# Roughness Spectra Derived from Multi-Scale LiDAR Point Clouds of a Gravel Surface: A Comparison and Sensitivity Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Related Work and Objectives

- To analyze the influence of basic DTM interpolation methods on TLS, ULS, and ALS roughness spectra;
- To analyze how the number of TLS scans affects the roughness spectrum;
- To compare TLS, ULS, and ALS roughness spectra.

## 2. Study Site and Data

#### 2.1. ALS Data

^{2}for a single strip (i.e., two scans). The pulse repetition rate was 400 kHz and the flying speed was approximately 56 m/s (110 knots). The plot was covered with four strips, resulting in an average point density of 36 points per m

^{2}. This corresponds to an average point sampling of approximately 17 cm. As the flying height was 640 m above the terrain, the footprint diameter was also approximately 16 cm (the beam divergence being 0.25 mrad). This meant that the footprints of the individual ALS samples were (on average) touching, or just slightly overlapping one another.

#### 2.2. ULS Data

^{2}for a single ULS strip. The effective measurement rate was 350 kHz (cf. [29]) and the flying speed was approximately 8 m/s (15.5 knots). The plot was covered with ten strips (the effective FoV: 230°, cf. [29]), resulting in an average point density of 11.8 points per dm

^{2}. This corresponds to an average point sampling of ~2.9 cm. As the flying height was 50 m above the terrain, the footprint diameter was ~2.5 cm (the beam divergence being 0.5 mrad). Thus, the footprints of the individual ULS samples were (on average) not overlapping with one another. As in the case of the ALS data, the raw ULS data had been preprocessed before (cf. [28,29]), providing a georeferenced point cloud in the same global coordinate system. The standard deviation of the point-to-plane residuals was reported to be below 2 cm for both the ULS strips only and the ULS and ALS strips combined [28].

#### 2.3. TLS Data

^{®}5010c mounted on a high tripod. This scanner utilizes phase shift ranging and has a small beam divergence (0.3 mrad). The beam diameter at the exit (0.1 m range) is 3.5 mm, which allows for a high-resolution scanning of close objects (maximum range < 130 m). The ranging precision of this instrument is 0.3 mm, which is specified by the manufacturer for ranges smaller than 10 m and for grey targets [33].

^{2}, which corresponds to an average point sampling of 0.7 mm. As the scanner height above the terrain was between 2.4 m and 2.6 m and the range was not larger than 4.5 m, the TLS footprint diameter was smaller than 5 mm within the plot. This meant that the TLS footprints were largely overlapping one another. For a single scan, the average sampling distance was ~2.4 mm (i.e., ~17 points per cm

^{2}) and thus, the TLS footprints are largely overlapping one another even for the single scan samples. Such a scan maximizes the resolution of the TLS data and that scanning scheme is also known as correlated scanning [34]. Furthermore, scanning from a high tripod ensured that the incidence angle was smaller than 52° for 90% of the data, which fulfilled the recommendations for soil roughness scanning [3].

^{®}software (version 8.6) [36]. The georeferencing of the TLS scans was done indirectly, using four ground control points (GCPs) and a ‘point block adjustment’ method implemented in the above software. The GCPs were located outside the plot and scan positions, ensuring target visibility in all scans and a favorable network geometry for the co-registration. The coordinates of the GCPs were derived from the total station measurements and in the global coordinate system of the ALS and ULS data. After georeferencing the individual TLS scans, the standard deviation of the 3D distance residuals at the GCPs was 1.6 mm. The same statistic derived for individual TLS scans was 0.8 mm. The georeferenced TLS scans were then exported for an improved, global co-registration with a version of the Iterative Closest Point (ICP) algorithm [28]. The standard deviation of the point-to-plane distances between all scans after this global co-registration was 0.6 mm. This suggests a good co-registration as the ranging noise of the TLS scanner alone was ~0.3 mm [33].

#### 2.4. Handheld Images

## 3. Methods

#### 3.1. Derivation of the Roughness Spectra

#### 3.1.1. Co-Registration and Detrending

#### 3.1.2. DTM Interpolation

#### 3.1.3. Roughness Spectra and Periodograms

#### 3.1.4. Windowing, Variance Reduction, and Confidence Intervals

#### 3.2. Comparison of Roughness Spectra

#### 3.2.1. Roughness Spectra and the DTM Interpolation Method

#### 3.2.2. Roughness Spectra and the Number of TLS Scans

#### 3.2.3. Roughness Spectra and Multi-Scale LiDAR Data

## 4. Results

#### 4.1. Overview of the LiDAR DEMs and Their Quality

#### 4.2. Sensitivity to the DTM Interpolation Method

#### 4.3. Sensitivity to the Number of TLS Scans

#### 4.4. Comparison of TLS and ULS Roughness Spectra

#### 4.5. Multi-Scale Spectra and DTMs

## 5. Discussion

#### 5.1. Interpretation of the Results

#### 5.1.1. DTM Interpolation

#### 5.1.2. TLS Scan Setup

- a single TLS scan can measure the roughness spectra at wavelength scales of either 4 dm–20 m or 5 cm–10 m, with a maximum spectral difference less than 0.5 dB (the dB threshold value).

#### 5.1.3. Multi-Scale Data and Spectral Differences

#### 5.2. Spectral Analysis

#### 5.3. The dB Threshold

#### 5.4. ALS and ULS Data

#### 5.5. Limitations and Suggested Further Experiments

## 6. Conclusions

## Supplementary Material

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Study site properties: (

**a**) location of the study site and the Pielach river; (

**b**) an aerial nadir photo of the gravel plot and the point bar; (

**c**) a photo of the terrestrial laser scanning (TLS) setup and the gravel plot; (

**d**) microtopography of the plot visualized by the color-coded digital terrain models (DTM) of the plot; (

**e**) typical pebble sizes within the plot.

**Figure 2.**The workflow applied to process the georeferenced point clouds to their roughness spectra. The bottom workflow is applied for processing the handheld images.

**Figure 6.**The influence of the nearest neighbor (NN), triangular irregular network (TIN), and four-point moving planes (MP) interpolations on the roughness spectra. The figures in columns (from left to right) refer to the TLS-, ULS-, ALS- datasets, respectively. The top-row plots, (

**a**,

**d**,

**g**), show the TLS-, ULS-, and ALS- roughness spectra, respectively. The middle-row plots, (

**b**,

**e**,

**h**), show the TLS-, ULS-, and ALS- spectral ranges, respectively. The bottom-row plots, (

**c**,

**f**,

**i**), show the absolute differences between the DIM spectrum and TLS-, ULS, and ALS spectra referring to each interpolation method, with the horizontal extent being adapted to the respective column. The vertical solid line shows the footprint wavelength.

**Figure 7.**The influence of the number of TLS scans on roughness spectra; (

**a**) the measurement setups when TLS scans are distributed on both sides of the plot; (

**b**) the measurement setups with a single scan placed at the middle of one of the logger plot sides; (

**c**) the measurement setups with a single scan placed at one of the plot corners. The vertical solid line shows the footprint wavelength. For comparison, the spectra for the 7+7 Scan Setup and the DIM DTM are also shown.

**Figure 8.**Comparison of the TLS and ULS spectra: (

**a**) the spectra, (

**b**) their spectral differences, and (

**c**) the relative errors of the spectral slope derived from TLS and ULS dataset.

**Figure 9.**(

**a**) The spectra from the multi-scale data, and (

**b**–

**e**) the color-coded height values of the corresponding DTMs. The black and red rectangles in the DTMs mark the sub-areas shown in the following figure.

**Table 1.**An overview of the data used in this paper. The data characteristics refer to the gravel plot area.

Technique | Sensor | Slant Range | Footprint Diameter | Mean Sampling Distance | Overlapping Footprints | Processing Stage |
---|---|---|---|---|---|---|

(m) | (mm) | (mm) | ||||

ALS | RIEGL LSM-Q1560 | 640–720 | ~160 | 167 | No | Georef. and block adj. strips |

ULS | RIEGL VUX-1 UAV | 50 ^{2} | ~25 | 29 | No | Georef. and block-adj. strips |

TLS | Z+F Imager 5010c | 2.5–4.5 | ~5 | 0.7 | Yes | Raw data |

DIM ^{1} | Nikon D800 | 1.5–2.7 | 0.5 ^{3} | 0.5 ^{3} | No | Raw data |

^{1}Dense Image Matching;

^{2}Flying height above ground;

^{3}Ground Sampling Distance (GSD).

Precision | Co-Reg. Error | PointDensity | ||||
---|---|---|---|---|---|---|

Ranging | Angular | Planar | Vertical | |||

(mm) | (°) | (mm) | (mm) | (mm) | (pts/dm^{2}) | |

TLS | 0.3 | 0.007 | 0.4 | 0.5 | 0.6 | 19,030 |

ULS | 5 | 0.001 | 2.6 | 5 | 14 | 11.8 |

ALS | 20 | 0.001 | 13 | 20 | 10 | 0.36 |

**Table 3.**The general statistics (mean and standard deviation) derived from the vertical residuals (DoD) between each LiDAR digital terrain model (DTM) and the DIM DTM.

DoD NN | DoD TIN | DoD MP | ||||
---|---|---|---|---|---|---|

Mean | Std. | Mean | Std. | Mean | Std. | |

(mm) | (mm) | (mm) | (mm) | (mm) | (mm) | |

TLS | –0.2 | 3.2 | –0.2 | 2.8 | –0.2 | 2.8 |

ULS | 6.6 | 10.8 | 6.6 | 8.8 | 6.7 | 9.1 |

ALS | 8.9 | 20.1 | 9.1 | 16.0 | 8.8 | 16.4 |

**Table 4.**Summary of the spectral range analysis based on the sensitivity of the basic interpolation methods (NN, TIN, and four-point MP). The dB threshold, ${\lambda}_{th}$ and $\Delta \mathrm{S}$ are introduced in Section 3.2. The values in the ${\lambda}_{\mathrm{footprint}}$ column are spatial wavelengths that correspond to the size of the laser footprint diameter. The $\Delta \mathrm{S}({\lambda}_{\mathrm{footprint}})$ column shows the spectral range values found at the ${\lambda}_{\mathrm{footprint}}$. The last two columns list the figures where the parameters are first shown and where they are later used, respectively.

Data | dB Threshold | ${\mathit{\lambda}}_{\mathit{t}\mathit{h}}$ | $\mathbf{\Delta}\mathbf{S}({\mathit{\lambda}}_{\mathit{t}\mathit{h}})$ | ${\mathit{\lambda}}_{\mathbf{footprint}}$ | $\mathbf{\Delta}\mathbf{S}({\mathit{\lambda}}_{\mathbf{footprint}})$ | Derived in | Used in |
---|---|---|---|---|---|---|---|

(dB) | (cm) | (dB) | (cm) | (dB) | |||

TLS | 0.5 | 1.8 | 0.5 | 0.5 | 3.2 | Figure 6b | Figure 8a and Figure 9a |

ULS | 0.5 | 100 | 0.5 | 2.5 | 8.5 | Figure 6e | Figure 8a–c and Figure 9a |

ALS | 0.5 | 500 | 0.5 | 16 | 7.9 | Figure 6h | Figure 9a |

**Table 5.**Summary of the spectral range analyses based on the sensitivity of the number of TLS scans. The spectral differences ($\Delta \mathrm{S}$) are calculated relative to the 7+7 TLS spectrum (when all 14 scans are used). The dB threshold, ${\lambda}_{th}$ and $\Delta \mathrm{S}$ are introduced in Section 3.2. The values in the $\Delta \mathrm{S}({\lambda}_{th})$ column are $\Delta \mathrm{S}$ values found at the wavelength ${\lambda}_{th}$. The values in the column are ${\lambda}_{\Delta \mathrm{S}>0.65\mathrm{dB}}$ spatial wavelengths where $\Delta \mathrm{S}$ exceeds 0.65 dB. The latter dB value is the rounded $\mathrm{max}\left[\Delta \mathrm{S}({\lambda}_{th})\right]$, which is introduced to show that the Single Scans Centre spectra, the 2+2 spectra, and the 1+1 spectra just slightly violate the dB threshold (0.5 dB), i.e., only around ${\lambda}_{th}$ wavelengths.

Setup | Number of Scans | dB Threshold | ${\mathit{\lambda}}_{\mathit{t}\mathit{h}}$ | $\Delta \mathbf{S}({\lambda}_{\mathit{t}\mathit{h}})$ | ${\mathit{\lambda}}_{\Delta \mathbf{S}>0.65\mathbf{dB}}$ | Derived in |
---|---|---|---|---|---|---|

(dB) | (cm) | (dB) | (cm) | |||

3+3 Scan setup | 6 | 0.5 | 0.4 | 0.5 | 0.3 | Figure 6 and Figure 7a |

2+2 Scan Setup | 4 | 0.5 | 3.6 | 0.57 | 0.5 | Figure 6 and Figure 7a |

1+1 Scan Setup | 2 | 0.5 | 4.1 | 0.62 | 0.8 | Figure 6 and Figure 7a |

Single Scan Centre 1 | 1 | 0.5 | 5.3 | 0.61 | 0.6 | Figure 7b |

Single Scan Centre 2 | 1 | 0.5 | 4.5 | 0.58 | 0.5 | Figure 7b |

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## Share and Cite

**MDPI and ACS Style**

Milenković, M.; Ressl, C.; Karel, W.; Mandlburger, G.; Pfeifer, N. Roughness Spectra Derived from Multi-Scale LiDAR Point Clouds of a Gravel Surface: A Comparison and Sensitivity Analysis. *ISPRS Int. J. Geo-Inf.* **2018**, *7*, 69.
https://doi.org/10.3390/ijgi7020069

**AMA Style**

Milenković M, Ressl C, Karel W, Mandlburger G, Pfeifer N. Roughness Spectra Derived from Multi-Scale LiDAR Point Clouds of a Gravel Surface: A Comparison and Sensitivity Analysis. *ISPRS International Journal of Geo-Information*. 2018; 7(2):69.
https://doi.org/10.3390/ijgi7020069

**Chicago/Turabian Style**

Milenković, Milutin, Camillo Ressl, Wilfried Karel, Gottfried Mandlburger, and Norbert Pfeifer. 2018. "Roughness Spectra Derived from Multi-Scale LiDAR Point Clouds of a Gravel Surface: A Comparison and Sensitivity Analysis" *ISPRS International Journal of Geo-Information* 7, no. 2: 69.
https://doi.org/10.3390/ijgi7020069