## Abstract

Radar echoes from insects, birds, and bats in the atmosphere exhibit both symmetry and asymmetry in polarimetric patterns. Symmetry refers to similar magnitudes of polarimetric variables at opposite azimuths, and asymmetry relegates to differences in these magnitudes. Asymmetry can be due to different species observed at different azimuths. It is shown in this study that when both polarized waves are transmitted simultaneously, asymmetric patterns can also be caused by insects of the same species that are oriented in the same direction. A model for scattering of simultaneously transmitted horizontally and vertically polarized radar waves by insects is developed. The model reproduces the main features of asymmetric patterns in differential reflectivity: the copolar correlation coefficient and the differential phase. The radar differential phase on transmit between horizontally and vertically polarized waves plays a critical role in the formations of the asymmetric patterns. The width-to-length ratios of insects’ bodies and their orientation angles are retrieved from matching the model output with radar data.

## 1. Introduction

Insects, birds, and bats in the atmosphere cause radar backscatter detectable with centimeter-wavelength weather radars. Such echoes are frequently observed in the absence of clouds and precipitation and therefore they are commonly referred to as reflections from “clear air.” Hardy and Katz (1969), using three radars with widely different wavelengths, concluded that such echoes are caused by insects and birds in the atmosphere. Mueller and Larkin (1985), studying echoes from clear air with two radars, have concluded that insects caused the echoes. Works by Glover et al. (1966), Vaughn (1985), Wilson and Schreiber (1986), and Achtemeier (1991) conducted with single-polarization weather radars, documented various features of radar echoes from insects. A large amount of data has been obtained with special entomological radars (Riley 1975; Drake 1983; Drake and Reynolds 2012, hereafter DR2012).

Polarimetric weather radars have demonstrated new opportunities in studying return signals from insects and birds. In echoes from bioscatterers, the values of differential reflectivity (the correlation coefficient) are typically larger (smaller) than those in rain (e.g., Wilson et al. 1994; Zrnić and Ryzhkov 1998; Lang et al. 2004). These two properties can be used to recognize echoes from bioscatterers. Data from weather radars are typically presented in a form of plan position indicators (PPIs). Some echoes from bioscatterers exhibit nearly symmetrical patterns as functions of azimuth on PPI displays (e.g., Riley 1975; Mueller and Larkin 1985; Wilson et al. 1994; Lang et al. 2004). Quasi-symmetrical reflectivity and differential reflectivity fields have been studied to obtain the orientation of species relative to the wind (e.g., Wilson et al. 1994; Lang et al. 2004; DR2012), whereas other polarimetric variables, such as the differential phase and correlation coefficient, have not been examined. We consider all available radar polarimetric moments for obtaining information on sizes and the orientation of insects in the atmosphere. The focus of this study is on asymmetric patterns of polarimetric variables, an example of which is presented in the next section.

Radar observations of bioscatterers can provide useful information for biologists studying insect behavior and ecology or for scientists who forecast the invasive movements of species that are agricultural pests (e.g., Chapman et al. 2002; Chilson et al. 2012; Leskinen et al. 2012; DR2012). Here, data on orientation, speed and direction of movement, number concentration, size, and any other characters that help identify the species are of interest. Birds are not small compared to the 10-cm (S band) radar wavelength, so their scattering properties are calculated via the -matrix method. These properties exhibit resonance features that complicate interpretations of radar patterns. However, most insects are Rayleigh scatterers at S band (DR2012, chapter 4) and our considerations in section 3 show that a Rayleigh scattering model allows for understanding of asymmetric patterns in echoes from insects. The developed model is applied in section 3 to estimate the parameters of the species.

## 2. Radar data

Fields of reflectivity factors (*Z*), Doppler velocity (*V*), and spectrum width (*W*) caused by bioscatterers are frequently observed with the S-band Weather Surveillance Radars-1988 Doppler (WSR-88Ds) up to distances of 150 km in range. These three parameters are measured at the horizontal polarization. Recently, the radars have been upgraded to dual polarization, and differential reflectivity (*Z*_{DR}), differential phase (Φ_{DP}), and correlation coefficient (*ρ*_{hv}) between signals in the horizontal and vertical channels have become available. These three polarimetric variables depend on the axis ratios and orientations of scatterers but do not depend upon number concentration.

The quasi symmetry of some radar echoes from insects has been noticed from the first observations and explained with collective orientations of species (Riley 1975; DR2012, Figs. 11.3 and 11.5). Extensive studies of collective orientation of insects have been conducted with entomological radars (DR2012, 140–143, 244–253). Collective orientation has been also deduced from the quasi symmetry of differential reflectivity patterns (e.g., Muller and Larkin 1985; Lang et al. 2004). Observations with the polarimetric WSR-88Ds show that patterns of Z_{DR}, Φ_{DP}, and *ρ*_{hv} of insect echoes are frequently asymmetric; that is, it is hard to draw a line about which the radar patterns are quasi symmetrical. An example of such patterns is presented in Fig. 1. The data were collected with WSR-88D KOUN located in Norman, Oklahoma. Figure 1f presents a field on phase *ψ*_{DP} instead of Φ_{DP} that will be discussed in section 3. All radar data analyzed in this study have been collected in cloudless weather. The images in Fig. 1 are typical for insect echoes for central Oklahoma for wind speeds of 5–10 m s^{−1} in a convective boundary layer capped by a stable layer. Strips in the fields are noticeable and may represent convective rolls (DR2012, p. 263). Despite visual granularities and strip features, overall asymmetry in the images is apparent except the *V* and *W* fields.

Azimuthal profiles of radar variables, shown in Fig. 2 with the blue lines, exhibit asymmetry clearly. The profiles present the mean values in an annular ring with radii of 50 and 55 km. The data were gathered using the radial resolution of 250 m, so that data from 20 radial range gates are used to calculate the means. Such averaging makes the profiles not so fluctuating. Some perturbations in the profiles are caused most likely by leftovers of ground clutter filtering. For instance, a drop in *Z*_{DR} around an azimuth of 280° (Fig. 2b) coexists with a deviation of the velocity azimuthal profile toward zero at the same azimuth (Fig. 2a). Despite the small-scale fluctuations, all the profiles but the *V* one are strongly asymmetric. This asymmetry could be explained with different species at different azimuths or with the same species oriented differently. But the longevity of the patterns (sometimes of several hours) and their azimuthal smoothness make such an explanation doubtful and raise questions about the possible impacts of radar parameters. Can similarly oriented insects exhibit asymmetric azimuthal patterns? This is possible with simultaneously transmitted and received (STAR) radars as shown in the next section.

## 3. The model

### a. General formulation

The dual-polarization WSR-88D radars employ a configuration with STAR horizontally and vertically polarized waves (Doviak et al. 2000). In the STAR radars, signal paths in the two radar channels with horizontally and vertically polarized waves are different, so the transmitted and received waves acquire hardware phase shifts on transmit (*ψ*_{t}) and on receive (*ψ*_{r}). A medium with nonspherical preferentially oriented scatterers shifts the phase between the polarized waves by the propagation differential phase Φ_{DP} and differential phase upon scattering *δ*, so that the measured phase is *ψ*_{DP} = *ψ*_{t} + *ψ*_{r} + Φ_{DP} + *δ*. Phase *ψ*_{sys} = *ψ*_{t} + *ψ*_{r} is called the system differential phase. Scattering of electromagnetic waves is described by the 2 × 2 scattering matrix with the coefficients *S*_{hh}, *S*_{hv}, and *S*_{vv} (e.g., Doviak and Zrnić 2006, sections 8.5.2.1 and 8.5.2.3). Propagation and scattering of polarized waves by a single scatterer can be described by the following matrix equation:

where the amplitudes of received waves are on the left side and the amplitudes of transmitted waves on the right side are denoted as *E*_{h} and *E*_{υ}. The matrix to the left of (*E*_{h }*E*_{υ}) describes the propagation of the waves from radar to the radar resolution volume. The matrix to the left of the scattering matrix *S*_{ij} describes the propagation of the waves on their way back to radar, and _{r} is a constant that depends on the radar parameters and range to the resolution volume. This constant is omitted in the following discussion because *Z*_{DR}, *ψ*_{DP}, and *ρ*_{hv} do not depend upon it.

The out-of-diagonal terms in the propagation matrices in (1) are set to zero, assuming weak depolarization of signals in the directions of propagation (forward and backward) in comparisons with the intensity of incident radiation. The phase *δ* is explicitly absent in the exponents in (1) because it is contained in the backscatter matrix as shown later. The calibration procedure equalizes the amplitudes *E*_{h} and *E*_{υ} so they can be replaced with unities. In echoes from insects at S-frequency band, the differential phase does not steadily increase along the radar radials as it does in rain. There are no noticeable radial changes in *ψ*_{DP} (Fig. 1f), so phase Φ_{DP} in (1) can be neglected. Small (if any) propagation differential phases in insects are due to their low number concentration. Negligible Φ_{DP} allows for neglecting differential attenuation in insects as well. In contrast to the radial changes, the azimuthal variations in *ψ*_{DP} are strong and should be explained.

The mean received powers 〈*P*_{h}〉 and 〈*P*_{υ}〉 in the channels and correlation function 〈*R*_{hv}〉 are obtained as

where the asterisk denotes the complex conjugate and the angle brackets stand for averaging over time. Because of ergodicity this averaging is equivalent to averaging over the orientation angles of scatterers and their sizes used in the model presented in sections 3b and 3c. The *Z*_{DR} (dB), phase *ψ*_{DP} (°), and *ρ*_{hv} are calculated as

Notation *ψ*_{DP} is used here for echoes from insects to distinguish it from Φ_{DP}. By neglecting Φ_{DP} in (1), the received voltages are obtained as

The system phase in receive *ψ** _{r}* enters in

*E*

_{vr}as a multiplicative exponent, so it does not affect the power in the vertical channel and the modulus of the correlation function. In contrast,

*ψ*

_{t}affects

*P*

_{h},

*P*

_{υ}, and

*R*

_{hv}. One can also see from (4) that if depolarization of the waves is weak (for instance, it is the case for almost spherical droplets in drizzle)—that is, if |

*S*

_{hv}| ≪ |

*S*

_{hh}| and |

*S*

_{hv}| ≪ |

*S*

_{vv}|—then the term with

*ψ*

_{t}becomes an exponential factor in the second equation in (4) and

*P*

_{h},

*P*

_{υ},

*Z*

_{DR}, and

*ρ*

_{hv}do not depend on this phase. However, insects do cause significant depolarization, so that

*S*

_{hv}is not negligible.

A critical scattering parameter is the ratio of the scatterer’s maximal size *l*_{max} and radar wavelength *λ*. If *π*|*m*| *l*_{max}/*λ* < 1, where *m* is the scatterer’s refractive index, then the Rayleigh scattering law can be applied. The wings of insects contain little water and therefore it is assumed herein they do not contribute significantly to the scattering properties. The insects’ bodies contain water and it is assumed here that the refractive index of the bodies is equal to that of water. So, for scattering of radar waves by insects, *l*_{max} should be applied to insects’ bodies, which are smaller than their total visible sizes.

In this scattering model, we approximate an insect’s body with a prolate spheroid (Fig. 3).

The radar cross section of a prolate spheroid depends upon its length, width (2*a* and 2*b* in Fig. 3), and orientation, which can be characterized with angles *θ* and *φ*. The direction of wave propagation is designated with vector **k**, which has angle *γ* with the horizontal plane X–O–Y. This is the elevation of the radar antenna. Angle *θ* is an angle between the spheroid’s symmetry axis O–O′ and the vertical axis O–Z. The amplitudes of the incident vertically and horizontally polarized radar waves are shown with vectors **E**_{υ} and **E**_{h}, respectively. Vector **E**_{h} is horizontal, whereas the direction of **E**_{υ} is not strictly vertical due to nonzero elevation angle *γ*.

The orientations of insects can be described with probability distributions that include two mean orientation angles *θ*_{m} and *φ*_{m} and also the deviations of actual *θ* and *φ* angles from these means; that is, the orientations should be characterized with four parameters. An insect’s body is described in the model with its major *a* and minor *b* semiaxes. So, there are six model parameters, whereas only three parameters—that is, *Z*_{DR}, *ψ*_{DP}, and *ρ*_{hv}, which do not depend on number concentration—are measured with radar. The large number of model parameters makes their retrievals uncertain for a single radar volume. To considerably decrease this uncertainty, the azimuthal dependences of radar variables are utilized here.

### b. Strictly aligned Rayleigh scatterers

The WSR-88D radars operate at wavelengths of about 10 cm. The Rayleigh scattering law can be applied for scatterers with sizes less than *l*_{max} = *λ/π|m|*, which results in 0.35 cm for *m* = 8.93–0.87*i* for water at 15°C (Lhermitte 2002, p.189). The size of 0.35 cm should be applied to insects’ abdomens. Rayleigh scattering has often been applied to backscatter by insects (e.g., Riley 1985; Wolf et al. 1993; DR2012, chapter 4). For a single Rayleigh scatterer at a given orientation, the scattering matrix coefficients can be represented as

where *α*_{a} and *α*_{b} are polarizabilities along the major and minor axes of the spheroid, respectively; *ε* is the dielectric permittivity of the scatterer, *ε* = *m*^{2}; and the matrix coefficients in (5a) are presented in the backscatter convention [e.g., Bringi and Chandrasekar 2001, (2.53)].

Consider identical spheroids aligned in the same direction, that is, all the scatterers have the same angles *θ* and *φ*_{m}, where the subscript is introduced here to distinguish this angle from *φ*, which enters in (5c) and (5d). Angle *φ* is the difference between *φ*_{m} and radar azimuth *φ*_{r}, that is, *φ **= **φ*_{r} − *φ*_{m}. Azimuthal dependences of *Z*, *Z*_{DR}, and *ψ*_{DP} for radar that scans over azimuth at constant elevation angle *γ* are of interest. Let *γ* = 0.5° as it is in the case shown in Fig. 1. Substitution of (5) with constant angles *θ*, *φ*_{m}, and *γ* into (4) and then into (3) yields the azimuthal profiles shown in Fig. 4. To generate the graphs, the following model parameters have been used: *b*/*a* = 0.25, *θ* = 80°, *φ*_{m} = 20°. Angle *θ* = 90° corresponds to the strictly horizontal orientation of the scatterer. So, *θ* = 80° can be viewed as a tilt (i.e., pitch angle) of 10° in elevation relative to the horizontal plane.

The model *ψ*_{DP} equals phase *δ* because no *ψ*_{sys} is introduced, so phase *δ* is shown in Figs. 4c,f. It can be seen that the azimuthal profiles exhibit asymmetry. For instance at *ψ*_{t} = 0°, the difference between the *Z*_{DR} peaks is larger than 10 dB. This difference is caused by strong depolarization of incident waves and by in-phase and antiphase interference of the main scattered and depolarized waves as shown below. Note a strong dependence of *Z*_{DR} and *δ* upon the differential transmit phase *ψ*_{t}: *Z*_{DR} changes by more than 10 dB at an azimuth of 110° when *ψ*_{t} changes from 0° to 150° (Fig. 4b). The *Z*_{DR} values are also affected by the elevation angle *γ* of the antenna (not shown) but that dependence is rather weak at elevations lower than 10°. The profiles in Fig. 4 appear to be antisymmetrical for *ψ*_{t} and *ψ*_{t} + 180°, but this is because elevation angle *γ* is small. Examinations of such profiles for larger *γ* reveal no antisymmetry in replacing *ψ*_{t} with *ψ*_{t} + 180°.

To demonstrate the significance of depolarization of electromagnetic waves upon scattering, consider some numerical values. The powers in the polarimetric channels are given by the first two equations in (2). They can be written in forms (8) and (9) presented below. The power in the horizontal channel (8) contains three contributions. The first one is the power of incident horizontal wave scattered back, *P*_{h1} = |*S*_{hh}|^{2}. The second contribution *P*_{h2} describes interference between this backscattered wave and the wave at horizontal polarization caused by depolarization of the incident vertically polarized wave. The third contribution *P*_{h3} in (8) is the power of the depolarized wave. At the right maximum in Fig. 4a (azimuth = 280°, *ψ*_{t} = 0°), these values in relative units are *P*_{h1} = 100.0 (taken as 100%), *P*_{h2} = 28.6, *P*_{h3} = 2.0, so that *P*_{h} = 130.6. For the vertical channel, *P*_{v1} = 4.6, *P*_{v2} = 6.1, *P*_{v3} = 2.0, *P*_{υ} = 12.7, and Z_{DR} = 10.1 dB. Term *P*_{h2} is about one-third of *P*_{h1} and has the same sign (in-phase interference). In the vertical channel, *P*_{v2} is larger than *P*_{v1}; that is, the depolarization contribution is stronger than the power of the vertically polarized wave scattered back to radar. At the left *Z*_{DR} maximum of the same curve, the magnitude of all contributions remain the same but *P*_{h2} and *P*_{v2} change their signs (antiphase interference), so *Z*_{DR} = 21.6 dB. The difference in *Z*_{DR} values at the peaks is 11.5 dB. For *θ* = 90°—that is, for strictly horizontal orientation—maximal *Z*_{DR} is 14.7 dB. So, a vertical tilt of scatterers relative to the horizontal plane can change *Z*_{DR} considerably.

The above-mentioned considerations are relevant to the STAR polarimetric configuration. If the alternate polarization (AP) is considered, then only the first contributions to the scattered signals—that is, *P*_{h1} and *P*_{v1}—are relevant. In this case, the *Z*_{DR} pattern would be symmetrical with a maximal value of 100/4.6 = 13.4 dB, that is, 8 dB lower that the maximum for the STAR mode. So, significant differences in *Z*_{DR} values measured with STAR and AP radars can be expected.

The differential phase is obtained according to the second equation in (3). Since Φ_{DP} is negligible in echoes from insects, phase *δ* is obtained as follows: *δ = ψ*_{DP} − *ψ *_{t} − *ψ*_{r}. Phase *δ* is measured in the interval from −180° to +180° and is shown in Figs. 4c,f as a function of azimuth for various *ψ*_{t}. The *δ* curves are not symmetrical about an azimuth of 180° (Figs. 4c,f): the curves’ amplitudes are different in intervals 0°–180° and 180°–360° if *ψ*_{t} is not equal to 90°. So, the model allows for understanding of asymmetry in measured differential phases. Figure 4 also demonstrates that *Z*_{DR} and *ψ*_{DP} depend on not only the physical parameters of the scatterers but also *ψ*_{t}. This finding helps with understanding of the differences in measured *ψ*_{DP} in echoes from insects observed with adjacent radars in the same spatial areas: the values of *ψ*_{DP} can be different because of different viewing angles and different *ψ*_{t} in the radars.

It is hard to expect that this simple model is capable of reproducing details of radar data and indeed some model *Z*_{DR} values are quite different from the radar data: the minimal *Z*_{DR} in Fig. 4 are negative, whereas *Z*_{DR} values from radar are positive. The negative model *Z*_{DR} values can be explained with simple geometry: at those azimuths, the vertical projection of scatterer is larger than its horizontal projection. Negative *Z*_{DR} in insect echoes are frequently observed at elevations above 3° (Rennie et al. 2010; Melnikov et al. 2010; Leskinen et al. 2012). It is clear that some variance in angle *φ*_{m} can eliminate the negative *Z*_{DR} values (see the next subsection).

For strictly aligned scatterers, *ρ*_{hv} equals unity always, whereas the radar data lie in the interval from 0.4 to 0.9 (Fig. 2d). This simple model reproduces asymmetry in *Z*_{DR} and *ψ*_{DP}, but it is not capable of reproducing *ρ*_{hv} radar profiles. To generalize the model, variances in angular alignments of scatterers are introduced next.

### c. Not strictly aligned Rayleigh scatterers

Both orientation angles—that is, *θ* and *φ*—of insects change in flight. Assume independence of orientations in *θ* and *φ* and consider first fluctuations in angle *φ*. Orientation of a scatterer on the horizontal plane is characterized with the mean angle *φ*_{m} and the standard deviation *σ*_{φ}. Assume the distribution in *φ* to be the truncated Gaussian, that is,

In (6) and (7), *D* = *D*(*φ*_{m}, *σ*_{φ}) because the integration limits are finite (*D* can be represented via the error function). For narrow distributions, *σ*_{φ} equals the standard deviation in *φ* and *D* = 1/(2*π*)^{1/2}*σ*_{φ}. The mean powers in the horizontal and vertical channels and correlation function can be obtained by inserting (4) into (2) and averaging

where Re(*x*) stands for the real part of *x*, and the angle brackets denote averaging over *φ* and *θ*. Let the distribution in *θ* be also the truncated Gaussian with the mean alignment angle *θ*_{m} and standard deviation *σ*_{θ}, that is,

with *G* = exp(*σ*_{θ}^{2}/2)/[(2π)^{1/2}*σ*_{θ} sin*θ*_{m}] for small *σ*_{θ} ( appendix B).

To calculate the mean radar moments 〈*P*_{h}〉, 〈*P*_{υ}〉, and 〈*R*_{hv}〉, the mean products of matrix coefficients have to be obtained. For instance, 〈*P*_{h}〉 in (8) depends upon 〈|*S*_{hh}|^{2}〉, which can be written as

Terms , , and for distributions (6) and (11) are calculated in the appendixes. Other mean products of the matrix coefficients in (8)–(10) are calculated similarly.

The model has the following variables: *φ*_{m}, *σ*_{φ}, *θ*_{m}, *σ*_{θ}, *b*/*a*, and *ψ*_{t}. To estimate these six parameters, azimuthal profiles of *Z*_{DR}, *ψ*_{DP}, and *ρ*_{hv} have been produced with the model and compared with the radar profiles shown in Figs. 2b–d. A reflectivity azimuthal profile has been produced as well for a constant number concentration. We match the model profiles of *Z*_{DR}, *ψ*_{DP}, and *ρ*_{hv} with radar data and then, using the model parameters from the best match, generate a reflectivity profile.

The results of such a match are presented in Fig. 5, where the model red and green curves have been generated with two sets of parameters shown in the second and third columns of Table 1, respectively. The best match has been obtained subjectively; no procedures like the least squares fit have been applied so far because not one but the three model profiles (*Z*_{DR}, *ψ*_{DP}, and *ρ*_{hv}) have to be matched with the radar data. One can see from Fig. 5 that the agreement between the model and radar profiles is good. The azimuthal profile of reflectivity (Fig. 5a) closely follows the observed data that signifies nearly constant number concentration of insects at those altitudes. This demonstrates the impact of orientation of insects on observed reflectivity: lowering of reflectivity at azimuths of about 0° and 150°–170° (Fig. 5a) is probably caused not by a lower number concentration of insects but by their orientations along the radar radial; this is the conventional explanation (e.g., Drake 1983; Lang et al. 2004; among others). A sufficiently good match between the model and radar profiles signifies that the species are approximate Rayleigh scatterers; this conclusion supports our assumption on insects as the primary source of the echo.

The dashed lines in Fig. 5 have been generated with deviated axis ratios by 0.02 to examine the sensitivity of the obtained *b*/*a* = 0.15. The black dash curves correspond to *b*/*a* = 0.15 + 0.02, and the magenta dashed line is for *b*/*a* = 0.15–0.02. One can see that the *ψ*_{DP} and *ρ*_{hv} profiles are more sensitive to changes in *b*/*a* than the *Z* and *Z*_{DR} values. Therefore, consideration of all polarimetric variables delivers a more accurate estimation of *b*/*a* than that which could be obtained from *Z*_{DR} alone.

Two sets of model parameters shown for case 1 in the second and third columns of Table 1 produce similar profiles as shown in Fig. 5 with the red and green lines, respectively. Both model sets retrieve the same *b*/*a*, *σ*_{φ}, *φ*_{m}, and *σ*_{θ}. The big differences in *θ*_{m} and *ψ*_{t} are explained as follows. Angle *θ*_{m} can be presented in the following form: *θ*_{m} = 90° + *β*_{m}, where *β*_{m} is the vertical tilt of the scatterer. The solid curves in Fig. 5 remain nearly the same if we simultaneously replace 90° − *β*_{m} with 90° + *β*_{m} and *ψ*_{t} with 180° + *ψ*_{t}. From Table 1 we have *θ*_{m} = 79° for case 1, set 1, which is 90° − 11° (*β*_{m} = 11°), and *θ*_{m} = 101° = 90° + 11° for set 2. Angle *ψ*_{t} = 27° in set 1 corresponds to *ψ*_{t} = 180° + 25° = 205° in set 2. Phase 25° is close to 27° for set 1. Since *θ*_{m} = 90° corresponds to the horizontal alignment of the scatterers, it can be concluded that the mean inclination angle to the horizon is 11° in both model sets but for set 1, the tilt is negative relative to the line of sight and the tilt is positive for set 2. Resolving this ambiguity could be important for entomologists because this shows the vertical orientation of insects. So, the knowledge of *ψ*_{t} even with an accuracy of 180° can be informative for interpretations of insects’ flights. Phases *ψ*_{t} in the WSR-88Ds are unknown.

The model is capable of estimating the mean axis ratio *b*/*a* = 0.15, mean vertical inclination *β*_{m} = 11°, and mean horizontal orientation *φ*_{m} = 350°. The spread of alignments on the horizontal plane is *σ*_{φ} = 20° for both model sets. The scatterers are oriented in the vertical plane with a spread of *σ*_{θ} = 10°. DR2012 (246–247) indicate large variations in *σ*_{φ} obtained with various methods. The insects flew from an azimuth of 335° that has been obtained from the best match of Doppler velocity in Fig. 2a and a sinusoid. So, the track direction of insects is 155° (335° − 180°). Since *φ*_{m} = 350°, the insects’ bodies are oriented at ~15° to their track direction and their heading deviates to the right from the direction. Such differences between mean headings and track directions have often been observed (DR2012, 248–250). The parameters of flights in our cases are listed in Table 2. The temperatures at the heights of data analysis are the bottom entry in Table 2.

Monarch butterflies (*Danaus plexippus*) were continuing to actively migrate southward across Oklahoma around the date of 1 November 2013 and may have represented the principal radar bioscatterers (case 1). This assessment is further supported by previous observations of soaring monarch butterflies in convective updrafts (Gibo and Pallett 1979) at altitudes as high as 1250 m AGL (Gibo 1981), and by the radar display of convective strips that were aligned with the wind direction (Fig. 1). The abdomen of the monarch butterfly is 30.2 mm long and 3.7 mm wide (Hobbs and Aldhous 2006). Most likely the abdomens contribute the most to radar cross sections due to their relatively high water content. The monarch butterfly body axis ratio can be obtained as 3.7/30.2 = 0.12 mm, which is the visual *b*/*a*. The radar-derived value is 0.15, which differs from the visual value because radar “sees” parts of the body filled with water. The body length exceeds the Rayleigh size, so these are estimations. To use radar-derived *b*/*a* in recognition of species, laboratory measurements are needed.

### d. Two more cases

Data analyzed in the previous subsection were collected with a research WSR-88D that has no limitations on data representation. In the network WSR-88Ds, radar variables are represented with a 1-byte format (except Φ_{DP}) that makes the measurement interval of *Z*_{DR} to be from −7.9 to 7.9 dB. All values outside this interval are clipped to its boundaries. Our observations with KOUN show that *Z*_{DR} in insect echoes sometimes exceeds 20 dB. The maximal *Z*_{DR} measured in the case analyzed above is 20 dB. Such values cannot be measured with the network WSR-88Ds.

The second feature of the network WSR-88Ds is that their initial system differential phase is set to 20°–40°. Since *ψ*_{sys} = *ψ*_{t} + *ψ*_{r} and *ψ*_{r} can be changed with signal processing, *ψ*_{sys} can be set at any given value. This is done in such a way that *ψ*_{sys} lies in the interval 20°–40°, which prevents the phases from aliasing at small intrinsic Φ_{DP} and preserves a large interval (320°–340°) for Φ_{DP} in precipitation.

An example of asymmetric radar patterns from insects observed with WSR-88D KLWX at Sterling, Virginia, is shown in Fig. 6. This case will be referred to as case 2. Bird migration is not significant at this location in July, so the echoes are primarily insects, although some contribution from birds cannot be excluded. Two echo layers are apparent in Figs. 6a–d. The low layer is seen as a circle with a radius of about 35 km and negative reflectivity factors. In the upper layer, which is in a form of a ring, strong asymmetry in *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} values is apparent. Multilayered insect echoes have been documented in many publications (DR2012, 235–242 and references therein). The lower layer in Fig. 6b contains patches of low *Z*_{DR} that suggest the presence of Bragg scatter, that is, scattering by turbulent air free from insects (Melnikov et al. 2011). Volume coverage pattern (VCP) 31 (Crum et al. 1993, their Table 1) has been used to collect the data. In Figs. 6e–h, the mean values of several radar variables for an annular ring of radii 40 and 50 km as functions of azimuth are depicted with the blue curves. The radial range resolution in this VCP is 0.5 km, so that the mean values in Figs. 6e–h have been obtained by averaging the data from 20 range gates. The 10-km width of the annular data ring was chosen to reduce fluctuations in the radar profiles. Note that the profiles exhibit strong asymmetry. The *Z*_{DR} profile has a “saturated” plateau at azimuths from 65° to 150°, where actual values are very likely larger than 7.9 dB. The influence of the wind rotation in the upper echo can be seen in Figs. 6b,d, where interleaved colors along the radar radials are apparent (e.g., azimuths of 50°–60° and 230°–240° in Fig. 6b and 90°–120° in Fig. 6d). At azimuth near 120° in Fig. 6d, one can see that Φ_{DP} increases along the radial to some distance and then decreases. This decrease means that it is not due to propagation effects but changing orientations of the scatterers that can be due to wind rotation.

The differential phase is denoted as Φ_{DP} in Figs. 6d,h because this is the usual notation for data from the WSR-88Ds. This should not be confused with the propagation differential phase that enters in (1) and is neglected in insect echoes. With this in mind, the differential phase in echoes from insects is denoted as Φ_{DP} in this subsection.

The developed model with the parameters shown in the fourth and fifth columns of Table 1 produced the profiles depicted in Figs. 6e–h with the red and green lines. One can see a good match for the *Z*_{DR} and *ρ*_{hv} profiles, taking into consideration the clipped *Z*_{DR} values. The model Φ_{DP} profiles (Fig. 6h) reproduce the measured phase’s magnitude, but they seem to be shifted right from the data profile. Our attempts to shift them to the left by varying the model parameters led to left shifts in the *Z*_{DR} profiles, that is, to mismatched *Z*_{DR}. Our attempts to achieve a better match in Φ_{DP} by introducing a spread in the axis ratios of the scatterers and varying the dielectric permittivity of the scatterers were unsuccessful. To achieve a better Φ_{DP} match, solutions to (1) with the matrix coefficients calculated via the -matrix method were also studied. Such calculations are usually done for birds (e.g., Zrnić and Ryzhkov 1998; Melnikov et al. 2012; Vivekanandan et al. 2013). Lang et al. (2004) applied the -matrix method to scattering by insects. The results of our calculations using the -matrix method (not shown) did not produce a better match. The origin of this discrepancy remains unclear. One possible source of this mismatch could be the presence of two (or more) species in the radar volume. The second species requires a second set of parameters, which makes the scattering problem too uncertain.

The minima in *Z* profiles are different (Fig. 6e). The conventional explanation of the minima is that insects are oriented along the radial at those directions. But the difference in the minima is usually explained with the difference in “head-on” and “tail-on” cross sections of scatterers. For STAR radars, a second mechanism for such differences arises: symmetrical ellipsoidal scatterers can produce the difference also due to in-phase and antiphase interferences of the scattered waves.

The model produces high values of the correlation coefficients at some azimuths. The *ρ*_{hv} values exceed 0.98 in the azimuthal interval from 265° to 310° (Fig. 6g). Such values are indicative of weather but *Z*_{DR} in this azimuthal interval are too high for raindrops (>5.5 dB at low *Z*) and an automated classifier recognizes these scatterers as biota. In the azimuthal intervals 180°–210° and 350°–30°, the values of *ρ*_{hv} exceed 0.96 and *Z*_{DR} < 3 dB. The latter parameters can belong to clouds and precipitation as well and insects could be classified as weather by automatic radar target classification algorithms.

The insects flew from an azimuth of 47°, which was obtained from the Doppler velocity. So, the insect track direction is 227°. The mean alignment angle on the horizontal plane is *φ*_{m} = 20°. Thus, the insects’ bodies are oriented at ~27° to their track direction and their heading deviates to the left from the direction.

Another case observed with KLWX on 27 June 2012 (case 3) is shown in Figs. 7a–d. The azimuthal profiles of radar moments calculated for an annular ring with radii of 22 and 28 km are depicted in Figs. 7e–h with the blue lines. All radar moments exhibit asymmetry. Clipping of *Z*_{DR} values in the azimuthal interval from 125° to 170° is apparent.

A sufficiently good agreement between the radar and model profiles is achieved at model parameters shown in the sixth and seventh columns of Table 1. One can see a good match of the *Z*_{DR} and *ρ*_{hv} profiles and deviations in the Φ_{DP} profiles in their negative slopes, which are similar to those in Fig. 6h. The model reflectivity profile deviates considerably from radar data, which signifies large variations in the number concentration of insects over azimuth.

The model parameters of the match are close to those obtained in case 2 except the orientations of scatterers on the horizontal plane *φ*_{m} = 67° and a wider standard deviation *σ*_{φ} of 25°. The track direction, obtained from the Doppler velocities, is 193°. So, the insects are oriented to their track direction at 54° and their heading deviates to the right from that direction. The track directions, the mean body orientations relative to these directions, and some other parameters for cases 1–3 are listed in Table 2.

Although the identity of insect scatterers in cases 2 and 3 is unclear, an insect species that is dispersing within Virginia and states along the Atlantic coast is a possible candidate. Kudzu bug (*Megacopta cribraria*) was detected about that time in Virginia (D. Ames Herbert, Virginia Polytechnic Institute and State University, 2014, private communication). Adults are 3.5–6 mm long and have a round body shape. Note that *b*/*a* is about 0.4 in cases 2 and 3, whereas in case 1 it is 0.15, that is, more prolate. The body of the monarch butterfly (Fig. 2) is more prolate than that of the kudzu bug, which qualitatively corresponds to the difference in the radar-retrieved width and length of 0.15 and 0.4, respectively (Table 1). To make quantitative comparisons, reference laboratory radar measurements are required.

## 4. Discussion

A model for scattering of electromagnetic waves by insects has been developed herein for radars with the STAR polarimetric configuration such as the WSR-88Ds. Insect wings do not contain a significant amount of water, so their contributions to the returned signals are ignored in the model. An insect is represented with a prolate spheroid that approximates its body. The model produces asymmetric polarimetric echo patterns observed with the WSR-88Ds on PPI displays. The strength of asymmetry is governed by the axis ratio of insects’ bodies, orientations of the bodies relative to the radar beam, and radar differential phase on transmit. Depolarization of the horizontally and vertically polarized waves and their in-phase and antiphase interference with the main scattered waves cause the azimuthal asymmetry in radar patterns. The *Z*_{DR} measured with STAR and alternate polarization radars in insects can differ significantly.

The Rayleigh scattering approach is capable of reproducing the main features of radar patterns in considered cases. By matching the model output with the radar azimuthal profiles of differential reflectivity, the differential phase, and correlation coefficient, the axis ratios of scatterers and the parameters of their orientations can be estimated. The developed model allows for estimating the mean orientation angles of insects and the spread in orientations. The model assumes one insect species oriented at the same angle relative to the track direction for all azimuths.

The azimuthal *Z*_{DR} and *ρ*_{hv} profiles have been matched well (Figs. 5b,d, 6f,g, and 7f,g). The model Φ_{DP} profiles in case 1 is in good agreement with radar data (Fig. 5c), whereas in cases 2 and 3 (Figs. 6h, 7h) the model profiles deviate noticeably from the radar data. Our attempts to obtain a better match by changing the dielectric permittivity of insects, introducing a spread into axis ratios, and considering non-Rayleigh scattering were unsuccessful. The source of this discrepancy in these cases remains unclear. The presence of various insect species in the radar volume could be the cause. Another cause could be due to imperfect ellipsoids that approximate the insects’ bodies in the model. The difference in head-on and tail-on scattering (DR2012, 55–57) should be studied. Also, the possible influence of the propagation differential phase should be studied in more detail.

The model allows for estimations of four orientation parameters: mean angle *φ*_{m} in the horizontal plane, elevation *θ*_{m} of the scatterers relative to the horizon, and the standard deviations *σ*_{φ}, *σ*_{θ} in these two angles, respectively. Parameters *φ*_{m} and *σ*_{φ} have been studied extensively (DR2012, chapters 7 and 10); we have not found radar studies on *θ*_{m} and *σ*_{θ}, although it is known that many species of insects fly with their bodies tilted in the horizontal plane (Dudley and Ellington 1990; Fischer and Kutsch 2000; Taylor 2001; Willmott and Ellington 1997; Achtemeier 1991).

The main contributor to the scattering properties of an insect is its abdomen, which occupies a part of the insect’s body. This could be a reason that the Rayleigh approximation works fairly well at X frequency band (wavelength ~3 cm; Riley 1985; Hobbs and Aldhous 2006). Some polarization measurements have been made at X band (Wolf et al. 1993; Hobbs and Aldhous 2006). We are not aware of such measurements at S band, and we are not sure that the X-band results can be applied to S band because insects at X band may not be Rayleigh scatterers. To be applicable to the radar observations we discuss herein, the differential phase should be measured along with the radar cross sections.

The radar differential phase on transmit *ψ*_{t} strongly affects the amplitudes of azimuthal profiles of the polarimetric variables. So, it is desirable to know this phase. It is possible to estimate *ψ*_{t} from the radar data as it is done in sections 3c and 3d herein, but such an estimation has an ambiguity of 180°. For accurate retrievals of orientations of insects, phase *ψ*_{t} should be known with an accuracy of about ±5°. The system differential phase *ψ*_{sys} is measured in the edges of weather echoes nearest to radar. It is not simple to measure *ψ*_{r} and *ψ*_{t} separately. Phase *ψ*_{sys} is measured in the WSR-88Ds, but *ψ*_{t}, which is important for insect observation, is unknown.

The presented results also demonstrate that echoes from bioscatterers should not be used for comparing reflectivity factors from adjacent radars. To verify the reflectivity calibration in the adjacent WSR-88Ds, reflectivity factors are compared in areas common for the radars. Such areas should be chosen in precipitation not in echoes from bioscatterers, where the reflectivity factors depend on the viewing angle (e.g., Figs. 5a, 6e). So, measured reflectivity factors can be different for adjacent radars even though they scan the same area. This conclusion is also applicable to differential reflectivity; for example, Figs. 5b, 6f, and 7f exhibit different *Z*_{DR} at different viewing angles. Measured *Z*_{DR} is also depends on the differential phase on transmit that can be different in adjacent radars.

## Acknowledgments

The authors thank the anonymous reviewers for their constructive comments, which improved the presentation. Funding for this study was provided in part by NOAA’s Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce.

### APPENDIX A

#### Calculations of the Moments of *φ* Distribution

*φ*

In this appendix, the mean values of trigonometric moments of the *φ* distribution in (6) are calculated. The value of 〈*P*_{h}〉 is given by (8) and depends upon , , and . The mean is given by (12). To obtain 〈*A*〉 and 〈*A*^{2}〉 in (12), the mean values 〈sin^{2}*φ*〉 and 〈sin^{4}*φ*〉 have to be calculated. The powers of the trigonometric functions can be expressed via functions of multiple arguments. Calculate first the moments 〈sin*mφ*〉 using distribution (6) for small *σ*_{φ}:

We made the substitution *x* = *φ − φ*_{m} in the first line of (A1), enlarged the limits of integration to infinity in the third line assuming small *σ*_{φ}, and utilized a table integral [e.g., Gradshteyn and Ryzhik 1980, (3.896.4)]. The value of 〈cos*mφ*〉 is calculated analogously; the result is

The values of 〈sin^{2}*φ*〉 and 〈sin^{4}*φ*〉 can be obtained as follows:

So, the first term in 〈*P*_{h}〉 in (8) is expressed via (A2) and (A3). The second term in (8) depends on , which can be written as

The value of 〈*B*〉 depends on 〈sin*φ*〉 and 〈sin*φ* cos*φ*〉. The first moment was calculated in (A1), and the second moment can be represented as 〈sin*φ* cos*φ*〉 = 0.5〈sin2*φ*〉, which is obtained from (A1) also. The value of 〈*AB*〉 depends on 〈sin^{3}*φ*〉 and 〈sin^{3}*φ* cos*φ*〉. These terms can be represented as

which can be expressed via (A1).

The third term in (8) is = The value of 〈*B*^{2}〉 depends upon 〈sin^{2}*φ*〉, 〈sin^{2}*φ* cos*φ*〉, and 〈sin^{2}*φ* cos^{2}*φ*〉. The first function is obtained from (A3). The remaining two functions can be represented as

The value of 〈*P*_{υ}〉 is given by (9) and depends upon , , and . The third term was considered previously. The first term is

The values of 〈*C*〉 and 〈*C*^{2}〉 depend on 〈cos*φ*〉, 〈cos^{2}*φ*〉, 〈cos^{3}*φ*〉, and 〈cos^{4}*φ*〉. The first term was calculated in (A2) and the remaining terms can be represented as

and can be obtained from (A2).

The correlation function (10) depends upon , , , and . The first and third terms were obtained already. The second term is

### APPENDIX B

#### Calculations of the Moments of *θ* Distribution

*θ*

In this appendix, the mean values of trigonometric moments of the *θ* distribution are calculated. For the truncated Gaussian distribution (11), parameter *G* is obtained from normalization, defined as

where factor sin*θ* in the first integral is from the solid angle and we enlarged the integration limits to infinity in the second line of (B1) assuming small *σ*_{θ}. Angle *θ*_{m} should not be close to zero or *π*, which is satisfied for spheroids oriented nearly horizontally, that is, for *θ*_{m} close to *π*/2. Term *G* is obtained from (B1) as

Calculate 〈sin*mθ*〉 and 〈cos*mθ*〉 at small *σ*_{θ}:

and

The values of 〈*P*_{h}〉, 〈*P*_{υ}〉, and 〈*R*_{hv}〉 depend upon the mean values of *A*, *B*, *C*, given by (5c) and (5d), their squares, and products. These functions depend on the following mean values: 〈sin^{2}*θ*〉, 〈cos^{2}*θ*〉, 〈sin*θ* cos*θ*〉, 〈sin^{2}*θ* cos^{2}*θ*〉, 〈sin^{4}*θ*〉, 〈cos^{4}*θ*〉, 〈sin^{3}*θ* cos*θ*〉, and 〈sin*θ* cos^{3}*θ*〉. The seven first functions can be represented via functions of multiple arguments as it is done in appendix A. The last function can be written as

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