ARTICLE IN PRESS

0032-0633/$ - se

doi:10.1016/j.ps

�CorrespondE-mail addr

Planetary and Space Science 54 (2006) 78–86

www.elsevier.com/locate/pss

Analytical and graphical determination of the trajectory of a fireballusing seismic data

Jose Pujola,�, Paul Rydelekb, Yoshiaki Ishiharac

aDepartment of Earth Sciences, University of Memphis, Memphis, TN 38152, USAbCenter for Earthquake Research and Information, University of Memphis, Memphis, TN 38152, USA

cResearch Center for Prediction of Earthquakes and Volcanic Eruptions, Tohoku University, Aoba-ku, Sendai, 980-8578, Japan

Received 28 March 2005; received in revised form 10 August 2005; accepted 29 August 2005

Available online 9 November 2005

Abstract

In this paper we discuss two methods, one analytical and the other graphical, to determine the trajectory of a fireball using the arrival

times of atmospheric shock waves recorded by a seismic network. In the analytical method the trajectory and the raypaths are assumed to

be straight and we solve for the fireball velocity, the azimuth (j) and elevation angle (d) of the trajectory, the coordinates of the

intersection of the trajectory with the earth’s surface, and the corresponding intersection time (t0). Because the problem is nonlinear, we

solve it iteratively. The fireball velocity cannot be determined uniquely, and trades off with t0. The graphical method is based on the

drawing of contours of arrival times, which should be elliptical for fireball shock waves. If the distribution of seismic stations is

appropriate, the horizontal projection of the fireball is given by the axis of symmetry of the contours, which allows the estimation of j,while d can be estimated from the spacing between contours along the symmetry axis. Application of the two methods to data from four

fireballs shows that the graphically derived parameters can be within a few degrees of the analytical parameters. In addition, a fireball

recorded in the Czech Republic has reliable trajectory parameters derived from video recordings, which allows an independent

assessment of the quality of the parameters determined analytically. In particular, j and d have errors of 1:7 � and 1:3 �, respectively,which are not particularly large considering that the station distribution was not favorable.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Fireball; Trajectory; Seismic; Shock waves

1. Introduction

Borovicka et al. (2003) noted that of more than eighthundred documented meteorite falls, only six had a reliabledetermination of the velocity and trajectory of the fireballthat preceded the fall. More recently, ReVelle et al. (2004)added another meteorite to the list. Given the importanceof this information for studies of the solar system,increasing the number of well-determined fireball trajec-tories is highly desirable. Interestingly, some of thisinformation is already being provided by analysis of thefireball shock waves recorded by seismic networks (e.g.,Brown et al., 2003; Ishihara et al., 2003, 2004; Pujol et al.,2005). In recent years, the number and quality of seismic

e front matter r 2005 Elsevier Ltd. All rights reserved.

s.2005.08.003

ing author. Tel.: +1901 6784827; fax: +1 901 6782178.

ess: jpujol@memphis.edu (J. Pujol).

networks around the world has increased significantly and,as a consequence, the network data have the potential tobecome an important source of information to theastronomical community. In this paper we briefly describetwo methods, one analytical and the other graphical, todetermine the trajectory of a fireball using the arrival timesof the atmospheric shock waves recorded at seismicstations (Pujol et al., 2005). The analytical method isiterative and has been designed to compute the velocity ofthe fireball, the azimuth and elevation angle of thetrajectory (assumed straight), the coordinates of theintersection of the trajectory with the earth’s surface, andthe corresponding intersection time. The sound speed isassumed to be known and constant. Under appropriateconditions the azimuth, elevation angle, and the coordi-nates of the intersection are well constrained. The fireballvelocity, however, cannot be determined uniquely; it

ARTICLE IN PRESSJ. Pujol et al. / Planetary and Space Science 54 (2006) 78–86 79

depends on the initial value assigned to it and trades offwith the intersection time. The graphical method requiresthe drawing of the isochrones, which may allow a quickcomputation of the azimuth and elevation angle of thetrajectory. The two methods have been applied to datafrom fireballs recorded in the United States (Arkansas),Japan (Miyako and Kanto) and the Czech Republic(Moravka), and comparison of the results obtained usingthe two methods gives a good idea of the capabilities andlimitations of the graphical method. In addition, theMoravka fireball has reliable trajectory parameters derivedfrom video recordings, which allows an independentassessment of the quality of the parameters determinedanalytically.

(a)

(b)

Fig. 1. (a) Local spherical coordinate system used to define the trajectory

of a fireball. The system is centered at the intersection of the trajectory

with the earth’s surface (point P). The angle d is the elevation of the

trajectory and u is a unit vector. (b) Geometry for the derivation of Eq. (1).

The dashed lines are as in (a). The wave that arrives at a point S on the

earth’s surface originates at the point T on the trajectory that makes the

line TS perpendicular to the Mach cone, identified by its angle b. FromPujol et al. (2005).

2. Methods

2.1. Analytical method

Before applying this method it is necessary to derive theequation for the arrival time of the shock waves at pointson the surface of the earth. This in turn requires a numberof assumptions regarding the propagation of the waves,which usually are the following: (1) the fireball trajectory isa straight line, (2) its velocity ðvÞ and the speed of sound (c)are both constant, and (3) there are no atmospheric winds.Because c is actually a function of the fireball height, anaverage value is used and the raypaths are assumedstraight. As noted in the references given above, theseassumptions are reasonable for the distances involved inthe examples discussed here, which also allow neglectingthe earth’s curvature. As distance increases the assumptionof straight rays is violated and the propagation model is nolonger valid. It is possible, however, to improve on theapproximations of constant c and absence of winds bycomputing average values of c depending on the fireballheight (e.g., Brown et al., 2003) and by moving the stationsby an amount opposite to the wind drift (Borovicka andKalenda, 2003). The effect of some of the approximationswe used will be assessed in the context of the Moravkafireball, for which reliable parameters derived from opticaldata are available. As Fig. 1 shows, the trajectory is definedby the following parameters: the angles j and W and thecoordinates ðx0; y0; 0Þ of its intersection with the earth’ssurface (point P). The elevation angle (d) is equal to 90� W.Two other parameters to be determined are the velocity v

and the time t0 at point P. The wave generated at point T

on the trajectory arrives at a station S with coordinatesðxs; ys; zsÞ at a time t given by

t ¼ t0 �dt

vþ

dp cos bc

, (1)

(Pujol et al., 2005) where d t and dp are distances along andperpendicular to the trajectory, respectively, equal to

d t ¼ PQ; dp ¼ SQ (2)

(see Fig. 1) and b is the Mach angle, equal to

b ¼ sin�1c

v. (3)

Eq. (1) generalizes that used by Ishihara et al. (2003, 2004),which is valid for a rotated system with the z-axis along thetrajectory and the x-axis in the vertical plane that containsthe trajectory. For the computation of d t and dp thefollowing relations are used:

u ¼ ðcosj sin W; sinj sin W;� cos WÞ ¼ ðu1; u2; u3Þ, (4)

b ¼ SP�!¼ ðxs � x0; ys � y0; zsÞ, (5)

dt ¼ jbj cos a ¼ jb � uj, (6)

dp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijbj2 � d2

t

q. (7)

Before proceeding, it is worth noting that shock wavescannot be observed at all points on the ground. Withreference to Fig. 1b, the waves cannot reach points to theleft of a line passing through the point P and perpendicularto the projection of the trajectory. On the other hand, not

ARTICLE IN PRESSJ. Pujol et al. / Planetary and Space Science 54 (2006) 78–8680

all the points to the right of this line will receive waves, astheir existence depends on the energy of the fireball. Inaddition, because a fireball decelerates during its passagethrough the atmosphere, its velocity will become subsonicwell before the point P in Fig. 1, which actually representsa limit point. The Moravka fireball will be used as anexample of this fact. For the first three fireballs discussedhere, the point P was beyond the network, but for thefourth one (Moravka) it was inside, and in this case someof the stations did not record the shock waves (Brownet al., 2003).

As Eq. (1) shows, the time t is a nonlinear function of allthe parameters except t0, which means that given a set ofobserved arrivals times it is not possible to solve for theparameters in a single step. For this reason we solved theproblem using a standard linearization approach based onthe Taylor expansion of t about initial estimates of theparameters, which gives

r ¼ tobs � t ¼qt

qvdvþ

qt

qx0dx0 þ

qt

qy0

dy0

þ dt0 þqt

qjdjþ

qt

qWdW, ð8Þ

where r stands for arrival time residual, tobs is the observedarrival time at a given station, and t is the correspondingtheoretical time, computed for the initial estimates usingEq. (1). Eq. (8) is linear in the parameter adjustments dv,dx0, dy0, dt0, dj, and dW, and was solved using dampedleast squares and the generalized inverse technique. Bothsolutions have been discussed in detail in Pujol et al. (2005)and produce similar results. Once the adjustments havebeen computed, the initial estimates are updated and theprocess is repeated until a best-fit solution is found. Toquantify the goodness of the fit we use the root-mean-square residual, defined by

RMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N � 6

XN

i¼1

r2i

vuut , (9)

where N is the number of stations and ri is the residualcorresponding the ith station. The 6 in the denominator isthe number of parameters to be computed, and isintroduced to make RMS and unbiased estimate of thestandard deviation (Jenkins and Watts , 1968). If the valuesof one or more of the parameters are fixed, the 6 should bereplaced by the appropriate number.

Once the best-fit solution has been computed, for eachstation the following quantities are also computed. One isthe height of point T in Fig. 1b, equal to

hT ¼ jTPj sin d (10)

which is useful because it is expected that the size of ameteorite will increase as the penetration of the fireballincreases. Also computed are the coordinates of the surfaceprojection of the point T, given by

xT ¼ jTPju1; yT ¼ jTPju2. (11)

This information is important because it can be used toestimate the reliability of the computed parameters whenthe stations have three-component seismometers. In such acase the two horizontal components can be used to find thedirection of the arrival of the waves (i.e., the horizontalpolarization) at each station. In the absence of winds, thisdirection should agree with the direction of a vector fromthe station location to the point with coordinates ðxT ; yT Þ.This comparison was done for the Arkansas fireball withsatisfactory results (Pujol et al., 2005). In addition, a linethrough the points ðxT ; yT Þ for all the recording stationsidentifies the part of the trajectory that generated the shockwaves. This line will be plotted for each of the fireballsconsidered below.

2.2. Graphical method

To introduce the method it is convenient to work in arotated coordinate system ðx0; y0; zÞ with the x0-axis alongthe horizontal projection of the trajectory. This requires acounterclockwise rotation of angle j about the z-axis,which means that in the rotated system j ¼ 0 and y00 ¼ 0.We will also assume that all the stations are at the sameheight above sea level, which for convenience will be takenas zero. Under these conditions Eqs. (4)–(7) become

u ¼ ðsin W; 0;� cos WÞ, (12)

b ¼ ðx0s � x00; y0s; 0Þ, (13)

dt ¼ jb � uj ¼ jðx0s � x00Þ sin Wj, (14)

dp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx0s � x00Þ

2 cos2 Wþ y02s

q(15)

(Pujol et al., 2005).Introducing Eqs. (14) and (15) into Eq. (1) shows that

tðy0sÞ ¼ tð�y0sÞ, as expected. In other words, the theoreticalisochrones are symmetric with respect to the projection ofthe trajectory. Therefore, if the distribution of the seismicstations is adequate (see the examples below), to determinethe projection of the trajectory using observed arrival timestwo steps are needed. First, draw the isochrones. Second,find the symmetry axis. This in turn allows for theimmediate determination of j. It must be noted, however,that the assumption of equal height for all the stations maynot be valid, in which case it will be necessary to includeheight corrections. For example, the stations that recordedthe Kanto fireball (see below) have heights ranging betweenabout 0 and 1.5 km, which for vertical rays translates into adifference of 4.5 s in arrival times for a sound speed of0.33 km/s near the earth’s surface. Therefore, ignoring thestation heights has the potential to distort the shape of thecontours. For this reason we add to the observed arrivaltimes a correction equal to the station height divided by0.33 km/s. Although this correction is only approximate, itmade the graphical contours noticeably closer to theanalytical contours. If the stations are underground thecorrection is not applied.

ARTICLE IN PRESS

Fig. 2. Stations used for the study of the trajectory of the Arkansas

fireball (solid and open circles). All the stations were used for the

generation of the graphical contours (bold lines) while those identified by

solid circles were used with the analytical method. Station and arrival time

information are provided in Table A1. The dashed lines were generated by

contouring of arrival times generated with Eq. (1) and the parameters in

Table 1. The bold straight line corresponds to the projection of the part of

the trajectory that generates the waves reaching the stations used with the

analytical method. The segment associated with each of those stations

indicates the direction of the projection of the ray from the trajectory to

the station. The dashed straight line represents the symmetry axis for the

graphical contours determined by visual inspection. Matlab software was

used to interpolate the irregularly spaced data onto a regular grid and to

generate the contours.

J. Pujol et al. / Planetary and Space Science 54 (2006) 78–86 81

The determination of W (and thus, d) is based on thefollowing considerations. First, note that

qt

qx0s¼ �

sin Wv

Sþ1

dp

cos bcðx0s � x00Þ cos

2 W, (16)

where

S ¼ sgnðb � uÞ. (17)

Second, along the trajectory y0s ¼ 0 and in this case Eq. (16)reduces to

qt

qx0s¼ �

sin Wv

Sþcos b

ccos W; y0s ¼ 0. (18)

Finally, for fireballs, vbc, cos b � 1 and from Eq. (18) weobtain

cos W � cDt

Dx0; y0s ¼ 0, (19)

where Dx0 is the distance between two given isochronesalong the projection of the trajectory and Dt is thecorresponding time difference (Pujol et al., 2005). Thepossibility of deducing W and j from the isochrones wasalso pointed out by Tatum (1999).

3. Applications

We applied the analytical and graphical methods to afireball observed over the northeastern part of the state ofArkansas in the USA, two fireballs in Japan (Miyako andKanto), and another fireball in the Czech Republic(Moravka). The fireballs have been investigated by otherresearchers and here we use their arrival times and theirvalues of c (except for the Moravka fireball). The stationlocations, arrival times and time residuals for all thefireballs are given in the tables in the Appendix. The tablesalso include the height hT defined above.

3.1. Arkansas fireball

The fireball was observed on November 3, 2003, at about10 pm local time (equal to UT � 6). It was reported to bebrighter than Venus, moved in a roughly east-to-westdirection, and was associated with loud sounds. The shockwaves generated by the fireball were clearly recorded bymore than twenty stations of the University of Memphisseismic network, but the stations used in the determinationof the trajectory (Fig. 2) excluded a few others to the northand east, which had very large residuals when the analyticalmethod was applied. As noted above, the assumptionsbehind the analytical method impose a limit to thedistances involved and ignoring the more distant stationsis justified. Interestingly, the residuals for two of thestations in Fig. 2 are considerably larger than for the otherstations in the figure, although in this case it was notpossible to identify the cause of these large residuals (Pujolet al., 2005). The results from the analytical method withthese two stations removed from the data set are given in

Table 1. The isochrones computed using these parametersas well as those determined by contouring of the arrivaltimes are shown in Fig. 2. The agreement between the twosets of contours is excellent, and far surpassed ourexpectations. There are minor differences near the twostations not used with the analytical method, which seemsto confirm that the arrival times for these two stations aresomewhat anomalous.As noted above, the drawing of the isochrones is the first

step in the graphical method. The second step is to find thesymmetry axis. Whether this step is feasible or not dependson the distribution of stations with respect to the projectionof the trajectory. For the Arkansas fireball, although mostof the stations are on one side of the trajectory, the changein the curvature of the contours is sufficient to infer theposition of the axis. To avoid bias, in all the examplesdiscussed here the symmetry axes shown in the correspond-ing figures were determined visually without reference tothe analytical results. In this case the difference between thegraphical and analytical azimuths is about 2 �, while thedifference in W is only 1 � (Table 2). For all the fireballs Dx0

ARTICLE IN PRESS

Table 1

Fireball parameters determined using the analytical method

Fireball v x0 y0 t0 j W RMS F Y c

Arkansas 19.5 �3.3 �97.7 �165.2 �92.6 49.1 0.27 �90.5 35.5 0.32

Miyako 27.6 �80.9 186.4 68.0 105.1 69.3 0.47 141.0 39.0 0.32

Miyako 26.2 �80.6 189.4 65.4 104.7 69.3 0.30 141.0 39.0 0.32

Kanto 18.4 78.2 108.5 �151.2 51.3 73.4 0.54 141.0 36.0 0.31

Kanto 13.8 81.4 111.5 �151.3 51.2 73.3 0.27 141.0 36.0 0.31

Moravka 21.0 48.5 7.9 �69.6 6.2 70.9 0.19 18.5 49.6 0.32

Moravka 12.0 45.1 7.7 �67.0 6.4 70.1 0.17 18.5 49.6 0.32

x0 and y0 are measured in km with respect to a reference point with longitude and latitude given by F and Y. v and c are in km/s and t0 and RMS in

seconds. t0 is measured with respect to the reference time indicated in the corresponding table in the Appendix. The elevation angle is d ¼ 90� W. Thefireball velocity ðvÞ depends on the initial value, and trades off with t0. The two sets of values given for the Miyako and Kanto fireballs were obtained using

the stations listed in Tables A2 and A3. The two sets of values for the Moravka fireball show the effect of different values of v (the value of 12.0 was fixed).

Table 2

Graphical values of j and W

Fireball j W Dt Dx0 c

Arkansas �91.0 50.0 60 29.9 0.32

Miyako 99.5 70.0 70 65.8 0.32

Kanto 45.0 74.5 60 69.4 0.31

Dx0 measured along the symmetry axis determined graphically. Wcomputed using Eq. (19).

Fig. 3. Stations used for the study of the trajectory of the Miyako fireball.

Symbols and other features as in Fig. 2. Station and arrival time

information are provided in Table A2. For the analytical contours and

trajectory information the parameters in Table 1 corresponding to v ¼

27:6m=s were used.

J. Pujol et al. / Planetary and Space Science 54 (2006) 78–8682

is measured along the graphical symmetry axis and ischosen so that it goes over as many well-defined contoursas possible.

3.2. Miyako fireball

This bright fireball was observed on March 30, 1998, atabout 3:20 am local time (equal to UT + 9) and its shockwaves were recorded by stations of a portable seismicnetwork installed by a consortium of Japanese universities,and by permanent stations installed by Tohoku Universityand the Japan Meteorological Agency. Ishihara et al.(2003) determined the trajectory using an equationequivalent to Eq. (1) and a forward grid search method.Their optimal values of j and W are equal to 107 � and71:5 � (in the convention used here). The correspondingvalues computed using two photographic records are 116 �

and 71:7 �. For the analytical method we used thedata from all the stations except one with a large residual(Fig. 3). Our values of j and W (105 � and 69 �, Table 1) arevery close to those of Ishihara et al. (2003). However,because the values of j are considerably different fromthose determined using the photographic records werepeated the computations using a subset of fifteen stations(see Table A3) with much smaller residuals. The corre-sponding results show only small differences in both j andW (Table 1) and considerably larger differences for some ofthe other parameters. However, in spite of these differencesthe values of hT for the second case are larger than for the

previous case by relatively small amounts, which varybetween 1.1 km for station JOM and 0.9 km for stationGNY. On the other hand, the coordinates of the projectionof the point T for each station in the two subsets areessentially the same.For the graphical method all the stations were used. The

agreement between the graphical and analytical contours

ARTICLE IN PRESS

Fig. 4. Stations used for the study of the trajectory of the Kanto fireball.

Symbols and other features as in Fig. 2. Station and arrival time

information are provided in Table A3. For the analytical contours and

trajectory information the parameters in Table 1 corresponding to v ¼

18:4m=s were used. The station identified by a crossed circle controls the

shapes of the 20–50 s graphical contours.

J. Pujol et al. / Planetary and Space Science 54 (2006) 78–86 83

(Fig. 3) is good, but not as close as for the Arkansasfireball. This is puzzling because the corresponding stationdistributions seem to be equally adequate to constrain thegraphical contours in both cases and the arrival timeresiduals are only somewhat larger for the Miyako fireball(Tables A1 and A2). Therefore, one would expect no majordifferences for the two cases. To investigate this differencewe generated synthetic arrival times for all the stations inFig. 3 assuming that their heights were equal to zero andusing the parameters in Table 1. Surprisingly, the syntheticgraphical contours are very similar to the actual ones,which means that their irregularity must be an artifactof the station distribution and/or the contouringsoftware. Fig. 3 also shows that the graphical contoursdo not have a straight axis of symmetry, so that thereis a range of possibilities for the selection of a trajectoryaxis. The one shown in the figure has W within one degree ofthe analytical value but j has a difference of about 6 � withthe analytical axis. Interestingly, regardless of the methodused, the seismically determined values of j differsignificantly from the photographic value.

3.3. Kanto fireball

It was observed on June 16, 2003 at about 22:10 localtime (equal to UT + 9) and its shock waves were recordedby stations installed by the Japan Meteorological Agency,the National Institute of Earth Science and DisasterPrevention, the University of Tokyo and Tohoku Uni-versity. Ishihara et al. (2004) determined the trajectoryusing the technique applied to the Miyako fireball. Becauseof heavy clouds, the only available optical recording comesfrom a video camera, which only constrains the fireball’sangular velocity. This means that the seismic data are themajor source of information for this fireball. The optimalvalues of j and W determined by Ishihara et al. (2004) areequal to 49:5 � and 74:5 �. For application of the analyticalmethod we ignored four stations that had large residuals(Fig. 4). Our values of j and W are equal to 51:3 � and 73:4 �

(Table 1). Removing three additional stations withrelatively large residuals produces very close values of t0,j and W, larger differences in the other parameters, andvalues of hT larger than before by amounts that varybetween 1.5 km for station EINB and 2.0 km for stationASHI. As for the Miyako fireball, the horizontal coordi-nates of the point T for each station remain essentiallyunchanged.

The analytical and graphical contours are shown inFig. 4. The differences in the two sets of contours isremarkable, but the contouring of synthetic data shows apattern very similar to that observed for the actual data.This includes the kink in the 50 s contour, although lesspronounced. Therefore, we conclude that the observedcontours depend to a large extent on the station distribu-tion and/or contouring software. Moreover, the stationidentified by a crossed circle in Fig. 4 controls the shape ofthe contours between 20 and 50 s; if that station is removed

the contours bend sharply to the south, thus making itdifficult to identify the trajectory. For this fireball thesymmetry axis is well constrained, but has a difference ofabout 6 � with the analytical axis. However, the differencein W is about 1 �.

3.4. Moravka fireball

This was a very bright fireball observed on May 6, 2002at about 11:50 UT by a large number of people inthe Czech Republic and in some neighboring countries.In addition, three video recordings are available, whichafter careful calibration allowed the determination ofthe fireball velocity and trajectory (Borovicka and Kalen-da, 2003; Borovicka et al., 2003). The following is asummary of results relevant to our work. The valuesof j and W are 4:5 � and 69:6 �, respectively. The fireballsuffered significant fragmentation along its trajectory.Investigation of the dynamics of the main fragmentindicates that its velocity decreased from 21.9 km/s at45.7 km height to 3.8 km/s at 21.2 km height. At the latterheight the fireball was no longer visible on the videorecordings, but this height is probably close to the terminalheight (at which the velocity of the fireball becomes lessthan the speed of sound). The fireball reached its maximumluminosity at a height of 33 km, some time after the

ARTICLE IN PRESS

Fig. 5. Stations used for the study of the trajectory of the Moravka fireball. Symbols and other features as in Fig. 2. Station and arrival time information

are provided in Table A4. For the analytical contours and trajectory information, the parameters in Table 1 corresponding to v ¼ 21:0m=s were used. Thestation identified by a crossed circle controls the shapes of all the graphical contours. Only a limited number of contours is shown.

J. Pujol et al. / Planetary and Space Science 54 (2006) 78–8684

fragmentation had begun. This video-derived informationwill be used to assess the quality of the results describedbelow.

The shock waves generated by this fireball were recordedby a seismic network installed in a mining district and by afew other nearby stations (Fig. 5). Three of the stationswere underground. The surface stations were used byBrown et al. (2003) for additional investigation of thefireball. An iterative approach was used, with the speed ofsound computed as an average value that depended on thefireball height. Their computed values of j and W are equalto 8 � and 70:2 �, which differ by 3:5 � and 0:6 � from thevalues determined from the video recordings.

Application of our analytical method to the data inBrown et al. (2003) produced the following results. Usingthe surface data and the arrival times from the shallowestunderground station (KVE) we obtained the parameters inTable 1 corresponding to a value of v equal to 21 km/s. Thevalues of j and W are 6:2 � and 70:9 �, which differ by 1:7 �

and 1:3 � from the video-derived values. When the datafrom the two other underground stations were included inthe computations the residuals were much larger. However,when only the surface data were used the new solutiondiffered only slightly from that in Table 1. Finally, to seethe effect of v on the results of the analytical method wefixed the value of v to 12 km/s, which is equal to the averageof the values of v (about 19 and 5 km/s, see Borovicka andKalenda, 2003) for the maximum and minimum values ofhT in Table A4. The corresponding values of j and W(Table 1) are very close to the previous ones and becausethe fireball moved in a roughly N–S direction the value ofy0 remained essentially unchanged. The only significantchanges are in x0 and t0, as expected. More important, thevalues of hT range between 21.8 and 34.2 km, which arevery close to those in Table A4, and the horizontalprojection of the segment of the trajectory that generatedthe shock waves (Fig. 5) is essentially the same in the twocases. Interestingly, these two extreme values of hT areclose to the terminal height and the height for maximum

luminosity, which supports our results. Regarding theactual impact point of the main fragment, Borovicka andKalenda (2003) estimated it at 49:48 �N and 18:55 �E,which is about 13 km to the south of the reference pointgiven in Table 1. On the other hand, the value of x0 is atleast 45 km to the south, which would misplace the impactpoint by at least 32 km, but as noted earlier, the pointwith coordinates ðx0; y0Þ should be considered a limitingcase. For the Moravka fireball the lowest value of hT

(22 km) is a more useful piece of information because atthat height any fireball will be highly decelerated, reachingsubsonic velocities at a horizontal distance of a fewkilometers.Although the analytical method performed extremely

well for this fireball, the graphical method could not beapplied in this case because the station distribution isinadequate. In fact, for practical purposes all the stationsbut one (identified by a crossed circle) are along a line, withthe contours controlled by the off-line station. If thecorresponding arrival time is removed, the contoursbecome very narrow and confined to the vicinity of theline defined by the remaining stations.

4. Conclusions

In this paper we discussed two methods, one analyticaland the other graphical, to determine the trajectoryof a fireball using the arrival times of atmospheric shockwaves recorded by a seismic network. Its application todata from four fireballs allows us to draw the followingconclusions.(1) The Moravka fireball trajectory has been reliably

determined thanks to the availability of carefully calibratedvideo recordings. Comparison of the corresponding valuesof j and d with those determined using the analyticalmethod shows that the latter are determined with errors of1:7 � and 1:3 �, respectively, which are not particularlylarge considering that the station distribution was notfavorable. This good agreement means that the simplifying

ARTICLE IN PRESS

Table A1

Arkansas fireball. Station information, relative arrival times and time

residuals

Station Lon. ð�Þ Lat. ð�Þ zs (km) tr (s) Res (s) hT (km)

BVAR �90.677 35.443 0.101 0.00 �0.22 41.2

HBAR �90.657 35.555 0.074 3.32 0.25 41.8

QUAR �90.649 35.644 0.115 10.17 0.21 42.0

TWAR �90.560 35.361 0.061 26.99 0.21 46.8

JHAR �90.524 35.606 0.063 29.56 0.00 47.7

TMAR �90.489 35.695 0.065 43.27 �0.10 49.2

BLAR �90.449 35.369 0.063 45.59 �0.03 51.8

NHAR �90.544 35.786 0.067 45.91 0.03 46.5

NFAR �90.393 35.448 0.063 51.19 �0.26 54.1

TYAR �90.292 35.509 0.066 68.83 �0.18 58.6

LPAR �90.300 35.602 0.067 69.86 �0.09 57.9

RVAR �90.286 35.690 0.066 78.47 �0.20 58.4

CPAR �90.236 35.556 0.067 79.96 0.08 61.0

BOAR �90.287 35.823 0.068 93.97 �0.44 58.1

HTAR �90.185 35.655 0.067 94.52 0.52 63.1

MSAR �90.147 35.784 0.069 113.34 0.19 64.5

LVAR �90.222 35.915 0.071 119.65 �0.04 60.8

BFAR �90.084 35.873 0.071 136.72

DLAR �90.008 35.810 0.067 142.78

HOVM �90.067 36.044 0.071 168.05 0.05 67.6

zs: station height (referred to sea level). tr: observed arrival time minus

reference time, equal to 3:52:40.34 UT (November 4, 2003). Res: residual,

equal to tr minus time computed using the parameters for this fireball in

Table 1. hT : height of the point T in Fig. 1b. The stations without

information in the last two columns were not used to determine the fireball

parameters.

J. Pujol et al. / Planetary and Space Science 54 (2006) 78–86 85

assumptions behind the method are not too severe,although the assumption of constant v was clearly notsatisfied. On the other hand, that assumption leads to anoverestimation of the distance that the fireball will travelafter it has reached the terminal height. This fact will haveto be taken into account when considering the values of x0

and y0 computed with the analytical method.(2) The Arkansas fireball is important for two reasons.

First, because the contouring of the arrival times does notmake any assumption about wave propagation, theexcellent agreement between the graphical and analyticalcontours provides additional independent evidence that thesimplifying assumptions in the analytical method arewarranted with the caveats above. On the other hand, thesame agreement shows that when the station distribution isadequate, then the graphical method can produce estimatesof the azimuth and elevation angle of the trajectory close tothose determined using the analytical method.

(3) The Miyako fireball is interesting because althoughthe station distribution seems adequate to constrain thecontours well, they are somewhat irregular, which mightsuggest that the data are affected by errors. However,analysis of synthetic data shows that this irregularity is anartifact related to the station distribution and/or thecontouring software. For the Kanto fireball this problemis even more pronounced, with some of the contourscontrolled by one station. In any case, the information inthe two data sets is sufficient to determine the azimuth andelevation angle within a few degrees of the values computedwith the analytical method.

(4) Comparison of the analytical and graphical contoursfor the Miyako, Kanto and Moravka fireballs (particularlyfor the last one) may make the reader wonder why theanalytical method is capable of producing ellipticalcontours, while the graphical method cannot. The reasonis that in the first method the contours are a directconsequence of the model represented by Eq. (1), whilethere are no constraints on the shapes of the contoursproduced by the contouring software.

(5) In some instances, meteors suffer explosive fragmen-tation during their passage through the atmosphere, whichbecomes another source of waves. This point sourcegenerates spherical waves, with circular contours on theearth surface (e.g., Qamar, 1995; Tatum, 1999; Tatum etal., 2000). As long as the distribution of stations isadequate, contouring of the arrival times provides a simpleway to discriminate between spherical waves and shockwaves.

(6) In summary, as the number and quality of the seismicnetworks around the world increase, the number ofdetected fireballs is likely to increase, and the graphicalmethod can be used as a tool for quick identification ofthe passage of a fireball and for a potentially reliabledetermination of its azimuth and elevation. Thesetwo angles, in turn, can be used as initial estimatesfor the analytical method, which is capable of providingmore accurate information without much additional

effort (the software is available from the first author onrequest).

Acknowledgements

The data used here were provided by the followingsources. Arkansas fireball: Center for Earthquake Researchand Information, The University of Memphis. Miyakofireball: consortium for the 1997–1998 joint seismicobservations in the Tohoku Backbone Range. Kantofireball: Japan Meteorological Agency, National Instituteof Earth Science and Disaster Prevention, EarthquakeResearch Institute (University of Tokyo) and ResearchCenter for Prediction of Earthquakes and VolcanicEruptions (Tohoku University). Moravka fireball: SeismicPolygon OKD (operated by the DPB Paskov firm), PolishAcademy of Science and Masaryk University. Dr. P.Brown gave permission for the use of the data for thisfireball. We thank the two reviewers, Drs. J. Borovicka andJ. Tatum, for their very useful comments.

Appendix

In this Appendix we provide station coordinates,arrival time information, time residuals and the heighthT of point T for the fireballs discussed in the text(Tables A1–A4).

ARTICLE IN PRESS

Table A2

Miyako fireball. Station information, relative arrival times and time

residuals

Station Lon. ð�Þ Lat. ð�Þ zs (km) tr (s) Res (s) hT (km)

KGJ 141.570 39.387 0.375 222.74

JOMa 141.290 39.473 0.210 247.83 0.37 54.1

MNS 141.200 39.355 0.065 253.78 �0.46 57.7

NTMa 141.300 39.632 0.311 262.06 0.49 52.4

THR 141.260 39.118 0.165 268.93 0.70 58.4

NAMa 141.000 39.466 0.245 277.34 0.05 62.2

HAN 140.940 39.374 0.300 278.35 �0.56 64.8

GTO 140.910 39.237 0.610 281.29 �0.58 67.0

YHBa 141.080 39.618 0.300 283.85 �0.36 58.6

SAWa 140.770 39.403 0.280 297.55 0.12 69.2

JMKa 141.220 38.952 0.070 299.12 �0.34 61.1

SWUa 140.790 39.486 0.445 301.14 0.36 67.9

HRQa 141.040 38.984 0.123 301.50 �0.06 65.7

OSKa 140.900 39.616 0.270 303.98 �0.13 63.6

HMKa 141.240 39.848 0.650 306.76 �0.18 52.1

KT44 140.720 39.086 0.400 307.38 �0.61 73.6

HRNa 140.630 39.256 0.170 308.64 0.45 74.5

KT43a 140.660 39.130 0.280 309.76 0.11 74.9

JRGa 140.630 39.396 0.200 312.09 0.26 73.1

KT48a 140.590 39.061 0.235 320.42 �0.12 77.5

GNYa 140.720 38.857 0.440 339.11 0.51 75.9

Definitions as in Table A1. Reference time: 18:20:00 UT (March 29, 1998).

The stations with values of Res were used to get the parameters with

v ¼ 27:6 in Table 1.aThe stations correspond to the other set of parameters in Table 1.

Table A3

Kanto fireball. Station information, relative arrival times and time

residuals

Station Lon. ð�Þ Lat. ð�Þ zs (km) tr (s) Res (s) hT (km)

EINBa 140.859 35.702 0.060 14.96 �0.13 33.9

TENNa 140.859 35.703 0.062 15.00 0.09 33.9

NMOT 140.217 36.554 0.140 44.79 �0.75 62.4

HITA 140.569 36.611 0.215 45.22

IYASa 140.194 36.228 0.031 54.81 0.41 56.7

ETSKa 140.110 36.210 0.280 67.99 0.40 58.0

TUYMa 140.248 36.924 0.555 86.18 0.72 68.9

NFJW 139.696 36.983 0.665 100.06

NNOM 139.977 37.126 1.515 111.31 �0.52 78.0

EKBH 139.528 36.655 0.750 111.71 �0.39 77.6

EKROa 139.498 36.687 0.865 114.11 0.21 78.8

NOHRa 139.691 36.357 0.105 115.15 �0.43 68.9

ENIKa 139.491 36.621 1.310 117.56 0.28 77.8

ESEKa 139.491 37.088 0.705 119.90 0.11 86.5

EASOa 139.460 36.649 0.720 121.55 0.36 78.8

NASOa 139.465 36.631 0.755 122.02 0.41 78.4

EGNZ 139.412 36.653 0.880 127.82

TUMAa 139.925 37.217 0.885 127.93 �0.06 80.6

ASHIa 139.453 36.425 0.241 143.43 �0.68 74.8

KATA 139.249 36.767 0.933 143.84

Definitions as in Table A1. Reference time: 13:10:00 UT (June 16, 2003).

The stations with values of Res were used to get the parameters with

v ¼ 18:4 in Table 1.aThe stations correspond to the other set of parameters in Table 1.

Table A4

Moravka fireball. Station information, relative arrival times and time

residuals

Station Lon. ð�Þ Lat. ð�Þ zs (km) tr (s) Res (s) hT (km)

CSM 18.561 49.800 0.278 0.00 0.15 22.0

KVE 18.501 49.800 �0.141 0.30 �0.10 22.1

MAJ 18.471 49.824 �0.365 2.85

CSA 18.493 49.853 �0.497 5.16

RAJ 18.582 49.851 0.272 6.30 �0.18 23.7

LUT 18.415 49.883 0.217 10.80 0.18 25.2

PRS 18.553 49.914 0.205 12.70 0.04 25.9

BMZ 18.141 49.834 0.250 39.00 �0.08 24.3

RAC 18.191 50.083 0.214 47.60 �0.04 32.6

MORC 17.546 49.775 0.742 155.40 0.03 23.9

OJC 19.797 50.219 0.300 248.80 0.00 33.8

Definitions as in Table A1. Reference time: 11:53:20.1 UT (May 6, 2002).

The computations were carried out using the parameters corresponding to

v ¼ 21:0 in Table 1.

J. Pujol et al. / Planetary and Space Science 54 (2006) 78–8686

References

Borovicka, J., Kalenda, P., 2003. The Moravka meteorite fall: 4.

Meteoroid dynamics and fragmentation in the atmosphere. Meteorit.

Planet. Sci. 38, 1023–1043.

Borovicka, J., Spurny, P., Kalenda, P., Tagliaferri, E., 2003. The Moravka

meteorite fall: 1. Description of the events and determination of the

fireball trajectory and orbit from video records. Meteorit. Planet. Sci.

38, 975–987.

Brown, P., Kalenda, P., ReVelle, D., Borovicka, J., 2003. The Moravka

meteorite fall: 2. Interpretation of infrasonic and seismic data.

Meteorit. Planet. Sci. 38, 989–1003.

Ishihara, Y., Tsukada, S., Sakai, S., Hiramatsu, Y., Furumoto, M., 2003.

The 1998 Miyako fireball’s trajectory determined from shock wave

records of a dense seismic array. Earth Planets Space 55, e9–e12.

Ishihara, Y., Furumoto, M., Sakai, S., Tsukada, S., 2004. The 2003 Kanto

large bolide’s trajectory determined from shockwaves recorded by a

seismic network and images taken by a video camera. Geophys. Res.

Lett. 31, L14702 odoi:10.1029/2004GL0202874.

Jenkins, G., Watts, D., 1968. Spectral Analysis. Holden-Day, Oakland,

CA.

Pujol, J., Rydelek, P., Bohlen, P., 2005. Determination of the trajectory of

a fireball using seismic network data. B. Seismol. Soc. Am. 95,

1495–1509.

Qamar, A., 1995. Space shuttle and meteoroid—tracking supersonic

objects in the atmosphere with seismographs. Seism. Res. Lett. 66 (5),

6–12.

ReVelle, D., Brown, P., Spurny, P., 2004. Entry dynamics and acoustics/

infrasonic/seismic analysis for the Neuschwanstein meteorite fall.

Meteorit. Planet. Sci. 39, 1605–1626.

Tatum, J., 1999. Fireballs: interpretation of airblast data. Meteorit.

Planet. Sci. 34, 571–585.

Tatum, J., Parker, L., Stumpf, L., 2000. Sound from a fireball—

distinguishing between the hypersonic shock front and the terminal

burst. Planet. Space Sci. 48, 921–923.