# 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

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**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