Path integration (PI) method for the parameter-retrieval of aircraft wake vortex by Lidar

Jianbing Li; Jianbing Li; Chun Shen; Chun Shen; Hang Gao; P. W. Chan; K. K. Hon; Xuesong Wang

doi:10.1364/OE.382968

1. Introduction

Wake vortex is an inevitable by-product of lift for a flying aircraft. It is generally composed of two strongly counter-rotating trailing vortices and is regarded as one of the most severe hazards in aviation. The wake vortex generated by a large aircraft may cause a following aircraft to roll out of control, particularly during the take-off and landing phases (see Fig. 1). With the development of modern society, air traffic has almost doubled every 15 years in the past decades, which has brought about an urgent demand for the detection and parameter-retrieval of wake vortex in air traffic management (ATM) to ensure the safety of flights or relax the restriction of separations between flights. The existing monitoring sensors for wake vortex include Radar and Lidar, and they are applicable for different weather conditions due to their different scattering mechanisms. Under clear air condition, the scattering is mainly determined by the aerosols involved in the wake vortex, and the widely-used wake vortex sensor is Lidar [1–5]. But under rainy/foggy/snowy conditions, Lidar doesn’t work well due to the heavy attenuation and the proposed sensor is radar [6–14]. In this paper, we only study the Lidar detection of wake vortices under dry air condition.

Fig. 1. The impact of wake vortex to a following aircraft [15].

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In aviation safe area, two parameters are used to assess the hazard of wake vortex, say vortex-core positions and velocity circulations. The vortex-core positions indicate where the vortices are, and the velocity circulations well describe the strength of a wake.

With the efforts of many research institutes in the last decades, some wake vortex parameter-retrieval algorithms based on Lidar detection have been proposed. Holzäpfel et al. [16] uses the tangential velocity of the wake vortex to estimate the core positions and calculate circulations (TV method). Rahm et al. [17,18] estimated the vortex-core positions by setting a fixed threshold to the Doppler spectral envelope, but its robustness could vary for different noise environments. To improve the applicability, Wassaf et al. [19] proposed to use a floating threshold instead. Basically, these methods are sensitive to the estimation bias of the vortex-core positions. Frehlich et al. [20] established two maximum likelihood (ML) algorithms based on the comparison of vortex model and spectrum data. Jacob et al. [21–23] estimated the vortex-core positions and circulations based on the spectrum-matched filters and maximum likelihood (ML) estimators. Hallermeyer et al. [24] estimated the vortex-core positions from velocity envelopes, and applied the ML algorithm to estimate circulations. Yoshikawa et al. [25] also used the similar ML algorithm to estimate the wake vortex parameters, and improvement was made on modeling the return signal of a wake. In the above methods, the researchers assumed that the measured Doppler velocity corresponds to the velocity at the range bin center of interest, but there is actually a difference between them. I. N. Smalikho et al. [26–28] analyzed this difference and proposed a new parameter-retrieval method based on this difference, say the radial velocity (RV) method. This is a very interesting method, but its relatively high computation load is a problem deserves special concern. K. K. Hon and P. W. Chen [5] tried to use the Doppler velocity integration method to calculate the circulation over a fan region around a vortex core, but they did not take into account the contribution of arc path to the circulation, which will introduce additional errors to the retrieved results. H. Gao et al. [29] proposed to retrieve the wake vortex parameters with an optimization method (Opt method), it has good accuracy and robustness, but the efficiency is still a problem of concern.

According to the above description, there is a great need to make a trade-off among robustness, accuracy, and efficiency. This paper proposes a new parameter-retrieval method for wake vortex, i.e, the Path Integration method (PI method). The method uses Doppler velocity profiles to locate the vortex cores, and then the path integrations of Doppler velocities are used to retrieve the circulations of the two vortices. The integration is a low-passing filter, so the new method has the benefits of being robust. Numerical examples and field detection data have verified the good performances of the new method.

2. Methodology

In this paper we are considering of using a Lidar to detect the wake vortex in clear air. For a Lidar working with Mie scattering mechanism, the scattering is mainly caused by the aerosols involved in the moving air. The aerosols can be assumed as weak inertia particles entrained by air moving at a normal velocity that is smaller than tens of meters per second (the Stokes Number $\def\oiint{\unicode[Times]{x222F}}S_{\textrm{t}}$ is much smaller than 1) [10], and the Doppler velocity obtained by a Lidar can well represent the background wind’s project on the Lidar’s line-of-sight (LOS). For more details about obtaining the Doppler velocity from raw data, please refer to [30].

2.1 Geometry configuration

In wake vortex detection campaigns, the proposed geometry set-up is side-looking mode. A Lidar is set on one side of the runway, and the Lidar beam scans up and down alternately in the vertical plane perpendicular to the runway. The coordinate is set up as shown in Fig. 2. The origin is the Lidar, the $x$ axis is the horizontal line originated from the Lidar and orthogonal to the runway’s center-line, and the $y$ axis is upright from the Lidar. The beam scanning speed is $\pm \omega$ and the interval of elevation angle $\alpha$ is $\left [\alpha _{\textrm{min}},\alpha _{\textrm{max}}\right ]$, where $\alpha _{\textrm {min}}$ and $\alpha _{\textrm {max}}$ respectively represent the minimum and maximum elevation angle; this results in a series of Doppler velocity distributions in a series of RHI (Range Height Indicator) scans. When an aircraft flies by, it will generate a pair of wake vortices. For the case shown in Fig. 2, the vortex cores of the two vortices are denoted by $O_{ci} (i=1,2)$, and the circulations are $\Gamma _{ci} (i=1,2)$. For the observed data on a Lidar’s line-of-sight, the range bins are located at $R_{k}=R_{0}+k\Delta R, k=0,1,2,\ldots$, where $\Delta R$ is the range bin interval and $R_{0}$ is the minimal measurement range. In this paper, we mainly concern the components of wind field on the RHI plane, and the positive and negative components are away from and toward the Lidar, respectively.

Fig. 2. Proposed geometry setup for Lidar detection of wake vortex. $P$ is a point inside the wake, $V_{L}$ and $V_{R}$ are the velocities deduced by the two vortices, $l_{n}$ is the length of measurement bins under consideration, $l'_{n}$ is a part of circle centering at the left vortex core $O_{c1}$, $l''_{n}$ is a part of radius connecting $A_{n}$ and $C_{n}$, and $S_{n}$ is the area surrounded by lines $l''_{n}$, $l'_{n}$ and $l_{n}$, $V_{c}$ is the cross wind, $V_{\epsilon }$ is the turbulence wind.

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To describe the characteristics of wake vortex, two very important parameters are usually employed: vortex-core positions and velocity circulations. The vortex-core position is a parameter showing the location of the vortex; the circulation is the integration of the velocity along a closed path enclosing one of the vortex core, which well describes the strength of a vortex. This paper proposes a new method to retrieve these two parameters robustly.

2.2 Locating of vortex cores

A wake vortex is composed of two counter-rotating vortices, and the locating of vortex cores is crucial to the estimation of their circulations. The velocity distribution of wake vortex is generally quite different to the background wind; from this difference it is possible to separate the wake from the background wind.

Normally, the range resolution of a Lidar is generally somewhat coarse, and a typical value of it for wake detection is 15m, which is larger than the vortex-core size. If no overlap method is used in determining the radial velocity, the range grid is too coarse to properly locate the vortex cores. Moreover, for a "stay-go" scanning mode Lidar, the time interval between two neighboring LOS should be long enough to get good SNR (Signal to Noise Ratio) [31] by data accumulation, resulting in a coarse transverse resolution at a relatively long range bin. In this case, the Doppler velocity should be interpolated to finner range grids, for example, 1 meter in between. If the maximum range of interest is $R_{\textrm {max}}$, the angular and range step for interpolation can be set as $\Delta \alpha = 1/R_{\textrm {max}}$(rad) and 1m, respectively. The interpolation is proceeded by triangulation-based cubic interpolation method based on the existing Doppler velocity, which is the average effect of the particles’ velocity in a volume depending on the laser pulse shape, window function, etc. The locating of vortex core will be processed on the interpolated Doppler velocity.

① Locating of the ranges of vortex cores

For a given range $R$, the difference between the maximum and minimum Doppler velocity along the direction of elevation $\alpha$ is

(1)$$\Delta V(R)=\max _{\alpha} V_{\textrm{d}}(R,\alpha)-\min _{\alpha} V_{\textrm{d}}(R,\alpha), \alpha \in[\alpha_{\textrm{min}},\alpha_{\textrm{max}}],$$

where $V_{\textrm {d}}(R,\alpha )$ is the Doppler velocity at range $R$ and elevation $\alpha$. This difference is also called the "Doppler velocity range". According to the velocity distribution of wake vortex, the "Doppler velocity range" should have two pronounced peaks that correspond to the two vortex cores, i.e. $R_{c1}$ and $R_{c2}$ (see Fig. 3).

Fig. 3. Determination of vortex-cores’ radial distances by Doppler velocity range distribution, where the Doppler velocity range is the difference between the maximum and minimum of Doppler velocity along the elevation.

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② Locating of elevation angles of vortex cores

For a single vortex, the velocity is symmetric to the vortex core. In this manner, at the radial distance of a vortex core, the elevation angle of this core should correspond to the average angle of the maximum and minimum velocity along the elevation direction.

(2)$$\hat{\alpha}_{ci}= \frac{\mathop{\textrm{argmax}}\limits_{\alpha} V_{\textrm{d}}(R_{ci},\alpha)+\mathop{\textrm{argmin}}\limits_{\alpha} V_{\textrm{d}}(R_{ci},\alpha)}{2}, \alpha \in[\alpha_{\textrm{min}},\alpha_{\textrm{max}}],$$

where $i=1,2$ respectively represent left and right vortex core.

2.3 Path integration method for the parameter-retrieval of wake vortex

2.3.1 Velocity of a point inside the wake vortex

We assume the background wind is composed of cross wind and turbulence. When an aircraft flies by, it will generate two counter-rotating vortices, and the velocity vector ${\mathbf {V}}$ of a point inside the wake vortex can be expressed as follows

(3)$${\mathbf{V}}={\mathbf{V}}_{L}+{\mathbf{V}}_{R} +{\mathbf{V}}_{c}+{\mathbf{V}}_{\epsilon},$$

where ${\mathbf {V}}_{L}$ and ${\mathbf {V}}_{R}$ are the velocity components induced by the left and right vortex respectively, ${\mathbf {V}}_{c}$ is the cross wind and ${\mathbf {V}}_{\epsilon }$ is the turbulence velocity component.

In a RHI, the cross wind ${\mathbf {V}}_{c}$ is composed of two components in $\overrightarrow {{\mathbf {x}}}$ and $\overrightarrow {{\mathbf {y}}}$ directions:

(4)$${\mathbf{V}}_{c} = V_{c}^x\cdot \overrightarrow{{\mathbf{x}}} + V_{c}^y \cdot \overrightarrow{{\mathbf{y}}}.$$

Normally, the $x$ component is assumed to be with linear shear, i.e.,

(5)$$V_{c}^{x} = V_{c}^{0} + \beta\cdot y,$$

where $V_{c}^{0}$ is the x-component cross wind at $y=0$, $\beta$ is the wind shear rate, $y$ is the height above ground. The cross wind is assumed to not change too much in one RHI, so we can usually estimate the cross wind ${\mathbf {V}}_{c}$ from the regions free of wake vortex (see Fig. 4). For a point in these regions, the radial velocity should be

(6)$$V_{r} = V_{c}^{0}\cdot \textrm{cos}\alpha +\beta\cdot y \cdot \textrm{cos}\alpha + V_{c}^{y}\cdot \textrm{sin}\alpha.$$

Fig. 4. Regions free of wake vortex that was used to estimate the background wind.

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When $N$ points($N\geqslant 3$) are employed, we can get the parameters $V_{c}^{0}$, $\beta$ and $V_{c}^{y}$, and then the cross wind can be determined. Consequently the cross wind can be eliminated from the observed velocity, resulting in a velocity $\tilde {{\mathbf {V}}}$ as follows:

(7)$$\tilde{{\mathbf{V}}}={\mathbf{V}}-{\mathbf{V}}_{c}={\mathbf{V}}_{L}+{\mathbf{V}}_{R} + {\mathbf{V}}_{\epsilon}.$$

Based on this velocity $\tilde {{\mathbf {V}}}$, we try to retrieve the circulation of wake vortex with a path integration method.

We first present the path integration of each velocity component in the following section.

2.3.2 Path integrations of the velocity components

① The path integration about velocity ${\mathbf {V}}_{L}$

As seen in Fig. 2, a Lidar beam scans the wake vortex up and down alternately in the RHI plane. When the Lidar beam illuminates the wake vortex at an angle of $\alpha _{n}$, $l_{n}=\overrightarrow {A_{n} B_{n}}$ is the length of measurement bins under consideration, $l'_{n}$ is a part of circle centering at the left vortex core $O_{c1}$, $l''_{n}$ is a part of radius connecting $A_{n}$ and $C_{n}$, and $S_{n}$ is the area surrounded by lines $l''_{n}$, $l'_{n}$ and $l_{n}$, the Stokes’ theorem gives the following equation about the velocity path integration:

(8)$$\int_{l'_{n}+ l^{''}_{n}+ l_{n} }{\mathbf{V}}_{L}{\bullet}{\textrm{d}} {\mathbf{l}} =\oiint_{S_{n}}\nabla\times{\mathbf{V}}_{L}{\bullet}{\textrm{d}} {\mathbf{S}}.$$

There have been some existing tangential velocity models for the roll-up wake vortex behind an aircraft [32], for example the Rankine model, Hallock-Burnham model [33], Lamb-Ossen model [34], Adapted vortex (Proctor [35]), Smooth blending vortex (Winckelmans et al. [36]), Multiple scale vortex (Jacquin et al. [37]), etc. Plots of these velocity models are presented in Fig. 5, where $b_{0}$ is the distance between the two vortices.

Fig. 5. Different velocity profile models for a vortex, where the circulation $\Gamma = 400 m^2/s$, the distance between the two vortices $b_{0}=50m$.

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If the Rankine vortex is taken into account, the tangential velocity is expressed as

(9)$$V_{\textrm{t}}(r)=\left\{ \begin{array}{ll} \frac{\Gamma}{2\pi r_{c}}\frac{r}{r_{c}}, & r\leqslant r_{c},\\ \frac{\Gamma}{2\pi r}, & r\;>\;r_{c}, \end{array} \right.$$

where ${\Gamma }$ and $r_{c}$ are respectively the vortex circulation and core radius. It is known that a Rankine vortex is irrotational outside the vortex core and has a uniform vorticity inside the vortex core [38]:

(10)$$\nabla\times{\mathbf{V}}= \left\{ \begin{array}{ll} \frac{{\Gamma}}{\pi {r_{ c }}^2}, & r\leqslant r_{c}\\ 0, & r\;>\;r_{c}. \end{array} \right.$$

Normally the dynamics inside a vortex core could be very complex due to the synthesis effect of vorticity and turbulence, we do not recommend to retrieve the parameters of wake vortex based on the Doppler velocity observations in the vortex core. If the concerned line $l_{n}=\overrightarrow {A_{n} B_{n}}$ does not cross the vortex core, the right hand side of Eq. (8) vanishes, i.e.,

(11)$$\oiint_{S_{n}}\nabla\times{\mathbf{V}}_{L}{\bullet}{\textrm{d}} {\mathbf{S}}=0.$$

Theoretically, this formula is only valid for Rankine velocity profile model. But as shown in Fig. 5, if the concerned region is somewhat far to the vortex core ($r\geqslant 0.2b_{0}$), all the models almost coincide, and then the Eq. (11) would also holds well.

Moreover, the integration about $l''_{n}$ also vanishes if the velocity component ${\mathbf {V}}_{L}$ is orthogonal to radius $l''_{n}$:

(12)$$\int_{l^{''}_{n}}{\mathbf{V}}_{L}{\bullet}{\textrm{d}} {\mathbf{l}} =0.$$

The integration of velocity ${\mathbf {V}}_{L}$ along $l'_{n}$ can be simply obtained from the velocity model because the magnitude of the tangential velocity remains unchanged along the circle part $l'_{n}$:

(13)$$\int_{l'_n}{\mathbf{V}}_{L}{\bullet}{\textrm{d}} {\mathbf{l}} =\frac{\theta_{c1}}{2\pi}\Gamma_{c1},$$

where $\theta _{c1}$ is the angle from vector $\overrightarrow {O_{c1}B_n}$ to vector $\overrightarrow {O_{c1}A_n}$. If the two vectors are rewritten as complex numbers: $\left (\overrightarrow {O_{c1} B_n}\right )_x+\textrm{i} \left (\overrightarrow {O_{c1} B_n}\right )_y$, $\left (\overrightarrow {O_{c1} A_n}\right )_x+\textrm{i} \left (\overrightarrow {O_{c1} A_n}\right )_y$, the angle $\theta _{c1}$ is simply obtained as

(14)$$\theta_{c1}=\arg\frac{\left(\overrightarrow{O_{c1} A_n}\right)_x+\textrm{i} \left(\overrightarrow{O_{c1} A_n}\right)_y}{\left(\overrightarrow{O_{c1} B_n}\right)_x+\textrm{i} \left(\overrightarrow{O_{c1} B_n}\right)_y},$$

where $\arg (\cdot )$ means the argument of a complex number.

Therefore, substituting Eqs. (11)–(13) into Eq. (8) gives

(15)$$\int_{ l_n }{\mathbf{V}}_{L}{\bullet}{\textrm{d}} {\mathbf{l}} =-\frac{\theta_{c1}}{2\pi}\cdot\Gamma_{c1}.$$

② The path integration about ${\mathbf {V}}_{R}$

When the line $l_n=\overrightarrow {A_n B_n}$ does not cross the right vortex core, the integration about ${\mathbf {V}}_{R}$ can be obtained in a similar way as ${\mathbf {V}}_{L}$:

(16)$$\int_{ l_n }{\mathbf{V}}_{R} {\bullet}{\textrm{d}} {\mathbf{l}} =-\frac{\theta_{c2}}{2\pi}\cdot\Gamma_{c2}.$$

③ The path integration about ${\mathbf {V}}_{\epsilon }$

For fully developed turbulence, most of the turbulence energy remains in the high frequency range (inertial subrange and dissipation range). It is known that the integration algorithm is a low-pass filter, so the path integration of fully developed turbulence velocity (${\mathbf {V}}_{\epsilon }$) will be approximately zero [39], i.e.,

(17)$$\int_{ l_n }{\mathbf{V}}_{\epsilon}{\bullet}{\textrm{d}} {\mathbf{l}} \approx 0.$$

For non-fully developed turbulence, there might be a sort of deviation in this approximation, but it will holds well if a long integration path is taken into account.

As a result, combination of the above three velocity integrations in ①, ② and ③ (Eqs. (15)–(17)) gives the following velocity integration for fully developed background turbulence:

(18)$$\int_{ l_n }\tilde{{\mathbf{V}}}{\bullet}{\textrm{d}} {\mathbf{l}} =\int_{ l_n }({\mathbf{V}}-{\mathbf{V}}_c){\bullet}{\textrm{d}} {\mathbf{l}} =\int_{ l_n }({\mathbf{V}}_{L}+{\mathbf{V}}_{R}+{\mathbf{V}}_{\epsilon}){\bullet}{\textrm{d}} {\mathbf{l}} =-\frac{1}{2\pi}\left(\theta_{c1}\Gamma_{c1}+\theta_{c2}\Gamma_{c2}\right).$$

2.3.3 Parameter-retrieval based on the path integrals

For the Lidar detection, what we can obtain is the Doppler velocity which is the projection of wind vector on the LOS. When the background wind is removed from the observed Doppler velocity, we get the distribution of $\tilde {{\mathbf {V}}}$, and the integration in Eq. (18) can be approximated as follows:

(19)$$\int_{ l_n }\tilde{{\mathbf{V}}}{\bullet}{\textrm{d}} {\mathbf{l}} \approx \sum_{m=1}^{M_n}\tilde{V}(\alpha_n,m)\cdot\delta,$$

where $\delta$ is the interpolation range interval of the observation bins, $M_n$ is the number of bins used on the $n^{\textrm{th}}$ LOS.

When $N$ integration paths at different elevation angles are taken into account, we have the following observation equation based on Eqs. (18) and (19):

(20)$$\left[\begin{matrix} \mathop{\sum}\limits_{m=1}^{M_1}\tilde{V}(\alpha_1,m)\\ \mathop{\sum}\limits_{m=1}^{M_2}\tilde{V}(\alpha_2,m)\\ \vdots\\ \mathop{\sum}\limits_{m=1}^{M_N}\tilde{V}(\alpha_N,m)\\ \end{matrix} \right]\approx-\frac{1}{2\pi\delta}\left[\begin{matrix} \theta^{(1)}_{c1} & \theta^{(1)}_{c2}\\ \theta^{(2)}_{c1} & \theta^{(2)}_{c2}\\ \vdots &\vdots\\ \theta^{(N)}_{c1} & \theta^{(N)}_{c2}\\ \end{matrix}\right]\cdot\left[\begin{gathered} \Gamma_{c1}\\ \Gamma_{c2} \end{gathered}\right]\triangleq {\mathbf{\Theta}} \left[\begin{gathered} \Gamma_{c1}\\ \Gamma_{c2} \end{gathered}\right],$$

where the matrix ${\mathbf {\Theta}}$ is a collection of variables about the flare angles $\theta _{ci}$ ($i=1,2$). From Eq. (20), the circulations $\Gamma _{ci}$ $(i=1,2)$ can be obtained with the least square method if the number of integration paths is larger than 2:

(21)$$\left[\begin{gathered} \hat{\Gamma}_{c1}\\ \hat{\Gamma}_{c2} \end{gathered}\right]\approx\left[{\mathbf{\Theta}}^{\textrm{H}}{\mathbf{\Theta}}\right]^{-1}{\mathbf{\Theta}}^{\textrm{H}} \left[\begin{matrix} \mathop{\sum}\limits_{m=1}^{M_1}\tilde{V}(\alpha_1,m)\\ \mathop{\sum}\limits_{m=1}^{M_2}\tilde{V}(\alpha_2,m)\\ \vdots\\ \mathop{\sum}\limits_{m=1}^{M_N}\tilde{V}(\alpha_N,m)\\ \end{matrix} \right].$$

2.3.4 Adjusting of vortex core positions to compensate the compressing/expanding effects of wake vortex caused by the scanning of Lidar beam

During the detection, the vortices descend while the Lidar beam scans up and down, which will result in a compressing effect for upward scanning, and an expanding effect for downward scanning. In this case, each LOS corresponds to a different scanning time and the vortex-core positions for each LOS should vary with time. Because the flare angles $\theta _{c1}$ and $\theta _{c2}$ in Eqs. (14)–(16) greatly depend on the vortex-core positions, it is important to well estimate these positions at the instant time of each LOS.

For an asymmetric wake, the two vortices descend at the following velocities due to the induction effect:

(22)$$V_{c1,\textit{descent}}=-\frac{|\Gamma_{c2}|}{2\pi b_0},V_{c2,\textit{descent}}=-\frac{|\Gamma_{c1}|}{2\pi b_0}.$$

By using the method mentioned in section 2.2, we can preliminarily obtain two vortex-core positions $O_{c1}(t_{c1})$ and $O_{c2}(t_{c2})$, where $t_{c1}$ and $t_{c2}$ correspond to the time samples for $\alpha _{c1}$ and $\alpha _{c2}$. Consequently, the vortex cores at time $t_n$ (corresponding to elevation angle $\alpha _n$) are estimated as

(23)$$\begin{gathered} \hat{\mathbf{O}}_{c1}(t_n)=\mathbf{O}_{c1}(t_{c1})+ \left[\begin{gathered} V_{c}^{x}\\ V_{c}^{y}-\frac{|\Gamma_{c2}|}{2\pi \hat{b}_0} \end{gathered}\right]\cdot(t_n-t_{c1}), \\ \hat{\mathbf{O}}_{c2}(t_n)=\mathbf{O}_{c2}(t_{c2})+ \left[\begin{gathered} V_{c}^{x}\\ V_{c}^{y}-\frac{|\Gamma_{c1}|}{2\pi \hat{b}_0} \end{gathered}\right]\cdot(t_n-t_{c2}), \end{gathered}$$

where $V_{c}^{x}$ and $V_{c}^{y}$ are the cross wind in $x$ and $y$ direction. The estimated vortex-core positions in Eq. (23) are related to the circulations $\Gamma _{c1}$ and $\Gamma _{c2}$, so substituting Eq. (23) into Eqs. (20) and (21), we get the expressions of ${\mathbf {\Theta}}$ about $\Gamma _{c1}$ and $\Gamma _{c2}$. In this manner, Eq. (21) becomes the following nonlinear equation:

(24)$$\left[\begin{matrix} \mathop{\sum}\limits_{m=1}^{M_1}\tilde{V}(\alpha_1,m)\\ \mathop{\sum}\limits_{m=1}^{M_2}\tilde{V}(\alpha_2,m)\\ \vdots\\ \mathop{\sum}\limits_{m=1}^{M_N}\tilde{V}(\alpha_N,m)\\ \end{matrix} \right] = {\mathbf{\Theta}}(\hat{\Gamma}_{c1},\hat{\Gamma}_{c2})\left[\begin{gathered} \hat{\Gamma}_{c1}\\ \hat{\Gamma}_{c2} \end{gathered}\right],$$

which can be further rewritten as

(25)$$\left[\begin{gathered} \hat{\Gamma}_{c1}\\ \hat{\Gamma}_{c2} \end{gathered}\right]\approx\left[{\mathbf{\Theta}}(\hat{\Gamma}_{c1},\hat{\Gamma}_{c2})^{\textrm{H}}{\mathbf{\Theta}}(\hat{\Gamma}_{c1},\hat{\Gamma}_{c2})\right]^{-1}{\mathbf{\Theta}}(\hat{\Gamma}_{c1},\hat{\Gamma}_{c2})^{\textrm{H}} \left[\begin{matrix} \mathop{\sum}\limits_{m=1}^{M_1}\tilde{V}(\alpha_1,m)\\ \mathop{\sum}\limits_{m=1}^{M_2}\tilde{V}(\alpha_2,m)\\ \vdots\\ \mathop{\sum}\limits_{m=1}^{M_N}\tilde{V}(\alpha_N,m)\\ \end{matrix} \right].$$

This nonlinear equation can be well solved with iteration method. The following iteration procedure is applied to obtain $\hat {\Gamma }_{c1}$ and $\hat {\Gamma }_{c2}$.

(1) Obtain the two vortex-core positions $O_{c1}(t_{c1})$ and $O_{c2}(t_{c2})$ by using the method mentioned in section 2.2, and assume $\Gamma ^{(0)}_{c1}=0$ and $\Gamma ^{(0)}_{c2}=0$.
(2) The vortex-core positions $O_{c1}(t_{c1})$ and $O_{c2}(t_{c2})$ are used to calculate $\theta ^{(j)}_{c1}$ and $\theta ^{(j)}_{c2}$ with Eq. (14), and then the circulations $\Gamma ^{(n)}_{ci}$ $(i=1,2, n=1,2,\ldots )$ are obtained according to Eq. (24). Then use $\Gamma ^{(n)}_{ci}$ to adjust the vortex-core positions for the $j^{\textrm{th}}$ LOS with Eq. (23), resulting in $O_{ci}(t_j), i=1,2$.
(3) Repeat step (2) until the residual error is below a given threshold, for example $\left |\frac {\Gamma ^{(n+1)}_{ci}-\Gamma ^{(n)}_{ci}}{\Gamma ^{(n)}_{ci}}\right |\;<\;1\%$ $(i=1,2)$. Generally, the iteration can converge in just a few iterations, and the computation cost related to it is fairly small.

Moreover, when the circulations are retrieved, the vortex-core positions at the center time of a RHI, $\tilde {t}$ (corresponding to elevation angle $\alpha = (\alpha _{\textrm{max}}+\alpha _{\textrm{min}})/2$), can be obtained as

(26)$$\begin{gathered} \hat{\mathbf{O}}_{c1}(\tilde{t})=\mathbf{O}_{c1}(t_{c1})+ \left[\begin{gathered} V_{c}^{x}\\ V_{c}^{y}-\frac{|\Gamma_{c2}|}{2\pi \hat{b}_0} \end{gathered}\right]\cdot(\tilde{t}-t_{c1}), \\ \hat{\mathbf{O}}_{c2}(\tilde{t})=\mathbf{O}_{c2}(t_{c2})+ \left[\begin{gathered} V_{c}^{x}\\ V_{c}^{y}-\frac{|\Gamma_{c1}|}{2\pi \hat{b}_0} \end{gathered}\right]\cdot(\tilde{t}-t_{c2}). \end{gathered}$$

These positions in the consecutive RHIs are then taken as the vortex-core trajectories.

2.3.5 Ground effect of wake vortex

When the height of wake vortex-core above ground is smaller than $1.5b_0$, the ground effect should be considered [40]. The ground effect can be modeled by using two image vortices that rotate inversely to the real vortices [41] (see Fig. 6), and the circulations of the image vortices are $\Gamma _{\textrm {img},c1} = -\Gamma _{c1}$ and $\Gamma _{\textrm {img},c2} = -\Gamma _{c2}$.The integral about $\tilde {{\mathbf {V}}}$ in Eq. (18) is then modified as

(27)$$\int_{ l_n }\tilde{{\mathbf{V}}}{\bullet}{\textrm{d}} {\mathbf{l}} =-\frac{1}{2\pi}\left[(\theta_{c1}-\theta_{\textrm{img},c1})\Gamma_{c1}+(\theta_{c2}-\theta_{\textrm{img},c2})\Gamma_{c2}]\right..$$

The rest processes are the same as those mentioned in section 2.3.3 and section 2.3.4.

Fig. 6. Ground effect modeled by image vortices.

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2.3.6 Suggestions for the selection of integration paths

For this method, the selection of integration path is a problem of concern. A basic principle is that the selected range bins should have comparatively big velocities so that the method’s performance could be less sensitive to the background wind and turbulence. Based on this principle, we can present several suggestions (see Fig. 7):

Fig. 7. Proposed integration path configuration for circulation retrieval, where the thick lines are the integration paths to be used.

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(1) The distance from the vortex core to the integration path is suggested to be larger than $0.2 b_0$; this is derived from the variation of vortex formula. As shown in Fig. 5, big variation is observed in and around the vortex core, but the velocity outside vortex core is stable, especially when $r\geqslant 0.2 b_0$. Moreover, the present method is theoretically based on Rankie model, so it would be rational to regulate the integration paths to $r\geqslant 0.2 b_0$, where all the velocity models become almost identical. At the same time, the distance should be not larger than $0.5b_0$; otherwise the vortex velocity tends to be overwhelmed by the background wind.
(2) An integration path is suggested to be longer than $0.5 b_0$, but not longer than $1.2b_0$. If the path is too short, the turbulence effect shown in Eq. (17) would not be well mitigated; but if the path is too long, part of the integration path may reach far regions where the vortex velocity is too small to compete with the background wind and turbulence.
(3) The integration path is suggested to be approximately symmetric to the radial position of a vortex core, i.e., $\overline {A_n D_n} = \overline {D_n B_n}$. It is known that the Doppler velocity around the radial position of a vortex core is normally bigger than that far to this range, so the selection scheme in this suggestion can make an integration path to occupy more big Doppler velocity bins with a limited integration length.

3. Performance verification of the method

3.1 Numerical experiments

To verify the performance of the proposed method, a simulation toolbox about the dynamics, scattering, echo signal and Doppler spectra of wake vortex, has been established.

The first work is to obtain the wind field velocity, which is assumed to be the superposition of wake vortex velocity and background wind velocity. The wake vortex velocity is obtained by using the Hallock-Burnham model [33], and its evolution is characterized with D2P model [42]. Meanwhile, the ground effect is considered when the vortex core trajectories is near the ground. The initial circulations of the two vortices are both $\Gamma _0 = 400 \textrm {m}^2/\textrm {s}$, and the initial positions for the left and right vortices are $O_{c1}(t_0) = (550, 107)$m and $O_{c2}(t_0) = (610, 105)$m respectively. The background wind velocity contains wind shear and turbulence. More details about simulation of wake vortex and background wind can be found in [43].

The Lidar receives back scattered signals caused by a large number of randomly distributed particles, so the echo of a range bin can be regarded as the superposition of scattering signals of all the particles in this range bin. According to the signal model proposed by Zrnic [44], the echo of a range bin containing $N$ particles can be obtained as:

(28)$$z(t_s) = \mathop{\sum}\limits_{k=1}^{N}(\rho_k S_0\textrm{exp}\left[\begin{gathered}\frac{(t_s-2r_k/c)^2}{2\sigma^2_w}\end{gathered}\right] \times \textrm{exp}(4\pi \textrm{j}t_s v_k/\lambda)+n_k),$$

where $\lambda$ is the wavelength, $n_k$ is the random noise signal, $v_k$ is the velocity of the $k^{\textrm {th}}$ particle which combine with wake vortex velocity and background wind velocity, $\rho _k$ is the scattering intensity of the $k^{\textrm {th}}$ particle, $S_0$ is the amplitude of transmitted signal, $t_s$ is the sampling time, $r_k$ is the distance from Lidar to the $k^{\textrm {th}}$ particle, $c$ is the light speed, $\sigma _w$ is the effective window length. In the simulation, the parameters $\rho _k$ and $S_0$ are assumed to be unified to all particles, and $n_k$ is a parameter related to the SNR of a single shot. The Lidar’s parameters for the simulations are listed in Table 1.

Table 1. Lidar parameters used in the simulations.

View Table | View all tables in this article

Consequently, the Doppler spectrum $S_{\textrm {D}}(V_k)$ is calculated from several consecutive echo samples (7 for example) with FFT (Fast Fourier Transform). In order to improve the resolution of the signal spectrum, zero padding the samples to a longer series (1024 for example) is always applied before FFT. Moreover, since the SNR of a single measurement is limited, the spectra accumulation between PRTs (pulse repetition time) is recommended to improve the SNR. Based on the accumulated spectrum, the Doppler velocity can be obtained from the weighted average around the spectral peak [30].

Based on the above process, the Doppler velocity distribution of consecutive RHIs can be simulated. One of them is shown in Fig. 8 as an example.

Fig. 8. Simulated radial velocity distribution of wake vortices in a RHI.

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3.1.1 Locating of vortex-core positions

We use Hallock-Burnham velocity model to simulate the dynamics of wake vortex, the background wind velocity is set as $-2 \textrm {m/s}$ (from right to left, toward the Lidar) and the eddy dissipate rate (EDR) is $0.05 \textrm {m}^2/\textrm {s}^3$ [45]. The locating results of the two vortex cores are showed in Fig. 9.

Fig. 9. Comparison between the theoretical and estimated vortex core trajectories.

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The standard deviation $E_s$ is introduced as follows:

(29)$$E_{s}=\frac{1}{N}\mathop{\sum}\limits_{j=1}^{N}\left|\frac{P_{ci}^{(j)}-P_{ci}^{j}}{b_0}\right|\times100\%,i=1,2,$$

where $P_{ci}^{(j)}$ is the estimated vortex-core position for the $j^{\textrm{th}}$ RHI, $P_{ci}^{j}$ is the corresponding theory value, $N$ is the total number of consecutive scans of interest, $i=1,2$ respectively represent the left and right vortex. The standard deviation of the left and right vortex core locating results are only $4.9\%$ and $4.3\%$, which indicates that the vortex core are well located. The small biases may be caused by the scanning of Lidar beam, background wind field (shear and turbulence), and other abnormal interferences.

3.1.2 Evaluation of the retrieved velocity circulation

According to section 2.3.4, the retrieved circulations, together with the results by tangential velocity (TV) method [16], are shown in Fig. 10. The TV method is processed as follows: when the vortex cores are determined, the distance of a vortex core to a LOS is the radius $r_i$, and the maximum of absolute radial velocity along this LOS is adopted as the tangential velocity $V_t(r_i)$. The circulation is then estimated as $\hat {\Gamma } = \frac {1}{N}\sum _{i=1}^{N}2\pi r_i|V_t(r_i)|$. During this process, only the radii between 5m and 15m are used to calculate the circulation. It is noted that, if even number of points above and below the vortex core are used in TV method to estimate the circulation, the impact of background wind can be mitigated, especially for uniform background wind case. From the results, it is observed that the path integration (PI) method basically gives better estimations than the TV method.

Fig. 10. Circulation Results obtained by PI method and TV method.

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To better analyze the performance quantitatively, the relative error $E_r$ is introduced as follows:

(30)$$E_{r}=\frac{1}{N}\mathop{\sum}\limits_{j=1}^{N}\left|\frac{\Gamma_{ci}^{(j)}-\Gamma_{ci}^{j}}{\Gamma_{ci}^{j}}\right|\times100\%,i=1,2,$$

where $\Gamma _{ci}^{(j)}$ is the estimated circulation for the $j^{\textrm{th}}$ RHI, $\Gamma _{ci}^{j}$ is the corresponding theory value, $N$ is the total number of consecutive scans of interest, $i=1,2$ respectively represent the left and right vortex.

By using the same simulation parameters mentioned in Table 1, the relative errors that consider the ground effect for the left and right vortex are $11.1\%$ and $8.88\%$ respectively when the PI method is used, while the relative errors that do not consider the ground effect for the left and right vortex are $13.6\%$ and $10.03\%$. This shows an improvement of using the image vortices to deal with the near ground effect. Also the corresponding errors for the TV method are $37.43\%$ and $66.25\%$. This means that the PI method can get more robust result than TV method. The TV method may bias sharply from the theory circulations, this is mainly caused by having neglected the turbulence, the background wind and the influence between two vortices. The PI method has a slight bias from the theory circulations, which is caused by the spatial averaging of velocity in a range bin and the bias of vortex core location. But as a whole, the results of PI method is much less fluctuant than the TV method; this shows the benefit that the noise and abnormal effects can be well mitigated by the integration process.

To better show the performance of this method under more complex situations, different background wind and turbulence intensities have been employed in the simulations (see Table 2). Besides TV method, the optimization method (Opt method) [29] is also adopted as a comparison to verify the good performance of the new method. The results show that both the optimization method and the present method have fairly good accuracy and robustness under different turbulence levels and background wind intensities. But the TV method is much more sensitive to the background parameters.

Table 2. Circulation results with PI method, Opt method and TV method under different turbulence and background wind simulation parameters.

View Table | View all tables in this article

We have also made further detection simulations under different scan rates. The EDR is 0.05$\textrm {m}^{2}/\textrm {s}^{3}$, background wind is -2m/s. When the scan rates are $\pm 1.5^\circ /\textrm {s}$, $\pm 2^\circ /\textrm {s}$, and $\pm 3^\circ /\textrm {s}$, the circulation relative errors of the left vortex are $11.1\%$, $10.39\%$ and $11.62\%$ respectively, and those of the right vortex are $8.88\%$, $8.41\%$ and $9.13\%$ respectively. This phenomenon indicates that the present method is not sensitive to the scan rate of Lidar and the compressing/expanding effects have been well compensated.

3.2 Field campaigns

3.2.1 Field detection in Hong Kong international airport (HKIA)

A series of detection experiments were conducted at Hong Kong international airport with a Doppler Lidar, WindCube 200s, whose pulse duration is 200 ns, and the wavelength is 1.54 $\mu$m. The Lidar was set on the roof of the Asia World-Expo (about 19 m above the ground) to observe the north way of HKIA at a distance of 400m from south (see in Fig. 11). More details about the campaign can be found in [29]. One of the obtained RHIs of radial velocity in the campaign is shown in Fig. 12.

Fig. 11. Geometry setup of the detection campaign in HKIA, 2014 [29].

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Fig. 12. An example of radial velocity distribution of wake vortices observed in Hong Kong in 2014.

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In the field measurement, we did not record the aircraft parameters, so the reference parameters of circulation and vortex cores are not available. Figure 13 shows the vortex-core trajectories of two aircrafts ((a) is Boeing B777-300ER, (b) is Boeing B747-400) estimated by the present method. From which we can find that the vortex-core positions can be well located. Also a certain level of fluctuations to the vortex core trajectories is observed. This is mainly caused by the sudden change of background wind in different RHIs, which might deform the vortex’s structure accordingly.

Fig. 13. Vortex core locating results for two cases observed at Hong Kong international airport in 2014.

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The initial theoretical circulation of vortices generated by the aircrafts can be calculated by $\Gamma _0=M_Ag/\rho _ab_0V_A$, where $M_A$, $g$, $\rho _a$, $V_A$ are respectively the aircraft mass, gravity acceleration, air density, and aircraft speed [46]. We did not record the exact flight information, so we can take some empirical values of $V_A$ and $M_A$ to estimate the circulation. The empirical landing speed $V_A$ of these two aircrafts is 250km/h, $M_A$ is a value between the maximum and minimum landing weight of the aircraft, where $M_A$ of B777-300ER is between 167800kg and 351500kg and B747-400 is between 185000kg and 396000kg. Figure 14 shows the evolution of circulation obtained by the PI method. It is observed that the estimated circulations in the initial period are within the lower and upper limits; this indicates the good fidelity of the retrieval results to reality. By the way, bigger variations are observed in the circulation plots in comparison with the simulation results; this is mainly caused by the fact that the complex ambient wind and surrounding environment may deform the vortex structure to a certain degree.

Fig. 14. Circulation retrieval results for two cases observed at Hong Kong international airport in 2014.

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3.2.2 Field detection in China Tsingtao Liuting international airport

A series of detections were conducted at Tsingtao Liuting international airport with a Doppler Lidar (WindPrint S4000) in November 2018 by Tsingtao HuaHang Lidar company. The Lidar’s pulse duration is 200 ns, and the wavelength is 1.55 $\mu$m. It was set on the balcony of the meteorological observatory (about 10m above the ground) to observe the runway at a distance of 390 m (see in Fig. 15). One of the obtained RHIs of radial velocity in the campaign is shown in Fig. 16.

Fig. 15. Geometry setup of the observation in Tsingtao Liuting international airport, 2018.

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Fig. 16. An example of radial velocity distribution of wake vortices observed in Tsingtao in 2018.

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Vortex-core trajectories and circulations of the PI method for two aircraft landing cases ((a) is Airbus A321-100, (b) is Airbus A330-300) are shown in Fig. 17 and Fig. 18. From Fig. 17, it is observed that the vortex core are well located and the trajectories are fairly smooth.

Fig. 17. Vortex core locating results for two cases observed at Tsingtao international airport in 2018.

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Fig. 18. Vortex core circulation results for two cases observed at Tsingtao international airport in 2018.

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In this field measurement, the details of aircraft parameters, such as aircraft mass, landing speed, were not recorded either, so we the method in section 3.2.1 to estimate the lower and upper boundaries of the initial circulations. The empirical landing speed $V_A$ of A321-100 and A330-300 are 250km/h, $M_A$ of A321-100 is between 48200kg and 93500kg and A330-300 is between 122000kg and 230000kg. The circulation results were shown in Fig. 18. We can find that the circulations estimated in the initial period are within lower and upper limits. The fluctuations of vortex-core positions and circulations are also considered to be affected by the changing background wind in different RHIs.

4. Conclusion and discussion

This paper proposes a parameter-retrieval method of wake vortex based on the Doppler velocity integration along some paths. The method locates the vortex cores according to the Doppler velocity profiles and uses the path integration of the measurement bins along the line-of-sight(LOS) above and below the vortex cores to obtained the circulations of the two vortices. Also, an iteration process is proposed to mitigate the impact of compressing and expanding effects of wake vortex caused by the scanning of Lidar beam in a RHI.

Numerical examples and field detection campaigns show that the proposed method has the benefits of being accurate, efficient and robust. The main reasons of these good performances are: 1. the integration of velocity can well mitigate the noise and other abnormal effects involved in the Lidar detection. 2. the use of more observing data introduces more information to the algorithm than traditional methods which only use the observing data at the ranges of vortex cores. In this sense, this method can provide good support for air traffic controller to assess the hazards of wake vortex in real-time.

It is noted that the performance of this method depends on the accuracy of vortex-core locating which may be impacted by the compressing and expanding effects of vortices caused by the scanning of Lidar beam. In the near future, efforts will be made to deal with these effects better, and then the performance of this method is expected to be further improved.

Funding

National Natural Science Foundation of China (41375040, 61490649, 61625108, 61771479); Hunan Natural Science Foundation for Distinguished Young Scholars (2018JJ1030).

Acknowledgments

The authors would like to express their gratitude to Tsingtao HuaHang Lidar company for providing the wake vortex detection data for analysis.

Jianbing Li and Chun Shen contributed equally to this work.

Disclosures

The authors declare no conflicts of interest.

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Parameter	Values
Wavelength	1.55 $μ$ m
Pulse duration	170 ns
Time window	120 ns
Sampling rate	50 MHz
Pulse accumulation number	1500
Signal to noise ratio	-5 dB

condition		PI method		Opt method		TV method
condition		$E_{r} (Γ_{c 1})$	$E_{r} (Γ_{c 2})$	$E_{r} (Γ_{c 1})$	$E_{r} (Γ_{c 2})$	$E_{r} (Γ_{c 1})$	$E_{r} (Γ_{c 2})$
EDR 0.05 $m^{2} / s^{3}$	background wind -1m/s	7.07%	9.67%	6.31%	7.24%	24.34%	34.73%
	background wind -2m/s	11.1%	8.88%	8.65%	7.27%	37.43%	66.25%
	background wind -3m/s	10.33%	12.62%	8.15%	9.74%	56.06%	77.50%
background wind -2m/s	EDR 0.01 $m^{2} / s^{3}$	8.11%	9.65%	7.63%	6.69%	34.95%	65.45%
	EDR 0.02 $m^{2} / s^{3}$	10.62%	9.23%	9.14%	8.27%	38.52%	69.08%
	EDR 0.05 $m^{2} / s^{3}$	11.1%	8.88 %	10.15%	9.89%	47.43%	76.25%

Parameter	Values
Wavelength	1.55 $μ$ m
Pulse duration	170 ns
Time window	120 ns
Sampling rate	50 MHz
Pulse accumulation number	1500
Signal to noise ratio	-5 dB

condition		PI method		Opt method		TV method
condition		$E_{r} (Γ_{c 1})$	$E_{r} (Γ_{c 2})$	$E_{r} (Γ_{c 1})$	$E_{r} (Γ_{c 2})$	$E_{r} (Γ_{c 1})$	$E_{r} (Γ_{c 2})$
EDR 0.05 $m^{2} / s^{3}$	background wind -1m/s	7.07%	9.67%	6.31%	7.24%	24.34%	34.73%
	background wind -2m/s	11.1%	8.88%	8.65%	7.27%	37.43%	66.25%
	background wind -3m/s	10.33%	12.62%	8.15%	9.74%	56.06%	77.50%
background wind -2m/s	EDR 0.01 $m^{2} / s^{3}$	8.11%	9.65%	7.63%	6.69%	34.95%	65.45%
	EDR 0.02 $m^{2} / s^{3}$	10.62%	9.23%	9.14%	8.27%	38.52%	69.08%
	EDR 0.05 $m^{2} / s^{3}$	11.1%	8.88 %	10.15%	9.89%	47.43%	76.25%

Path integration (PI) method for the parameter-retrieval of aircraft wake vortex by Lidar

Abstract

1. Introduction

2. Methodology

2.1 Geometry configuration

2.2 Locating of vortex cores

2.3 Path integration method for the parameter-retrieval of wake vortex

2.3.1 Velocity of a point inside the wake vortex

2.3.2 Path integrations of the velocity components

2.3.3 Parameter-retrieval based on the path integrals

2.3.4 Adjusting of vortex core positions to compensate the compressing/expanding effects of wake vortex caused by the scanning of Lidar beam

2.3.5 Ground effect of wake vortex

2.3.6 Suggestions for the selection of integration paths

3. Performance verification of the method

3.1 Numerical experiments

3.1.1 Locating of vortex-core positions

3.1.2 Evaluation of the retrieved velocity circulation

3.2 Field campaigns

3.2.1 Field detection in Hong Kong international airport (HKIA)

3.2.2 Field detection in China Tsingtao Liuting international airport

4. Conclusion and discussion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (18)

Tables (2)

Equations (30)

Optics Express