The performance of the quadratic kinematic deformation analysis model

Journal Pre-proofs The performance of the quadratic kinematic deformation analysis model Utkan Mustafa Durdag, Bahattin Erdogan, Taylan Ocalan, Serif Hekimoglu, Ali Hasan Dogan PII: DOI: Reference:

S0263-2241(20)30144-5 https://doi.org/10.1016/j.measurement.2020.107607 MEASUR 107607

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Measurement

Received Date: Revised Date: Accepted Date:

12 November 2019 4 February 2020 7 February 2020

Please cite this article as: U. Mustafa Durdag, B. Erdogan, T. Ocalan, S. Hekimoglu, A. Hasan Dogan, The performance of the quadratic kinematic deformation analysis model, Measurement (2020), doi: https://doi.org/ 10.1016/j.measurement.2020.107607

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The performance of the quadratic kinematic deformation analysis model

Utkan Mustafa Durdag Methodology Formal analysis Writing - original draft Softwarea, Bahattin Erdogan Conceptualization Project administration Writing – review & editingb,*,[email protected], Taylan Ocalan Resources Project administration Methodologyb, Serif Hekimoglu Supervision Writing - original draftb, Ali Hasan Dogan Software Data Curationb aArtvin

Coruh University, Faculty of Engineering, Department of Geomatic Engineering, Seyitler, Artvin, Turkey bYildiz

Technical University, Faculty of Civil Engineering, Department of Geomatic Engineering, Esenler, İstanbul, Turkey *Corresponding

author.

Highlights The coordinated differences estimated by Bernese v5.2 are close to the true values. Quadratic model is mostly preferred in Kinematic Deformation Analysis (KDA). The Least Square Estimation smears the effects of the displacements to all points. Smearing effect decrease the reliability of the quadratic KDA method.

Abstract Kinematic Deformation Analysis (KDA) models are preferred to estimate displacement, velocity and acceleration parameters. The quadratic models are often used in KDA. To estimate the reliability of the model, Linear Motion Systems (LMS) integrated with GPS were established at four object points located properly in University Campus. Moreover, five continuously operating GPS sites (off-campus) were predefined as reference points. The object points were displaced along three different planes: 1st Vertical direction (Up), 2nd Horizontal plane (Northeast) and 3rd Horizontal plane and vertical direction for intended scenarios as considering the different magnitudes for velocity and acceleration. Each deformation scenario contains four epochs of GPS observations with eight hours of session duration. According to the obtained results, the displacements estimated by the Bernese v5.2 scientific software are close to their real values. However, the quadratic KDA model spreads the effects of the displaced points to other nondisplaced points. Keywords: Kinematic Deformation Analysis; Reliability; Linear Motion Systems; Simulation 1. Introduction Monitoring the movements of the plate tectonics or engineering structures has vital importance for geosciences. For this reason, deformation monitoring networks are established and observed periodically for monitoring the behaviors of natural hazards or man-made structures. The classical geodetic approach in deformation analysis considers a 1

purely geometrical comparison between two different epochs of observations without explicitly regarding "time" and "loads" (Welsch and Heunecke 2001). Different deformation analysis models have been considered in the literature as static model (Pelzer 1971; Koch 1985; Niemeier 1985; Schaffrin 1986; Caspary,1988; Caspary et al. 1990; Schaffrin and Bock 1994; Betti et al. 1999; Snow and Schaffrin 2007; Cai et al., 2008; Hekimoglu et al. 2010; Erdogan and Hekimoglu 2014; Baselga et al. 2015, Zienkiewicz 2017) and kinematic model (Pelzer 1986; Heunecke 1995; Liu 1998; Yalcinkaya and Bayrak 2005; Acar et al. 2008, Yi et al. 2013, Durdag et al. 2018, Kim et al. 2018, Xi et al. 2018). The Kinematic Deformation Analysis (KDA) model that considers the velocity and acceleration parameters to estimate the displacements of the points is preferred especially for monitoring of the plate tectonics. Depending on the geological structure of the area, the points can be separated as reference points that are assumed to be located outside of the deformable area and object points that are not stable and undergo deformation. The KDA model involves determination of the coordinates of the reference and the object points as functions of time (Pelzer 1985, 1987). In the literature, the models in KDA may be built either linear or quadratic (Ghitau 1970, Holdahl 1975, Mälzer et al., 1979; Mälzer et al., 1983, Zippelt 1986, Yu and Segall, 1996; Kenner and Segall, 2000; Shahar and Tzur, 2014, Durdag et al. 2018). Ghitau (1970) and Holdahl (1975) defined the points’ displacements depending on the velocity and acceleration parameters. Zippelt (1986) investigated the stepwise model fitting for KDA models by using Karlsruhe approach that was defined by Holdahl (1975). This model was also controlled by statistical tests. Moreover, the expansion or reduction of the KDA model can be determined statistically with a recursive adjustment procedure (Yalcinkaya and Bayrak, 2005). Kenner and Segal (2000) reanalyzed the historical triangulation data following the 1906 San Francisco earthquake by using the methods of Yu and Segall (1996). They intended to increase both temporal and spatial resolution of applicable data in northern California. Acar et al. (2008) applied the KDA model based on Kalman Filtering at a landslide area. Yi et al. (2013) investigated the reliability of using highrate carrier phase Global Positioning System (GPS) receivers to characterize dynamic oscillations of bridges. Shahar and Tzur (2014) redefined the datum congruency and presented the dynamic datum concept. They offered an algorithm of extended free net adjustment constraints solutions for the analysis of a geodetic control network. Baselga et al. (2015) developed hypothesis of the maximum number of stable points for the relative deformation monitoring network. Durdag et al. (2018) suggests the quadratic KDA model for reliable results when both linear and quadratic models are considered for the same simulated working samples. Least Squares Estimation (LSE) is widely used for estimating the unknown parameters in the KDA model. LSE is the optimal estimator when observations do not include any outlier. But if there are deviations from models assumed, i.e. if there are displaced points in the deformation monitoring network, the results obtained from the LSE may not be correct. LSE spreads the spoiling effects of deflection from the assumed model on the residuals of good observations; also it spreads the effects of displaced points on the other estimated points coordinates that are not displaced. Thus, obtained results deviate from their optimal values (Chen et al. 1987; Schwarz and Kok 1993; Kuang 1996; Prószyñski 2000; Hekimoglu et al. 2

2010, Erdogan and Hekimoglu 2014, Durdag et al. 2018). Therefore, the reliabilities of the results from the quadratic KDA model should be investigated. To examine the reliability of the estimation method or deformation analysis model, the real values of the displacement should be known. Martin et al. (2015) set up a mobile test platform and investigated the performance of the real-time kinematic precise point positioning technique in deformation analysis. Also, Durdag et al. (2018) simulated the displacements and obtained reliabilities of the KDA models. In this study, a GPS network that consisted of five reference points and four object points was established. The GPS observations were carried out for twelve different scenarios. Four linear motion system (LMS) devices with a precision of 1 micrometer along the horizontal plane and of 0.1 mm along the vertical direction were built. The displacements were loaded by using the LMS devices to object points. The GPS data were analyzed by Bernese v5.2 scientific software. When the estimated coordinates from scientific software were analyzed; it was seen that the coordinate differences determined based on the reference epoch are close to their real values. Also, the quadratic KDA model was applied to the simulated working samples and the results have been compared with the real values. According to the obtained results, the LSE method in KDA model spreads the effect of the displaced points and wrong results can be obtained especially for the large values of velocity and acceleration parameters. 2. Quadratic Kinematic Deformation Analysis Model At the first step of the KDA model, the global congruency test is applied. The global congruency test is based on comparison of the coordinate differences between two different observation epochs. Classically, each epoch is adjusted separately as a free network considering the Gauss-Markov Model given below (Koch 1999): 𝐥𝟏 + 𝐯𝟏 = 𝐀𝟏𝐱𝟏,

𝐂𝐥𝟏𝐥𝟏 = 𝜎20𝐏𝟏―𝟏

(1)

𝐥𝟐 + 𝐯𝟐 = 𝐀𝟐𝐱𝟐,

𝐂𝐥𝟐𝐥𝟐 = 𝜎20𝐏𝟐―𝟏

(2)

where 𝐀𝟏 and 𝐀𝟐 are the coefficient matrices of the first and second observation epochs, respectively and 𝐀𝟏 = 𝐀𝟐 in case of the configuration of the network is not changed for different epochs. 𝐱𝟏 and 𝐱𝟐 are the unknown parameters’ vectors, 𝐥𝟏 and 𝐥𝟐 are the observation vectors, 𝐯𝟏 and 𝐯𝟐 are the residual vectors, 𝐏𝟏 and 𝐏𝟐 are the weight matrices of the epoch 𝑡1 and epoch 𝑡2, respectively. The estimated coordinates are obtained by using the LSE method as follows: +

+

(3)

+

+

(4)

𝐱𝟏 = (𝐀𝐓𝟏𝐏𝟏𝐀𝟏) 𝐀𝐓𝟏𝐏𝟏𝐥𝟏; 𝐐𝐱𝟏𝐱𝟏 = (𝐀𝐓𝟏𝐏𝟏𝐀𝟏) 𝐱𝟐 = (𝐀𝐓𝟐𝐏𝟐𝐀𝟐) 𝐀𝐓𝟐𝐏𝟐𝐥𝟐; 𝐐𝐱𝟐𝐱𝟐 = (𝐀𝐓𝟐𝐏𝟐𝐀𝟐)

(5)

𝛺 = 𝐯𝐓𝟏𝐏𝟏𝐯𝟏 + 𝐯𝐓𝟐𝐏𝟐𝐯𝟐 𝛺

(6)

𝑠20 = 𝑓1 + 𝑓2

3

where 𝐐𝐱𝟏𝐱𝟏 and 𝐐𝐱𝟐𝐱𝟐 are the cofactor matrices of unknown parameters, (+) denotes the pseudo-inverse, 𝑓1 and 𝑓2 denote the degrees of the freedom of the first and second epochs, respectively and 𝑠20 is the estimated pooled variance. In this study, to estimate the 𝐱𝟏 and 𝐱𝟐 Bernese v5.2 scientific software was used. The cofactor matrices and standard deviations for each epoch are determined from the software. To investigate whether there are any statistically significant displacements or not between two epochs, the null (𝐻0) and alternative (𝐻𝐴) hypotheses are written as; 𝐻0:𝐸(𝐱𝟏) = 𝐸(𝐱𝟐)

(7)

𝐻𝐴:𝐸(𝐱𝟏) ≠ 𝐸(𝐱𝟐)

(8)

where 𝐸 denotes the expectation. Then, to confirm whether the point/s is/are displaced significantly or not, in case of the absence of correlations, the test statistic is given between two epochs as follows (Pelzer 1971, 1985; Niemeier 1985; Koch 1985; Cooper 1987): (9)

𝐝 = 𝐱𝟐 ― 𝐱𝟏

(10)

𝐐𝐝𝐝 = 𝐐𝐱𝟏𝐱𝟏 + 𝐐𝐱𝟐𝐱𝟐

(11)

+ 𝑅 = 𝐝𝐓𝐐𝐝𝐝 𝐝 𝑅

(12)

𝑇 = 𝑠2ℎ ~ 𝐹ℎ,𝑓,1 ― 𝛼 0

where 𝐝 is the difference vector of the estimated coordinates, 𝐐𝐝𝐝 is the cofactor matrix of + the 𝐝 and its pseudo-inverse is 𝐐𝐝𝐝 , ℎ is the rank of 𝐐𝐝𝐝, 𝑓 = 𝑓1 + 𝑓2 is the sum of the degrees of freedom and 1 ― 𝛼 is the chosen confidence level of F-distribution. If 𝑇 > 𝐹ℎ,𝑓,1 ― 𝛼, the null hypothesis is rejected. Hence, the differences in the coordinates between two epochs are interpreted as a result of displacements. When the coordinates’ differences are verified as displacements, the quadratic KDA model can be applied to estimate the velocity and the acceleration parameters. Mathematical model of the quadratic approach, which consists of position, velocity and acceleration parameters for x, y and z coordinates is given for the jth point as follows; 1

𝑥𝑖𝑗 + 1 = 𝑥𝑖𝑗 + 𝑣𝑥𝑗(𝑡𝑖 + 1 ― 𝑡𝑖) + 2𝑎𝑥𝑗(𝑡𝑖 + 1 ― 𝑡𝑖)2 1

𝑦𝑖𝑗 + 1 = 𝑦𝑖𝑗 + 𝑣𝑦𝑗(𝑡𝑖 + 1 ― 𝑡𝑖) + 2𝑎𝑦𝑗(𝑡𝑖 + 1 ― 𝑡𝑖)2

(13)

1

𝑧𝑖𝑗 + 1 = 𝑧𝑖𝑗 + 𝑣𝑧𝑗(𝑡𝑖 + 1 ― 𝑡𝑖) + 2𝑎𝑧𝑗(𝑡𝑖 + 1 ― 𝑡𝑖)2 for 𝑗 = 1,2,⋯,𝑝 where p is the number of points, 𝑥𝑖𝑗, 𝑦𝑖𝑗, 𝑧𝑖𝑗 are the coordinates of the points at epoch 𝑖 in the deformation network, 𝑣𝑥𝑗, 𝑣𝑦𝑗, 𝑣𝑧𝑗 are the velocity parameters between epoch 𝑡𝑖 + 1 and 𝑡𝑖; 𝑎𝑥𝑗, 𝑎𝑦𝑗, 𝑎𝑧𝑗 are the acceleration parameters of three corresponded components of the coordinates in deformation network for epochs 𝑡𝑖 + 1 and 𝑡𝑖, 𝑝 is the number of points (Pelzer 1985, 1987). To estimate the unknown parameters in the quadratic KDA model at least four 4

epochs must be analyzed. The coefficient matrix (𝐀𝒌𝒊𝒏) of the quadratic KDA model is given as;

𝐀𝒌𝒊𝒏 =

[

𝐈

𝐈 × (𝑡𝑖 + 1 ― 𝑡𝑖)

𝐈×

𝐈

𝐈 × (𝑡𝑖 + 2 ― 𝑡𝑖)

𝐈×

𝐈

𝐈 × (𝑡𝑖 + 3 ― 𝑡𝑖)

𝐈×

𝐈

𝐈 × (𝑡𝑖 + 4 ― 𝑡𝑖)

𝐈×

]

(𝑡𝑖 + 1 ― 𝑡𝑖)2 2 (𝑡𝑖 + 2 ― 𝑡𝑖)2 2 (𝑡𝑖 + 3 ― 𝑡𝑖)2

(14)

2 (𝑡𝑖 + 4 ― 𝑡𝑖)2 2

where 𝐈 is the 𝑝 × 𝑝 identity matrix (Yu and Segall, 1996; Kenner and Segall, 2000; Shahar and Tzur, 2014). The estimated coordinates for each observation epoch are defined as the observation vector for the quadratic KDA model. (15)

𝒙𝟎 = 𝐱𝟏

The coordinate differences are calculated by subtracting each epoch from the first epoch as given below (in this study only four epochs have been considered for the KDA model); 𝐱′𝒊 = 𝐱𝒊 ― 𝐱𝟎, 𝑖 = 1, 2, 3, 4. 𝐥𝑻𝒒𝒖𝒂𝒅 = [𝐱′𝟏 𝐱′𝟐 𝐱′𝟑 𝐱′𝟒]

(16) (17)

where 𝐱′𝟏, 𝐱′𝟐, 𝐱′𝟑 and 𝐱′𝟒 are the coordinate differences between 𝐱𝒊 that is the estimated coordinates of the 𝑖th epoch and 𝐱𝟎 that is the estimated coordinates of the first epoch, 𝐥𝒒𝒖𝒂𝒅 is the observation vector of the quadratic KDA model. The equations for the adjustment of the models are given as follows: 𝐐𝒒𝒖𝒂𝒅 = (𝐀𝐓𝒒𝒖𝒂𝒅𝐏𝐪𝐮𝐚𝐝𝐀𝒒𝒖𝒂𝒅)

―𝟏

(18)

𝐱𝒒𝒖𝒂𝒅 = 𝐐𝒒𝒖𝒂𝒅(𝐀𝐓𝒒𝒖𝒂𝒅𝐏𝐪𝐮𝐚𝐝𝐥𝒒𝒖𝒂𝒅)

(19)

𝐯𝒒𝒖𝒂𝒅 = 𝐀𝒒𝒖𝒂𝒅𝐱𝒒𝒖𝒂𝒅 ― 𝐥𝒒𝒖𝒂𝒅

(20)

s0quad = (𝐯𝐓𝒒𝒖𝒂𝒅𝐏𝐪𝐮𝐚𝐝𝐯𝒒𝒖𝒂𝒅)/𝑓𝑞𝑢𝑎𝑑

(21) (22)

𝑓𝑞𝑢𝑎𝑑 = 𝑛𝑞𝑢𝑎𝑑 ― 𝑢𝑞𝑢𝑎𝑑

where 𝐱𝒒𝒖𝒂𝒅 is the unknown parameters vector that includes displacement, velocity and acceleration parameters, 𝐐𝒒𝒖𝒂𝒅 is the cofactor matrix of the unknown parameters, 𝐯𝒒𝒖𝒂𝒅 is the vector of residual, s0quad is a posteriori standard deviation, 𝑓𝑞𝑢𝑎𝑑 is the degrees of freedom, 𝑛𝑞𝑢𝑎𝑑 is the number of observation, 𝑢𝑞𝑢𝑎𝑑 is the number of the unknown parameters and full covariance matrices for each epoch is combined; so 𝐏𝐪𝐮𝐚𝐝 is derived as a block-diagonal weight matrix. The full covariance matrices, which consist of the knowledge about the accuracy of the estimated coordinates for each epoch, obtained from Bernese v5.2 software were used for the derivation of 𝐏𝐪𝐮𝐚𝐝 matrix which states the stochastic model of the adjustment step.

5

3. Statistical Analysis of the results of the Quadratic Kinematic Deformation Analysis Model For the deformation analysis, each estimated parameter should be statistically tested whether the displacements or the parameters are significant or not. In this study, the displacements have been simulated and loaded to the object points by using LMS devices. After this step, 8 hours GPS observations have been carried out and Bernese v5.2 scientific software was used for the analyses (Dach et al. 2015). For the next step, the velocity and the acceleration parameters were estimated depending on the quadratic KDA model given in Section 2, and then two different statistical tests have been applied. At first, to test whether the estimated parameters are significant or not, the null hypothesis and the alternative hypothesis are formed as follows: 𝐻0:𝐸(𝑣) = 0

𝐻𝐴:𝐸(𝑣) ≠ 0

(23)

𝐻0:𝐸(𝑎) = 0

𝐻𝐴:𝐸(𝑎) ≠ 0

(24)

where 𝑣 and 𝑎 are the estimated velocity and acceleration parameters, respectively. The test value 𝑡𝑖 is calculated as dividing the estimated value by its standard deviation (Cooper, 1987). 𝑡𝑖 =

|𝐱𝑞𝑢𝑎𝑑,𝑖| s0quad 𝐐𝒒𝒖𝒂𝒅𝒊,𝒊

~ 𝑡𝑓𝑞𝑢𝑎𝑑,1 ― 𝛼 2; 𝑖 = 1⋯𝑢𝑞𝑢𝑎𝑑

(25)

where 𝛼 is the significance level that has been chosen as 0.001. If 𝑡𝑖 > 𝑡𝑓𝑞𝑢𝑎𝑑,1 ― 𝛼 2 it means that the parameter is statistically significant. Secondly, the differences between estimated values and loaded values have been tested. The null and alternative hypotheses are given below: 𝐻0:𝐸(𝑣) = 𝐸(𝜇0𝑣)

𝐻𝐴:𝐸(𝑣) ≠ 𝐸(𝜇0𝑣)

(26)

𝐻0:𝐸(𝑎) = 𝐸(𝜇0𝑎)

𝐻𝐴:𝐸(𝑎) ≠ 𝐸(𝜇0𝑎)

(27)

where 𝜇0𝑣 and 𝜇0𝑎 are the values of the loaded velocity and acceleration parameters by using LMS devices. The test value 𝑇𝑖 is estimated as (Cooper, 1987): 𝑇𝑖 =

|𝐱𝑞𝑢𝑎𝑑,𝑖 ― 𝜇0| s0quad 𝐐𝒒𝒖𝒂𝒅𝒊,𝒊

(28)

~ 𝑡𝑓𝑞𝑢𝑎𝑑,1 ― 𝛼 2

If 𝑇𝑖 ≤ 𝑡𝑓𝑞𝑢𝑎𝑑,1 ― 𝛼 2 it means that the value of the estimated parameter is statistically equal to value of the loaded parameter. 4. The Scenarios and analysis In order to obtain the reliability of the quadratic KDA model, a GPS deformation network that consisted of five reference points (TERK, BEYK, ISTN, ISTA, TUZL) and four object points (AYDN, ARBY, ERBD, UZEL) was established (Fig. 1). ISTA is the point of the International Global Navigation Satellite Systems Service (IGS) network; ISTN is the point of the CORS-TR (Continuously Operating Reference Stations – Turkey); TERK, BEYK and TUZL are the points of the Istanbul Water and Sewerage Administration CORS network. AYDN, ARBY, ERBD and 6

UZEL are the terrace pillars that are constructed at the top of the Faculty Building and the LMS devices are established on these points (Fig. 2). The baseline lengths depending on the ISTA point are given at the Table 1. Thirty-seven sessions of GPS observations with 8-hours session duration were carried out. Twelve different scenarios have been generated. These scenarios are given in Table 2. The GPS observations have been implemented between 3rd July 2018 and 10th August 2018. It is assumed that each scenario contains four GPS campaigns and the interval among campaigns is one year. Depending on these assumptions the magnitudes of the displacements that were imposed on the object points by using LMS devices have been calculated as given: 1

(29)

𝑑ℎ𝑗 = 𝑑ℎ0 + 𝑣𝑑ℎ (𝑡𝑗 ― 𝑡0) + 2𝑎𝑑ℎ(𝑡𝑗 ― 𝑡0)2 1

(30)

𝑑𝑦𝑗 = 𝑑𝑦0 + 𝑣𝑑𝑦 (𝑡𝑗 ― 𝑡0) + 2𝑎𝑑𝑦(𝑡𝑗 ― 𝑡0)2

where 𝑑ℎ is the displacement on the horizontal plane, 𝑑𝑦 is the displacement on the vertical direction, 𝑡0 is the first epoch that is assumed 0, 𝑡𝑗 is the jth period, 𝑑ℎ0 and 𝑑𝑦0 are the displacements on the horizontal plane and vertical direction, respectively, at reference period that are assumed 0, 𝑣𝑑ℎ and 𝑎𝑑ℎ are the velocity and acceleration parameters on the horizontal plane, respectively, 𝑣𝑑𝑦 and 𝑎𝑑𝑦 are the velocity and acceleration parameters on vertical direction, respectively. The values of the 𝑣𝑑ℎ, 𝑎𝑑ℎ, 𝑣𝑑𝑦 and 𝑎𝑑𝑦 are taken from Table 2.

The displacements on the horizontal plane and vertical direction were imposed on the points by using LMS devices. The coordinates and their covariance matrices used for deformations analysis were obtained with Bernese v.5.2 computing single-session solutions in multibaseline mode; that is, processing all the data of a session together. The phase doubledifferences linear combination ionosphere-free was used to process baselines, ambiguities were resolved when possible. Moreover, precise ephemeris and antenna phase center variation files provided by CODE (Center for Orbit Determination in Europe) of Bern University were used. Finally, tropospheric parameters were estimated for one parameter for each site every 4 h. All of the GPS data processing were performed by constraining reference points (TERK, BEYK, ISTN, ISTA, TUZL), and the estimated coordinates together with their covariance matrices are referred to ITRF2008 datum and 2018.504 epoch. Although the accuracies of the estimated coordinates from Bernese v5.2 software are very high, the Variance-Covariance (VCV) matrices are very optimistic. Since it takes into account only mathematical correlation, neglects the physical correlations and the a priori assigned measurement errors are often arbitrary, the results of the VCV matrices are not realistic (Han and Rizos 1995a; 1995b; Mao et al. 1999; McClusky et al. 2000; Nocquet et al. 2002; ElRabbany and Kleusberg 2003; Geirsson 2003; Kashani et al. 2004; Cetin et al. 2018; Erdogan and Dogan 2019). In this study, the VCV matrices were multiplied by 22 depending on Kashani et al. (2004). For the quadratic KDA model, the coordinates and the VCV matrices at the Earth Centered Earth Fixed Coordinate System (X, Y, Z) were transformed to the Topocentric Coordinate System (north, east, up). The relations between the estimated displacements that differ from the first epoch and loaded displacements are given in Figs. 37

18. The first GPS observation on 3rd July 2018 (GPS Day – 184) was considered as the reference epoch (first epoch). In Figs. 3-18, “Estimated” expresses the estimated values of the coordinate differences by using Bernese v5.2 software and “Loaded” states the true values of the coordinate differences.

Figs. 3-6 consist of the 1st, 2nd, 3rd and 4th scenarios that consider only the displacement in the vertical direction for ARBY, AYDN, ERBD and UZEL object points, respectively. The explanations about the scenarios are given in Table 2. Fig.3 shows the coordinate differences for point ARBY. The estimated and loaded displacement values are close to each other. Also, the correlation coefficient (ρ) that shows the relationship with estimated and loaded values is bigger than 0.98 which means that the estimated values and loaded values are close to each other. Fig.4 shows the coordinate differences in the vertical direction for point AYDN. Similar to the point ARBY the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.98 which means that the estimated values and loaded values are close to each other.

Fig.5 shows the coordinate differences in the vertical direction for point ERBD. Similar to the points ARBY and AYDN the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.98. Fig.6 shows the coordinate differences in the vertical direction for point UZEL. Similar to the points ARBY, AYDN and ERBD the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is 0.99. The estimated and loaded coordinate differences in 5th, 6th, 7th and 8th scenarios that consider only the displacements in the horizontal plane are given in Figs. 7-10 for ARBY, AYDN, ERBD and UZEL object points, respectively. Fig.7 shows the coordinate differences in the horizontal plane for point ARBY. The correlation coefficient is bigger than 0.97 which means that the estimated values and loaded values are close to each other. Fig.8 shows the coordinate differences in the horizontal plane for point AYDN. Similar to the point ARBY the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.95.

8

Fig.9 shows the coordinate differences in the horizontal for point ERBD. Similar to the points ARBY and AYDN the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.96. Fig.10 shows the coordinate differences in the horizontal for point UZEL. Similar to the points ARBY, AYDN and ERBD the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is 0.97. In 9th, 10th, 11th and 12th scenarios, the displacements have been loaded for both horizontal plane and vertical direction, also the estimated and the loaded displacements are given in Figs. 11-18 for ARBY, AYDN, ERBD and UZEL object points, respectively. Fig.11 shows the coordinate differences in the horizontal plane for point ARBY and the correlation coefficient is bigger than 0.95. Fig.12 shows the coordinate differences in the vertical direction for point ARBY and the correlation coefficient is bigger than 0.98. Fig.13 shows the coordinate differences in the horizontal plane for point AYDN and similar to the point ARBY the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.91. Fig.14 shows the coordinate differences in the vertical direction for point AYDN and similar to the point ARBY the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.98. Fig.15 shows the coordinate differences in the horizontal plane for point ERBD and similar to the points ARBY and AYDN the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.92. Fig.16 shows the coordinate differences in the vertical direction for point ERBD and similar to the points ARBY and AYDN the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.98. Fig.17 shows the coordinate differences in the horizontal plane for point UZEL and similar to the points ARBY, AYDN and ERBD the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.94. Fig.18 shows the coordinate differences in the vertical direction for point UZEL and similar to the points ARBY, AYDN and ERBD the estimated and loaded displacement values are close to each other. Also, the correlation coefficient is bigger than 0.98. When Figs. 3-18 are analyzed for twelve scenarios, generally it can be said that the loaded displacements have been determined by using Bernese v5.2 software reliably. Especially for the scenarios that contain displacements in the vertical direction the correlation coefficient is very high that means the estimated values and loaded values are close to each other. 5. Quadratic kinematic deformation analysis results

9

In this study, to show the reliabilities of the results of the quadratic KDA model, displacements have been simulated, loaded to the object points by LMS devices and estimated. After processing the GPS data with Bernese v5.2 software, the coordinates and the full-cofactor matrices for thirty-seven epochs have been obtained. Depending on these epochs twelve different scenarios were simulated and the quadratic KDA model was applied. The quadratic KDA model generally uses the LSE method and it spreads the effect of the displaced points to other points, that’s why, the results of the analysis may be wrong. At the first step of the quadratic KDA model, the velocity and the acceleration parameters were estimated and then two different statistical tests were applied. At first, whether the estimated parameters are significant or not has been tested, secondly, the differences between estimated values and loaded values have been tested. The estimated parameters and test statistics are given in Table 3. The absolute values of the estimated parameters were given; the bold, italic and underlined value in Table 3 states the correct test results. Also, the identified object points for each scenario are presented in Tables 4-6. In Tables, the dashed lines mean that the displacements have not been loaded and underlined object name means that these points are identified as displaced but estimated values of these points are not statistically equal to loaded values.

In Table 4; the 1st and 2nd scenarios, which are the linear loaded model in the only vertical direction, consider merely velocity parameter and the magnitude of the velocities are small and large, respectively. Although the velocity has been loaded to the four object points, only the AYDN and ERBD object points were identified in the 1st scenario and ARBY, AYDN and ERBD object points were identified in the 2nd scenario. The 3rd and 4th scenarios of quadratic loaded model consider both velocity and acceleration parameters. For the 3rd scenario in the vertical direction both velocity parameters have been verified as insignificant and the acceleration parameters have been identified as significant in all object points; also LSE method smeared the effect of the displacements from the vertical direction to the horizontal plane. The velocity parameters and the acceleration parameters were identified for object points in the horizontal plane although the displacement was not loaded. This situation is valid for the 4th scenario, too. In the 4th scenario, four object points’ velocity and acceleration parameters were identified as significant although the magnitudes of the estimated values are bigger than the loaded values of ARBY, AYDN and ERBD object points.

In Table 5; the 5th and 6th scenarios involved in linear loaded models, consider only velocity parameters; the 7th and 8th scenarios involved in quadratic loaded model, contain both velocity and acceleration parameters in only horizontal plane. Although the velocity has been loaded to the four object points, three of them AYDN, ERBD and UZEL were identified in the 5th scenario and only UZEL object point were identified in the 6th scenario. For the 7th scenario in the horizontal plane both velocity and the acceleration parameters have been identified in all object points; but the LSE method smeared the effect of the displacements from horizontal plane to the vertical direction, then the velocity and the acceleration 10

parameters for UZEL were identified in vertical direction although the displacement were not loaded. In the 8th scenario, all parameters for all object points have been identified reliably; but only for UZEL object points, the magnitude of the estimated values are bigger than the loaded values. In Table 6; the 9th and 10th scenarios contain only velocity parameters; the 11th and 12th scenarios contain velocity and acceleration parameters as well, both in vertical direction and horizontal plane. Although the velocity in both horizontal plane and vertical direction has been loaded to the four object points, only the parameters in the horizontal plane were identified in the 9th scenario. Also, the estimated velocity parameters for ERBD and UZEL object points were not equal to the loaded values. For the 10th scenario; the parameters for object points could not be identified as significant, though the estimated values of the parameters are equal to the loaded values statistically. For the 11th and 12th scenarios; all object points for the horizontal plane have been identified correctly; but the estimated values of the parameters of AYDN, ERBD and UZEL object points were not equal to the loaded values, statistically. Also, in the vertical direction for the 11th and 12th scenarios, the quadratic KDA model can identify only AYDN points and AYDN and ERBD object points, respectively. In Table 7, the quadratic KDA model performance was calculated for each scenario 𝑆

considering the Eqs. 23 and 24 by 𝑃 = 𝑇 ∗ 100, where 𝑆 is the number of correct identified object point and 𝑇 represents the number of total loaded object points. It can be seen in Table 7, the performance rate of the quadratic KDA model of both linear and quadratic loaded models are the same in the horizontal plane (see North-East row header) when displacements magnitude is small (5th and 7th scenarios). If large magnitude displacements were loaded, the performance rate of quadratic KDA model of the quadratic loaded model is better than the linear loaded one in the North-East (6th and 8th scenarios) and North-East-Up (10th and 12th scenarios) directions.

6. Conclusions 11

In this study, to investigate the reliabilities of the results of the quadratic KDA model, a real GPS deformation monitoring network that consists of four object points and five reference points was established and twelve different scenarios have been generated. The scenarios contain both velocity and acceleration parameters for small and large magnitudes in the horizontal plane and the vertical direction. Two different tests have been applied to identify the parameters. According to the obtained results, the values of the coordinated differences estimated by Bernese v5.2 are close to the loaded values; but for the quadratic KDA model some problems may be happened. When the significances of the parameters tested for the scenarios that contain only velocity parameters, the number of the identified object points are smaller than the real number of the displaced object points; for the scenarios that contain both velocity and acceleration parameters, the magnitudes of the estimated parameters are larger than the real values for most of the object points. Also, the more reliable results have been obtained for the scenarios that contain the displacements only in horizontal plane. The main reason for the wrong results can be based on the spreading effect of the LSE method. LSE methods smear the effects of the displaced points to the other nondisplaced points. For the 3rd, 9th and 10th scenarios none of the object points can be identified correctly; the most reliable results have been estimated for the 2nd, 5th, 7th and 8th scenarios. When the magnitudes of the parameters decreases, the reliabilities of the quadratic KDA model decrease, too. Acknowledgements This study was supported by the Scientific and Technological Research Council of Turkey (TUBITAK-CAYDAG Project Number 117Y263). We are thankful to International GNSS Service (IGS), Istanbul Water and Sewerage Administration (ISKI), Scripps Orbit and Permanent Array Center (SOPAC), Center for Orbit Determination in Europe (CODE) and Continuously Operating Reference Stations – Turkey (CORS-TR) for the GPS data and IGS precise orbits. Also, the Fig.1 was plotted by using Generic Mapping Tools (GMT) (Wessel and Smith, 1998) References Acar, M., Ozludemir, M. T., Erol, S., Celik, R. N., Ayan, T. (2008). Kinematic landslide monitoring with Kalman filtering. Natural Hazards and Earth System Science, 8(2), 213-221. Baselga, S., García-Asenjo, L., Garrigues, P. (2015). Deformation monitoring and the maximum number of stable points method, Measurement, Vol. 70, pp. 27-35. Betti, B., Biagi, L., Crespi, M., Riguzzi, F. (1999). GPS sensitivity analysis applied to non-permanent deformation control networks. Journal of Geodesy, 73(3), 158-167. Cai, J., Wang, J., Wu, J., Hu, C., Grafarend, E., Chen, J. (2008). Horizontal deformation rate analysis based on multiepoch GPS measurements in Shanghai. Journal of Surveying Engineering, 134(4), 132137. Caspary, W. (1988). A robust approach to estimating deformation parameters. In Proc. of the 5th Canadian Symp. on Mining Surveying and Rock Deformation Measurements (pp. 124-135). Dept. of Surveying Engineering Univ. of New Brunswick, Fredericton, NB.

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Figure 1. The points of the GPS deformation monitoring network

Figure 2. The object points AYDN, ARBY, ERBD and UZEL

Figure 3. The estimated and loaded displacement in vertical direction at the object point ARBY for the scenarios 1, 2, 3 and 4 15

Figure 4. The estimated and loaded displacement in vertical direction at the object point AYDN for the scenarios 1, 2, 3 and 4

Figure 5. The estimated and loaded displacement in vertical direction at the object point ERBD for the scenarios 1, 2, 3 and 4

Figure 6. The estimated and loaded displacement in vertical direction at the object point UZEL for the scenarios 1, 2, 3 and 4

Figure 7. The estimated and loaded displacement in horizontal plane at the object point ARBY for the scenarios 5, 6, 7 and 8

Figure 8. The estimated and loaded displacement in horizontal plane at the object point AYDN for the scenarios 5, 6, 7 and 8

Figure 9. The estimated and loaded displacement in horizontal plane at the object point ERBD for the scenarios 5, 6, 7 and 8

Figure 10. The estimated and loaded displacement in horizontal plane at the object point UZEL for the scenarios 5, 6, 7 and 8

Figure 11. The estimated and loaded displacement in horizontal plane at the object point ARBY for the scenarios 9, 10, 11 and 12

Figure 12. The estimated and loaded displacement in vertical direction at the object point ARBY for the scenarios 9, 10, 11 and 12

Figure 13. The estimated and loaded displacement in horizontal plane at the object point AYDN for the scenarios 9, 10, 11 and 12

Figure 14. The estimated and loaded displacement in vertical direction at the object point AYDN for the scenarios 9, 10, 11 and 12

16

Figure 15. The estimated and loaded displacement in horizontal plane at the object point ERBD for the scenarios 9, 10, 11 and 12

Figure 16. The estimated and loaded displacement in vertical direction at the object point ERBD for the scenarios 9, 10, 11 and 12

Figure 17. The estimated and loaded displacement in horizontal plane at the object point UZEL for the scenarios 9, 10, 11 and 12

Figure 18. The estimated and loaded displacement in vertical plane at the object point UZEL for the scenarios 9, 10, 11 and 12

Table 1. Approximate baseline lengths Baseline Length (km) ISTA – ISTN 20.2 ISTA – TUZL 38.5 ISTA – TERK 36.4 ISTA – BEYK 10.2 ISTA – ARBY 14.3 ISTA – AYDN 14.3 ISTA – ERBD 14.2 ISTA – UZEL 14.2 Table 2. The scenarios and the magnitudes of the velocity and acceleration parameters Number of Scenarios

Magnitude Velocity (mm/year) Acceleration (mm/year2) Horizontal Vertical Horizontal Vertical Plane Direction Plane Direction

Explanation

1

Only vertical direction, small magnitude, without acceleration (GPS Days – 184, 185, 187, 188)

-

2

Only vertical direction, large magnitude, without acceleration (GPS Days – 184, 189, 190, 191)

-

3

Only vertical direction, small magnitude, with acceleration (GPS Days – 184, 192, 193, 194)

-

4

Only vertical direction, large magnitude, with acceleration (GPS Days – 184, 195, 196, 197)

-

5

Only horizontal plane, small magnitude, without acceleration (GPS Days – 184, 198, 200, 202)

ARBY – 5.13 AYDN – 5.00 ERDB – 5.18

17

ARBY – 16.90 AYDN – 17.30 ERDB – 17.15 UZEL – 17.00 ARBY – 29.90 AYDN – 30.30 ERDB – 30.15 UZEL – 30.00 ARBY – 16.90 AYDN – 17.30 ERDB – 17.15 UZEL – 17.00 ARBY – 29.90 AYDN – 30.30 ERDB – 30.15 UZEL – 30.00 -

-

-

-

ARBY – 0.00 AYDN – 0.00 ERDB – 0.00

ARBY – 0.00 AYDN – 0.00 ERDB – 0.00 UZEL – 0.00 ARBY – 0.00 AYDN – 0.00 ERDB – 0.00 UZEL – 0.00 ARBY – 16.00 AYDN – 16.00 ERDB – 16.00 UZEL – 16.00 ARBY – 32.00 AYDN – 32.00 ERDB – 32.00 UZEL – 32.00 -

6

Only horizontal plane, large magnitude, without acceleration (GPS Days – 184, 204, 206, 209)

7

Only horizontal plane, small magnitude, with acceleration (GPS Days – 184, 211, 213, 215)

8

Only horizontal plane, large magnitude, with acceleration (GPS Days – 184, 217, 219, 221)

9

Both horizontal plane and vertical direction, small magnitude, without acceleration (GPS Days – 184, 199, 201, 203)

10

Both horizontal plane and vertical direction, large magnitude, without acceleration (GPS Days – 184, 205, 208, 210)

11

Both horizontal plane and vertical direction, small magnitude, with acceleration (GPS Days – 184, 212, 214, 216)

12

Both horizontal plane and vertical direction, large magnitude, with acceleration (GPS Days – 184, 218, 220, 222)

UZEL – 4.91 ARBY – 8.00 AYDN – 7.90 ERDB – 8.19 UZEL – 8.11 ARBY – 5.00 AYDN – 5.20 ERDB – 5.10 UZEL – 4.90 ARBY – 8.30 AYDN – 8.10 ERDB – 8.20 UZEL – 8.00 ARBY – 5.13 AYDN – 5.00 ERDB – 5.18 UZEL – 4.91 ARBY – 8.00 AYDN – 7.90 ERDB – 8.19 UZEL – 8.11 ARBY – 5.13 AYDN – 5.00 ERDB – 5.18 UZEL – 4.91 ARBY – 8.00 AYDN – 7.90 ERDB – 8.19 UZEL – 8.11

-

-

ARBY – 16.90 AYDN – 17.30 ERDB – 17.15 UZEL – 17.00 ARBY – 29.90 AYDN – 30.30 ERDB – 30.15 UZEL – 30.00 ARBY – 16.90 AYDN – 17.30 ERDB – 17.15 UZEL – 17.00 ARBY – 29.90 AYDN – 30.30 ERDB – 30.15 UZEL – 30.00

UZEL – 0.00 ARBY – 0.00 AYDN – 0.00 ERDB – 0.00 UZEL – 0.00 ARBY – 5.70 AYDN – 5.90 ERDB – 5.80 UZEL – 5.60 ARBY – 8.10 AYDN – 8.00 ERDB – 8.30 UZEL – 8.20 ARBY – 0.00 AYDN – 0.00 ERDB – 0.00 UZEL – 0.00 ARBY – 0.00 AYDN – 0.00 ERDB – 0.00 UZEL – 0.00 ARBY – 6.00 AYDN – 6.00 ERDB – 6.00 UZEL – 6.00 ARBY – 8.00 AYDN – 8.00 ERDB – 8.00 UZEL – 8.00

-

-

ARBY – 0.00 AYDN – 0.00 ERDB – 0.00 UZEL – 0.00 ARBY – 0.00 AYDN – 0.00 ERDB – 0.00 UZEL – 0.00 ARBY – 16.00 AYDN – 16.00 ERDB – 16.00 UZEL – 16.00 ARBY – 32.00 AYDN – 32.00 ERDB – 32.00 UZEL – 32.00

Table 3. Estimated parameters and test statistics Estimated Values (mm/year) - (mm/year2) Horizontal Plane Vertical Direction Velocity Acceleration Velocity Acceleration

Test Statistic (Test 1) – Critical Value 1.782 Horizontal Plane Vertical Direction Velocity Acceleration Velocity Acceleration

Test Statistic (Test 2) – Cri Horizontal Plane Velocity Acceleration Ve

Number of Scenarios

Object Points

1

ARBY AYDN ERBD UZEL

1.20 2.78 0.56 1.74

1.33 2.28 0.72 1.65

15.05 21.57 20.45 10.99

6.50 1.13 1.62 3.72

0.34 0.78 0.19 0.50

0.51 0.87 0.30 0.67

1.11 2.31 2.19 0.83

0.66 0.12 0.17 0.38

0.34 0.78 0.19 0.50

0.51 0.87 0.30 0.67

0 0 0 0

2

ARBY AYDN ERBD UZEL

5.52 2.27 1.43 4.03

3.21 1.05 0.41 0.92

41.76 37.13 32.40 18.24

8.02 4.68 2.41 4.50

1.35 0.55 0.39 1.04

1.08 0.36 0.15 0.31

2.71 2.44 2.14 1.19

0.71 0.42 0.22 0.40

1.35 0.55 0.39 1.04

1.08 0.36 0.15 0.31

0 0 0 0

3

ARBY AYDN ERBD UZEL

5.40 5.27 6.69 8.83

2.79 2.87 3.64 4.47

6.00 10.31 5.77 8.92

25.37 21.47 23.75 31.60

2.29 2.41 3.03 3.84

1.57 1.74 2.17 2.56

0.64 1.12 0.63 0.97

3.71 3.18 3.53 4.69

2.29 2.41 3.03 3.84

1.57 1.74 2.17 2.56

1 0 1 0

4

ARBY AYDN ERBD UZEL

6.84 10.76 9.51 4.91

8.13 10.12 8.79 6.51

53.94 54.65 49.86 36.07

14.85 13.90 16.85 23.33

2.70 4.19 3.68 1.89

4.34 5.33 4.61 3.45

5.58 5.79 5.28 3.81

2.05 1.95 2.36 3.27

2.70 4.19 3.68 1.89

4.34 5.33 4.61 3.45

2 2 2 0

5

ARBY AYDN ERBD UZEL

4.55 8.13 7.73 7.98

2.02 4.19 3.81 1.83

3.90 6.42 1.61 16.85

3.72 5.82 0.88 8.82

1.29 2.27 2.14 2.23

0.65 1.40 1.24 0.66

0.25 0.40 0.10 1.07

0.32 0.50 0.07 0.75

0.16 0.87 0.70 0.86

0.65 1.40 1.24 0.66

0 0 0 1

6

ARBY AYDN ERBD UZEL

8.42 7.25 8.53 9.88

2.01 3.18 2.87 2.47

13.72 6.25 1.27 7.12

10.57 3.91 2.06 0.70

1.61 1.44 1.69 1.91

0.52 0.77 0.70 0.65

0.62 0.28 0.06 0.32

0.68 0.25 0.13 0.04

0.08 0.13 0.07 0.34

0.52 0.77 0.70 0.65

0 0 0 0

7

ARBY AYDN ERBD UZEL

5.33 5.27 6.42 7.18

5.15 6.58 4.96 4.62

2.84 0.43 5.66 14.09

2.78 1.50 4.58 6.87

5.94 6.08 7.34 8.13

7.29 9.13 7.12 6.49

0.76 0.12 1.51 3.78

0.98 0.53 1.62 2.44

0.22 0.31 1.41 2.57

1.21 0.81 1.49 1.93

0 0 1 3

8

ARBY AYDN

12.34 11.13

9.89 11.78

2.63 11.09

2.85 7.05

3.18 2.86

3.09 3.67

0.15 0.65

0.23 0.57

1.12 0.83

0.59 1.18

0 0

18

ERBD UZEL

15.60 16.28

8.99 9.93

14.06 13.72

9.93 6.79

4.04 4.19

2.81 3.09

0.83 0.80

0.80 0.54

1.72 2.10

0.31 0.60

0 0

9

ARBY AYDN ERBD UZEL

13.52 15.53 18.58 16.12

9.35 10.73 12.41 10.85

9.19 7.39 9.01 1.91

6.58 7.73 4.85 10.36

2.30 2.60 3.10 2.75

2.04 2.32 2.68 2.38

0.38 0.31 0.38 0.08

0.37 0.43 0.27 0.58

1.43 1.76 2.23 1.91

2.04 2.32 2.68 2.38

0 0 0 0

10

ARBY AYDN ERBD UZEL

4.45 6.41 6.09 6.44

3.67 5.82 4.85 3.89

11.22 6.53 3.06 5.69

12.13 16.36 16.46 20.56

0.99 1.33 1.31 1.44

0.99 1.56 1.30 1.05

0.59 0.34 0.16 0.30

0.89 1.20 1.21 1.52

0.79 0.31 0.45 0.37

0.99 1.56 1.30 1.05

0 1 1 1

11

ARBY AYDN ERBD UZEL

8.59 8.29 10.78 10.88

7.31 9.64 7.04 6.81

12.13 19.79 10.62 4.78

20.73 15.40 22.44 29.42

4.40 4.15 5.53 5.58

4.44 5.85 4.27 4.14

1.44 2.36 1.26 0.57

3.30 2.46 3.58 4.70

1.77 1.65 2.88 3.06

0.80 2.21 0.63 0.49

0 0 0 1

12

ARBY AYDN ERBD UZEL

13.48 11.79 18.45 16.17

8.04 10.17 7.74 8.15

20.63 29.88 29.67 2.14

38.74 32.21 31.26 48.27

4.14 3.67 5.73 4.99

2.94 3.72 2.86 3.00

1.47 2.13 2.12 0.15

3.75 3.12 3.03 4.66

1.68 1.21 3.18 2.48

0.02 0.79 0.10 0.06

0 0 0 1

Table 4. The correctly identified object points for the scenarios 1, 2, 3 and 4 Identified Points

Direction Vertical Direction

1.Scenario

2.Scenario

3. Scenario

4. Scenario

AYDN, ERBD

ARBY, AYDN, ERBD

-

ARBY, AYDN, ERBD, UZEL

Horizontal Plane

Table 5. The correctly identified object points for the scenarios 5, 6, 7 and 8 Identified Points Direction

5.Scenario

6.Scenario

7. Scenario

8. Scenario

AYDN, ERBD, UZEL

UZEL

ARBY, AYDN, ERBD, UZEL

ARBY, AYDN, ERBD, UZEL

Vertical Direction Horizontal Plane

Table 6. The identified object points for the scenarios 9, 10, 11 and 12 Direction

9.Scenario

10.Scenario

11. Scenario

12. Scenario

Vertical Direction

-

-

AYDN

AYDN, ERBD

Horizontal Plane

-

-

ARBY, AYDN, ERBD, UZEL

ARBY, AYDN, ERBD, UZEL

Table 7. The performance of quadratic KDA 19

Linear loaded model

Performance (%)

Scenario

Performance (%)

Scenario

Performance (%)

Large Magnitude

Scenario

Small Magnitude

Performance (%)

Large Magnitude

Scenario

Small Magnitude

Quadratic loaded model

Up

1st

50

2nd

75

3rd

0

4th

25

North-East

5th

75

6th

25

7th

75

8th

75

North-East-Up

9th

0

10th

0

11th

0

12th

25

Loaded Component

20