Resolving the heading‐direction ambiguity in vertical‐beam radar observations of migrating insects

Abstract Each year, massive numbers of insects fly across the continents at heights of hundreds of meters, carried by the wind, bringing both environmental benefits and serious economic and social costs. To investigate the insects' flight behavior and their response to winds, entomological radar has proved to be a particularly valuable tool; however, its observations of insect orientation are ambiguous with regard to the head/tail direction, and this greatly hinders interpretation of the migrants' flight behavior. We have developed two related methods of using wind data to resolve the head/tail ambiguity, and we have compared their outputs with those from simply assigning the heading direction to be that which is closer to the track direction. We applied all three methods to observations of Australian plague locust migrations made with an insect monitoring radar. For the study dataset, some of the headings selected by the simpler method are shown to be clearly incorrect. The two new methods generally agree and reveal a significantly different, and presumably more accurate, relationship of heading direction to track direction. However, use of these methods leads to quite a large proportion of the sample being lost because the wind values, which derive from a regional‐scale numerical model, are shown to be incompatible with the radar observations. This exploratory study has moreover demonstrated that locusts are frequently oriented at a large angle to their track and that quite often their movement is at least slightly tailfirst. Both new methods appear to be a significant improvement on the simpler method. As well as providing an accurate representation of migratory flight behavior, they allow occasions when the model wind values are unreliable to be eliminated from the data sample.


| INTRODUC TI ON
Migratory behavior in insects has evolved as an adaptation to environmental variability across space and through time (Drake & Gatehouse, 1995). Throughout the year, massive numbers of insects migrate across the globe. These migrations may lead to invasions of growing crops and consequent economic losses (Drake & Reynolds, 2012, pp 312-346;Koul, Cuperus, & Elliott, 2008), but also to huge seasonal exchanges of biomass and nutrients (Hu et al., 2016). Insect migration patterns and strategies have evolved differently among insect species, but the vehicle of movement is usually the wind.
However, recent studies on large nocturnal migrants have shown that they are not undiscriminating passengers on the wind, but appear to optimize their migration trajectories by selecting favorable heights (Wood et al., 2006) or flying with a crosswind heading direction (or orientation-we will use these two terms interchangeably) to optimize their flight trajectories . Mediumsized (10-70 mg) nocturnal insect migrants are predicted to orient toward the right of the downwind direction in the northern hemisphere as they respond to turbulence cues resulting from the change of wind direction with altitude due to the effect of surface friction (the "Ekman spiral"; . On the other hand, a larger moth, the silver Y (Autographa gamma; ~150 mg), was shown to have seasonally changing preferred migration directions, to compensate for wind drift, and to initiate migratory flight predominantly under tailwind conditions in order to gain ground speed and be carried toward their destination . They fly with a significantly higher degree of drift compared to nocturnal songbird migrants, thus exhibiting lower precision in orientation (Chapman et al., 2015). These findings and recent investigations of the orientation mechanisms used by high-flying insects in darkness (Dreyer et al., 2018) are leading to improved understanding of flight behavior, and how these behaviors guide insects of a variety of species to regions with seasonally favorable climates and habitat resources (Chapman et al., 2012).
Radar observation has been a major tool for developing our understanding of insect migrations (Chapman, Drake, & Reynolds, 2011;Drake & Reynolds, 2012, pp 74-99). Recently, several studies have achieved insights by combining radar observations with detailed meteorological information from high-resolution atmospheric models (Boulanger et al., 2017;Reynolds, Chapman, & Drake, 2017;Westbrook & Eyster, 2017). The radar automatically records the insects' track directions and speeds of travel relative to the ground (the track vector T) and also their body alignments (Drake, 2002).
The insect track direction, the direction toward which the insect is traveling, is measured as an azimuthal angle between 0° and 360° (clockwise from north). However, insect body alignment is an axial direction between 0° and 180°, also clockwise from north, that is, the head and tail of the insects cannot be distinguished so there is a 180° ambiguity in the orientation. Some scanning radars do provide additional clues suggesting how the ambiguity should be resolved, but these are not clear-cut or easily interpreted (Drake, 1983;Melnikov, Istok, & Westbrook, 2015).
To investigate the flight behavior of migrating insects and their response to the wind, it is essential to know their orientations; this requires that the head/tail ambiguity be resolved. We present here two methods of doing this that rely on the vector triangle of velocities that applies to animals moving in environmental flows (Chapman et al., 2008). The heading vector H, incorporating the insect's heading direction and speed relative to the air (the airspeed), is the vectorial difference of the track vector (T) and the wind vector (W; the speed and direction of the wind relative to the ground; Figure 1a). We also consider a simpler method for resolving the ambiguity, one that relies on radar data alone: The heading direction that is closest to the track direction is selected. In some recent studies, radar-measured insect speeds were 3-6 m s -1 faster than wind speeds from an atmospheric model (Chapman et al., 2008, and use of this method is then supported. Earlier work combining scanning entomological radar observations of insect migrations with pilot-balloon ascents also never detected overall backward displacement movement (Riley & Reynolds, 1986). However, with our observations (see below), selecting the heading closest to the track direction sometimes produces implausible variations with time or height, and track speeds are only slightly greater than wind speeds on average (with some negative differences). Thus, an alternative method of heading selection appears necessary.
Winds experienced by insects flying near the surface are relatively easily measured (Riley & Osborne, 2001;Srygley, 2001), but at altitude, in the absence of a co-located wind profiler or frequent pilotballoon observations, it is necessary to rely on values derived from an atmospheric model driven by routine meteorological observations.
A model provides a near-continuous series of wind values and combining these with the similarly continuous observations from a radar allows stronger inferences about orientation behavior, especially its continuity and variation, to be drawn-as the analyses presented here will demonstrate. However, model winds are sometimes incorrect, radar measurements have limited precision, and there is a degree of uncertainty about the insects' airspeeds. Therefore, an ambiguity-resolution method based on the triangle of velocities must incorporate recognition of uncertainty in the various input quantities and include some form of "quality control" to eliminate occasions when the wind estimates are incompatible with the radar observations.
In this paper, we present our two methods and assess their effectiveness, and that of the simpler method, by applying them to radar observations of migrations of Australian plague locusts (Chortoicetes terminifera) obtained with an insect monitoring radar (IMR; Drake, Wang, & Harman, 2002) at Bourke, New South Wales, Australia.

| Circular statistics
Circular statistics have been used to calculate mean values of directional quantities and two measures of variance, the mean resultant vector length (R), and the circular standard deviation (Berens, 2009).
R ranges from 0 to 1, with larger values of R indicating greater deviations of the population from a uniform distribution, that is, more clustering about the mean. The Rayleigh test (Fisher, 1995) was used to test the uniformity of angular distributions, at a significance level of p < 0.05.

| Radar observations
The radar observation data are from an IMR at Bourke airport (30.0392°S, 145.952°E, 107 m above sea level) and were acquired from September 2010 to April 2011: This period comprises spring (September-November), summer (December-February), and autumn (March-May) of a single insect-flight year. Australian plague locusts, which were relatively plentiful over this year, were identified from the characteristics of their radar returns (Drake & Wang, 2013;Hao, 2016). The radar observations were recorded every night in three periods of ∼8 min during each hour from 18:00 hr to 06:00 hr Australian Eastern Standard Time (AEST = UTC+10 hr; all times in this paper are AEST). Radar echoes were recorded from heights between 175 and 1,300 m and collated into "units" 150 m deep and 1 hr in duration. For a unit to be included in the analysis, it had to contain at least 12 targets identified as C. terminifera. In each unit, outliers of the track speed and the track direction were discarded; the acceptable ranges were two standard deviations from the mean for speed and either two circular standard deviations or 81° (the maximum possible value for a circular standard deviation) from the angular mean for direction. Units were also required to exhibit significant nonuniformity (Rayleigh test, p < 0.05) in both track and alignment directions. Only nights of "strong migration"-those with more than 1,000 locust targets and with 25 or more included units-were used in the main analysis.

| Wind data
The air pollution model (TAPM; Hurley, 2008) was used to estimate the wind vector W at the heights the insects were flying. TAPM is designed for high-resolution weather simulation and has been tested in a wide range of locations, both in Australia and overseas (Hurley, Edwards, & Luhar, 2008). TAPM simulation of boundarylayer winds has been verified by comparing its outputs with winds measured by sodar and electromagnetic wind profilers at Wagga Wagga, NSW (Hao, 2016;Taylor, Zawar-Reza, Low, & Aryal, 2005).
The results showed that the root-mean-square errors are smaller than Australian plague locust airspeeds (approximately 4 m/s; Clark, 1969;Hao, Drake, Sidhu, & Taylor, 2017), so should not dominate our vector-triangle calculations. For consistency with insect track and heading directions, wind directions in this paper are stated as the direction the wind was traveling toward (rather than the direction it was coming from, as is usual in meteorology).

| Method using only radar data
As discussed above, the true heading direction of the insect may be either the alignment value provided by the IMR, or 180° from that value. The simplest method to determine the orientation (method A hereafter) would be to select the heading direction closer to the track direction. This method is considered applicable only to strongflying insects and implicitly assumes that these do not head into winds stronger than their airspeed and thus do not exhibit overall backward displacement. (We will refer to such movements as tailfirst, and this is to be understood as including the full 180° sector of orientations for which the along-track component of the heading vector is negative.) Two variants of method A appear possible: Either the "closest" criterion is applied to each echo in a unit individually, and a distribution of heading directions is then obtained (method A i ) or mean track and alignment directions are calculated for all the echoes in the unit, and the criterion is applied once to these means (method A u , Figure 1b).
Method A confines the inferred heading directions to within a 180°-wide sector around the track directions, and if the assumption of downwind flight is incorrect, this will produce spurious results. An example (from the locust dataset of this paper) where this appears to have occurred is shown in Figure 2. While the alignment varied gradually through the night, the heading direction inferred from method A reversed suddenly at around midnight and the results are still problematic as flipping of the inferred heading directions still occurs (and in fact is more frequent). The failure of both variants of method A in this example is due to the almost perpendicular track and alignment directions a phenomenon that occurs commonly in Australian locust migrations (Drake, 1983;Rennie, 2013).
When method A is applied to the strong migration nights of 2010-2011, the track directions and heading directions are seen to be positively related in all three seasons (Figure 3), but this is an inevitable consequence of not allowing tail-first movement. The diagonal white (i.e., unpopulated) regions are where tail-first movements would appear (Figure 3a). The sharp edges to the distribution of points adjacent to these diagonals are clearly artifactual and suggest the true distribution extends into these regions, indicating that some tail-first movement did occur. When the procedure uses units rather than individuals (method A u rather than method A i ), the sharp edges are less apparent ( Figure 3b); however, they are evident in a larger dataset (with 7 years of radar observations; see Fig. 7.5a in Hao, 2016). This longer study provides additional indications of tailfirst movement as locust track speeds were found to exceed the corresponding wind speeds by only 1.2 m/s on average (both mean and median), which is much less than the 4 m/s airspeed of C. terminifera or the 3-6 m/s speed difference seen by Chapman et al. (2010), and there was a proportion of negative differences (Hao, 2016, Fig. 7.1).
Thus, while method A may give reasonable results in some circumstances, it does not appear reliable for use with Australian locusts, or indeed for any migration in which alignments are at large angles to tracks-that is, in which there is strong drift.

| Methods incorporating wind data
Our two methods are fundamentally equivalent, drawing on the triangle of velocities, but the procedures employed differ considerably. As the atmospheric model provides winds at hourly intervals, there is only one wind value for each unit and it seems sensible to relate this to single unit-average values of insect tracks and alignments, doing so also avoids the generation of split distributions. With these methods, the inferred heading directions can range over 360°, that is, they may determine that the insects are moving tailfirst. In some instances, however, a vector triangle that is consistent with the radar observations of insect alignment, and that has reasonable values for insect airspeeds, cannot be formed, and there is then no basis for inferring a heading direction. Our procedures therefore incorporate quality control, so that unreliable results are identified and eliminated from samples. This of course reduces the size of the sample and could possibly introduce bias. The tests allow a wide 1-7 m/s range for the airspeed, to account for variance in, and a degree of uncertainty about, this 4 m s −1 value.
As mentioned, an insect's movement over the ground, expressed by the track vector T, is the vector sum of the heading vector H and the wind vector W (Figure 1a). We assume that the direction of the vectorial difference between T and W, that is, of the insect's "heading vector by subtraction" (H S , blue arrow), is aligned with the insects' body axis, that is, that the insects fly without sideslip. We can estimate H S from radar measurements of T and values of W from an atmospheric model and use it to resolve the head/tail ambiguity by choosing the alignment direction that is closer to the H S direction (H B in Figure 1c). This constitutes our first new method (method B hereafter). We apply method B only to units, because if applied to individuals, split distributions ( Figure 2c, plot H Bi ) arise just as in method A i . Quality control is implemented through tests that the H S speed is within the range 1-7 m/s, and the difference between the direction of H S and the closer of the two mean heading directions for the unit (α in  Figure 1d), and the heading direction that produced this putative wind is selected. In this method, quality control is applied through the ratio where |D L | is the length of the difference vector which is lesser and ΔW′ L is the magnitude of the uncertainty in the selected putative wind vector, which is estimated as the distance to a corner of the uncertainty trapezium (Figure 1d; this being the quadrature sum of the lateral and longitudinal uncertainties). If R W′ ≤ 2, then that putative wind and the selected heading direction are considered to be reliable; if 2 < R W′ ≤ 3, the selection will be considered as tentative (and will be plotted in different colors, see e.g. Figure 4f); and if R W′ > 3, the unit fails quality control. For some units, both putative winds may have passing R W′ values, and to resolve the selection, we then require that one is clearly in better agreement with the model wind than the other. This test is implemented as that is, the rejected distance must be at least 50% greater than the selected one.

| RE SULTS
Analysis of the 2010-2011 radar observations revealed that there were 46 nights of strong migration (as defined previously). On one of these nights, track speeds were very low and orientations changed apparently randomly before midnight, so this case has been excluded from the sample. From the remaining nights, we have selected four cases to represent different scenarios and for each of  Figure 1(b,c) for key to symbols. Note speed scale at top left, which is same for all vectors; the length of the hourglass triangles indicate an airspeed of 4 m/s these, we show how the simpler method and the two new methods performed. We also present a summary of their performances over all 45 nights.

| Case studies
For the night of 13/14 March 2011 (case 1), all three methods produce similar results (Figure 4a,b). The wind maintained a southwest direction through time and height, while the locust track directions were toward the northwest all night at lower heights but shifted to westward later at higher levels. Body alignments are approximately north/south early in the night and low down, and northwest/southeast later and higher up. Both body alignments and track directions were highly concentrated throughout the night and exhibited only gradual variations. According to all three methods, the locusts were heading north at 21:00 and gradually changed to a northwestward heading (Figure 4a,b). All quality-control tests (methods 2 and 3) passed.
For 12/13 October 2010 (case 2), the new methods select headings that disagree with those obtained by method A (Figure 4c,d). The wind was blowing to the southwest at ~10 m/s from 19:00, gradually increasing in speed to ~15 m/s after midnight. Alignments are north/south initially, but then northwest/ southeast for most of the night. Tracks are toward the southwest, and so are at a large angle to the body alignments. When selected by the track direction (method A), all but one of the headings are to the southeast. (the exception, at 23 hr, represents a flip followed immediately by a second reversal.) In contrast, method B consistently indicates that the locusts were flying toward the northwest, and the computed airspeeds are reasonable (Figure 4c). The same result is obtained with method C, as the putative winds derived from the track vectors and heading directions to the north or northwest are quite closely aligned with the model winds while their counterparts with southward or southeastward headings are at much larger angles to them (magenta arrows, Figure 4d). There are no flips, and after the first hour, the quality-control requirements are met for all units. Both new methods show that the locusts were heading toward the northwest but were being blown sideways and slightly backward by the strong winds, so that they were undertaking tail-first movement.
Case 3, the night of 11/12 November 2010, provides an example of simulated winds that cannot be reconciled with the track and body alignment information from the radar. During the early part of the night, the wind simulations appear to have been poor as the quality-control tests fail; however, after midnight, the model winds become compatible with the radar data and the two new methods produce mostly consistent heading selections (Figure 4e,f). A similar scenario, of TAPM outputs being wrong initially but improving later, occurs also on other nights. As in case 2, the headings selected by method A often differ from those obtained with the new methods.
The method-1 headings flip across both time and height early in the night, whereas the new methods eliminate these units from the sample. However, on this occasion, method C also produces a flip, at 23.00.
For the night of 6/7 November 2010 (case 4), both new methods indicate that the simulated winds are inaccurate at lower levels during much of the night (Figure 4g,h). Method A is not reliable, as shown by its heading selections flipping through both time and F I G U R E 5 (a) Scatter plot for method B, showing H S speed against the angle α (see Figure 1b), with the angle scaled by the uncertainty on it (Equation 1). (b) Scatter plot for method C, of the length of the difference vector D L for the closer putative wind (see Figure 1d) against the corresponding length D G for the more distant putative wind, with both lengths scaled by their uncertainties. The red crosses indicate the units in case 4 that were accepted by method B but rejected by method C. The insets identify regions referred to in the text height. Method B rejects 6 units and method C rejects these and 4 more; but they give consistent headings for those remaining. The units rejected by both new methods are largely at lower heights, especially between 325 and 475 m, a zone where a low-level jet wind (see Discussion) is quite likely to have developed. The four units for which the two new methods give different rejection statuses (red crosses in Figure 5) are mostly in situations close to the boundary between acceptance and rejection (see next section).

| Effect of quality control
For our 1-year sample, 53.7% of units were rejected by quality control when using method B, and 49.2% with method C. The manner in which these rejections occur is indicated in the scatter plots of In Figure 5b, the data appear as a band with restricted width; this is due to the difference between the two putative winds being limited by the assumed locust airspeed of 4 m/s (which falls to ~3 m/s with vector averaging over the various individual heading directions).

| Full-year dataset
Some statistics for performance of the methods over the 45 nights of strong migration are presented in Table 1. Model winds and radar observations were incompatible for ~45% of units, rendering ~40% of nights unusable for analyses of locust migratory behavior.
Rather more units were found to be compatible by method C than by method B, suggesting the method C quality control may be less stringent, or perhaps that this method works better. When both methods B and C give a result, these differ for only ~1% of units. There was agreement between all three methods for ~20% of units, and disagreement between the simpler method and two new methods for a similar proportion; the equivalent proportions for nights were similar. These two categories correspond to forward and tail-first movement, and in this dataset, it appears that these occur with approximately equal frequency. Overall, the heading directions selected by the two new methods are very similar (compare Figure 6a,b) but differ significantly from those selected by method A (Figure 3b). The track direction is related to heading to some degree, with this relationship being slightly different across seasons. The summer pattern appears to be a mixture of the spring and autumn patterns. During spring, three clusters of points are revealed by both new methods, although two of these (X and Y in Figure 6) could be combined as the headings wrap around from 360° to 0°. This cluster, with headings to the NW and N, indicates a preferred direction which is not evident in Figure 3b (spring) where many of the headings are incorrectly assigned. In autumn, this cluster is still present but the track directions have shifted from SW to WSW and NNW, so tail-first movements are not so frequent; however, there is now also a smaller cluster (W) with heading directions still toward the NW but tracks to the SE. The third cluster in spring (Z) has heading directions toward the SE and tracks to the SE group. This cluster is also present in summer and (though weaker) in autumn. In summer, but not the other two seasons, there is a diffuse cluster (V) with headings and tracks to the N and E; this exhibits no tail-first movement and in consequence is also evident in Figure 3b.

| D ISCUSS I ON
In this paper, the simpler method of using track direction to select heading direction from the ambiguous radar observations of alignment and two new methods that also draw on wind informa- Our results show that the two new methods, which differ only in how they implement the same fundamental test, produce similar results and rejection rates. There thus appears to be little reason to prefer one over the other. Method B is more intuitive and readily interpretable, method C more akin to a formal statistical test. Methods A and B can be applied either to individual insect targets or to samples (such as our one unit), but we have shown that application to individuals leads to artifactual splits and should not be implemented.
Method C can only be based on samples as it requires estimates of uncertainties, which are derived from sample statistics.
From the analyses presented here, and building on the earlier examination of Figure 2, five principles for assessing heading-selection  (1) and (2) form the basis of the quality-control tests incorporated into our methods, and principle (3) leads to our rejection of individual-based heading selection. Principles (4) and (5) provide a basis for validating methods, though we note that some degree of expert judgment will remain necessary to distinguish artifactual effects from real, and perhaps unforeseen, migrant responses to changing winds and to other varying environmental cues.

ACK N OWLED G M ENTS
We thank Leesa Sidhu for advice during preliminary investigations and with statistics. Zhenhua Hao is employed through European Research Council Advanced Grant (GAP-741298), and previously held short-term appointments funded by UNSW.

CO N FLI C T O F I NTE R E S T
None declared.

AUTH O R CO NTR I B UTI O N S
All authors contributed to conception of the study, design of the procedures, and analysis and interpretation of the data.
Zhenhua led the writing of the manuscript. All authors critically reviewed drafts, contributed revisions, and gave final approval for publication.