# On the Estimation of Vector Wind Profiles Using Aircraft-Derived Data and Gaussian Process Regression

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Deriving Wind Velocity from ADS-B and Mode S Data

#### 2.1.1. ADS-B and Mode S

#### 2.1.2. Wind Velocity Derivation

#### 2.2. Vector Wind Profile

#### 2.3. The Kalman Filter-Based Models

- If $k=0$, the algorithm is initialized. An initial state vector is given as ${\mathbf{x}}_{0}^{\mathrm{f}}$, where the superscript f indicates forecast. An error covariance matrix of this estimation, ${P}_{0}^{\mathrm{f}}$, is also given as an input.
- If $k\ge 1$:
- (a)
- Analysis
- The Kalman gain matrix is computed with the formula:${K}_{k}={P}_{k}^{\mathrm{f}}{H}_{k}^{t}{({H}_{k}{P}_{k}^{\mathrm{f}}{H}_{k}^{t}+{R}_{k})}^{-1}$. This matrix is based in the uncertainties on the current state and the new measurements.
- The state vector is updated using the new observations and ${K}_{k}$:${\mathbf{x}}_{k}^{\mathrm{a}}={\mathbf{x}}_{k}^{\mathrm{f}}+{K}_{k}({\mathbf{y}}_{\mathbf{k}}-{H}_{k}{\mathbf{x}}_{k}^{\mathrm{f}})$, where the superscript a stands for analysis.
- The covariance matrix of the analysis estimation is computed as:${P}_{k}^{\mathrm{a}}=(I-{K}_{k}{H}_{k}){P}_{k}^{\mathrm{f}}$.

- (b)
- Forecast
- The forecast of states for the next time step is calculated as:${\mathbf{x}}_{k}^{\mathrm{f}}={M}_{k+1}{\mathbf{x}}_{k}^{\mathrm{a}}$.
- The error covariance matrix of this estimation is calculated as:${P}_{k}^{\mathrm{f}}={M}_{k+1}{P}_{k}^{\mathrm{a}}{M}_{k+1}^{t}+{Q}_{k+1}$.

#### 2.3.1. Adapted Kalman Filter

#### 2.3.2. The Smooth Adapted Kalman Filter

#### 2.4. Gaussian Process Regression

- $f\left(\mathbf{x}\right)$ is a Gaussian process. Any sample, $f\left({\mathbf{x}}_{1}\right),f\left({\mathbf{x}}_{2}\right),f\left({\mathbf{x}}_{3}\right),\dots ,f({\mathbf{x}}_{n})$, is jointly Gaussian-distributed with zero-mean and some covariance function $k(\mathbf{x},{\mathbf{x}}^{\prime})$.
- h is a basis function that projects the input $\mathbf{x}$ into a p-dimensional space and allows the trend to be, in general, non-linear.

#### Adaptation of GPR to Wind Velocity Output

## 3. Results

#### 3.1. Model Set Up

#### 3.1.1. Kalman Filters

#### 3.1.2. GPR

#### 3.1.3. Baseline Vector Wind Profile

#### 3.2. Performance Evaluation

- 1.
- The AKF, SAKF, and baseline estimators initiate at 13:45 UTC, 15 min before validation. This time interval acts as a burn-in period.
- 2.
- The GPR model is initially trained using a previous 1 h dataset selected from 12:55 to 13:55 UTC. All landing data passing through the considered waypoints are excluded from the train dataset, and are only used for validation.
- 3.
- The validation phase starts at 14:00 UTC. The vector wind profiles are compared with the testing data coming from the landing aircraft when passing through the waypoints, namely RILKO IAF and FAF. Every 15 min, a new GPR model is trained to detect potential trend changes in the wind behaviour. Validation finishes at 15:00.
- 4.
- Different performance measures are considered in order to assess the VWP estimations. The root mean square error (RMSE) and the mean absolute error (MAE) of the wind components and the wind speed are computed. In addition, boxplots of wind speed and direction errors are also obtained.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**VWP estimation framework. The cyan trajectory represents the final approach path of an aircraft landing. Blue vertical lines represent the locations where wind profiles are estimated.

**Figure 3.**Realisations of different $1D$ Gaussian processes generated by diverse covariance functions or kernel and represented as wind profiles.

**Figure 4.**Estimated vector wind profiles obtained using different assimilation techniques along with the training and testing data available at 14:10 UTC. (

**a**) Vector wind profile at RILKO IAF. (

**b**) Vector wind profile at FAF waypoint.

**Figure 7.**Comparison of estimated and observed wind vector when aircraft 3442CC flies through the waypoints RILKO IAF and FAF. The east-west and south-north directions are represented in the x and y axes, respectively.

**Figure 8.**Confidence intervals for the VWP estimated using the GPR technique along with the training and testing data available at time 14:10 UTC. (

**a**) Confidence interval for the VWP estimated at RILKO IAF waypoint. (

**b**) Confidence interval for the VWP estimated at FAF waypoint.

Type | Variable | Baseline | AKF | SAKF | GPR |
---|---|---|---|---|---|

RMSE (m/s) | u | 6.2 | 4.8 | 5.0 | 3.1 |

v | 6.0 | 5.1 | 5.1 | 2.9 | |

Wind speed | 6.5 | 5.0 | 5.2 | 3.0 | |

MAE (m/s) | u | 4.8 | 3.7 | 3.9 | 2.4 |

v | 4.4 | 3.6 | 3.5 | 2.4 | |

Wind speed | 5.0 | 3.8 | 3.9 | 2.2 |

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**MDPI and ACS Style**

Marinescu, M.; Olivares, A.; Staffetti, E.; Sun, J.
On the Estimation of Vector Wind Profiles Using Aircraft-Derived Data and Gaussian Process Regression. *Aerospace* **2022**, *9*, 377.
https://doi.org/10.3390/aerospace9070377

**AMA Style**

Marinescu M, Olivares A, Staffetti E, Sun J.
On the Estimation of Vector Wind Profiles Using Aircraft-Derived Data and Gaussian Process Regression. *Aerospace*. 2022; 9(7):377.
https://doi.org/10.3390/aerospace9070377

**Chicago/Turabian Style**

Marinescu, Marius, Alberto Olivares, Ernesto Staffetti, and Junzi Sun.
2022. "On the Estimation of Vector Wind Profiles Using Aircraft-Derived Data and Gaussian Process Regression" *Aerospace* 9, no. 7: 377.
https://doi.org/10.3390/aerospace9070377