# A Method for Detecting Atmospheric Lagrangian Coherent Structures Using a Single Fixed-Wing Unmanned Aircraft System

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Lagrangian-Eulerian Analysis

#### 2.2. Gradient Approximation from UAS Flight

- A scalar field measured along a circular arc, $f\left(\theta \right)$, as an $n\times 1$ array input, where n is the number of measurements taken;
- The angle $\theta $ as a monotonically decreasing $n\times 1$ array input;
- And the radius of the circular arc, r, which is assumed to be constant, as a scalar input.

Algorithm 1 Circle Gradient approximates the gradient of a scalar from samples along an arc |

#### 2.3. Model Data

## 3. Results

#### 3.1. Approximating Local Eulerian Metrics from UAS Flights

#### 3.2. Using Eulerian Metrics to Infer Lagrangian Dynamics

#### 3.3. Inferring Lagrangian Dynamics from UAS Measurements

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

UAS | Unmanned aircraft systems |

ASL | Above sea level |

AUC | Area under the curve |

FTLE | Finite-time Lyapunov exponent |

LCS | Lagrangian coherent structure |

iLCS | Infinitesimal-time LCS |

OECS | Objective Eulerian coherent structure |

WRF | Weather research and forecasting |

NAM | North American Mesoscale model |

OSSE | Observing system simulation experiment |

ROC | Receiver Operating Characteristic |

## Appendix A. Flight Dynamic Model

- Earth is a flat, inertial reference.
- The aircraft is a rigid body, symmetric about its longitudinal plane, with constant mass m.
- For wind-aircraft interaction, the aircraft is a point “located” at it’s center-of-mass.
- The wind is described by a ${C}^{1}$-smooth kinematic vector field.
- Aircraft thrust ${f}_{th}$ is an instantaneously-controllable force acting nose-forward from the center-of-mass.
- All parameters are invariant with altitude. (e.g., no altitudinal variation of density $\rho $, gravity g, ground-effect, etc.)

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**Figure 2.**Schematic of the effect of a trough of the attraction rate, ${s}_{1}$, field on a fluid parcel.

**Figure 3.**Schematic of the trajectory divergence rate: Where $\dot{\rho}<0$, trajectories are instantaneously converging; where $\dot{\rho}>0$ trajectories are instantaneously diverging.

**Figure 4.**Schematic showing positions where velocity measurements were made and the position of the circle gradient frame to the reference frame.

**Figure 5.**Simulated 3D unmanned aircraft systems (UAS) flight path (solid blue line) along with a 2D perfect circle (dotted black line). The 2D circle is at a constant altitude, while the 3D UAS flight tracks the 850 mb isosurface. The vertical dimension is shown highly exaggerated.

**Figure 6.**Comparison of measurements of the trajectory divergence rate $\dot{\rho}$ at the center point of the circular sampling orbit (

**red**), along the path of a simulated UAS flight (

**black**), and along a circular arc (

**blue**). The radius of the circle and flight path is shown in the lower right hand corner of each subplot. The circular arc and the simulated UAS flight are nearly on top of each other.

**Figure 7.**Comparison of measurements of the attraction rate ${s}_{1}$ at the center point of the circular sampling orbit (

**red**), along the path of a simulated UAS flight (

**black**), and along a circular arc (

**blue**). The radius of the circle and flight path is shown in the lower right hand corner of each subplot. The circular arc and the simulated UAS flight are nearly on top of each other.

**Figure 8.**Comparison of the trajectory divergence rate with the $0.5,1,$ and 2 h backward-time finite-time Lyapunov exponent (FTLE) from $t=4$ to $t=215$ h. FTLE fields have been multiplied by $-1$ to offer better comparison of attraction.

**Figure 9.**Comparison of the attraction rate with the $0.5,1,$ and 2 h backward-time FTLE from $t=4$ to $t=215$ h. FTLE fields have been multiplied by $-1$ to offer better comparison of attraction.

**Figure 10.**Depiction of a true positive (

**left**) and a false positive (

**right**), from approximately 43.5 h and 20.67 h, respectively. The true attraction rate field over Kentland Farm Va. is displayed; darker greens are lower values of the attraction rate. Grid points where the value of the attraction rate is above the threshold value are masked (white). Attracting LCS with an integration time of $-1$ h are shown as blue lines. The threshold radius is shown by the black dashed circle and has a radius of 5 km. In both examples, the attraction rate at the center of the sampling domain is below the threshold value, thus meeting the criterion for a positive identification of an LCS passing within the threshold radius. In the true positive example, there is an LCS passing within the threshold radius. Meanwhile, in the false positive, there is no LCS within the threshold radius. The ROC plot point for the particular case shown in this figure (i.e., threshold value and 5.0 km radius) is depicted by a red “+" marker in Figure 14.

**Figure 11.**Schematic of Lagrangian coherent structure (LCS) detection showing two examples of threshold radii as dashed lines. An attracting LCS falls within the larger threshold radius but does not fall within the smaller radius. The instantaneous wind field is depicted as the background vector field.

**Figure 12.**Schematic of an idealized receiver operating characteristic (ROC) curve and with threshold percentiles.

**Figure 13.**ROC curves for the trajectory divergence rate, $\dot{\rho}$, as measured at the center point ability to detect 90th percentile LCSs with integration times of 0.5 (

**green**), 1 (

**red**), and 2 (

**blue**) h. Threshold radius for each subplot is displayed in the upper left hand corner. The area under the curve (AUC) for each integration time is given in the legend.

**Figure 14.**ROC curves for the attraction rate, ${s}_{1}$, as measured at the center point ability to detect 90th percentile LCSs with integration times of 0.5 (

**green**), 1 (

**red**), and 2 (

**blue**) h. The threshold radius for each subplot is displayed in the upper left hand corner. The AUC for each integration time is given in the legend. The red “+” marker corresponds to the plot point for the case (i.e., threshold value and radius) that was shown in Figure 10.

**Figure 15.**ROC curves for the trajectory divergence rate, $\dot{\rho}$, as measured from a 2 km radius UAS simulation ability to detect 90th percentile LCSs with integration times of 0.5 (

**green**), 1 (

**red**), and 2 (

**blue**) h. The threshold radius for each subplot is displayed in the upper left hand corner. The AUC for each integration time is given in the legend.

**Figure 16.**ROC curves for the attraction rate, ${s}_{1}$, as measured from a 2 km radius UAS simulation ability to detect 90th percentile LCSs with integration times of 0.5 (

**green**), 1 (

**red**), and 2 (

**blue**) h. The threshold radius for each subplot is displayed in the upper left hand corner. The AUC for each integration time is given in the legend.

**Table 1.**Pearson correlation coefficients for trajectory divergence rate, $\dot{\rho}$, measurements. Coefficients range from 0.730 to 0.965.

2 km Circle | 5 km Circle | 10 km Circle | 15 km Circle | 2 km Flight | 5 km Flight | 10 km Flight | 15 km Flight | |
---|---|---|---|---|---|---|---|---|

center point | 0.955 | 0.854 | 0.790 | 0.730 | 0.931 | 0.827 | 0.781 | 0.730 |

2 km circle | -- | 0.946 | 0.815 | 0.751 | 0.981 | 0.923 | 0.811 | 0.765 |

5 km circle | -- | 0.866 | 0.768 | 0.935 | 0.981 | 0.865 | 0.784 | |

10 km circle | -- | 0.928 | 0.804 | 0.836 | 0.974 | 0.902 | ||

15 km circle | -- | 0.745 | 0.738 | 0.904 | 0.955 | |||

2 km flight | -- | 0.944 | 0.824 | 0.783 | ||||

5 km flight | -- | 0.870 | 0.793 | |||||

10 km flight | -- | 0.937 |

**Table 2.**Pearson correlation coefficients for attraction rate, ${s}_{1}$, measurements. Coefficients range from 0.577 to 0.939.

2 km Circle | 5 km Circle | 10 km Circle | 15 km Circle | 2 km Flight | 5 km Flight | 10 km Flight | 15 km Flight | |
---|---|---|---|---|---|---|---|---|

center point | 0.939 | 0.838 | 0.677 | 0.590 | 0.910 | 0.821 | 0.675 | 0.577 |

2 km circle | -- | 0.932 | 0.742 | 0.644 | 0.980 | 0.917 | 0.739 | 0.627 |

5 km circle | -- | 0.898 | 0.789 | 0.916 | 0.980 | 0.887 | 0.760 | |

10 km circle | -- | 0.908 | 0.729 | 0.881 | 0.978 | 0.864 | ||

15 km circle | -- | 0.637 | 0.788 | 0.907 | 0.965 | |||

2 km flight | -- | 0.936 | 0.746 | 0.644 | ||||

5 km flight | -- | 0.900 | 0.791 | |||||

10 km flight | -- | 0.909 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Nolan, P.J.; McClelland, H.G.; Woolsey, C.A.; Ross, S.D. A Method for Detecting Atmospheric Lagrangian Coherent Structures Using a Single Fixed-Wing Unmanned Aircraft System. *Sensors* **2019**, *19*, 1607.
https://doi.org/10.3390/s19071607

**AMA Style**

Nolan PJ, McClelland HG, Woolsey CA, Ross SD. A Method for Detecting Atmospheric Lagrangian Coherent Structures Using a Single Fixed-Wing Unmanned Aircraft System. *Sensors*. 2019; 19(7):1607.
https://doi.org/10.3390/s19071607

**Chicago/Turabian Style**

Nolan, Peter J., Hunter G. McClelland, Craig A. Woolsey, and Shane D. Ross. 2019. "A Method for Detecting Atmospheric Lagrangian Coherent Structures Using a Single Fixed-Wing Unmanned Aircraft System" *Sensors* 19, no. 7: 1607.
https://doi.org/10.3390/s19071607