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On the impact of correlations on the congruence test: a bootstrap approach

On the impact of correlations on the congruence test: a bootstrap approach The detection of deformation is one of the major tasks in surveying engineering. It is mean- ingful only if the statistical significance of the distortions is correctly investigated, which often underlies a parametric modelization of the object under consideration. So-called regression B-spline approximation can be performed for point clouds of terrestrial laser scanners, allowing the setting of a specific congruence test based on the B-spline surfaces. Such tests are known to be strongly influenced by the underlying stochastic model chosen for the observation errors. The latter has to be correctly specified, which includes account- ing for heteroscedasticity and correlations. In this contribution, we justify and make use of a parametric correlation model called the Matérn model to approximate the variance covariance matrix (VCM) of the residuals by performing their empirical mode decomposi- tion. The VCM obtained is integrated into the computation of the congruence test statistics for a more trustworthy test decision. Using a real case study, we estimate the distribution of the test statistics with a bootstrap approach, where no parametric assumptions are made about the underlying population that generated the random sample. This procedure allows us to assess the impact of neglecting correlations on the critical value of the congruence test, highlighting their importance. Keywords Terrestrial laser scanner · Deformation · Statistical test · Congruency test · Correlation · Matérn covariance function · Variance component estimation · Bootstrap * Kermarrec Gaël kermarrec@gih.uni-hannover.de Kargoll Boris kargoll@hs-anhalt.de Alkhatib Hamza alkhatib@gih.uni-hannover.de Geodetic Institute, Leibniz Universität Hannover, Nienburger Str. 1, 30167 Hannover, Germany Institut für Geoinformation und Vermessung Dessau, Anhalt University of Applied Sciences, Seminarplatz 2a, 06846 Dessau, Germany 1 3 Vol.:(0123456789) 496 Acta Geodaetica et Geophysica (2020) 55:495–513 1 Introduction Statistical tests for deformation provide a way to detect potential risks linked with natural hazards or the collapse of artificial objects, such as bridges. One of their main applications in geodesy is the estimation and minimization of risks. Combined with, for example, the concept of utility theory, they allow their consequences and costs to be estimated (Zhang and Neumann 2014). Terrestrial laser scanners (TLS) capture a large amount of three-dimensional (3D) points rapidly, with high precision and spatial resolution. In addition to the visualization of 3D topography or buildings, TLS are, thus, increasingly used to detect and quantify displacements or deformations. Software such as CloudCompare (www.danie lgm.net/cc), 3DReshaper (Hexagon Metrology) or Geomagic Studio (3DSystems) provide maps of deformation that allow one to perform a first deformation analysis. Such software, how - ever, does not perform congruence tests, which necessitate an adequate parametric approx- imation of the point clouds (Niemeier 2002). Such an approximation can be obtained by fitting B-spline surfaces to the TLS point clouds with a least-square (LS) adjustment (Koch 2009). Once the mathematical surface has been obtained, an adapted version of the congru- ence test based on the gridded surface points can be applied (Zhao et al. 2019). The congruence test is known to be the most powerful test in Gauss-Markov models with normally distributed random deviations and a correctly specified stochastic model (Pelzer 1971). To make use of its full potential for risk assessment, the setting of a cor- rect variance covariance matrix (VCM) of the errors has, thus, to be carried out as cor- rectly as possible. One way is to use an empirical model for computing the variances of the errors of the raw TLS observations. The datasheet from the manufacturers provides a good approximation for the angle variances (Boehler and Marbs 2002; Cosarca et al. 2009). The range variance can be modelled with an empirical intensity model, as proposed in Wujanz et  al. (2017) and Wujanz et  al. (2018) and simplified in Kermarrec et  al. (2018). Math- ematical correlations must be accounted for because TLS point clouds approximated with B-spline surfaces have to be transformed into Cartesian co-ordinates. Their impact on the test statistic for deformation between B-spline surface approximations has been studied in previous contributions of the authors: it could be shown that neglecting mathematical correlations affects the congruence test in the case of small deformation and unfavourable scanning geometries (see Zhao et al. 2019 for simulations; Kermarrec et al. 2019a for real data analysis). These results confirmed theoretical derivations (Hoog and Craig 1978) by additionally quantifying the expected impact on the test statistics of neglecting mathemati- cal correlations. Temporal correlations between TLS raw observations are often disregarded due to the asso- ciated computational burden (Kauker and Schwieger 2017). In the same way as mathematical correlations, they have to be taken into consideration to avoid biased test statistics. A usual approach would make use of variance component estimation (Amiri-Simkooei 2007; Searle 1995; Teunissen and Amiri-Simkooei 2008) based on the residuals of the LS adjustment from the B-spline surface fitting. Unfortunately, this procedure may lead to numerical inaccuracies, particularly for small samples: matrices inversion inaccuracies translate into the detection of a non-existing deformation. An alternative is to make use of the residuals by modelling their correlation structure, i.e. estimating the parameters of a known function, such as the Matérn function (Mátern 1960; Stein 1999). Such a statement suggests approximating geometrical and physical effects simultaneously within one function only: we are aware that this strong assumption necessitates deep foundations to identify when and if it holds true. This is the first 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 497 aim of the present contribution, which provides simple and understandable tools to analyse the to analyse the spectral content of the residuals using an empirical mode decomposition (EMD; Flandrin et al. 2004, 2005; Huang et al. 1998). Once the temporal correlation of the residuals is modelled, the assumption of the Chi square distributed test statistics of the congruence test can be investigated. This is our sec- ond goal: deviations from the theory are likely to happen, particularly when temporal correla- tions are neglected. The correct critical value under a given confidence level can be empiri- cally derived following Efron and Tibshirani (1993) using a bootstrapping approach. In a first approximation, we propose fitting the distribution of the test statistic with a Gamma distribu- tion, which includes the Chi square distribution as a particular case. We will make use of real TLS observations from a bridge under load to highlight the deviations from the expected theo- retical distribution (Kermarrec et al. 2019a). The paper is organised as follows: in the second section, the principle of B-spline approxi- mation is presented briefly, along with the test statistics for deformation, with a focus on the bootstrapping procedure. The third section is devoted to the case study, where the theoretical results are applied to observations from a bridge under load. The distribution of the test sta- tistics for deformation is derived with a Monte Carlo approach. We conclude this contribu- tion with some recommendations on how to account for correlations for deformation analysis based on B-spline surfaces. 2 Mathematical background 2.1 B‑spline surfaces Scattered and noisy raw point clouds from TLS can be approximated mathematically by means of a linear combination of basic functions, such as B-splines. The mathematical surface in its parametric formulation can be expressed as n m (u, v) = x , y , z = N (u)N (v) i,j i,j i,j i,p j,q i,j (1) i=0 j=0 where u, v ∈ [0, 1] × [0, 1] are the parameters in the two directions, so that a B-spline ( ) surface maps the unit square to a rectangular surface patch. The choice of the parameters affects the shape of the surface. The uniformly spaced or chord length methods (Bureick et al. 2016; Piegl and Tiller 1997) provide good results to determine u and v for regularly and rectangular-shaped point clouds with rows and w columns. Once a set of parameters is obtained, the knot vector can be obtained over which the basic functions N and N i,p j,q of degree p and q, respectively, are computed. Different methods can be employed; the simplest one being to use a uniformly spaced knot vector or to average the parameters (de Boor 2001). The basic function N u is, a composite curve of degree p polynomials ( ) i,p with joining points at knots in the interval u , u and can be evaluated by means of i i+p+1 a recurrence relationship.  is called the control points vector, with i varying from 0 to i,j n and j from 0 to m. Given the degree of the basic functions, approximating a surface is simply looking to find the control points such that the distance of the data points to the approximated surface is minimized. This step can be performed within the context of a LS of size n + 1 × m + 1 ,3 is the matrix containing the co-ordinates approximation. If (( ) ( ) ) 1 3 498 Acta Geodaetica et Geophysica (2020) 55:495–513 of the control points to be estimated and  3n ,3 n + 1 m + 1 the deterministic ( )( ) obs design matrix containing the values of the tensor product of the B-spline functions,  is the difference between the Cartesian co-ordinates of the point cloud  of size n ,3 and obs . Interested readers should refer to Zhao et al. (2017) for the description of the design matrix. We assume E() =  , where E(⋅) is the expectation operator. We further call  , the ll 2 2 VCM of the errors in Cartesian co-ordinates with  =   , where  is an a priori vari- ll 0 0 ance factor and  the identity matrix of size 3n ,3n . The estimated co-ordinates of the obs obs control points are expressed by the unbiased ordinary LS estimator (Koch 1999) −1 −1 −1 −1 T T T T ̂ =       = (2) ll ll The a priori VCM of the estimated parameters is given by −1 −1 𝚺 = 𝐀 𝚺 𝐀 (3) 𝐱̂𝐱̂ ll The number of control points to estimate can be iteratively adjusted by using informa- tion criteria (e.g. Alkhatib et al. 2018 for the specific application of information criteria to B-spline surface approximation and the references inside). Two possibilities exist: (1) the Akaike information criterion (AIC), which minimizes the Kullback–Leibler divergence of the assumed model from the data-generating model, or (2) the Bayesian information crite- rion (BIC), which assumes that the true model exists and is, thus, more adequate for large samples. 2.2 Stochastic model The estimated co-ordinates of the control points are not affected by the choice of the esti- mated  instead of  , due to the unbiasedness of the LS estimator. However, the ordinary ll ll LS estimator is no longer the most efficient within the class of linear unbiased estima- tors when  deviates from the true VCM and hypothesis tests, such as the global test, ll outlier tests or congruence tests, become invalid (Williams et  al. 2013). The assumption of homoscedasticity has to be adapted in order to derive realistic test statistics based on Eq.  (3). In this contribution, we propose to make use of the residuals from the B-spline surface approximation to approximate  . The VCM of the Cartesian observation errors ll , with 𝐯 ̂ = 𝐥 − 𝐀 𝐱̂ the residuals will, thus, be considered as the VCM of the residuals 𝚺 𝐯̂𝐯̂ of the LS adjustment. Because the residuals are correlated and heteroscedastic, 𝚺 is fully 𝐯̂𝐯̂ populated. The global VCM of the three components is set up by neglecting the correla- tions between components: 𝚺 𝟎𝟎 ⎡ ⎤ ̂ ̂ 𝐯 𝐯,x ̂ ⎢ ̂ ⎥ 𝚺 = 𝟎 𝚺 𝟎 (4) 𝐯̂𝐯̂ 𝐯̂𝐯̂,y ⎢ ⎥ ⎣ 𝟎𝟎 𝚺 ⎦ ̂ ̂ 𝐯 𝐯 ,z As a powerful alternative to empirical VCE, we will assume that the correlation structure of the residuals can be approximated with a parametric Matérn covariance function (“Appendix  1”). We assume, thus, that only a coloured noise is present in the residuals, which is a strong assumption. The impact of geometrical misspecifications on the residuals has to be investigated prior to a fitting with such a parametric covariance function. One way is to make use of maximum likelihood-based model selection, which, 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 499 as indicated by its name, is based on predefined models (often a combination of col- oured and white noise). To our point of view, such procedures answer only conditional the question whether a parametric covariance function is a relevant choice or not; their result may be biased in case of discrepancies from the noise assumption due to model misspecification. As a less computational demanding alternative, we propose making use of the analysis of the power spectral decomposition of the residuals combined with an EMD (“Appendix  2”) to validate the use of such a model, similarly to denoising strategies (Kopsinis and Mclanglin 2009). We propose for didactical purposes to high- light our methodology with a practical example in Sect. 3. Three parameters will have to be estimated: the variance, the range and the smooth- ness of the correlation function from the residuals. A possibility to make a computationally efficient estimation of the parameters in the case of small samples is described in “ Appen- dix 3”. The most important parameter is the smoothness, which it impacts strongly on the first values of the inverse of the VCM and, thus, the value of the test statistics (Kermarrec and Schön 2016). 2.3 Congruence test for deformation analysis In order to test for deformation between two epochs, we follow Zhao et  al. (2019) and make use of an adapted form of the global congruency test derived by Pelzer (1971). No difference between the two vectors of control points co-ordinates (the “parameters”) can be computed because the number of control points estimated at different epochs may vary. Therefore, we test the deformation at the level of the estimated surfaces and not at the parameter level (the control points). Using gridded observations and assuming a deformation only in the z-direction, we only consider 𝚺 from Eq.  (4) and determine the surface difference. The surface points at a 𝐯̂𝐯,̂z given epoch are estimated by a corresponding matrix product 𝐅 𝐱̂ from Eq. (1), where the is evaluated from the tensor product of B-spline functions estimated at the parameters of the gridded points. The uniformly most powerful invariant test for deformation under the null hypothesis can be shown to be based on the test statistic: 𝐓 −𝟏 𝐓 −𝟏 2 ̂ ̂ ̂ ̂ T = 𝚫 𝚺 𝚫 = 𝚫 𝐐 𝚫 ∼ 𝜒 apriori ̂ ̂ ̂ ̂ p 𝚫 𝚫 2 𝚫 𝚫 (5) 𝚺 = 𝐇𝛴 𝐇 ̂ ̂ ̂ ̂ 𝚫 𝚫 𝛃 𝛃 where 𝚺 is the VCM of the estimated CP for both epochs and  = contains the ̂ ̂ 𝛃 𝛃 ̂ LS estimates of the CP at epoch 1 and 2. The matrix  is split into two parts to determine a difference at the level of the surface points:  = −  with p rows. The dif- ̂ ̂ ference between the two surfaces is given by 𝚫 = 𝐇 𝛃 , where we here only consider the dif- ference in the z-direction. We define the hypotheses for deformation analysis as follows: 1 3 500 Acta Geodaetica et Geophysica (2020) 55:495–513 H ∶  = 0 vs H ∶  ≠ 0 0 1 The null hypothesis states that no deformation happened. The test decision involves the quantile k based on the test distribution and the significance level  . In par- test test 1− ∕ ticular, H is accepted if T ≤ k . 0 apriori test 1− ∕ The test statistic T is strongly dependent on the VCM of the CP estimates, which apriori is based on 𝚺 , see Eq. (2). Consequently, any misspecification will affect the decision ̂ ̂ 𝛃 𝛃 step, potentially leading to the erroneous conclusion that no deformation occurs or, con- trarily, that a deformation occurs (error of type I and II). The results are particularly sensitive in the presence of correlated errors: they may lead to the detection of deforma- tions which are not the consequence of an existing deformation but only the result of the correlated noise structure of the observations. We will consider two cases for the VCM in the following: (i) (i) 𝚺 leading to T 𝐯̂𝐯̂ apriori diag (ii) (ii) 𝚺 leading to T 𝐯̂𝐯̂ apriori Fig. 1 Methodology of the congruence test using an estimation of the covariance structure from the residu- als of the LS approximation 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 501 diag ̂ ̂ The VCM 𝚺 is determined by multiplying diag 𝚺 component-wise with the 𝐯̂𝐯,̂z 𝐯̂𝐯,̂z identity matrix of corresponding size. The full methodology is summarized in a flowchart form in Fig. 1 . Additional comment Our approach is not based on a physical model for the raw (polar) TLS observations. In such cases, mathematical correlations would have to be additionally considered due to the transformation from potentially correlated and heteroscedastic polar raw measurements and Cartesian co-ordinates. 2.4 V alidation of the test distribution The defined a priori test statistic is expected to follow a  distribution. This result remains theoretical and may not hold when an approximated stochastic model is used or when cor- relations are present. A way to validate the assumed distribution is to make use of a boot- strapping-based hypothesis test. This elegant procedure allows one to estimate the param- eters of a general Gamma distribution, which includes the  as a particular case (Kargoll et al. 2018; Appendix A.2). Fig. 2 Flowchart of the computational steps to derive the distribution of the test statistics using a Monte Carlo approach 1 3 502 Acta Geodaetica et Geophysica (2020) 55:495–513 In the following, we will briefly present the methodology used for validation, which is further summarized in a flowchart in Fig.  2. Our procedure involves three steps, in the same way as Kargoll et al. (2019): 1. Testing: the first step starts with the approximation of scattered (TLS) observations from 2 epochs with B-spline surfaces. 2. Generating: in the kth-generating step, we add a noise vector, whose structure corre- sponds to the positive definite 𝚺 , to the two approximated surfaces. We use a Cholesky 𝐯̂𝐯̂ ̂ ̂ decomposition and decompose 𝚺 as 𝚺 = 𝐆 𝐆 . We further generate a Gaussian ran- 𝐯̂𝐯̂ 𝐯̂𝐯̂ dom vector  , i = 1, 2 for the two epochs with mean 0 and variance 1 from the Mat- i,k lab random number generator randn. The correlated noise vector reads:  =   . i,k i,k We add this generated noise vector to its corresponding surface and approximate it by a B-spline surface. We compute the a priori test statistic T from the two estimated apriori surfaces for each step. For one iteration k, we call the corresponding test statistic T . apriori 3. Evaluation: we perform k = 1000 iterations. The mean of the k test statistics total total values in the evaluation step are statistically evaluated using the function fitdist from Matlab. We compute the parameters of a Gamma distribution—the  being a particular case corresponding to a scale parameter of 2. In the next section, we propose applying this theoretical derivation to a case study. We will make use of the same TLS observations as in Kermarrec et al. (2019a). The focus of this contribution is on the impact of neglecting the correlations of the Cartesian residuals on the test statistics. We investigate particularly how they may affect (i) the test statistics and (ii) their corresponding test distribution. 3 Case study In this section, we use TLS observations from a bridge under load and test for the defor- mation between two steps of load. We start from a gridded TLS point cloud, which we approximate with B-spline surfaces. Following the methodology described in the previ- ous section, we estimate the correlation structure and the variance from the residuals of the B-spline approximation, which we use to set up the corresponding Toeplitz VCM, assuming stationarity of the variance. The test statistics are computed using both the fully populated VCM of the residuals (case (i)) and the diagonal simplification (case (ii)), i.e. neglecting correlations. After having briefly described the B-splines surface approxima- tion we investigate the impact on the a priori test statistics of the two stochastic models. In a last step, we validate the distribution of the test statistics with the proposed Monte Carlo approach. The justification of the fitting with a parametric covariance function will be didactically explained. 3.1 Description of the data set We use a real data set of a historic masonry arch bridge over the river Aller near Verden in Germany made of circular brick arches. The aim of the experiment was the combination of numerical models and experimental investigations for model calibration (Paffenholz et al. 2018). 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 503 Fig. 3 A bridge under load: localization of the points L10 and L8 Fig. 4 Top left: the gridded point clouds (yellow) and the approximation (green) for L8. Bottom left: the z-residuals. Top right: scaled Toeplitz VCM 𝚺 estimated from the z-component. Bottom right: first ten 𝐯̂𝐯,̂ z −1 values of the first row of 𝚺 𝐯̂𝐯̂,z The standard load of 1.0 MN (100 t) was defined and further loadings up to five times the standard load were realised in the scope of the load testing. Thus, a maximum load of approximately 6.0 MN was defined, produced by four hydraulic cylinders mounted on the arch. In the following, we will call the different load steps E00 (reference), E11, E22, E33, E44 and E55. The deformations will always be computed regarding the reference load, which we will denote E11-00, E22-00, E33-00, E44-00, E55-00. The 3D point cloud acquisition was carried out using TLS sensors of type Zoller + Fröh- lich (Z + F) Imager 5006/h in periods of a constant load on the bridge. The 3D point clouds for different load scenarios ranging from 1 up to 6 MN were captured and finally processed. A filtering regarding objects on the arch surface was performed to eliminate interfering objects, i.e. other sensor installations, such as prisms for the laser tracker and strain gauges. Following Kermarrec et al. (2019a), we selected small patches of observations in order to have an optimal functional model. Using the software CloudCompare, the same surfaces were chosen for each load. The patches chosen were located around two reference Laser Tracker points L8 and L10, as shown in Fig. 3. L10 can be considered as optimally scanned at a short distance in the Up-direction, whereas L8 was scanned with a less favourable geometry. Moreover, L10 has a stronger deformation magnitude than L8 (approximately 10 vs. 4 mm, see Paffenholz et al. 2018). 1 3 504 Acta Geodaetica et Geophysica (2020) 55:495–513 3.2 B‑spline surface approximation The TLS observations were gridded in 15 cells in both directions leading to a total of 225 gridded observations. This number of observations allows for a reliable estimation of the correlation parameters using the Whittle likelihood methods, as presented in “Appendix 3”. The two surfaces are shown in Fig.  4 (left), where it can be seen that they correspond approximately to a plane. All observations falling in one cell were averaged for the grid- ding. The parameterization was carried out using a uniform method, which is efficient for the uncomplicated geometries under consideration. Similarly, the knot vector was taken as equidistant. The number of CP was determined using the Bayesian information crite- rion, which gave similar results to the Akaike information criterion. For each dataset, 4 and 3 CP in the u- and v-direction, respectively, were found to be optimal for L8; 5 CP in both directions were considered for L10. The same values were found for all epochs under consideration. 3.3 Estimation of the VCM We wish to estimate the VCM of the residuals in the z-direction from the residuals of the B-spline surface approximation and assume that a Matérn model can describe the stochas- ticity of the residuals. As mentioned in “Appendix 1”, any Matérn processes will have a typical power spectral decomposition with two different regimes (Lilly et al. 2017): • at high frequencies, a power law decay governed by the smoothness parameter  , and at low frequencies, a constant value, i.e. a “locally white” behaviour governed by the parameter . The power spectral decomposition obtained with the Matlab function pwelch is plot- ted in Fig. 5 (top left). Due to the small sample size under consideration, the identification of the classic Matérn spectral is far from evident, although two different modes can be guessed. In order to justify the use of a Matérn model, we made use of the EMD of the residuals (“Appendix  2”). Since the latter are not smooth, we use ‘pchip’ as an interpola- tion method for the Matlab function EMD and decompose them in five modes. The corre- sponding decomposition is shown exemplarily for L8 in Fig. 5 (bottom left). We define E from the energy-based variance of the IMF, following Flandrin et  al. (2004). It was proven from Flandrin et al. (2005) or Wu and Huang (2004) that the mean energy of the different intrinsic model function (IMF) E from a pure power law noise fol- lows a linear decrease in a semi-log diagram, i.e. log E regarding the IMF index k . The 2 k corresponding plot is shown in Fig.  5 (top right). We observed that after the fourth IMF, the energies diverge significantly from the theoretical model, indicating the presence of significant amounts of “no-coloured noise” signal. This finding is supported by the shape of the corresponding EMD (Fig. 5, bottom left). Following Flandrin et  al. (2005) further, we filtered the noise by excluding the fourth and fifth IMF and interpret these as corresponding to some unknown model misspecifica- IMF1+IMF2+IMF3 tions. Since the ratio of the variances ≈ 25 , we, furthermore, chose to neglect IMF4+IMF5 these two IMF and concentrated on the third first one, which corresponds to the coloured noise. The first IMF carries primarily the white noise, which we validated with the 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 505 Fig. 5 Top left: the power spectral decomposition for L8 and L10 residuals (red and green line, respec- tively) and the decomposition for the IMF2 + 3 of the same residuals as the dashed line. Bottom: the EMD for the L8 residuals. Top right: log E versus the IMF k 2 k Kolmogorov–Smirnov test. Consequently, and in order to facilitate the estimation of the Matérn parameters from IMF2 + IMF3 , we modelled 𝚺 as ̂ ̂ 𝐯 𝐯,z 2 2 ̂ ̂ 𝚺 = 𝜎 𝐈 + 𝜎 𝚺 , with  as the identity matrix. We estimated the cor- 𝐯 ̂𝐯 ̂,z IMF2+IMF3 IMF1 IMF2+IMF3 2 2 responding variances  ,  from IMF1 and IMF2 + IMF3, respectively. IMF1 IMF2+IMF3 is modelled as a Toeplitz VCM and its corresponding Matérn parameters are IMF2+IMF3 estimated using the methodology presented in “Appendix 3”. The power spectral decompo- sition of IMF2 + IMF3 is shown as a dotted line in Fig.  5 (top left): the two different regimes are now clearly identifiable and support an unbiased estimation of the parameters. We summarize our methodology in 4 steps: compute the z-residuals from the B-spline approximation; • perform the EMD. Plot the mean energy with respect to the IMF in a log diagram. Identify the power law noise IMF (straight line). If at that step, model misspecifications dominate, no straight line will be identifiable (LS fitting can be used for that purpose). We can still consider the second IMF as being the coloured noise. However, in that case, a power law model only, such as the Matérn covariance function, will be subopti- mal, particularly if the variance of the third IMF is still high; • separate the white noise IMF from the power-law noise and estimate the corresponding Matérn parameters; and • compute 𝚺 assuming stationarity of the variance, which can be checked with IMF1. ̂ ̂ 𝐯 𝐯,z 1 3 506 Acta Geodaetica et Geophysica (2020) 55:495–513 Table 1  , ,  ,  for L8 and L10 are computed for the first load step E00. The standard devia- IMF1 IMF2+3 tion is given in [mm] L8 L10 , ,  ,   , ,  , IMF1 IMF2+3 IMF1 IMF2+3 E00 [0.5, 0.3, 2.2, 9.1] E00 [0.7, 0.3, 1.58, 6.5] Fig. 6 Values of the test statistics (i) (ii) T and T for the five apriori apriori deformations E00-11, E00-22, E00-33, E00-44 and E00-55. Top: L8, Bottom L10. The red line corresponds to the diagonal and the blue line to the fully populated VCM, respectively The Toeplitz structure of 𝚺 is shown exemplarily in Fig.  4 (top right) for L8 E00. 𝐯̂𝐯,̂z The small level of correlation is highlighted by the sparse structure of the VCM. Addition- ally, we show the structure of the inverse of the VCM with the first ten values of one row of the inverse. After a large value in the diagonal (three times higher than the second one), the following nine values oscillate around and close to 0. Compared with results obtained from diagonal matrices, this particular structure has a strong effect on the test statistics, as mentioned in Kermarrec and Schön (2016). The first value is directly linked with the smoothness of the corresponding process: the power spectral decomposition at high fre- quency should permit a good estimation of this parameter, i.e. a linear decay should be found (Fig. 5, top left, dashed line). The approximation done by neglecting the IMF4 and 5, which corresponds to the low frequency noise, is, thus, fully justified, as well as the splitting between the white noise IMF and the “coloured noise” IMF. We present the values of the Matérn parameters obtained for L8 and L10 (E00) in Table 1. For the sake of brevity, the other similar values are not shown. Larger values of the smoothness and range parameters are found for L8, which is scanned under an unfavourable geometry, compared to L10, for which the correlations are less pronounced. The standard deviations of the power law noise are, however, similar for both points. The amount of white noise is slightly higher for L10, which we justify by the optimal scanning geometry. 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 507 3.4 Impact on the test statistic Having computed the estimated fully populated VCM from the residuals for each load step following the previous methodology, we highlight their impact on the test statis- tics in Fig.  6. To that aim, we follow the methodology proposed in Sect.  2.3 and com- pare the results with the one obtained with a diagonal VCM. We notice that a low level of correlation (L10) combined with a high deformation magnitude leads to a difference between the two stochastic models smaller than for L8. In the latter case, a smaller defor- mation magnitude together with a higher correlation level makes the difference between (i) (ii) (ii) (i) T and T larger. Whereas for L10, T ≈ 1.2T , the factor increases to apriori apriori apriori apriori (ii) (i) T ≈ 1.35T for L8. We interpret the small difference as due to the relatively low apriori apriori level of correlations. However, the factor for L8 is still slightly higher than for L10, as a consequence of the parameter estimates presented in Table 1. (ii) (i) The main result is the overestimation of the test statistics T regarding T : in all apriori apriori (ii) apriori cases, the ratio is higher than 1. As mentioned previously, this has to be linked with (i) apriori the particular structure of the inverse of the VCM, which acts as a variance factor: account- ing for correlations in the test statistics is similar to having decreased the variance of a cor- responding diagonal VCM (Kermarrec et al. 2019b). Even if in our particular case study, the a priori test statistics are in all cases far over the critical value of the Chi square distri- bution with a confidence level of 0.05, this behaviour should not be underestimated for more subtle cases, where the deformation to detect is much smaller. When correlations are present and a correct VCM is used, the test statistics will become smaller and, thus, more realistic: this will lead to accepting the null hypothesis “no deformation”. This is an impor- tant point, as correlations present in the error term should not be considered as true defor- mation. We, thus, strongly encourage the use of a realistic VCM which can be estimated from the residuals. However, these conclusions assume that the Chi square distribution can be used, which needs to be validated or not. Fig. 7 (Left) histogram and corresponding Gamma distribution for case (i) corresponding to an optimal sto- chastic model including correlations. Right: the same for case (ii), i.e. a diagonal VCM neglecting correla- tions. The observations are taken from L8, deformation E00-11 1 3 508 Acta Geodaetica et Geophysica (2020) 55:495–513 3.5 Test distribution (i) (ii) Using the methodology described in Sect.  2.4., the ratio about T and T is com- apriori apriori puted using a Monte Carlo approach. The corresponding histograms and a fitted gamma distribution are presented in Fig. 7. The parameters of the distribution are estimated using the Matlab function fitdist. We justify the choice of the gamma distribution as it includes the assumed Chi squared distribution as a particular case. Further analysis will test other member of the distribution family, such as the exponential. For the sake of brevity, the results correspond to the particular case of L8 and the defor- mation step E00-11. The shape parameters of the Gamma distribution were found to be 3.4 and 3.5 for cases (i) and (ii), respectively. The scale parameters were 575 and 370, respec- tively. Thus, these parameters are different from those of a Chi square distribution, for which the scale parameter would equal 2. Correspondingly, the critical values for a given significance level do not correspond to the assumed one: they are much higher. Exempla- rily, we found a critical value of 5731 with a significance level of 0.05 for (i), whereas a value of 3750 has to be considered for (ii). The ratio of the two critical values is, thus, 1.5 (ii) apriori and larger than . Therefore, the risk of making a wrong testing decision by neglecting (i) apriori correlations is slightly decreased but still exists. Because the spread of the test statistics is, however, much higher in that case, we encourage not disregarding correlations in the test- ing procedure. The other cases (deformation steps as well as L10) led to similar conclusions. The parameters of the Gamma distribution, however, changed from case to case and needed to be revaluated for each data set to determine the correct critical value for a given confidence level. The parameters of the Gamma distribution for L10 were found to be similar for both cases due to the low level of correlations, which are in line with the conclusion of the pre- vious section. The values of the scale parameters were found to be 515 and 585 for case (ii) and (i), respectively, for the deformation step E00-11. We note that they remain between 500 and 600, the same as for L8. A generalization of this finding necessitates further inves- tigations that we propose to carry out in a next step. To conclude, we briefly resume the proposed methodology to perform a rigorous test for deformation when correlations are suspected: 1. from two TLS point clouds, estimate the control points of the two B-spline surfaces with a LS adjustment (Sect. 2.1); 2. compute the residuals; 3. validate whether a Matérn model is adequate (Sect. 3.3); 4. if yes, estimate the corresponding Toeplitz VCM; 5. compute the a priori test statistic following Eq. (5); and 6. perform a Monte Carlo analysis to derive the correct critical values of the congruence test (Fig. 2). 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 509 4 Conclusion An accurate and realistic knowledge of the stochastic properties of TLS errors is of great importance to avoid untrustworthy test decisions. Standard software allows the visualiza- tion of deformation by means of maps of distances, but not for the rigorous statistical test- ing of deformation. This latter can only be achieved when the point clouds are mathemati- cally modelled with B-spline surfaces so that an adapted congruence test at the level of the estimated surface can be performed. However, test statistics for deformation derived from the surface differences between two epochs are known to be strongly influenced by the VCM of the errors chosen. In this contribution, we proposed a simplified stochastic model with data-driven parameters. We made use of the Matérn covariance function, whose parameters are obtained from the residuals of B-spline approximation. Using the EMD, we justified the use of a power-law model from the linear relationship between the mean energy of the IMF and its IMF index. In addition to being simple, the main advantage of this methodology is in its computational stability and the easy identification of model mis- specification. By means of the de-biased Whittle likelihood, the model parameters, includ- ing the variance, can be trustworthily estimated, even for small samples. This procedure allowed us to investigate the impact of correlations on the test statistics and the effect of neglecting them. As a follow-up to a previous contribution dealing with mathematical correlations, we used a real data set from a bridge under load. Two areas cor- responding to different scanning geometries and magnitudes of deformation were approxi- mated with B-spline surfaces using optimal fitting parameters. It could be shown that the test statistics obtained by neglecting correlations were overoptimistic: accounting for cor- relations acts as decreasing the variance in an equivalent diagonal matrix. As expected, the correlation structure was found to depend strongly on the scanning geometry, i.e. an optimized geometry with short range and optimal incidence angle decreased the level of correlations of the LS residuals and, thus, the impact of neglecting correlations in the test statistics for deformation. Correlations should not be neglected for a trustworthy test deci- sion: they provide a more trustworthy test decision for deformation. A Gamma distribu- tion could be fitted from the test statistics using a Monte Carlo approach. Since the critical values of the distribution—as well as the corresponding test statistics—were larger when correlations were neglected, the risk of making a wrong decision that a deformation occurs was decreased. However, the optimal test distribution should be investigated further. This will be the topic of our next contribution. Acknowledgements Open Access funding provided by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. 1 3 510 Acta Geodaetica et Geophysica (2020) 55:495–513 Appendix 1 The Matérn process is a three-parameter continuous Gaussian process defined by an unali- ased spectrum W : 2−1 2 W() = (6) 2 2 where  is the frequency. In this formulation,  is the variance of the process, c a normal- izing constant,  the range parameter and  the smoothness parameter (Lilly et al. 2017). Thus, whereas the smoothness determines the behaviour of the spectral density as  → ∞ , the range parameter has a strong impact at low frequency as  → 0 . The corresponding autocovariance function is found to be (Mátern 1960): C() =  M () (7) where we introduce the Matérn function as M () =    K () 1 − ∕ (8) − ∕ 2 − ∕ 2 with  the gamma function, K the decaying modified Bessel function of the second − ∕ kind of order  − ∕ . 𝜏> 0 is the time difference,  a range parameter that controls how quickly C decreases as  increases, whereas  is related to the mean-square differenti- matern ability of the process at the origin, as  → 0. Figure  8 (left) displays C() scaled to 1 (i.e. the correlation function) for different choices of the shape parameter  . The smoothness parameter determines the behaviour of the covariance function as  approaches 0. It has a great importance since it determines how fully populated the inverse of the VCM will be and how large its impact is. The following cases are well-known for the one-dimensional field and should be mentioned: Fig. 8 Matérn correlation function. Left: variation of the smoothness parameter  with  = 0.05 . Right: var- iation of the range parameter  by keeping  fixed to 2 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 511 •  = ∕ corresponds to the exponential covariance function, i.e. a short memory pro- cess •  = 1 to the Markov process of first order, and •  =∞ gives the squared exponential covariance function, which corresponds to a phys- ically less plausible infinitely differentiable Gaussian process at the origin. Figure 8 (right) shows the different correlation functions obtained by varying the range parameter  . As can be seen,  is linked to the decay of the covariance function. Please note that other parametrizations of the Matérn function are possible (Stein 1999) Appendix 2 Huang et  al. (1998) developed the EMD to decompose a given time series into a set of components of different time scales called the Intrinsic Mode Functions (IMF). Combined with the Hilbert Transform, this decomposition provides a time–frequency-energy repre- sentation of a signal and is intuitively based on the wave form of the data; it is also called the Hilbert-Huang transform in the literature. The EMD is independent of mother wave- lets, such as in wavelet analysis or trigonometric functions, for example, in Fourier trans- form. Thus, it can be used on non-stationary signals and allows a characterization by direct extraction of the local energy associated with the IMF. The decomposition is, however, not unique and depends exemplarily on the particular interpolation scheme for envelope extraction, the stopping criterion (usually 1e-6) used in the sifting process and the number of IMF to determine. The EMD algorithm can be summarized as the following loop (Flandrin et al. 2004): 1. Identification of the extrema of the signal 2. Interpolation between minima (maxima) to get an envelope of the signal 3. Computation of the average of the envelope 4. Extraction of the detail as difference between the envelope and the original signal 5. Iteration on the residual The IMF should have the following properties: (i) zero mean, (ii) all the maxima and all the minima of h(i)(t) are correspondingly positive and negative (iii) narrowband. The num- ber of zero-crossings can be either N(i) or N(i) − 1. Each IMF has fewer extrema than all the lower order IMFs, leading to fewer and fewer oscillations as the IMF order increases. The EMD components of coloured noise (i.e. having a spectral decomposition following a power law) have the same rescaled Fourier spectrum. Thus, apart from the first noise- only IMF which carries the main energy and is considered as the white noise component, the energies of the other IMF decrease linearly in a semi-log diagram regarding the IMF index. We understand under “energy” the empirical variance estimate of a given IMF. Any discrepancies from the linear relationship can be used to denoise a signal, see, for example, Kopsinis and McLaughlin (2009). In this contribution, the noise itself is of interest as it is used to estimate the Matérn parameters: we make use of this important property to extract the noise from the other IMF, considered as model misspecifications. 1 3 512 Acta Geodaetica et Geophysica (2020) 55:495–513 Appendix 3 Assuming Gaussian data, the parameters of the Matérn model—including the variance— can be estimated with Maximum Likelihood, eventually by fixing one parameter to reduce the computational burden (Kaufman et al. 2008). This standard approach has a computa- tional efficiency of the order of n for regularly spaced observations. It produces estimates obs that are asymptotically efficient, i.e. for smaller samples alternatives such as Restricted Maximum Likelihood have to be employed. Another powerful approach is referred as “quasi-likelihood”, as introduced in Sykulski et  al. (2019) using the Whittle likelihood (Percival and Walden 1993). This method for estimating the Matérn parameters of any time series modelled as a second-order stationary stochastic process is employed in the freely available software package www.ulc.ac.uk/stati stics /resea rch/spg/softw are. Exact maximum likelihood inference can be performed by maximizing the log-likeli- hood. This strategy, however, involves matrix inversion so that Whittle (1953) proposed approximating the autocovariance matrix in the frequency domain. The “Whittle likeli- hood” makes use of the periodogram of the observations, which may be a biased measure for finite samples (Percival and Walden 1993) due to blurring and aliasing effects. Sykulski et  al. (2019) proposed an alternative called the “de-biased Whittle likelihood”, which is particularly interesting for discrete-time processes of finite length, such as the residuals of the LS adjustment used in this contribution. 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Chapman and Hall/CRC, Boca Raton Flandrin P, Rilling G, Gonçalvés P (2004) Empirical mode decomposition as a filter bank. IEEE Signal Process Lett 11:112–114 Flandrin P, Rilling G, Gonçalvés P (2005) EMD equivalent filter banks, from interpretations to appli- cation. In: Huang NE, Shen S (eds) Hilbert-Huang transform and its application, 1st edn. World Scientific, Singapore Hoog RV, Craig AT (1978) Introduction to mathematical statistics. Macmillan Publishing Co., New York Huang NE, Shen Z, Long SR, Wu ML, Shih HH, Zheng Q, Yen NC, Tung CC, Liu HH (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc A Math Phys 454:903–995 Kargoll B, Omidalizarandi M, Loth I, Paffenholz JA, Alkhatib H (2018) An iteratively reweighted least- squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and t-distributed deviations. J Geod 92:271–297 Kargoll B, Omidalizarandi M, Paffenholz JA, Neumann I, Kermarrec G, Alkhatib H (2019) Bootstrap tests for model selection in robust vibration analysis of oscillating structures. In: Proceedings 4th joint international symposium on deformation monitoring (JISDM), 15–17 May, Athens, Greece Kaufman CG, Schervish MJ, Nychka DW (2008) Covariance tapering for likelihood-based estimation in large spatial data sets. J Am Stat Assoc 103:1545–1555 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 513 Kauker S, Schwieger V (2017) A synthetic covariance matrix for monitoring by terrestrial laser scan- ning. A synthetic covariance matrix for monitoring by terrestrial laser scanning. J Appl Geod 11:77–87 Kermarrec G, Schön S (2016) Taking correlation into account with a diagonal covariance matrix. J Geod 90:793–805 Kermarrec G, Alkhatib H, Neumann I (2018) On the sensitivity of the parameters of the intensity-based stochastic model for terrestrial laser scanner. Case study: B-spline approximation. Sensors 18:2964 Kermarrec G, Bureick J, Alkhatib H (2019a) On mathematical correlations for detecting deformations using B-splines approximation: a bridge under loading. In: Proceedings 4th joint international sym- posium on deformation monitoring (JISDM), 15–17 May, Athens, Greece Kermarrec G, Neumann I, Alkhatib H, Schön S (2019b) The stochastic model for global navigation sat- ellite systems and terrestrial laser scanning observations: a proposal to account for correlations in least squares adjustment. J Appl Geod 13:93–104 Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, Berlin Koch KR (2009) Fitting free-form surfaces to laserscan data by NURBS. AVN 116:134–140 Kopsinis Y, McLaughlin S (2009) Development of EMD-based denoising methods inspired by wavelet thresholding. IEEE Trans Signal Process 57(4):1351–1362 Lilly JM, Sykulski AM, Early JJ, Olhede SC (2017) Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion. Nonlinear Proc Geophys 24:481–514 Mátern B (1960) Spatial variation-stochastic models and their application to some problems in forest surveys and other sampling investigation. Medd Statens Skogsforskningsinstitut 49:5 Niemeier W (2002) Ausgleichungsrechung. Gruyter Verlag, Berlin Paffenholz JA, Huge J, Stenz U (2018) Integration von Lasertracking und Laserscanning zur optimalen Bestimmung von lastinduzierten Gewölbeverformungen. AVN 125:75–89 Pelzer H (1971) Zur analyse geodatischer deformations-messungen, München, Verlag der Bayer. Akad. d. Wiss., Munchen, Beck, 400: 164 Percival DB, Walden AT (1993) Spectral analysis for physical applications: multitaper and conventional univariate techniques. Cambridge University Press, Cambridge Piegl LA, Tiller W (1997) The NURBS book. 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ISPRS 125:146–155 Wujanz D, Burger M, Tschirschwitz F, Nietzschmann T, Neitzel F, Kersten TP (2018) Determination of intensity-based stochastic models for terrestrial laser scanners utilising 3D-point clouds. Sensors 18:2187 Zhang Y, Neumann I (2014) Utility theory as a method to minimize the risk in deformation analysis deci- sions. J Appl Geod 8:283–293 Zhao X, Alkhatib H, Kargoll B, Neumann I (2017) Statistical evaluation of the influence of the uncertainty budget on B-spline curve approximation. J Appl Geod 11:215–230 Zhao X, Kermarrec G, Kargoll B, Alkhatib H, Neumann I (2019) Influence of the simplified stochastic model of TLS measurements on geometry-based deformation analysis. J Appl Geod 13(3):199–214 1 3 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Acta Geodaetica et Geophysica" Springer Journals

On the impact of correlations on the congruence test: a bootstrap approach

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Abstract

The detection of deformation is one of the major tasks in surveying engineering. It is mean- ingful only if the statistical significance of the distortions is correctly investigated, which often underlies a parametric modelization of the object under consideration. So-called regression B-spline approximation can be performed for point clouds of terrestrial laser scanners, allowing the setting of a specific congruence test based on the B-spline surfaces. Such tests are known to be strongly influenced by the underlying stochastic model chosen for the observation errors. The latter has to be correctly specified, which includes account- ing for heteroscedasticity and correlations. In this contribution, we justify and make use of a parametric correlation model called the Matérn model to approximate the variance covariance matrix (VCM) of the residuals by performing their empirical mode decomposi- tion. The VCM obtained is integrated into the computation of the congruence test statistics for a more trustworthy test decision. Using a real case study, we estimate the distribution of the test statistics with a bootstrap approach, where no parametric assumptions are made about the underlying population that generated the random sample. This procedure allows us to assess the impact of neglecting correlations on the critical value of the congruence test, highlighting their importance. Keywords Terrestrial laser scanner · Deformation · Statistical test · Congruency test · Correlation · Matérn covariance function · Variance component estimation · Bootstrap * Kermarrec Gaël kermarrec@gih.uni-hannover.de Kargoll Boris kargoll@hs-anhalt.de Alkhatib Hamza alkhatib@gih.uni-hannover.de Geodetic Institute, Leibniz Universität Hannover, Nienburger Str. 1, 30167 Hannover, Germany Institut für Geoinformation und Vermessung Dessau, Anhalt University of Applied Sciences, Seminarplatz 2a, 06846 Dessau, Germany 1 3 Vol.:(0123456789) 496 Acta Geodaetica et Geophysica (2020) 55:495–513 1 Introduction Statistical tests for deformation provide a way to detect potential risks linked with natural hazards or the collapse of artificial objects, such as bridges. One of their main applications in geodesy is the estimation and minimization of risks. Combined with, for example, the concept of utility theory, they allow their consequences and costs to be estimated (Zhang and Neumann 2014). Terrestrial laser scanners (TLS) capture a large amount of three-dimensional (3D) points rapidly, with high precision and spatial resolution. In addition to the visualization of 3D topography or buildings, TLS are, thus, increasingly used to detect and quantify displacements or deformations. Software such as CloudCompare (www.danie lgm.net/cc), 3DReshaper (Hexagon Metrology) or Geomagic Studio (3DSystems) provide maps of deformation that allow one to perform a first deformation analysis. Such software, how - ever, does not perform congruence tests, which necessitate an adequate parametric approx- imation of the point clouds (Niemeier 2002). Such an approximation can be obtained by fitting B-spline surfaces to the TLS point clouds with a least-square (LS) adjustment (Koch 2009). Once the mathematical surface has been obtained, an adapted version of the congru- ence test based on the gridded surface points can be applied (Zhao et al. 2019). The congruence test is known to be the most powerful test in Gauss-Markov models with normally distributed random deviations and a correctly specified stochastic model (Pelzer 1971). To make use of its full potential for risk assessment, the setting of a cor- rect variance covariance matrix (VCM) of the errors has, thus, to be carried out as cor- rectly as possible. One way is to use an empirical model for computing the variances of the errors of the raw TLS observations. The datasheet from the manufacturers provides a good approximation for the angle variances (Boehler and Marbs 2002; Cosarca et al. 2009). The range variance can be modelled with an empirical intensity model, as proposed in Wujanz et  al. (2017) and Wujanz et  al. (2018) and simplified in Kermarrec et  al. (2018). Math- ematical correlations must be accounted for because TLS point clouds approximated with B-spline surfaces have to be transformed into Cartesian co-ordinates. Their impact on the test statistic for deformation between B-spline surface approximations has been studied in previous contributions of the authors: it could be shown that neglecting mathematical correlations affects the congruence test in the case of small deformation and unfavourable scanning geometries (see Zhao et al. 2019 for simulations; Kermarrec et al. 2019a for real data analysis). These results confirmed theoretical derivations (Hoog and Craig 1978) by additionally quantifying the expected impact on the test statistics of neglecting mathemati- cal correlations. Temporal correlations between TLS raw observations are often disregarded due to the asso- ciated computational burden (Kauker and Schwieger 2017). In the same way as mathematical correlations, they have to be taken into consideration to avoid biased test statistics. A usual approach would make use of variance component estimation (Amiri-Simkooei 2007; Searle 1995; Teunissen and Amiri-Simkooei 2008) based on the residuals of the LS adjustment from the B-spline surface fitting. Unfortunately, this procedure may lead to numerical inaccuracies, particularly for small samples: matrices inversion inaccuracies translate into the detection of a non-existing deformation. An alternative is to make use of the residuals by modelling their correlation structure, i.e. estimating the parameters of a known function, such as the Matérn function (Mátern 1960; Stein 1999). Such a statement suggests approximating geometrical and physical effects simultaneously within one function only: we are aware that this strong assumption necessitates deep foundations to identify when and if it holds true. This is the first 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 497 aim of the present contribution, which provides simple and understandable tools to analyse the to analyse the spectral content of the residuals using an empirical mode decomposition (EMD; Flandrin et al. 2004, 2005; Huang et al. 1998). Once the temporal correlation of the residuals is modelled, the assumption of the Chi square distributed test statistics of the congruence test can be investigated. This is our sec- ond goal: deviations from the theory are likely to happen, particularly when temporal correla- tions are neglected. The correct critical value under a given confidence level can be empiri- cally derived following Efron and Tibshirani (1993) using a bootstrapping approach. In a first approximation, we propose fitting the distribution of the test statistic with a Gamma distribu- tion, which includes the Chi square distribution as a particular case. We will make use of real TLS observations from a bridge under load to highlight the deviations from the expected theo- retical distribution (Kermarrec et al. 2019a). The paper is organised as follows: in the second section, the principle of B-spline approxi- mation is presented briefly, along with the test statistics for deformation, with a focus on the bootstrapping procedure. The third section is devoted to the case study, where the theoretical results are applied to observations from a bridge under load. The distribution of the test sta- tistics for deformation is derived with a Monte Carlo approach. We conclude this contribu- tion with some recommendations on how to account for correlations for deformation analysis based on B-spline surfaces. 2 Mathematical background 2.1 B‑spline surfaces Scattered and noisy raw point clouds from TLS can be approximated mathematically by means of a linear combination of basic functions, such as B-splines. The mathematical surface in its parametric formulation can be expressed as n m (u, v) = x , y , z = N (u)N (v) i,j i,j i,j i,p j,q i,j (1) i=0 j=0 where u, v ∈ [0, 1] × [0, 1] are the parameters in the two directions, so that a B-spline ( ) surface maps the unit square to a rectangular surface patch. The choice of the parameters affects the shape of the surface. The uniformly spaced or chord length methods (Bureick et al. 2016; Piegl and Tiller 1997) provide good results to determine u and v for regularly and rectangular-shaped point clouds with rows and w columns. Once a set of parameters is obtained, the knot vector can be obtained over which the basic functions N and N i,p j,q of degree p and q, respectively, are computed. Different methods can be employed; the simplest one being to use a uniformly spaced knot vector or to average the parameters (de Boor 2001). The basic function N u is, a composite curve of degree p polynomials ( ) i,p with joining points at knots in the interval u , u and can be evaluated by means of i i+p+1 a recurrence relationship.  is called the control points vector, with i varying from 0 to i,j n and j from 0 to m. Given the degree of the basic functions, approximating a surface is simply looking to find the control points such that the distance of the data points to the approximated surface is minimized. This step can be performed within the context of a LS of size n + 1 × m + 1 ,3 is the matrix containing the co-ordinates approximation. If (( ) ( ) ) 1 3 498 Acta Geodaetica et Geophysica (2020) 55:495–513 of the control points to be estimated and  3n ,3 n + 1 m + 1 the deterministic ( )( ) obs design matrix containing the values of the tensor product of the B-spline functions,  is the difference between the Cartesian co-ordinates of the point cloud  of size n ,3 and obs . Interested readers should refer to Zhao et al. (2017) for the description of the design matrix. We assume E() =  , where E(⋅) is the expectation operator. We further call  , the ll 2 2 VCM of the errors in Cartesian co-ordinates with  =   , where  is an a priori vari- ll 0 0 ance factor and  the identity matrix of size 3n ,3n . The estimated co-ordinates of the obs obs control points are expressed by the unbiased ordinary LS estimator (Koch 1999) −1 −1 −1 −1 T T T T ̂ =       = (2) ll ll The a priori VCM of the estimated parameters is given by −1 −1 𝚺 = 𝐀 𝚺 𝐀 (3) 𝐱̂𝐱̂ ll The number of control points to estimate can be iteratively adjusted by using informa- tion criteria (e.g. Alkhatib et al. 2018 for the specific application of information criteria to B-spline surface approximation and the references inside). Two possibilities exist: (1) the Akaike information criterion (AIC), which minimizes the Kullback–Leibler divergence of the assumed model from the data-generating model, or (2) the Bayesian information crite- rion (BIC), which assumes that the true model exists and is, thus, more adequate for large samples. 2.2 Stochastic model The estimated co-ordinates of the control points are not affected by the choice of the esti- mated  instead of  , due to the unbiasedness of the LS estimator. However, the ordinary ll ll LS estimator is no longer the most efficient within the class of linear unbiased estima- tors when  deviates from the true VCM and hypothesis tests, such as the global test, ll outlier tests or congruence tests, become invalid (Williams et  al. 2013). The assumption of homoscedasticity has to be adapted in order to derive realistic test statistics based on Eq.  (3). In this contribution, we propose to make use of the residuals from the B-spline surface approximation to approximate  . The VCM of the Cartesian observation errors ll , with 𝐯 ̂ = 𝐥 − 𝐀 𝐱̂ the residuals will, thus, be considered as the VCM of the residuals 𝚺 𝐯̂𝐯̂ of the LS adjustment. Because the residuals are correlated and heteroscedastic, 𝚺 is fully 𝐯̂𝐯̂ populated. The global VCM of the three components is set up by neglecting the correla- tions between components: 𝚺 𝟎𝟎 ⎡ ⎤ ̂ ̂ 𝐯 𝐯,x ̂ ⎢ ̂ ⎥ 𝚺 = 𝟎 𝚺 𝟎 (4) 𝐯̂𝐯̂ 𝐯̂𝐯̂,y ⎢ ⎥ ⎣ 𝟎𝟎 𝚺 ⎦ ̂ ̂ 𝐯 𝐯 ,z As a powerful alternative to empirical VCE, we will assume that the correlation structure of the residuals can be approximated with a parametric Matérn covariance function (“Appendix  1”). We assume, thus, that only a coloured noise is present in the residuals, which is a strong assumption. The impact of geometrical misspecifications on the residuals has to be investigated prior to a fitting with such a parametric covariance function. One way is to make use of maximum likelihood-based model selection, which, 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 499 as indicated by its name, is based on predefined models (often a combination of col- oured and white noise). To our point of view, such procedures answer only conditional the question whether a parametric covariance function is a relevant choice or not; their result may be biased in case of discrepancies from the noise assumption due to model misspecification. As a less computational demanding alternative, we propose making use of the analysis of the power spectral decomposition of the residuals combined with an EMD (“Appendix  2”) to validate the use of such a model, similarly to denoising strategies (Kopsinis and Mclanglin 2009). We propose for didactical purposes to high- light our methodology with a practical example in Sect. 3. Three parameters will have to be estimated: the variance, the range and the smooth- ness of the correlation function from the residuals. A possibility to make a computationally efficient estimation of the parameters in the case of small samples is described in “ Appen- dix 3”. The most important parameter is the smoothness, which it impacts strongly on the first values of the inverse of the VCM and, thus, the value of the test statistics (Kermarrec and Schön 2016). 2.3 Congruence test for deformation analysis In order to test for deformation between two epochs, we follow Zhao et  al. (2019) and make use of an adapted form of the global congruency test derived by Pelzer (1971). No difference between the two vectors of control points co-ordinates (the “parameters”) can be computed because the number of control points estimated at different epochs may vary. Therefore, we test the deformation at the level of the estimated surfaces and not at the parameter level (the control points). Using gridded observations and assuming a deformation only in the z-direction, we only consider 𝚺 from Eq.  (4) and determine the surface difference. The surface points at a 𝐯̂𝐯,̂z given epoch are estimated by a corresponding matrix product 𝐅 𝐱̂ from Eq. (1), where the is evaluated from the tensor product of B-spline functions estimated at the parameters of the gridded points. The uniformly most powerful invariant test for deformation under the null hypothesis can be shown to be based on the test statistic: 𝐓 −𝟏 𝐓 −𝟏 2 ̂ ̂ ̂ ̂ T = 𝚫 𝚺 𝚫 = 𝚫 𝐐 𝚫 ∼ 𝜒 apriori ̂ ̂ ̂ ̂ p 𝚫 𝚫 2 𝚫 𝚫 (5) 𝚺 = 𝐇𝛴 𝐇 ̂ ̂ ̂ ̂ 𝚫 𝚫 𝛃 𝛃 where 𝚺 is the VCM of the estimated CP for both epochs and  = contains the ̂ ̂ 𝛃 𝛃 ̂ LS estimates of the CP at epoch 1 and 2. The matrix  is split into two parts to determine a difference at the level of the surface points:  = −  with p rows. The dif- ̂ ̂ ference between the two surfaces is given by 𝚫 = 𝐇 𝛃 , where we here only consider the dif- ference in the z-direction. We define the hypotheses for deformation analysis as follows: 1 3 500 Acta Geodaetica et Geophysica (2020) 55:495–513 H ∶  = 0 vs H ∶  ≠ 0 0 1 The null hypothesis states that no deformation happened. The test decision involves the quantile k based on the test distribution and the significance level  . In par- test test 1− ∕ ticular, H is accepted if T ≤ k . 0 apriori test 1− ∕ The test statistic T is strongly dependent on the VCM of the CP estimates, which apriori is based on 𝚺 , see Eq. (2). Consequently, any misspecification will affect the decision ̂ ̂ 𝛃 𝛃 step, potentially leading to the erroneous conclusion that no deformation occurs or, con- trarily, that a deformation occurs (error of type I and II). The results are particularly sensitive in the presence of correlated errors: they may lead to the detection of deforma- tions which are not the consequence of an existing deformation but only the result of the correlated noise structure of the observations. We will consider two cases for the VCM in the following: (i) (i) 𝚺 leading to T 𝐯̂𝐯̂ apriori diag (ii) (ii) 𝚺 leading to T 𝐯̂𝐯̂ apriori Fig. 1 Methodology of the congruence test using an estimation of the covariance structure from the residu- als of the LS approximation 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 501 diag ̂ ̂ The VCM 𝚺 is determined by multiplying diag 𝚺 component-wise with the 𝐯̂𝐯,̂z 𝐯̂𝐯,̂z identity matrix of corresponding size. The full methodology is summarized in a flowchart form in Fig. 1 . Additional comment Our approach is not based on a physical model for the raw (polar) TLS observations. In such cases, mathematical correlations would have to be additionally considered due to the transformation from potentially correlated and heteroscedastic polar raw measurements and Cartesian co-ordinates. 2.4 V alidation of the test distribution The defined a priori test statistic is expected to follow a  distribution. This result remains theoretical and may not hold when an approximated stochastic model is used or when cor- relations are present. A way to validate the assumed distribution is to make use of a boot- strapping-based hypothesis test. This elegant procedure allows one to estimate the param- eters of a general Gamma distribution, which includes the  as a particular case (Kargoll et al. 2018; Appendix A.2). Fig. 2 Flowchart of the computational steps to derive the distribution of the test statistics using a Monte Carlo approach 1 3 502 Acta Geodaetica et Geophysica (2020) 55:495–513 In the following, we will briefly present the methodology used for validation, which is further summarized in a flowchart in Fig.  2. Our procedure involves three steps, in the same way as Kargoll et al. (2019): 1. Testing: the first step starts with the approximation of scattered (TLS) observations from 2 epochs with B-spline surfaces. 2. Generating: in the kth-generating step, we add a noise vector, whose structure corre- sponds to the positive definite 𝚺 , to the two approximated surfaces. We use a Cholesky 𝐯̂𝐯̂ ̂ ̂ decomposition and decompose 𝚺 as 𝚺 = 𝐆 𝐆 . We further generate a Gaussian ran- 𝐯̂𝐯̂ 𝐯̂𝐯̂ dom vector  , i = 1, 2 for the two epochs with mean 0 and variance 1 from the Mat- i,k lab random number generator randn. The correlated noise vector reads:  =   . i,k i,k We add this generated noise vector to its corresponding surface and approximate it by a B-spline surface. We compute the a priori test statistic T from the two estimated apriori surfaces for each step. For one iteration k, we call the corresponding test statistic T . apriori 3. Evaluation: we perform k = 1000 iterations. The mean of the k test statistics total total values in the evaluation step are statistically evaluated using the function fitdist from Matlab. We compute the parameters of a Gamma distribution—the  being a particular case corresponding to a scale parameter of 2. In the next section, we propose applying this theoretical derivation to a case study. We will make use of the same TLS observations as in Kermarrec et al. (2019a). The focus of this contribution is on the impact of neglecting the correlations of the Cartesian residuals on the test statistics. We investigate particularly how they may affect (i) the test statistics and (ii) their corresponding test distribution. 3 Case study In this section, we use TLS observations from a bridge under load and test for the defor- mation between two steps of load. We start from a gridded TLS point cloud, which we approximate with B-spline surfaces. Following the methodology described in the previ- ous section, we estimate the correlation structure and the variance from the residuals of the B-spline approximation, which we use to set up the corresponding Toeplitz VCM, assuming stationarity of the variance. The test statistics are computed using both the fully populated VCM of the residuals (case (i)) and the diagonal simplification (case (ii)), i.e. neglecting correlations. After having briefly described the B-splines surface approxima- tion we investigate the impact on the a priori test statistics of the two stochastic models. In a last step, we validate the distribution of the test statistics with the proposed Monte Carlo approach. The justification of the fitting with a parametric covariance function will be didactically explained. 3.1 Description of the data set We use a real data set of a historic masonry arch bridge over the river Aller near Verden in Germany made of circular brick arches. The aim of the experiment was the combination of numerical models and experimental investigations for model calibration (Paffenholz et al. 2018). 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 503 Fig. 3 A bridge under load: localization of the points L10 and L8 Fig. 4 Top left: the gridded point clouds (yellow) and the approximation (green) for L8. Bottom left: the z-residuals. Top right: scaled Toeplitz VCM 𝚺 estimated from the z-component. Bottom right: first ten 𝐯̂𝐯,̂ z −1 values of the first row of 𝚺 𝐯̂𝐯̂,z The standard load of 1.0 MN (100 t) was defined and further loadings up to five times the standard load were realised in the scope of the load testing. Thus, a maximum load of approximately 6.0 MN was defined, produced by four hydraulic cylinders mounted on the arch. In the following, we will call the different load steps E00 (reference), E11, E22, E33, E44 and E55. The deformations will always be computed regarding the reference load, which we will denote E11-00, E22-00, E33-00, E44-00, E55-00. The 3D point cloud acquisition was carried out using TLS sensors of type Zoller + Fröh- lich (Z + F) Imager 5006/h in periods of a constant load on the bridge. The 3D point clouds for different load scenarios ranging from 1 up to 6 MN were captured and finally processed. A filtering regarding objects on the arch surface was performed to eliminate interfering objects, i.e. other sensor installations, such as prisms for the laser tracker and strain gauges. Following Kermarrec et al. (2019a), we selected small patches of observations in order to have an optimal functional model. Using the software CloudCompare, the same surfaces were chosen for each load. The patches chosen were located around two reference Laser Tracker points L8 and L10, as shown in Fig. 3. L10 can be considered as optimally scanned at a short distance in the Up-direction, whereas L8 was scanned with a less favourable geometry. Moreover, L10 has a stronger deformation magnitude than L8 (approximately 10 vs. 4 mm, see Paffenholz et al. 2018). 1 3 504 Acta Geodaetica et Geophysica (2020) 55:495–513 3.2 B‑spline surface approximation The TLS observations were gridded in 15 cells in both directions leading to a total of 225 gridded observations. This number of observations allows for a reliable estimation of the correlation parameters using the Whittle likelihood methods, as presented in “Appendix 3”. The two surfaces are shown in Fig.  4 (left), where it can be seen that they correspond approximately to a plane. All observations falling in one cell were averaged for the grid- ding. The parameterization was carried out using a uniform method, which is efficient for the uncomplicated geometries under consideration. Similarly, the knot vector was taken as equidistant. The number of CP was determined using the Bayesian information crite- rion, which gave similar results to the Akaike information criterion. For each dataset, 4 and 3 CP in the u- and v-direction, respectively, were found to be optimal for L8; 5 CP in both directions were considered for L10. The same values were found for all epochs under consideration. 3.3 Estimation of the VCM We wish to estimate the VCM of the residuals in the z-direction from the residuals of the B-spline surface approximation and assume that a Matérn model can describe the stochas- ticity of the residuals. As mentioned in “Appendix 1”, any Matérn processes will have a typical power spectral decomposition with two different regimes (Lilly et al. 2017): • at high frequencies, a power law decay governed by the smoothness parameter  , and at low frequencies, a constant value, i.e. a “locally white” behaviour governed by the parameter . The power spectral decomposition obtained with the Matlab function pwelch is plot- ted in Fig. 5 (top left). Due to the small sample size under consideration, the identification of the classic Matérn spectral is far from evident, although two different modes can be guessed. In order to justify the use of a Matérn model, we made use of the EMD of the residuals (“Appendix  2”). Since the latter are not smooth, we use ‘pchip’ as an interpola- tion method for the Matlab function EMD and decompose them in five modes. The corre- sponding decomposition is shown exemplarily for L8 in Fig. 5 (bottom left). We define E from the energy-based variance of the IMF, following Flandrin et  al. (2004). It was proven from Flandrin et al. (2005) or Wu and Huang (2004) that the mean energy of the different intrinsic model function (IMF) E from a pure power law noise fol- lows a linear decrease in a semi-log diagram, i.e. log E regarding the IMF index k . The 2 k corresponding plot is shown in Fig.  5 (top right). We observed that after the fourth IMF, the energies diverge significantly from the theoretical model, indicating the presence of significant amounts of “no-coloured noise” signal. This finding is supported by the shape of the corresponding EMD (Fig. 5, bottom left). Following Flandrin et  al. (2005) further, we filtered the noise by excluding the fourth and fifth IMF and interpret these as corresponding to some unknown model misspecifica- IMF1+IMF2+IMF3 tions. Since the ratio of the variances ≈ 25 , we, furthermore, chose to neglect IMF4+IMF5 these two IMF and concentrated on the third first one, which corresponds to the coloured noise. The first IMF carries primarily the white noise, which we validated with the 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 505 Fig. 5 Top left: the power spectral decomposition for L8 and L10 residuals (red and green line, respec- tively) and the decomposition for the IMF2 + 3 of the same residuals as the dashed line. Bottom: the EMD for the L8 residuals. Top right: log E versus the IMF k 2 k Kolmogorov–Smirnov test. Consequently, and in order to facilitate the estimation of the Matérn parameters from IMF2 + IMF3 , we modelled 𝚺 as ̂ ̂ 𝐯 𝐯,z 2 2 ̂ ̂ 𝚺 = 𝜎 𝐈 + 𝜎 𝚺 , with  as the identity matrix. We estimated the cor- 𝐯 ̂𝐯 ̂,z IMF2+IMF3 IMF1 IMF2+IMF3 2 2 responding variances  ,  from IMF1 and IMF2 + IMF3, respectively. IMF1 IMF2+IMF3 is modelled as a Toeplitz VCM and its corresponding Matérn parameters are IMF2+IMF3 estimated using the methodology presented in “Appendix 3”. The power spectral decompo- sition of IMF2 + IMF3 is shown as a dotted line in Fig.  5 (top left): the two different regimes are now clearly identifiable and support an unbiased estimation of the parameters. We summarize our methodology in 4 steps: compute the z-residuals from the B-spline approximation; • perform the EMD. Plot the mean energy with respect to the IMF in a log diagram. Identify the power law noise IMF (straight line). If at that step, model misspecifications dominate, no straight line will be identifiable (LS fitting can be used for that purpose). We can still consider the second IMF as being the coloured noise. However, in that case, a power law model only, such as the Matérn covariance function, will be subopti- mal, particularly if the variance of the third IMF is still high; • separate the white noise IMF from the power-law noise and estimate the corresponding Matérn parameters; and • compute 𝚺 assuming stationarity of the variance, which can be checked with IMF1. ̂ ̂ 𝐯 𝐯,z 1 3 506 Acta Geodaetica et Geophysica (2020) 55:495–513 Table 1  , ,  ,  for L8 and L10 are computed for the first load step E00. The standard devia- IMF1 IMF2+3 tion is given in [mm] L8 L10 , ,  ,   , ,  , IMF1 IMF2+3 IMF1 IMF2+3 E00 [0.5, 0.3, 2.2, 9.1] E00 [0.7, 0.3, 1.58, 6.5] Fig. 6 Values of the test statistics (i) (ii) T and T for the five apriori apriori deformations E00-11, E00-22, E00-33, E00-44 and E00-55. Top: L8, Bottom L10. The red line corresponds to the diagonal and the blue line to the fully populated VCM, respectively The Toeplitz structure of 𝚺 is shown exemplarily in Fig.  4 (top right) for L8 E00. 𝐯̂𝐯,̂z The small level of correlation is highlighted by the sparse structure of the VCM. Addition- ally, we show the structure of the inverse of the VCM with the first ten values of one row of the inverse. After a large value in the diagonal (three times higher than the second one), the following nine values oscillate around and close to 0. Compared with results obtained from diagonal matrices, this particular structure has a strong effect on the test statistics, as mentioned in Kermarrec and Schön (2016). The first value is directly linked with the smoothness of the corresponding process: the power spectral decomposition at high fre- quency should permit a good estimation of this parameter, i.e. a linear decay should be found (Fig. 5, top left, dashed line). The approximation done by neglecting the IMF4 and 5, which corresponds to the low frequency noise, is, thus, fully justified, as well as the splitting between the white noise IMF and the “coloured noise” IMF. We present the values of the Matérn parameters obtained for L8 and L10 (E00) in Table 1. For the sake of brevity, the other similar values are not shown. Larger values of the smoothness and range parameters are found for L8, which is scanned under an unfavourable geometry, compared to L10, for which the correlations are less pronounced. The standard deviations of the power law noise are, however, similar for both points. The amount of white noise is slightly higher for L10, which we justify by the optimal scanning geometry. 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 507 3.4 Impact on the test statistic Having computed the estimated fully populated VCM from the residuals for each load step following the previous methodology, we highlight their impact on the test statis- tics in Fig.  6. To that aim, we follow the methodology proposed in Sect.  2.3 and com- pare the results with the one obtained with a diagonal VCM. We notice that a low level of correlation (L10) combined with a high deformation magnitude leads to a difference between the two stochastic models smaller than for L8. In the latter case, a smaller defor- mation magnitude together with a higher correlation level makes the difference between (i) (ii) (ii) (i) T and T larger. Whereas for L10, T ≈ 1.2T , the factor increases to apriori apriori apriori apriori (ii) (i) T ≈ 1.35T for L8. We interpret the small difference as due to the relatively low apriori apriori level of correlations. However, the factor for L8 is still slightly higher than for L10, as a consequence of the parameter estimates presented in Table 1. (ii) (i) The main result is the overestimation of the test statistics T regarding T : in all apriori apriori (ii) apriori cases, the ratio is higher than 1. As mentioned previously, this has to be linked with (i) apriori the particular structure of the inverse of the VCM, which acts as a variance factor: account- ing for correlations in the test statistics is similar to having decreased the variance of a cor- responding diagonal VCM (Kermarrec et al. 2019b). Even if in our particular case study, the a priori test statistics are in all cases far over the critical value of the Chi square distri- bution with a confidence level of 0.05, this behaviour should not be underestimated for more subtle cases, where the deformation to detect is much smaller. When correlations are present and a correct VCM is used, the test statistics will become smaller and, thus, more realistic: this will lead to accepting the null hypothesis “no deformation”. This is an impor- tant point, as correlations present in the error term should not be considered as true defor- mation. We, thus, strongly encourage the use of a realistic VCM which can be estimated from the residuals. However, these conclusions assume that the Chi square distribution can be used, which needs to be validated or not. Fig. 7 (Left) histogram and corresponding Gamma distribution for case (i) corresponding to an optimal sto- chastic model including correlations. Right: the same for case (ii), i.e. a diagonal VCM neglecting correla- tions. The observations are taken from L8, deformation E00-11 1 3 508 Acta Geodaetica et Geophysica (2020) 55:495–513 3.5 Test distribution (i) (ii) Using the methodology described in Sect.  2.4., the ratio about T and T is com- apriori apriori puted using a Monte Carlo approach. The corresponding histograms and a fitted gamma distribution are presented in Fig. 7. The parameters of the distribution are estimated using the Matlab function fitdist. We justify the choice of the gamma distribution as it includes the assumed Chi squared distribution as a particular case. Further analysis will test other member of the distribution family, such as the exponential. For the sake of brevity, the results correspond to the particular case of L8 and the defor- mation step E00-11. The shape parameters of the Gamma distribution were found to be 3.4 and 3.5 for cases (i) and (ii), respectively. The scale parameters were 575 and 370, respec- tively. Thus, these parameters are different from those of a Chi square distribution, for which the scale parameter would equal 2. Correspondingly, the critical values for a given significance level do not correspond to the assumed one: they are much higher. Exempla- rily, we found a critical value of 5731 with a significance level of 0.05 for (i), whereas a value of 3750 has to be considered for (ii). The ratio of the two critical values is, thus, 1.5 (ii) apriori and larger than . Therefore, the risk of making a wrong testing decision by neglecting (i) apriori correlations is slightly decreased but still exists. Because the spread of the test statistics is, however, much higher in that case, we encourage not disregarding correlations in the test- ing procedure. The other cases (deformation steps as well as L10) led to similar conclusions. The parameters of the Gamma distribution, however, changed from case to case and needed to be revaluated for each data set to determine the correct critical value for a given confidence level. The parameters of the Gamma distribution for L10 were found to be similar for both cases due to the low level of correlations, which are in line with the conclusion of the pre- vious section. The values of the scale parameters were found to be 515 and 585 for case (ii) and (i), respectively, for the deformation step E00-11. We note that they remain between 500 and 600, the same as for L8. A generalization of this finding necessitates further inves- tigations that we propose to carry out in a next step. To conclude, we briefly resume the proposed methodology to perform a rigorous test for deformation when correlations are suspected: 1. from two TLS point clouds, estimate the control points of the two B-spline surfaces with a LS adjustment (Sect. 2.1); 2. compute the residuals; 3. validate whether a Matérn model is adequate (Sect. 3.3); 4. if yes, estimate the corresponding Toeplitz VCM; 5. compute the a priori test statistic following Eq. (5); and 6. perform a Monte Carlo analysis to derive the correct critical values of the congruence test (Fig. 2). 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 509 4 Conclusion An accurate and realistic knowledge of the stochastic properties of TLS errors is of great importance to avoid untrustworthy test decisions. Standard software allows the visualiza- tion of deformation by means of maps of distances, but not for the rigorous statistical test- ing of deformation. This latter can only be achieved when the point clouds are mathemati- cally modelled with B-spline surfaces so that an adapted congruence test at the level of the estimated surface can be performed. However, test statistics for deformation derived from the surface differences between two epochs are known to be strongly influenced by the VCM of the errors chosen. In this contribution, we proposed a simplified stochastic model with data-driven parameters. We made use of the Matérn covariance function, whose parameters are obtained from the residuals of B-spline approximation. Using the EMD, we justified the use of a power-law model from the linear relationship between the mean energy of the IMF and its IMF index. In addition to being simple, the main advantage of this methodology is in its computational stability and the easy identification of model mis- specification. By means of the de-biased Whittle likelihood, the model parameters, includ- ing the variance, can be trustworthily estimated, even for small samples. This procedure allowed us to investigate the impact of correlations on the test statistics and the effect of neglecting them. As a follow-up to a previous contribution dealing with mathematical correlations, we used a real data set from a bridge under load. Two areas cor- responding to different scanning geometries and magnitudes of deformation were approxi- mated with B-spline surfaces using optimal fitting parameters. It could be shown that the test statistics obtained by neglecting correlations were overoptimistic: accounting for cor- relations acts as decreasing the variance in an equivalent diagonal matrix. As expected, the correlation structure was found to depend strongly on the scanning geometry, i.e. an optimized geometry with short range and optimal incidence angle decreased the level of correlations of the LS residuals and, thus, the impact of neglecting correlations in the test statistics for deformation. Correlations should not be neglected for a trustworthy test deci- sion: they provide a more trustworthy test decision for deformation. A Gamma distribu- tion could be fitted from the test statistics using a Monte Carlo approach. Since the critical values of the distribution—as well as the corresponding test statistics—were larger when correlations were neglected, the risk of making a wrong decision that a deformation occurs was decreased. However, the optimal test distribution should be investigated further. This will be the topic of our next contribution. Acknowledgements Open Access funding provided by Projekt DEAL. 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To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. 1 3 510 Acta Geodaetica et Geophysica (2020) 55:495–513 Appendix 1 The Matérn process is a three-parameter continuous Gaussian process defined by an unali- ased spectrum W : 2−1 2 W() = (6) 2 2 where  is the frequency. In this formulation,  is the variance of the process, c a normal- izing constant,  the range parameter and  the smoothness parameter (Lilly et al. 2017). Thus, whereas the smoothness determines the behaviour of the spectral density as  → ∞ , the range parameter has a strong impact at low frequency as  → 0 . The corresponding autocovariance function is found to be (Mátern 1960): C() =  M () (7) where we introduce the Matérn function as M () =    K () 1 − ∕ (8) − ∕ 2 − ∕ 2 with  the gamma function, K the decaying modified Bessel function of the second − ∕ kind of order  − ∕ . 𝜏> 0 is the time difference,  a range parameter that controls how quickly C decreases as  increases, whereas  is related to the mean-square differenti- matern ability of the process at the origin, as  → 0. Figure  8 (left) displays C() scaled to 1 (i.e. the correlation function) for different choices of the shape parameter  . The smoothness parameter determines the behaviour of the covariance function as  approaches 0. It has a great importance since it determines how fully populated the inverse of the VCM will be and how large its impact is. The following cases are well-known for the one-dimensional field and should be mentioned: Fig. 8 Matérn correlation function. Left: variation of the smoothness parameter  with  = 0.05 . Right: var- iation of the range parameter  by keeping  fixed to 2 1 3 Acta Geodaetica et Geophysica (2020) 55:495–513 511 •  = ∕ corresponds to the exponential covariance function, i.e. a short memory pro- cess •  = 1 to the Markov process of first order, and •  =∞ gives the squared exponential covariance function, which corresponds to a phys- ically less plausible infinitely differentiable Gaussian process at the origin. Figure 8 (right) shows the different correlation functions obtained by varying the range parameter  . As can be seen,  is linked to the decay of the covariance function. Please note that other parametrizations of the Matérn function are possible (Stein 1999) Appendix 2 Huang et  al. (1998) developed the EMD to decompose a given time series into a set of components of different time scales called the Intrinsic Mode Functions (IMF). Combined with the Hilbert Transform, this decomposition provides a time–frequency-energy repre- sentation of a signal and is intuitively based on the wave form of the data; it is also called the Hilbert-Huang transform in the literature. The EMD is independent of mother wave- lets, such as in wavelet analysis or trigonometric functions, for example, in Fourier trans- form. Thus, it can be used on non-stationary signals and allows a characterization by direct extraction of the local energy associated with the IMF. The decomposition is, however, not unique and depends exemplarily on the particular interpolation scheme for envelope extraction, the stopping criterion (usually 1e-6) used in the sifting process and the number of IMF to determine. The EMD algorithm can be summarized as the following loop (Flandrin et al. 2004): 1. Identification of the extrema of the signal 2. Interpolation between minima (maxima) to get an envelope of the signal 3. Computation of the average of the envelope 4. Extraction of the detail as difference between the envelope and the original signal 5. Iteration on the residual The IMF should have the following properties: (i) zero mean, (ii) all the maxima and all the minima of h(i)(t) are correspondingly positive and negative (iii) narrowband. The num- ber of zero-crossings can be either N(i) or N(i) − 1. Each IMF has fewer extrema than all the lower order IMFs, leading to fewer and fewer oscillations as the IMF order increases. The EMD components of coloured noise (i.e. having a spectral decomposition following a power law) have the same rescaled Fourier spectrum. Thus, apart from the first noise- only IMF which carries the main energy and is considered as the white noise component, the energies of the other IMF decrease linearly in a semi-log diagram regarding the IMF index. We understand under “energy” the empirical variance estimate of a given IMF. Any discrepancies from the linear relationship can be used to denoise a signal, see, for example, Kopsinis and McLaughlin (2009). In this contribution, the noise itself is of interest as it is used to estimate the Matérn parameters: we make use of this important property to extract the noise from the other IMF, considered as model misspecifications. 1 3 512 Acta Geodaetica et Geophysica (2020) 55:495–513 Appendix 3 Assuming Gaussian data, the parameters of the Matérn model—including the variance— can be estimated with Maximum Likelihood, eventually by fixing one parameter to reduce the computational burden (Kaufman et al. 2008). This standard approach has a computa- tional efficiency of the order of n for regularly spaced observations. It produces estimates obs that are asymptotically efficient, i.e. for smaller samples alternatives such as Restricted Maximum Likelihood have to be employed. Another powerful approach is referred as “quasi-likelihood”, as introduced in Sykulski et  al. (2019) using the Whittle likelihood (Percival and Walden 1993). This method for estimating the Matérn parameters of any time series modelled as a second-order stationary stochastic process is employed in the freely available software package www.ulc.ac.uk/stati stics /resea rch/spg/softw are. Exact maximum likelihood inference can be performed by maximizing the log-likeli- hood. This strategy, however, involves matrix inversion so that Whittle (1953) proposed approximating the autocovariance matrix in the frequency domain. The “Whittle likeli- hood” makes use of the periodogram of the observations, which may be a biased measure for finite samples (Percival and Walden 1993) due to blurring and aliasing effects. Sykulski et  al. (2019) proposed an alternative called the “de-biased Whittle likelihood”, which is particularly interesting for discrete-time processes of finite length, such as the residuals of the LS adjustment used in this contribution. 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