# Temporal Up-Sampling of Planar Long-Range Doppler LiDAR Wind Speed Measurements Using Space-Time Conversion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Planar LiDAR Data

#### 2.1. Measurement Trajectories

#### 2.2. Synthetic LiDAR Data

## 3. Method

#### 3.1. Wind-Speed Reconstruction

#### 3.2. Wind-Field Propagation

#### 3.3. Temporal Correction and Data Synchronisation

## 4. Results

#### 4.1. Calculation of Synthetic LiDAR Data

#### 4.1.1. Error Case Discrimination

- Error at the centreline in the wake: $\gamma {D}^{-1}=0$ and $\zeta {D}^{-1}=0$.
- Planar wake error: $-1\le \gamma {D}^{-1}\le 1$ and $-1\le \zeta {D}^{-1}\le 1$.
- Error in the free stream outside of the wake (opposite the planar wake case).

#### 4.2. Time-Resolution Improvement

#### 4.3. Influence of the Interpolation Time Step, Δt, on the Statistical Error

- Inside the wake for $-1\le \gamma {D}^{-1}\le 1$ and $-1\le \zeta {D}^{-1}\le 1$.
- Along the centreline for $\gamma {D}^{-1}=0$ and $\zeta {D}^{-1}=0$.
- In the free flow.

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Visualisation of the normalised 10-min averaged wind-speed component $\overline{u}$ of the original RHI data and (

**d**–

**f**) and (

**j**–

**k**) the corresponding flow deviations, ${\epsilon}_{\overline{u}}$, from (

**i**) the normalised 10-min averaged wind-speed component, $\overline{u}$, of LES data.

**Figure A2.**Visualisation of the normalised 10-min standard deviation, ${\sigma}_{u}$, of the wind-speed component, $u$, of the original RHI data and (

**d**–

**f**) and (

**j**–

**k**) the corresponding deviations, ${\epsilon}_{{\sigma}_{u}}$, from (

**i**) the normalised 10-min standard deviation of wind-speed component, $u$, of LES data.

## Appendix B

**Table A1.**Average wind speed and standard deviation error against the LES of PPI scans for different angular velocities, ${\omega}_{\varphi}$.

1°/s | 2°/s | 3°/s | 4°/s | 5°/s | 6°/s | 8°/s | 12°/s | 19.1°/s | 27.2°/s | 33.3°/s | |
---|---|---|---|---|---|---|---|---|---|---|---|

wake region$\mathbf{-}\mathbf{1}\mathbf{\le}\mathit{\gamma}{\mathit{D}}^{\mathbf{-}\mathbf{1}}\mathbf{\le}\mathbf{1}$ | |||||||||||

${\mathit{\epsilon}}_{\overline{\mathit{u}}}$ | 1.68% | 0.72% | 0.36% | 0.29% | 0.31% | 0.35% | 0.49% | 0.99% | 1.99% | 3.17% | 4.21% |

${\mathit{\epsilon}}_{{\mathit{\sigma}}_{\mathit{u}}}$ | 12.61% | 5.71% | 4.81% | 5.00% | 5.93% | 7.06% | 9.43% | 14.83% | 21.04% | 25.63% | 27.85% |

wake centreline$\mathit{\gamma}{\mathit{D}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\mathbf{0}$ | |||||||||||

${\mathit{\epsilon}}_{\overline{\mathit{u}}}$ | 1.71% | 0.81% | 0.54% | 0.46% | 0.49% | 0.81% | 1.54% | 2.77% | 4.91% | 4.27% | 0.65% |

${\mathit{\epsilon}}_{{\mathit{\sigma}}_{\mathit{u}}}$ | 9.81% | 5.54% | 3.59% | 2.36% | 2.29% | 5.23% | 11.87% | 20.65% | 27.09% | 20.72% | 2.01% |

free flow | |||||||||||

${\mathit{\epsilon}}_{\overline{\mathit{u}}}$ | 0.77% | 0.43% | 0.25% | 0.19% | 0.18% | 0.17% | 0.18% | 0.23% | 0.38% | 0.74% | 1.17% |

${\mathit{\epsilon}}_{{\mathit{\sigma}}_{\mathit{u}}}$ | 11.78% | 7.19% | 4.91% | 3.77% | 3.68% | 3.90% | 5.00% | 8.34% | 14.22% | 18.15% | 16.69% |

**Table A2.**Average wind speed and standard deviation error against the LES of RHI scans for different angular velocities, ${\omega}_{\theta}$.

1°/s | 2°/s | 3°/s | 4°/s | 5°/s | 6°/s | 8°/s | 12°/s | 19.1°/s | 27.2°/s | 33.3°/s | |
---|---|---|---|---|---|---|---|---|---|---|---|

wake region$\mathbf{-}\mathbf{1}\mathbf{\le}\mathit{\zeta}{\mathit{D}}^{\mathbf{-}\mathbf{1}}\mathbf{\le}\mathbf{1}$ | |||||||||||

${\mathit{\epsilon}}_{\overline{\mathit{u}}}$ | 2.15% | 0.76% | 0.40% | 0.26% | 0.31% | 0.30% | 0.54% | 1.04% | 2.21% | 2.84% | 3.59% |

${\mathit{\epsilon}}_{{\mathit{\sigma}}_{\mathit{u}}}$ | 11.87% | 6.15% | 4.66% | 4.96% | 5.70% | 6.94% | 9.24% | 12.87% | 18.43% | 21.02% | 29.30% |

wake centreline$\mathit{\zeta}{\mathit{D}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\mathbf{0}$ | |||||||||||

${\mathit{\epsilon}}_{\overline{\mathit{u}}}$ | 1.58% | 0.73% | 0.39% | 0.31% | 0.32% | 0.50% | 1.11% | 0.56% | 4.13% | 2.11% | 8.20% |

${\mathit{\epsilon}}_{{\mathit{\sigma}}_{\mathit{u}}}$ | 10.28% | 5.29% | 3.97% | 2.78% | 2.87% | 5.47% | 9.37% | 7.58% | 19.78% | 10.00% | 28.98% |

free flow | |||||||||||

${\mathit{\epsilon}}_{\overline{\mathit{u}}}$ | 2.01% | 0.72% | 0.45% | 0.42% | 0.51% | 0.53% | 0.81% | 1.44% | 2.39% | 3.00% | 3.58% |

${\mathit{\epsilon}}_{{\mathit{\sigma}}_{\mathit{u}}}$ | 12.02% | 6.27% | 4.98% | 5.51% | 6.48% | 7.92% | 10.60% | 14.26% | 19.26% | 22.48% | 30.87% |

## Appendix C

**Figure A3.**Visualisation of the normalised 10-min averaged wind-speed component, $\overline{u}$, of the propagated RHI data and (

**d**–

**f**) and (

**j**–

**k**) the corresponding flow deviations, ${\epsilon}_{\overline{u}}$, from (

**i**) the normalised 10-min averaged wind-speed component, $\overline{u}$, of LES data.

**Figure A4.**Visualisation of the normalised 10-min standard deviation, ${\sigma}_{u}$, of the wind-speed component, $u$, of the propagated RHI data and (

**d**–

**f**) and (

**j**–

**k**) the corresponding deviations, ${\epsilon}_{{\sigma}_{u}}$, from (

**i**) the normalised 10-min standard deviation of wind-speed component, $u$, of LES data.

## Appendix D

**Figure A5.**

**Figure**

**A5.**Effects of different numbers of interpolation steps (${\mathsf{\Pi}}_{\theta}$) on the error in (

**a**–

**c**) the average wind speed (${\epsilon}_{\overline{u}}$) and (

**d**–

**f**) the error in the standard deviation (${\epsilon}_{{\sigma}_{u}}$) for (

**a**,

**d**) wakes in the range of $-1\le \zeta {D}^{-1}\le 1$, (

**b**,

**e**) along the centreline, $\zeta {D}^{-1}=0$, and (

**c**,

**f**) in the free flow of propagated RHI scans.

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**Figure 1.**Illustration of cross-measurement trajectories. A horizontal PPI scan is shown in dark blue, and a vertical RHI scan is shown in pink. The wake volume is indicated in light blue.

**Figure 2.**Illustration of the polar measurement grid of a PPI scan in the Cartesian χ-γ coordinate system. Each sub-figure plots the grid density for eight out of a total of eleven different angular velocities (${\omega}_{\varphi}$).

**Figure 3.**Illustration of the measurement geometry, including the wind vector, $\overrightarrow{u}$, the azimuth angle, ($\varphi $), and the elevation angle, ($\theta $), of (

**a**) a horizontal PPI scan and (

**b**) a vertical RHI scan. The red line indicates the LiDAR laser beam.

**Figure 4.**Example of the weighting function, ${w}_{n}\left(t\right)$, for forward propagation (blue) and the function, $1-{w}_{n}\left(t\right)$, for backwards propagation (red).

**Figure 5.**Temporal alignment of a PPI scan at time $t$. Dark-blue lines indicate LiDAR measurements, light-blue lines indicate the propagation steps, green lines indicate the measurement reset time (${t}_{r}$), and the red line marks the time of interpolation. This example shows ${\mathrm{N}}_{\varphi}=11$.

**Figure 6.**Example of the temporal correction for a time shift with a PPI scan. The first column (

**a**,

**d**,

**g**) shows the original LiDAR scan in the time interval, ${T}_{n}=\left[160\text{}\mathrm{s},\text{}200\text{}\mathrm{s}\right]$. The second column (

**b**,

**e**,

**h**) shows the temporal improved and corrected data at (

**b**) $t=160\text{}\mathrm{s}$, (

**e**) $t=185\text{}\mathrm{s}$, and (

**h**) $t=200\text{}\mathrm{s}$. The third column (

**c**,

**f**,

**i**) illustrates the instantaneous wind-speed deviations of the data in the first and second columns. The green circles mark the characteristic wake structure within the original PPI scan. The red circles indicate the propagated position of the same characteristic wake structure within the time-aligned data at the defined time points.

**Figure 7.**(

**a**–

**c**,

**g**,

**h**) Visualisation of the normalised 10-min averaged wind-speed component ($\overline{u}$) of the original PPI data and (

**d**–

**f**,

**j**,

**k**) the corresponding flow deviations (${\epsilon}_{\overline{u}}$) in comparison to (

**i**) the normalised 10-min averaged wind-speed component ($\overline{u}$) of the LES data.

**Figure 8.**(

**a**–

**c**,

**g**,

**h**) Visualisation of the normalised 10-min standard deviation (${\sigma}_{u}$) of the wind-speed component ($u$) of the original PPI data and (

**d**–

**f**,

**j**,

**k**) the corresponding deviations (${\epsilon}_{{\sigma}_{u}}$) from (

**i**) the normalised 10-min standard deviation of the wind-speed component ($u$) in the LES data.

**Figure 9.**Velocity error (${\epsilon}_{\overline{u}}$, blue) and standard-deviation error (${\epsilon}_{{\sigma}_{u}}$, red) of synthetic LiDAR data as compared against the LES for the planar section (crosses), centreline (circles), and free-flow region (squares) of (

**a**) PPI scans and (

**b**) RHI scans.

**Figure 10.**Visualisation of the normalised 10-min averaged wind-speed component ($\overline{u}$) of the propagated PPI data and (

**d**–

**f**,

**j**,

**k**) the corresponding flow deviations (${\epsilon}_{\overline{u}}$) in comparison to (

**i**) the normalised 10-min averaged wind-speed component ($\overline{u}$) of the LES data.

**Figure 11.**Visualisation of the normalised 10-min standard deviation (${\sigma}_{u}$) of the wind-speed component ($u$) of the propagated PPI data and (

**d**–

**f**,

**j**,

**k**) the corresponding deviations (${\epsilon}_{{\sigma}_{u}}$) in comparison to (

**i**) the normalised 10-min standard deviation of the wind-speed component ($u$) of the LES data.

**Figure 12.**Effect of different numbers of interpolation steps, ${\mathsf{\Pi}}_{\varphi}$, on the error of (

**a**–

**c**) the average wind speed, ${\epsilon}_{\overline{u}}$, and (

**d**–

**f**) the error of the standard deviation, ${\epsilon}_{{\sigma}_{u}}\text{}$, for (

**a**,

**d**) the wake in the range of $-1\le \gamma {D}^{-1}\le 1$, (

**b**,

**e**) along the centreline, $\gamma {D}^{-1}=0$, and (

**c**,

**f**) in free flow of the propagated PPI scans.

**Table 1.**Simulated LiDAR trajectories of cross-measurements for ${n}_{r}=180$ and $\mathrm{T}=600\text{}\mathrm{s}$.

${\omega}_{\varphi}$, ${\omega}_{\theta}$ | $\Delta \varphi $, $\Delta \theta $ | ${\mathrm{N}}_{\varphi}$, ${\mathrm{N}}_{\theta}$ | ${n}_{pnt}$ | ${n}_{\varphi}$, ${n}_{\theta}$ | ${\mathcal{R}}_{\varphi}$, ${\mathcal{R}}_{\theta}$ | ${\mathrm{T}}_{\varphi}$ ${\mathrm{T}}_{\theta}$ | ${f}_{s}$ | ${\eta}_{m}$ |

1°/s | 40° | 15 | 36000 | 200 | 0.2° | 40.0 s | 0.024 Hz | 97.2 % |

2°/s | 40° | 29 | 18000 | 100 | 0.4° | 20.0 s | 0.047 Hz | 94.2 % |

3°/s | 40° | 42 | 11880 | 66 | 0.6° | 13.3 s | 0.068 Hz | 91.0 % |

4°/s | 40° | 54 | 9000 | 50 | 0.8° | 10.0 s | 0.089 Hz | 89.2 % |

5°/s | 40° | 66 | 7920 | 40 | 1.0° | 8.0 s | 0.011 Hz | 86.8 % |

6°/s | 40° | 77 | 6840 | 33 | 1.2° | 6.7 s | 0.127 Hz | 84.6 % |

8°/s | 40° | 97 | 4500 | 25 | 1.6° | 5.0 s | 0.161 Hz | 80.6 % |

12°/s | 40° | 133 | 2880 | 16 | 2.5° | 3.3 s | 0.221 Hz | 73.4 % |

19.11°/s | 40° | 183 | 1800 | 10 | 4.0° | 2.1 s | 0.303 Hz | 63.4 % |

26.22°/s | 40° | 221 | 1260 | 7 | 5.7° | 1.5 s | 0.370 Hz | 55.8 % |

33.33°/s | 40° | 250 | 1080 | 6 | 6.7° | 1.2 s | 0.417 Hz | 50.0 % |

**Table 2.**Combinations of $\omega $ and $\mathsf{\Pi}$ that give local minima for different error regions in the PPI and RHI data.

PPI | RHI | |||||
---|---|---|---|---|---|---|

Wake Region | Wake Centreline | Free Flow | Wake Region | Wake Centreline | Free Flow | |

$\omega $ | 5°/s | 5°/s | 5°/s | 5°/s | 5°/s | 3°/s |

$\mathsf{\Pi}$ | 16 | 18 | 9 | 14 | 16 | 7 |

${\mathit{\epsilon}}_{\overline{\mathit{u}}}$ | 0.24% | 0.15% | 0.14% | 0.20% | 0.13% | 0.40% |

$\omega $ | 4°/s | 5°/s | 4°/s | 4°/s | 5°/s | 3°/s |

$\mathsf{\Pi}$ | 33 | 21 | 16 | 33 | 26 | 13 |

${\mathit{\epsilon}}_{{\mathit{\sigma}}_{\mathit{u}}}$ | 2.89% | 0.60% | 3.02% | 2.79% | 1.61% | 4.53% |

**Table 3.**Combinations of $\omega $ and $\mathsf{\Pi}$ that optimize ${\eta}_{\overline{u}}$ and ${\eta}_{{\sigma}_{u}}$ for different cases of PPI and RHI data.

PPI | RHI | |||||
---|---|---|---|---|---|---|

Wake Region | Wake Centreline | Free Flow | Wake Region | Wake Centreline | Free Flow | |

$\omega $ | 1°/s | 2°/s | 2°/s | 1°/s | 1°/s | 1°/s |

$\mathsf{\Pi}$ | 16 | 21 | 13 | 13 | 16 | 8 |

${\eta}_{\overline{u}}$ | 35.5% | 31.0% | 36.9% | 30.5% | 31.2% | 46.5% |

$\omega $ | 1°/s | 1°/s | 1°/s | 1°/s | 1°/s | 1°/s |

$\mathsf{\Pi}$ | 33 | 42 | 37 | 67 | 60 | 26 |

${\eta}_{{\sigma}_{u}}$ | 42.8% | 43.6% | 45.6% | 43.9% | 42.0% | 60.5% |

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**MDPI and ACS Style**

Beck, H.; Kühn, M.
Temporal Up-Sampling of Planar Long-Range Doppler LiDAR Wind Speed Measurements Using Space-Time Conversion. *Remote Sens.* **2019**, *11*, 867.
https://doi.org/10.3390/rs11070867

**AMA Style**

Beck H, Kühn M.
Temporal Up-Sampling of Planar Long-Range Doppler LiDAR Wind Speed Measurements Using Space-Time Conversion. *Remote Sensing*. 2019; 11(7):867.
https://doi.org/10.3390/rs11070867

**Chicago/Turabian Style**

Beck, Hauke, and Martin Kühn.
2019. "Temporal Up-Sampling of Planar Long-Range Doppler LiDAR Wind Speed Measurements Using Space-Time Conversion" *Remote Sensing* 11, no. 7: 867.
https://doi.org/10.3390/rs11070867