Squared M_{split(q)} S-transformation of control network deformations
Abstract
Identification of stable potential reference points (PRPs) is the most critical stage of computations in conventional deformation analysis of geodetic control networks. An appropriate matching of two adjusted networks at stable PRPs plays a key role in this task. Unfortunately, the geodetic control networks are free networks suffering from datum defect and can realize infinitely many possible matchings at PRPs. Therefore, accurate estimation of PRP displacements and later efficient identification of stable PRPs is a quite difficult task. This study makes some step forward in this field and presents a new approach to deformation analysis, including the identification of stable PRPs. The idea behind this approach is inspired by the theory of squared M_{split(q)} estimation and lies in the non-conventional assumption that estimated displacements of PRPs can be the realizations—not of one but—of many congruence models, which simultaneously realize many different matchings. Displacements of unstable PRPs in such a multi-split congruence model do not have such a negative effect on expected matching at stable PRPs as in the conventional robust S-transformation. Here, these displacements can be realizations of other congruence models and their attention can be absorbed by other, unexpected, matchings. Thanks to this, the robustness of the suggested approach can be relatively high. To establish what the number of congruence models is in a given case, which model is the one expected and whether the chosen model is valid, the statistical hypothesis tests were proposed. The experiments performed on 1D and 2D simulated control networks showed that the presented approach can provide more accurate values of estimated displacements than conventional approaches, and in consequence, more efficient results of stable PRPs identification, especially when there exist more unstable PRPs than stable ones. In light of the above, the correct identification of stable PRPs and, in consequence, the correct final estimation of controlled object point displacements are possible in cases when it has not been possible so far.
Keywords
Geodetic deformation analysis Point displacement Robust M-estimation Squared M_{split(q)} estimation S-transformation1 Introduction
One of the important functions of surveying and geodesy is the deformation monitoring of engineering structures and the surface of Earth’s crust. The object or area under investigation is usually represented by a geodetic control network which is measured in two or more epochs of time, and the results of these measurements are then analyzed. The models used in this analysis may be categorized into two groups: descriptive models (Pelzer 1971; Niemeier 1981; Chen 1983; Caspary 2000), which are employed in conventional deformation analysis (CDA), and dynamic models (Papo and Perelmuter 1991, 1993; Shahar and Even-Tzur 2014). The descriptive models may also be divided into congruence models (also called static models) which geometrically describe the deformations by means of displacement vectors and kinematic models which temporally describe the deformations by means of displacement velocities and accelerations. Generally, the descriptive models exclusively examine deformations without regard to their influencing factors and the object’s physical properties. While the dynamic models are the extension of the descriptive models which link deformations to their influencing factors (causative forces, internal and external loads) and the object’s physical properties (material constants, extension coefficients, etc.) (Welsch and Heunecke 2001), the typical geodetic models are congruence models which are employed in CDA. In many geodetic applications, congruence models provide completely sufficient information about a deformable body, its change in shape and dimension, as well as rigid body movements and local deformations (Chrzanowski and Chen 1990; Caspary 2000).
The simultaneous analysis of observations from two epochs is performed in congruence models. It is usually done between any two consecutive epochs and, additionally, between the first and the current one. The matching of objects (represented by control network points) between both epochs is carried out on the group of mutually stable points (the group which has a congruent/rigid geometrical structure at both considered epochs). However, the main problem is the identification of such a group in the usually specified group of potential reference points (PRPs). The geodetic control networks are free networks suffering from datum defect (e.g., Chen 1983, ch.4), whereby correct identification of mutually stable points is rather difficult and it can be even impossible with a large number of unstable points. An erroneous identification leads to erroneous defining of datum for estimated object deformations and, in consequence, to disinformation. The identification of mutually stable points is the only serious problem in congruence models, and it is still the subject of interest for surveyors and geodesists (Baselga and García-Asenjo 2016; Amiri-Simkooei et al. 2016; Aydin 2017).
Today, mutually stable points can be identified iteratively with a global congruency test (GCT) or with a robust M-estimation with a high level of reliability. The first iterative step is the same for both approaches, and it involves the least squares estimation of PRPs displacements (the so-called raw displacements). A minimum trace datum is here defined for all PRPs. In the first approach, unstable PRPs are successively removed from the computational base until all the unstable points are identified. The test statistic can be based on estimated displacements of PRPs (Pelzer 1971, 1974; Niemeier 1981; Denli and Deniz 2003) or residuals (Kok 1982; Heck 1983; Gründig and Neureither 1985; Hekimoglu et al. 2010). A functional model forces zero displacements of PRPs in the second method; hence, any point displacements and measurement errors are “pushed out” in values of residuals. Despite the completely different algorithms, both methods give equivalent results. In a robust approach, which is especially useful in the deformation analysis of large and very large control networks, unstable PRPs are not removed from the computational base, but their participation in datum is iteratively suppressed. The PRPs displacements from the final S-transformation are then tested, one by one, to identify unstable points. This approach can be implemented by two methods: robust S-transformation of coordinates differences (Chen 1983; Caspary and Borutta 1987; Chen et al. 1990; Caspary et al. 1990; Caspary 2000, p.130–132) or robust M-estimation of observation differences (Nowel and Kamiński 2014; Nowel 2015, 2016a). The literature also provides new interesting approaches, such as the modern non-iterative GCT approach, where all considered subsets of non-congruent point patterns are tested (Velsink 2015, 2018; Lehmann and Lösler 2017), or the R-estimation approach (Duchnowski 2010, 2013). Furthermore, a completely different, non-conventional approach was recently presented by Zienkiewicz (2014), Zienkiewicz and Baryla (2015), Wiśniewski and Zienkiewicz (2016). The authors do not identify mutually stable points and start an estimation of an object deformation straight away. Datum is defined on all PRPs with the zero displacement condition (the so-called rigid datum instead of minimum trace datum). Any displacements of PRPs appear in residuals and are separated with M_{split} estimation. However, this approach is quite limited in practical applications because it does not tolerate a large number of displaced object points.
Generally, the GCT approach and the robust approach yield very comparable results (e.g., Caspary 2000, p. 149–154). These approaches are universal, most mature and commonly regarded as the most effective tools used to identify mutually stable points. It is noteworthy that these approaches dominate not only in a deformation analysis, but also in other similar issues of geodesy, e.g., in the identification of observations without outliers. Unfortunately, certain situations are known to exist in which even these methods fail. For example, if the number of outliers is much larger than that of the expected realizations in a set, then such methods—especially the robust approach—may fail (Hampel et al. 1986, p. 12; Koch 1996, 2010, p. 263). This also applies to identification of mutually stable points in congruence models, where displaced PRPs may be treated as outliers, and stable PRPs as expected realizations. Whereas this paper presents and pretests a completely different approach to the identification of mutually stable points, unlike in the conventional congruence model, unstable points are not suppressed or deleted from a computational base. This approach is based on the concept of the M_{split(q)} estimation and allows for the simultaneous existence of many congruence models which differ by the datum parameters. The unstable points do not have such a negative effect on datum parameters of expected congruence model as in conventional approaches because these points can be the realizations of other congruence models and their attention can be absorbed by other datum parameters. Thanks to this, the robustness of the suggested approach can be relatively high. The presented idea of applying M_{split(q)} estimation in deformation analysis is completely different from the one presented in the literature and referenced earlier in this section. Most of all, that simple concept, based on the M_{split} estimation, does not deal with the identification of stable PRPs and focuses only on observation residuals disclosing some information about unstable points, which can be treated as outliers.
The next part of the paper is organized as follows: Sect. 2 describes the conventional robust approach to the S-transformation of deformations. Particular attention is paid to a mathematical congruence model, an optimization problem which corresponds with this model and a solution for it. Section 3 demonstrates the general idea of M_{split(q)} estimation. Section 4 presents, with reference to Sects. 2 and 3, the suggested approach to the S-transformation of deformations. A research motivation is presented in Sect. 4.1. The next subsections focus on a suggested multi-split mathematical congruence model, an optimization problem which corresponds with this model and a suggested strategy for its solution. A validation method for the solution is presented as well. Section 5 is devoted to numerical experiments to demonstrate the suggested concept against conventional ones. Section 6 gives the conclusions from the study.
2 Robust S-transformation of deformations
The approach proposed will be described in Sect. 4 in some reference to robust S-transformation; therefore, this section will be devoted to this approach.
Robust S-transformation, especially the iterative weighted similarity transformation (IWST) (Chen 1983; Caspary and Borutta 1987), was used for deformation analysis of the Tevatron atomic particle accelerator complex at the Fermilab laboratory in the USA (Bocean et al. 2006). This method was also implemented in the automated ALERT monitoring system developed by the Canadian Centre for Geodetic Engineering (Wilkins et al. 2003) and in the universal GeoLab geodetic computation software (Chrzanowski et al. 2011). A robust S-transformation is especially useful in the deformation analysis of large and very large control networks, such as the network of Tevatron accelerator, which comprised nearly 2000 control points. With such networks, calculations by this method are much more convenient than with GCT, and the quality of results is at a similar, satisfying level.
2.1 Congruence model
In a geometrical sense, the idea behind the robust S-transformation approach is based on the matching of both adjusted networks at mutually stable points to disclose the unbiased values of possible single point displacements. It means that one estimates such a vector of datum parameters for displacement vector, \( {\hat{\mathbf{t}}} \), which realizes a network congruency at mutually stable points and, in consequence, clearly discloses possible single point displacements in the residual vector, \( {\hat{\mathbf{d}}} \). In other words, the raw displacement vector, \( {\varvec{\Delta}}_{x} \), is here transformed to such datum (by means of estimated datum parameters, \( {\hat{\mathbf{t}}} \)) which most clearly shows a deformation pattern, \( {\hat{\mathbf{d}}} \). However, the question remains: What estimation/matching method should be used?
2.2 Estimation
Obviously, the matching which is based on the ordinary S-transformation fails to detect the single point displacements. With the robust S-transformation, no problems arise as long as the majority of points conform to the congruence model (2), i.e., are mutually stable. However, when the majority of points will not conform to model (2) the robust S-transformation also might fail.
2.3 Statistical testing and final S-transformation
After stable PRPs are identified a final S-transformation for all the network points and their covariance matrices should be conducted to the minimum trace datum defined on the previously identified stable PRPs. The estimator of displacement vector still has the form of (6.2), but the weight matrix now fulfills the role of datum selector matrix and has the form \( {\mathbf{W}} = {\text{diag}}\left( {{\mathbf{I}},{\mathbf{0}}} \right) \), where ones concern the datum points and zeros other points. The deformations of the object points (including also the unstable PRPs), as computed from final S-transformation, form the basis for all further deformation analyses which may concern rigid body displacement, deformation tensor and/or polynomial deformation model. More information on the robust S-transformation approach can be found in the papers: Chen (1983), Chen et al. (1990), Caspary and Borutta (1987), Caspary (2000, p. 130–132).
3 The general idea of M _{split(q)} estimation
Let us assume that the above random sample contains “good” values: the points in the middle, and two extreme groups of outliers: the other points. The robust M-estimation treats two extreme groups as outliers and the middle group as “good.” However, the estimator of location parameter is not fitted well into the “good” values because it is slightly absorbed by the left outliers. A similar solution comes from the LS estimation, whereas the M_{split(q)} estimation allows to treat the random sample as the realizations of three random variables. In a simultaneous and joint optimization process, each estimator finds its values. Other values do not attract the estimator, because the other estimators absorb their attention. Note that the M_{split(q)} estimation for two random variables (q = 2) is treated as a special case of the presented theory and is called the M_{split} estimation (Wiśniewski 2009).
More information, e.g., the solutions of the above optimization problems using Newton’s iterative procedure, can be found in Wiśniewski (2009, 2010).
4 Squared M _{split(q)} S-transformation of deformations
4.1 Motivation
However, a question arises: How would the M_{split(q)} estimation behave in such critical situations?
Keep in mind that the raw displacement vector forms a random sample in deformation congruence model (2). It has been pointed out that the attention of a given M_{split(q)} estimator is absorbed only by the values of a random sample which are realizations of a given random variable (values of random sample which well—in some sense—fit to this estimator) and other values do not have a negative effect on this estimator. Therefore, it can be expected that when the right calculation strategy is followed, one of the M_{split(q)} estimators should identify good datum parameters for displacement vector (matching both adjusted networks at stable points), even in critical situations. The thought experiments results suggest that this hypothesis may be plausible, and they inspire this research.
4.2 Congruence model
In the robust S-transformation approach one assumes that the majority of points in the PRP group are stable and only individual points may be unstable. This is a sufficient condition to obtain an expected datum for displacement vector (expected matching). In a probabilistic sense, one assumes that the raw displacement vector of PRPs consists mainly of values which realize the random variable with accepted normal distribution (stable points) and there are only some outlying values which realize the random variable with some other unaccepted probability distributions (unstable points), (2) and (3).
In deformation reality, the multi-split model (15) might describe the scenario where several subgroups of PRPs may regularly displace each other and the points are mutually stable inside each subgroup. Then, each j local model realizes the congruency of one subgroup of such points. It is the most intuitive scenario for model (15). However, this model may also describe the extended scenario where additionally single point displacements exist. Then, the congruences of single points may also be realized by means of additional models. For example, if the group of PRPs consists of two different displaced subgroups with mutually stable points inside each subgroup and three unstable single points, model (15) should be theoretically split into five local congruence models (q = 5).
It is also worth noting that the multi-split model (15) can be interpreted as the collection of mean shift (MS) models which are considered in the statistical testing theory which is also widely applied in geodetic deformation analysis. Of course, these models play a quite different role in the suggested approach.
4.3 Estimation
The algorithm for the displacement vectors (22.2), in a more detailed version, has the following form (k = 1,…).
4.4 Number of models
In general, it is assumed in the theory of M_{split(q)} estimation that the number of competitive mathematical models, q, is known a priori (Wiśniewski 2010). However, the number of competitive models, differing by datum parameters, \( {\mathbf{t}}_{(j)} \), is not known a priori in our multi-split congruence model (15). Although stable PRPs realize one congruence model, displaced PRPs can realize many models. Theoretically, each displacement may even realize a different congruence model. Now, the crucial question is: How many congruence models should a split model (15) contain in a given case?
Establishing—not known a priori—the appropriate number of potential mathematical models is still one of not properly solved problems of M_{split(q)} estimation. Only one solution to this problem has been presented so far (Wiśniewski and Zienkiewicz 2016). In most general terms, the authors propose that the appropriate number of models should be determined with a certain control value, added to values of the original random sample. However, this solution does not yield satisfying results for a congruence model (15); therefore, a different solution is proposed in this study, based on hypothesis testing.
If the multi-split null hypothesis (26) passes for q = 2, it may be assumed that all estimated PRP displacements realize two congruence models and additional absorbing models are not needed. Otherwise, it may be assumed that estimated PRP displacements realize at least three congruence models, q ≥ 3, and now three models should be tested. If the multi-split null hypothesis (26) passes for q = 3, it may be assumed that all estimated PRP displacements realize three congruence models and additional absorbing models are not needed. Otherwise, the splitting into four congruence models should be tested, etc. Finally, if the multi-split null hypothesis (26) passes, it may be assumed that all the estimated PRP displacements have found their absorbing congruence models. Splitting the congruence model (15) can now be completed.
4.5 Choice of the best model and its validation
If it turns out that there are unstable points in the group of PRPs and, in consequence, the number of congruence models is greater than one, q > 1, then one more question emerges: Which congruence model is expected, i.e., which realizes the matching at mutually stable PRPs?
The answer to this question can be trivial if it is assumed that a subgroup of stable PRPs is the most populated subgroup of mutually stable points in the whole group of PRPs. With this assumption, the congruence model with the largest number of statistically insignificant estimated displacements (28) should be the one which realizes the matching at mutually stable PRPs. It must be noted that the above sufficient condition for the proper solution of the squared M_{split(q)} S-transformation is less restrictive than in the robust S-transformation. This is because the conventional approach requires that the number of stable points should be greater than unstable ones. However, the approach presented here only requires that stable points should make up the most populated subgroup of mutually stable points. If this condition is met, then the presented approach should be effective, even if the number of unstable points is larger than stable ones.
A detailed explanation on the aforementioned derivations can be found in Teunissen (2006, ch.3-4).
The acceptance of the null hypothesis (29) indicates that the preliminary solution of squared M_{split(q)} S-transformation is valid, i.e., preliminarily identified stable PRPs can be treated as mutually stable. Otherwise, the preliminary solution cannot be valid and it should be rejected. It can be due to the collapse of the squared M_{split(q)} S-transformation, e.g., there can exist more congruence models than necessary. In this case, also the second-best congruence model and other models can give invalid solutions. Hence, for example, the solution of robust S-transformation may be recommended in such cases. However, the problem of collapse of squared M_{split(q)} S-transformation may be treated as an open issue and it deserves further research.
4.6 Final S-transformation
After a stable reference base is identified, it is suggested here that—like in the conventional robust S-transformation— the final S-transformation of all the network points and their covariance matrix should be conducted to the minimum trace datum, defined on the reference base, as described in Sect. 2. This will result in a better matching of both networks at stable PRPs than the original matching from the best congruence model of the squared M_{split(q)} S-transformation.
5 Numerical experiments
The following hypothesis was put forward in an earlier study: “If the subgroup of stable PRPs is the most populated subgroup of mutually stable points, then the squared M_{split(q)} S-transformation will identify stable PRPs.” This hypothesis was examined numerically in the paper based on simulated and real control networks. All sets of simulated observations were free of outliers. The displacements of PRPs were estimated with the squared M_{split(q)} S-transformation (22.2) and, additionally, with the robust S-transformation (6.2). The minimal L_{1}-norm of the vector of estimated displacements (5) was taken as an objective function in the robust S-transformation. The cofactor matrices of estimated displacements were always the one which realizes the minimum trace. For simplicity and without loss of generality, the significance levels for global and local hypothesis testing were assumed as α = 0.05 and α_{i} = 0.001, respectively, in each experiment.
5.1 Experiment 1
Results of the stable PRPs identification
Point | Simulated displacements, \( {\mathbf{d}} \)[mm] | Estimated displacements, \( {\hat{\mathbf{d}}} \)[mm] | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Robust S-transformation | Squared M_{split(q)} S-transformation (results of the best model) | |||||||||||
Variant | ||||||||||||
1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | |
Scenario 1: Displacements have different signs | ||||||||||||
1 | 0 | 0 | 0 | 0 | − 0.2 | − 0.2 | − 0.1 | − 0.1 | − 0.2 | − 0.1 | − 0.1 | 0.0 |
2 | 0 | 0 | 0 | 0 | − 0.2 | − 0.2 | 0.0 | − 0.1 | − 0.2 | − 0.1 | − 0.1 | 0.0 |
3 | 0 | 0 | 0 | 2 | 0.1 | 0.1 | 0.2 | 2.2 | 0.1 | 0.2 | 0.2 | 2.3 |
4 | 0 | 0 | − 4 | − 4 | 0.0 | 0.0 | − 3.9 | − 4.0 | 0.0 | 0.1 | − 3.9 | − 3.8 |
5 | 0 | 6 | 6 | 6 | 0.3 | 6.3 | 6.4 | 6.4 | 0.3 | 6.4 | 6.4 | 6.5 |
6 | − 8 | − 8 | − 8 | − 8 | − 8.2 | − 8.2 | − 8.1 | − 8.2 | − 8.2 | − 8.2 | − 8.1 | − 8.0 |
7 | 20 | 20 | 20 | 20 | 20.7 | 20.7 | 20.8 | 20.7 | 20.6 | 20.7 | 20.7 | 20.8 |
11 | 0 | 0 | 0 | 0 | − 0.2 | − 0.2 | − 0.1 | − 0.2 | − 0.3 | − 0.2 | − 0.2 | − 0.1 |
12 | 0 | 0 | 0 | 0 | 0.7 | 0.7 | 0.8 | 0.7 | 0.7 | 0.8 | 0.8 | 0.9 |
Number of congruence models, q | ||||||||||||
3 | 4 | 5 | 6 | |||||||||
Scenario 2: All displacements have positive sign | ||||||||||||
1 | 0 | 0 | 0 | 0 | − 0.4 | − 1.0 | − 4.2 | − 4.2 | − 0.2 | − 0.1 | − 0.1 | 0.0 |
2 | 0 | 0 | 0 | 0 | − 0.3 | − 1.0 | − 4.2 | − 4.2 | − 0.2 | − 0.1 | − 0.1 | 0.0 |
3 | 0 | 0 | 0 | 2 | − 0.1 | − 0.7 | − 3.9 | − 1.9 | 0.1 | 0.2 | 0.2 | 2.3 |
4 | 0 | 0 | 4 | 4 | − 0.2 | − 0.8 | 0.0 | 0.0 | 0.0 | 0.0 | 4.1 | 4.2 |
5 | 0 | 6 | 6 | 6 | 0.1 | 5.5 | 2.3 | 2.3 | 0.3 | 6.4 | 6.4 | 6.5 |
6 | 8 | 8 | 8 | 8 | 7.6 | 7.0 | 3.8 | 3.8 | 7.8 | 7.8 | 7.9 | 7.9 |
7 | 20 | 20 | 20 | 20 | 20.5 | 19.9 | 16.7 | 16.7 | 20.7 | 20.7 | 20.8 | 20.8 |
11 | 0 | 0 | 0 | 0 | − 0.4 | − 1.0 | − 4.2 | − 4.2 | − 0.2 | − 0.2 | − 0.1 | − 0.1 |
12 | 0 | 0 | 0 | 0 | 0.5 | − 0.1 | − 3.3 | − 3.3 | 0.7 | 0.8 | 0.8 | 0.9 |
Number of congruence models, q | ||||||||||||
3 | 3 | 4 | 5 |
For the displacements with different signs, very similar values of estimated displacements and always correct results of the identification of stable PRPs were obtained in both approaches. Both approaches yielded good results even in the critical variants 3 and 4, when there were 4/7 and 5/7 unstable PRPs, respectively. It was different for displacements with the same sign. When there existed more displaced PRPs than stable ones (variants 3 and 4), the robust S-transformation failed. Estimated and simulated values of displacements were considerably different and, in consequence, the identification of stable points gave completely wrong results. However, the squared M_{split(q)} S-transformation was reliable in each variant.
One can note that—unlike the conventional approach (red line)—the best congruence model in the squared M_{split(q)} S-transformation (green line) located the datum correctly, at actually stable PRPs (1, 2, 3) and these points were also identified as stable by means of local F-tests (Fig. 6 or Table 1). Thanks to this, the values of estimated displacements (4.1, 6.4, 7.9, 20.8 [mm], Fig. 6 or Table 1) of actually unstable PRPs (4, 5, 6, 7), i.e., the distances between these points and the green line (datum) in the above figure, were very similar to the simulated ones (4.0, 6.0, 8.0, 20.0 [mm], Table 1). Theoretically, the results would be even better for the final S-transformation to the minimum trace datum defined on the previously identified stable PRPs.
Furthermore, it is worth noting that only four congruence models turned out sufficient in the squared M_{split(q)} S-transformation, instead of five expected. (One subgroup of mutually stable points and four different displaced points give together five congruence models.) However, the values of simulated displacements are quite similar in the experiment under discussion; hence, the estimated displacements of points 5, 6 can be assigned to one congruence model (j = 3), and this is why the null hypothesis (26) has been accepted for already four congruence models in this variant.
5.2 Experiment 2
Average values of mean absolute true errors of estimated PRP displacements
Number of stable PRPs | Value of error, \( \bar{e}_{{\hat{d}}} \) [mm] | Number of valid solutions of squared M_{split(q)} | |
---|---|---|---|
Robust | Squared M_{split(q)} | ||
Scenario 1: Displacements have different signs | |||
4/7 | 0.52 | 0.41 | 964 |
3/7 | 1.35 | 0.52 | 956 |
Scenario 2: Displacements have positive sign | |||
4/7 | 0.98 | 0.43 | 989 |
3/7 | 6.89 | 0.74 | 987 |
Generally, the experiment results are similar to those of experiment 1. For displacements with different signs, both approaches gave similar, satisfying values of errors; only for the critical case, the robust S-transformation gave significantly higher errors, \( \bar{e}_{{\hat{d}}} = 1.35 \) mm. However, the robust S-transformation—more or less—failed in the case of displacements with the same sign. Average value of mean absolute true errors of all simulations was relatively large: \( \bar{e}_{{\hat{d}}} = 6.89 \) mm. Since the number of unstable PRPs was always greater than stable ones, the conventional approach always failed. However, the squared M_{split(q)} S-transformation failed only in 39/1000 cases (Fig. 7; the last part); hence, the average value of mean absolute true errors of all simulations was only slightly larger than earlier, in scenario 1, \( \bar{e}_{{\hat{d}}} = 0.74 \) mm. Those 39 cases, when the solution breaks down, can be explained by the cases not meeting the sufficient condition for the correct solution. Let us recall that the squared M_{split(q)} S-transformation requires that stable points should make up the most populated subgroup of mutually stable points. It is conceivable that in 1000 simulations, there were about a several dozen times when displacements of at least three PRPs were similar enough to be statistically regarded as a subgroup of mutually stable points and, in consequence, the datum parameter (here, only shift) chose this unstable location.
Results of the stable PRPs identification
Method | How many times was identified as stable: | |||
---|---|---|---|---|
4 points | 3 points | 2 points | 1 point | |
Scenario 1: Displacements have different signs | ||||
4/7 points are stable | ||||
Robust | 921 | 64 | 9 | 5 |
Squared M_{split(q)} | 982 | 16 | 2 | 0 |
GCT | 967 | 16 | 0 | 0 |
3/7 points are stable | ||||
Robust | – | 834 | 46 | 12 |
Squared M_{split(q)} | – | 962 | 28 | 3 |
GCT | – | 934 | 18 | 0 |
Scenario 2: Displacements have positive sign | ||||
4/7 points are stable | ||||
Robust | 712 | 215 | 47 | 14 |
Squared M_{split(q)} | 964 | 32 | 4 | 0 |
GCT | 958 | 28 | 1 | 0 |
3/7 points are stable | ||||
Robust | – | 4 | 7 | 33 |
Squared M_{split(q)} | – | 940 | 29 | 5 |
GCT | – | 835 | 15 | 0 |
It can be claimed that the results are as expected. The results of identification of stable points in scenario 1 are satisfying and quite similar for the robust and squared M_{split(q)} S-transformation; the conventional approach is only slightly less effective. However, the robust S-transformation proved completely ineffective in the critical case from scenario 2. This approach identified all the three stable points in 4/1000 cases, and it did not identify any stable points in as many as 956/1000 cases. In these cases, the minimum trace datum in final S-transformation would be surely defined on an unstable reference base; in consequence, deformations of object points would be completely misinforming. However, the efficiency of squared M_{split(q)} S-transformation is still satisfying and only slightly lower than in scenario 1. Despite a critical displacement scenario, this approach identified all three stable points in 940/1000 cases.
5.3 Experiment 3
Results of the stable PRPs identification
Number of the point | Simulated | Estimated | |||||
---|---|---|---|---|---|---|---|
Robust S-transformation | Squared M_{split(q)} S-transformation | ||||||
Model j = 1 | Model j = 2 (the best model) | ||||||
\( {\mathbf{d}} \) [mm] | \( {\hat{\mathbf{d}}} \)[mm] | T _{ i} | \( {\hat{\mathbf{d}}} \)[mm] | T _{ i} | \( {\hat{\mathbf{d}}} \)[mm] | T _{ i} | |
PRPs | |||||||
1 | 0.10 | 1.22 | 0.72 | 45.67 | 0.09 | 0.64 | |
0.16 | 0.57 | 0.03 | |||||
2 | − 0.08 | 0.25 | 0.47 | 13.77 | − 0.13 | 0.78 | |
− 0.01 | 0.48 | − 0.10 | |||||
3 | − 0.50 | − 0.39 | 13.48 | − 0.09 | 0.50 | − 0.57 | 22.08 |
− 0.60 | − 0.60 | − 0.03 | − 0.65 | ||||
4 | 0.16 | 0.75 | 0.29 | 8.54 | − 0.11 | 0.35 | |
0.03 | 0.56 | − 0.04 | |||||
6 | 0.18 | 2.54 | 0.56 | 7.35 | 0.04 | 3.61 | |
− 0.31 | 0.01 | − 0.49 | |||||
7 | 0.00 | 1.66 | 0.58 | 11.72 | − 0.03 | 0.25 | |
0.26 | 0.59 | 0.09 | |||||
9 | − 0.04 | 0.03 | 0.13 | 7.32 | − 0.28 | 1.86 | |
0.00 | 0.62 | − 0.02 | |||||
Validation test (34) is accepted: T = 1.13 ≤ F_{α=0.05}(9, 58) = 2.05 | |||||||
Object points | |||||||
10 | 0.04 | – | 0.74 | – | 0.07 | – | |
0.28 | 0.71 | 0.16 | |||||
11 | 0.60 | 0.31 | – | 0.95 | – | 0.31 | – |
− 0.75 | − 0.73 | − 0.13 | − 0.76 | ||||
12 | 1.10 | 1.57 | – | 2.08 | – | 1.50 | – |
0.50 | 0.41 | 1.14 | 0.44 | ||||
13 | 1.00 | 1.34 | – | 1.67 | – | 1.18 | – |
0.30 | 0.33 | 1.07 | 0.37 | ||||
14 | 0.10 | – | 0.26 | – | − 0.15 | – | |
0.08 | 0.74 | 0.08 |
It can be noted that very similar values of estimated displacements and correct results of identification of stable PRPs were obtained in both approaches. It means that both approaches identified points 1, 2, 4, 6, 7, 9 as reference base. Since for both approaches the final S-transformation to the minimum trace datum defined on the reference base is still recommended, therefore, the final results of the analysis of deformation of all the network points would be exactly the same for both approaches.
5.4 Experiment 4
Average values of mean absolute true errors of estimated PRP displacements
Number of stable PRPs | Value of error, \( \bar{e}_{{\hat{d}}} \) [mm] | Number of valid solutions of squared M_{split(q)} | |
---|---|---|---|
Robust | Squared M_{split(q)} | ||
Scenario 1: Displacements have different signs | |||
5/7 | 0.14 | 0.13 | 951 |
4/7 | 0.50 | 0.30 | 845 |
3/7 | 1.36 | 0.93 | 514 |
Scenario 2: Displacements have positive sign | |||
5/7 | 0.13 | 0.13 | 948 |
4/7 | 1.48 | 0.45 | 815 |
3/7 | 2.62 | 1.81 | 446 |
Results of the stable PRPs identification
Method | How many times was identified as stable: | |||
---|---|---|---|---|
5 points | 4 points | 3 points | 2 points | |
Scenario 1: Displacements have different signs | ||||
5/7 points are stable | ||||
Robust | 791 | 125 | 61 | 21 |
Squared M_{split(q)} | 874 | 80 | 35 | 11 |
4/7 points are stable | ||||
Robust | – | 441 | 108 | 179 |
Squared M_{split(q)} | – | 665 | 131 | 143 |
3/7 points are stable | ||||
Robust | – | – | 185 | 136 |
Squared M_{split(q)} | – | – | 412 | 210 |
Scenario 2: Displacements have positive sign | ||||
5/7 points are stable | ||||
Robust | 934 | 49 | 16 | 1 |
Squared M_{split(q)} | 912 | 59 | 23 | 6 |
4/7 points are stable | ||||
Robust | – | 0 | 0 | 24 |
Squared M_{split(q)} | – | 622 | 106 | 82 |
3/7 points are stable | ||||
Robust | – | – | 0 | 0 |
Squared M_{split(q)} | – | – | 278 | 123 |
Additionally, as before, Table 6 shows how many times a given method identified five, four, three and two stable PRPs.
Generally, it can be concluded from the above table that the results are consistent with previous ones. Hence, a detailed study of stable PRPs identification (Table 6) is left to the reader.
One can see that only three congruence models (q = 3) turned out sufficient in the squared M_{split(q)} S-transformation. It means the null hypothesis (26) has already passed for three models. The estimated displacements of all seven PRPs are located inside their confidence regions here; model j = 1 contains two statistically insignificant estimated displacements (points 4, 7), model j = 2 contains three such displacements (points 1, 2, 3), and model j = 3 contains two such displacements (points 6, 9). The model j = 2 is the best model because it contains the largest number of statistically insignificant estimated PRP displacements. This model turned out also valid (34).
As can be seen from the right graphs—unlike the conventional approach (red line)—the best congruence model in the squared M_{split(q)} S-transformation realizes the expected matching at actually stable PRPs. In consequence, these points were identified as stable by means of local F-tests (28); the estimated displacements of these points are inside their confidence regions (green ellipses). The actually unstable points do not have such a negative effect on matching/datum parameters of expected congruence model as in robust S-transformation because these points are realizations of other congruence models (j = 1 and j = 3) and their attention is absorbed by those matching/datum parameters. Thanks to this, the vectors of estimated displacements of actually unstable points are very similar to the simulated ones. Theoretically, the results would be even better for the final S-transformation to the minimum trace datum defined on the previously identified stable points.
6 Summary and conclusions
The identification of stable PRPs is a key issue in conventional deformation analysis. Since geodetic control networks have a datum defect (free networks), the accurate estimation of PRP displacements and later efficient identification of stable PRPs is a quite difficult task. For example, when there are more unstable PRPs than stable ones this task is not often possible by means of the conventional robust S-transformation. It is still a challenge for surveyors and geodesists.
This paper presents and pretests a new approach to the S-transformation of control network deformations. The idea behind this approach comes from the theory of squared M_{split(q)} estimation and lies in the non-conventional assumption that between the control networks adjusted in two considered epochs can simultaneously exist—not one as in the conventional robust S-transformation but—many congruences (matchings) which differ by the datum/transformation parameters. It is assumed that one model realizes the expected congruence, i.e., the congruence at a subgroup of stable PRPs, and other models can realize different congruences at unstable PRPs. Thanks to this, the robustness of the presented approach can be very high, because the unstable PRPs can be absorbed by other models. To establish what the number of congruence models is in a given case and whether the chosen/best model is valid, the statistical hypothesis tests were suggested.
The paper proves the hypothesis that if stable points make up the most populated subgroup of mutually stable points in the group of PRPs and measurement errors do not mask or generate displacements, then the presented approach can transform the estimated raw displacements to an expected datum (in a geometrical sense, can realize the expected matching at stable PRPs) and the results of identification of stable PRPs can be correct. It is noteworthy that the above sufficient condition is much less restrictive than the sufficient condition in the conventional robust S-transformation. Numerical experiments showed that the suggested S-transformation—unlike the conventional approach—can be effective even in critical cases, when there are more unstable PRPs than stable ones, and the sign of all displacements is the same. Owing to this, correct identification of stable PRPs and, in consequence, the correct final estimation of controlled object point displacements are possible in cases when it has not been possible so far. Hence, the squared M_{split(q)} S-transformation presented here seems to be an interesting and useful alternative to the more conventional robust S-transformation and, as such, deserves further research. For example, since the presented method allows to identify many subgroups of mutually stable points (there are no outlying points in the multi-split congruence model), it can have a wider application in geodetic deformation analysis.
Finally, it is advisable to know that the presented approach to deformation analysis, by means of M_{split} estimation, is completely different from the one presented in the literature and referenced in Introduction section. Most of all, that simple concept, based on the M_{split} estimation, does not deal with the identification of stable PRPs and focuses only on observation residuals which disclose some information about unstable points, which can be treated as outliers.
Notes
Acknowledgements
The author thanks the anonymous reviewers and the responsible Editor Prof. Wolf-Dieter Schuh for their constructive comments and suggestions. The author also feels greatly indebted to Prof. Zbigniew Wiśniewski for his time and valuable discussions.
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