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Practical Deformation Detection and Analysis Via Matlab

Roya Olyazadeh & Halim Setan Department of Geomatic Engineering, Faculty of Geoinformation Science and Engineering,

Universiti Teknologi Malaysia (UTM), 81310 UTM Skudai, Johor, Malaysia. Email: [email protected], [email protected]

ABSTRACT

Network adjustment and deformation monitoring are amongst the main activities of engineering surveying. Generally, the deformation measurement can be divided into geotechnical and geodetic methods. There are two types of geodetic networks, namely the absolute and relative networks. In an absolute network, some of the points are assumed to be located out of the deformable body. However, in a relative network, all points are assumed to be located on the deformable body. There are several known Deformation Monitoring System software packages, from several universities research groups. Specialized computer programs are required because most commercial software like STARNET does not provide the required analysis for deformation detection. This study focuses on 2D absolute geodetic deformation detection. MATLAB is a mathematical computer programming that covers all mathematical functions required for many applications. The main purpose of this research is to develop a program using MATLAB for 2D deformation detection and analysis. Consequently, the results to date show the practicality of the program.

Key words: Deformation detection and analysis, IWST, MATLAB

1.0 INTRODUCTION

Deformation analysis is one of the main research fields in geodesy and geomatic. The deformation measurement techniques can be divided into geotechnical, structural and geodetic methods. In the geodetic method there are two basic types of geodetic monitoring networks; namely the reference (absolute) and relative networks (Chrzanowski, 1986). In a reference network, some of the points or stations are assumed to be located outside of the deformable body or object, thus serving as reference points for the determination of the absolute displacements of the object points. However, in a relative network, all surveyed points are assumed to be located on the deformable body (Setan and Singh, 2001).

In general, a deformation monitoring system involves four fundamental steps, namely measurement collection, least squares estimation (LSE) and statistical testing, reference point

stability checking, and deformation detection and analysis. In the first step, a deformation monitoring network and observation scheme are established to achieve some specified accuracy and precision (Caspary, 1987). Measurements are collected during different epochs (i.e. different time period). In the second step, each epoch of measurements is separately adjusted using a LSE procedure, which may be checked using a global test (Chi-square), local test (Baarda’s Data Snooping), or residual testing. It is essential to perform initial checking on the raw data and test the posteriori variance factors of both epochs before deformation analysis (Setan and Singh, 2001). Datum and deformation network analysis is necessary before designing the LSE, because of issues such as rank deficiency and the internal geometry of the monitoring network. This study implemented rigorous method for analysis, iterative weighted similarity transformation (IWST) for a two dimensional (2D) network.

2.0 METHOD FOR DEFORMATION ANALYSIS

There are two common methods used for analyzing the stability of reference points, namely the congruency testing method and the robust method (IWST and LAS). In this work IWST is considered.

2.1 LSE for each epoch

The method of least squares network adjustment or LSE tries to solve for an optional estimate of both the coordinates and residuals by minimizing the sum of squares of the weighted residuals (v) (USACE, 2002):

Minimize:

Subject to observation equation:

Where, l is observation matrix, A is design matrix and x is unknown matrix (Setan, 2008). Calculate:

Where, v is residual matrix and , coefficient matrix

And

l is vector of actual observations and lo is vector of computed observations

, The updated parameters

, cofactor matrix of a xˆ

After first step of LSE, iteration must be done. At the end of step one, approximate coordinates must be updated and lo and A should be calculated again. Limit for iteration is (Setan, 2008):

1. close to zero 2. close to zero 3. Stable

In this study, the first limit is used. In each computation step, matrices A & L are changed, matrices P and l0 unchanged. During LSE, other important aspects that need to be considered are the global test (Chi-square), local test (Tau test or Baarda Method), precision and accuracy analysis. For further details on LSE the interested reader are referred to Wolf and Ghilani (1997), Singh (1997), and Caspary (1987).

2.2 Iterative Weighted Similarity Transformation (IWST)

Chen (1983) has proposed a robust method known as an iterative weighted similarity transformation (IWST). This robust method was developed at the University New Brunswick, Canada. This study focuses on IWST robust method for estimating the trend of movements of a monitoring network (USACE, 2002). This method is followed by test on variance ratio, trend analysis, S-transformation, iterations and single point test.

Initial checking of data and test on Variance Ratio

In this research, the number of points in the first epoch must be equal and same to the number of points in second epoch. So initial checking of data is performed for all points in both epochs. The number of datum defects depends on the union of the datum defects in both epochs. For example, if the first epoch is a trilateration network which has three datum defects (i.e., two translations and one rotation), while the second epoch is a triangulation network with a datum defect of four (i.e., two translations, one rotation and one scale), then the union of datum defects is equal to the datum defects in the second epoch (i.e. four). The a posteriori variance factors of both epochs are then tested for their compatibility (Setan and Singh, 2001).

With j and i represent the larger and smaller variance factors, F is the Fisher’s distribution, a is the chosen significance level (typically a = 0.05) and dfi and dfj are the degrees of freedom for epochs i and j respectively.

The above test is accepted if T < F (a , dfj, dfi) at a significance level a. The failure of the above test may be caused by incompatible weighting between the two epochs or incorrect weighting scheme and any further analysis should be stopped at this stage.

Trend analysis

After the test on the variance ratio is accepted, the displacement vector (coordinates differences) and its cofactor matrix can be computed as:

d = x 2 - x1

Qd = Qx 1 + Qx 2 Where, x1 and x2 are the estimated coordinates of all points in the first and second epochs respectively, Qx 1 and Qx2 are the cofactor matrix of the estimated coordinates x1 and x2, d is the displacement vector and Qd is the cofactor matrix of d (Setan and Singh, 2001).

S-transformation

S-transformation (similarity or Helmert transformation) is defined as below (Singh and Setan, 1999).

Where; I = identity matrix k = number of iterations G=inner constraint matrix d = displacement vector Qd= Covariance vector S = S-transformations matrix W = weight matrix

In the first transformation (k = 1) the weight matrix is taken as identity (W (k) = I) for all the common points, then in the (k+1) transformation the weight matrix is defined as

Matrix GT (Inner Constraint Matrix) for 2-D network

The first two rows of GT define the translation along x and y axes respectively. The third row defines the rotations z axis and the last row defines the scale of the network. Then if the observations contain datum information, the rank deficiencies are less than 4 and the corresponding rows of GT are omitted (Setan and Singh, 2001).

Where

and

are the coordinates of point pi which are reduced to the centroid or center of gravity of the network.

Datum analysis defines the fixed parameters in a deformation network whose stability can be analyzed at single epochs, two epoch adjustment and/or multi-epoch adjustment (Caspary, 1987). Datum defects can be removed by an S-transformation (similarity transformation) which can facilitate the change of datum definition without repeating the LSE process (Caspary, 1987).

Single point test

The stability information of each common point j can then be determined through a single point test as below (Setan and Singh 1999).

Where; dj, Qdj are displacement vector and its cofactor matrix respectively for each common point j

s 2 is pooled variance factor

df1, df2 are degrees of freedom of epochs 1 and 2 df = df1 + df2, sum of degrees of freedom of epochs 1 and 2 a = significance level (usually chosen as 0.05) F referred to Fisher distribution.

If the above test passes (i.e., Tj < F (a , 2, df)) then the point is assumed to be stable at a significance level a. Otherwise, if the test fails (i.e., Tj = F (a , 2, df)) then the point is assumed to be deformed (moved) (Setan and Singh, 2001).

Final transformation After single point test, final S-transformation is performed. In this S-transformation, moved station should be removed from fixed station in weight matrix (W) and S-transformation shall be carried again. Finally Single Point test is performed and displacements (d) for each point are computed again and results and graphics are displayed.

3.0 RESULTS AND ANALYSIS

The geodetic method of deformation measurement was carried out from a known data of a dam monitoring network by Caspary (1987). It has been used to test the developed procedure and the program package. All computations for geodetic method have been solved by utilizing a program package developed in MATLAB7. ADJMATDEF is developed using MATLAB7 and can handle only 2-D monitoring networks.

MATLAB is a powerful language that simplifies the process of solving technical problems in a variety of disciplines by including a very extensive library of predefined functions. It also offers many special-purpose tool boxes and provides Graphical User Interface (GUI) tools that make it suitable for application development. Because of these capabilities, MATLAB has been recognized as the premier program for numerical computations and data visualization and has been used by the educational, industrial, and government communities all over the world (Mathworks, 2001).

The program package consists of two modules known as Adjustment Program and Deformation Program. The LSE employs minimum constraint datum definition and the graphic presentation which is performed by Adjustment program in this package. While Deformation program is

designed for deformation analysis and Graphical display in the form of vector of displacement together with standard ellipses.

3.1 Input data

Input data for this package is as follows:

1. Two-epoch observations for adjustment 2. Deformation file for deformation detection

These programs can read data in text format and station numbers must be in numeric format.

Adjustment File

Data format for adjustment is quite similar to STARNET format. In this project, ‘-‘is removed from STARNET format to make programming and converting observation easy. The STAR*NET software suite handles general purpose, rigorous least squares analysis and adjustments of 2D, 3D and 1D land survey networks (Starplus, 2000). It employs a rigorous simultaneous least squares adjustment using the variation of coordinate method. Data input includes angles, bearings, azimuths, coordinate differences, directions, distances, elevation differences, zenith angles, etc. For LSE, all these data can be independently or globally weighted. Approximate coordinate can be entered manually or computed automatically. STARNET V6 performs LSE or adjustment of network, data checking, blunder detection and pre analysis.

Example of input data for adjustment program:

C 1 10000 10000 1 1 C 2 10002.7140 19998 0 0 C 3 20000.7140 19999.26 0 0 D 1 2 9997.930 0.038 D 2 3 9998.030 0.038 D 1 3 14142.110 0.038 A 1 2 3 44 59 19 2.4 A 2 3 1 90 18 22.1 2.4 A 3 1 2 44 59 18.1 2.4 Z 3 1 225 0 15 1 Z 1 2 0 0 58 1 Where, C: coordinate, D: Distance, A:Angle, Z: Azimuth and 1 is for fixed station and 0 for non-fixed station.

Deformation File Deformation file is automatically obtained from Adjustment Program, and contains the adjusted data and covariance matrix for each epoch as follows:

Example of deformation input data:

I 1 1.447429 27 6 3 1 1 1 0 P 1 1000.000000 1000.000000 P 2 938.857000 894.100000 P 3 814.459000 956.706000 P 4 878.926000 1021.071000 P 5 911.951000 959.432000 P 6 893.034000 954.470000 E 1 1000.000000 999.999992 1 1 E 2 938.861918 894.097295 1 1 E 3 814.461696 956.702070 1 1 E 4 878.926000 1021.071000 1 1 E 5 911.956069 959.430468 0 0 E 6 893.036206 954.465949 0 0 V 1 .000000000000000E+00 V 2 .000000000000000E+00 V 3 .226602567955182E-04 V 4 .000000000000000E+00 . . . V 77 .591407360765137E-05 V 78 .000000000000000E+00

Where, first line (I) shows Epoch number, posteriori variance factor, No of degrees of freedom, No of points, No of datum defects, Types of datum defects P means approximate parameters (coordinates) E means adjusted parameters (coordinates). 1 is for datum station and 0 is for no datum station. V is lower triangle for Cofactor matrix Qx (variance covariance) for adjusted parameters (coordinates).

3.2 Results of data set

The monitoring network consists of two-epochs of angle and distance observations with 12 stations, 7 reference points (1, 2, 3, 4, 6, 7 and 9) and 5 object points (10,11,12,13 and 14) were done on a two-epochs basis. Each measurement epoch consists of 44 angles and 6 horizontal distances. Network adjustment of each point was carried out by minimum constraints. The significance level for Chi-square (global) and Baarda (local) test were chosen as 0.05. The result of each epoch passed both global and local tests.

The Deformation program calculated displacement in X axis (Dx), Y axis (Dy) and displacement vector (Disp.). Ftab in table shows value from fisher distribution and Fcom are calculated from covariance matrix as described in single point test module (Table 1).

Deformation results leads to the conclusion that points 3, 6, 12 and 13 are significantly deformed while all others remained stable. Capary (1987) also confirmed stations 3, 6, 12 and 13 as unstable. Caspary (1987) calculated deformation analysis by another method called Congruency while Deformation program employs IWST. This result is emphasized by the plot of the deformation vectors together with 95%-confidence ellipses in Figure 1. Figure 2 displays the displacement vectors. The maximum movement is in station 12 (1.4 mm).

The final results of the analysis are listed in Table 1 and the results comparison between Caspary (1987) and Deformation Program are given in Table 2. The significant difference for displacement vector can be found in station 12 with critical value -0.266 mm.

Figure 1: Displacement vector and error ellipse graphic by Deformation program

4.0 CONCLUSIONS

The determination of deformation of control points is very useful and can be applied for the economical planning of alignment surveys of the machine and monitoring deformation trends in sites like Dam, Tunnel and etc. In this research the IWST method obtains the displacement vector after LSE for each epoch of observations. A 2-D deformation analysis was made because of a lower accuracy of the Z component with respect to the X and Y components. Therefore, for further work, in order to determine the movement in the vertical dimension, the use of a precise leveling or another method that has the potential to provide a vertical accuracy equal to or greater than the horizontal accuracy of GPS is recommended.

To implement this procedure a software package called ADJMATDEF has been developed in MATLAB7 and successively used to analyze adjustment for each epoch separately and the deformation detection for both epochs without any interfere program; because such program like STARNET does not provide the necessary information (especially variance covariance matrix) for deformation detection purposes.

In order to monitor and measure possible displacements and to check this package, a known data set is used from Caspary (1987). Deformation results for this data set are compared with results from Caspary (1987) and no significant difference is achieved for point movement and single point test.

As a future work, other methods (e.g., LAS and Congruency) can be applied for deformation detection. In addition they will be launched in web server that anyone can use them.

It is hoped that surveyors might be beneficial from this package with other software for deformation surveying applications.

-1-0.5

00.5

11.5

2

1 3 6 9 11 13

DX

DY

DISP.

Figure 2: Displacement in X axis (Dx), Y axis (Dy) and displacement vector (Disp.) in mm

Table 1: Displacement vector (mm) and single point test for Deformation program and Caspary (1987)

(mm) Deformation

Program (mm) Results

From Caspary

STN DX DY DISP.

FCOM FTAB DX DY DISP. Info

REFERENCES

Caspary. W.F. (1987). Concepts of Network and Deformation Analysis, 1st. ed., School of Surveying, The University of New South Wales, Monograph 11, Kensington, N.S.W.

Chen, Y.Q. (1983). Analysis of Deformation Surveys – A Generalized Method. Department of Surveying Engineering Technical Report No. 94. University of New Brunswick, Fredericton.

Chrzanowski A. (1986). Geotechnical and other non-geodetic methods in deformation measurements, Proceedings of the Deformation Measurements Workshop, Boston, Massachusetts, 31 October-1 November, Massachusetts Institute of Technology, Cambridge, M.A., pp. 112-153.

Mathworks (2001). MATLAB® The Language of Technical Computing. Retrieved on May 17, 2010, from fttp://www.mathworks.com

1 0.213 0.124 0.247 1.680343

3.155932

0.19 0.15 0.242074

2 0.088 -0.025 0.092 1.081632

3.155932

0.09 0.03 0.094868

3 -0.791 -0.498 0.935 23.96812

3.155932

-0.58 -0.39 0.698928

Moved

4 -0.14 0.037 0.145 0.742038

3.155932

0.13 0.11 0.170294

6 -0.401 0.241 0.468 3.031361

3.155932

-0.25 0.14 0.286531

Moved

7 0.325 0.039 0.327 1.92086 3.155932

0.31 -0.01 0.310161

9 -0.161 -0.135 0.21 1.1721 3.155932

0.11 -0.03 0.114018

10 0.313 0.086 0.324 0.716348

3.155932

0.18 0.13 0.222036

11 -0.589 0.284 0.654 2.72997 3.155932

-0.65 0.48 0.808022

12 0.461 1.325 1.403 11.30803

3.155932

0.54 1.58 1.669731

Moved

13 0.182 1.151 1.165 5.740104

3.155932

0.4 1.34 1.398428

Moved

14 -0.071 -0.002 0.071 0.081368

3.155932

0.2 0.12 0.233238

Setan. H. and R. Singh. (1999). Comparison of Different Strategies for Trend Analysis of the Displacement Field in Deformation Surveys, Unpublished.

Setan. H. and Singh. R. (2001). Deformation analysis of a geodetic monitoring network. Geomatica. Vol. 55, No. 3, 2001

Setan, H. (2008). Lecture notes for Adjustment computations,Slide.1-24

Singh. R. (1999). Pelarasam dam ama;isis jaringan pengawasan untuk pengesanan deformasi secara geometri. M.Surv.Sc (Land) thesis.

Starplus Software (2000). STARNET V6 least squares Survey Adjustment Program. Reference Manual.

USACE (2002). Structural Deformation Surveying (EM 1110-2-1009). US Army Corps of Engineers, Washington, DC.

Wolf. P.R. and Ghilani, C.D. (1997). Adjustment Computations: Statistics and Least Squares in Surveying and GIS. John Wiley & Sons, Inc., New York.

AUTHORS

Roya Olyazadeh is a M.Sc. student at the Department of Geomatic Engineering, Faculty of Geoinformation Science & Engineering, Universiti Teknologi Malaysia (UTM). She holds a B.Sc. degree in Civil-surveying from Zanjan University in Zanjan, Iran in 2007. Her master project focuses in the area of deformation monitoring.

Dr. Halim Setan is a professor at the Faculty of Geoinformation Science and Engineering, Universiti Teknologi Malaysia (UTM). He holds B.Sc. (Hons.) in Surveying and Mapping Sciences from North East London Polytechnic (England), M.Sc. in Geodetic Science from Ohio State University (USA) and Ph.D from City University, London (England). His current research interests focus on precise 3D measurement, deformation monitoring, least squares estimation and 3D modeling.

Table 2: Difference in displacement vector (Delta Disp.) for data set two between Deformation program and Caspary (1987)

STN(mm)

Delta DX

Delta DY

Delta DISP.

1 0.023 -0.026 0.0049256

2 -0.002 -0.055 -0.002868

3 -0.211 -0.108 0.2360722

4 -0.27 -0.073 -0.025294

6 -0.151 0.101 0.181469

7 0.015 0.049 0.0168388

9 -0.271 -0.105 0.0959825

10 0.133 -0.044 0.101964

11 0.061 -0.196 -0.154022

12 -0.079 -0.255 -0.266731

13 -0.218 -0.189 -0.233428

14 -0.271 -0.122 -0.162238