# Methodologies for Wind Field Reconstruction in the U-SPACE: A Review

## Abstract

**:**

## 1. Introduction

## 2. AWEA—Airborne Wind Estimation Algorithm

- Definition of a flight trajectory;
- Generation of a first-attempt solution. It is produced through a standard logarithmic profile [24] starting from a measurement performed onboard.$${V}_{w}(h)={V}_{w0}{\left(\frac{h}{{h}_{0}}\right)}^{p}$$
_{w}(m/s) changes with the altitude h (m) and depends on the wind velocity V_{w}_{0}measured at an assigned height h_{0}. The numerical value of the p exponent is empirically obtained and is equal to 1/7. This power law equation yields a very general approximation of the wind profile in the Planetary Boundary Layer (PBL). - Kalman filter update at the frequency 1 Hz.

- (1)
- The filter starts to predict the state estimate x and estimate covariance P.
- (2)
- When new observations are available, they are used to construct the observation model matrix C. The observations determine the measurement noise covariance matrix R too. The coefficients of R depend on K
_{w}, which is a scaling parameter varying between 0 and 1. K_{w}determines the influence of an observation based on the distance between the measurement and the trajectory at the same altitude. - (3)
- The innovation covariance matrix S is calculated using the C and R matrices. In this way, the current estimate for the altitude of the observation can be calculated and compared to the measured value to determine the innovation.
- (4)
- The Kalman gain K is determined by using the observation model, the state covariance and innovation covariance matrices.
- (5)
- The innovation is multiplied by the Kalman gain, providing the updated state estimate.

_{w}from 0.1 to 1 in steps of 0.1. The analysis showed that the RMSE value is reduced when K

_{w}is increased, from about 3.5 knots for K

_{w}= 0.1 to 2.2 knots for K

_{w}= 1, confirming the importance of distance information.

## 3. Wind Field Reconstruction with Random Fourier Features

_{d}. This process for the definition of a model f is defined as “training” and is performed by minimizing a loss function. There is an important distinction between f and f

_{d}, since the function f represents a model that is trained on the original data and produces a vector field f

_{d}that approximates the velocity vector $\overrightarrow{u}$. In the classical Fourier series models, the coefficients are defined as:

_{k}. In this view, the random Fourier features are an example of a neural network with a hidden layer and a trigonometric activation function [25]. In particular, ω are the weights connecting the inputs x to the hidden layer, and β are the weights connecting the nodes in the hidden layer to the output layer. In other words, it is the training algorithms that distinguish the random Fourier features.

## 4. The Meteo Particle Model (MPM)

#### 4.1. Selection of Input Data

- A Gaussian probabilistic function is built starting from the current field, once that mean μ e variance σ has been calculated:$$p=\mathrm{exp}\left[-\frac{1}{2}\frac{{\left(x-\mu \right)}^{2}}{{k}_{1}\sigma}\right]$$In which k
_{1}is a control parameter defined as the acceptance probability factor. - New data selection: each new data has a p probability of being accepted, in such a way that data related to extreme values have a low probability of being selected. The numerical value assigned to k
_{1}is defined by the user in an empirical way, and its value can be augmented to allow a larger tolerance (increase in the number of accepted measurements). The value proposed by the authors of the method is 3.

#### 4.2. Construction of Particles

_{p}, y

_{p}, z

_{p}(m) of the N particles (x

_{p,i}, y

_{p,i}, z

_{p,i}, with i = 1, … N) at the new time step t + Δt are evaluated on the basis of the positions at the previous step t using the following expressions:

_{2}factor (particle random walk factor). Along the vertical direction (z), the propagation follows a zero-average Gaussian track. All the particles are re-sampled at the end of each step. The time step Δt is chosen according to criteria of numerical stability, considering the time frame of the specific application, e.g., the size of the geographical domain, the time interval between two successive measurements, and the time step of the NWP (if the MPM is coupled with an NWP, see Section 6). The particles that for their motion fall outside the domain (both in horizontal and vertical directions) are removed, while the remaining ones are classified according to their age, according to the following probability function:

_{α}is a control parameter (aging parameter). This resampling ensures a periodic particle renewal, in such a way that the oldest ones are removed.

#### 4.3. Evaluation of Variables Value in a Generic Point

_{p}assigned to each particle is calculated as a product of two exponential functions:

_{0}of the particle from its origin:

_{d}is a control parameter (weighting parameter). These formulae are based on the IDV technique (Inverse Distance Weighted) [28].

## 5. Examples of Application of the MPM

#### 5.1. The Metsis Project

_{i}represent the MPM estimates and o

_{i}are observational values. MAE was evaluated both for wind speed and for the three direction angles, for each scenario considered.

#### 5.2. Wind Field Reconstruction at Delft (the Netherlands)

#### 5.3. Model Extension at Different Heights

## 6. NWP Models

## 7. Discussion and Conclusions

- AWEA estimates wind profiles using measurements of wind data taken from nearby aircraft, providing high-fidelity and high-resolution user-tailored wind profiles, which can be used for predicting wind at locations farther along the trajectory with small errors. The power law equation used in AWEA only yields a very general approximation of the wind profile in the PBL. Especially since the PBL is characterized by the interaction between the free stream wind at higher altitudes and the disturbing forces of friction caused by the Earth’s roughness, this approximation can differ from the true wind profile. However, in order to derive a useful model not dependent on local surface roughness or latitude, this power law is used as a first approximation of the wind profile and could also be used to fill the gaps when measurement data is sparse. For this reason, the dependence on the surface roughness length was neglected. In future developments, the usage of historical data and cross-correlations between time series provided by different aircraft would improve the accuracy of the results.
- The random Fourier features is a novel interpolation model that results competitively with respect to other statistical interpolation models, such as kriging or modern machine learning methods, e.g., random forests and neural networks. The authors found that the model can be extended to new areas of research, including, for example, the possibility of incorporating data over multiple times and including more terrain-specific features. Since the definition of interpolation models was restricted to a two-dimensional range (only the horizontal wind vector is considered), better accuracy would be provided by the introduction of a modified coordinate system, defined to follow the terrain.
- The MPM concept demonstrated the possibility of using drones as a large sensor network to construct a global scale real-time meteorological measurement system. However, the accuracy for wind direction did not meet the WMO standards [31]. On the other side, the effects of obstacles on wind can be considered without affecting performances, as long as wind measurements are available near the obstacles. The near-surface turbulence is a source of difficulty in wind field reconstruction. For example, Kiessling et al. [13] found non-negligible interpolation errors related to this kind of turbulence; in particular, they found that the error is lower during nighttime: in the summer period this can be explained by the observed reduction in wind speed during night time; instead, in winter, even if wind speeds are higher, the atmosphere tends to be more stable at nighttime [38], which might explain this decrease in error.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagram block describing the usage of the Kalman filter in AWEA (adapted from de Jong et al., 2014 [12]).

**Figure 2.**Position of the reference and measurement drones. Arrows indicate the flight direction of measurement drones (adapted from Sunil et al., 2021 [29]).

**Figure 3.**(

**a**) Retrieval map of wind field during 17:59:10~18:00:50. (

**b**) Wind field map of ECMWF ERA-5 data at 18:00:00 (from Zhu et al., 2021 [35]).

Method | Main Idea | Authors | Ref. |
---|---|---|---|

AWEA | Estimation of wind profiles using measurements of wind data taken from nearby aircraft. | De Jong et al. | [12] |

random Fourier features | Interpolation model based on a machine learning approach. | Kiessling et al. | [13] |

MPM | Estimation of atmospherical variables using a Monte Carlo approach, using surveillance data from aircraft. | Sun et al. | [14] |

NWPs | Delivery of weather forecasts by solving the full set of prognostic equations of atmosphere. | Various authors | e.g., [5,6,15] |

**Table 2.**List of different implementations of wind estimation (adapted from de Jong et al., 2014 [12]).

Name | Description | Details | RMSE (Knots) |
---|---|---|---|

Charts | Wind estimation using wind charts | The simulated wind is flattened into a single static wind profile | 3.2 |

Ground 10 | Ground-based AWEA with ADS-B broadcast rate of 10 s | Broadcast wind measurements from aircraft within their own TMA | 1.0 |

Ground 30 | Ground-based AWEA with ADS-B broadcast rate of 30 s | As in Ground 10 | 1.1 |

Air 10 | Aircraft-based AWEA with ADS-B broadcast rate of 10 s | Combines the received ground-based profile with measurements from other aircraft | 0.8 |

Air 30 | Aircraft-based AWEA with ADS-B broadcast rate of 30 s | As in Air 10 | 0.9 |

**Table 3.**Differences in the values of ε(f) and Q(f) related to the random Fourier features and the other models (data from Kiessling et al., 2021 [13]).

Interpolation Model | Δ ε(f) | Δ Q(f) |
---|---|---|

Nearest neighbors | 0.258 | 4.576 |

Inverse distance weighting | 0.037 | 0.651 |

Universal kriging | 0.018 | 0.318 |

Random forest | 0.017 | 0.293 |

Neural network | 0.011 | 0.192 |

Fourier series | 0.010 | 0.171 |

**Table 4.**List of variables considered in the experiments by Sunil et al., 2021 [29].

Obstacle | Triangle Size (m) | Altitude (m) |
---|---|---|

Baseline (None) | 60, 40, 20 | 5, 10, 20, 100 |

Trailer | 60, 40, 20 | 5, 10 |

Trees | 40 | 5, 10, 20 |

**Table 5.**Mean average error (MAE) of wind speed and directions considering the three different obstacles, for (

**a**) static and (

**b**) dynamic scenarios (data extracted from Figure 9 of Sunil et al., 2021 [29]).

(a) Static scenario | |||

Baseline | Trailer | Tree | |

w_{mae} | 0.65 m/s | 0.81 m/s | 0.71 m/s |

ϑ_{xy} | 18° | 17° | 19° |

ϑ_{xz} | 19° | 39° | 62° |

ϑ_{yz} | 20° | 8° | 10° |

(b) Dynamic scenario | |||

Baseline | Trailer | Tree | |

w_{mae} | 0.65 m/s | 0.79 m/s | 0.81 m/s |

ϑ_{xy} | 62° | 72° | 60° |

ϑ_{xz} | 30° | 48° | 58° |

ϑ_{yz} | 105° | 78° | 65° |

**Table 6.**Original and optimized values of some control factors defined in MPM (data from Zhu et al., 2021 [35]).

Factor | Original Value | Optimized Value |
---|---|---|

Acceptance probability factor k_{1} | 3 | 7 |

Particle Random walk factor k_{2} | 10 | 8 |

Number of particles per aircraft N | 250 | 300 |

Aging parameter α | 180 | 500 |

Weighting parameter C_{0} | 30 | 21 |

**Table 7.**Original and optimized values of some control factors defined in MPM (data from Zhu et al., 2021 [35]).

U Component | V Component | |||
---|---|---|---|---|

Original Value | Optimized Value | Original Value | Optimized Value | |

MAE (m/s) | 1.37 | 1.24 | 1.76 | 1.54 |

RMSE (m/s) | 2.21 | 1.89 | 2.68 | 2.36 |

COR | 0.95 | 0.96 | 0.93 | 0.95 |

R | 0.99 | 0.99 | 0.93 | 0.95 |

COMBINE | 0.97 | 0.98 | 0,93 | 0.95 |

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**MDPI and ACS Style**

Bucchignani, E.
Methodologies for Wind Field Reconstruction in the U-SPACE: A Review. *Atmosphere* **2023**, *14*, 1684.
https://doi.org/10.3390/atmos14111684

**AMA Style**

Bucchignani E.
Methodologies for Wind Field Reconstruction in the U-SPACE: A Review. *Atmosphere*. 2023; 14(11):1684.
https://doi.org/10.3390/atmos14111684

**Chicago/Turabian Style**

Bucchignani, Edoardo.
2023. "Methodologies for Wind Field Reconstruction in the U-SPACE: A Review" *Atmosphere* 14, no. 11: 1684.
https://doi.org/10.3390/atmos14111684