3D) deformation analysis are shown in Figure 1 and Figure. The regional velocities of the sites for both approaches are shown in detail in appendix, where (D) deformation vectors are shown with black arrows and (3D) vectors transformed to local plane are shown in gray. Fig. 1 Differences of the deformation vectors between (d) and (3d) deformation analysis in mm/year Fig. Differences of the deformation vectors in direction between D and 3D deformation analysis in gon 5. Conclusions This study presents potential difference between a (D) and (3D) deformation analysis carried out with the same sample GPS data. The global test for the (3D) approach shows that there are 4 stable points within the network. With a global test for the (D) approach, the network gained more points as stable beside the points obtained from the (3D) approach. These points are located at the northeastern part of the network. The average magnitude for the (D) results is about 14.6 mm/year and the average magnitude for the (3D) results is about 16 mm/year. The deformation vectors are in agreement proposed with the fault geometry given in past studies [13]. The length of the semiaxes in the relative confidence ellipse varies between mm for the semimajor axes and between mm for the semi-minor axes. The largest differences between the (D) and (3D) approaches have been detected at the western part of the network with about 3.8 mm/year. The direction of the deformation vectors does not show any systematic behavior and varies between gon and gon except of one point with a change of 9.5 gon. References: [1] Rothacher, M., Beutler G., Gurtner W., Brockmann E., Mervart L., Bernese GPS Software Version 3.4 Dokumentation, Bern, [] Denli, H. H., Bernese yazılımı ile GPS veri analizi, (Data analyzing using the Bernese Software), Journal of Yildiz Technical University, 4:16-5, 001. [3] Soler, T., Densifying 3D GPS Networks by Accurate Transformation of Vector ISBN:

5 Components, GPS Solutions, 4:7-33, 001. [4] Baarda, W., S - Transformations and Criterion Matrices, Netherland Geodetic Commission, Publications on Geodesy, Delft, [5] Leick, A., GPS Satellite Surveying, John Wiley & Sons, New York, [6] Caspary, W. F., Concepts of Network Analyses, School of Surveying, University of New South Wales, Kensington, [7] Denli, H. H., Stable Point Research on Deformation Networks, Survey Review, 40:74 8, 008. [8] Koch K. R., Parameterschätzung und Hypothesentests, Dümmler, Bonn, [9] Demirel, H., S - Transformasyonu ve Deformasyon Analizi, Türkiye 1. Harita Bilimsel ve Teknik Kurultayı, , [10] Denli, H. H., Crustal Deformation Analysis in the Marmara Sea Region, ASCE Journal of Surveying Engineering, 4: , 004. [11] Barka A. A., The North Anatolian fault zone, Anales Tectonicae, Special Issue, Supplement to Volume 6, , 199. [1] Westaway, R., Evidence for dynamic coupling of surface processes with isostatic compensation in the lower crust during active extension of western Turkey, Journal of Geophysical Research, 99:003-03, [13] Straub, C., Recent Crustal Deformation and Strain Accumulation in the Marmara Sea Region inferred from GPS Measurements, Institute für Geodaesie und Photogrammetrie, Mitteilungen Nr.58, Zürich, Appendix Regional velocities according to (D) and (3D) deformation analysis ISBN: