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Estimating animal densities in the aerosphereusing weather radar:To Z or not to Z?

PHILLIP B. CHILSON,1,2,� WINIFRED F. FRICK,3 PHILLIP M. STEPANIAN,1,2 J. RYAN SHIPLEY,4

THOMAS H. KUNZ,5 AND JEFFREY F. KELLY6,7

1School of Meteorology, University of Oklahoma, Norman, Oklahoma 73072 USA2Advanced Radar Research Center, University of Oklahoma, Norman, Oklahoma 73072 USA

3Department of Ecology and Evolutionary Biology, University of California, Santa Cruz, California 95064 USA4Center for Spatial Analysis, University of Oklahoma, Norman, Oklahoma 73072 USA

5Center for Ecology and Conservation Biology, Boston University, Boston, Massachusetts 02215 USA6Oklahoma Biological Survey, University of Oklahoma, Norman, Oklahoma 73019 USA

7Department of Biology, University of Oklahoma, Norman, Oklahoma 73072 USA

Citation: Chilson, P. B., W. F. Frick, P. M. Stepanian, J. R. Shipley, T. H. Kunz, and J. F. Kelly. 2012. Estimating animal

densities in the aerosphere using weather radar: To Z or not to Z? Ecosphere 3(8):72. http://dx.doi.org/10.1890/

ES12-00027.1

Abstract. Weather radars provide near-continuous recording and extensive spatial coverage, which is a

valuable resource for biologists, who wish to observe and study animal movements in the aerosphere over

a wide range of temporal and spatial scales. Powerful biological inferences can be garnered from radar data

that have been processed primarily with the intention of understanding meteorology. However, when

seeking to answer certain quantitative biological questions, e.g., those related to density of animals,

assumptions made in processing radar data for meteorological purposes interfere with biological inference.

In particular, values of the radar reflectivity factor (Z ) reported by weather radars are not well suited for

biological interpretation. The mathematical framework we present here allows researchers to interpret

weather radar data originating from biological scatterers (bioscatterers) without relying on assumptions

developed specifically for meteorological phenomena. The mathematical principles discussed are used to

interpret received echo power as it relates to bioscatterers. We examine the relationships among

measurement error and these bioscatter signals using a radar simulator. Our simulation results

demonstrate that within 30–90 km from a radar, distances typical for observing aerial vertebrates such

as birds and bats, measurement error associated with number densities of animals within the radar

sampling volume are low enough to allow reasonable estimates of aerial densities for population

monitoring. The framework presented for using radar echoes for quantifying biological populations

observed by radar in their aerosphere habitats enhances use of radar remote-sensing for long-term

population monitoring as well as a host of other ecological applications, such as studies on phenology,

movement, and aerial behaviors.

Key words: radar; aeroecology; bioscatter; phenology.

Received 2 February 2012; revised 4 May 2012; accepted 24 May 2012; final version received 11 July 2012; published 16

August 2012. Corresponding Editor: D. P. C. Peters.

Copyright: � 2012 Chilson et al. This is an open-access article distributed under the terms of the Creative Commons

Attribution License, which permits restricted use, distribution, and reproduction in any medium, provided the original

author and sources are credited.

� E-mail: [email protected]

v www.esajournals.org 1 August 2012 v Volume 3(8) v Article 72

INTRODUCTION

There is a long tradition of incorporating radartechnology into biological studies as a means ofobserving movements of airborne animals andquantifying their numbers in the lower atmo-sphere (Gauthreaux 2006). Much of this researchhas been conducted using radar systems thathave been specifically adapted for observationsof volant species (e.g., Bruderer et al. 1999,Harmata et al. 2003, Chapman et al. 2011,Alerstam et al. 2011). However, some investiga-tors have chosen to incorporate data fromoperational radar systems (those designed forroutine operation) into their biological studies.One example is the use of weather radar, such asthose deployed by national weather services, tostudy birds, bats, and insects (Gauthreaux et al.2008, van Gasteren et al. 2008, Dokter et al. 2011,Chilson et al. 2012, Kelly et al. 2012). Indeed,Gauthreaux (1970) began using weather radars inthe US for biological research not long after thesefacilities were established in 1959. Radar tech-nology has greatly advanced since that time, andit has become increasingly easier to access andprocess weather radar data with the advent offaster processing and data storage capabilities.Consequently, we are witnessing a rapidlygrowing interest in integrating radar data intobiological studies.

Weather radars are designed to providecontinuous observations over extended spatialdomains. Often several weather radar stationsare networked together to further extend thespatial coverage. For example, the US NationalWeather Service operates over 150 weathersurveillance Doppler radars (WSR-88D) withthe majority of these located in the continentalUS (Crum et al. 1998, Serafin and Wilson 2000,Kelleher et al. 2007). Collectively the network ofradars is known as NEXt generation RADar(NEXRAD) and data from this network are freelyavailable from the National Climate Data Center(NCDC) via the Internet. As such, researcherscan readily request radar observations andincorporate them into their studies of animalbehavior in the aerosphere (lower atmosphere)over a wide range of spatial and temporal scales(Chilson et al. 2012). Researchers now haveaccess to a wealth of data through numerousarchives: meteorological, climatological, topo-

graphical, land cover, land usage, and others.The effective utilization of these data requires theintegration of atmospheric science, earth science,geography, ecology, computer science, computa-tional biology, and engineering. This transdisci-plinary approach to further understanding ofbiological patterns and processes in the aero-sphere is known as aeroecology (Kunz et al.2008).

As with any cross-disciplinary research enter-prise, to effectively utilize the technology andunderstanding offered by other subject areas, onemust become familiar with their particularparlance and terminology. This is certainly truewhen integrating weather radar data into aero-ecology. To make optimal use of weather radardata for biological studies, it is important tounderstand how radio waves interact withbiological scatterers (bioscatterers) in the aero-sphere. For inferential studies of animal behaviorand movements, it is sufficient to be able todiscriminate bioscatter from weather signals andthen use the bioscatter to track the presence andmovements of animals (Horn and Kunz 2008,Buler and Moore 2011, Kelly et al. 2012).However, if the goal of a study is to quantifythe number of animals in the aerosphere (Russelland Gauthreaux 1998, Gauthreaux 1970, Liechtiet al. 1995, Gauthreaux and Belser 1998, Diehl etal. 2003, Nebuloni et al. 2008) then a morefundamental understanding of bioscatter is re-quired.

The intensity of backscattered signals reportedby weather radars is typically given in terms of aradar reflectivity factor. The radar reflectivity factorhas been specifically devised to facilitate inter-pretation of radar echoes received from precip-itation. Moreover, calculation of the radarreflectivity factor relies on several simplifyingassumptions. When dealing with radio wavescatter from biological entities (bioscatter), how-ever, a more meaningful and intuitive measure ofecho power is radar reflectivity. Under theappropriate conditions, radar reflectivity can bedirectly related to the number of animals presentwithin a sampled volume of space. Fortunately,reported values of the radar reflectivity factorfrom weather radars can be readily converted toradar reflectivity.

Our purpose is to describe mathematicalrelationships between bioscatterers and radar


CHILSON ET AL.

reflectivity based on conventional weather radarproperties and to illustrate how this remotesensing capability can be used when observingand measuring biological entities in the aero-sphere. We (1) discuss the relationship betweenthe radar reflectivity factor and radar reflectivity,(2) highlight some of the assumptions involvedwhen calculating these two parameters, (3)demonstrate why the radar reflectivity factor isnot well suited for biological studies, and (4)explain how to calculate radar reflectivity fromthe radar reflectivity factor. We then use resultsfrom a Monte-Carlo based radar simulation todemonstrate how much variance one shouldexpect when estimating radar reflectivity frombioscatter under various conditions.

Increased access to archived weather radarproducts has sparked interest in various biolog-ical uses of radar data (e.g., Russell andGauthreaux 1998, Gauthreaux and Belser 1998,Diehl et al. 2003, Horn and Kunz 2008, Buler andDiehl 2009, Dokter et al. 2011, Chilson et al. 2012,Kelly et al. 2012). Advanced investigations thatuse these remotely sensed data to quantifypopulation densities of animals in the aerosphererequire a rigorous understanding of the basicmathematical properties and inherent assump-tions in analyzing radar echo power. Severalradar tutorials for biologists already exist (e.g.,Bruderer 1997a, b, Gauthreaux and Belser 2003,Diehl and Larkin 2005, Larkin and Diehl 2012)and much of the mathematical content providedbelow in the Background Section can be found invarious papers and books that deal with radarhardware, radar signal processing, or the inter-pretation of weather radar signals (Battan 1973,Sauvageot 1992, Doviak and Zrni�c 1993, Rinehart2004). To clarify the mathematical properties ofradar data and demonstrate their utility forecological investigations, we extract and synthe-size those components most germane for analyz-ing bioscatter.

BACKGROUND

Simplified radar equationTo demonstrate the importance of the distinc-

tions between radar reflectivity and the radarreflectivity factor and their relationships tobioscatter, we examine the basic radar equation,which is used to calculate the power of the

backscattered electromagnetic radiation receivedby a radar. Radars transmit electromagneticradiation in the form of radio waves throughthe use of an antenna. The transmitted radiationinteracts with an object or objects and thescattered energy is received by either the sameantenna used for transmission (monostatic) or aseparate antenna (bistatic). The expected powerof the received radiation is calculated using thebasic radar equation. For the case of scatter froma single object located at a distance r (m) from amonostatic radar, the received power Pr (W) isgiven by a simplified form of the radar equation:

Pr ¼ Pt

G2k2r64p3r4

ð1Þ

where Pt (W) is the transmit power of the radar,G is the gain of the antenna, k (m) is thewavelength of the radio waves, and r (m2) isthe radar cross section (RCS) of the scatterer(Rinehart 2004). Antenna gain is basically ameasure of an antenna’s capacity to amplifysignal power (sensitivity) as a function oforientation and can be related to the radarwavelength and the effective area of the antenna.The RCS of an object is a measure of howreflective it is to radio waves of a givenwavelength and has units of area. RCS valuescan be different than the physical area of theobject intercepting radio waves. The value of rgenerally is a function of the scattering anglewith respect to incident angle and since alloperational weather radars are monostatic, thescattering angle is oriented in the oppositedirection of the incident angle for backscatter.

In the treatment that follows, we consider twovariants of the radar equation based on theunderlying density and distribution of theanimals sampled by the radar. In the first case,which we call discrete bioscatter, it is assumed thatthe received power Pr can be attributed to scatterfrom individual animals. We refer to the secondcase as distributed bioscatter. Here we assume thatthe sampled animals are uniformly distributed inspace and that they are sufficiently abundant tobe treated as a ‘‘continuum’’ of scatterers. Thesecases are not unique to bioscatter. It is common toconsider discrete and distributed scatter whenusing radar to study geophysical phenomenasuch as precipitation and ionized media (plasma)(Doviak and Zrni�c 1993). A thorough under-


CHILSON ET AL.

standing of the differences between discrete anddistributed bioscatter is needed when attemptingto quantify animal densities in the aerosphereusing radar. Although the focus of the followingdiscussion is predominantly on weather radar,the treatment applies to most radars used forbiological studies.

Radar equation for discrete bioscatterThe radar equation given by Eq. 1 is funda-

mentally correct; however, it does not explicitlyaccount for certain sensitivity effects introducedby the radar, such as those attributed to rangeand beam weighting as described below. For thecase of weather radar, the gain of the antenna(and therefore its sensitivity) is greatest along theradial component corresponding to the antennapointing direction and decreases as a function ofangle measured relative to that radial; ignoringthe effects of antenna side lobes, which is beyondthe scope of our discussion here. That is, thecontribution of a scatterer to the total receivedpower is weighted according to its angularlocation taken with respect to the pointingdirection of the antenna. The so-called one-waybeam weighting function can be expressedapproximately as

f 2ðh;/Þ ¼ exp � h2

2r2h

� /2

2r2/

!ð2Þ

where h and / are angles oriented along thehorizontal and vertical plane of the transmittedradio wave, respectively. These angles are mea-sured with respect to the antenna pointingdirection (ho, /o) (Probert-Jones 1962). Further-more, rh ¼ h1=

ffiffiffiffiffiffiffiffiffi2ln2p

and r/ ¼ /1=ffiffiffiffiffiffiffiffiffi2ln2p

with h1and /1 being the one-way half-power (3 dB)beamwidths in the horizontal and vertical planes,respectively. The half-power beamwidths areonly reference points; however, they are oftenused when discussing the extent of a radarsampling volume transverse to the beam direc-tion.

Weather radars do not transmit radio wavescontinuously, but rather send out pulses ofenergy as a means of determining the range ofdetected scatterers. Since the propagation speedof radio waves in the atmosphere is known,range can be determined from the measured timedelay between pulse transmission and reception

of the scattered energy. We refer to this distanceas ro. Owing to hardware constraints andlimitations in frequency allocation, radars mustoperate within finite bandwidths (frequencybounds), which dictate the minimum pulseduration allowed. Furthermore, best perfor-mance of the radar can be achieved when thereceiver’s filter is matched to the transmit pulse(Doviak and Zrni�c 1993). A matched filter isdesigned to maximize a radar’s signal-to-noiseratio. For a pulsed radar of a given bandwidthand corresponding matched filter, a simplifiedform of the one-way range weighting functioncan be written approximately as

WðrÞ ¼ exp �ðr � roÞ2

4r2r

" #ð3Þ

(Doviak and Zrni�c 1984). Here r is the magnitudeof the vector directed from the radar antenna to aregion of space being probed by the radar, rr ¼0.35 Dr, and Dr ¼ cs/2, with c being the speed atwhich the electromagnetic radiation propagatesthrough the atmosphere (approximately equal tothe speed of light in a vacuum) and s is theduration of the transmitted wave. The quantityDr is often referred to as the range resolution.Note that W(r) attains its maximum value at ro.

The backscattered power resulting from asingle bioscatterer having an RCS of r andlocation given by (r, h, /) with respect to theradar antenna is given by

Pr ¼ Pt

G2k2

64p3r4W2ðrÞf 4ðh;/Þr: ð4Þ

Therefore, the power received by a radar for asingle discrete bioscatterer (e.g., individual ani-mal) depends on its RCS value, range from theantenna, and angular location with respect to thepointing direction of the antenna. In the mostgeneral case, r also exhibits an angular depen-dence, which we are not explicitly consideringhere. Although we are able to determine ro andthe pointing direction of the antenna (ho, /o), theprecise location of the bioscatter within thesampling volume cannot be retrieved. The degreeof uncertainty in determining the bioscatterer’slocation is prescribed by W2(r) and f 4(h, /).

Typically, the bioscatter received by weatherradars is not attributed to a single animal butrather a collection of animals. If we consider the


CHILSON ET AL.

backscattered power for a collection of scatterers(e.g., group of animals aloft) then the receivedpower becomes

Pr ¼ Pt

G2k2

64p3

Xi

W2ðriÞf 4ðhi;/iÞri

r4i

ð5Þ

where the summation includes all bioscatterers.Although the summation is taken, in principle,over all bioscatterers, only those in the vicinity of(ro, ho, /o) will contribute appreciably to thereceived power. As for the case involving a singlebioscatterer, we cannot resolve the actual loca-tions of the individuals nor their correspondingcontributions to the total received power.

Radar equation for distributed bioscatterIf the bioscatterers are uniformly distributed in

space within the region being probed by theradar and the number of scatterers underconsideration is sufficiently large then we cansimplify Eq. 5 by introducing the concept of aradar sampling volume. Clearly a radar pulsetransmitted by an antenna will only effectivelyilluminate a finite region of space. A radarsampling volume produced by the beam andrange weighting functions given by Eq. 2 and Eq.3, respectively, can be expressed as

Vrad ¼ r2o

Zr

W2ðrÞdr

ZX

f 4ðh;/ÞdX ð6Þ

where X is the solid angle subtended by thecenter of the sampling volume. As the number ofbioscatterers being observed becomes large, wecan assume that they are uniformly distributedthroughout the sampling volume and thereforetreat the summation of discrete entities as anintegral over a continuum of volume scattermultiplied by the number density of the entitiesweighted by their individual RCS values andtheir ranges. This is referred to as the volume-filling assumption. In this case

Xi

W2ðriÞf 4ðhi;/iÞri

r4i

! 1

DV

Xvol

ri

r4o

" #Vrad ð7Þ

whereP

vol represents the summation over allscatterers within a unit volume DV and we haveused ri ’ ro. That is, the total number ofscatterers being considered is the number ofscatterers contained within a unit volume times

the radar sampling volume given by Eq. 6. Theeffect of the weighting functions are accountedfor by our definition of the radar samplingvolume.

The next task is to express Eq. 6 in ananalytic form. It was shown by Probert-Jones(1962) that

r2o

ZX

f 4ðh;/ÞdX ’pr2

oh1/1

8lnð2Þ ð8Þ

and it can easily be shown thatZr

W2ðrÞdr ¼ffiffiffiffiffiffi2pp

0:35Dr: ð9Þ

Therefore, the radar sampling volume is givenby

Vrad ¼0:35

ffiffiffiffiffiffi2pp

2ln2

pr2oh1/1Dr

4

� �ð10Þ

where the quantity in parenthesis is equivalentto truncated oval-based cone having diametersalong the horizontal and vertical axes of roh1and ro/1, respectively and a length of Dr. This isdepicted in Fig. 1. We should note that thefactor of

ffiffiffiffiffiffi2pp

0:35 resulting from the rangeweighting function shown in Eq. 10 is some-times included into the radar equation as a lossfactor caused by having a finite receiverbandwidth (Doviak and Zrni�c 1993).

As shown in Eq. 10, the radar samplingvolume increases with range. Consider a collec-tion of bioscatterers uniformly distributed inspace or at least uniformly distributed within alocalized region of interest. In this case we candefine a bioscatter number density Nbio as thenumber of bioscatterers per unit volume. Thenthe effective number of bioscatterers contributingto the receiver power in the radar equation issimply Vrad � Nbio. In the next section we discusshow a non-uniform distribution of bioscattererscan be treated.

Radar reflectivity and radarreflectivity factor

The cumulative backscattering cross sectionper unit volume is referred to as the radarreflectivity or sometimes simply the reflectivity g.The value of g is defined as


CHILSON ET AL.

g [1

DV

Xvol

ri ð11Þ

(Doviak and Zrni�c 1993, Rinehart 2004). Com-bining the equations discussed above we finallyarrive at a form of the radar equation useful forbiological applications

g ¼ 64p3r4o

G2k2Vrad

Pr

Pt

¼ 512ðln2Þp2r2o

0:35ffiffiffiffiffiffi2pp

G2k2h1/1Dr

Pr

Pt

: ð12Þ

This equation assumes a large number ofbioscatterers uniformly distributed within theradar sampling volume. If the bioscatter can bedescribed through a number density Nbio witheach animal having an RCS of r then the radarreflectivity simply becomes

g ¼ 1

DV

Xvol

r ¼ Nbior: ð13Þ

Therefore, under these conditions, it is possible tocalculate Nbio directly from the radar estimateof g.

The magnitude of the backscattered signal forweather radars is not reported directly as radarreflectivity but rather as a radar reflectivity

factor. Whereas radar reflectivity naturally lendsitself to applications involving the quantitativeanalysis of bioscatter, this is not the case for theradar reflectivity factor. The radar reflectivityfactor has been specifically developed for thestudy of precipitation and as such, containsmany assumptions that are customized for thisapplication, but could inhibit biological inter-pretability of radar signals.

In the beginning of the 20th century, GustavMie formulated a rigorous theory (Mie scatteringtheory) to describe the scatter of electromagneticwaves by suspended particles (Mie 1908). Tocreate a tractable theoretical framework, heassumed the particles to be spherical andcomposed of a dielectric material having acomplex permittivity (Mie 1908). If you assumethat the diameter of the particle D is smallcompared to the wavelength of the electromag-netic wave, e.g., D � k/16 (Doviak and Zrni�c1993), then

r ¼ p5

k4jKmj2D6 ð14Þ

where Km¼ (m2� 1)/(m2þ 2) and m¼nþ ij is thecomplex refractive index of the material. Here, n

Fig. 1. Illustration depicting the radar sampling volume discussed in the text. The one- way range and beam

weighting functions are represented by f 2(h, /) and W(r), respectively.


CHILSON ET AL.

is the conventional refractive index related to thephase speed of waves in a dielectric mediumcompared to its speed in a vacuum and j is theabsorption coefficient (Battan 1973, Doviak andZrni�c 1993). The condition described by Eq. 14 isknown as the Rayleigh approximation andrepresents a special case of the full Mie scatteringtheory. For weather radars, the Rayleigh approx-imation can typically be applied for precipitationparticles and in this case

g ¼ p5

k4jKmj2Z ð15Þ

where the radar reflectivity factor Z is defined as

Z [1

DV

Xvol

D6i : ð16Þ

For the case of precipitation, we can furtherassume that the dielectric material of the scatter-ers is water; however, the actual value that weassign to m depends on the water’s phase andtemperature and the wavelength of the radar.Since these properties of the water are not ingeneral known, values of the radar reflectivityfactor reported by weather radars are given asthe equivalent radar reflectivity factor Ze definedas

Ze ¼k4

p5jKmj2g ð17Þ

where g is obtained from Eq. 12. Here, we notonly assume that the Rayleigh approximationholds but also that all scatterers consist of liquidwater, resulting in jKmj2¼ 0.93, which is valid forwavelengths used by most weather radars. Thatis, the equivalent radar reflectivity value repre-sents the value that would be associated with theobserved backscatter if it were attributed tovolume-filling liquid precipitation in the Ray-leigh regime.

Because the radar reflectivity factor Z (orequivalent radar reflectivity Ze) can have such awide dynamic range, values are commonlyreported and discussed using a logarithmic scale

Z½dBZ� ¼ 10log10

Z

1 mm6=m3

� �: ð18Þ

Values of Ze for meteorological phenomenamight range from about 10–20 dBZ for light rainto 60–70 dBZ for severe weather events involving

hail. As we can see, the assumptions used toderive values of radar reflectivity factor havelittle relevance for quantitative studies of bio-scatter.

Reflectivity as a biological parameterUnlike the radar reflectivity factor Z, radar

reflectivity g is a quantity that can be directlyrelated to bioscatter. The units of g are inverselength, e.g., 1/m since we are dealing with thetotal effective scattering area per unit volume.There are far fewer assumptions associated withradar reflectivity as compared to the radarreflectivity factor. Fortunately, we can readilyconvert values of Ze to g if we know thewavelength of the radar that was used to makethe measurements.

Consider as an example an equivalent radarreflectivity factor of 15 dBZ reported for NEX-RAD. From Eq. 18 we see that 15 dBZ corre-sponds to a value of 32 mm6/m3 (or equivalently323 10�18 m3) in linear units. NEXRAD operatesat a wavelength of k¼ 0.107 m. Therefore, usingEq. 17 we find that Ze¼ 15 dBZ translates to g¼6.9 3 10�11/m. A more meaningful biologicalinterpretation of g results if we express thisquantity in units of square centimeters per cubickilometer (cm2/km3), a convention already usedby some (e.g., Dokter et al. 2011, Shamoun-Baranes et al. 2011). That is, a value of Ze ¼ 15dBZ as measured using NEXRAD corresponds tog ¼ 690 cm2/km3. A value of 690 cm2/km3 couldbe produced, for example, by about 86 willowwarblers per cubic kilometer (9 g and r¼ 8 cm2)or 345 song thrushes per cubic kilometer (70 gand r ¼ 2 cm2) or 17 woodpigeons per cubickilometer (500 g and r¼ 40 cm2), where the massand RCS values are taken from (Alerstam 1990).These RCS values were calculated using Miescattering theory for equivalent-mass spheres ofwater. That is, the scattering properties of ananimal are assumed to be equivalent to those of asphere of water having the same mass as theanimal (Eastwood 1967). Note that the RCSvalues given in the example do not increasemonotonically with mass or size. This is becausethe sizes of the equivalent-mass spheres fallwithin the resonant regime of the Mie scatteringtheory for NEXRAD (Martin and Shapiro 2007).We should also note that Mie theory does notaccount for the shape or orientation of a scatterer,


CHILSON ET AL.

since a spherical dielectric is assumed.Just as 1 mm6/m3 is used as a reference value

when reporting radar reflectivity factor, we coulduse 1 cm2/km3 as a reference value for g and thenplot or discuss reflectivity for bioscatter inlogarithmic units. That is, our g of 690 cm2/km3

would be 28 dB (calculated using 10 log10(g/g0)),where g0¼ 1 cm2/km3. For a given wavelength, itthen becomes easy to convert from a radarreflectivity factor in dBZ to radar reflectivity indB using

g½dB� ¼ Z½dBZ� þ b ð19Þ

where b ¼ 10log10(103p5jKmj2/k4) with k ex-

pressed in cm and jKmj2 ¼ 0.93 and the factor of103 is needed to account for unit conversion. Forexample, b ¼ 14.54 for k ¼ 10 cm (S-band), b ¼26.58 for k¼ 5 cm (C-band), and b¼ 35.46 for k¼3 cm (X-band), which are common values forweather radars. When using Eq. 19, one shouldideally use the exact wavelength of the radar,since the calculation of b is sensitive to the valueof k.

When calculating the radar reflectivity factorfrom radar reflectivity, it is assumed that thevolume-filling assumption applies. If this is notthe case, then g calculated using Eq. 12 must beinterpreted as the weighted sum of bioscattererswithin the radar sampling volume normalized bythe radar sampling volume:

g! 1

Vrad

Xi

W2ðriÞf 4ðhi;/iÞri: ð20Þ

As we discussed earlier, under these conditionsthere is no means of knowing the locations of thebioscatterers within radar sampling volume ortheir corresponding weights, which results insome inherent patterns in the variance aroundthe estimate g even if r and Nbio are constant. Toquantify this relationship we present a Monte-Carlo simulation study of these patterns invariation and interpret the results relative toour ability to quantify the density of bioscatterersin the aerosphere.

METHODS

Radar simulator for bioscatterersIf the radar is sampling a sufficiently large

number of bioscatterers, then the volume-fillingassumption allows us to simply relate the

bioscatter to the product of a number densityand radar sampling volume. Although thevolume-filling assumption greatly simplifies theradar equation and makes it possible to quanti-tatively utilize bioscatter observations, the ques-tion arises: what is sufficiently large? Moreover,what are the consequences of assuming volume-filling even though the number of sampledbioscatterers is not sufficiently large or notuniformly distributed?

To address these questions we developed asimple radar simulator to assess uncertainties inestimating radar reflectivity from bioscatter. Thesimulator is written in MATLAB and utilizes aMonte Carlo simulation approach (Eckhardt1987) to account for variability in the locationsof bioscatterers. The primary scattering routine inthe simulator calculates backscattered power Pr,radar reflectivity g, and effective radar reflectiv-ity factor Ze for an assumed radar configurationand specified values of the range ro, bioscatternumber density Nbio, and radar cross section r.This routine operates as follows:

� Get assigned input parameters related to thetype of radar being simulated and thebioscatterers to be observed (discussed be-low and in Tables 1 and 2).

� Based on the input parameters and for eachassigned value of range ro, create a one-waybeam weighting function f 2(h, /) (Eq. 2) andone-way range weighting function W(r) (Eq.3).

� Using the weighting function, define a radarsampling volume (Eq. 6).

� Create a domain within the simulator basedon the radar sampling volume. Whereas theradar sampling volume theoretically extendsover all space, bioscatterers located far fromits center are not expected to contributeappreciably to the collective bioscatter. Thesimulator domain is defined such that itencompasses values of r, h, and / largeenough to allow f 4(h, /) and W2(r) to achievevalues of �12 dB or greater.

� Randomly populate the simulator domainwith bioscatterers consistent with a specifiednumber density Nbio and assign each with aradar cross section. The same value of r isused for all bioscatterers.

� Calculate the backscattered power from the


CHILSON ET AL.

sum of the range- and beam-weightedcontributions from all bioscatterers withinthe simulator domain (Eq. 5).

� Calculate radar reflectivity (Eq. 12) andeffective radar reflectivity factor (Eq. 17).

The radar simulator as a whole is designed tocall the primary scattering routine repeatedlywith different values of ro and Nbio. An ensembleof samples is generated by running the simulatorrepeatedly using the sample input parameters.The only aspect of the simulator that changes inthis case is the randomly assigned locations ofthe bioscatterers.

Simulator setup and implementationThe simulator was configured according to the

technical specifications for one of the WSR-88Dsthat make up the NEXRAD network. A partiallisting of the technical specifications of a WSR-88D is provided in Table 1. Parameters used inthe simulation are indicated in bold. The simu-lator developed for this study focuses on thedistribution and number density of the bioscat-terers. It does not include the effects of systemnoise, interference, or other factors that limit thebioscatter detectability. If this were the case thenthe gain, transmit power, bandwidth, and min-imum detectable signal would need to beconsidered. More information on the WSR-88Dsand NEXRAD can be found in (Doviak and Zrni�c1993, Committee on Weather Radar TechnologyBeyond NEXRAD 2002, Rinehart 2004).

Having established the radar specifications, wenow consider parameters used by the simulatorthat describe operation of the radar, the bio-scatterers, and ensemble averaging. These values

are summarized in Table 2. Weather radarstypically scan the environment using severalsettings for the antenna elevation angle. Sincemost bioscatterers are located near the Earth’ssurface, we are only simulating an elevationangle of 0.58, which is a common value for thelowest elevation scan for NEXRAD. As discussedabove, the simulator steps through a series ofvalues of range ro. The values used to initiate theloops to generate ro for the present study areprovided in Table 2. The combination of theantenna elevation angle and range is used todetermine the height above ground of thesampling volume.

When calculating the height of the radar beamabove the Earth’s surface, the 4/3 Earth radiusmodel was used (Doviak and Zrni�c 1984). Thismodel takes into account the effects of refractionbased on a standard model of the Earth’satmosphere. Shown in the upper panel of Fig. 2is a depiction of the beam geometry for a WSR-88D scanning at a 0.58 elevation angle. The upperand lower bounds for the beam are based on the3-dB beam width of the radar. See Table 1.

We next consider those parameters used tocontrol the bioscatterers and the ensembleaveraging. As for the range ro, the simulatorsteps through a series of values of the bioscatternumber density Nbio. Values of Nbio are given interms of the number of bioscatterers containedwithin a one cubic kilometer volume. That is,Nbio¼ 100/km3 designates 100 animals per cubickilometer. From Nbio we can then calculate thenumber of bioscatterers within a radar samplingvolume using

nbio ¼ NbioVrad: ð21Þ

In the lower panel of Fig. 2 we show nbio as afunction of range for various assumed number

Table 1. Typical technical specifications for the WSR-

88D.

Parameter Value Unit

Antenna diameter 8.53 mBeam width (3-dB, one-way) 0.96 degAntenna gain 45 dBWavelength 10.7 cmTransmit power 1000 kWPulse width 1.57 lsReceiver bandwidth 0.63 MHzMinimum detectable signal �113 dBmRange gate spacing 250 m

Note: Parameters used in the simulation are indicated inbold.

Table 2. Parameters used in the radar simulator.

Parameter Value Unit

Antenna elevation angle 0.5 degMinimum range 10 kmMaximum range 200 kmRange step size 5 kmMinimum number density 50 1/km3

Maximum number density 250 1/km3

Number density step size 50 1/km3

Radar cross section 10 cm2

Number of simulated realizations 5000 ...


CHILSON ET AL.

densities. The plot corresponds to the specifica-tions for a WSR-88D with a pulse width of 1.57ls. The selected values of Nbio are representativeof migrating or foraging birds and bats (Bruderer1999, Dokter et al. 2011). Much larger valueswould be needed to represent number densitiesfor insects or during some emergences ofroosting birds or bats.

For the present analysis, a single value of r isused for all calculations and it is assumed that rhas no angular dependence. Representative RCSvalues for birds reported in the literature includer ¼ 11 cm2 (Dokter et al. 2011) and r ¼ 15 cm2

(Diehl et al. 2003). Here we use r¼10 cm2, whichrepresents a small passerine songbird or bat. Thesimulator was run 5000 times for each realizationof the different ranges and number densitiesshown in Table 2 to create an ensemble of results

for statistical calculations. For each individual

realization of the Monte-Carlo type simulation

described we present means and standarddeviations calculated for both the radar reflec-

tivity (Eq. 12) and effective radar reflectivityfactor (Eq. 17).

Effects of non-uniform distributionof bioscatter

We briefly consider the case of non-uniformly

distributed bioscatterers in the aerosphere andhow non-uniformity might impact our simula-

tion and interpretation of radar reflectivity.

Whereas it may be safe to assume that bioscat-terers are uniformly distributed horizontally

within the domain of a radar sampling volume,this is less likely to be the case along the vertical

Fig. 2. Upper panel: Vertical cross-section of the radar beam at an elevation angle of 0.58 calculated using the 4/

3 Earth radius model. The displacement from the lower to upper edge of the sampling domain for a given range

ro as defined by the 3-dB beam width is given by ro/. See Fig. 1. Lower panel: Number of bioscatterers contained

in a radar sampling volume as a function of range. Curves for five different assumed number densities of

bioscatterers are shown. Values of Nbio are given as bioscatterers per cubic kilometer. That is, 50 bioscatterers per

cubic kilometer is 50/km3. These calculations are valid for NEXRAD and for a transmitted pulse width of 1.57 ls.


CHILSON ET AL.

extent. Clearly the number of animals in theaerosphere must decrease with height. Moreover,their distribution in height can vary dependingon wind conditions, time of year, location, and soforth (Bruderer and Liechti 1998, Dolbeer 2006,Schmaljohan et al. 2008, Bruderer et al. 2008,Buler and Diehl 2009, Dokter et al. 2011).Therefore Nbio should ideally be treated as afunction of height. However, it is neither feasiblenor possible in the present study to fully accountfor the wide range of height variability in Nbio.As an alternative, we present a means of partlyaccounting for this height dependence.

Let us reconsider the effective radar samplingvolume. Earlier we showed that the radarsampling volume is tapered in space as a resultof the beam and range weighting functionsthrough Eqs. 2 and 3, respectively. For the sakeof mathematical expediency, let the combinedeffects of these functions be expressed in Carte-sian coordinates as a one-way, three-dimensionalvolume weighting function Wvol (x, y, z) such that

Vrad ¼Z ‘

�‘

Z ‘

�‘

Z ‘

�‘

W2volðx; y; zÞdxdydz: ð22Þ

This is equivalent to Eq. 6. As before, nbio ¼NbioVrad provided Nbio is uniformly distributedand the volume-filling assumption applies. IfNbio is not uniformly distributed in height thenthe effective number of bioscatterers containedwithin a radar sampling volume can be ex-pressed as

nbio ¼Z ‘

�‘

Z ‘

�‘

Z ‘

�‘

W2volðx; y; zÞNbioðzÞdxdydz ð23Þ

where Nbio(z) is now written explicitly as afunction of height (z).

To account for the height dependence ofNbio wecan assume that the distribution of bioscatterers inthe aerosphere can be modeled using a normal-ized probability distribution function (pdf ) p(z)such that Nbio(z) ¼ Nop(z), where No is a scalingfactor. The quantity No has units of Nbio and canbe adjusted to reflect temporal variations in thebioscatter. Using this convention, we can write

nbio ¼ No

Z ‘

�‘

Z ‘

�‘

Z ‘

�‘

W2volðx; y; zÞpðzÞdxdydz

¼ NoV 0rad

ð24Þ

where

V 0rad ¼Z ‘

�‘

Z ‘

�‘

Z ‘

�‘

W2volðx; y; zÞpðzÞdxdydz: ð25Þ

That is, using Eqs. 24 and 25 we can treat thenumber density of bioscatterers as a ‘constant’ andaccount for the height dependence by modifyingthe effective radar sampling volume.

As an example, consider a collection ofbioscatterers that are log-normally distributedin height. In this case the pdf is expressed as

pðzÞ ¼ 1

zffiffiffiffiffiffiffiffiffiffiffi2pr2p exp �ðln z� lÞ2

2r2

" #ð26Þ

where l and r are the mean and standarddeviation of the underlying normal distribution,respectively. Using values No¼ 100000/km3 , r¼0.8, and l¼ 6.34 for the pdf shown in Eq. 26 anda uniform distribution along the horizontalextent, we have generated locations of bioscatterswithin a 1 km 3 1 km 3 1 km box. This can beseen in Fig. 3. The chosen value of No corre-sponds to a vertically integrated value of 100bioscatterers per km2 and the values of l and rresult in a mode of 300 m for the pdf. Thesevalues were motivated in part by the resultsshown in Fig. 3 of Dokter et al. (2011). Forreference, we show upper and lower bounds ofthe two-way beam pattern for NEXRAD (definedby the 3-dB points) that would result using a 0.58

elevation angle and assuming that the center ofthe box is located 50 km and 100 km from theradar.

When deriving the proposed method toaccount for Nbio(z) it has been implicitly assumedthat the volume filling assumption applies. Aswe have already discussed, when that assump-tion breaks down, a degree of uncertainty isintroduced into our estimates of radar reflectiv-ity. It should be noted that there is a distinctdifference between the actual weighting func-tions used for the radar sampling volume and theproposed weighting function to account forNbio(z). Bioscatterers further from the center ofthe radar sampling volume will actually contrib-ute less to the collective received power by theradar. They are being illuminated with lesspower from the radar and the radar is lesssensitive to the power they reflect. However, forthe case of Nbio(z), the actual numbers of animals


CHILSON ET AL.

are decreasing with height. We are simplyapproximating the effect by shaping the sam-pling volume with a height-dependent weightingfunction given by the pdf. The more scatterers inthe sampling volume, the better the approxima-tion becomes.

RESULTS

Results of the study show that estimated meanvalues of g calculated for all ranges from theradar approach the expected values used in thesimulation, demonstrating the feasibility of usingmean g to estimate animal densities. However,the variance of the estimates exhibits a notablerange dependence. A range dependence isanticipated since the radar sampling volumeincreases with increasing distance from the radarand we have assumed a constant number density

of bioscatterers in the simulation. This effect isillustrated in Fig. 4 by plotting the mean, median,and confidence interval (CI) of the mean fromour estimates of the radar reflectivity obtainedfrom the simulator. The expected value of radarreflectivity in this case is 1000 cm2/km3. Sincenumber density of bioscatterers is fixed, sam-pling volumes at greater ranges encompass alarger number of bioscatterers. Consequently, thelikelihood of the volume-filling assumption beingfulfilled for a given range improves and thestandard deviation of the estimate of reflectivitydecreases. Keep in mind that Nbio is being heldconstant. To account for a potential decrease Nbio

with height, one could use Eq. 24 to adjust theeffective size of the radar sampling volume.

Note that the confidence intervals shown inFig. 4 become increasingly asymmetric about themean with decreasing range from the radar.

Fig. 3. Left panel: A depiction of the positions of bioscatterers with an assumed uniform distribution in space

horizontally and a log-normal distribution in height. Also shown are the upper and lower bounds of a radar

beam for a radar located at 50 and 100 km away. Right panel: A histogram height dependence in bioscatter

number density for the case shown in the left panel. The bin sizes in the histogram are 20 m. The assumed pdf

(sampled at 1-m intervals) is shown for comparison.


CHILSON ET AL.

Moreover, the median of the estimated radarreflectivity correspondingly decreases. Theseeffects are indicative of a transition from havinga few discrete bioscatterers within a radarsampling volume to the case of distributed (andpossibly volume-filling) bioscatterers. In theformer case, Eq. 5 should be applied, whichpredicts that the received power exhibits a 1/r4

dependence in range. For distributed bioscatter-ers, we expect a 1/r2 dependence as indicated inEq. 12. Therefore when the number of bioscat-terers is small, estimates of radar reflectivity willbe biased low for most samples (decreasingmedian); however, some of the samples willresult in an overestimation of g, e.g., if a singlebioscatterer is located at or near the center of theradar sampling volume. As the number ofbioscatterers in a sampling volume increases,estimates of the median approach the mean andthe confidence intervals become more symmetric

about the mean, which can be interpreted as ameasure of the degree to which the volume-filling assumption is fulfilled. For the case shownin Fig. 4, this appears to occur at a range of about50 km.

The validity of the volume-filling assumptionand our ability to accurately estimate radarreflectivity is ultimately related to the numberof bioscatterers present within a given radarsampling volume, which is affected by both thesize of the sampling volume and the numberdensity of bioscatterers. We have shown that thestandard deviation in our estimates of radarreflectivity increases as the radar samplingvolume decreases, resulting in a break down ofthe volume-filling assumption. However, whathappens when the number density of bioscatter-ers increases? For a given sampling volume, thiscondition should increase the likelihood ofmeeting the volume-filling assumption. Howev-

Fig. 4. Upper panel: Estimates of the mean, median, and confidence interval of the mean for observed radar

reflectivity using an assumed number density of Nbio¼ 100/km3 and RCS of r¼ 10 cm2. Lower panel: Coefficient

of variation of the radar reflectivity estimates for different values of the number density of bioscatterers. Both

plots were calculated using the parameters provided in Table 1.


CHILSON ET AL.

er, the value of g increases with an increase in thenumber density (Eq. 13), which also leads to anincrease in the standard deviation. Therefore, weuse the coefficient of variation (CV; standarddeviation divided by the mean) of g whenexploring the impacts of sampling volume sizeand number density on the uncertainty of ourradar reflectivity estimates. A plot of the CV of gas a function of range and number density isprovided in the lower panel of Fig. 4. The CVdecreases both for increasing range and increas-ing number density.

The results presented in Fig. 4 are valid forNEXRAD; however, we would like to generalizethe analysis for other weather radar systems. Tothis end, we consider how CV of g varies inresponse to the number of bioscatterers persampling volume (Eq. 21). For each of the datapoints shown in the lower panel of Fig. 4, wecalculated a value of nbio. A linear relationshipemerges when log10(CV of g ) is plotted versuslog10(nbio) as shown in the upper panel of Fig. 5.When fitted using a linear regression algorithm,we find that

CV of g ¼ 0:59n�0:5bio : ð27Þ

This result should be general according to themathematical treatment provided above. More-over, since g and Nbio are linearly relatedthrough r (Eq. 13), if r is constant, then we canuse Eq. 27 to calculate the STD of Nbio.

Based on the antenna characteristics andoperating parameters of a particular radar, thenumber of bioscatterers per sampling volumenbio can easily be found for different assumedvalues of the animal number density Nbio usingEq. 10 together with Eq. 21. An example of suchcalculations for NEXRAD is shown in the lowerpanel of Fig. 5. Having estimated nbio, Eq. 27 canthen used to find the corresponding value of CVfor g. Although Nbio is not known from the radarobservations alone, it can be estimated using thecalculated value of g and using an assumedvalue of r.

DISCUSSION

Past efforts to quantify aerial densities basedon radar observations have typically relied oncomparing relative values of Z (Horn and Kunz2008, Buler and Moore 2011) or used linear

regression to calibrate densities estimated fromindividual bioscatterers from marine radar toreported values of Z from NEXRAD installations(Diehl et al. 2003, van Gasteren et al. 2008, Bulerand Diehl 2009). While these efforts have proveduseful for answering ecological questions incertain contexts (Russell et al. 1998, Bonter et al.2009), they have limited generality and rely onthe radar reflectivity factor, which assumesproperties inherent to meteorological entities.Using a ‘first principles’ approach, we show thatg relates more directly to biologically derivedradar signal and therefore should be used as thebasis for interpreting densities of biologicalentities from radar signals in future efforts.Transformation from Z to g is easily accom-plished (Eq. 19) and should not impede futureefforts. We demonstrate the mathematics ofinterpreting received echo power in terms ofdensity of bioscatterers and examine the rela-tionships among measurement error and thesebioscatter signals. We also present a frameworkfor using radar echoes for quantifying biologicalpopulations observed by radar in their aero-sphere habitats. We argue that this frameworkwill enhance use of radar remote-sensing forlong-term population monitoring as well as ahost of other ecological applications, such asstudies on phenology, movement, and aerialbehaviors. Advances in radar aeroecology willbe facilitated by a solid understanding of theradar equation in its different forms and thevarious assumptions involved when derivingthem.

One of the limitations of using radar data forquantifying population densities and its subse-quent ability for long-term population monitor-ing is lack of sufficient understanding of themeasurement error associated with reportedreflectivity values for a given radar samplingvolume and how that measurement error isrelated to the number of scatterers within agiven volume at different ranges from a radar(Horn and Kunz 2008). While these relationshipshave been considered in detail for meteorologicalscatter, how such radar theory applies tobiological entities is in its nascency. Typically,the radar equation is applied to one of two cases:echo power comes from a single entity or echopower is from a large collection of entities forwhich the volume-filling assumption is valid. For


CHILSON ET AL.

the case of bioscatter, neither of these conditionsmay apply, making an interpretation of echopower difficult. A confounding measurementerror arises from uncertainties in the RCS valuesof the animals observed. One does not typicallyknow a priori the type of animal present.Moreover, shape and orientation affects theRCS of a biological entity. In the presenttreatment we have not considered these effects.

We have shown that estimates of radarreflectivity will inherently contain a certainamount of error whenever the volume-fillingassumption does not apply. When volume-fillingdoes apply, measurement error decreases withincreased densities and range from a radar (Fig.2). These results provide the first attempt at ageneral framework for understanding how tocompare relative values of radar reflectivity thatrepresent signal received from numbers ofanimals aloft. By explicitly accounting for error

and knowing the effects of range and density onmeasurement error, radar data can be convertedto number densities of animal populations withknown precision, which permits use for compar-ative studies and long-term monitoring of pop-ulations regularly observed with radar, e.g.,Brazilian free-tailed bat (Tadarida brasiliensis) orpurple martin (Progne subis) colonies (Horn andKunz 2008, Kelly et al. 2012).

Based on results from our Monte-Carlo basedradar simulator, we have been able to show howerror estimates of radar reflectivity are impactedwhen certain assumptions used to calculate g arenot valid. By introducing the coefficient ofvariation of g, we relate the expected variancein g to the number of bioscatterers nbio presentwithin a given radar sampling volume, whichprovides a basis for developing unbiased esti-mators for measuring the number of animalsaloft from radar observations. Estimates of nbio

Fig. 5. Upper panel: Coefficient of variation of the radar reflectivity estimates for different values of the number

density of bioscatterers shown with the resulting fitted line given by Eq. 27. The coefficient of determination for

the fit is given as R2. Lower panel: Reference plot valid for NEXRAD to be used when interpreting the result

shown in the upper panel.


CHILSON ET AL.

can be found in turn for a particular set ofmeasurements if the radar sampling volume Vrad

and the number of density of animals Nbio areknown. The former can be calculated if certainspecifications of the radar and how it wasoperated are known, which is generally the casefor most radars. The latter can be estimated usingobserved values of g if one can assume arepresentative value of r for the observations.Alternatively, one might use a representativevalue of Nbio for the particular biological phe-nomena being observed. In Fig. 6, we demon-strate how calculations described in this papercan be used to derive biologically meaningfulestimates of number densities of aerial organismsdirectly from radar products if associated pa-rameters such as the RCS of an organism are

known.

The practical applications of our results arebest demonstrated through a biological example.Brazilian free-tailed bats form large maternityaggregations in caves throughout the Americansouthwest during the summer, which are easilyobservable on NEXRAD. Two large coloniesreadily detected by NEXRAD when bats emergein the evening are Bracken and Frio Caves insouthern Texas. Frio Cave is located approxi-mately 60 km from a WSR-88D site (KDFX) andBracken is 31 km from another WSR-88D site(KEWX). To illustrate effects of range on CVs,assume that bats emerge from both caves withequal number densities (Nbio¼ 300/km3) that areuniformly distributed in space. At Bracken Cavewe expect 9.5 bats per sampling volume, which

Fig. 6. Diagram illustrating the workflow for calculating biologically relevant parameters from radar products,

where Ze is the equivalent radar reflectivity factor reported from NEXRAD, k is radar wavelength (e.g., 10 cm for

a S-band NEXRAD radar), r is the RCS for a given organism, ro is the range (distance) from the radar, h1 , /1 and

Dr are the horizontal and vertical the one-way half-power beamwidths and range resolution, respectively, g is

radar reflectivity, Nbio is the number of bioscatterers per unit volume, Vrad is the radar sampling volume, nbio is

the number of bioscatterers per radar sampling volume, and CV is the coefficient of variation associated with an

estimate of nbio.


CHILSON ET AL.

corresponds to CV of g ¼ 0.19 (Eq. 27). At FrioCave the expected number of bats per samplingvolume is 35 and CV of g ¼ 0.10. Therefore,estimates of population sizes based on radarreflectivity at Frio Cave will be more precise thanthose at Bracken Cave because of this rangeeffect. However, proximity to a radar stationconfers an advantage in that most biologicalactivity in the aerosphere occurs at loweraltitudes. At sites that are much greater than 90km from a NEXRAD station, biological entitiesare typically flying quite literally ‘‘under theradar’’ and therefore estimates of population sizemay be biased by limited detectability based onbeam height (Fig. 2).

In summary, we contend that radar reflectivityis a more meaningful measure of echo powerthan the equivalent radar reflectivity factorreported by weather radars when consideringbioscatter. There are clear advantages of workingwith weather radar data for aeroecologicalstudies. In addition to providing continuous datastreams with wide spatial coverage, users benefitfrom the efforts of the weather services tomaintain and calibrate the radars and preprocessthe received echo powers. This preprocessinginvolves (1) calculating and removing noise fromthe echo signal, (2) correcting the power toaccount for range squared effects, (3) calculatingeach sampling volume Vrad, and finally (4)calculating the effective radar reflectivity factorZe. However, we have shown how g expressed indB can be easily related to Ze expressed in dBZ(Eq. 19). To that end, it may be worthwhile toconsider a new dB value that specifically appliesto bioscatter. There are many examples of dBvalues that have special designators such as thosefor the radar reflectivity factor relative to 1 mm6/m3 (dBZ), RCS relative to one square meter(dBsm), power relative to one mW (dBm ordBmW), gain of an antenna relative an isotropicantenna (dBi), and so forth. Perhaps an appro-priate decibel unit of radar reflectivity frombioscatter relative to 1 cm2/km3 would be thedBg.

As we continue to draw upon weather radardata for the study of animal movements in theaerosphere and use them to derive useful andquantifiable biological parameters, it becomesincumbent upon us to draw inspiration fromclassic literature and ponder for a moment the

age-old message:To Z or not to Z? That is the question:Whether ‘tis nobler in the mind to sufferThe slings and arrows of outrageous assumptionsOr to take arms against a sea of inappropriate

nomenclatureAnd by opposing end them.When dealing with and interpreting weather

radar data for quantitative biological studies, wefeel that the answer to this question is clear: it isbetter not to Z.

ACKNOWLEDGMENTS

PBC and PMS were supported through internalfunding from the University of Oklahoma. JFK and JRSwere supported by NSF Grants IOS-0541740 and EPS-0919466. WFF was supported by NSF DBI-0905881.

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