# Evaluation of a LIDAR Land-Based Mobile Mapping System for Monitoring Sandy Coasts

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## Abstract

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## 1. Introduction

**Figure 1.**A real example of dune erosion on the Dutch coast on Ameland and the possible consequence; photo taken by Johan Krol [Archive Natuurcentrum Ameland].

## 2. An Overview of Processing Steps

## 3. Quality of Laser Point Heights

#### 3.1. Reconstructing the Scanning Geometry

_{fp}, can be computed for each measured laser point using the point and trajectory position as follows.

_{fp}, is computed from the laser beam divergence, β, the incidence angle, α, and range, R, as written in Equation (1):

#### 3.2. Theoretical Quality of Laser Points

_{Zi,m}.

**Figure 4.**The range error δR due to the non-perpendicular scanning geometry and the influence of δR on vertical and horizontal laser point positioning error.

_{Zi,δR}(geometrical precision) is computed.

_{ΔZ}, of the height difference, ΔZ, between two laser points q and r is computed as:

#### 3.3. Empirical Quality of Laser Points

- As the footprint diameter goes to infinity when the incidence angle is 90° only laser points that have an incidence angle less than 89.9° are considered: α
_{P}< 89.9°. - Because just the vertical component of two points is compared, points should lie on an almost horizontal plane in order to avoid the influence of surface slope on the height difference. This requirement is considered to be fulfilled if the vertical component of the unit normal vector n
_{p,z}computed at each laser point, as explained in Section 3.1, is close to 1; that is: n_{p,z}≈ 1.

_{i,j}between laser point P

_{i}and its nearest neighbor P

_{j}is smaller than the minimal size of their footprint radii r

_{i}and r

_{j}respectively. Additionally a second threshold is needed, because the footprint size is here approximated with a circle. In general, it is expected that for bigger incidence angles in combination with the increasing range the footprint takes an elliptical shape. For more rough surfaces, the footprint can have any shape in 3D. Therefore, the circle approximation results in an overestimated footprint size when the incidence angle increases. To reduce the influence of this effect, a threshold of 5 cm for the radii of the footprints was experimentally found suitable. Thus:

- Identical points (IP) from the complete data set.
- Identical points (IP) belonging to different scanners (scanner overlap).
- Identical points (IP) belonging to overlapping drive-lines (drive-line overlap).

## 4. DTM Interpolation and Quality

_{DTM}. In other words, the systematic errors are assumed to be zero [13]. First a grid of 1 × 1 m size is laid over the terrain laser points. For grid cells, which include four or more terrain laser points, a tilted plane is fitted in a Least Squares fashion by a first order polynomial as given in Equation (7):

_{0}, a

_{1}and a

_{2}are the unknown plane coefficients. The graphical representation of each term in Equation (7) is shown in Figure 5.

**Figure 5.**A graphic representation of terms given in Equation (7); after [25].

_{G},Y

_{G}) as the origin; therefore, the method is called Moving Least Squares (MLS) adjustment [26]. This simplifies the plane equation as the plane coefficient a

_{0}becomes the elevation of the grid point itself:

_{i}, Y

_{i}, Z

_{i}for i = 1 … n are the coordinates of the n original laser terrain points included in the plane computation. Then the unknowns in vector $\widehat{x}$ (in the short form) and their variance-covariance matrix ${\sum}_{\widehat{x}\widehat{x}}$ are computed in a least squares adjustment as written in Equation (10) and Equation (11) respectively:

_{yy}is the covariance matrix of observations, in which the theoretical height precision of the laser points σ

_{Zi}computed in Section 3.2 is used.

_{DTM}, a mathematical model after [25] as written in Equation (13) is used.

_{a0}represents the quality of the original data and accounts for the precision of the original laser points (FD2), their density (FD1) and distribution (FD3). The second term σ

_{e}represents the quality loss due to the representation of the terrain surface by a plane. In this research, the RMSE is considered as a measure of the terrain surface roughness (FR) with respect to the plane modeled by the chosen random-to-grid moving least squares interpolation (FI). Therefore, σ

_{e}simply equals the RMSE as computed in Equation (12).

## 5. Results and Discussion

#### 5.1. Data Description

_{fp}, range error due to the scanning geometry δR, measuring precision σ

_{Zm}and geometrical precision σ

_{Z,δR}. Besides the 3D laser point data, the data of trajectories given in *.trj file is used. The positions of eight trajectories within the test area are shown in Figure 6.

**Figure 6.**Left, location of the test area at the west coast of The Netherlands. Right, aerial view of the test area [GoogleMaps]. Coordinates are in RDNAP. The black dashed lines mark the trajectories driven downward, i.e., from the north to the south, and the solid lines mark the trajectories driven in the opposite direction.

#### 5.2. Results of Theoretical Precision

_{Zi,m}and geometrical precision σ

_{Zi,δR}are shown. Laser points within the whole test area are considered. Because the geometrical precision σ

_{Zi,δR}increases to infinity when the incidence angle approaches 90°, the computation of statistical measures in Figure 7(b) considers just points that have incidence angle α smaller than 89.9°. The median measuring precision σ

_{Zi,m}is 2.63 cm, while the median geometrical precision σ

_{Zi,δR}is much smaller and is 0.86 cm. On the other hand the dispersion of σ

_{Zi,δR}is as expected much bigger than for the σ

_{Zi,m}. The minimum measuring precision σ

_{Zi,m}is 2.54 cm, which is due to the main error contributor that is the GPS error.

**Figure 7.**Theoretical precision of the laser point heights, computed by the first random error model.

**(a)**Measuring precision,

**(b)**Geometrical precision.

#### 5.3. Results of Height Differences of Identical Points

ALL | Scanner overlap | Drive-line overlap | ||
---|---|---|---|---|

No. of identical point pairs | 17,754 | 608 | 5,473 | |

Height difference ΔZ [mm] | Min | −47 | −20 | −47 |

Max | 46 | 36 | 46 | |

Avg | 0.1 | 0.2 | 0.0 | |

Std | 3.1 | 2.5 | 3.5 |

**Figure 8.**The relation of the absolute height differences abs(ΔZ) and scanning geometry attributes.

**(a)**Range R

**(b)**Incidence angle α.

#### 5.4. Comparison of Empirical and Theoretical Height Precision

_{Δz}is computed for each height difference between identical points, according to Equation (5). In Table 2 the statistics of empirical and theoretical precision measures are given. The comparison of theoretical and empirical precision of height differences between laser points shows large differences. The theoretical RMSE is approximately 28 times larger. This could partly be expected, because the estimation of the theoretical height precision relies on many assumptions (e.g., about calibration parameters, scan angle error, error due to non-perpendicular scanning geometry).

min | max | mean | std | RMSE | ||
---|---|---|---|---|---|---|

Empirical | ΔZ_{ALL} [m] | −0.0470 | 0.0460 | 0.0001 | 0.0031 | 0.0031 |

Theoretical | σ_{Δz} [m] | 0.0376 | 4.8515 | 0.0573 | 0.0658 | 0.0872 |

#### 5.5. Results of DTM Interpolation and Precision Estimation

_{MLS}. The grid point elevation is changing from −0.19 m at the coastline to up to 22 m in the dunes (see Table 3). The white holes in the DTM are results of the shadow-effect (white holes in green area) and most probably of the presence of water bodies on the beach (white holes in the blue area).

n (FD1) | $\overline{{\sigma}_{Zi}}$ (FD2) | Z_{MLS} | σ_{a0} | σ_{e} | σ_{DTM} | |
---|---|---|---|---|---|---|

[m] | [m] | [m] | [m] | [m] | ||

min | 4 | 0.026 | −0.19 | 0.0015 | 0.0000 | 0.0018 |

max | 333 | 0.85 | 21.88 | 2.9 | 0.1001 | 2.9 |

med | 69 | 0.033 | 1.25 | 0.0042 | 0.0008 | 0.0047 |

rstd | 76 | 0.008 | 1.28 | 0.0030 | 0.0006 | 0.0030 |

**Figure 10.**Number of points n per 1 × 1 m grid cell. The black lines mark the trajectories driven by the scanning vehicle, compare Figure 6.

_{a0}is presented in Figure 11(a). The size of σ

_{a0}depends mostly on the number of points n (FD1) and the quality of the individual terrain laser point σ

_{zi}(FD2). The standard deviation σ

_{a0}is the smallest, i.e. below 2 mm, within the driving path of drive-line DL5 (light blue strip on the right side of the figure). The main reason is the high number of terrain laser points, which is more than 250 points along this drive-line (see Figure 10). The green color indicates grid cells that have a standard deviation σ

_{a0}of about 4.2 mm. The black colored points correspond to 10% of the analyzed grid cells with a standard deviation σ

_{a0}of larger than 2.32 cm. These lie mostly in the dune area and on the edges of the drive-lines.

_{e}. It is represented by the root mean square error (RMSE) of the vertical residuals between terrain laser points and the fitted planes. With a given grid size, which is here 1 × 1 m, and a functional model to represent the surface, which is here the tilted plane, σ

_{e}depends mainly on the complexity of the terrain surface. The quality of the measurements is assumed to be high enough. The spatial variability of the terrain roughness component σ

_{e}is shown in Figure 11(b). This pattern is almost independent of the laser point height precision, although vertical stripes are recognizable in the direction of the trajectories. The pattern shows more distinctly the morphology of the terrain (see also Figure 6). In the pre-dune area higher values of the terrain roughness component σ

_{e}are present. About 10 % of the grid cells have a terrain roughness component σ

_{e}larger than 4.5 mm. These are depicted in black color and are located mostly in the pre-dune and dune area. Therefore, these parts of the test area are considered to have a rougher topography, and are thus more difficult to model with a planar surface of 1 × 1 m size.

_{DTM}over the test area is shown. The height precision σ

_{DTM}of the grid points, as computed by Equation (13), varies between 0.0018 and 2.9 m. The average precision of grid points σ

_{DTM}equals to 4.7 mm. For comparison, the precision of the observations σ

_{Zi}is on average 2.4 cm. The grid cells having a standard deviation σ

_{DTM}higher than 2.56 cm are colored dark red and black. They represent approximately 10 % of all the grid cells. The green color shows grid points having a height precision σ

_{DTM}smaller than 1 cm. Most of the beach area has a high DTM quality, which decreases with increasing distance from the trajectory. For example, the precision at the edges of the drive-line DL11 (the leftmost line) decreases and is in some areas worse than 2.56 cm, mostly due to the number of points n (compare to the dark blue areas in Figure 10). The DTM quality is lower also in the dune area, due to the low quality of the terrain laser points (compare to Figure 11(a)) and the high terrain roughness (compare to Figure 11(b)).

_{DTM}(y-axis), the number of points n (x-axis) and the data quality component σ

_{a0}(colorbar), is presented. A comparison of the colorbar and the y-axis scale shows, that the size of the grid point height precision σ

_{DTM}depends mainly on the data quality component σ

_{a0}. Besides, one can observe that if approximately 50 or more points are included in the grid point computation, the standard deviation of the grid point heights σ

_{DTM}drops below 1 cm.

**Figure 11.**The two components directly employed in the computation of the grid point height precision and the final grid point height precision. Values are shown per 1 × 1 m grid cell.

**(a)**Data quality component σ

_{a0},

**(b)**Terrain roughness component σ

_{e},

**(c)**Grid point height precision σ

_{DTM}.

**Figure 12.**Correlation between the grid point height precision σ

_{DTM}and the number n of terrain laser points; color-coded by the data quality component.

## 6. Summary, Conclusion and Recommendations

#### 6.1. Summary

- The LMMS measurements resulting in the measurement precision of laser points.
- The non-perpendicular scanning geometry resulting in the geometrical precision of laser points.

#### 6.2. Conclusions

#### 6.3. Recommendations

## Acknowledgments

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**MDPI and ACS Style**

Bitenc, M.; Lindenbergh, R.; Khoshelham, K.; Van Waarden, A.P.
Evaluation of a LIDAR Land-Based Mobile Mapping System for Monitoring Sandy Coasts. *Remote Sens.* **2011**, *3*, 1472-1491.
https://doi.org/10.3390/rs3071472

**AMA Style**

Bitenc M, Lindenbergh R, Khoshelham K, Van Waarden AP.
Evaluation of a LIDAR Land-Based Mobile Mapping System for Monitoring Sandy Coasts. *Remote Sensing*. 2011; 3(7):1472-1491.
https://doi.org/10.3390/rs3071472

**Chicago/Turabian Style**

Bitenc, Maja, Roderik Lindenbergh, Kourosh Khoshelham, and A. Pieter Van Waarden.
2011. "Evaluation of a LIDAR Land-Based Mobile Mapping System for Monitoring Sandy Coasts" *Remote Sensing* 3, no. 7: 1472-1491.
https://doi.org/10.3390/rs3071472