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The Application of GPS Sensor for Indo-Australia and Eurasia Tectonic Plates Deformation Analysis
Yul Amri & Halim Setan Department of Geomatic Engineering Faculty of Geoinformation Science and Engineering Universiti Teknologi Malaysia 81310 UTM Skudai Johor, Malaysia Email: y_amri@yahoo.com, halim@fksg.utm.my ABSTRACT The study of tectonic earth-quake becomes more interesting after Acheh-Tsunami disaster. The movement accumulation (or deformation) of two tectonic-plates (Eurasia and Indo-Australia) leads to the accumulation of stresses that causes earth failure or earth-quake. The deformation status of the tectonic plates can be provided by geodetic method via Global Positioning System (GPS). GPS can be used to measure or determine the spatial position of discrete points on earth surfaces. The continuous or episodic measurements are used to determine the actual movement vector throughout the computation procedure and related statistical analysis. This paper describes the application of GPS to determine the actual spatial displacement vector of the tectonic-plate deformation. The paper also discusses the GPS network design for Indo-Australia and Eurasia tectonic plates study. 1. Introduction Global Positioning System (GPS) is satellite-based positioning system developed by USA. The invention of GPS has given great contribution in earth science, including plate-tectonic study. GPS operates day and night (24 hours), in all weather and worldwide. Moreover, inter-visibility between survey stations is not required. The only constraint is that the signals of at least 4 satellites must be received simultaneously at any survey point. For international civil users, GPS sensor provides high precision observable in differential mode, i.e. spatial baseline vector between two points (?X, ?Y, ?Z), instead of point positioning mode that directly gives latitude, longitude and height (f, ?, h). Higher precision baseline vector can be achieved by using more comprehensive mathematical model (and specialized scientific software such as BERNESE) in processing the signal of GPS. The high-precision baseline vector is quite important if mm-level magnitude of earth movement is required. Pre-campaign and post-campaign quality controls are required to evaluate the network performance for specific monitoring purpose. Post-campaign statistical assessment is required to ensure whether the displacement vector given by two epochs of GPS campaign represents the real movement. The movement of tectonic plates is usually represented as horizontal displacement vector component (dx, dy) and vertical displacement component (dh). The GPS network observable and displacement vectors directly derived from it are based on WGS 84 coordinate system. Then it is required to convert the displacement vectors and their stochastic properties derived from GPS network into horizontal and vertical components. 2 GPS Network for Tectonic Plate Deformation and Mathematical formulation At initial stage, even for pre-campaign analysis or post-campaign analysis, the mathematical model of GPS network has to be formulated. Generally, mathematical model of the networks consists of functional, stochastic and datum constraint model. Functional model relates the observation data and unknown parameters. The observation data of GPS are baseline vectors (?Xij ?Yij ?Zij )T. If (vxij vyij vzij )T is residual vector corresponding to the observation baseline vector, and (Xi , Yi , Zi ) and (Xj , Yj , Zj ) are the unknown parameters, then the functional model for each observation data can be written as: 
In matrix form the functional the model can be written as: A x + v = b (2) Since the observation is subjected to the stochastic properties, the stochastic model of the observation should be involved. Covariance matrix S represents the stochastic properties. The level of importance of the observation as represented by weight matrix W, is actually the most usable in computational process. The relationship between covariance matrix S and weight matrix W is: 
In terestrial observation, it is generally assumed that the observations are uncorrelated, and the covariance and weight are diagonal matrix (i.e. off-diagonal elements are zero). In the case of GPS observation, it is proven that observations (baseline vector) are correlated to each other, covariance matrix and weight matrix are full matrix. Datum definition is required to define the coordinate system of the GPS network. Datum definition can be included in full-rank matrix A. The full-rank design matrix A can be always provided if one of the GPS station is defined as datum point, or assumed to be fixed or has zero-variance. However, since the matrix A is datum variant then the derived quantities are also datum variant. In deformation analysis, more than one control stations may proven as stable points. These stable points act as datum points, and have to be included in datum definition. Datum definition can also be formulated as inner constraint, and written as Gx = 0 (5) where 
The first three columns of matrix G define translation, and the second three columns define rotation. The row of matrix G is equal to 3n where n is the number of stations. More detailed explanation of inner-constraint can be found in Leick (1995), Halim (1995), Kuang (1996), Caspary (1987), among others. 3. Pre-campaign Quality Control The quality performances of the GPS network can be measured by:
3.1 Precision If matrix A is full-rank, then the precision of network is represented by covariance matrix of the parameters, Cx. It should be noted that the expected precision of the network can be determined even in pre-campaign, as: 
Since full-rank matrix A is datum variant then the precision of network determined via Equation (7) is also datum variant. The most suitable one to represent precision is using inner-constraint datum definition, then precision of the network is given by 
The scalar value of the network precision can be represented by 
3.2 Reliability The observation data, including the high precision GPS data are subject to errors, not only small magnitude random allowable error but also gross and systematic errors. All marginal errors need to be detected or eliminated. The smallest (or marginal) error that can be detected in the network represents the internal reliability and the influence of each marginal error to the unknown parameter represents the external reliability. The criteria of GPS network should include these reliability measures. The smallest (or marginal) error that can be detected (MDGE-Marginal Detectable Gross Error) is determined as below: 
The influence of each marginal error to the unknown parameter Ñx, is determined as below: 
4. Spatial Trend Analysis Spatial displacement vector d can be computed from two epochs as: d = (dx dy dz) = x2 - x1 (16) In equation (16), x2 and x1 are the estimated coordinates of epoch one and two respectively. Generally, x2 and x1 must refer to the same common datum definition. In case of full-rank matrix A (or minimum constraint solution), the least squares estimated coordinates for each epoch are obtained by: x = (ATWA)-1ATW b (17) In most cases, there are more that minimum number of datum points available in the network. The estimated coordinates have to be referred to the same (common) datum definition. The common stable point to be used to find the final displacement vector can be determined by the use of congruency analysis. The congruency analysis is iterative in manner. The previous datum point with the largest displacement is removed from next datum definition. The S-transformation (similarity transformation) facilitates the change of datum definition without repeating the least squares estimation process. More detailed on S-transformation can be found in Halim (1995) and Caspary (1987). As an alternative to the congruency analysis, the spatial displacement trend can be provided by the iterative weighted similarity transformation (IWST) scheme. The scheme of IWST can be briefly explained as below (Chen, 1983). At the first iteration (k=1), matrix W is identity or W =I. And the next iteration is performed by the schematic below: 
and, matrix Wk is computed as 
Before the spatial trend analysis (as formulated by Equation (14) to (22)), statistical test of each epoch must be performed. The statistical test includes the compatibility test of a posteriori variance factor to a priori variance factor. The compatibility test is to ensure whether the assumption of the normality of the observation is correct. The complementary of such test is test of outlier. It should be noted that the existence of any outlier will distort the result of least squares estimation. The least squares estimation and the test of outlier are the iterative processes. Due to the sensitivity of the least squares estimation, it is recommended to use the least absolute estimation (or L1-norm estimation). Yul (2000) has explored and implemented the concept of L1-norm estimation to the data of GPS network. The next statistical test is to evaluate the variance factor of first epoch is equal to variance factor of the second epoch. The statistic to be tested is variance ratio according to Fisher distribution. More detailed of the test can be found in Halim (1995), Mikhail (1976), and Leick (1995). 5. The Design of GPS Network Geodetic networks, including GPS network for deformation monitoring purpose can be divided into two configuration, i.e. reference network and relative network. Reference network contains reference points and object points (to be monitored) The reference points are located on the expected stable zone. Object points are located on the surface of the object under study. The reference network is commonly used for deformation monitoring of engineering structures. In this case, the object ponts are located on the engineering structure. The relative network is implemented to monitor two zones where each zone has its own movement pattern. Since the assumption that the tectonic plates are not static, it is suitable to implement the relative network for tectonic plates deformation monitoring. The typical reference and relative networks are shown in Figure 1 and Figure 2 respectively.  Figure 1. Reference network  Figure 2. Relative network As shown in Tectonic Map of Indonesia in Figure 3 (BMG, 2005), Eurosia plate covers Nicobar islands, Andaman islands (India Teritory), Sumatra (including Nias and Mentawai islands), Java, Bali, Sumba, Timor, Ambon and Sulawesi (Indonesia Teritory). Indo-Australia plate covers Maldevis, Cocos island and Christmas island (Australia Teritory), Ceylon and India continent. The Simeuleu island, Nias island, Mentawai islands and Enggano island are located on the subduction line of the Indo-Australian plate. To study the dynamic behaviour of Indo-Asuralia plate and Eurasia plate, since none of the plates can be benchmarked as reference zone, then the relative network is adopted. The points of GPS network on the Eurasia plate are distributed along the shore line of Sumatra, Java, Bali, Lombok, Flores and Timor. Some points are also located on Thailand, Andaman, and Nicobar. The points on Indo-Australia plate are distributed on Ceylon, Cocos, Christmas and northern-west shore-line territory of Australian. To study the dynamic behavior of the subduction zone, it is proposed to distribute the points along Simeuleu, Nias, Mentawai and Enggano. Figure 3 illustrated the proposed network for movement monitoring purpose. It is noted that there are five of Asia Pacific countries involved in this monitoring campaign, i.e. Indonesia, Thailand, India, Ceylon and Australia.  Figure 3. Proposed network 6. Data Conversion All the geometric quantities explained above such as displacement vector are represented in 3D Cartesian coordinate system of WGS '84. However, the horizontal movement interpretation of tectonic plate is usually performed in mapping plane while the vertical component is represented by ellipsoid height or height above mean sea level. The results in WGS '84 reference frame need to be converted onto the mapping surface and height system. The scheme of data conversion is shown in Figure 4 and the detailed explanation is given in Leick (1995). 7. Concluding Remarks The paper has discussed the application of data provided by GPS sensor to determine the spatial displacement trend in two separated dimension, i.e. two-dimensional trend and vertical trend. The quality measures of the networks have been discussed to ensure that the GPS network is suitable for the movement monitoring purpose. Further work that can be done is complete deformation modeling of two tectonic plates or part of them. The deformation model can be used to confirm the delineation zone of the tectonic plates including the theory of subduction zone. Multilateral GPS campaign of at least five Asia-Pacific countries suggests the international agreement to realize the monitoring scheme.  Figure 4. The scheme of data conversion References
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