1. Introduction
The short-term prediction of disturbances encountered by aircraft in the last minutes of landing provides safer and more comfortable flight for passengers. Here we evaluate the performance of a recently proposed Lagrangian (i.e., particle based) algorithm for the detection of such disturbances using historical landing data and pilot reports from Hong Kong International Airport.
Topography-induced wind shear accounts for about 70% of reported disturbances on landing aircraft at Hong Kong International Airport (HKIA) (Shun and Chan 2008). While radars detect wind shear due to thunderstorms, they may miss high levels of wind shear in nonrainy weather. Such clear-air turbulence, by contrast, is well captured by Doppler lidars currently in operation at HKIA (Shun and Lau 2002; Chan et al. 2006; Shun and Chan 2008).
Lidars measure air velocity from reflections of a laser beam on aerosols using the Doppler effect. Although individual lidar returns only provide line-of-sight velocity components, variational schemes are available to reconstruct full two-dimensional velocity components in the conical surface spanned by the rotating lidar beam (Chan and Shao 2007). This velocity field is updated every 2 min, providing a time-resolved input for automated algorithms that seek to detect aerial turbulence.
One such algorithm locates critical values of the velocity fluctuation, defined as the difference of the measured air velocity from an averaged background velocity (Chan et al. 2006; Chan and Lee 2012). Another approach is based on eddy dissipation rate, an internationally adopted metric for turbulence intensity (Chan 2010). Finally, the autocorrelation of scanned velocity in different sectors has also been employed in wind shear detection (Chan and Lee 2012).
These wind shear detection methods are fundamentally Eulerian, inferring flow features from instantaneous air velocity. As such, they are sensitive to transient errors in velocity measurement and reconstruction. To de-emphasize such errors and anomalies, one may instead look for the signatures of sustained wind shear in the two-dimensional virtual particle dynamics generated by the reconstructed velocity field on the cone of the lidar scan. This Lagrangian averaging approach has been developed in a series of papers by Tang et al. (Tang et al. 2010, 2011a,b), serving as a basis for the present study.
Tang et al. located attracting Lagrangian coherent structures (LCSs)—that is, locally most attracting material lines—in the Lagrangian flow over the lidar cone. They obtained LCSs as select ridges of the backward-time finite-time Lyapunov exponent (FTLE) field, which characterizes the rate of convergence of virtual air particles. These results confirm the ability of backward-time FTLE ridges to pinpoint the location of intense three-dimensional material updrafts between rolling vortices from two-dimensional lidar scans (Tang et al. 2011a). The extracted attracting LCSs were insensitive to instantaneous velocity fluctuations, and correlated well with wind shear jumps measured on landing fixed-wing research aircraft.
Here we present a more systematic study to assess the power of LCS analysis in real-time wind shear detection for commercial aircraft. We analyze roughly 11 months of continuous velocity data from long-range lidars based at HKIA as well as actual pilot reports of wind shear from commercial aircraft during the same period. Using receiver operating characteristic (ROC) graphs, we assess various metrics for the aviation hazard posed by a given LCS. Out of this comparison, the gradient of the FTLE field projected along the runway emerges as a reliable wind shear detection tool, with high hit rates and relatively low alert durations.
For the spring-season part of the available dataset, our FTLE-based wind shear detection scheme is found to compare very favorably with the algorithm currently in operation at HKIA, even though the latter also uses additional information from dedicated short-range glide-path scans and radars. Our method shows consistent but more modest performance for the remaining two seasons. In these seasons, the current operational algorithm prevails because of its additional sources of input, which continue to provide reliable velocity scans in the presence of rain as well.
The organization of this paper is as follows. In section 2, we review the topography of HKIA, its lidar installation, and the dataset used in our study. Section 3 summarizes the main elements of LCS theory and FTLE analysis, and outlines the Lagrangian metrics we test for quantifying the aviation hazard of a given FTLE ridge. In section 4, we employ ROC graphs to evaluate these metrics, and compare them with the wind shear detection scheme currently in use at HKIA. We present a summary and outlook in section 5.
2. Data used in this study
a. Topography of Hong Kong International Airport
Shown in Fig. 1, HKIA is built on Chek Lap Kok Island, surrounded by ocean on three sides and by Lantau Island to the south. This island has a mountainous terrain, with peaks of 1000 m above sea level. Prevailing easterly winds tend to generate rolling vortices off the peaks, creating major aerial disturbances in the form of updrafts, downdrafts, and close-to-ground planar shear. Indeed, pilot reports recorded at HKIA show that one in about 500 flights experiences significant wind shear (Shun and Chan 2008).
Topography of HKIA and its surroundings. Also shown are the lidar installations (blue dots) and the runway 25RA (red) considered in this study. Axis unit are in meters; the origin is the midpoint between the two lidars.
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
Figure 1 also indicates the location of HKIA’s northern runway (25RA) and those of two long-range lidars. A second runway in the southern part of the island is not considered in this study.
b. Lidar scans, reconstructed velocity, and pilot reports
The two long-range lidars at HKIA have been in round-the-clock operation since 2002, providing three types of scans: plan position indicator (PPI) scans, glide-path scans, and vertical scans (Shun and Lau 2002). In a PPI scan, the elevation angle is fixed while the lidar beam performs a full 360° rotation, providing updated velocities every 2 min. To stay close to common flight paths in the last minutes of landing, the scans use the elevation angles 1.4° and 3°.
Figure 2 shows the PPI scan geometry for the northern lidar, whose scans are used in the present study. This lidar operates at a wavelength of 2 μm, with pulse energy of about 2 mJ. Its range resolution of 100 m with 1024 range bins is appropriate for the detection of typical wind shear features, reported to be between 400 m and 4 km in length (Fujita 1978; Proctor et al. 2000; ICAO 2005).
Geometry of scans by the northern lidar at HKIA, with elevation angles of 1.4° (green) and 3° (blue). All spatial dimensions are in meters, with the origin of the coordinate system based at the midpoint between the two lidars. For better visibility, only parts of the full conical scans are shown.
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
Lidars provide line-of-sight (LOS) velocity components along the rays in their conical span. To reconstruct the two-dimensional projection of the full air velocity field onto the cone of the scan, we use the variational algorithm discussed in Chan and Shao (2007). Our study period comprises historical velocity data from the year 2010 for the following three time intervals:
spring: 1 January–8 April,
summer: 1 May–31 August, and
fall: 1 September–29 November.
We also utilize reports of observed wind shear cases, as recorded at HKIA during our study period. In aviation meteorology, wind shear refers to a sustained change (one lasting more than a few seconds) in the headwind or tailwind of the aircraft. Conventionally, a velocity change of 15 kt (1 kt = 0.51 m s−1) or more is considered significant wind shear. In our study period, the following numbers of significant wind shear were reported by pilots for runway 25RA: spring: 79 reports; summer: 90 reports; fall: 38 reports.
Since rain is known to negatively impact lidar performance, the amount of continuously available velocity data in a given season correlates with the amount of clear days in that season. As a result, all three study periods contain time periods (ranging from minutes to a couple of days) with no reliable lidar scans because of rain. Another cause for missing data was lidar component malfunction. All in all, the highest-quality data were available for the spring period, but we did carry out our study for the summer and fall periods to uncover any potential significant seasonal dependence in our analysis.
3. LCS analysis
a. Mathematical setup
By contrast, visualizing the organizing centers of Lagrangian behavior in temporally aperiodic flow data is a challenge. LCSs have been found to form such centers, providing a generalization of stable and unstable manifolds for nonperiodic flows. Attracting LCSs are locally the most attracting material surfaces in the flow, along which tracer blobs elongate in forward time (Haller 2001). Similarly, repelling LCSs are locally most repelling material surfaces, acting as dividing lines between regions of different particle behavior.
Mathematical theories and numerical methods for detecting LCSs have been constantly improving (Peacock and Dabiri 2010). The currently most advanced methods take a variational approach, providing high detail but also requiring high spatial resolution in the velocity data (Haller 2011; Haller and Beron-Vera 2012). Here we detect LCSs from FTLEs, which are well suited to handle the relatively low spatial and temporal resolution of the available lidar scans.
Recent results show (Haller 2011; Tang et al. 2011b; Farazmand and Haller 2012) that ridges of the FTLE field computed in backward time (i.e., for t < t0) mark attracting LCSs if they are approximately normal to the dominant strain eigenvector field ξ2(x0). Computations in Tang et al. (2011b) confirmed that most LCSs relevant in the detection of updrafts at HKIA tend to satisfy this near-orthogonality condition. The low number of FTLE ridges not satisfying the near-orthogonality condition corresponds to horizontal shear, which may impose a horizontal jolt on landing aircraft. Here we only focus on identifying the FTLE ridges accurately and assessing the level of aviation hazard they pose.
b. Computation of LCSs
The two-dimensional velocity field (2), reconstructed from LOS velocities via the variational algorithm of Chan and Shao (2007), is available on a Cartesian grid of 15 km × 15 km, centered at the midpoint between the two lidars shown in Fig. 1.
We calculate the FTLE distribution 
(left) Computational domain near runway 25RA. Red dots denote the initial grid domain for FTLE calculations. Within this, cyan dots mark the domain for LD. Units are in meters. (right) Magnified computational grid containing the initial conditions x0 for the functions 
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
The end time of backward integration is t = t0 − 300 s for each present time t0. Given that the velocity field is updated every 2 min, about three scans are used in each advection. We have found empirically that this time scale is short enough to reveal transient Lagrangian effects while still yielding well-pronounced FTLE ridges. Trajectories leaving the computational domain during the integration are advected further by a smooth, linear, and divergence-free extension of the original velocity field. As shown in Tang et al. (2010), this method eliminates artificial FTLE ridges created by edge effects that arise if trajectories are stopped at the boundary.
We apply finite differencing to the discretized 
A sample FTLE distribution over the initial grid shown in Fig. 3.
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
c. Lagrangian and Eulerian metrics for wind shear intensity
The FTLE field 
Correspondingly, 
(left) FTLE distribution at 1222 UTC 2 Feb 2010, when no wind shear was reported. (Parallel horizontal lines on the FTLE plot indicate a typical glide path.) Note how projection on the glide path reduces the peak in the FTLE gradient along the rightmost ridge that encloses a small angle with the flight path. (right) As in (left), but for 1718 UTC 8 Feb 2010, when a missed approach due to wind shear was reported. Note how projection onto the glide path preserves the high peak in the FTLE gradient at ridges that are nearly orthogonal to the flight path.
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
While the two events shown in Fig. 5 confirm basic intuition, we will still assess the efficacy of 
Figure 6 shows a comparison of the various wind shear metrics on data from the early morning hours of 15 January 2010. These time series illustrate that Lagrangian metrics are insensitive to isolated, instantaneous jumps in the velocity field. All FTLE-based criteria show good performance, peaking near the wind shear reports and staying near zero otherwise.
Time evolution of the normalized maximum values of (top to bottom) Lagrangian and Eulerian metrics along the glide path, including a period of repeated wind shear reports on 15 Jan 2010. (All quantities are normalized by their maximum values within the time interval considered.)
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
We discuss a more systematic evaluation of these metrics over the full dataset in the next section.
4. Performance comparison for wind shear metrics
We evaluate the Lagrangian and Eulerian wind shear metrics discussed in section 3 using the spring, summer, and fall datasets described in section 2b. For all times and for each metric, we generate a 5-min wind shear alert if the given metric returns a maximum value exceeding a critical threshold (model parameter) over the glide path.
a. Definition of ROC graphs
True positive: The pilot reported wind shear and there was a 5-min alert in effect some time during the landing.
True negative: The pilot reported no wind shear and there was no 5-min alert in effect any time during the landing.
False positive: The pilot reported no wind shear, but there was a 5-min alert in effect some time during the landing.
False negative: The pilot reported wind shear, but there was no 5-min alert in effect any time during the landing.
The ROC graph of a wind shear metric is obtained by plotting the hit rate as a function of the false alarm rate for each tested value of the model parameter. The resulting graph shows the trade-off between benefits (hit rate) and costs (false alarm rate), enabling a comparison between different wind shear metrics, and the optimization of the model parameter for each metric under various performance objectives.
In our study, negative instances would be obtained from null reports—that is, pilot reports of wind shear not exceeding 15 kt. In practice, however, pilots rarely report the lack of wind shear after an uneventful landing, which results in an artificially low number of null reports. This necessitates a modification in the ROC curve: we replace the false alarm rate with the alert duration as the independent variable of the graph (Chan and Lee 2012). The alert duration is defined as the percentage of time in which a wind shear alert would have been generated by the given wind shear metric. This is just the sum of 5-min windows around wind shear alerts, expressed as a percentage of the length of the total study period. Using alert duration instead of the false alarm rate eliminates the issue caused by the lack of a statistically significant number of null reports.
b. Computation of ROC graphs
For each wind shear metric introduced in section 3c, we compute the metric along the flight path for the 5-min window before each reported wind shear event. If, within these 5 min, the value of the metric exceeds the model parameter, then the wind shear report (instance) is marked as a true positive. The total number of these true positives divided by the total number of wind shear reports in the study period gives the hit rate, as defined in (6).
For alert duration, we calculate each metric in every minute of the entire study period (this is achieved by linearly interpolating between two actual lidar scans, which are 2 min apart). If the value of the metric exceeds the model parameter, then a wind shear alert is generated for the following 5 min. The alert duration is then obtained as the total length of these 5-min alerts divided by the length of the entire study period.
For each metric, these computations yield a single point of the ROC graph for a fixed value of the model parameter. Varying the model parameter of the metric and recomputing the hit rate and alert duration will yield further points in the graph. To restrict the analysis to feasible values of the model parameter, we normalize each metric by the historical mean of its maximal values (computed for the given season) during each landing event. This historical mean of maxima is extracted from an initial sample period, the first two weeks of our study period. After normalization by this mean, we only vary the model parameter for each metric in the [0, 1] interval.
The historical means of maxima obtained in this fashion for the spring are as indicated in Table 1 (with bars referring to spatially averaged values along the glide path and with the gradients nondimensionalized by the length of the runway).
Historical means of maxima for various wind shear metrics.
c. Results
Figure 7 shows the ROC graphs obtained by evaluating our metrics over the summer period (from 1 January to 8 April 2010), containing 79 reported cases of wind shear. As seen in the figure, FTLE-based metrics outperform Lagrangian divergence (LD) in the detection of aerial disturbances. This extends the conclusions of Tang et al. 2011b (established for vertical accelerations on fixed-wing research aircraft from 20 data points) to over 3 months of actual pilot reports from commercial aircraft. Indeed, LD maxima can only mark three-dimensional stretching, whereas FTLE ridges can also be created by two-dimensional strain or shear, both of which may produce jolts on landing aircraft.
ROC graphs derived for the spring study period (3 months and 8 days, starting from 1 Jan 2010). The brown triangle marks the performance of the GLYGA, currently in operational use at HKIA. The brown circle shows the closest point obtained from the PFG metric. The corresponding dimensionless model parameter is 39.67.
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
Notably, the performance of the Eulerian velocity jump metric is the weakest among all metrics considered, even though it mimics the highly effective glide-path shear alert generation algorithm (GLYGA) currently in operation at HKIA. The reason for this discrepancy is that GLYGA has been developed and optimized for dedicated glide-path scans. By contrast, the velocity jump algorithm used here is implemented on PPI scans that deviate from actual flight paths. The poor performance of the velocity jump algorithm in our setting underlines the robustness of Lagrangian detection methods, which perform well even on PPI scans.
Within the class of FTLE-based metrics plotted in Fig. 7, the projected FTLE gradient emerges as the statistically most efficient wind shear indicator. This is in agreement with our discussion in section 3c, extending our conclusions from the single daily time series of Fig. 6 to the entire spring period.
For comparison, the brown triangle in Fig. 7 marks the current working point of the GLYGA, computed for the spring period. With an alert duration of 2.5% and a hit rate of 68.9%, this algorithm is currently used in the operational wind shear alerting system at Hong Kong International Airport (Chan and Lee 2012). In addition to detecting velocity jumps from long-range lidar scans, the well-tested and optimized GLYGA also incorporates dedicated glide-path scans as well as scans from ground-based radars.
In the absence of more specific performance objectives, a natural choice for our optimal model parameter value is the one that produces the ROC graph point closest to the current operating point of the GLYGA at HKIA. In Fig. 7, this closest point is encircled, with an alert duration of 4.2% and a hit rate of 67.6%. The corresponding model parameter (i.e., nondimensionalized critical threshold for 
Remarkably, the projected FTLE gradient (PFG) algorithm developed here produces performance levels closely approaching that of the operational GLYGA. The latter algorithm has been optimized for years, and relies on additional measurements (glide-path scans, radar scans, and anemometer readings) unavailable for our study.
Finally, in Fig. 8, we compare the ROC curves for the PFG algorithm for the scanning angles 1.4° and 3.0°. In the same figure, we also show the ROC curve for a simple combined PFG algorithm that monitors both scans, and triggers an alert if the PFG exceeds the model parameter in either scan.
The performance of the PFG metric, 
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
We conclude from Fig. 8 that the PFG algorithm applied to the 1.4° scan gives significantly better performance than the 3° scan. This result is expected, because lower-angle scans are closer to typical flight paths, and hence will be more in sync with pilot reports. In addition, updrafts between two rolling vortices generate more pronounced attracting LCSs in lower-angle scans, given the intense convergence of particles near the ground [cf. Tang et al. (2011b) for a more detailed description of the three-dimensional geometry of such updrafts].
Also shown in Fig. 8, the quality of the PFG metric computed from both scanning angles approaches and ultimately surpasses that of the 1.4° scan for increasing alert durations. A combined PFG metric with different threshold parameters for each scanning angle might ultimately outperform the 1.4° scan across the board, but the design of such a metric will not be pursued in our present study.
Given the above-mentioned findings for the best-quality spring data period, we focus our study on FTLE-based detection metrics and 1.4° scans for the somewhat lower-quality summer and fall datasets. In each of these datasets, we had intervals of no coverage totaling more than 3 days. Missing data intervals longer than 10 min were left out of the analysis. Those lasting less than 10 min were handled by interpolation from available scans.
The results for these datasets are shown in Fig. 9. We conclude that the efficiency of our PFG algorithm relative to the current operational algorithm at HKIA was slightly lower in the summer than in the spring dataset. The efficacy further decreases in the fall data, with the somewhat surprising emergence of the FTLE maximum as the best metric. Overall, lidar-based Lagrangian wind shear detection shows the highest accuracy for the spring period. The performance of two remaining seasons are also promising, given that no additional velocity data beyond lidar scans was used in these alerts.
ROC graphs derived for the summer and fall study periods. Again, brown triangle marks the performance of the operational GLYGA detection algorithm at HKIA. Brown circles show the closest point obtained from the PFG and maximum FTLE metrics, respectively. The corresponding dimensionless model parameters are 16.0 and 0.374, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-12-00186.1
The results of Figs. 7 and 9 are better understood in the context of regional climate. The spring season in Hong Kong is relatively dry, securing mostly reliable operating conditions for lidars. While fog spreads often from the South China Sea along eastern and southeasterly winds, these winds are blocked by Lantau Island, leaving the skies over the airport clear. The resulting high viewing quality for lidar scans is clearly reflected in the performance of our algorithm over the spring dataset. By contrast, the summer section of the data fully overlaps with Hong Kong’s rainy season, which starts in May and ends in September. Degraded lidar performance due to frequent rain in this period explains the somewhat weaker performance of our algorithm on the summer data. Finally, the weakest performance in the dry months of October and November should be due to the low number of wind shear reports available from this period. This is expected to weaken the ability of ROC to assess the performance of our algorithm reliably.
5. Conclusions
Using over three seasons of historical data from Hong Kong International Airport (HKIA), we have assessed the power of Lagrangian coherent structure (LCS) analysis in detecting aerial turbulence for landing aircraft. The data used include 11 months of time-resolved long-range lidar scans over runway 25RA as well as time-stamped reports of 207 significant wind shear events reported by pilots in the same period.
We have found that the projection of the gradient of the finite-time Lyapunov exponent (FTLE) field along the runway is an efficient metric for quantifying the LCS signature of reported aerial disturbances. Specifically, calling a wind shear alert when the projected FTLE gradient (PFG) exceeds a critical threshold value, we achieve a historical hit rate of 67.6% over an alert duration of 4.2% for our spring study period. For the summer, the same approach yielded a hit rate of 65% over a 13.0% alert duration. In the fall season, FTLE magnitude emerged as the leading indicator, yielding a more modest 58% hit rate, but from a very short 3.8% alert duration.
For the spring season, the current operational algorithm at HKIA produced only a slightly higher hit rate of 68.9% over a slightly lower alert duration of 2.5%. This operational algorithm, however, also incorporates multiple sources of input, including dedicated scans along the glide path, radar scans, and anemometer readings. These additional inputs all act to mitigate the effect of lidar equipment failure and rain. Given that the present study did not benefit from these inputs, our results show significant potential for the PFG algorithm.
Parallel to lidar measurements, HKIA also operates an anemometer-based wind shear–alerting system called Anemometer-Based Wind shear Alerting Rules—Enhanced (AWARE). With an alert duration of 0.4%, AWARE reached a hit rate of 26% over the period of January–November 2010. This low value indicates the strength of lidar-based wind shear assessment technologies over anemometer-based ones.
Improvements to the PFG algorithm are mainly expected from more frequent and more accurate velocity updates. Indeed, the current scanning frequency of 2 min misses short-term turbulent features that have recently been observed from short-range lidar scans targeting the glide path (Chan and Lee 2012). Further increase in the frequency of velocity updates is expected from future advances in lidar technology, from the deployment of several lidars at the same time and from backup velocity scans from radars in case of rain.
Additional improvements should result from a more careful processing of the reconstructed velocity field. Indeed, sector blanking of the lidar beam and optical anomalies in the lidar return lead to velocity jumps that cause large Lagrangian particle separation over the lidar cone. This in turn may lead to spurious FTLE ridges, some of which trigger false alarms under the PFG algorithm. The development of a high-end filtering methodology that eliminates spurious velocity jumps is currently underway.
Acknowledgments
H. A. K. was partially supported by a McGill MEDA. G. H was partially supported by the Canadian National Science and Engineering Research Council under Grant 401839-11. This work builds on an initial study supported by the U.S. Air Force Office of Scientific Research under Grant FA9550-06-1-0092.
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