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Journal of Atmospheric and Solar-Terrestrial Physics 80 (2012) 208–229

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Journal of Atmospheric and Solar-Terrestrial Physics

1364-68

doi:10.1

n Corr

E-m

Wayne.

elizabet

journal homepage: www.elsevier.com/locate/jastp

Infrasound production by bolides: A global statistical study

T.A. Ens a,n, P.G. Brown a, W.N. Edwards b, E.A. Silber a

a Department of Physics and Astronomy, University of Western Ontario, London, ON, Canada N6A 3K7b NRCAN, Canadian Hazard Information Service, Ottawa, ON, Canada

a r t i c l e i n f o

Article history:

Received 20 September 2011

Received in revised form

17 January 2012

Accepted 27 January 2012Available online 10 February 2012

Keywords:

Bolide

Energy

Shock

Infrasound

26/$ - see front matter & 2012 Elsevier Ltd. A

016/j.jastp.2012.01.018

esponding author. Tel.: þ1 5197652087.

ail addresses: [email protected] (T.A. Ens), pbrown

[email protected] (W.N. Edwards),

[email protected] (E.A. Silber).

a b s t r a c t

We have examined a dataset consisting of 71 bolides detected by satellite sensors, which provide

energy and location estimates, with simultaneous measurements of the same events on 143 distinct

waveforms. These bolides have total source energies ranging from 0.02 kt TNT equivalent yield to

� 20 kt and probable diameters of order a few meters on average. We find that it is possible to detect

large events with energies of � 20 kt or more globally. Infrasonic detections of these events for

stratospheric arrivals have ranges between 350–17,000 km and show clear wind-related amplitude

modifications. We find that our period–yield relations are virtually identical to that found from AFTAC

nuclear test data with the most robust period–yield correlation found for those events having multiple

station averaged periods. We have also found empirical expressions relating maximum expected

detection range for infrasound as a function of energy and low and high frequency cut-off as a function

of energy. Our multi-variate fits suggest that 12 yield-scaling is most appropriate for long range bolide

infrasound measurements with a distance scaling exponent of � 1:1 best representing the data. Our

best-fit wind correction exponent is a factor of � 3 smaller than found by previous studies which we

suggest may indicate a decrease in the value of k with range. We find that the integral acoustic

efficiency for bolides is Z0:01% with a best lower limit estimate nearer 0.1%. Finally, we conclude that

a range independent atmosphere implementation of the normal-mode approach to simulate bolide

amplitudes is ineffective at large ranges due to the large change in atmospheric conditions along

source-receiver paths.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Large meteoroids colliding with the Earth generate light, heatand shock waves as they pass through the atmosphere. The non-linear shock produced during hypersonic entry decays ultimatelyto an acoustic-gravity wave and may be detected infrasonically(Edwards, 2009) at large ranges. Infrasound refers to any lowfrequency sound waves in the atmosphere with frequencyroughly in the range 0.01–20 Hz depending on the exact condi-tions of the atmosphere (Beer, 1974; Blanc et al., 2009). The lowerfrequency cut-off corresponds to the Brunt–Vaisala frequency ofthe atmosphere while the top cut-off is roughly the lowerthreshold of human hearing. Infrasonic waves, like all other soundwaves, travel through the atmosphere at approximately 0.3 km/s,the exact value depending on the time varying properties of theatmosphere. There are many natural and artificial sources ofinfrasound, including volcanic eruptions, nuclear and chemicalexplosions, microbaroms generated by the interaction of ocean

ll rights reserved.

@uwo.ca (P.G. Brown),

waves, and meteors (Brachet et al., 2009). Since attenuation ofsound is proportional to the square of frequency (Bass et al.,1972), infrasound can travel for long distances without beingsignificantly attenuated, in some cases even for tens of thousandsof kilometers. This makes infrasound an ideal (and often the onlyavailable) method for non-local observation of meteor eventsoccurring over oceans or other remote locations.

Meteoroids may generate atmospheric shocks and henceinfrasonic waves in several ways. First, since the pre-atmosphericspeed at which meteoroid travel is greater than the speed ofsound, upon collision with the atmosphere a hypersonic shockwave is generated that travels approximately perpendicular tothe path of the meteor (Fig. 1(a))—this is termed a ballisticwave. Further, since the meteoroid speed is always greater than11.2 km/s due to the gravitational acceleration of the Earth aloneand thus much larger than the speed of sound, the shock conebecomes nearly parallel and a cylindrical line-source is a goodapproximation (Edwards, 2009) to this ballistic wave. A meteor-oid may also fragment in the atmosphere producing a largeincrease in its rate of ablation and depositing a large fraction ofits total kinetic energy over a short segment of its path. Thiscreates a nearly point source of sound waves propagating quasi-omnidirectionally (Fig. 1(b)). In the rare event that a meteoroid

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T.A. Ens et al. / Journal of Atmospheric and Solar-Terrestrial Physics 80 (2012) 208–229 209

impacts the ground further impact waves may be generated aswell (Fig. 1(c)), though this has only been definitively recorded inone instance, the Carancas impact event of September 15, 2007(Le Pichon et al., 2008).

The detailed theory relating source parameters (meteoroidsize, speed, source height, etc.) to the infrasound signal detectedat the ground has been developed by ReVelle (1974) and sum-marized in ReVelle (1976). However, this detailed theory is mostapplicable at short ranges ðr300 kmÞ where the meteor infra-sound signal is relatively unmodified by the atmosphere. Foranalytical tractability and simplicity, the existing theory does nottake into account intrinsic signal dispersion, effects of atmo-spheric turbulence, caustics or geometrical dispersion of thesignal caused by spatial or temporal changes in atmosphericparameters. As a result, at large ranges, the meteor infrasoundsignal is heavily modified due to the conditions in the atmospherealong the propagation path; often these conditions are not knownwith sufficient accuracy to correct for atmospheric effects.

Fig. 1. A schematic of a meteoroids passage through the atmosphere. (a) Generation

of hypersonic shockwaves perpendicular to the meteor trail. (b) An explosive

fragmentation event generating a point source of infrasonic waves. (c) Occasionally

an impact with the Earth occurs.

Fig. 2. A global map of the locations of the IMS infrasound stations used in this stu

At present, inference of meteor source parameters based oninfrasound from bolides (highly energetic meteors) at largeranges must be done based on empirical correlations betweenprevious events with known source yields, wind fields and ranges.Such signals at large ranges are of interest for several reasons.First, entry of large bolides, which most often occur in areaswithout eyewitnesses or ground-based instruments, are onlydetectable on global scales using infrasound (and seismicallycoupled infrasound) or satellite (Tagliaferri et al., 1994) measure-ments. Correlation of large impacts with pre-impact imagery,such as was done for the asteroid 2008 TC3 (Jenniskens et al.,2009) as well as for the Villalbeto de la Pena fall of January 4th,2004 (Llorca et al., 2005; Trigo-Rodrıguez et al., 2006) is particu-larly valuable as it offers the possibility of linking astronomicalobservations of small near-Earth objects and fireballs withground-truth data (more details can be found in Edwards et al.,2006). Rapid identification of large meteoroid impacts can alsolead to airborne sampling of the associated debris cloud (Zolenskyet al., 1997) and additionally, when such large impacts occur,infrasound is often the only means both to validate that an eventhas occurred, to geolocate its impact point and provide anestimate of its energy (Brown et al., 2002b). Finally, in somecases, meteor infrasound may be useful for trajectory estimationand constraints on entry speed (Le Pichon et al., 2002).

Apart from defining source parameters for the meteor, under-standing and characterizing meteor infrasound is useful to poten-tially distinguish fireballs from other types of infrasoundproducing events. This follows from the general considerationthat in order to be useful for analysis, infrasound recordings mustbe identifiable with their sources which may be thousands ofkilometers away. A single instrumental signal recording, whilecontaining accurate pressure information, is difficult to correlatewith sources in the atmosphere, if no geographical location forthe event can be determined by other means. Providing the signalis recorded by an array of infrasound sensors spaced such that theaperture of the array is large compared to the infrasonic wave-length, multi-sensor analysis allows for the extraction of usefulsignal arrival information (Young and Hoyle, 1975) such as back

dy, as well as the non-IMS stations NVAR, SGAR, PDIAR, DLIAR, NTS and WSRA.

1000

100

10

1

0.1

0.01

0.0011 2 3 4 5 6 7 8 910 20 30 40

Period (s)

Ene

rgy

(Kilo

tons

of T

NT

Equ

ival

ent)

AFTAC(E<200kt)AFTAC(E<80kt)

Fig. 3. The empirical AFTAC period–yield relation relates the dominant observed

infrasonic period with source yield.

T.A. Ens et al. / Journal of Atmospheric and Solar-Terrestrial Physics 80 (2012) 208–229210

azimuth (the azimuthal angle of the arrival of the signal) andtrace velocity (the apparent velocity at which the wave propa-gates across the array). Further, two or more arrays detecting acommon signal may utilize cross-azimuth bearings for simplegeo-location (Brown et al., 2002b). As well, the range to infrasonicsources may also be broadly constrained from a single array usingstatistical estimates of the expected frequency content as afunction of range (Brachet et al., 2009).

The Comprehensive Nuclear Test Ban Treaty Organisation(CTBTO), an international organization set up to monitor fornuclear explosions world-wide, has implemented a worldwidemonitoring network, the International Monitoring System (IMS).The system comprises four technologies: seismic, hydroacoustic,infrasonic and radionuclide monitoring. The infrasonic portion of thesystem includes a proposed worldwide network of 60 infrasoundarrays, 43 of which are currently (as of early 2011) certified andoperational, located in countries around the world (see Fig. 2).

The network is designed to have at least two-station detectioncapabilities globally for a 1 kt TNT equivalent atmospheric nuclearexplosion (Christie and Campus, 2009). Due to the close analogbetween nuclear explosions and meteor events, this system iscapable of globally detecting large meteors, and currently providesmulti-station detections of most sizable bolides. As such, the IMS isan ideal source of infrasonic meteor data and allows a large datasetof energetic bolide events to be identified and analyzed as a singlepopulation for the first time.

With this in mind, in an earlier work, Edwards (2007) identi-fied bolide infrasound signals from IMS stations which were thencorrelated with satellite bolide detection, providing an indepen-dent estimate for source location and total energy. In whatfollows we build on this earlier work. Our goals are similar tothose of Edwards (2007), namely to establish empirical relationsbetween infrasound signal properties and bolide source charac-teristics, particularly yield.

2. Previous yield relations

There have been a number of empirical studies which havedeveloped relations between explosive yield and infrasonicamplitude and period using known explosive sources for calibra-tion. It is well known from blast theory (Sakurai, 1965) that forexplosions in the atmosphere, the amplitude derived yield scaleswith range following a power law of the form � 1

2 or � 13 power

depending on whether the wavefront is cylindrical or sphericalrespectively. These earlier studies have assumed such a powerlaw range dependence and then computed the relationshipbetween explosions of known yield and measured infrasonicwave properties (such as period or amplitude).

A commonly used empirical energy relation, the AFTAC (U.S.Air Force Technical Applications Center) period–yield relation,was developed from measurements of the dominant infrasoundperiod produced by ground-level nuclear explosions at rangesfrom 1300rRr8500 km. The regression to these data are givenby ReVelle (1997)

logW

2¼ 3:34 log t�2:58,

W

2r100 kt ð1Þ

logW

2¼ 4:14 log t�3:61,

W

2Z40 kt ð2Þ

where W is the source yield in kt of equivalent TNT ð1 kt TNT¼4:185� 1012 JÞ and t is the period in seconds of the observedinfrasonic signal at maximum amplitude (Fig. 3 displays theserelations).

Similar relations were found from AFTAC nuclear explosiondata relating source yield W, to infrasonic signal amplitude P

(measured in Pa) for a distance D, the range from source toreceiver in degrees (Clauter and Blandford, 1998)

log W ¼ 2 log Pþ2:94 log D�1:84 ð3Þ

Blanc et al. (1997) performed a comparable analysis on Frenchnuclear tests, resulting in the following relation between source-receiver range (R in km), yield (W in kt TNT) and signal amplitudeP (in Pa):

log W ¼ 2 log Pþ3:52 log R�10:62 ð4Þ

Similar empirical energy relations were derived using Sovietatmospheric nuclear explosion infrasound recordings from theInstitute for the Dynamics of the Geospheres in Moscow, Russia(Stevens et al., 2006). Here the dataset was split into downwindand crosswind returns with separate yield relations given by

log W ¼ 3:03 log Pþ3:03 log R�9:09 Crosswind ð5Þ

log W ¼ 3:03 log Pþ3:03 log R�10 Downwind ð6Þ

Another commonly used relation between yield, range andinfrasonic amplitude was developed using known explosionyields of ammonium nitrate fuel oil (ANFO) rather than nuclearmaterials (Mutschlecner and Whitaker, 2009). In this study itwas recognized that since infrasonic waves propagate through amoving medium, a correction must be applied based on windspeed. This produced

log W ¼ 1:49 log Awþ2:00 log R�4:18 ð7Þ

where W is the source yield in tons of ANFO equivalent, Aw is thewind corrected amplitude in microbars and R is the range in km.

The standard overpressure–distance curves for nuclear and highexplosive yields at low amplitudes suggested by the AmericanNational Standards Institute are (Eq. (6) in ANSI (1983))

Dp¼ 6:526W0:3667A R�1:1 p

p0

� �0:6333

ð8Þ

where Dp is the overpressure in Pa, WA is the yield in kt of TNTequivalent, R is the range in km, p0 is the air pressure at the point ofobservation and p is the ambient atmospheric air pressure at thesource, both in Pa.


While these chemical and nuclear explosions are analogues ofmeteors, it is not clear that meteors should result in identicalenergy scaling relations, as they occur both at higher altitudesand display directional sound generation due to the ballisticshock geometry (see Fig. 1(a)). These two characteristics aloneimply in some cases that sound generated from a range of heightsalong the meteor’s trail may be received simultaneously at anyone station. It is desirable, therefore, to independently generate aset of empirical relations between infrasonic signal properties andenergy for a suite of large meteors (bolides) having independentenergy estimates.

Such a study was undertaken by Edwards et al. (2006) andEdwards (2007). In order to do this, however, a means of ground-truth is required to calibrate meteor source energies since this isunknown for most meteor events. This was first accomplished inBrown et al. (2002a) using optical sensor satellite observations ofmeteors in the atmosphere to estimate total radiated energy. Anempirical wind correction factor was then introduced to take theeffects of atmospheric winds into account, following the proce-dure used in Mutschlecner and Whitaker (2009). The final rela-tions derived in that earlier (Edwards, 2007) study were

log W ¼3

bða�kvÞþ3 log R�

3

blog A ð9Þ

a¼ 3:3670:60, b¼�1:7470:24, k¼�0:0177 s=m for Wo7 kt

a¼ 2:5870:41, b¼�1:3570:18, k¼�0:0018 s=m for W47 kt

where a and b are regression coefficients, k is an empirical wind-correction constant and v is the average component of the windvelocity in m/s along the source to receiver wave propagationdirection in the stratospheric duct. Here A is the maximumobserved peak to peak amplitude of the wave. Similar resultswere obtained for maximum amplitude, integrated energy andsignal to noise ratios (see Fig. 4 for a comparison of the variousamplitude–energy relations discussed). The present study builds

1000

100

10

1

0.1

0.01100 1000 10000

Range (km)Stevens et al. (2006)-CrosswindStevens et al. (2006)-DownwindANSI (1983)Blanc et al. (1997)Edwards (2007)Mutschlecner & Whitaker (2009)Clauter & Blandford (1998)

Am

plitu

de (p

a)

Fig. 4. Comparison of AFTAC, French Nuclear, ANSI, Los Alamos HE and bolide

relations from Edwards (2007) showing observed infrasound amplitude with

range for an explosion of 1 kt yield (4.185�1012 J).

on this earlier work adding many more events and recordedwaveforms worldwide for analysis as well as optimizing themethods of analysis used.

As can be seen in all the above equations (Fig. 4), theserelations have a similar form; however, the exact values of theconstants vary between them for many complex reasons (seeReed, 1972 for a discussion of the need to rely on empirical, ratherthan purely theoretical approaches for long range amplitudedecay relations). Since these equations are linear in log-spacerather than linear space a small difference in a constant cantranslate into a large uncertainty in energy, so they must beapplied with care to obtain reliable information. In particular, thedetails of the fits (and weightings used) to establish the aboverelations are poorly documented and uncertainty bounds are notdefined.

The present work expands on Edwards (2007) by includinga larger dataset of satellite–infrasound bolide events (63 totalcompared to 31 total from Edwards, 2007) and employing a newapproach for objectively establishing the appropriate frequencybandpass to use for infrasound measurement. Further, it is knownthat since the atmosphere is a moving medium, waves travelingthrough it will be Doppler shifted (Morse and Ingard, 1987) so acorrection will be applied to determine whether this has asignificant effect on period measurements, a correction notexamined in Edwards (2007). The effects of wind on amplitudewill also be taken into account. Detailed uncertainty bounds forour relations using a multi-variate covariance approach to our fitswill be established. We will also examine the empirical characterof the frequency roll-off with distance and yield for bolide eventsto provide a discrimination tool for bolide infrasound signalassociation for those instances where range constraints areavailable. Finally, we examine the possibility of using a normal-mode approach to estimate bolide source heights, the remainingmajor uncertainty in bolide infrasound production, for individualbolide events under the assumption that most large bolides havedominant infrasound produced from fragmentation events occur-ring at discrete heights.

3. Theory and background

The detailed physics of the entry and ablation of largemeteoroids into planetary atmospheres is a complex topic wellcovered in the literature and will not be repeated in detail here.The interested reader is referred to Ceplecha et al. (1998) (andreferences therein) for more details. We briefly summarize onlythe major concepts pertinent to the present work.

As the meteoroid interacts with gas molecules in the Earth’satmosphere, energy and momentum is transferred between themeteoroid and atmosphere leading to light emission. It is nor-mally assumed that the luminosity associated with a meteor isproportional to the meteoroid kinetic energy-loss. The observedluminosity of the meteor as a function of time is known as thelight curve of the meteor and can be converted into a measure-ment of the source energy if the fraction of the total kineticenergy lost at any instant and radiated into the passband ofinterest is known (the luminous efficiency). For satellite measure-ments, the radiation efficiency (the fraction of the total energywhich is radiated across all frequencies, assumed to be in theform of a 6000 K blackbody) is given by Brown et al. (2002a)

tI ¼ 0:1212E0:115opt ð10Þ

where tI is the radiation efficiency and Eopt is the total radiatedenergy over the entire duration of the bolide event in kt of TNTequivalent. Eopt is found by integrating the light curve of themeteor (i.e. its radiant intensity as a function of time). Fig. 5

Fig. 5. An example of a satellite detected bolide light curve. The light intensity in the silicon sensor passband of the satellite is given as a function of time. This bolide

occurred on May 29, 1994 over the Southern U.K. The peak radiant intensity corresponds to an absolute astronomical magnitude of �21.5. Integration under the lightcurve

produces a total estimated energy for the event of 1 kt TNT equivalent.

�

R0

Fig. 6. (a) Ballistic shock cone of an object travelling at much greater speeds than

the speed of sound such as a meteor. The angle b is small for these events. (b) The

geometry is well-approximated by a cylinder with radius R0 and L being the trail

length of the meteor.


shows an example bolide lightcurve detected by satellite sensors.The total source energy of the bolide can then be calculated usingthis efficiency via

E¼Eopt

tIð11Þ

As shown in Edwards et al. (2006) the functional form of thisradiative efficiency has been validated on a limited number ofbolides where energy was independently determined by othermeans. The typical difference between the ‘‘ground-truth’’ energyand the total energy derived from satellite measurements wasfound to be 20–30%. Several recent bolide events having satelliteenergies and other independent energy estimates generallyshow similar agreement within their uncertainty bounds (e.g.Jenniskens et al., 2009; Welten et al., 2010; Klekociuk et al., 2005).

As meteoroids travel well above the local adiabatic soundspeed at all heights, they produce hypersonic shock cones whichare very close to being cylindrical in shape. Thus for the purposesof infrasound generation, the meteoroid ballistic wave can bemodeled as a cylindrical line source explosion (see ReVelle, 1976).While this cylindrical shock is always present, in some instances asudden, gross fragmentation of the object may occur as well. Suchevents are expected to produce quasi-spherical point sources ofacoustic radiation termed ablational shocks (Bronsthen, 1983).For the large bolides in this study, the latter mechanism isexpected to be common and given the sharp increase in energydeposition (and hence energy partitioned to the shock) duringsuch a point explosion (associated with a change in the blastradius) (ReVelle, 2005) it is reasonable to suppose that many (butcertainly not all) long range bolide infrasound detections could beassociated with these point source airbursts, particularly as manyof these will occur at lower altitude, often inside the stratosphericatmospheric waveguide.

The radius of the cylindrical line source, shown as R0 in Fig. 6,is physically the zone in which the atmosphere is very stronglyshocked by the meteoroid’s passage leading to non-linear wavepropagation. In this region, the wave is traveling at a speed muchgreater than that of the local sound speed and the energy of theshock wave is much higher than that of the surrounding atmo-sphere. Physically, the radius of this shock can be roughly definedas the radial distance from the trail where the energy density ofthe ambient surrounding atmosphere is equal to the energy

density released per unit length of the trail (Edwards, 2007).In other words, this radius is the distance at which all of thedeposited explosion energy per unit length would just equal theequivalent expansion work required by the shock to move thesurrounding atmosphere to this radius (Few, 1969).

Numerically, we can calculate R0 using line source blast wavetheory (Tsikulin, 1970) as

R0 ¼E0

p

� �1=2

ð12Þ

where E0 is the energy deposited per unit length along thecylindrical shock, and p is atmospheric pressure. For a meteor,the E0 term arises entirely due to collisions with atmosphericmolecules.

This blast radius R0 can be related to the fundamentalfrequency f0 of the bolide infrasonic wave via (ReVelle, 1976)

f 0 ¼cs

2:81R0ð13Þ

where cs is the local sound speed. This relation applies after thewave is no longer a highly non-linear shock but rather a weaklynon-linear wave.


At close enough ranges where the propagating wave is stillnon-linear, the dominant period of the wave is not constant (seeReVelle, 1976 or Edwards, 2009 for detailed accounts of theproperties of waves in the initial non-linear phases) however,once a transition from the weak shock regime to the linear regimeoccurs, the period remains essentially unchanged over the courseof propagation. The precise point at which this occurs is difficultto define and so is usually simply defined as the point after whichthe period of a sinusoidal wave will change by only 10% until itreaches the source. A detailed discussion of this topic is given inEdwards (2009). After this point, the period of the wave can beassumed to be a constant and is approximated by invertingthe fundamental frequency of the wave (Eq. (13)). This yieldsa particularly simple power law relation between energy andinfrasonic period, assuming atmospheric pressure is roughlyconstant.

At long ranges, we generally appeal to empirical measure-ments as described in the previous section. It is known that rangeand source yield can often be related by a cube-root type yieldscaling, known as Hopkinson scaling (Baker, 1973). ReVelle andWhitaker (1996b) have noted that the form of this scaling foroverpressure can be given by

Dp¼ Cp0

p

� �ðq�3Þ=3

R�qEq3 ð14Þ

where Dp is the overpressure, C is a constant, p0 is the pressure at thesource altitude, p is the pressure at the observation altitude, q¼1 inthe linear regime, 1:1rqr2 in the weak-shock regime and q¼3 inthe non-linear regime. For long distance bolide observations (whereq� 1�1:5) the height of the source is not known and pressure termsare held constant so that the relationship

Rs ¼R

Ec ð15Þ

describes how energy is expected to scale with range. The exponent,c, is generally taken to be between 1

2 and 13, with classical Hopkinson

scaling suggesting 13.

From the standpoint of the energetics of shock production, therate of energy loss by the meteoroid (through mass loss anddeceleration) to the atmosphere is the ‘‘source’’ for the initialenergy deposited in a cylindrical line source blast wave. At longranges, the bolide infrasound signal is usually assumed to behaveeffectively as if generated from a point source. This can meaneither that most of the energy is deposited in a short length oftrail (approximating a quasi-spherical point source)—a conditionfound to hold generally for very large bolide events (Hills andGoda, 1998) and/or that the range to the point of observation ismuch larger than the trail length. However, only a fraction of thetotal available energy goes into acoustic radiation. Using thebolide point source model, the total energy released by the bolidecan be expressed in terms of the perturbation pressure from theexplosive signal as detected at any infrasound station, via (Cox,1958; ReVelle and Whitaker, 1996a)

e¼ 2pR2ðIEÞ

Eopt

ð1�f Þ

ð1þ f Þf bR=250;000c

ð16Þ

where e is the acoustic efficiency, IE is the integrated acousticenergy flux of the signal at the receiver, Eopt is the satellite derivedenergy, R is the distance in which the blast wave has traveled inmeters and f is the fraction of incident energy reflected at eachinterface, taken to be the stratospheric duct. Note that we haveassumed that atmospheric density and the local sound speed areconstant over the time duration of the signal at the receiver andthat the maximum skip distance is constant at 250 km.

This expression applies to low altitude sources only underconditions of a near-surface temperature inversion. However,for the large bolides examined here, presuming similar heightsof burst apply, we may evaluate the above equation with theunderstanding that the resulting acoustic efficiencies may becompared in a relative sense only. The resulting values will becloser to a lower limit to the true integral acoustic efficiency. Thatthe most energetic bolides generating long range infrasoundhave comparable equivalent burst heights as was found byEdwards et al. (2006), justifies this latter approximation.

Previous estimates of bolide integral acoustic efficiency(ReVelle and Whitaker, 1996a) have ranged from 0.01% to nearly10% with the largest events having the smallest efficiency.Modeling results of ReVelle (2005) suggest that typical valuesfor larger events are of the order of a few percent.

4. Data and analysis

4.1. Construction of dataset

To perform our study, we first require a database of meteorevents for which there exist simultaneous satellite and infrasounddata. We have used as a starting point the database of existingsatellite meteor detections as described in Edwards et al. (2006). Foreach satellite detected fireball, we have the geographic location ofthe meteor as detected by the satellite, the time of the event and atotal integrated optical energy, which we convert to total eventenergy as described previously. For each event in this list, we thenperform a search of infrasound stations within � 10;000 km rangeextending up to a global scale for the largest events ð41 ktÞ. Theinfrasound stations searched were those of the International Mon-itoring System (IMS) of the Comprehensive Nuclear Test Ban TreatyOrganization (CTBTO) (Christie and Campus, 2009).

Infrasound data from each station was examined within a timewindow delimited by assumed celerities from 0.21 to 0.35 km/s andbased on the satellite measured bolide position and source time.These infrasound waveforms were then analyzed using the Infratool

component of the Matseis software package (documentation as wellas the software itself can be obtained at https://na22.nnsa.doe.gov/cgi-bin/prod/nemre/matseis.cgi) with the objective of identifyingbolide infrasound corresponding to the satellite-observed meteor.Coherent infrasound signals in the timeblock in question were firstidentified by searching the cross-power spectrum of time windowsof order 1 min in duration for correlations among array elementsexceeding the noise floor by at least one standard deviation. If suchsignals persist for at least five window intervals and show aconsistent backazimuth from window to window during this time,a potential signal is declared.

Signal backazimuths are computed using the time differentialbetween the onset of the infrasound signal at each microphone inthe array. This time offset can be used to calculate the direction inwhich the infrasonic wave is traveling across the array, includingboth the azimuth of the arrival and the apparent speed at which itmoves across the array, known as the trace velocity, providingan estimate of the vertical angle of arrival of the signal (see Eversand Haak, 2001 for details). The onset of the signal is then shiftedby this time delay at each array element and the waveformsat each individual microphone are compared statistically in agiven user-defined time window using a significance indicatorsuch as the Fisher F-statistic (Melton and Bailey, 1957). Fig. 7shows an example of an infrasonic detection made using thismethod.

As seen in Fig. 7, if there is no signal and just background noisebeing recorded, the azimuth and the trace velocity are quiteerratic. However, the onset of a signal, a coherent waveform from

https://na22.nnsa.doe.gov/cgi-bin/prod/nemre/matseis.cgi

https://na22.nnsa.doe.gov/cgi-bin/prod/nemre/matseis.cgi

Fig. 7. Infratool detection of a meteor signal. The display windows, from top down, are the cross-correlation of waveforms from the different elements as a function of

time, the trace velocity of the signal a function of time, the azimuth of the waveform as a function of time and the waveform as filtered by the given bandpass. Here the

azimuths and trace velocities are those that maximize the correlation of the different channels in a time window of length given in the ‘‘Window Parameters’’ box. The

‘‘Windowed Values & S.D.’’ dialogue box displays the averaged valued over the shaded time selection in the upper three panels.


a single source traveling across the array rather than the con-tinuous random pressure fluctuations caused by winds and otherbackground noise produces azimuth and trace velocity valueswhich are nearly constant. These are approximately the singlevalue corresponding to the propagation characteristics of theincoming meteor infrasound signal and the statistical correlationrises. If the trace velocity is near the local speed of sound(generally around 300 m/s accounting for the fact that the wavevelocity has a vertical component as well as a horizontal compo-nent), then the wave is moving nearly horizontally, consistentwith stratospheric ducting.

Once a probable signal is located in the time series, we check ifthe azimuth is consistent with where the satellite observedmeteor was georegistered (to within better than 251 in azimuth)and the timeframe of arrival of the signal to the infrasound stationis appropriate for a signal traveling at the speed of sound in astratospheric duct (i.e. from 0.21 to 0.35 km/s; Ceplecha et al.,1998) given possible wind variations. If all of these conditions aremet then it is highly likely that the observed signal is infrasoundgenerated by the meteor in question.

This analysis is often done several times per station per eventas there is constant background infrasound noise and so an


appropriate bandpass must be chosen to remove as much of thenoise as possible while keeping the signal intact. This bandpassdepends on the energy of the meteor (see Eqs. (12) and (13) inSection 3) so an estimate is made based on the satellite opticalenergy (usually � 0:223 Hz for a typical meteor event), however,this does vary with each event. Finding the signal in the noise isa trial and error process, making and refining estimates forthe bandpass and window parameters, a process guided byexperience.

In most cases showing fireball infrasound, the signal wasobvious and required no more than two or three bandpassiterations with Matseis. In some select cases where more rigoroussearching for weak signals seemed appropriate, we also made useof the Progressive Multi-Channel Correlation tool (PMCC) (Cansi,1995) (see Fig. 8 for an example).

Once a confirmed infrasound signal is positively associatedwith a satellite detected fireball using this methodology, astandard analysis is undertaken to extract signal metrics forfurther comparisons. The process is built around the 7 stepmethod used in Edwards et al. (2006) but with modificationswe hope will improve the quality of measurements.

The major difference from the earlier measurement methodol-ogy is in the choice of bandpass. The original (Edwards et al.,2006) approach used static bandpasses, with a high bandpassused for most signals and a lower bandpass used for some veryenergetic events. Since the bandpass frequencies were required inadvance of completing STEPS 1–4 in Edwards et al. (2006), in ourimplementation these steps were performed iteratively using thefrequencies as determined in STEP 4.3, described below, untiloptimal frequencies were obtained. For our first iteration, thesewere taken to be 0.2 Hz and 3 Hz with the exception of veryenergetic meteors for which a lower cut-off frequency was used.

We have also omitted the application of instrument responsesas originally done by Edwards et al. (2006) as these were found toproduce negligible differences (less than two percent in ampli-tude or period) for even the largest events (lowest frequencywhere instrument sensitivity begins to decrease) in our dataset.

Fig. 8. PMCC detection of the same event as in Fig. 7. The display windows show (from

across the array (as defined in Cansi, 1995), the statistical correlation of the signal across

waveform at one of the individual microphones. Each block of pixels corresponds to a

variable and frequency the dependent variable. Note that PMCC can be tuned such that

The following two steps, our modifications of the previousanalysis, are intended to be inserted within the 7 step methodgiven in Edwards et al. (2006) in numerical order.

In contrast to the generally fixed bandpass used in Edwardset al. (2006) here we adopt a more rigorous approach to definingthe bandpass containing the most signal energy. All determina-tions of the top and bottom of the frequency bandpass wereextracted from a power spectral density (PSD) of the signal whichmeasures the power per unit frequency carried by the signal. Inorder to obtain the most accurate measurements, the followingprocedure was followed twice, once where the PSD was taken ofthe entire signal as defined in STEP 1 to ensure all frequencycontent of the signal was included, and once with a smaller timewindow centered around the maximum signal amplitude asdefined in STEP 4 to get the best signal to noise ratio possible.The length of the smaller window was taken to be as small aspossible without causing low frequency cut-off problems; thisensures a satisfactory spectral resolution at the frequency of thesignal. The exact value used was slightly adjusted depending onthe properties of each signal.

As a result, very energetic (and thus lower frequency) bolideevents required a longer window than did less energetic events.A PSD was taken of the total infrasonic wave signal and also of awindow in the noise just before and after the signal of the samelength as the signal. Specifically, a Hann window was applied tothe first and last 10% of the signal time window to reduce cut-offerrors in the PSD and then the PSD was calculated using Bartlett’smethod (Proakis and Manolakis, 1996). Bartlett’s method wasused rather than a method that reduces noise like Welch’smethod (Proakis and Manolakis, 1996) since these involve split-ting the signal into smaller windows which in turn reduces thespectral resolution of the PSD. Since our goal was to measure thefrequency bandpass containing signal energy as accurately aspossible, we chose to sacrifice some SNR to get better spectralresolution. The PSD of the noise was then defined to be theaverage of equivalently sized time windows before and after thesignal. This was subtracted from the earlier PSD measurement of

the top moving down), the filtered and stacked signal, the consistency of the signal

the array, the azimuth of the signal, the trace velocity of the signal and the filtered

detection family above the threshold set by the user with time the independent

signals may often be followed for longer intervals than is the case with Infratool.

Fig. 9. Frequency content of a strong signal. The black curve, calculated by

subtracting the average PSD of the noise before and after the signal from the

PSD of the signal itself, shows the power spectral density of the signal alone. The

dominant frequency of the signal, defined as the frequency at which the maximum

of the signal PSD curve occurs is notated on the plot. fL and fU (b) are the lower and

upper frequencies of the bandpass, defined as the frequency before and after the

maximum of the PSD curve where the signal PSD drops down to the level of the

noise PSD, shown in red. (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)


the total signal in order to isolate the PSD of the signal with aslittle noise as possible.

To calculate the bandpass frequencies discussed in STEP 1 ofEdwards et al. (2006), the lower cut-off, fL, was defined to be thefrequency before the maximum at which the signal PSD droppeddown to the level of the noise (see Fig. 9). Likewise, the upper orhigh cut-off, fU, was defined as the frequency after the maximumat which the signal PSD dropped down to the level of the noise forthe last time (see Fig. 9). The above four steps were then repeatedusing these new frequencies until convergence was reached. Thedominant frequency of the signal was defined to be the frequencyat the maximum signal PSD after noise subtraction. In order tocalculate the uncertainty of the peak frequency measurement, thefrequencies at which the PSD peak dropped down to less thantwice the PSD of the averaged noise windows before and after themaximum were calculated and the uncertainty of the maximumfrequency measurement was defined to be the difference betweenthese two frequencies. Finally the largest two frequencies abovethe signal PSD maximum at which the signal PSD dropped belowthe level of the noise were calculated in order to estimate thehighest frequency content of the signal.

4.1.1. STEP 4.6: average noise measurements

The average noise at each station was found by calculating thestation PSDs in 10 windows, five before the signal and five afterthe signal, each with the same length as the signal. These werethen averaged together to get an overall PSD of the noise ofthe station at the time of the event. The value of the PSD wascalculated at the dominant frequency as determined in STEP 4.3,and also at fixed frequencies of 0.25 Hz, 0.5 Hz, 0.75 Hz and 1.0 Hzfor later quality checks.

A further change made in this study is to calculate theintegrated energy flux by summing the squares of the signaloverpressure and dividing by the local acoustic impedence at thereceiver (the product of the local sound speed and atmosphericmass density) converting this measurement to a true energy flux.This was not done in STEP 6 of Edwards et al. (2006) whereintegrated energy is simply taken to be the sum of the squaresof signal overpressure since the acoustic impedence was notexpected to vary greatly from station to station; however, weinclude it here for completeness.

A final step in this process is to extract a single representativevalue of the average wind speed between the bolide source andreceiver. This is done following the technique outlined in Edwardset al. (2006). In this approach wind data is derived from the dailyUnited Kingdom Meteorological Office (UKMO) (Swinbank andO’Neill, 1994) assimilated dataset. Wind values are extracted atrange steps of 125 km along the great circle path between thesource and receiver and the wind values at each range slice arefurther averaged over the height interval from 40 to 60 km tocapture a mean value for the stratospheric wind. This producesone height averaged mean value per range increment; the finalsource-receiver stratospheric wind value is computed from theaverage of all range slice values. We emphasize that the standarddeviation of any one estimate often exceeds the average windvalue, particularly for longer range events. This highlights thepitfall of using a single wind value to characterize the wind fieldover long source-receiver paths.

At the end of this measurement process at any single station,we have the following quantities extracted from the infrasoundbolide signal:

1.
Signal onset time (s); 2. Signal duration (s); 3. Signal mean backazimuth (1); 4. Maximum amplitude (Pa); 5. Maximum peak-to-peak amplitude (Pa); 6. Maximum frequency determined from PSD peak (Hz); 7. Period at maximum amplitude from zero crossings of
waveform (s);
8. Integrated signal energy flux (J m�2); 9. Integrated signal energy to noise ratio (dimensionless);
10.
Lower bandpass frequency (Hz); 11. Upper bandpass frequency (Hz).
4.2. Correlations among variables and fit errors

As infrasound attenuates with range, any relation involvingmeasurements of amplitudes must include a distance scalingcomponent (Mutschlecner et al., 1999). Depending on the geo-metry assumed, it is expected that an energy scaling exponent inthe range � 1

3 to � 12 will also appear in infrasound observations

(see Eqs. (14) and (15)). Thus a scaled range which is normalizedto the energy of the bolide is given by

Rs ¼ REcð17Þ

where R is the range in km, E is the energy in tons of TNTequivalent as computed by applying the luminous efficiencyvalue to the measured optical signal from sensors on the satellitesand c is the yield-scaling exponent. From the examination of plotsof maximum signal envelope amplitude, peak to peak amplitude,total integrated signal energy flux, and integrated signal energyflux to noise ratio as functions of yield-scaled range, it is clear thatthe data is best modeled by a power-law distribution. Thus aregression equation of the form

A¼ 10aRbEc0 ð18Þ

was chosen to model the data. Here A is the value of themeasurement, R is the range, E is the satellite determined sourceenergy and a, b and c0 are the regression constants to be fit to thedata. Here, c0 ¼ c=b where c is the yield-scaling exponent in Eq.(17). Note that we are estimating c from the data by fitting c0 andb in this study rather than fixing it to � 1

3 as was done in Edwardset al. (2006).

Following Edwards et al. (2006), to account for the effects ofatmospheric winds on signals, which tend to focus and increase theamplitude of with-wind propagation and inhibit upwind


propagation, we applied a wind correction of the form(Mutschlecner and Whitaker, 1990, 2009)

Aw ¼ 10kvh A ð19Þ

where A is the quantity to be corrected, k is an empirical constantmeasured in s/m, and vh is the horizontal component of thestratospheric wind directed toward the receiver in m/s.

Thus the full equation to be fit is

log A¼ aþb log Rþc0 log E�kvh ð20Þ

This suggests that a multi-variate linear regression model ofthe form y¼Xbþe is appropriate, where for n observations ofrange, wind velocity, source energy and waveform propertyfAi,ðRÞi,Ei,ðvhÞig

ni ¼ 1,

X¼

1 logðRÞ1 logðEÞ1 ðvhÞ1

^ ^ ^ ^

1 logðRÞn logðEÞn ðvhÞn

264

375

y¼

log An

^

log An

264

375, b¼

a

b

c0

�k

26664

37775

and the random errors e are assumed to have a multi-variatenormal distribution with mean 0 and variance s2.

From multi-variate least squares regression estimation theory(Stapleton, 1995) the fit constants b and the standard deviationdb of these constants are given by

b¼ ðXT XÞ�1XT y ð21Þ

dbi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS2ðXT XÞ�1

ii

q¼

ffiffiffiffiffiffiRii

pð22Þ

where S2¼ 1=ðn�4ÞJðy�XbÞJ2 is an unbiased estimator of s2 so

that R¼ S2ðXT XÞ�1 is the covariance matrix.

Confidence intervals for the regression function can beobtained by viewing evaluation of the regression function at R,E and vh as forming a linear combination of the regressionparameters. If d¼ ða,b,c0,�kÞ is the point at which we wish tocalculate the confidence interval, then since the errors wereassumed to be gaussian, it follows that the values of a, b, c0 and�k will also be normally distributed. However, the variance ofthese parameters is unknown a priori so using the estimator S2 ofthe variance leads to a t-distribution for the regression functionand we can define a 100ð1�aÞ% confidence interval as (Stapleton,1995)

½log AðdÞ7S2tn�4;1�a

ffiffiffiffiffiffiffiffiffiffiffiffidTRd

q� ð23Þ

where tn�4;1�a is the inverse of the t-distribution with n�4degrees of freedom at ð1�aÞ (i.e. the distance from the mean ofthe distribution such that the probability of being within thisregion is 100ð1�aÞ%).

The above interval can be calculated at any desired d todetermine the confidence interval at that point alone. However,it would also be useful to calculate the simultaneous confidenceinterval that holds for all d simultaneously. An upper bound to thesize of this confidence interval can be calculated using theBonferroni inequality which gives a lower bound on the prob-ability that the confidence interval will hold for all d simulta-neously. This results in the simple inequality (Stapleton, 1995)X

ajra ð24Þ

that must be satisfied if we want to calculate a confidence intervalof 100ð1�aÞ% where 100ð1�aiÞ% is the confidence interval

evaluated pointwise at each d. Thus the simultaneous confidenceinterval can simply be calculated by determining the pointwise100ð1�a=nÞ% confidence interval at each d value.

Note that this is a different method of calculating the con-stants a, b, c0 and k than was used by Edwards (2007). In thatstudy, k was chosen separately from a and b while c was fixed to� 1

3. The r2 coefficient of the wind-corrected infrasonic measure-ment and scaled range was determined as a function of k and thebest fit k value was chosen by maximizing the r2 value of each fit.The values of a and b were then determined using a simple linearregression model of the form log Aw ¼ aþb log Rs. Thus r2 was thecoefficient of the determination of only the wind correctedmeasurement and the scaled range. In the method used for thisstudy, r2 is instead the coefficient of multiple correlation betweenthe uncorrected measured quantity (A), the wind velocity (vh),source energy (E) and the range (R) so the values of r2 determinedby these two studies are not directly comparable.

A final possible correction to be examined is the Doppler shiftof the infrasound waves. To determine whether this causes anappreciable error in measurement of periods, a correction of theform

o¼o0 þk � V ð25Þ

where o is the angular frequency in the frame of the infrasoundstation, o0 is the angular frequency in the reference frame of thecomoving system, k is the wavevector and V is the velocity of themoving frame with respect to the frame of the infrasound station(i.e. the wind velocity) was examined (Morse and Ingard, 1987) asdescribed in the next section.

5. Results and discussion

5.1. General properties of the dataset and quality checks

For this study a total of 211 satellite detected bolides wereexamined for possible infrasound signals. Of these, we wereable to positively identify infrasound signals from 71 individualevents. These 71 events were recorded as 143 distinct infrasonicwaveforms—each bolide was, on average, detected at twostations, though in reality some 54% of all bolides were detectedat one station only. However, some of these infrasonic recordingsmay not be stratospherically ducted returns as our initialcelerity constraint (see Section 4) is not very strict. To eliminatesuch returns from the dataset, we apply further filtering basedon celerity, removing signals with celerity below 0.23 km/s aslikely thermospheric in origin. The one exception applied to thissecond stage celerity filtering is for shorter range ðr500 kmÞreturns. Since the satellite position and the actual positionwhere infrasound is produced on the trail may differ by up to� 100 km, these short range returns may show anomalously lowcelerity but still be stratospherically ducted arrivals. We alsoremove returns having trace velocities in excess of 0.5 km/s if thecorresponding celerity is below 0.25 km/s as being likely thermo-spheric returns.

Fig. 10(a)–(c) shows the trace velocities and celerities as afunction of range and wind speed of our remaining events after allpotentially non-stratospherically ducted signals are removed.

The final resulting dataset has 63 distinct meteor eventsrecorded as 113 waveforms. Fully, 54% of all our events are singlestation detections while our most well documented event wasdetected at seven stations. The smallest infrasonically recordedsatellite event is 0.02 kt while the largest is � 20 kt and the modeof the dataset is 0.3 kt. Most of the infrasound detections aremade at a distance of about 2000 km from the source, with a

0.50

0.45

0.40

0.35

0.30

Trac

e Ve

loci

ty (k

m/s

)0.36

0.34

0.32

0.30

0.28

0.26

0.24

0.22

0.20

Cel

erity

(km

/s)

1000 10000

Range (km) Range (km)

020

0040

0060

0080

0010

000

1200

014

000

1600

018

000

0.36

0.35

0.32

0.30

0.28

0.26

0.24

0.20

Cel

erity

(km

/s)

-100 -50 0 50 100Wind Speed (m/s)

Fig. 10. (a) The trace velocity of all events in the final dataset as a function of range. The one apparent outlier showing a trace velocity of over 0.5 km/s is slightly higher

than expected for a stratospherically ducted signal; it was left in the dataset since it passed all other selection criteria. (b) The celerity (averaged signal speed over entire

propagation path) of all events remaining in the dataset as a function of range. The convergence of celerities to an average value near 0.29 km/s as range increases occurs

due to the fact that over these large ranges, the random local variations in the atmospheric winds tend to cancel out leading to average celerities very close to the mean

expected value of a stratospherically ducted signal. (c) The celerity as a function of average wind speed; negative values indicate counter-wind returns. The one outlier

near 0.21 km/s was retained as it had a small range from source to receiver and may have had a relatively large range error while all other characteristics suggested a

stratospheric return.


maximum distance of 17,250 km and a minimum distance of320 km.

As a final check on how well the infrasound event can beassociated with the satellite measured bolide event, Fig. 11 showsthe difference between the azimuth at which the infrasonic wavepropagated across the array compared to the azimuth calculatedgeometrically using the satellite location and the infrasoundstation location for the final dataset of accepted events as afunction of the average of the stratospheric cross-winds. Mostshow agreement to better than 101 with a trend in range andcross-wind deviation as expected.

A further measure of the self-consistency of the periodmeasurements made in this study can be obtained by comparingthe measured period at maximum amplitude to the inverse of thefrequency at which the PSD is a maximum. As expected, thesevalues are close to being in a 1:1 correspondence.

5.2. Empirical infrasound discriminators of bolides

A common issue when examining infrasound records is con-fusion between sources. However, it may be possible to discrimi-nate among signal types using distinct signal properties,

30

20

10

0

-10

-20

-30-80 -60 -40 -20 0 20 40 60 80 100

Crosswind Speed (m/s)

Azi

mut

h D

iffer

ence

(Deg

rees

)

Fig. 11. The difference d Azimuth (observed–expected) of the signal azimuth as

described in the text calculated using Infratool and the predicted azimuth

calculated geometrically using the great circle path connecting the satellite-based

location of the bolide and the infrasound station as a function of the average cross-

wind from the source to the receiver. Negative values indicate cross-winds

directed to the north of the source-receiver great circle path.

10000

1000

Ran

ge (k

m)

0.01 0.1 1 10Energy (kilotons of TNT Equivalent)

Fig. 12. The range at which infrasonic signals of bolide sources are observed as

compared to the energy of the event. The outer envelope of this data, shown as the

superposed curve which was fit to the points highlighted by shaded triangles,

indicates a power law relationship between the maximum range at which an

infrasound signal is observable and the energy of the source producing the wave.


potentially unique to a specific type of source (Brachet et al.,2009). We have examined our complete dataset of bolide infra-sound observations and found several empirical relations whichmay provide a means of distinguishing bolide (or atmosphericexplosions more generally) from other types of signals.

Fig. 12 shows the satellite estimated yield as a function ofrange to each infrasound station with a positive signal detection.We have overlaid a relation of the form log R¼ 2:80þ0:33 log E asthe apparent upper range R in km where a bolide of energy E intons of TNT can be expected to show a detectable signal. The

averaged windspeed over the propagation path of the largeoutlier (0.04 kt, 3700 km) is 52 m/s. This strong with-wind couldexplain how this low energy event was observed at a larger thanexpected distance. Interestingly, this relation suggests thatbolides with yield 420 kt should be detectable globally, anestimate validated by the recent event described by Silber et al.(2011), while a 1 kt event is maximally detectable at a range ofapproximately 7000 km. This is not an artifact of our searchstrategy as all events with energies in excess of 1 kt had globalsearches for signals and all stations had searches out to ranges of10,000 km at least. That infrasound detection efficiency variesspatially and temporally has been modeled in detail (Le Pichonet al., 2009) and comparison of such models to the present datasetwould be particularly valuable as a means of ground-truth.

Another potential bolide infrasound signal discriminant is tomake use of the fact that frequency attenuation increases withpropagation distance. As described in Section 4, we found both alow and high-frequency cut-off for each event based on themeasured PSD. Since higher frequencies attenuate more rapidlythan lower frequencies, the plausible frequency ranges can beconstrained based on the distance from the source. Fig. 13(a) andb shows the top frequency cut-off from our PSD analysis forevents in our dataset as a function of range and also as a functionof scaled range to normalize for the differing energies. Further-more, Fig. 13(a) and b can be used as a means to discriminatelarge explosions versus distance via frequency content. Thebest fit curves to the outer envelopes of the data shown inFig. 13(a) and (b) are

f U ¼ 0:51þ20:9 exp �R

1883

� �ð26Þ

f Us¼ 1:09þ13:4 exp �

Rs

114

� �ð27Þ

for the maximum frequency (in Hz) content fU expected in aninfrasonic signal from a bolide at a range R (in km) or scaled rangeRs (in km=ton1=2).

Similarly, the lower frequency cut-off is shown in Fig. 14(a) asa function of range and Fig. 14(b) as a function of scaled range.These relations can be used to dismiss events as bolide generatedif it is known they are long range recordings yet only a highfrequency signal remains. The functional forms of the outerenvelopes for the lower frequency cut-off are given by

f L ¼ 0:06þ102 exp �R

679

� �ð28Þ

f Ls¼ 0:31þ4:25 exp �

Rs

60:1

� �ð29Þ

5.3. Period–yield estimates for bolides

The most common method currently used for infrasonicdetermination of the energy of meteor events is an empiricalrelationship between the dominant period of the signal and thekinetic energy of the meteor given by the AFTAC nuclear explo-sion relations (Eqs. (1) and (2)) (Ceplecha et al., 1998). It is theleast complex method for yield estimation as it requires noadditional knowledge of the actual meteor event, such as sourceheight, range, or atmospheric conditions between the sourceand receiver. Its simplicity has led to its widespread use, but ontheoretical grounds we expect differences in periods due tovariations in source heights and dispersion effects with range(ReVelle, 1974).

One effect which can be characterized in principle is the windDoppler shift influence on measured period. An attempt was

10000

12

10

8

6

4

2

00 5000 15000 20000

Range (km)0 5000 10000 15000 20000

Hig

h Fr

eque

ncy

Cut

off (

Hz)

12

10

8

6

4

2

0

Hig

h Fr

eque

ncy

Cut

off (

Hz)

Scaled Range (km/E(1/2))

Fig. 13. The observed drop-off in the upper frequency content (fU) of bolide associated infrasonic signals as a function of range and scaled range. (a) The outer envelope of

the high frequency cut-off-range dataset is approximately an exponential decay as given by Eq. (26). Here, the size of the data points indicate the satellite measured energy

of the bolide event and the black triangles indicate those points used in determining the outer envelope function. (b) A similar drop-off in frequency is observed with

scaled range (Rs); this decay curve is given by Eq. (27).

Low

Fre

quen

cy C

utof

f (H

z)

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Low

Fre

quen

cy C

utof

f (H

z)

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Range (km) Scaled Range (km/E(1/2))0 5000 10000 15000 20000 0 5000 10000 15000 20000

Fig. 14. The observed drop-off in the lower frequency content of a bolide signal as a function of range and scaled range. (a) The outer envelope of the low frequency cut-

off-range dataset is approximately an exponential decay as given by Eq. (28). The size of the data points indicate the satellite measured energy of the bolide event and the

yellow triangles indicate those points used in determining the outer envelope function. (b) A similar drop-off in frequency is observed with scaled range and the decay

curve is given by Eq. (29).


made to correct for this by applying a Doppler correction to theobserved period using the previously computed UKMO windvalue per event. Fig. 15 shows that only for a few events is theDoppler effect significant, producing a period change morethan 10%. Given that the standard deviation of the wind can becomparable to the magnitude of the wind, the result of thisanalysis suggests that the period–energy relation is not mean-ingfully changed so we ignore the wind Doppler shift.

To compare the AFTAC period–yield relations to our dataset, allthe events that passed through the quality check were used to

find a linear regression relationship of the form:

log E¼ a log tþb ð30Þ

where E is the bolide kinetic energy measured by satellite sensorsand t is the dominant period of the infrasonic wave as measuredat the infrasound array. The regression curve has an r2 value ofr2¼0.51 (see Fig. 16(a) which also compares our curve to that of

AFTAC).Since many of the detected bolides were recorded on multiple

infrasound stations, we also analyzed this multi-station data


subset by taking the periods measured at each individual stationfor the same event and averaging to produce a single mean periodassociated with each bolide. This serves to reduce the error in theperiod measurement since the effects of inhomogeneities inthe atmosphere and other such effects are also averaged. Theresulting regression equation, using only multi-station eventsthat have been averaged in this way, shows a much betterregression correlation, r2 ¼ 0:71 (Fig. 16(b)).

In so far as the infrasonic period remains approximatelyconstant and independent of range once the signal has becomelinear, we expect the above period–energy relations to be more

40

35

30

25

20

15

10

5

00 10 20 30 40 50

Uncorrected Period (s)

8

6

4

2

0

-2

-4

-6

-8

Dop

pler

Cor

rect

ed-U

ncor

rect

ed P

erio

d (s

)

Dop

pler

Cor

rect

ed P

erio

d (s

)

Period Difference

Fig. 15. A comparison of the Doppler corrected period (ordinate) using the

averaged stratospheric UKMO wind speeds to the uncorrected period (abscissa).

While a few of the events show a significant difference in period after Doppler

correction, the majority do not. The corrected Doppler values did not significantly

change the calculated period energy relations nor did they increase the r2 value of

the regression.

100

10

1

0.1

Per

iod

(s)

0.01 0.1 1 10Energy (Kilotons of TNT Equivalent)

Regression95% ConfidenceAFTAC (E<200kt)

Fig. 16. The period of the infrasonic signal at maximum amplitude versus the satellite-

in log-log space. (a) Each datapoint represents one recording of an event at a particular s

units of tons of TNT equivalent and s). (b) Only multi-station events with each datapoint

and the error bars the standard deviation among measurements. For this dataset, a¼3

given by Eqs. (1) and (2).

robust than those depending on the amplitude of the signal wherewind and attenuation effects are more severe. We note that theoverall least-squares relationship we find is remarkably similarto the AFTAC values, the AFTAC curve falling within the95% confidence intervals of the regression, validating its use inbolide-energy estimation. Some evidence exists for greater var-iance at larger energies/longer periods as was also notedby Edwards et al. (2006) but the statistical significance of thisis questionable. Table 1 presents a summary of the bolideperiod–yield relations for single and multi-station events,together with energy prediction limits at the one, two andthree-sigma level.

5.4. Empirical yield estimates using amplitude and infrasonic signal

energy flux

In addition to the period, the maximum pressure amplitudeand total integrated energy flux are easily measured quantities foreach bolide infrasound signal. In our dataset, we find that there isa natural gap in the satellite determined bolide energy between3.5 and 7 kt, a feature also found in Edwards et al. (2006). Forconsistency with the earlier study by Edwards, we derivedseparate relations for energies greater than 7 kt, and for energiesless than 3.5 kt.

By computing the mean stratospheric wind between sourceand receiver (see Section 4) and using the known range to thesource together with the satellite measured energy we are able tofit an equation of the form of Eq. (20). This fit is a multi-variatefit where we estimate the best-fit value for each of the fourcoefficients in Eq. (20) together with their variance. Table 2summarizes our results for the entire dataset, the low and highenergy events separately, and those events with source-receiverranges r3000 km. This is a significantly different approach thanwas used in Edwards et al. (2006) where cube-root yield scalingwas adopted (i.e. c¼� 1

3 was imposed), thus decreasing thedegrees of freedom used in those earlier fits. We note thatprevious work suggests that the yield scaling exponent (c in

100

10

1

0.1

Per

iod

(s)

0.01 0.1 1 10Energy (Kilotons of TNT Equivalent)

Regression95% ConfidenceAFTAC (E<200kt)

measured energy of the event shows a power law type relationship which is linear

tation. Here the regression coefficients are a¼3.7570.19 and b¼0.5070.29 (with

representing the average of the periods measured at the various stations per event

.2870.12 and b¼0.7170.18. Also shown for comparison are the AFTAC curves as

Table 1Empirical relationships relating bolide infrasound period/energy (in s/t of TNT equivalent), maximum detection range (in km) and

frequency cut-off (in Hz) with range (in km) or scaled range (in km=ton1=2).

Range discriminators

log R¼ 2:80þ0:33 log E Maximum distance R an event of energy E is observed

f U ¼ 0:51þ20:9 expð�R=1883Þ Maximum frequency content f of a signal at a range R

f U ¼ 1:09þ13:4 expð�Rs=114Þ Maximum frequency content f of a signal at scaled-range Rs

f L ¼ 0:06þ102 expð�R=678Þ Maximum lower frequency cut-off f of a signal at range R

f L ¼ 0:31þ4:25 expð�Rs=60:1Þ Maximum lower frequency cut-off f of a signal at scaled range Rs

Energy E as a function of period s for entire dataset

log E1sU ¼�1:34þ2:46 exp

log t1:06

� �Upper bound at confidence level 1s


log t0:85



log t0:72


log E¼ 3:75 log sþ0:50 Best fit regression equation

log E1sL ¼�3:45þ7:31 log t0:40 Lower bound at confidence level 1s



Energy E as a function of period s for averaged multi-station dataset


log t0:98


log E2sU ¼ 0:32þ1:14 exp

log t0:32


log E3sU ¼ 0:84þ0:74 exp

log t0:53


log E¼ 3:28 log sþ0:71 Best fit regression equation




Table 2Estimates of regression coefficients and r2 values for various fits (see text for details) following Eq. (20).

Energy Estimator Measurement k (s/m) a b c r2

This study: all ranges

Low energy ðo3:5 ktÞ Max amp �0.008470.0015 4.0670.78 �0.9970.13 �0.4370.07 0.55

P2P amp �0.008470.0014 4.3370.78 �1.0070.13 �0.4370.07 0.56

Int. energy �0.012170.0046 8.6172.50 �1.9270.40 �0.4470.21 0.29

IE SNR �0.000970.0016 3.2270.87 �0.4870.14 �0.2370.07 0.12

High energy ð47 ktÞ Max amp 0.001470.0041 10.7573.38 �1.1570.36 0.7870.99 0.56

P2P amp 0.001370.0040 11.3573.30 �1.2070.35 0.7770.97 0.59

Int. energy 0.015670.0132 24.21710.75 �3.3171.15 0.0270.01 0.47

IE SNR �0.002070.0053 13.6774.34 �1.1370.46 1.2371.27 0.48

This study: all energies

All ranges Max amp �0.006870.0014 4.3870.74 �1.0670.12 �0.4770.05 0.56

P2P amp �0.006870.0014 4.7170.73 �1.0870.12 �0.4670.05 0.57

Int. energy �0.007370.0045 10.3472.32 1.0270.16 �2.2270.38 0.34

IE SNR �0.001670.0017 4.4170.85 �0.6970.14 �0.2370.06 0.18

Within 3000 km Max amp �0.008670.0018 4.9771.19 �1.1870.19 �0.4570.06 0.61

P2P amp �0.008470.0018 5.3071.18 �1.2070.19 �0.4570.06 0.62

Int. energy �0.016170.0045 7.7573.03 �1.9270.49 �0.6370.15 0.51

IE SNR �0.002570.0019 2.5971.24 �0.4270.20 �0.5170.06 0.17

Edwards et al. (2007)

Low energy ðo3:5 ktÞ Max amp �0.0174 3.2170.59 �1.7570.24 0.49

P2P amp �0.0177 3.3670.60 �1.7470.24 0.48

Int. energy �0.0380 9.2071.30 �3.6470.53 0.41

IE SNR �0.0100 4.1270.51 �1.4470.20 0.43

High energy ð47 ktÞ Max amp �0.0024 2.1870.39 �1.2670.17 0.75

P2P amp �0.0018 2.5870.41 �1.3570.18 0.78

Int. energy 0.0010 5.9870.99 �1.9970.43 0.41

IE SNR �0.0023 4.5370.88 �1.6270.38 0.34

Mutschlecner et al. (1999) �0.4670.05

Reed (1972) �1.28 �0.43


Eq. (17) or bc0 in Eq. (20)) should be between � 13 and � 1

2, dependingon whether the observation is in the near-field or far-field, respec-tively and also the form of ducting which applies. The range scalingexponent (b in Eq. (20)), based on high explosive and nuclear tests,

has been found experimentally to lie between roughly �1.1 and�1.5 (Mutschlecner et al., 1999; Reed, 1972).

The results for the entire dataset show that c¼�0:4770:05and b¼�1:0670:12 are the best overall yield and distance


scaling exponent fits for pressure amplitude, consistent (within onesigma uncertainty) of far-field observations following a nearlystandard ‘‘R�1.2’’ law for overpressure drop-off with range (seeReed, 1972 for a discussion of airblast overpressure decay relations).We note that our best-fit wind k-value, �0.006870.0014, is halfthe value found in Edwards et al. (2006). This is unsurprising as ouryield-scaling and distance scaling exponents are quite different thanwere used in that earlier study. The coefficient values found for onlythe shorter range events ðr3000 kmÞ do not differ significantlyfrom the dataset as a whole, in all cases agreeing at the one sigmalevel, providing confidence that unaccounted for range effects donot unduly bias our results. Interestingly, the integrated signalenergy flux measurements show a much steeper distance scalingexponent (near �2.0), potentially reflecting a higher signal attenua-tion for the broader-pressure pulse, consistent with it being com-posed of proportionately more high frequency energy (whichexperiences more attenuation than that found at the peak ampli-tude). The integrated signal energy flux to noise ratio shows verylow correlation values and no coefficient fit values which arephysically meaningful, possibly reflecting the dominance of localnoise conditions on this measure.

Our yield-scaling exponent can be compared with that foundby Mutschlecner et al. (1999) who derived an equivalent valueof c¼�0:45670:05 for stratospheric infrasound returns fromnuclear tests, while Reed (1972) found c¼�0:425 and b¼�1:275for large high-explosive tests, both of which are almost identicalto our result.

Comparison of the larger/smaller energy data subsets iscomplicated by the paucity of higher energy events, which resultsin the two datasets differing in size by a factor of 5. The smallerevents generally have regression coefficients which agree withinone sigma to the dataset as a whole, indicating that the smallernumber of large energy (typically long range) events also do notunduly bias the overall fits.

The wind coefficient for the high energy bolide subset has avery large uncertainty and is consistent within uncertaintybounds with essentially no wind correction. This is the same

50

45

40

35

30

25

20

150.05 0.10 0.15 0.20 0.25 0.30

Peak to Peak Amplitude (Pa)

Hei

ght (

km)

SourceReceiverAveraged

Fig. 17. Sensitivity of the normal mode modeling to adopted wind speed. Each colored

and (b) Period as a function of height. The various colored curves show the normal mode

the source (red curve), averaged over the entire propagation path (blue curve), and wh

line represents the measured value of the peak-to-peak amplitude/period for IS52. Here

energy was 4.61 kt TNT. (For interpretation of the references to color in this figure leg

result found by Edwards et al. (2006). This may indicate that thesimple wind correction used in both studies does not adequatelymodel the long-range effects of wind on infrasound propagation.The major weakness of the model is that it averages the actualwind profile over a range of altitudes and further averages overthe entire great circle path from the source to the receiverresulting in just one wind velocity number that is expected tocharacterize the wind structure over the entire path of propaga-tion. For short distances this may be a valid approximation,however, for long distances such as those in this study, the windgenerally varies significantly over the propagation path. We note,for example, that at least one event in our dataset has a windvalue that changed by almost 100 m/s over the propagation path.Thus characterizing the atmosphere by a solitary value may notprovide a sufficiently detailed account of atmospheric winds.Performing individual ray tracings per event may provide a bettermeans of estimating the cumulative effect of the wind, thoughperforming this for bolide sources with unknown source altitudewould be problematic.

Another complication in comparing the results in Table 2 withthose found in Edwards et al. (2006) is the difference in the waybandpasses were chosen in the two studies. As described in Section 4,in the present work a different bandpass was chosen for each stationper event based on the power spectral density of the signal, whereasin Edwards et al. (2006) a standard bandpass of 0.2–3 Hz was chosenfor all but the large events, for which the low frequency cut-off had tobe decreased since the frequency content of very large events isknown to contain frequencies much lower than 0.2 Hz. This providesa more consistent way of processing the data, however, differentevents have different frequency content while the infrasonic noise ateach station has differing frequency content as well. We feel that themethod used in this study better isolates the frequencies associatedwith the meteor infrasound signal. A shortcoming of this newapproach, however, is that spectral power has a strongly decreasingfrequency dependance (see Fig. 9 in Section 4 for an example).Thus changing the bandpass can alter the amplitudes of the wavesignificantly.

4 6 8 10 12 14 16Period (s)

50

45

40

35

30

25

20

15

Hei

ght (

km)

SourceReceiverAveraged

curve represents the normal-mode calculated value for (a) peak to peak amplitude

predictions at the receiver for the cases where the vertical wind profile is taken at

ere the wind profile is taken at the receiver (green curve). The solid black vertical

the satellite measured source energy was 18.5 kt TNT and the normal-mode input

end, the reader is referred to the web version of this article.)


5.5. Source height estimation

An implicit assumption common to all the above methodsof analysis is that the mechanical energy deposition height isconstant for all meteor events. This assumption, given the largevariation in meteoroid sizes, speeds and compositions, is certainlyerroneous (see Ceplecha et al., 1998). Indeed, when regarded froma distance as a point source, the observed height of detonationfrom any given station will vary depending on how deeply intothe atmosphere the meteor penetrates as well as where in theatmosphere the meteor first begins producing sound. Further-more, the meteor may deposit sound at different points along thetrail (either as a ballistic shock or an ablational shock (Bronsthen,1983)) which then propagate as point sources from differentaltitudes in the atmosphere. Further complicating this picture,some source heights may not be able to produce detectablesignals in some directions given particular wind fields and atmo-spheric temperature profiles. We hypothesize that these differ-ences in detectable source height may lead to much of the scatterseen in the observed data. In order to examine the effect of bolidesource height on calculated energy from infrasonic measurements

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100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

Fig. 18. Energy relations for all events. The datapoint are the wind corrected infrason

versus the scaled range with the regression equation superposed. (a) Maximum ampl

(d) Integrated energy signal to noise ration relation.

alone a method of estimating source height was tested usingnormal mode modeling of the infrasonic signal. Our approach wasmotivated by the earlier successful reproduction of source energyfor the Moravka meteorite fall (Borovicka et al., 2003) using anormal mode matching of infrasound amplitude as a function ofsource height.

In this approach, we generate different discrete signals at arange of possible bolide source heights and energies and thenpropagate the resulting infrasonic signal through the atmosphereusing a normal mode approach (see Pierce and Posey, 1970 fordetails) as implemented in the Inframap software package (Gibsonand Norris, 2000). This produces a predicted synthetic amplitude–time waveform at the receiver. The approach assumes thereceived waveform is well represented by a discrete, sphericalsource signal emitted at a fixed altitude, a model we expect willbe approximately correct for bolides with fragmentation episodesproducing ablational shocks.

The synthetic waveform is then analyzed by the same proce-dure as applied to true signals and the synthetic peak amplitudeand period saved. The goal is to find a combination of heights andsource energy which simultaneously match the observed period

Win

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SN

R

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100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

ic measurements using the k value as determined by the multi-variate regression

itude relation. (b) Peak to peak amplitude relation. (c) Integrated energy relation.

Table 3Comparison of regression coefficients following Eq. (20) and r2 values calculated in this study for various subsets of the data to

those from Edwards (2007) using one-half yield-scaling (i.e. with c fixed at � 12).

Energy Estimator Measurement k (s/m) a b r2

This study: all rangesLow energy ðo3:5 ktÞ Max amp �0.008370.0015 3.5570.51 �0.9270.10 0.55

P2P amp �0.008270.0014 3.7970.51 �0.9370.10 0.55Int. energy �0.017470.0041 4.4271.43 �0.0270.00 0.35IE SNR �0.000770.0016 2.2970.57 �0.3670.11 0.10

High energy ð47 ktÞ Max amp �0.000670.0043 5.7971.63 �1.3970.36 0.48P2P amp �0.000870.0042 6.2471.60 �1.4470.35 0.51Int. energy 0.008570.0110 6.5474.20 �1.9570.92 0.26IE SNR �0.004670.0055 7.0972.10 �1.4470.46 0.39

This study: all energiesAll ranges Max amp �0.006870.0014 4.0470.44 �1.0170.09 0.56

P2P amp �0.006870.0014 4.2870.44 �1.0270.09 0.57Int. energy �0.012270.0040 6.1471.24 �1.9670.25 0.38IE SNR �0.001570.0017 2.6570.53 �0.4470.11 0.14

Within 3000 km Max amp �0.008670.0018 4.4470.53 �1.1070.11 0.61P2P amp �0.008470.0018 4.6870.52 �1.1170.11 0.62Int. energy �0.016770.0045 7.3671.32 �2.2370.27 0.51IE SNR �0.002570.0019 2.6170.55 �0.4270.11 0.17

Edwards et al. (2007)Low energy ðo3:5 ktÞ Max amp �0.0174 3.2170.59 �1.7570.24 0.49

P2P amp �0.0177 3.3670.60 �1.7470.24 0.48Int. energy �0.0380 9.2071.30 �3.6470.53 0.41IE SNR �0.0100 4.1270.51 �1.4470.20 0.43

High energy ð47 ktÞ Max amp �0.0024 2.1870.39 �1.2670.17 0.75P2P amp �0.0018 2.5870.41 �1.3570.18 0.78Int. energy 0.0010 5.9870.99 �1.9970.43 0.41IE SNR �0.0023 4.5370.88 �1.6270.38 0.34

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100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

Fig. 19. Energy relations for small ðo3:5 ktÞ events at all ranges. The datapoints are the wind corrected infrasonic measurements using the k value as determined by

the multi-variate regression versus the scaled range with the regression equation superposed. (a) Maximum amplitude relation. (b) Peak to peak amplitude relation.

(c) Integrated energy relation. (d) Integrated energy signal to noise ration relation.



and amplitude at the receiver. We performed this analysis on fourdifferent infrasonically recorded bolides with multiple stationdetections having simultaneous satellite data. For none of thesefour bolides, at any station, could we simultaneously match theobserved period and amplitude using the normal mode approachfor a series of physically plausible source heights and energies lessthan the total recorded satellite yield.

The failure of this approach we suspect is mainly because thenormal mode method used assumes a range-independent atmo-sphere. As a result, the wind must be averaged. Three differentmethods of reducing the range-dependent wind profile to one whichis range-independent were tried, namely taking the vertical wind-profile at the source, at the receiver and finally by averaging overthe propagation path. In Fig. 17 we show a representative examplesimulation fit using the identical set of source parameters but withthe three different wind averaging methods. The spread in predic-tions between the three curves due to the change in the wind valuealone is larger than the spread in the values of the model predictionsalong the individual curves due to changing source height. Thus theerror due to the assumption of a range-independent wind structurealone in the normal mode method limits its utility for estimatingsource heights in this fashion. That this approach worked in the case

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100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

Fig. 20. Energy relations for large ð47:0 ktÞ events at all ranges. The datapoint are the

multi-variate regression versus the scaled range with the regression equation supe


of the Moravka fall, may be a consequence of the relatively smallrange between the bolide and the receiver. In order to moreeffectively apply modeling techniques to determine the effect ofenergy deposition height and wind variability, it is important tohave more complete information from other techniques, as was thecase for the Moravka fall which had a well-determined trajectoryand range of heights at which energy deposition occurred.

5.6. Final fit results for 12 yield scaling

Appealing to our earlier multi-variate fit results which showedthat c�� 1

2 we adopt this yield scaling and plot our final setof infrasonic measurements versus scaled-range. These are alsobroken down into our small energy, large energy and low rangeðr3000 kmÞ subsets respectively.

The scaled-range fits for all events in the dataset are shown inFig. 18. The r2 values are poor for the integrated energy flux and SNRvalues, again likely reflecting large variations in local noise condi-tions. Amplitude versus scaled-range shows much better correlation,though substantial scatter in individual points remains.

Our best fit k-values (Table 3) remain almost a factor of threelower than found in Edwards et al. (2006) (who used � 1

3 scaling) or

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100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

wind corrected infrasonic measurements using the k value as determined by the

rposed. (a) Maximum amplitude relation. (b) Peak to peak amplitude relation.


the almost identical value determined in Mutschlecner et al. (1999),who employ� 1

2 scaling. Our k-value may be lower due to the highernumber of long range observations as compared to Mutschlecneret al. (1999) (suggesting a possible range dependence of k) and ouruse of a steeper range-scaling as compared to Edwards et al. (2006).

Low-energy event fits are shown in Fig. 19. These fits mimickthose of the full dataset, reflecting the small size of the high-energy subset, which is shown in Fig. 20. Here a substantialdifference is found in the k-value, being much smaller (withinerror zero). This may reflect the much larger average range forevents in this dataset (together with its small size); if the formerit further suggests there is a range dependence in k. Thispossibility is strengthened when only those bolides (all energies)with ranges r3000 km are analyzed as shown in Fig. 21. Thek-values here are higher than for the dataset as a whole or for thelarge energy subset suggesting that k may indeed vary with range,a suggestion also made by Mutschlecner et al. (1999).

5.7. Bolide acoustic efficiency

Using our dataset we may also estimate the distribution ofbolide integral acoustic efficiencies (see ReVelle, 2005) following

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100 200 500 1000 2000 5000 10000 20000Scaled Range (km/E(1/2))

Fig. 21. Energy relations for events within 3000 km at all energies. The datapoint are

the multi-variate regression versus the scaled range with the regression equation su


Eq. (16). Fig. 22 shows an estimate for the acoustic efficiency forall observations as a function of range using a reflection value off¼0.9, chosen as it produces physically reasonable values for allour events and is comparable to the value suggested by Cox(1958). From this plot the average bolide efficiency is near 1%,with the geometric average, more representative of the data as awhole, being 0.1%. The lowest values approach 0.001% and thelargest value we find is near 20%, likely unrealistic but still aphysically possible value. There is substantial scatter with noclear trend as a function of energy. A more noticeable trend withincreasing apparent efficiency with range is notable suggestingour assumed form of variation (R�2) is likely too steep. Using theshallower distance-scaling exponent (b) found in the previoussection decreases the efficiency by several orders of magnitudeand produces an opposite trend with range, suggesting thatdistance-scaling values of b��1:1 are too small. That thefunctional form of Eq. (16) does not lead to more range-indepen-dent efficiencies is not surprising as we are applying this equationoutside the original atmospheric conditions for which it wasderived. Our analysis would suggest that values for integral bolideacoustic efficiency vary by at least several orders of magnitude,possibly reflecting differences between ballistic and ablational source

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the wind corrected infrasonic measurements using the k value as determined by

perposed. (a) Maximum amplitude relation. (b) Peak to peak amplitude relation.

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Fig. 22. The acoustic efficiency of all events in the dataset as a function of the

range from source to receiver assuming a reflection coefficient of f¼0.9. The mean

and geometric mean of the efficiencies over all events are 1% and 0.1%

respectively.

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Stevens et. al. (2006)-CrosswindStevens et. al. (2006)-DownwindASI (1983)Blanc et al. (1997)Edwards (2007)Mutschlecner & Whitaker (2009)Clauter & Blandford (1998)This Study3-Sigma Bound

Am

plitu

de (

Pa)

Fig. 23. Comparison of the AFTAC, French Nuclear, ANSI, Los Alamos HE and bolide

relations from Edwards (2007) described in the Introduction as well as the new

bolide relation determined in this study using 12 scaling comparing observed

infrasound amplitude with range for an explosion of 1 kt yield (4.185�1012 J).


functions and hence differences in sampled source position alonga particular bolide trail as seen by different stations. Representativevalues we find are in the range of approximately a few tenths to apercent; our general result is a bit lower on average than thepredictions by Cox (1958) and ReVelle and Whitaker (1996a) andthe theory presented in ReVelle (2005) all of which suggest values oforder a few percent should apply. These studies are generally atvariance, however, with Astapovich (1946) who estimated theacoustic efficiency for bolides to be 0.01%, though we find someevents with apparent efficiencies this low and even lower.

6. Conclusions

Using a dataset consisting of over 70 bolides detected simulta-neously by satellites and recorded as more than 140 infrasonicwaveforms, we have been able to establish quantitative relationshipbetween observed infrasonic period and yield, together with uncer-tainty bounds for both. This period–yield relationship is remarkablysimilar to the AFTAC period–yield equation derived from nucleartests. Additionally, we have examined the influence of winds oninfrasound period and found these to produce negligible changes toperiods in practice. We have also established, through a set of fullmulti-variate fits, that infrasonic amplitude at long ranges follows a� 1

2 yield scaling and is best described by a range scaling exponentnear �1.1. Fig. 23 compares our amplitude–range fit (for a 1 ktnominal yield), together with the fit uncertainty bounds to othersimilar relations. As expected there is much deviation at shorterranges, but at ranges of � 5000 km our results and all (except theANSI scaling and Soviet nuclear relations) other overpressure-rangerelations agree to within a factor of two. Our fits also suggest thatthe wind exponent, k, may have a range dependence, falling in valueas ranges increase.

An empirical analysis of our dataset suggests that bolides withenergies in excess of 20 kt should be detected globally by thecurrent IMS system, while 1 kt events are detectable at ranges upto 7000 km under typical conditions. From our observations wehave established range–detection curves as a function of bolideenergy for the current IMS network as well as expected maximumupper and lower frequency content for bolide signals as a functionof range (summarized in Table 1).

We also find lower limits to the integral acoustic efficiencyfor bolides of order 0.1% with a variation of several orders ofmagnitude, in broad agreement with earlier results. It is clear,however, that any particular bolide event may have a wide rangeof possible efficiency values from our analysis, emphasizing theindividual nature of each event.

Finally, we have attempted to apply a normal-mode approachto estimate source height for bolides without success, which weattribute to the long range observations involved and the range-independent atmosphere required in the implementation ofnormal mode computation we adopted. For relatively short rangeevents, normal mode modeling may still be applicable, especiallyif applied in case by case studies where each event has detailedinformation from other sources as was successfully done in theMoravka event. In future, bolide heights might be determinedusing similar methods as in this study, however, with a moreaccurate time domain parabolic equation with a range dependentatmosphere implementation as well as other modeling techni-ques as was done recently in Silber et al. (2011).

A future extension of this work would be to examine a largesuite of infrasound returns from bolides having known trajec-tories and high quality ground truth for energy and energydeposition as a function of height beyond simple satellite energies(such as video/photographic records or recovered meteorites).Such a higher quality dataset would allow for detailed propaga-tion modeling and isolate which portion of a meteor trail isproducing acoustic radiation at each station.

Acknowledgments

The authors offer their thanks to the careful reviews of theoriginal manuscript from J.M. Trigo-Rodrıguez and one anonymous


reviewer. P.G.B. also thanks Natural Resources Canada, NSERC andthe Canada Research Chairs Program for funding support. T.N.E.and E.A.S. also thank Natural Resources Canada and NSERC foradditional funding for this study.

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