# Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach

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## Abstract

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## 1. Introduction

- Fluctuations in the flow field: Changes in terrain roughness upstream of the measurements cause large variations and changes in wind speed with height lead to uncertainty in the wind speed [6].
- Lidar technology: The weighting function of the lidar signal processing also leads to over- or under estimation of the wind speed.

- Is the length of the trajectory circulation time important in the comparisons with the met mast?
- Does the half opening-angle $\varphi $ have an influence on the accuracy of the correlations?
- Is the local resolution of the measurement points important for the result?

## 2. Materials and Methods

#### 2.1. Test Site and Measurement Campaign

#### 2.1.1. Local Conditions

#### 2.1.2. Met Mast

#### 2.1.3. Lidar

#### 2.1.4. Data Filtering and Selection

**Lidar:**The high-resolution recorded lidar data are processed with the wind field reconstruction methods: The last N data points are used for each time step. These methods determine both the wind velocity components $u,\phantom{\rule{4pt}{0ex}}v,\phantom{\rule{4pt}{0ex}}w$ and statistical parameters ($CN{R}_{mean}$, $CN{R}_{m}in$ and $CN{R}_{max}$) for the carrier-to-noise ratio (CNR). The CNR parameters are still required for later data processing. The horizontal wind speed is calculated from the wind speed components $u,\phantom{\rule{4pt}{0ex}}v$ (Equation (2)). These data are now subjected to a CNR filter to exclude samples outside of the CNR range. It is important that both the $CN{R}_{m}in$ and the $CN{R}_{max}$ are within the CNR limits. Values outside these limits are not taken into account for further consideration. From these data the 10 min statistics are calculated and selected.

**Met mast:**The recorded data is first subjected to a plausibility test where the system checks that values are within a realistic range. If data is available from a second sensor at nearly the same height a comparison of the data from those sensors is also made, e.g., between the cup anemometer at 100 m and the horizontal wind speed from the sonic at 98 m. Then the 10 min statistics are calculated, and the data is selected according to the selection criteria in Table 1.

#### 2.2. Wind Field Reconstruction Methods

#### 2.2.1. Continuous Least-Square

**A**of the equation system consists of the components of the normal vectors $\overrightarrow{n}{(x,\phantom{\rule{4pt}{0ex}}y,\phantom{\rule{4pt}{0ex}}z)}_{i}$ of the used N data points. To solve Equation (3), the vector $\mathbf{b}$ is multiplied by the inverse Matrix ${\mathbf{A}}^{-\mathbf{1}}$. If the trajectory has more than three data points (which are at least necessary) the Moore-Penroe pseudoinverse ${\mathbf{A}}^{-\mathbf{1}}$ will be used to solve the system of equations at the time step t. In order to solve this system of equations, it is not necessary to assume the homogeneous flow, since a solution is estimated which has the smallest absolute error. As a result, measurement errors (unrealistic data (e.g., due to bad CNR) strongly distort the solution of the equation system (Equation (3)). These unrealistic data must then be filtered in post-processing.

#### 2.2.2. Predicted Residual Error Sum of Squares (PRESS)

#### 2.2.3. Linear VAD Model

- ${n}_{x}y=xy/{f}_{d}=x{n}_{y}$ with ${n}_{x}=x/{f}_{d}$ (${f}_{d}$ is the distance of the lidar measuring point).

- The components are only evaluated along the vertical axis of the lidar device (coordinate origin), which means that ${x}_{0}={y}_{0}=0$.
- The gradient of the vertical wind speed components in x- and y-direction ${w}_{x}$ and ${w}_{y}$ would be zero.

## 3. Results

#### 3.1. Directional Dependency

#### 3.2. Wind Speed Comparisons

**SWE Scanner:**Figure 8 shows the results of the comparison between the SWE Scanner and the cup anemometer installed at 100 m using the CLS and PRESS methods.

**Galion:**The results from the Galion system are shown in Figure 9. For the CLS method all 18 trajectory points were used for the evaluation. A multiple overdetermined system of equations has to be solved. This lidar and method shows a better agreement between the lidar measurements and the met mast than the SWE Scanner measurements (Figure 9a; offset 0.018 m·s${}^{-1}$, slope 1.033, ${R}^{2}$ 0.974).

#### 3.3. Comparison of the Statistical Parameters of Methods

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The test site topography. An aerial picture (©2016 Google) is overlain with elevation contours (©2009 GeoBasis–DE/BKG, LGL (Geobasisdaten ©LGL Landesamt für Geoinformation und Landentwicklung Baden-Württemberg, Az.: 2851.9-1/19)). The locations of the met mast (▼), lidars (●), wind turbines (⋆), and vertical profiles from [15,16] are also shown. The Albtrauf is between the two thicker elevation contours. (

**b**) Terrain cross section. The locations of the met mast (▼) and lidars (●) are shown. The experiment site. (

**a**) Topography and structures (

**b**) Terrain section along red dashed line.

**Figure 3.**The vertical wind profile near the escarpment and on the plateau. M4 is at the escarpment, M2 is closest to the lidar, and M1 is closest to the met mast (Figure adapted and translated from Figure 5.8 in Ref. [16]).

**Figure 6.**Overview of the workflow to estimate the combination of data points the maximized the coefficient of determination ${R}^{2}$, based on the input variables ${v}_{los}$, the estimated wind velocity components $u,\phantom{\rule{4pt}{0ex}}v,\phantom{\rule{4pt}{0ex}}w$, the model predicted ${\widehat{v}}_{los}$ and the total sum of squares ($S{S}_{tot}$) and the regression sum of squares $S{S}_{reg}$.

**Figure 7.**Results of the projection of the USA onto the five lidar beam normal vectors. Each polar plot represents a combination of azimuth-elevation angles of the laser beam. The positions of the plots correspond to the azimuth angle of the scanner laser beam: the upper left graph is equivalent to the azimuth angle at 288${}^{\circ}$, the lower left graph is equivalent to 216${}^{\circ}$, the middle graphic is the vertical measurement, etc. The axes of the graphs are rotated in the respective angles of the laser beam direction. The azimuth is the wind direction from the met mast. The distance of the point from the origin corresponds to horizontal wind speed (8, 12, 16mper s). The absolute differences between met mast and lidar (${\overrightarrow{v}}_{usa}\xb7{\overrightarrow{n,i}}_{lidar}-{\overline{v}}_{los,i}$) are represented in colour and the size of points shows the relative differences between met mast and lidar. Positive absolute differences are marked in red, very small differences (magnitude $<0.001$) are shown in grey, and negative differences in blue.

**Figure 8.**Correlation between SWE Scanner and the cup anemometer at 100 m above ground. The black dots represent the 10 min data points, the black line is the regression line, and the blue lines are the 5th and 95th quantiles.

**Figure 9.**Correlation between Galion lidar and the cup anemometer at 100 m above ground. The black dots represent the 10 min data points, the black line is the regression line, and the blue lines are the 5th and 95th quantiles.

**Figure 10.**Overview of the distribution of ${R}^{2}$ and the reduction of the 5th and 95th percentile.

**Figure 11.**The error in the wind field reconstruction using the CLS method, versus the error due to the PRESS method. Negative values indicate that the CLS method had lower error compared to the met mast than the PRESS for a particular 10-minute period.

**Table 1.**Selection criterion used in Figure 5.

Parameter | Acceptable Range | |
---|---|---|

Minimum | Maximum | |

Lidar measurements | ||

CNR (min, mean) [dB] | −22 | 10 |

Weather conditions | ||

Wind direction [${}^{\circ}$] | 245 | 315 |

Temperature [${}^{\circ}$] | >2 | - |

Mast function | ||

Std. cup anemometer | >0.01 | - |

Data availability | ||

lidar [%] | 90 | 100 |

met mast [%] | 100 | - |

Lidar System | Method | Slope [-] | Offset [m·s${}^{-1}$] | ${\mathit{R}}^{2}$ |
---|---|---|---|---|

SWE Scanner | CLS | 1.074 | 0.219 | 0.949 |

SWE Scanner | PRESS | 1.103 | 0.385 | 0.938 |

SWE Scanner 2nd position | CLS | 1.046 | 0.537 | 0.942 |

SWE Scanner 2nd position | PRESS | 1.09 | 0.674 | 0.936 |

Galion 18 Pts. | CLS | 1.033 | 0.02 | 0.974 |

Galion 20${}^{\circ}$ | CLS | 1.00 | 0.293 | 0.93 |

Galion 39.2${}^{\circ}$ | CLS | 1.013 | 0.067 | 0.962 |

Galion 55${}^{\circ}$ | CLS | 1.032 | 0.03 | 0.960 |

Galion | linear | 1.026 | 0.03 | 0.976 |

SWE Scanner | Galion | ||||||
---|---|---|---|---|---|---|---|

VAD 5+1 | VAD 18 | ${\mathit{\varphi}}_{20}$ | ${\mathit{\varphi}}_{39.2}$ | ${\mathit{\varphi}}_{55}$ | linear | ||

$\varphi \left[{}^{\circ}\right]$ | 15 | 15 | 20, 39.2, 55 | 20 | 39.2 | 55 | 20, 39.2, 55 |

scan duration [m·s${}^{-1}$] | 8.8 | 4.78 | 49 | 16.3 | 16.3 | 16.3 | 49 |

$\kappa $ [-] | 5.78 | 5.78 | 2.02 | 4.30 | 2.00 | 1.26 | 120.66 |

Nb. of Points [-] | 6 | 6 | 18 | 6 | 6 | 6 | 18 |

Location [-] | 1 | 2 | 1 | 1 | 1 | 1 | 1 |

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**MDPI and ACS Style**

Hofsäß, M.; Clifton, A.; Cheng, P.W.
Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach. *Remote Sens.* **2018**, *10*, 1465.
https://doi.org/10.3390/rs10091465

**AMA Style**

Hofsäß M, Clifton A, Cheng PW.
Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach. *Remote Sensing*. 2018; 10(9):1465.
https://doi.org/10.3390/rs10091465

**Chicago/Turabian Style**

Hofsäß, Martin, Andrew Clifton, and Po Wen Cheng.
2018. "Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach" *Remote Sensing* 10, no. 9: 1465.
https://doi.org/10.3390/rs10091465