# A Robust Adaptive Unscented Kalman Filter for Floating Doppler Wind-LiDAR Motion Correction

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Instrumental Setup

^{TM}300 LiDARs. The fixed LiDAR was set up in stand-still configuration on PdP pier, as the reference device. The FDWL and the fixed LiDAR were located 50 m apart (see Figure 1). Both LiDARs were calibrated onshore, 1 m apart and over a period of 3 h, to ensure identical measurements on a 1 s and 10 min time basis. PdP is located on the coastline of Badalona (Barcelona, Spain), in the Barcelona metropolitan area. The experiment location surroundings are defined by an urban topology of low-height buildings (up to 20 m), which follow the coastline in the west and north cardinal directions, while the rest is defined by a sea-type topology.

^{TM}), which is prepared for offshore operation [28]. The device can measure the wind at user-defined heights between 10 m to 200 m, in steps of 1 m [29]. The LiDAR uses the VAD algorithm to retrieve the wind vector by measuring 50 LoSs at equally spaced azimuth angles (7.2-deg azimuth step between LoSs) along a conical scan of 30-deg aperture width from zenith. The ZephIR 300 can reach up to 1 scan/s when there is no LiDAR re-focusing required or CPU dead-time internal processes [29].

#### 2.2. Basic Theoretical Definitions

#### 2.3. The Estimation Viewpoint

#### 2.4. The Measurement Model: FDWL Motion

#### 2.4.1. Rotational Motion

#### 2.4.2. Translational Motion

#### 2.4.3. VAD Algorithm

#### 2.5. State-Transition Model

#### 2.5.1. Wind Model

#### 2.5.2. Initial Scan-Phase Model

#### 2.6. State-Space Formulation of the Problem

**I**is the identity matrix. This enables us to identify state-transition function $f(\xb7)$, as:

**M**is a 50 × 6 matrix, where each row is a LoS attitude measurement, and each column is an attitude parameter.

- (i)
- retrieval of the motion-corrupted instantaneous LoS set, $\widehat{\mathit{r}}$;
- (ii)
- estimation of the associated LoS velocities, ${\mathit{v}}_{LoS}$; and
- (iii)
- VAD retrieval of the motion-corrupted observation wind vector, ${\mathit{z}}_{k|k-1}$;

#### 2.7. Estimation of State- and Observation-Noise Covariance Matrices

#### 2.8. Filter Initialization

## 3. Results

#### 3.1. Data Set

#### 3.2. Data Filtering

#### 3.3. Campaign Overview

#### 3.4. UKF Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Abbreviations

DoF | Degrees of Freedom |

DWL | Doppler Wind LiDAR |

EKF | Extended Kalman Filter |

FDWL | Floating Doppler Wind LiDAR |

HWS | Horizontal Wind Speed |

IMU | Inertial Measurement Unit |

KF | Kalman Filter |

LoS | Line-of-Sight |

LSQ | Least Squares |

MD | Mean Deviation |

metmast | meteorological mast |

PdP | El Pont del Petroli |

PSD | Power Spectral Density |

RAUKF | Robust Adaptive Unscented Kalman Filter |

RMSE | Root Mean Square Error |

RW | Random Walk |

SV | Spatial Variation |

TI | Turbulence Intensity |

UKF | Unscented Kalman Filter |

VAD | Velocity–Azimuth Display |

VWS | Vertical Wind Speed |

WD | Wind Direction |

WE | Wind Energy |

## Appendix A. Kalman Filter Review

#### Appendix A.1. The Unscented Transform

#### Appendix A.2. The Unscented Kalman Filter

## Appendix B. RAUKF Fault-Detection Mechanism

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**Figure 3.**Schematic of the motion geometry of the FDWL buoy. (

**a**) The moving-body coordinate system (red arrows) and the fixed coordinate system (blue arrows). (

**b**) The LiDAR scanning cone and LiDAR pointing direction (green arrows) in relation to the XYZ coordinate system.

**Figure 4.**Comparison between the HWS RW model presented in Section 2.5.1 and experimental data: (

**a**) Temporal series; and (

**b**) PSD.

**Figure 5.**Block diagram depicting the measurement function $h(.)$, as a chain process in which rotation, translation, and VAD retrieval are modelled as elementary functions. Equation numbers inside each block refer to pertinent equations in the text.

**Figure 6.**Wind rose representing the HWS and WD (after data filtering), measured during the PdP campaign, by the reference LiDAR (10 min) from June 6 to June 30 of 2013 (1875 records).

**Figure 7.**WD time-series measured by the FDWL at 100 m height, showing the so-called “granularity” effect.

**Figure 8.**HWS time-series measured at 100 m height between the fixed LiDAR and the FDWL, with and without correction (see legend). Inset: PSD comparison. Low HWS-variance scenario (7 June 2013, PdP).

**Figure 9.**Same as Figure 8. High HWS-variance scenario (22 June 2013, PdP).

**Figure 10.**Scatter plot comparing the TI measured by the FDWL with reference to the fixed LiDAR, with and without correction (Red, without motion correction; Black, with motion correction). The dashed line indicates the ideal 1:1 line. Dot-dashed lines indicate corresponding color-coded linear regressions.

**Table 1.**Statistical indicators evaluating the comparison between FDWL (with and without correction) and fixed LiDAR TI measurements at the 10 min level.

Uncorrected | Motion-Corrected | WD Filtered Motion-Corrected | |
---|---|---|---|

${R}^{2}$ | 0.85 | 0.90 | 0.93 |

RMSE | 2.01% | 1.01% | 0.86% |

MD | −1.70% | 0.29% | 0.36% |

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**MDPI and ACS Style**

Salcedo-Bosch, A.; Rocadenbosch, F.; Sospedra, J.
A Robust Adaptive Unscented Kalman Filter for Floating Doppler Wind-LiDAR Motion Correction. *Remote Sens.* **2021**, *13*, 4167.
https://doi.org/10.3390/rs13204167

**AMA Style**

Salcedo-Bosch A, Rocadenbosch F, Sospedra J.
A Robust Adaptive Unscented Kalman Filter for Floating Doppler Wind-LiDAR Motion Correction. *Remote Sensing*. 2021; 13(20):4167.
https://doi.org/10.3390/rs13204167

**Chicago/Turabian Style**

Salcedo-Bosch, Andreu, Francesc Rocadenbosch, and Joaquim Sospedra.
2021. "A Robust Adaptive Unscented Kalman Filter for Floating Doppler Wind-LiDAR Motion Correction" *Remote Sensing* 13, no. 20: 4167.
https://doi.org/10.3390/rs13204167