# Parametric, Semiparametric, and Semi-Nonparametric Estimates of the Kinetic Energy of Ordered Air Motion and Wind Outliers in the Atmospheric Boundary Layer from Minisodar Measurements

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## Abstract

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## Featured Application

**In the present work, the spatiotemporal dynamics of kinetic wind energy is analyzed with and without allowance being made for the kinetic energy of the outliers retrieved by the postprocessing of the minisodar measurements of three wind velocity components and their variances. The wind outliers are taken to mean wind velocities, including wind gusts, the distribution of which deviates from the prior distribution of the majority of observations. The minisodar data were processed using robust parametric, semiparametric, and semi-nonparametric algorithms developed by the authors. Allowing for the contribution of the wind outliers in the parametric estimates of the kinetic wind energy enabled its fine structure to be determined, along with an estimation of its effect on the landing and takeoff of airplanes, light flying objects, high-rise buildings, and bridges, and an evaluation of the energy potentials of wind turbines.**

## Abstract

## 1. Introduction

## 2. Problem Formulation

^{2}/s

^{2}(m

^{2}/s

^{2}= J/kg) [24]. It is natural that the regularities in the spatiotemporal behavior of the reduced kinetic energy will fully refer to the total kinetic energy. For this reason, we use the term kinetic energy for the kinetic energy per unit air mass. It is equal to the sum of two components: the mean kinetic energy E

_{M}, associated with the mean wind velocity $\overline{V}$, and the turbulence kinetic energy E

_{T}, associated with the wind velocity variance ${\mathsf{\sigma}}^{2}$. Following [24], we can write:

## 3. Robust Parametric Algorithms and Their Application for Estimating Kinetic Wind Energy

_{0}, the orange curve) and with an allowance for the contribution of the kinetic outlier energy (E, the blue curve) retrieved from measurements with the commercial triaxial Doppler monostatic minisodar AV4000 (Atmospheric Systems Corporation, Santa Clara, CA, USA) [4]; its sounding range was 5–200 m with vertical resolution Δz = 5 m. The acoustic antenna was an array of 50 loudspeakers used to both transmit and receive acoustic signals at a frequency of 4900 Hz. This loudspeaker array was electrically steered to generate three independent beams: one vertical and two others at elevation angles of 76° in two mutually orthogonal planes. The minisodar had a pulse repetition period of 4 s and a pulse duration of 60 ms. The minisodar provided one vertical signal profile in all three channels every 4 s, which was used to calculate the wind vector components V

_{x}(z, t), V

_{y}(z, t), and V

_{z}(z, t), and their variances, ${\mathsf{\sigma}}_{x}^{2}(z,t)$, ${\mathsf{\sigma}}_{y}^{2}(z,t)$, and ${\mathsf{\sigma}}_{z}^{2}(z,t)$ from the well-known formulas for the Doppler frequency shifts. To investigate their dynamics, we sampled and processed 150 vertical profiles recorded from the beginning of each hour from 00:00 till 23:00 to obtain 10 min averages and to estimate the total kinetic wind energy E(z, t) and its components caused by the stationary air movement E

_{0}(z, t) and the wind outliers E

_{out}(z, t).

_{T}without (the orange curve) and with allowance for the contribution of the kinetic outlier energy (the blue curve). The comparison of Figure 2a,b demonstrates that, in this situation, the altitude behavior of the orange and blue curves differs quantitatively only in the layer below 25 m; above this altitude, they qualitatively agree. This suggests that at these altitudes, the predominant contribution to the total kinetic wind energy comes from the turbulence kinetic energy component.

## 4. Robust Semiparametric Algorithms and Their Application for Estimating Kinetic Wind Energy

## 5. Robust Semi-Nonparametric Algorithms and Their Application for Estimation of Kinetic Wind Energy

_{N}of the parameter β. A mathematical analysis of these estimates is beyond the scope of the present work. Note only that they are asymptotically unbiased and efficient for the distribution $G(\mathsf{\zeta},\mathsf{\theta})\in {\tilde{P}}_{0\mathsf{\theta}}$, provided that the sample ${\overrightarrow{{\rm Z}}}_{N}=({\mathsf{\zeta}}_{1},\dots ,{\mathsf{\zeta}}_{N})$ is from $F(\mathsf{\zeta},\overrightarrow{\mathsf{\theta}})$.

_{0}value at z = 10 m is apparently caused by the effect of the underlying surface. The rate of E

_{0}growth increased with altitude starting from z = 150 m. Our statistical analysis of the altitude behavior of the total kinetic energy E with an allowance for the contribution of the kinetic energy of outliers revealed the following. The E values differred only slightly from the corresponding E

_{0}values at altitudes up to 50 m, that is, in this altitude range, the contribution of the kinetic outlier energy to the total kinetic energy was low. The situation radically changed above 50 m. An essential nonmonotonic increase in E values was observed with a further increase in the altitude, accompanied by considerable deviations from the monotonic dependence.

_{out}, it was low at altitudes z ≤ 50 m. Its increase at z = 10 m was caused by the effect of the underlying surface. From 50 to 170 m, the nonmonotonic increase in E

_{out}was observed together with increasing E

_{out}values between 100 and 110 m. Above 170 m, a certain decrease in E

_{out}was observed.

_{MKE}(b), and turbulence, E

_{TKE}(c), components; and the kinetic energy of the outliers ${E}_{\mathrm{out}}=E-{E}_{0}$ (d) retrieved by the postprocessing of the minisodar measurements. From Figure 4a, it can be seen that the maximum values (red color area) E

_{max}~ 500 m

^{2}/s

^{2}were observed at night, from 00:00 till 02:00, local time, and in the evening, from 19:00 till 24:00. However, whereas at night the lower boundary of the layer with enhanced kinetic energy was about 125 m, at night it descended to 75 m. From 02:00 till 19:00, E did not exceed 100 m

^{2}/s

^{2}in the lower layer. The upper boundary of this layer first ascended from about 50 m to 200 m at noon and then descended to 100 m.

_{M}(Figure 4b) demonstrated that from 04:00 till 19:00, its values were about 50 m

^{2}/s

^{2}and were practically independent of time of day and sensing altitude. From midnight till 02:30, E

_{M}underwent local variations, and reached 225 m

^{2}/s

^{2}(red area) at altitudes above 125 m. From 02:30 till 04:00, it increased to 100 m

^{2}/s

^{2}at altitudes from 25 to 75 m (light blue area). From 19:00 till 24:00, areas with enhanced values of E

_{M}~ 225 m

^{2}/s

^{2}(red areas) were observed. As a whole, the contribution of E

_{M}to the total kinetic wind energy E was low.

_{T}obtained by the postprocessing of the minisodar measurements. From the figure, it can be seen that E

_{T}increased with altitude z. In the lower layer, the altitude of which first increased from 75 m at 00:00 to 200 m at 01:00 and then decreased to 75 m at 24:00 undergoing significant altitude hourly variations, it did not exceed 80 m

^{2}/s

^{2}. Above 75 m, an area of enhanced E

_{T}was clearly visible.

^{2}/s

^{2}at altitudes above 100 m. In the evening and at night, the kinetic energy of the outliers (red areas) reached 120 m

^{2}/s

^{2}, that is, providing significant contribution to the total kinetic energy E~200 m

^{2}/s

^{2}in this altitude range (see Figure 4a), and its effect was more pronounced at altitudes between 50 and 100 m.

## 6. Conclusions

_{0}, and with allowance E, for the kinetic energy of the outliers, E

_{out}; its mean, E

_{M}, and turbulence, E

_{T}, components were analyzed by the postprocessing of the minisodar measurements of the three wind vector components and their variances in the lower 200 m layer of the atmosphere. The parametric, semiparametric, and semi-nonparametric robust algorithms for obtaining effective $\widehat{E}$, ${\widehat{E}}_{0}$, and ${\widehat{E}}_{\mathrm{out}}$ estimates with the model distributions $F(\mathsf{\zeta},\overrightarrow{\mathsf{\theta}}),G(\mathsf{\zeta},\overrightarrow{\mathsf{\theta}})$, and $H(\mathsf{\zeta},\overrightarrow{\mathsf{\theta}})$ used to analyze their spatiotemporal dynamics allowed us to reveal some important physical features. Without making allowances for the contribution of the kinetic energy of the outliers, the altitude profiles of the kinetic wind energy components in the ABL were depicted as quite smooth curves. The allowance for the contribution of the kinetic energy of the outliers revealed their thin-layered structure, thereby demonstrating the efficiency of the proposed robust algorithms. The nonmonotonic increase in the kinetic energy of the wind outliers with altitude sounding was established. Physically, this can be explained by the nonmonotonic increase in the turbulence kinetic energy of local air vortices in the ABL. The vertical extension of the outlier layers was of the order of 10–20 m.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Foken, T. Micrometeorology; Springer: Berlin/Heidelberg, Germany, 2008; 306p. [Google Scholar]
- Byzova, N.L.; Ivanov, V.N.; Garger, E.K. Turbulence in the Atmospheric Boundary Layer; Gidrometeoizdat: Leningrad, Russia, 1989; 264p. [Google Scholar]
- Monin, A.S. Structure of the atmospheric turbulence. Theory Probab. Its Appl.
**1958**, 3, 285–317. [Google Scholar] [CrossRef] - Doppler MiniSoDAR System: Operation and Maintenance Manual. Available online: https://home.chpc.utah.edu/~{}\{\}u0035056/5910_2010/Minisodarmanual.pdf (accessed on 30 December 2020).
- Bradley, S. Atmospheric Acoustic Remote Sensing: Principles and Applications; CRC Press Taylor & Francis Group: Boca Raton, FL, USA, 2007; 296p. [Google Scholar]
- Simakhin, V.A.; Cherepanov, O.S.; Shamanaeva, L.G. Experimental Data Processing under Conditions of A Priori Statistical Uncertainty; Scientific Technology Publishing House: Tomsk, Russia, 2021; 340p. [Google Scholar]
- Zou, M.; Diokic, S.Z. A review of approaches for the detection and treatment of outliers in processing wind turbine and wind farm measurements. Energies
**2020**, 13, 4228. [Google Scholar] [CrossRef] - Hampel, F.R.; Ronchetti, E.M.; Rousseeuw, P.J.; Stahel, W.A. The Approach Based on Influence Functions; John Wiley & Sons, Inc.: New York, NY, USA, 1986; 502p. [Google Scholar]
- Gill, S.; Stephen, B.; Galloway, S. Wind Turbine Condition Assessment through Power Curve Copula Modeling. IEEE Trans. Sustain. Energy
**2012**, 3, 94–101. [Google Scholar] [CrossRef] - Shurygin, A.M. Applied Statistics. Robustness. Estimation. Prediction; Finansy i Statistika: Moscow, Russia, 2000; 223p. [Google Scholar]
- Shulenin, V.P. Robust Methods of Mathematical Statistics; Scientific Technology Publishing House: Tomsk, Russia, 2016; 260p. [Google Scholar]
- Simakhin, V.A.; Shamanaeva, L.G.; Avdjushina, A.E. Robust semiparametric and semi-nonparametric estimates of inhomogeneous experimental data. Russ. Phys. J.
**2021**, 64, 355–366. [Google Scholar] [CrossRef] - De Menezes, D.Q.F.; Prata, D.M.; Secchi, A.R.; Pinto, J.C. A review on robust M-estimators for regression analysis. Comp. Chem. Eng.
**2021**, 147, 107254. [Google Scholar] [CrossRef] - Pitselis, G.A. review on robust estimators applied to regression credibility. J. Comput. Appl. Math.
**2013**, 239, 231–249. [Google Scholar] [CrossRef] - Yu, C.; Yao, W. Robust linear regression: A review and comparison. Commun. Stat. Simul. Comput.
**2016**, 46, 6261–6282. [Google Scholar] [CrossRef] - Baeldung. Robust Estimators in Robust Statistics. Math and Logic. Probability and Statistics. 2023. Available online: http://www.baeldung.com/cs/robust-estimators-in-robust-statistics (accessed on 11 April 2023).
- Fedorov, V.A. Measuring the parameters of the radial components of the wind velocity vector with the “Volna-3” sodar. Opt. Atm. Okeana
**2003**, 16, 151–155. [Google Scholar] - Gladkikh, V.A.; Nevzorova, I.V.; Mamyshev, V.P.; Odintsov, S.L. Skewness and kurtosis of the distribution of the outer scales of turbulence in the near-surface layer of the atmosphere. In Proceedings of the XXV International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics, Novosibirsk, Russia, 1–5 July 2019. [Google Scholar]
- Jaeckel, L.A. Robust estimates of location: Symmetry and asymmetric contamination. Ann. Math. Stat.
**1971**, 42, 1020–1034. [Google Scholar] [CrossRef] - Muthukrishnan, R.; Poonkuzhali, G. A comprehensive survey on outlier detection methods. Am. Eurasian J. Sci. Res.
**2017**, 12, 161–171. [Google Scholar] - Chandola, V.; Banerjee, A.; Kumar, V. Anomaly detection: A survey. ACM Comput. Surv.
**2009**, 41, 58. [Google Scholar] [CrossRef] - Dmitriev, Y.G.; Koshkin, G.M. Nonparametric estimators of probability characteristics using unbiased prior conditions. Stat. Pap.
**2018**, 59, 1559–1575. [Google Scholar] [CrossRef] - Dmitriev, Y.G.; Koshkin, G.M. Using additional data in nonparametric estimation of density functionals. Avtomat. Telemekh.
**1987**, 10, 47–59. [Google Scholar] - Potekaev, A.; Shamanaeva, L.; Kulagina, V. Spatiotemporal dynamics of the kinetic energy in the atmospheric boundary layer from minisodar measurements. Atmosphere
**2021**, 12, 421. [Google Scholar] [CrossRef] - Underwood, K.H.; Shamanaeva, L.G. Temporal dynamics of longitudinal and transverse velocity structure functions retrieved from the data of acoustic sounding. Russ. Phys. J.
**2011**, 54, 113–120. [Google Scholar] [CrossRef] - Shikhovtsev, A.Y.; Kiselev, A.V.; Kovadlo, P.G.; Kolobov, D.Y.; Lukin, V.P.; Tomin, V.E. Method for estimating the atmospheric layers with strong turbulence. Atmos. Ocean. Opt.
**2020**, 33, 295–301. [Google Scholar] [CrossRef] - Bolbasova, L.A.; Shikhovtsev, A.Y.; Kopylov, E.A.; Selin, A.A.; Lukin, V.P.; Kovadlo, P.G. Daytime optical turbulence and wind speed distributions at the Baikal Astrophysical Observatory. Mon. Not. R. Astron. Soc.
**2019**, 482, 2619–2626. [Google Scholar] [CrossRef]

**Figure 1.**Prior normal distribution: (

**a**) distribution densities $g(\mathrm{\zeta},\overrightarrow{\mathrm{\theta}}),\mathsf{\epsilon}=0$ (cruve 1); $f(\mathrm{\zeta},1,1),\mathsf{\epsilon}=0.1$–internal outlier (curve 2); $f(\mathrm{\zeta},5,1),\mathsf{\epsilon}=0.1$–external outlier (curve 3); (

**b**) weight functions $W(\mathrm{\zeta},0,1),\mathsf{\epsilon}=0$ (curve 1); MLE $W(\mathrm{\zeta},1,1),\mathsf{\epsilon}=0.1$ (curve 2); weight function $W(\mathrm{\zeta},0,1),\mathsf{\epsilon}=0.1$ for the robust highly efficient MD Hellinger distance estimate [6,7] (curve 3); (

**c**) estimation functions ${U}_{1}(\mathrm{\zeta},0,1),\mathsf{\epsilon}=0$ (curve 1); MLE ${U}_{1}(\mathrm{\zeta},1,1),\mathsf{\epsilon}=0.1$ (curve 2); estimation fuction ${U}_{1}(\mathrm{\zeta},0,1),\mathsf{\epsilon}=0.1$ for the robust highly efficient MD Hellinger distance estimate (curve 3); (

**d**) estimation functions ${U}_{2}(\mathrm{\zeta},0,1),\mathsf{\epsilon}=0$ (curve 1); MLE ${U}_{2}(\mathrm{\zeta},1,1),\mathsf{\epsilon}=0.1$ (curve 2); estimation function ${U}_{2}(\mathrm{\zeta},0,1),\mathsf{\epsilon}=0.1$ for the robust highly efficient MD Hellinger distance estimates (curve 3).

**Figure 2.**Vertical profiles of the parametric estimates of the kinetic wind energy in the atmosphere obtained by postprocessing of the 10 min AV4000 minisodar measurement series on 10 September 2003, started at 12:00, local time: (

**a**) the total kinetic energy without (E

_{0}, the orange curve) and with allowance for the contribution of the kinetic outlier energy (E, blue curve), and (

**b**) turbulence kinetic energy component without (${E}_{0\mathrm{T}}$, the orange curve) and with allowance for the contribution of the kinetic outlier energy (E

_{T}, the blue curve).

**Figure 3.**Model of the asymmetric internal outliers with $\mathsf{\theta}=0,\mathsf{\epsilon}=0.1,\mathsf{\mu}=1,\mathrm{and}\mathsf{\lambda}=0.2$: (

**a**) distribution density, where curve 1 shows the estimate ${f}_{N}(x)$ with normal kernel (7), and curve 2 shows the distribution density $f(x)$; (

**b**) weight function $W(x)$, where curve 1 shows the estimate ${W}_{N}(x)$ with normal kernel (7), and curve 2 shows the parametric estimate $W(x)$.

**Figure 4.**Diurnal dynamics at hourly intervals of the robust semi-nonparametric estimates of the total kinetic wind energy with allowance for the outliers E (

**a**); its mean, E

_{M}(

**b**); and turbulence, E

_{T}(

**c**); components; and the kinetic energy of the outliers ${E}_{\mathrm{out}}=E-{E}_{0}$ (

**d**); retrieved by postprocessing of minisodar measurements observed from midnight till 04:00, when E

_{T}changed from 400 to 200 m

^{2}/s

^{2}. One more area with enhanced ET values was observed from 14:00 till 24:00; here, E

_{T}also changed from 400 to 200 m

^{2}/s

^{2}, and its lower boundary descended from 175 m at 15:00 to 125 m at 24:00. A comparison of Figure 4a–c shows that the turbulence kinetic energy component defines exactly the local features of the daily dynamics at hourly intervals of the total kinetic energy, and the daily dynamics by the hour of the mean kinetic energy component forms the background for the turbulence kinetic energy.

Altitude z, m | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

E_{0}, m^{2}/s^{2} | 8.03 | 3.29 | 4.40 | 4.13 | 3.76 | 5.82 | 6.47 | 5.60 | 6.75 | 8.03 |

E, m^{2}/s^{2} | 9.62 | 3.34 | 4.62 | 4.33 | 4.17 | 9.25 | 11.50 | 10.60 | 12.68 | 23.68 |

E_{out}, m^{2}/s^{2} | 1.59 | 0.05 | 0.22 | 0.20 | 0.41 | 3.43 | 5.03 | 5.0 | 5.93 | 15.65 |

Altitude z, m | 110 | 120 | 130 | 140 | 150 | 160 | 170 | 180 | 190 | 200 |

E_{0}, m^{2}/s^{2} | 8.93 | 6.61 | 8.36 | 11.0 | 11.7 | 16.6 | 34.76 | 40.13 | 53.94 | 85.83 |

E, m^{2}/s^{2} | 20.73 | 15.46 | 16.8 | 28.83 | 28.09 | 35.39 | 80.35 | 76.71 | 92.58 | 121.80 |

E_{out}, m^{2}/s^{2} | 11.8 | 8.85 | 8.44 | 17.83 | 16.39 | 18.79 | 45.59 | 36.58 | 38.64 | 35.97 |

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**MDPI and ACS Style**

Simakhin, V.A.; Potekaev, A.I.; Cherepanov, O.S.; Shamanaeva, L.G. Parametric, Semiparametric, and Semi-Nonparametric Estimates of the Kinetic Energy of Ordered Air Motion and Wind Outliers in the Atmospheric Boundary Layer from Minisodar Measurements. *Appl. Sci.* **2023**, *13*, 6116.
https://doi.org/10.3390/app13106116

**AMA Style**

Simakhin VA, Potekaev AI, Cherepanov OS, Shamanaeva LG. Parametric, Semiparametric, and Semi-Nonparametric Estimates of the Kinetic Energy of Ordered Air Motion and Wind Outliers in the Atmospheric Boundary Layer from Minisodar Measurements. *Applied Sciences*. 2023; 13(10):6116.
https://doi.org/10.3390/app13106116

**Chicago/Turabian Style**

Simakhin, Valerii Anan’evich, Alexander Ivanovich Potekaev, Oleg Sergeevich Cherepanov, and Liudmila Grigor’evna Shamanaeva. 2023. "Parametric, Semiparametric, and Semi-Nonparametric Estimates of the Kinetic Energy of Ordered Air Motion and Wind Outliers in the Atmospheric Boundary Layer from Minisodar Measurements" *Applied Sciences* 13, no. 10: 6116.
https://doi.org/10.3390/app13106116