# How Significant Are Differences Obtained by Neglecting Correlations When Testing for Deformation: A Real Case Study Using Bootstrapping with Terrestrial Laser Scanner Observations Approximated by B-Spline Surfaces

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

#### 2.1. Approximation of TLS Point Cloud with B-Spline Surfaces

#### 2.2. Determination of the Optimal Number of CP by Information Criteria

- (i)
- the Akaike information criterion (AIC), which minimizes the Kullback-Leibler divergence of the assumed model from the data-generating model, or
- (ii)
- the Bayesian information criterion (BIC), which assumes that the true model exists and is thus more adequate for large samples.They are defined as:$$\begin{array}{l}AIC=-2\left[l\left(\widehat{p}\right)\right]+2{n}_{obs}\\ BIC=-2\left[l\left(\widehat{p}\right)\right]+\mathrm{log}\left(3\left(n+1\right)\left(m+1\right)\right){n}_{obs}\end{array}$$

#### 2.3. Stochastic Model for TLS Observations

#### 2.4. Mathematical Correlations

#### 2.4.1. Fully Populated VCM

#### 2.4.2. Simplification of the VCM

- (i)
- The approximated VCM ${\widehat{\mathsf{\Sigma}}}_{(i)}=F{\widehat{\mathsf{\Sigma}}}_{ori}{F}^{T}$ is used in the adjustment and further computation
- (ii)
- The diagonal values of (i) are only considered and the approximated diagonal VCM ${\widehat{\mathsf{\Sigma}}}_{(ii)}$ is built as follows: $diag\left({\widehat{\mathsf{\Sigma}}}_{(ii)}\right)=diag\left({\widehat{\mathsf{\Sigma}}}_{(i)}\right)$.
- (iii)
- The scaled identity matrix: ${\widehat{\mathsf{\Sigma}}}_{(iii)}={\sigma}_{mean}^{2}I$ is used. The scaling factor ${\sigma}_{mean}^{2}$ is computed as the mean of the diagonal values of ${\widehat{\mathsf{\Sigma}}}_{(ii)}$. $I$ is the identity matrix of size $\left(3{n}_{obs,}3{n}_{obs,}\right)$.

#### 2.5. Test for Deformation

#### 2.5.1. Test Statistic

#### 2.5.2. Bootstrap p-Values

- (1)
- Testing step: The first step starts with the approximation of scattered (TLS) observations from 2 epochs with B-spline surfaces. In the second step, a large number of observation vectors under ${H}_{0}$ have to be generated. We define a so-called bootstrap sample, as the mean of the surface differences, i.e., ${S}_{H0}=\frac{{S}_{2}-{S}_{1}}{2}$ considered as being generated under ${H}_{0}$ that no deformation occurs.
- (2)
- Generating step: The generating step begins by adding to the generated bootstrap surface a noise vector, which structure corresponds to ${\widehat{\mathsf{\Sigma}}}_{(i)}$. We use a Cholesky decomposition of the VCM ${\widehat{\mathsf{\Sigma}}}_{(i)}={G}^{T}G$ and generate a Gaussian random vector ${W}_{noise,i,k},i=1,2$ for the two epochs with mean 0 and variance 1 from the Matlab random number generator randn. The noise vector thus reads: ${N}_{i,k}={G}^{T}{W}_{noise,i,k}$ Added to ${S}_{H0}$, we generate consecutively two noised surfaces, which we approximate with B-splines surfaces. Finally, we compute the aposteriori test statistics ${T}_{post}$. For one iteration ${k}_{BS}$, we call the corresponding test statistics ${T}_{post}^{{k}_{BS}}$.
- (3)
- Evaluation step: ${K}_{BS}$ iterations are carried out. Following Davindson and McKinnon [25], we fixed ${K}_{BS}=999$. Finally, the p-value is estimated by $\widehat{p}{v}_{HD}=\frac{1}{{K}_{BS}}{\displaystyle \sum _{{k}_{BS}=1}^{{K}_{S}}I\left({T}_{post}^{{k}_{BS}}-{T}_{ref}\right)}$ according to McKinnon [23]. $I$ is an indicator function, which takes the value 1 when ${T}_{post}^{{k}_{BS}}>{T}_{ref}$ and 0, on the contrary.
- (4)
- Decision test: A large $\widehat{p}{v}_{HD}$ indicates a large support of ${H}_{0}$ by the observations. Assuming that all assumptions were correct, ${H}_{0}$ is rejected if $\widehat{p}{v}_{HD}<{\alpha}_{test2}$, where ${\alpha}_{test2}$ is the specified significance level, usually taken to 0.05.

#### 2.6. Interpreting the p-Values

## 3. Case Study

#### 3.1. Experiment Design and Data Acquisition

#### 3.1.1. Surface Approximation

#### 3.1.2. Stochastic Model for TLS

#### 3.1.3. Stochastic Model for LT

^{2}(i.e., at the submm level for the corresponding standard deviation) cannot be considered as significant enough to draw conclusions from them. The variance for the z-component is higher than for the other components, as it is strongly depending on the range variance of the raw measurements, which is higher than the angle variances (see Equation (9)). The points L13 and L8 have the same deformation magnitude under load. However, as these 2 points correspond to two different geometries (Figure 4, right), the corresponding observations have, thus, different variances for the x-, y- and z-components. Similarly, the intensity values of the TLS measurements led to different standard deviations of the range. These two results are coherent.

#### 3.2. Results

#### 3.3. Discussion

#### 3.3.1. Role of ${\sigma}_{\rho}$ on the p-Values

#### 3.3.2. Role of ${\sigma}_{\rho}$ on ${T}_{post}$

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Holst, C.; Kuhlmann, H. Challenges and Present Fields of Action at Laser Scanner Based Deformation Analysis. J. Appl. Geod.
**2016**, 10, 17–25. [Google Scholar] - Pelzer, H. Zur Analyse Geodatischer Deformations-Messungen; Verlag der Bayer. Akad. d. Wiss, Muenchen, Beck400: Muenchen, Germany, 1971; p. 164. [Google Scholar]
- Zhao, X.; Kermarrec, G.; Kargoll, B.; Alkhatib, H.; Neumann, I. Influence of the simplified stochastic model of TLS measurements on geometry-based deformation analysis. J. Appl. Geod.
**2019**, 13. [Google Scholar] [CrossRef] - Kermarrec, G.; Alkhatib, H.; Bureick, J.; Kargoll, B. Impact of mathematical correlations on the statistic of the congruency test case study: B-splines surface approximation from bridge observations. In Proceedings of the 4th Joint International Symposium on Deformation Monitoring (JISDM), Athens, Greece, 15–17 May 2019. [Google Scholar]
- Kermarrec, G.; Schön, S. Taking correlation into account with a diagonal covariance matrix. J. Geod.
**2016**, 90, 793–805. [Google Scholar] [CrossRef] - Soudarissanane, S.; Lindenbergh, R.; Menenti, M.; Teunissen, P. Scanning geometry: Influencing factor on the quality of terrestrial laser scanning points. ISPRS J. Photogramm. Remote Sens.
**2011**, 66, 389–399. [Google Scholar] [CrossRef] - Wujanz, D.; Burger, M.; Mettenleiter, M.; Neitzel, F. An intensity-based stochastic model for terrestrial laser scanners. ISPRS J. Photogramm. Remote Sens.
**2017**, 125, 146–155. [Google Scholar] [CrossRef] - Jurek, T.; Kuhlmann, H.; Host, C. Impact of spatial correlations on the surface estimation based on terrestrial laser scanning. J. Appl. Geod.
**2017**, 11, 143–155. [Google Scholar] [CrossRef] - Kermarrec, G.; Schön, S. Apriori fully populated covariance matrices in Least-squares adjustment—Case study: GPS relative positioning. J. Geod.
**2017**, 91, 465–484. [Google Scholar] [CrossRef] - Wasserstein, R.L.; Lazar, N.A. The ASA’s Statement on p-Values: Context, Process, and Purpose. Am. Stat.
**2016**, 70, 129–133. [Google Scholar] [CrossRef] - Kargoll, B.; Omidalizarandi, M.; Paffenholz, J.A.; Neumann, I.; Kermarrec, G.; Alkhatib, H. Bootstrap tests for model selection in robust vibration analysis of oscillating structures. In Proceedings of the 4th Joint International Symposium on Deformation Monitoring (JISDM), Athens, Greece, 15–17 May 2019. [Google Scholar]
- Bureick, J.; Alkhatib, H.; Neumann, I. Robust spatial approximation of laser scanner points clouds by means of free-form curve approaches in deformation analysis. J. Appl. Geod.
**2016**, 10, 27–35. [Google Scholar] [CrossRef] - Piegl, L.; Tiller, W. The NURBS Book; Springer: Berlin, Germany, 1997. [Google Scholar]
- Ma, W.; Kruth, J.P. Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces. Comput.-Aided Des.
**1995**, 27, 663–675. [Google Scholar] [CrossRef] - de Boor, C.A. Practical Guide to Splines, Revised ed.; Springer: New York, NY, USA, 2001. [Google Scholar]
- Alkhatib, H.; Kargoll, B.; Bureick, J.; Paffenholz, J.A. Statistical evaluation of the B-Splines approximation of 3D point clouds. In Proceedings of the 2018 FIG-Kongresses, Istanbul, Turkey, 6–11 May 2018. [Google Scholar]
- Boehler, W.; Marbs, A. 3D Scanning instruments. In Proceedings of the CIPA WG6 International Workshop on Scanning for Cultural Heritage Recording, Corfu, Greece, 1–2 September 2002. [Google Scholar]
- Zámêcníková, M.; Neuner, H.; Pegritz, S.; Sonnleitner, R. Investigation on the influence of the incidence angle on the reflectorless distance measurement of a terrestrial laser scanner. Österr. Z. Vermess. Geoinform.
**2015**, 103, 208–218. [Google Scholar] - Hebert, M.; Krotkov, E. 3D measurements from imaging laser radars: How good are they? Image Vis. Comput.
**1992**, 10, 170–178. [Google Scholar] [CrossRef] - Wujanz, D.; Burger, M.; Tschirschwitz, F.; Nietzschmann, T.; Neitzel, F.; Kersten, T.P. Determination of intensity-based stochastic models for terrestrial laser scanners utilising 3D-point clouds. Sensors
**2018**, 18, 2187. [Google Scholar] [CrossRef] [PubMed] - Kermarrec, G.; Alkhatib, H.; Neumann, I. On the Sensitivity of the Parameters of the Intensity-Based Stochastic Model for Terrestrial Laser Scanner. Case Study: B-Spline Approximation. Sensors
**2018**, 18, 2964. [Google Scholar] [CrossRef] [PubMed] - Teunissen, P.J.G. Testing Theory; An Introduction; VSSD Publishing: Delft, The Netherlands, 2000. [Google Scholar]
- McKinnon, J. Bootstrap Hypothesis Testing; Queen’s Economics Department Working Paper, No. 1127; Queen’s University: Kingston, ON, Canada, 2007. [Google Scholar]
- Efron, B. Bootstrap methods: Another look at the Jackknife. Ann. Stat.
**1979**, 7, 1–26. [Google Scholar] [CrossRef] - Davidson, R.; MacKinnon, J.G. Bootstrap Tests: How Many Bootstraps? Econom. Rev.
**2009**, 19, 55–68. [Google Scholar] [CrossRef] - Schacht, G.; Piehler, J.; Müller, J.Z.A.; Marx, S. Belastungsversuche an einer historischen Eisenbahn-Gewölbebrücke. Bautechnik
**2017**, 94, 125–130. [Google Scholar] [CrossRef] - Paffenholz, J.A.; Huge, J.; Stenz, U. Integration von Lasertracking und Laserscanning zur optimalen Bestimmung von lastinduzierten Gewölbeverformungen. Allg. Vermess.-Nachr.
**2018**, 125, 75–89. [Google Scholar] - Lenzmann, L.; Lenzmann, E. Strenge Auswertung des nichtlinearen Gauß Helmert-Modells. Allg. Vermess.-Nachr.
**2004**, 111, 68–73. [Google Scholar] - Kermarrec, G.; Neumann, I.; Alkhatib, H.; Schön, S. The stochastic model for Global Navigation Satellite Systems and terrestrial laser scanning observations: A proposal to account for correlations in least squares adjustment. J. Appl. Geod.
**2019**, 13, 93–104. [Google Scholar] [CrossRef]

**Figure 2.**Side view from west of the arch 4 of the historic masonry arch bridge. The whitewashed area indicates the area of the direct influence of the load application. On the bridge: four hydraulic cylinders for the load application (Paffenholz et al. [27]). Reproduced with permission from Paffenholz, AVN; published by Wichmann-Verlag, 2018.

**Figure 3.**Representation of the bridge under load with the localization of the three patches L13, L10 and L8. The load was positioned approximately in the middle of the bridge under which the TLS was positioned (image adapted from Paffenholz et al. [27]). Reproduced with permission from Paffenholz, AVN; published by Wichmann-Verlag, 2018.

**Figure 4.**Left: Localisation of the TLS surfaces (red rectangles) and corresponding LT points. Right: mean HA, VA in [°] and range for the surface under consideration.

**Figure 5.**Top: Euclidian distance in [mm] between the epochs under consideration for LT measurements (Def1, Def2, Def3, Def4, Def5). 3 LT points are considered: L8, L10 and L13. Bottom: corresponding apriori variance of the distance difference in [mm2]: * for x-, + for y- and o for z-component, respectively.

**Figure 6.**p-Values obtained for L8 (top), L10 (middle) and L13 (bottom) for the three stochastic model ((i), blue accounting for mathematical correlation), (ii), diagonal values of (i) and (iii), scaled identity matrix). The values obtained for the 5 deformation cases Def01, 02, 03, 04, 05 corresponding to increasing load are linked by a line for the sake of readability. The dotted line for L13 (bottom) corresponds to the values that would have been obtained by artificially increasing ${\sigma}_{\rho}$ to 5 mm.

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**MDPI and ACS Style**

Kermarrec, G.; Paffenholz, J.-A.; Alkhatib, H. How Significant Are Differences Obtained by Neglecting Correlations When Testing for Deformation: A Real Case Study Using Bootstrapping with Terrestrial Laser Scanner Observations Approximated by B-Spline Surfaces. *Sensors* **2019**, *19*, 3640.
https://doi.org/10.3390/s19173640

**AMA Style**

Kermarrec G, Paffenholz J-A, Alkhatib H. How Significant Are Differences Obtained by Neglecting Correlations When Testing for Deformation: A Real Case Study Using Bootstrapping with Terrestrial Laser Scanner Observations Approximated by B-Spline Surfaces. *Sensors*. 2019; 19(17):3640.
https://doi.org/10.3390/s19173640

**Chicago/Turabian Style**

Kermarrec, Gaël, Jens-André Paffenholz, and Hamza Alkhatib. 2019. "How Significant Are Differences Obtained by Neglecting Correlations When Testing for Deformation: A Real Case Study Using Bootstrapping with Terrestrial Laser Scanner Observations Approximated by B-Spline Surfaces" *Sensors* 19, no. 17: 3640.
https://doi.org/10.3390/s19173640