Detecting the Unstable Points in Deformation Monitoring Geodetic Networks in Analysis Method of Subnetwork

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Introduction
Nowadays behavior evaluation of big and sensitive structures such as dams, power plants and towers is of very high importance.Behavior survey of these structures is usually done in two geotechnical and geodetic ways.In geotechnical procedure, gauge tools of tension, shear and deflection inside the structure are installed during configuration and the data resulted from these gauges are continuously studied during and after optimization the structure in order to stability control.These tools provide the possibility of internal control of structure.In geodetic procedure, a network of points is created on the body and around the environment of structure and is monitored and controlled through geodetic observations mostly the length, angle and coordinates in different epochs.These observations provide the possibility of the deformation monitoring of the outer structure.
One of the most important geodetic network applications is deformation monitoring network or microgeodesy network.One of the important issues in these networks is the detection of stable and unstable points of network.First, by detection of stable points it can be reached to unit definition of coordinate system in two times epoch.Secondly, the deformation amount of unstable points can be calculated.If stable points are not properly identified, the calculated deformations will not valid for unstable points.In other words, the calculated deformations for network points due to the deformation of network coordinate system and isn't due to real deformation of these points.As a result, the calculated deformations can't be trusted.That is why the need for using appropriate productive, effective producer is felt to the detection of stable and unstable points.
In general, to consider the deformation of a network points, the awareness of points situated in two times periods and comparison of them is needed.To estimate the coordinates of the network points in each EPOCH, the least squares procedure is used.The Conventional Deformation Analysis (CDA) compares the difference of network points coordinates between two different epoch using statistical tests.If the coordinate difference is statistically significant, then the unstable point is detected.Besides, the Least Squares Estimation (LSE) is an optimal estimator, provided that the observations follow a normal distribution function.In other words, there was no wrong data in data set because LSE spread the effect of wrong data on the rest of the observations.Unfortunately, the unstable point in microgeodesy discussion also acts like error data.Therefore, the efficacy of the unstable point deformation on other spread stable points and estimated deformation will not be according to reality.In other words, when the point displaced in adjustment with the least squares procedure, the deformation effect of that point is not the only effects on that point but also effects on other points that is due to the spreading property in LSE (Chen, Kavouras & Chrzanowski, 1987), Kuang, 1996;Prószyñski, 2000;Schwarz & Kok, 1993).
Such as Conventional Deformation Analysis (CDA) it can be referred to Test method of global congruency test and Iterative Weighted Similarity Transformation (IWST) to find the stable points in networks.The iterative weighted similarity transformation (IWST) is done in two ways.A) Minimization L1-norm of deformation vector (LAS-L1).Minimization the second norm of deformation vector (LAS-L2).The L1-norm minimization is a powerful mathematical tool to identify the errors in mapping data.Calculation of minimization method had acceptable results.Sensitivity the first norm to the second one is in more deformation and deformations are clearer in this norm.The property of this procedure is in which it shows less sensitivity against apparent errors than the typical procedure of the least squares.In the L1-norm minimization method the efficacy of points is weighted in network datum and this weight will be appropriate to coordinate difference inverse of each point.These operations will continue because of being iterative as much the datum will be stable.It can be found by statistical test that which points have been moved at epoch time intervals.In another type of classic method i.e. the whole network stability test, first adjustment of the observations of two networks epoch is done separately.Then, it is determined by removing the efficacy of all points in datum of two networks epoch and doing statistical test that which points in datum is caused more stability.This result is obtained by testing of the whole network stability.Comparing two ways of whole network stability and minimization the first norm that was conducted by Jazayeri (1378) showed that minimization the L1-norm of the deformation vector, has more power in detection deformed points regarding to the whole network stability.
Such as new performed surveys in the field of microgeodesy, it can be referred to the performed studies in (Erdogan & Hekimoglu, 2014;(Hekimoglu, Demirel & Aydin, 2002;Hekimoglu, Erdogan & Butterworth, 2010).The results of performed simulations in these studies indicate that the detected points by Convention Deformation Analysis (CDA) procedure methods are not always properly diagnosed.Due to the nature of errors spreading at the least squares the results of this study also have shown that if the network is only included one unstable point, good results will be achieved and the efficacy of errors spreading in the least squares estimation will be minimized.
According to the obtained results, it is proposed to increase the efficiency of the classic methods in microgeodesy including global congruency test method and the L1-norm minimization dividing the network to several subnetworks is used that each subnetwork includes one subject point and other reference points.Then, each of subnetworks is analyzed separately.In the present study, the data has also been implemented on simulated data on a real network.This article contains five main parts and section.The first part discusses the introduction.The second section offer all common ways including overall network stability tests in (2-1) and the L1-norm minimization method in (2-2).The following part would be the one offered by the author himself, called subnetwork analysis method.Numerical results will be presented in section four.Section (4-1) will provide obtain results from simulated data.Section (4-2) is allocated to discuss results obtained from the real deformation monitoring network.The last section is a place to state comprehensive results and discussion

Conventional Deformation Analysis (CDA) Methods
In this section testing procedures of the global congruency test and L1-norm minimization method will be explained.Proceeding is in a way that adjusting the network in internal constraints method in two epoch observations is done separately.In this case assume that for first and the second epoch respectively we have ( ^ , ^ , , ) and ( ^, ^, , ) where ^ is adjusted coordinates, ^is variance covariance matrices of adjusted coordinates, is the freedom degree, is the estimated secondary variance factor and i is the considered epoch (first or second).In order to consider the survey consistency of two epoch observations, the amount of the secondary variance factor in two epoch should be tested.Therefore the following test estimation will be used (Cooper, 1987;Chen, Chrzanowski & Secord, 1990;Caspary, 1987): Where in the meaningful level , zero hypothesis will be accepted if < , , .Otherwise, zero hypothesis is rejected and represent the conflict between two epoch observations that often are the reason for this misinterpretation of variance factor or observations weight in adjustment.After doing the above test, in the adjusted coordinates in first epoch ^ with covariance matric ^ and second epoch ^ with covariance matrice ^ , the external deformation vector is defined as: = − (2) And covariance matric with correlation assumption ^, ^ is abtained from following equation: 3) The next phase is the testing of apparent deformation in network.So, the following statistical test is used.
The test estimation use to confirm the meaningful deformation is as follows: Where in this equation h = rank ( ^), = is the collective variance factor, = + is the whole freedom degree and , is fisher distribution.Besides, the sign of "+" indicates Moore-Penrose invers of covariance-variance matric.The statistical test (5) indicates the stability of network (Pelzer, 1971).If > , , , then E( ^)=0 i.e. the deformation is not meaningful, but if w> , , , then E( ^)≠ 0, so the zero hypothesis is rejected and the deformation is significant.In other words, the unstable datum will be detected.If in a network the deformation is significant using one of the methods of global congruency test and L1-norm minimization, first, the reference points will be detected and then the deformation values are calculated

Overall Network Stability Tests
This method is from the oldest methods of stable point's detection in deformation network that now is also used.For example, one of cases that this method has been used in order to the detection of points stability has been provided in reference (Hekimoglu, Demirel & Aydin, 2002) that in this paper, it has been used to analyze the deformation of vertical network after adjusting this method.
If datum detected unstable (test 5 rejected), according to share of all points in datum, it is cleared that at least one point of this network has displacement in an internal of two epochs.First, internal constraints datum is placed between all points on network.Now it is required to find a point that has the most deformation and delete its role from datum.Therefore the datum with internal constraints between the remaining points (except deleted point or points) is selected.Now the share of different points on estimation ω should be found.To identify the point that causes network instability, the deformation share of each point such as i in formation (5), is calculated using the following equation: Where ^ variance-covariance matric of point and deformation vector of each point is attained by following equation: Assuming the point 1 ≤ ≤ > that m is the number of network points has the highest amount for Ω, this point may be one of the deformed network points.The most important point is that necessarily the biggest Ω is not belonging to the biggest ^ or the biggest ^.The effect of this suspected point that has the biggest Ω should be removed from datum and ω retested.In the event of rejection again, ω on all remaining points of above steps is repeated and the point is removed from datum again.This process continues unit ω under the relationship test is not rejected.To consider other points that may have been displaced, first, it is required the point of j is removed from datum definition.To achieve this goal, instead of the columns related to this point in internal constraint datum matric, zero is place.To avoid the repetition of adjustment calculations, similarity transformation can be used (Cooper, 1987).A more detailed description is available in references (Baarda, 1981;Teunissen, 1985).Therefore, the deformation vector in new system and transformed variance-covariance matric respectively, are calculated by the following relations.
That in this equation the similarity transformation is = − ( ) .
In this equation, matric H is internal constraint datum.In relation ( 8), ^ is closer to real network deformation.
Now, the point that has the highest value in ^ (or the same point j) and has variance-covariance in matric is removed and returned again to the second step i.e. ω test and the test are as follows: where ℎ = ( ^).If , , there will be a significant deformation in network again.Therefore, the explained cases are repeated again unit ω test estimation is not rejected.The global congruency test method is explained completely by (Van Mierlo, 1978) and (Niemeier, 1981).Also, this method is performed by (Erdogan & Hekimoglu, 2014).An interested reader can refer to mentioned references or (Cooper, 1987).

The L1-norm Minimization Method
In this method, the coordinates system is selected that the length of deformation vector is minimized that is called stable coordinates system (Chen, 1983).The base of this method is based on the L1-norm minimization of deformation vector.In other words, in this method, between available datum, a datum is selected in which the first of deformation vector is minimized.In mathematical way: (11) Therefore, after the determination the vector d, this vector is transformed to different coordinates system to clear in which system the L1-norm deformation vector has the least value passible.Again, to avoid of adjusting after the changing of coordinates system, the similarity transformation of S is used as ^= ^ (relation 8) in which Datum matric D is determined as follows: Where ω is the weight matric for network coordinates system.As a result, by the placement of equation 12 in relation of similarity transformation S we will have: By repeating this procedure, the effect of unstable points in coordinates system is reduced.At first stage, weight matric is matric I named =I, where deformation vector will be ^= ^.Considering that it is necessary that the weight of datum points with displacement has inverse proportion, matric is a diagonal matric that components of main diameter is ^( ) .Due to it is possible that deformation components is some places or at least one of zero coordinates components is obtained, the denominator will be zero, therefore, it is necessary to avoid of zero denominator.In fact, the weight matric structure in the next irritations changes as follows: Where is a small positive number that avoid od zero denominator.By this definition of w, in fact, the points that has small deformation will acquire a larger share in Datum an reverse the points that are with larger deformation, has will the smaller share in Datum.i.e.
The important point in the provided process in 15 relations is the similarity transformation S on each of ^or first ^applied, it has no difference, because = , named the similarity transformation S is the transformation of that exponent.The repetition continues until the vector difference is reduced in two successive stages than a desired specified limit, i.e.
^( ) − ^( ) that usually a small number is selected for i.e 0.5 or 0.1 mm.finally after determining the weight points in network datum named completing the relations loop (15) that selected the final datum in a way that the L1-norm of the points deformation vector is minimized, the detection test of stable an unstable points as follows: Where k is the point number and is coordinates components of deformation vector related to each point after transformation by helping of the similarity transformation S each C is equal to network to network dimension.This test is done on the individual points to determine each point is stable or not?If f 1 >f c,df,a the point is displaced (is removed as unstable point), otherwise the point is stable.More details about this method is available in (Chen, 1983;Chen, Chrzanowski & Secord, 1990;Setan, 1995;Setan, & Sing, 2001;Taşçi, 2010) resources.

Subnetwork Analysis Method
The conventional deformation analysis (CDA), the coordinates statistical test is compared.Id coordinates difference shows deformation title.In the previous sections, two examples of CDA common methods were explained.The results of performed simulations in recent researches suggests that detected points by CDA is not always identified correctly (Hekimoglu, Erdogan & Butterworth, 2010).That is the reason can be the same as spreading property of LSE.Because the importance of this section and using of proposed idea in this research, the summary of research (Erdogan & Hekimoglu, 2014) and (Hekimoglu, Erdogan & Butterworth, 2010) is presented.
In paper (Hekimoglu, Erdogan & Butterworth, 2010), the obtained results for performed simulations (simulations in 1000 times) shown when applied deformation is small, the number of correct points detection using of CDA methods is less than the number of points that were really displaced.The reason of it can be LSE spreading property (Hekimoglu, Erdogan & Butterworth, 2010).Therefore, the optional applied deformation degree between r and 2r and again between r and 3r is selected.(r) is a number that by changing it, the success rate of correct detection from unstable points is 81.3 percent that is a value for efficiency measurement of CDA methods when a point is displaced and optional applied deformation is between rand 2r.These results are not satisfactory, the results of this study also indicated that if the network only has one stable point, good results is achieved and the efficacy of LSE spreading is minimized.Therefore, according to this point, the idea that has been presented in this paper is the division of overall network to multi-subnetwork so that each unstable point is placed on individual subnet and then is checked.In fact, every subnet includes one subject point and other referenced points.In this method, the relation between unstable points is interrupted from each other and finally effect of spreading the least squares is removed.Through this method, if there are 3 subject points (A, B, C) in a network, so we will have subnet i numbers of subject points.Subnetwork I is subnet that only includes one and other reference points (not include B, C points).At the same way, the second subnetwork includes B point and other reference points and the third subnetwork includes C point and other points.Each subnet is analyzed separately.What is clear from the obtained results is that in network division to subnet, better results have been obtained.Therefor subnet method is advised rather than the global congruency.In this stable to increase the efficiency the present classic method in micro geodesy such as the global congruency test, network division to multi subnet has been used.Then each is adjusted separately in internal constraints method and classic method of the overall stability test to detect stable and unstable points of network has been used.These calculations in each epoch, once for overall network and another time for three subnetworks are done and each of them has been compared to their first epoch.What is presented from the results and better results has been obtained from network division to subnet.According to obtained results is this paper, the efficiency survey of present classic method in microgeodesy such as overall stability test method and L1-norm minimization in two ways of overall analysis and subnet analysis using simulated date and real network observation will be discussed.

Numerical Results
In this section, the global congruency test methods and the L1-norm minimization in two ways of subnet analysis and the whole network analysis will be compared.Therefore, simulation observations of GPS three dimensional network have been used.Moreover, the possible displacement will be determined by using real network observations and GPS observations.

Simulated Data
In order to evaluate the performance of proposed method from subnet analysis compared to current method of overall network analysis, simulated data can be used in which first, deformation points, rates and deformation direction are applied optionally then the rate and percentage of correct detection of unstable points will be determined by using the algorithms of unstable points detection.According to common usage of GPS observations is microgeodesy, the applied simulated observation in this study, is GPS length-based observation.The network is a three-dimensional network including 8 points (5points as reference or stable points and 3 subject points).All present base lines are considered between points as the observations that total number of base lines is 28 (equal to 84 observations) in discussed 8 point network.To ensure the results, the observations are simulated 1000 times.In order to create simulated observations in two epochs, first the observations without error are calculated, and then the error rate with normal distribution to zero mean and standard deviation (SEM) 5 mm is added.Therefore, these observations are the first epoch observations.After making the first the first epoch observations, it is required to change the point and make simulated observations for second epoch according to explained method.It is considered different ways to simulate that related results to each simulated ways in sections 3.2.1,3.2.2,3.2.3,3.2.4 will be presented respectively.

Regular Network with Definite Deformations
In the first state, a regular network including 8 points (5 points as reference or stable points and 3 subject points) was presented.Reference points of this network (points 1-5) are placed on a circle with a radius 200 and in the center (500,500).The displacements of deformed points were supposed zero and the obtained were examined.Table 1 indicates the coordinates of network simulated points.Figure (1-A) indicates a view of simulated network.In this figure, the points 1-5 are reference or stable points and points OBJ1, OBJ2, OBJ3 are subject points.After the simulation of observations in two epoch in three different scenarios, individual adjusting of two epochs in order to usage of overall stability test methods and minimization of L1-norm in two ways of overall network analysis and subnetwork is done to detect displaced points.The approach is that first the network is adjusted in internal constraints method then classic method of global congruency test and L1-norm minimization have been used to detect stable and unstable points of network.These calculations in each epoch are done once for the whole network and another time for three subnetworks and each network is compared to its first epoch.Then the detected points by these two methods are compared with the points that are really displaced.In this way, the performance of these two methods will be compared with each other in unstable point's detection.As it said, in order to ensure the results, it will be repeated 1000 times.Table (3) indicates the summary of results in unstable point's detection in two global congruency test method and L1-norm minimization in two ways of overall network analysis and subnetwork in all three scenarios.As it is clear from table (3), in the first scenario in using of subnetworks instead of overall network, in both Global congruency test method and in the L1-norm minimization method, the improvement will be achieved.Especially in Global congruency test, the improvement is equal 20 percent and in the L1-norm minimization method the improvement equal to 3 percent will occurred.Also in second scenario in Global congruency test method the improvement will equal to 18 percent and in L1-norm minimization method it will equal to 2 percent.Finally, in the third scenario in Global congruency test method the improvement will be 22 percent and in the L1-norm minimization method it will be 2 percent

Regular Network with Random Deformations
In this part we will investigate the results on the same designed regular network of pervious section ended with random deformations between 8 to 10 mm (positive or negative that in each program running will be randomly selected).Here, three different scenarios are performed and compared with the first epoch.
As it is clear from table (5), in the first scenario in using subnetwork instead of overall network, the improvement will be occurred in both global congruency test method and L1-norm minimization method.However, this improvement in L1-norm minimization method is 5% percent but in global congruency test method it is increased 21 percent.In second scenario also using subnetwork instead of overall network is caused the improvement in the results or detection percentage in both global congruency test method and L1-norm minimization method.In this section, the improvement in L1-norm minimization is 8% but in global congruency test method is increased 15%.Therefore, it can be said that using subnetwork in global congruency test causes to results improvement.Finally, in third scenario also in using subnetwork instead of overall network in both global congruency test method and L1-norm minimization, the improvement will be achieved.The improvement of global congruency test method is about 22 percent and L1-norm minimization method is about 7 percent that is not so significant.

Irregular network with random displacements
In this section, an irregular network including 8 points (5 points as reference or stable points and 3 subject points) is designed.The deformations of random points will be placed between 8 to 10 mm (positive or negative).In this section also three different scenarios are performed according to 4-1-2 section.Table 6 indicates the results summary in unstable point's detection in two ways of global congruency test method and L1-norm minimization method in two methods of overall network analysis and subnetwork in all three scenarios.As it is clear from table 6, in first scenario in using of subnetwork instead of overall network, global congruency test method an improvement about 36 percent has been achieved.In second scenario also, the use of subnetwork instead of overall network will be about 37 percent only in global congruency test method and is caused an improvement in results and detection percentage.Finally, in third scenario in using subnetwork instead of overall network in global congruency test method, an improvement about 84 percent will be occurred while in L1-norm minimization, apparent difference is not considered.Finally, according to the tables (3,4,5,6) the following results were obtained: A) According to the results of other studies, the L1-norm minimization method is of better capability than Global congruency test method in overall network analysis (the current methods) in the detection of unstable points.
B) Using of subnetwork method instead of overall network is caused an improvement of results in global congruency test method especially when all three subject points are displaced, in all simulated ways (regular and irregular network with definite and random displacements), the improvement of correct percentage results of point's detection is This improvement is averagely about 35 percent in all methods.However, improvement in L1-mnorm minimization is about 1 percent that cannot be considered as a significant improvement.

The real deformation monitoring network
After presentation of the results related to the simulated data, now the performance of proposed method is examined using the real observations of deformation monitoring network.GPS created network around Kabudval dam is located in Golestan province.The used microgeodesy network in this study includes 6 pillars (KL1,KL2,KL3,KR1,KR2,KR3) that two pillars in left side, one pillar in right side, one pillar above the dam were created.The process is that observations between network points by GPS receiver is collected and processed.First, the observations of two epochs were adjusted separately and then were tested by classic methods of unstable point's detections of global congruency test method and L1-norm minimization method.-0.

Summary and Conclusion
In this study the practical survey of efficacy of two stable point's detection (global congruency test method and L1-norm minimization of deformation vector) in deformation monitoring networks as subnetwork analysis has been evaluated.For this purpose, global congruency test and L1-norm minimization were performed in two overall network and subnetwork analysis on some simulated data and the results were compared.The simulated observations, GPS length base observations were considered.As it is clear from the results taken from simulated network, the use of subnetwork analysis method instead of global congruency test method will cause to the improvement of the results.In subnetwork analysis method in all simulated forms (regular and irregular network with definite and random deformations), the improvement of correct results of points detection is considerable percent.This improvement in all simulated forms is averagely about 35 percent.According to the improvement of subnetwork method results is about 1 percent than L1-norm minimization that cannot be accounted it significant.In following unstable points detection algorithms, the current methods and subnetwork analysis method on observations of a real network around Kabudval dam located in Golestan province were performed that the obtained results are according to simulated network results and in the end, deformation level of detected unstable points was calculated

Figure
Figure 1. ( first subne Figure (2) indicates the location of these stations.The performed observations are GPS length based.The used receivers of two frequency satellite GPS receiver type are of system 500 and 1200 Leica.The observations of the first and second epoch respectively are done in the dates of 17 to 22 December 2012 and the early March 2013, i.e. at about three months interval.

Table 2 .
In

Table 2 .
The point's deformation in meters in different scenarios

Table 3 .
Correct detection percentage of unstable points in regular network with definite deformations

Table ( 6
) -correct detection percentage of unstable points in irregular network with random displacements