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Non-Gaussian Lagrangian Stochastic Model for Wind Field Simulation in the Surface Layer

Funds:

USDA-AFRI Foundational Grant (Grant No. 2012-67013-19687) and from the Illinois State Water Survey at the University of Illinois at Urbana—Champaign


doi: 10.1007/s00376-019-9052-7

  • Wind field simulation in the surface layer is often used to manage natural resources in terms of air quality, gene flow (through pollen drift), and plant disease transmission (spore dispersion). Although Lagrangian stochastic (LS) models describe stochastic wind behaviors, such models assume that wind velocities follow Gaussian distributions. However, measured surface-layer wind velocities show a strong skewness and kurtosis. This paper presents an improved model, a non-Gaussian LS model, which incorporates controllable non-Gaussian random variables to simulate the targeted non-Gaussian velocity distribution with more accurate skewness and kurtosis. Wind velocity statistics generated by the non-Gaussian model are evaluated by using the field data from the Cooperative Atmospheric Surface Exchange Study, October 1999 experimental dataset and comparing the data with statistics from the original Gaussian model. Results show that the non-Gaussian model improves the wind trajectory simulation by stably producing precise skewness and kurtosis in simulated wind velocities without sacrificing other features of the traditional Gaussian LS model, such as the accuracy in the mean and variance of simulated velocities. This improvement also leads to better accuracy in friction velocity (i.e., a coupling of three-dimensional velocities). The model can also accommodate various non-Gaussian wind fields and a wide range of skewness–kurtosis combinations. Moreover, improved skewness and kurtosis in the simulated velocity will result in a significantly different dispersion for wind/particle simulations. Thus, the non-Gaussian model is worth applying to wind field simulation in the surface layer.
    摘要: 准确模拟大气表层的风场能够有效帮助管理自然资源,例如空气质量,花粉基因流向,以及植物病毒传播等。拉格朗日随机模型是一种风场模拟模型,能够有效模拟风的随机行为。该模型假设风速服从高斯分布,然而实测大气表层风速分布具有很强的偏度和峰度。为了更加准确地模拟实测风场,本文提出了一种改进模型,称作非高斯拉格朗日随机模型。该模型以非高斯风速分布为目标,结合可控的非高斯随机变量,生成符合预期偏度和峰度的随机数序列。为了验证模型的正确性,本文利用CASES(Cooperative Atmospheric Surface Exchange Study)于1999年10月实测的大气表层风场数据,评估模型的模拟效果,并与传统的高斯模型进行对比。结果表明,非高斯模型能够显著提高风速偏度和峰度的模拟准确性,并具有较好的稳定性。于此同时,非高斯模型够维持传统模型的其它所有优点,例如在风速平均值和方差上的模拟准确性。此外,提高风速偏度和峰度准确性还使得非高斯模型能够生成更好的摩擦速度。非高斯模型还能够应用于不同类型的非高斯风场,广泛适应不同的偏度和峰度组合。再者,虽然非高斯模型与传统高斯模型仅仅在风速偏度和峰度模拟上有所差别,但两个模型模拟出的风/微粒传播轨迹却有显著差异,这意味着非高斯模型在大气表层风场模拟上可能具有良好的应用价值。
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  • Figure 1.  Rotated coordinate systems, and eight heights for measuring wind data in the CASES-99 project. Units: m.

    Figure 2.  Regression analysis on the mean (E), variance (V), skewness (S), kurtosis (K) of the simulated velocities by the Gaussian (g) and non-Gaussian (ng) models in three wind directions (u, v, w) at eight heights (z), where the simulated (sim) statistics are compared with the measured (mea) ones. Units of velocity: m s−1. This figure shares the same legend as Fig. 3.

    Figure 3.  Regression analysis on the friction velocity (u*) of the simulated velocities by the Gaussian (g) and non-Gaussian (ng) models at eight heights (z), where the simulated (sim) statistics are compared with the measured (mea) ones. Units: m s−1.

    Figure 4.  The range of simulated skewness–kurtosis combinations versus the measured ones. Blue line is from the Gaussian distribution.

    Figure 5.  Four statistical characteristics of wind displacement (Dis) distribution, including mean (M), variance (V), skewness (S), and kurtosis (K), with different number of total steps, driven by three types of velocity distributions (V1, V2, V3) in terms of the same mean/variance (E/V) and different skewness/kurtosis (S/K). Statistical differences between two displacement distributions are estimated by the two-sample Kolmogorov–Smirnov test at a 5% significance level, and any two displacement distributions with the same total steps are significantly different (p-value < 0.001). Units for the mean and variance of the displacement are m and m2, respectively, while the skewness and kurtosis of the displacement are dimensionless.

    Figure 6.  The average concentration percentage differences between Gaussian and non-Gaussian models with 448 different 30-min weather conditions at six different heights, where each dot in a figure represents the percentage of the concentration difference (non-Gaussian model simulated concentration minus the Gaussian’s, then normalized by the Gaussian’s); the p-value denotes the statistical difference between concentrations generated by two models by the Wilcoxon signed rank test at a 5% significance level; the delta value indicates the size effect of statistical difference.

    Table 1.  An example of the input-parameter relationship.

    No.mvdεμpσpSK
    11.02.4230.840.0221.1230.2894.383
    20.82.8260.840.0331.1490.2154.384
    30.92.4260.900.0271.1060.244.385
    DownLoad: CSV

    Table 2.  The ranges of measured atmospheric conditions for each height (z) in three wind directions (u, v, w).

    Height
    z (m)
    SkewnessKurtosisFriction Velocity
    u* (m s−1)
    Atmospheric Stability
    L (m)
    Wind Direction
    θ (°)
    Mean Wind Speed
    $\bar u$ (m s−1)
    Variance
    u (m s−1)v (m s−1)w (m s−1)u (m s−1)v (m s−1)w (m s−1)u (m s−1)v (m s−1)w (m s−1)
    1.5−0.49 to 1.37−1.23 to 1.03−0.16 to 0.722.00 to 6.571.99 to 7.012.97 to 6.480.07 to 0.77−883.91 to −0.271.74 to 359.990.26 to 7.220.08 to 3.660.07 to 3.200.01 to 0.73
    5−1.39 to 1.23−1.23 to 1.43−0.39 to 0.911.75 to 7.511.79 to 6.362.71 to 6.950.07 to 0.77−565.75 to 216.900.40 to 359.680.31 to 9.560.12 to 4.500.06 to 4.440.03 to 0.89
    10−0.79 to 1.22−1.30 to 1.52−1.53 to 1.051.67 to 6.241.67 to 6.142.44 to 6.550.07 to 0.91−508.85 to −0.180.41 to 359.580.28 to 9.790.06 to 3.770.04 to 4.270.04 to 1.29
    20−0.95 to 1.07−1.13 to 1.74−0.73 to 2.351.55 to 5.231.76 to 6.891.42 to 6.770.07 to 1.07−1234.40 to 21.100.03 to 359.800.18 to 15.230.08 to 9.510.08 to 5.810.04 to 1.28
    30−0.98 to 1.04−1.18 to 1.30−0.48 to 1.451.48 to 7.161.81 to 5.742.15 to 7.140.08 to 1.00−1082.30 to 189.700.01 to 359.500.13 to 13.440.08 to 4.710.07 to 5.300.04 to 0.98
    40−1.01 to 1.06−1.25 to 1.44−0.31 to 1.201.45 to 5.381.65 to 6.672.13 to 5.740.06 to 1.04−3859.30 to 553.700.65 to 359.090.25 to 14.430.06 to 5.110.07 to 6.050.04 to 1.21
    50−1.22 to 1.21−1.46 to 1.27−0.21 to 1.411.61 to 4.941.72 to 6.691.95 to 6.980.04 to 1.00−733.29 to 663.850.42 to 359.400.38 to 14.210.06 to 4.520.13 to 5.640.10 to 1.20
    55−1.29 to 1.13−1.30 to 1.25−0.56 to 1.371.55 to 5.941.67 to 5.751.92 to 7.640.09 to 1.12−1130.40 to 1018.700.16 to 359.180.52 to 14.940.07 to 4.630.07 to 5.860.03 to 1.47
    DownLoad: CSV

    Table 3.  Cliff’s delta and the effectiveness level.

    LevelCliff’s delta (|δ|)Effectiveness level
    10.000 ≤ |δ| < 0.147Negligible
    20.147 ≤ |δ| < 0.330Small
    30.330 ≤ |δ| < 0.474Medium
    40.474 ≤ |δ| < 1.000Large
    DownLoad: CSV

    Table 4.  Statistical comparisons of spectral density functions between measured wind velocities and simulated velocities by a Gaussian or non-Gaussian model.

    ModelStatistical difference (p < 0.05)No statistical difference (p ≥ 0.05)
    Negligible effect sizeSmall effect sizeNegligible effect sizeLarge effect size
    Gaussian2161418193222614925
    Non-Gaussian0002310 728
    DownLoad: CSV

    Table 5.  Performance comparison of non-Gaussian model with three different settings: S1, origin; S2, remove pw from u; S3, remove historical effects in three directions (e.g., qu=ru); S4, repeat the original setting 50 times. Units of velocity (u, v, w) and friction velocity (u*): m s−1.

    SettingMeanVariance
    uvwuvw
    S11.13 (0.56)1e12 (2e-4)−2e-2 (6e-4)0.58 (0.46)0.57 (0.39)1.36 (0.83)
    S21.13 (0.56)−2e12(5e-4)2e-2 (5e-4)0.57 (0.46)0.57 (0.39)1.36 (0.83)
    S31.13 (0.56)6e11 (1e-3)4e-3 (1e-3)0.58 (0.46)0.57 (0.39)1.37 (0.84)
    Mean of S41.13 (0.56)−3e9 (4e-4)4e-4 (3e-4)0.57 (0.46)0.57 (0.39)1.36 (0.83)
    Std of S47e-4 (5e-4)2e12 (5e-4)2e-2 (4e-4)2e-3 (2e-3)2e-3 (2e-3)4e-3 (2e-3)
    u*SkewnessKurtosis
    uvwuvw
    0.97 (0.95)0.95 (0.96)1.02 (0.97)1.03 (0.91)0.94 (0.92)1.02 (0.94)1.05 (0.89)
    0.42 (0.58)1.01 (0.97)1.02 (0.97)1.03 (0.91)1.01 (0.92)1.04 (0.93)1.05 (0.89)
    0.86 (0.97)1.01 (0.98)1.02 (0.97)1.01 (0.93)1.01 (0.94)1.04 (0.93)1.00 (0.89)
    0.97 (0.95)0.95 (0.96)1.02 (0.97)1.03 (0.91)0.95 (0.92)1.03 (0.93)1.04 (0.88)
    4e-3 (2e-3)3e-3 (1e-3)4e-3 (1e-3)7e-3 (3e-3)1e-2 (7e-3)1e-2 (6e-3)1e-2 (7e-3)
    DownLoad: CSV

    Table 6.  Statistical characteristics comparison of 448 pairs of displacement distributions in three coordinate directions (X, Y, Z) generated by Gaussian and non-Gaussian models. The statistical difference between each pair of displacement distributions are estimated by the two-sample Kolmogorov–Smirnov test at a 5% significance level, where *** indicates that the p-value is lower than 0.001. This table provides the average value (Avg) for distribution characteristics and the maximum value (Max) for the p-value. Units for the mean and variance of the displacement are m and m2, respectively, while the skewness and kurtosis of the displacement are dimensionless.

    MeasureGaussianNon-Gaussian
    XYZXYZ
    Avg[Mean]317.963169.621175.7564.082590.55130.01
    Avg[Variance]1.73e77.01e69.13e51.00e72.63e63.94e4
    Avg[Skewness]−0.12−0.380.760.05−0.362.40
    Avg[Skewness]3.403.663.722.563.4810.90
    Max[p-value]*********
    DownLoad: CSV
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Manuscript History

Manuscript received: 05 April 2019
Manuscript revised: 25 July 2019
Manuscript accepted: 26 August 2019
通讯作者: 陈斌, bchen63@163.com
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    沈阳化工大学材料科学与工程学院 沈阳 110142

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Non-Gaussian Lagrangian Stochastic Model for Wind Field Simulation in the Surface Layer

    Corresponding author: Junming WANG, wangjim@illinois.edu
  • 1. Key Laboratory of Dependable Service Computing in Cyber Physical Society Ministry of Education, Chongqing University, Chongqing 401331, China
  • 2. School of Big Data and Software Engineering, Chongqing University, Chongqing 401331, China
  • 3. Department of Natural Resources and Environment, University of Connecticut, Storrs, CT 06268, USA
  • 4. Climate and Atmospheric Science Section, Illinois State Water Survey, Prairie Research Institute, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA

Abstract: Wind field simulation in the surface layer is often used to manage natural resources in terms of air quality, gene flow (through pollen drift), and plant disease transmission (spore dispersion). Although Lagrangian stochastic (LS) models describe stochastic wind behaviors, such models assume that wind velocities follow Gaussian distributions. However, measured surface-layer wind velocities show a strong skewness and kurtosis. This paper presents an improved model, a non-Gaussian LS model, which incorporates controllable non-Gaussian random variables to simulate the targeted non-Gaussian velocity distribution with more accurate skewness and kurtosis. Wind velocity statistics generated by the non-Gaussian model are evaluated by using the field data from the Cooperative Atmospheric Surface Exchange Study, October 1999 experimental dataset and comparing the data with statistics from the original Gaussian model. Results show that the non-Gaussian model improves the wind trajectory simulation by stably producing precise skewness and kurtosis in simulated wind velocities without sacrificing other features of the traditional Gaussian LS model, such as the accuracy in the mean and variance of simulated velocities. This improvement also leads to better accuracy in friction velocity (i.e., a coupling of three-dimensional velocities). The model can also accommodate various non-Gaussian wind fields and a wide range of skewness–kurtosis combinations. Moreover, improved skewness and kurtosis in the simulated velocity will result in a significantly different dispersion for wind/particle simulations. Thus, the non-Gaussian model is worth applying to wind field simulation in the surface layer.

摘要: 准确模拟大气表层的风场能够有效帮助管理自然资源,例如空气质量,花粉基因流向,以及植物病毒传播等。拉格朗日随机模型是一种风场模拟模型,能够有效模拟风的随机行为。该模型假设风速服从高斯分布,然而实测大气表层风速分布具有很强的偏度和峰度。为了更加准确地模拟实测风场,本文提出了一种改进模型,称作非高斯拉格朗日随机模型。该模型以非高斯风速分布为目标,结合可控的非高斯随机变量,生成符合预期偏度和峰度的随机数序列。为了验证模型的正确性,本文利用CASES(Cooperative Atmospheric Surface Exchange Study)于1999年10月实测的大气表层风场数据,评估模型的模拟效果,并与传统的高斯模型进行对比。结果表明,非高斯模型能够显著提高风速偏度和峰度的模拟准确性,并具有较好的稳定性。于此同时,非高斯模型够维持传统模型的其它所有优点,例如在风速平均值和方差上的模拟准确性。此外,提高风速偏度和峰度准确性还使得非高斯模型能够生成更好的摩擦速度。非高斯模型还能够应用于不同类型的非高斯风场,广泛适应不同的偏度和峰度组合。再者,虽然非高斯模型与传统高斯模型仅仅在风速偏度和峰度模拟上有所差别,但两个模型模拟出的风/微粒传播轨迹却有显著差异,这意味着非高斯模型在大气表层风场模拟上可能具有良好的应用价值。

1.   Introduction
  • Lagrangian stochastic (LS) models are widely used for wind field simulation, incorporating the stochastic process and statistical information on wind velocities under different field conditions (Rossi et al., 2004). LS models play a particularly important role in managing atmospheric pollutants (Wang et al., 2008; Fattal and Gavze, 2014; Leelössy et al., 2016; Asadi et al., 2017), biological particles (e.g., weed pollen) (Wang and Yang, 2010a, b), and the like. These management efforts are achieved by simulating particle dispersion behaviors caused by the force of wind in the surface layer (Wilson and Shum, 1992; Aylor and Flesch, 2001). Besides, LS models are required to follow the well-mixed condition criterion (Thomson and Wilson, 2012), which states that if the particles of tracers are well mixed initially, they should remain so.

    Previously, Wilson and Shum (1992) assumed that the wind velocity in the surface layer follows the Gaussian distribution, and they successfully developed a Gaussian LS model that was proven to be satisfactory to the well-mixed condition criterion set by Thomson (1987). Their model also considers the inhomogeneity of the wind field—namely, the mean and variance of targeted Gaussian distribution changes at different heights.

    However, their Gaussian distribution assumption was not satisfied because the measured wind velocity distribution in the surface layer in general was observed to be non-Gaussian (i.e., skewness and kurtosis of the wind velocity distribution did not equal 0 and 3, respectively) (Legg, 1983; Flesch and Wilson, 1992). Therefore, the researchers were challenged to build a non-Gaussian LS model and to use higher-order statistics (skewness and kurtosis) to evaluate the accuracy of the wind field simulation (Du, 1997; Rossi et al., 2004).

    Since then, some non-Gaussian LS models have been proposed for wind field simulation in the surface layer (Legg, 1983; De Baas et al., 1986; Sawford and Guest, 1987) with the understanding that their generated velocity distribution cannot be consistent with the measured non-Gaussian distribution (Flesch and Wilson, 1992; Wilson and Sawford, 1996). Thus, their models are counter to the well-mixed condition criterion (Flesch and Wilson, 1992).

    Meanwhile, non-Gaussian LS models were largely studied for the wind field in the convective boundary layer (Bærentsen and Berkowicz, 1984; Luhar and Britter, 1989; Weil, 1990; Luhar et al., 1996; Cassiani et al., 2015). In general, they solved this issue of non-Gaussian field simulation by combining two random variables following Gaussian distributions, which reflects an interaction between an updraft and downdraft wind turbulent velocities (Luhar et al., 1996; Cassiani et al., 2015). Nevertheless, their models cannot be generalized for simulation in the surface layer (Flesch and Wilson, 1992).

    Later, Flesch and Wilson (1992) developed a surface-layer non-Gaussian LS model under the well-mixed condition framework. However, their dispersion simulation results were no better than results from the Gaussian LS model proposed by Wilson and Shum (1992) because of the difficulty in formulating correct non-Gaussian velocity distributions (Flesch and Wilson, 1992). This obstacle is due to the changing velocity distribution in the wind field (i.e., non-stationarity). Specifically, the wind field in the surface layer displays a strong and differing non-Gaussian behavior at each small-scale observation (Pope and Chen, 1990; Katul et al., 1994), e.g., in a 30-min period. Therefore, the non-stationarity of the wind field requires that the simulated velocity distribution of an LS model be adaptive to the measured wind velocity distribution in different observation periods.

    To solve this non-stationarity problem, a three-dimensional (3D) LS model (Wang et al., 2008; Wang and Yang, 2010b) was developed based on the Gaussian LS model proposed by Wilson and Shum (1992). This 3D LS model adjusts the mean and variance of the simulated wind velocity in different short time periods to adapt to the changing velocity distribution in the field. However, the 3D LS model still assumes that the field velocity distribution is Gaussian, and a non-Gaussian, non-stationary LS model remains to be developed.

    In this article, we present a non-Gaussian LS model that is built upon the Gaussian 3D inhomogeneous, non-stationary LS model (shortened to the Gaussian model hereafter) by Wang et al. (2008). We incorporate correct skewness and kurtosis in the simulated wind velocities by combining them with controllable non-Gaussian random variables. A non-Gaussian variable is supported by a combination of two Gaussian random number generators with parameters that are produced by a heuristic approach.

    The proposed model is verified by the Cooperative Atmospheric Surface Exchange Study, October 1999 (CASES-99) experimental data (Poulos et al., 2002) which contain 3584 sets of measured 3D velocities in the field at eight different heights. We simulated these measured velocities with our non-Gaussian model and the original Gaussian model. We also conducted linear regression analysis on the statistical characteristics (mean, variance, skewness, kurtosis, friction velocity) of simulated wind velocities that are counter to the measured characteristics. The model performance is estimated by the slope (k) of a linear regressed line between the simulated and measured characteristics and the coefficient of determination (R2). A more accurate model generates both k and R2 close to 1 on all statistical characteristics. The experimental results show that:

    ● The distribution accuracy of wind velocities simulated by the proposed non-Gaussian model substantially outperforms that generated by the Gaussian model. The reason is that the non-Gaussian model can produce more accurate skewness and kurtosis (their k and R2 are close to 1) in the simulated velocities as compared to those produced by the Gaussian model (k and R2 are close to 0) while maintaining the accuracy in mean and variance of the Gaussian model.

    ● The improved skewness and kurtosis better simulate friction velocities (a coupling of 3D velocities) that are comparable to those measured, where k increases from 0.86 to 0.97, and R2 enhances from 0.67 to 0.95.

    ● The improved accuracy in skewness, kurtosis, and friction velocity can be stably generated by the proposed non-Gaussian model since the standard deviations of their k and R2 are close to 0 for 50 simulations. This result also indicates that the non-Gaussian model can better simulate wind field velocities and trajectories, as required by the well-mixed condition criterion.

    ● The proposed model can adapt to numerous types of non-Gaussian wind fields, in terms of a wide range of skewness–kurtosis combinations. Therefore, the model can meet the non-stationarity requirement in the field.

    To analyze the necessity and utility of developing a non-Gaussian model, we investigated how the improved skewness and kurtosis (i.e., the third and fourth moments of the wind velocities, respectively) affect the wind/particle trajectory simulation. Results show that:

    ● In a one-dimensional (1D) homogeneous stationary field for wind trajectory simulation, the wind displacements (i.e., the final dispersion distances) converge to a Gaussian distribution. The skewness and kurtosis of simulated wind velocities can lead to a totally different displacement distribution because they heavily influence the variance of the displacement distribution, where velocity skewness and kurtosis have positive and negative effects, respectively, on the displacement variance.

    ● In a 3D inhomogeneous, non-stationary field for particle dispersion simulation, the non-Gaussian model with correctly simulated velocity skewness and kurtosis generates significantly different particle displacement distributions, leading to a substantial difference in particle concentrations as compared with the Gaussian model. The sensitivity of particle dispersion on velocity skewness and kurtosis implies the necessity of building a non-Gaussian model.

    The remainder of this paper is organized as follows: Section 2 presents the background of the traditional Gaussian LS model and the proposed non-Gaussian LS model. Section 3 describes the experimental setup for wind field simulation. Section 4 provides the experimental results and discussion. Section 5 summarizes the study and its findings.

2.   Methodology
  • In this section, we first present the background of the LS model in section 2.1, then describe the theory of our non-Gaussian LS model in section 2.2, and finally provide the implementation details in section 2.3.

  • In a wind field simulation, the Gaussian LS models by Wilson and Shum (1992), Wang et al. (2008), and Wang and Yang (2010b) regard wind behavior as a random process in a sequence of short time steps (dt). In each step, wind velocities in the horizontal, cross, and vertical wind directions are represented by u, v, and w, respectively, as follows:

    where $\bar u\left(z \right)$ is the mean velocity along the wind speed at height z; σu, σv, and σw represent the standard deviations for three wind directions; vs is the settling velocity in the vertical direction; and qu, qv, and qw are random terms that follow the standard Gaussian distribution. These random terms are formed by a Markov chain that describes the historical effect of wind velocities at time t + dt, as indicated below:

    where $\alpha = 1 - {\rm{d}}t/{\tau _L}$, $\beta = \sqrt {1 - {\alpha ^2}} $, and $\gamma = 1 - \alpha $ are coefficients and dt = 0.025τL; the passive fl