Reconstruction of lightning currents and return stroke model parameters using remote electromagnetic fields

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. D19, PAGES 24,469-24,481, OCTOBER 16, 2000 Reonstrution of lightning urrents an return stroke moel parameters using remote eletromagneti fiels M. Popov Division of Eletromagneti Theory, Royal Institute of Tehnology, Stokholm, Sween S. He Centre for Optial an Eletromagneti Researh, State Key Laboratory of Moern Optial Instrumentation, Zhejiang University, Hangzhou, People's Republi of China R. Thottgppillil Institute of High Voltage Researh, Uppsala University, Uppsala, Sween Abstrat. Estimating parameters of a lightning return stroke moel an the urrent at the base of the lightning hannel from measurements of remote eletromagneti fiels is an important pratial problem. In the present paper we apply a geneti algorithm (a global optimization tehnique) to the above problem. Firstly, we use the geneti algorithm to reonstrut the return stroke moel parameters using the hannel-base urrent an the remote eletromagneti fiel measure simultaneously. The hannel-base urrent is use to alulate the eletri fiel at the measurement point. The ifferene between the alulate an the measure fiels is minimize using the geneti algorithm, whih fins the optimal solution for the return stroke parameters. The transmission line an Dienorfer-Uman return stroke moels are use in the illustrations. Seonly, we evelop a metho to reonstrut the return stroke moel parameters an hannel-base urrent from eletri or magneti fiels measure at two ifferent istanes, one of whih is a far-fiel an the other is a near fiel. The Dienorfer-Uman moel is use in the reonstrution. Thirly, we evelop a metho to reonstrut the return stroke moel parameters an hannelbase urrent from two magneti fiels at two ifferent istanes, one of whih is an intermeiate fiel an the other is a near fiel. For oing the above with the Dienorfer-Uman moel we erive an expliit inversion formula using both the inution an the raiation terms of the magneti fiel. 1. Introution Determination of the urrent at the base of a lightning return stroke hannel from remotely measure eletri an magneti fiels is an interesting pratial problem. Most of Europe, North Ameria, an some parts of Asia an South Ameria are now overe by automati lightning loation networks, an these networks use remotely measure eletromagneti fiels from the lightning return stroke to estimate the peak return stroke urrent at the hannel base. Furthermore, etermination of the urrent an harge on the light- ning hannel is important in unerstaning the physis of lightning, an the most ommonly measure signals from natural lightning are the eletri an magneti fiels. Expressions relating the temporal an spatial istribution of the lightning urrent or harge in the return stroke hannel with remote eletri an magneti fiels proue by the return stroke are foun in the literature [e.g., Urnan 1985; Thottappillil et al., 1997]. Expres- sions for the vertial eletrial fiel E an horizontal magneti fiel B 0 at groun level are given as follows: Also at the Division of Eletromagneti Theory, Royal Institute of Tehnology, Stokholm, Sween. Copyright 2000 by the Amerian Geophysial Union. Paper number 2000JD /00/2000JD ,469

2 24,470 POPOV ET AL.' RECONSTRUCTION OF LIGHTNING CURRENTS z'- / / v i (z',t) Figure 1. Geometry of the return stroke moel. B (,t)_ lzo ]in(t) [.( Oi(z,t-7) where is the horizontal istane to the hannel from (2) the measuring point, is the spee of light, R = V/Z'2 q- 2 is the istane to the hannel at height z', an h(t) is the height of the return stroke wave front at time t as seen by an observer at the fiel measuring point (see Figure 1). Given a return stroke spee v(z'), h for a speifie time t is etermine from the following relation [f. ThottappilIil an Urnan, 1994] h V/h t---- q-, Fay(h) where Vav(h) is the average return stroke spee efine by h ' Return stroke moels that relate the hannel-base ur- rent to the temporal an spatial variation of urrent in the hannel an then be use onveniently to alulate eletri an magneti fiels using (1) an (2) [e.g., Brue an Cole, 1941; Uman an MLain, 1969; Lin et ai., 1980; HeiIer, 1985; Nui et ai., 1988; Rakov an DuIzon, 1991; Dienorfer an Uman, 1990; Cooray, 1993; ThottappiIIiI et al., 1991]. A review an omparison of various return stroke moels an be foun in the work of Nui et ai. [1990], ThottappiIIiI et ai. [1997], an Rakov an Uman [1998]. Previously, attempts were mae to test the valiity of some of the moels using simultaneously measure hannel-base urrent, return stroke spee an remote eletri fiels from return strokes in triggere lightning [Willerr et al., 1989; Thor- tappiiiii an Uman, 1993]. It was foun that all moels give reasonable peak eletri fiels, if proper parameters are use. However, there were large variations in the moels' ability to reproue the measure eletri fiel wave shapes for a given hannel-base urrent. In general, it is not possible to solve the remote fiel expressions (1) an (2) to fin the urrent istribution on the hannel or the hannel-base urrent. However, far from the return stroke hannel the main ontribution to the total fiel is the raiation fiel, whih is given by the last term of (1) or (2), involving the time erivative of the urrent. In this ase, for a given return stroke moel, relatively simple analytial expressions an be foun to etermine the hannel-base urrent from the far-fiel [e.g., Urnan an MLain, 1970; Rahii an ThottappiIIiI, 1993]. Far-fiel approximation an be use suessfully at any istane an time, as long as the total fiel at the groun level is mostly given by the raiatio n fiel. Return stroke wave front propagates upwar at spee about one thir of the spee of light. The maximum rate of hange of the urrent at the hannel base happens within the first few miroseons. Therefore fiels at groun level at a istane of a kilometer or so may be ominate by the raiation fiel for these first few miroseons. Most of the lightning loation networks use an analyti expression to estimate the return stroke peak urrent from the measure far-eletri or magneti fiel. The ommonly use relationship between lightning urrent an far-eletri fiel at groun level is given by i(o, t) 2 reø2 E( r+ where an are as efine earlier, an v is the upwar spee of the return stroke wave front. Note that it is assume that the istane is large enough for the raiation term of the vertial eletri fiel to be ominant. Equation (3) is appliable only for early times t, when the length vt of the hannel, whih signifiantly ontributes to the raiation fiel, is small ompare to. Equation (3) is erive by assuming so-alle "transmission line" (TL) moel in whih the urrent at the hannel-base is assume to travel upwar with a onstant spee v, without any istortion or attenuation. In lightning loation systems, istane in equation (3) is etermine using magneti iretion fining or time-of-arrival tehnique using multiple stations. The average spee of the lightning return stroke is typially assume (however, for some triggere lightning it an be measure) an usually put to be one thir of the spee of light. Ione et ai. [1998] has etermine that the experimental relationship between the return stroke peak urrent an the fiel peak use by the National Lightning Detetion Network is equivalento using an effetive average return stroke spee of 1.2 x 10 s m/s (see also Ione et al. [1993] an Cummins et ai. [1998]). However, the atual spee of the return stroke an be

3 POPOV ET AL- RECONSTRUCTION OF LIGHTNING CURRENTS 24,471 anywhere from 0.2 x 108 m/s to very near to 3 x 108 m/s (see Table 2 in Rakov et al. [1992]), an the spee varies from lightning to lightning. Therefore the theoretial error in iniviual peak urrent estimates using (3) an the assume return stoke spee 1 x x 108 m/s an be very large. If the spee of the return stroke, a parameter in the TL moel, an also be estimate, for example, from the simultaneous measurement of fiels at more than one istane, then the error in peak urrent estimation an be reue. In the metho propose in this paper the eletri or magneti fiel at istane from the return stroke is assume to be given by (1) or (2), respetively, with the approximation that the height of the hannel vt is small ompare to the istane. Then for a given return stroke moel the hannel-base urrent is alulate as a funtion of the moel parameters. Simultaneous fiels at two ifferent istanes are use to estimate the value of the moel return stroke parameters an hannel-base urrent that gives the best math to the fiels. The parameters are etermine by the geneti algorithm, a global optimization tehnique [e.g., Golberg, 1989]. Although the raiation fiel omponent is usually use in the inversion formulas to fin the hannel-base urrent, Urnan an MLain [1970] have erive expliit inversion formulas for the hannel-base urrent using both the inution an the raiation fiel omponents of the magneti fiel. These relationships are onsiere to be vali for any istane beyon a kilometer. They use the transmission line an Brue-Gole return stroke moels for the inversion. However, reent moel valiation stuies showe that for a given hannel-base urrent the Brue-Gole moel gives large errors in the peak fiels [f. Thottappillil an Urnan, 1993], whereas the transmission line moel gives the large errors in the alulation of the near-fiel waveshape [f. Thottappillil et al., 1997]. Therefore more omplex moels suh as the Dienorfer-Uman moel are preferre if both the fiel peak an the overall fiel wave shape are of interest. In this paper we erive an expliit inversion formula for the hannel-base urrent using the Dienorfer-Uman moel an both the inution an the raiation terms of the magneti fiel. Reently, Cooray an Gomes [1998] onsiere the return stroke moel propose by Cooray [1993] an use a proeure to estimate the peak return stroke urrents an return stroke spees from single-station measure fiels. The proeure was to hange the spee an other return stroke parameters of the moel until the best fit to the measure fiels are obtaine. They i not use any expliit inversion formula to alulate the hannel-base urrent from the measure fiels. The the return stroke moels use in this paper are ifferent from that in the work of Cooray an Gornes [1998]. 2. Theory an Appliation 2.1. Problem Formulation Consier a straight vertial return stroke hannel above a perfetly onuting plane. The geometry of the moel is shown in Figure 1. The return stroke is assume to start at groun level, z : 0, at time t = 0 an thereafter propagates upwar with spee v, whih is inepenent of height. The return stroke front is- harges the harge store in the hannel exponentially an the total isharge urrent travels own with a onstant spee u. Then the spatial-temporal istribution of the return stroke for the Dienorfer-Uman (DU) moel is [f. Dienorfer an Urnan, 1990] z! i(z', t) - i(o, t + -) - i(o, u t x xp( ), 7- D! Z,! Z,! Z! t > --, (4) where TD is a isharge time onstant. Stritly speaking, the original DU moel requires an arbitrary ivision of the urrent into two parts, one assoiate with a fast time onstant (the breakown part) an the other with a slower time onstant (orona part). In the present paper we use only one urrent omponent an its assoiate time onstant in alulating the fiels, with the justifiation that the moel fiels are proue essentially by the breakown omponent of the urrent for the first few miroseons of the return stroke [Dienorfer an Urnan, 1990]. When the moel urrent along the hannel is etermine, the vertial eletrial fiel Ez an horizontal magneti fiel B e at groun level an be alulate using (1) an (2). Typially, the value of the return stroke spee v is assume, although oasionally, for some triggere lightning, it is obtaine from the optial measurements. The values of other parameters are assume (u is usually assume equal to the spee of light an -D is hosen to be about lzs [Dienorfer an Uman, 1990]). In the present paper we onsier the ase when all three DU moel parameters ( -D, v, an u) are unknown an evelop a tehnique to reonstrut their values either from a remote fiel an the hannel-base urrent or from two remote fiels. In the latter ase the tehnique evelope allows us to simultaneously reonstrut the unknown parameters an the hannel-base urrent. In Appenix A we survey the geneti algorithm an in setion 2.2 apply it for the metho propose in this paper is ifferent from Cooray ase when both the hannel-base urrent an a meaan Gomes [1998] in that (1) expliit inversion formulas sure fiel (eletrial or magneti) at single point are are use to fin the hannel-base urrent from the fiel known. The unknown moel parameters are foun for an (2) the algorithm searhes automatially for the both the DU an the TL moels (in the ase of the optimum values of the moel parameters. Furthermore, TL moel, the unknown parameter is v). In setion

4 ., 24,472 POPOV ET AL.: RECONSTRUCTION OF LIGHTNING CURRENTS 2.3 we onsier a ase when the hannel-base urrent is unknown an the parameters an the hannel-base urrent are reonstrute using measure fiels at two points, one of whih shoul be in a far-fiel region. For the ase when both measurement points are relatively lose to the lightning, we erive in setion 2.4 (together with Appenix B) an improve inversion formula for the etermination of the hannel-base urrent from in- termeiate magneti fiel, using both terms of the fiel expression in equation (2) an implement a reonstrution of the hannel-base urrent an return stroke parameters using this formula in setion 2.5. In all setions the problem is solve by an optimization metho (the geneti algorithm) Reonstrution of Return Stroke Moel Parameters From the Measure Channel-Base Current an a Remote Fiel Consier a ase when the hannel-base urrent an eletrial or magneti fiel at a single point on the groun are known. The inverse problem is formulate as the problem of fining the values of return stroke moel parameters whih provie the best math between the measure an the ompute fiels. We solve the inverse problem using an optimization metho. In any optimization metho one first introues a suitable objetive funtional. Assume that the measure an alulate fiels from the hannel-base ur- rent are known with a time resolution At on the interval [0, Tmax] an efine the objetive funtional F by 1 N - -- (5) n where H(tn) an H(m)(tn) are, respetively, the values rent an remote eletri fiel. (a) The hannel-base of alulate an measure fiels at the time t, an urrent for return stroke 8715_10. (b) Comparison of measure vertial omponent of eletri fiel at 5.16 At + I is the number of sample points. We minimize this objetive funtional using a geneti algorithm, km (soli line) with reonstrute one using DU moel (long-ashe line) an with reonstrute one using TL whih survey is given in Appenix A. The output of moel (short-ashe line). The moel fiels are aluthe geneti algorithm is the reonstrute return stroke late from equation (1) using moel parameters reonparameters, whih then an be onveniently use for strute by far-fiel approximation an geneti algoalulation of the fiels at any arbitrary point. rithm. For onveniene, the polarity of the eletri fiel To illustrate the metho, we use the ata obtaine is reverse in all relevant pitures. The reonstrute from an artifiially initiate (triggere) negative lou- parameters are given in setion 2.2. to-groun lightning provie by J.Willet an esribe by Willet et al. [1989]. The experiment was onute at the NASA Kenney Spae Center in The hannel-base urrent for the lightning 8715_10 is shown erivative of the urrent was smoothe using the same in Figure 2a. Note that the urrent ata exhibit a highfrequeny noise with an amplitue of the orer of +1 filter as for the hannel-base urrent. The eletri fiel reore at the istane of 5.16 km with time resolution ka. To minimize this noise, a Savitzky-Golay smoothing 10 ns is shown by the soli line in Figure 2b. The return filter with parameters M - 6, nœ - 10, an nr - 10 stroke spee obtaine from the optial measurement is [e.g., Press et al., 1992] was applie to smooth the ur- 1.4 x 10 s m/s. rent ata use for the fiel alulation (the urrent For reonstrution using these experimental ata, the waveform in Figure 2a is shown without any smooth- following parameters (for both DU an TL moels) ing). The time erivative of the urrent is alulate are hosen in the geneti algorithm (see Appenix A): numerially from the measure hannel-base urrent, the number of hromosomes in the initial population sine the measure erivative was not available. The Pinit , the number of hromosomes in the work- o10! loo b O.5 i 1 i 1.5 i 2 Time (s) x 10 '6-80 / u : i/ --true... 20/,,,,?... reonstrut!on us!ng D_U moel /,:,;?... reonstrution using TL moel,o0' :2 ' ' ' Time (s) x 10 '6 Figure 2. Return stroke moel parameters reonstrution using simultaneously measure hannel-base ur-

5 POPOV ET AL.: RECONSTRUCTION OF LIGHTNING CURRENTS 24,473 ing population P = 400, the rossover an mutation rates are Pr= 0.9 an Prnut = 0.05, respetively, the number of iteration J - 50, the number of genes enoing v (in DU an TL moels), u an 'D (both only in DU moel) are Nv = 15, N = 8 an NrD : 12, respetively, an = 0.01 in equation (A1). The maximum time, 7 ax is hosen equal 1 H,s, beause the D U moel with one urrent omponent (assoiate with the breakown time onstant) is vali only for the first few miroseons [Dienorfer an Urnart, 1990], an the TL moel is vali only up to the peak fiel [Thottappilli! an Urnart, 1993]. To provie reliable reonstrution results, three minimization trials were arrie out an the values orresponing to the trial with the minimal ost funtion F value were hosen. The numerial reonstrution using the DU moel resulte in the following parameters: upwar return stroke spee 1.47 x l0 s m/s, ownwarisharge urrent spee 2.57 x l0 s m/s, an isharge time onstant 0.117/ s. The upwar return stroke spee resulte from the numerial reonstru- tion using the TL moel is l0 s m/s. As one an see, the values of the return stroke spee obtaine from the optimization using both moels are lose to eah other an in a goo agreement with that obtaine from the optial measurements (1.4 x 10 s m/s). The reon- strute eletrial fiels are shown in Figure 2b by the ashe an otte lines for the D U an TL moels, respetively. As one an see, the fiel reonstrute using the DU Inoel provies better fitting of the experimental (true) urve ompare with the fiel reonstrute using the TL moel. Perhaps, this an be interprete as a valiation that the DU moel is more aurate in the alulating of the fiels than the TL moel provie that both the hannel-base urrent an the return stoke parameters are known Simultaneous Reonstrution of Parameters an Channel-Base Current From Two Remote Eletri Fiels: One Far Fiel an One Near Fiel If the hannel-base urrent is unknown, one an use an analyti inversion formula erive for the various an the alulate fiels (at this seon point) is minireturn stroke moels [Rahii an Thottappillil, 1993] mize by the geneti algorithm (the objetive funtional B f t o./ovt Oi (z' t), (,t+-) = ' z'. (7) 27r Ot The solution of the inverse problem for (6) an (7)(for the DU moel) an be foun through the solution of the following ifferential equation [f. Rahii an Thottappillil, 1993]: given that where k- 1 +, v. 1 t i(t) + rd z(t) -- li rnoo 7f( 7)' (8) f(t) - lim i(k ) an for the eletrial fiel, an 1, i=1 : o, E? (,t+t) -f-td t ' Bof (,t + )] f (t) = ul2o (, + -)+ t (9) for the magneti fiel. If the parameters of the return stroke moel are known or assume, one an alulate the hannel-base urrent from a far-fiel using (8) with (9) or (10), respetively, for the eletri an magneti fiel. Note that the information ontaine in the fiel in the time perio [0,_T_, ] is neessary an su ient to alulate the hannel-base urrent in the time perio [0, Tmax] an vie versa. (see equations (1) an (2)). To reonstrut the hannel-base urrent an parameters -D, v, an u simultaneously, we propose the following metho: Assume one has measure fiels at two points an one of them is in the far-fiel region. Using equation (8) with some parameters 'D, v, an u, one alulates (using the far-fiel) the hannel-base urrent. The hannel base urrent, in turn, is use to alulate the fiel at the seon measurement point, whih is lose to the hannel. The ifferene between the measure to obtain the hannel-base urrent from the measure value is alulate using (5)). remote fiel. In the present paper we onsier the inver- For this ase the authors ha no appropriate experision proeure for the D U moel. At istanes far from mental ata, an therefore the reonstrution was arthe lightning hannel, one an assume that the fiel is rie out using the syntheti ata. First, the eletri essentially a raiation fiel an that the istane from fiels at two istanes (50 m, 100 km) are alulate any single ipole at height z' to the observation point is assuming the DU moel for the return stroke an the hannel-base urrent use in the work of Rahii an pratially equal to. Therefore one has the following expressions for the far fiels: Thottappillil, [1993]. The assume parameters are return stroke spee v = 1.3 x 10 s m/s, ownwar isharge urrent spee u = 3 x 10 s m/s, an isharge zf?r(, t -f- 7) : '/o 0t ' time onstant -D = 0.1 tzs. These fiels are onsi- (6) ere to be true or measure ones. The peak value an zero-to-peak risetime of hannel-base urrent use for the alulation are ka an 0.500/ s, respetively. The following parameters are hosen in the geneti

6 24,474 POPOV ET AL.- RECONSTRUCTION OF LIGHTNING CUPsRENTS loo an see, the reonstrute parameters an hannel-base urrent are very lose to the assume ones. Sine the syntheti ata are generate by the exat formula (1), while the inversion is base on the far-fiel approximation (6), one annot ahieve a perfet reonstrution, even though the omputational errors are negligible. In orer wors, the reonstrution errors (assuming that o 3o 4o 5o Number of the iterations true Figure 3. Evolution of the objetive funtional F (equation(5)) as the number of iterations inreases for 10 trials of two points reonstrution metho using fiels at two istanes, one of them is a far fiel. The best trial (minimum F) is shown by the ashe line. gorithm: the number of genes enoing v, u, an rd are Nv = 14, Nu - 9, an N D - 11, respetively, Truax = 2/ s (the other parameters are the same as for the alulations in setion 2.2). Ten minimization trials were arrie out an the graphs of the funtional values F evolution versus number of iteration for these 10 trials are shown in Figure 3. The reonstrution results orresponing to the trial with the minimal obtaine ost funtion value (the ashe line in Figure 3) are upwar return stroke spee x 108 m/s, own- war isharge urrent spee x 108 m/s an isharge time onstant its (the relative reonstrution errors are 0.5%, 1.3%, an 3.0%, respetively). The. 1 true an reonstrute hannel-base urrents are shown in Figure 4a. The obtaine peak hannel-base urrent value an zero-to-peak risetime are ka an / s, respetively (relative errors are 0.2% an 0.8%, respetively). The true an reonstrute eletrial fiels are shown in Figure 4b (for 50 m) an Figure 4 (for 100 kin) by the soli urves an irles, respetively. As one Figure 4. Reonstrution results for the hannel-base urrent an return stroke moel parameters for the trial represente by the ashe urve in Figure 3. The alulate eletri fiels at 50 m an 100 km (true fiels in Figure 4b an Figure 4) are use as inputs to the reonstrution problem. The fiels are alulate using equation (1), hannel-base urrent in Figure 4a (true urrent), an DU moel parameters given in setion 2.3. The reonstrute parameters are also given in setion 2.3. (a) Comparison of the true an reonstrute hannel-base urrents. (b) Comparison of the true fiel at 50 m an the fiel alulate from the reonstrute parameters an urrent. () Comparison of true fiel at 100 km an fiel alulate from the reonstrute parameters an urrent mol.5.., x 10 4 o true Time (s) x 10-6 i i _.., O Time (s) x 10 '6._.4,,, true ½3 ½/ o_ a reonstr Time (s) x 10 '6

7 POPOV ET AL.' RECONSTRUCTION OF LIGHTNING CURRENTS 24, lean -- no,sy "%. =100 km Time (s) x [ true hannel-base urrent an iffer onsierably from the true ones. However, the reonstrution shoul not be affete too muh by the presene of ranom noise in any measurement ata. Here we test the above numerial example with noisy ata to hek the stability of the reonstrution. To the eletrial fiels (50 m an 100 kin) we a a Gaussianoise of a variane 0.02 x E ax, where E z nax is the maximal value of the eletrial fiels in the interval [0, Tmax]. The noisy ata for the far eletrial fiel (100 kin) are shown by the ashe urve in Figure 5a (the soli urve is the lean ata). Note that the noisy ata are also preproesse before inversion by the Savitzky-(]olay filter with parameters M = 4, nœ = 20, an nr = 20. We hoose the parameters of the geneti algorithm to be the same as for the previous ase, when lean ata were use. The reonstrute parameters are upwar return stroke spee x l08 m/s, ownwar isharge urrent spee x l08 m/s, an isharge time onstant / s (the relative reonstrution errors are 3.2%, 1.5%, an 19.5%, respetively). The true an reonstrute hannel-base urrents are shown in Figure 5b. The obtaine peak hannel-base urrent value an zero-to-peak risetime are ka an Ms, respetively (relative errors are 1.4% an 0.3%, respetively). As one an see, the reonstrution is reasonably stable uner noise. O Time (s) x 10 '6 Figure 5. Effet of ranom noise in input fiels on the reonstrution of hannel-base urrent an DU return stroke moel parameters. (a) Noisy fiel ata at 100 km ompare with lean ata. Similar noise was also introue in the 50 m ata. (b) The reonstrute hannel-base urrent using noisy ata. The simultaneously reonstrute DU moel parameters are given in setion 2.3. the omputational errors are zero) epen on the re- moteness of the fiel hosen for inversion: the loser the fiel the larger the errors in the far-fiel approximation. Note that the magneti fiel an be use for the reonstrution in the same way an with the same restrition. In setion 2.4 we propose a metho that uses the exat expression (2) for the magneti fiel inversion an therefore an be applie to the fiels at signifiantly loser measurement points. Note that this reonstrution is mae using the syntheti ata an essentially is the valiation of the metho esribe above but not the valiation of the DU moel. If the remote fiel measurement has large measurement errors ue to inue interferene or other reasons, the reonstrute parameter values as well as the 2.4. Expliit Inversion of the Channel-Base Current Using Intermeiate Magneti Fiel Rahii an Thottappillil [1993] have erive an inversion formula. for the return stroke hannel-base urrent for the omplex DU moel from the raiation omponent of the eletri fiel. However, at lose istanes to the lightning hannel, one annot neglet the inution omponent of the magneti fiel or inution an stati omponents of the eletri fiel. In the present setion we erive an expression whih relates the remote magneti fiel an hannel-base urrent, using the DU moel an the full expression (both omponents) for the magneti file. In Appenix B, following Rahii an Thottappillil [1993], we erive an integral-ifferential equation whih relates the remote magneti fiel an hannel-base urrent, using the DU moel for the return stroke an both omponents for the magneti fiel. We reproue this equation below where B øa is the magneti fiel measure at the in-

8 _ 24,476 POPOV ET AL.' RECONSTRUCTION OF LIGHTNING CURRENTS termeiate istane. The integral-ifferential equation (11) annot be solve using a onventional numerial algorithm ue to the simultaneous presene of terms i(0, t) an i(o, kt). However, we show here that this equation an be solve iteratively using the solution obtaine from the far-fiel inversion formula as an initial guess. Rewrite (11) in the following form: [ (0. t)- i(0. t)] +. [i(0. kt) - i(0. t)] = f(t). (12) where 2 r [ B O( ' t + -) + r0 yø (.t + ;) ] Ul o t i(0, t )t - r [i(0, kt) - i(0, t)]. (13) Note that if one neglets the seon term in equation (13), it beomes equivalent to the far-fiel expression (10). s us t wlu /(t) ot l ul t a immeiately from (13), we first alulate an estimate tnl ) [ B;nO(, t q-) q- T ] - B ø(, f(t) = 2 r Ul o t of f(t). Then the numerial proeure for solution of the integral-ifferential equation (11), represente in 5,,,,, I... reonstrution using far fiel inversion formula... "- using improve formula -"- using improve formula an improvement algorithm i i i i i Time (s) x 0' the forms (12) an (13), an be esribe as follows: Step 1. Let m = I an alulate the hannel-base urrent estimate i(m)(0, t) using (8) with f(t) = f(t). Step 2. Calulate the estimate f(m)(t) using (13) with i(0, t) -- i ( ) (0, t). Step 3. Calulate the estimate i( +l)(0, t) using (8) with f(t) = f(" )(t). Let rn = rn + 1 an go to Step 2. After a number of iterations this proess onverges to some value i(0, t), whih is slightly ifferent from the true value i(0, t). This is beause the syntheti magneti fiel is alulate by the exat expression (2), while the above inversion algorithm is base on the approximatexpression (B2). In other wors, even using the improve inversion formulation, the perfet reonstrution of the hannel-base urrent (as in the far-fiel ase) annot be ahieve. However, a further improvement of the hannel-base urrent an be ahieve using the following algorithm: Write the expression for the intermeiate fiel as B ø(, t) = Bob(, t) --{B b(, t) -- B ø(, t)} - B (, t) -{ o oi( ', t - ) ] R 2 z 02 vt i(z, 2x t - -)z Figure 6. Channel-base urrent etermine from the using the far-fiel inversion formula (8), the otte line magneti fiel at 1 km. The soli line is the hannel- enotes the solution of equation (11) after three iterbase urrent use to alulate the magneti fiel at 1 ations, an the irles enote the urrent after three km using D U moel an equation (2). The long-ashe iterations of the algorithm before the substitution of line is the hannel-base urrent etermine by solving equation (8), the far-fiel inversion formula. The otte B (,t) by 2ø(,t) plus three iterations after the line is the hannel-base urrent etermine from solving substitution. Note that this proess with substitution equation (12), the improve inversion formula. The ir- of Br)(, t) by B2ø(, t) an onsequent les represent the hannel-base urrent etermine by of the hannel-base urrent, using the above iterative solving the improve inversion formula an improve tehnique, an be ontinue (i.e., loope) until the persolution algorithm given by equation (14). fet hannel-base reonstrution is ahieve. o vt Oi ( z t, t - ) 2 b '. (14) Let us alulate an estimate of B ø(,t), assuming that the first term in (14) is a measure fiel, an two others are alulate using the estimate (0, t). The estimate ø(, t) obtaine this way orrespons to the fiel alulate using the intermeiate fiel formula (B2), for whih one an ahieve perfet reonstrution. The estimate ø(,t) is then substitute in Step 2 instea of the previous value B (, t), an the iterative proess is repeate. To illustrate the propose metho, we use the magneti fiel alulate at a istane - I km assuming the same return stroke parameters an hannel-base urrent as in setion 2.3. The results are presente in Figure 6: the soli line enotes the true hannel-base urrent value, the ashe lines enotes the inversion improvement

9 POPOV ET AL.' RECONSTRUCTION OF LIGHTNING CURRENTS 24, Simultaneous Reonstrution of Parameters an Channel-Base Current From One Intermeiate an One Near Magneti Fiels 15 true Ultimately, we implement the simultaneous reonstrution of the return stroke parameters an the hannel-base urrent using the improve inversion tehnique in a similar way as for the far fiel. First, the magneti fiel at two istanes (50 m, 1 km) are alulate assuming the same return stroke parameters, the hannel-base urrent, an the parameters of geneti algorithm as use in setion 2.3 (exept for the number of hromosomes in the working population, whih is P here). During the inversion the number of iterations for the solution of equation (11) before an after the substitution of Bo(, t) by ] ø(,t) is put equal to 3. The reonstrute parameters are upwar return stroke spee x 108 m/s, ownwarisharge urrent spee x l0 s m/s, an isharge time onstant / s (the relative reonstrution errors are 0.1%, 3.5%, an 5.9%, respetively). The true an reonstrute hannel-base urrents are shown in Figure 7. The obtaine peak hannel-base urrent value an zero- to-peak risetime are ka an 0.512/ s, respetively (relativerrors are 0.6% an 2.4%, respetively). 3. Disussion an Summary In this paper we have shown how a ombination of analyti methos an numerial optimization tehniques an be use to reonstrut return stroke hannel-base urrent an moel parameters from remotely measure eletri or magneti fiels. A geneti algorithm is use as the optimization metho. First, it has been shown how the return stroke moel parameters oul be reonstrute from simultaneously measure hannel-base urrent an eletri fiel at groun level at a single station many kilometers away (setion 2.2). To illustrate this metho, we have use two return stroke moels: the simple TL moel having only one moel parameter (return stroke spee) an use wiely in lightning loation systems, an the more omplex DU moel, whih have three parameters (breakown time onstant, upwar return stroke spee, an ownwar isharge urrent spee). Then it has been shown how the hannel-base urrent an return stroke moel parameters oul be reonstrute if simultaneous eletri fiels at two istanes, at least one far away, are available (setion 2.3). Finally, a simple formula has been erive using the DU moel to relate the hannel-base urrent an remote magneti fiel (setion 2.4), whih is an improvement over the formula erive by Rahii an Thottappillil [1993]. The appliability of the improve formula, in ombination with a geneti algorithm, has been illustrate in preiting the hannel-base urrent an moel parameters if the simultaneous magneti fiels at 50 m an 1 km are available. i T i Time (s) x 10 '6 Figure 7. Simultaneous reonstrution of return stroke hannel-base urrent an D U moel parameters using the magneti fiels at 50 m an I kin. The input magneti fiels are alulate from equation (2) using hannel-base urrent shown by the soli line an DU moel parameters -r> = 0.1 / s, v = 1.3 x 108 m/s, an u = 3.0 x 108 m/s. The reonstrution is arrie out using the improve inversion formula (with improve solution algorithm) an geneti algorithm. The reonstrute hannel-base urrent is shown by the irles an the reonstrute moel parameters are -D: s, v = x 108 m/s, an u = x l0 s m/s. The metho evelope here to estimate the return stroke hannel-base urrent an moel parameters oul be a valuable researh tool in unerstaning the lightning hannel. Even though the DU moel is mostly use in this paper, the general metho an be applie to any return stroke moel, whih an esribe the hannel urrents by a set of parameters an for whih the expliit inversion formulas an be obtaine [f. Rahii an Thottappillil, 1993]. The truth of the reonstrute hannel-base urrent is ultimately etermine by the valiity an auray of the return stroke moel. In general, one an reonstrut the hannel-base urrent an the return stroke parameters from remote fiels measure at two ifferent istanes without using any expliit inversion formula. In this ase one has to fin the hannel-base urrent (a ontinuous funtion) an the return stroke parameters whih provie the best math between the measure an the alulate fiels at the both measurement points. However, a slow onvergene of the geneti algorithm in the reonstrution of the ontinuous funtions restrits its iret appliation for the solution of this problem. The use of the other optimization tehniques for the solution of this problem is a subjet of future researh. Lightning eletri an magneti fiels suffer from attenuation an istortion as they propagate along a finitely onuting groun. Furthermore, the return stroke hannel an be tortuous an nonvertial. There

10 24,478 POPOV ET AL.' RECONSTRUCTION OF LIGHTNING CURRENTS were many theoretial investigations on the influene of the finite groun onutivity on return stroke fiels [e.g., Wait, 1956; LeVine et al., 1986; Cooray an Lunquist, 1983], an some experimental investigations on the subjet an be foun in the work of Uman et al. [1976]. Effets of the nonvertial hannels on the return stroke fiels were treate by Uman et al. [1980], an effets of the hannel tortnosily on the raiate fiels from the return strokes were treate by Le Vine an Willeft [1995]. In general, the finite groun onutivity reues the peak an inreases the risetime of the remote return stroke fiels, ompare to the ieal ase with the infinite groun onutivity. The effet of the finite groun onutivity as well as a hannel inlination an hannel tortnosily are not taken into aount in equations (1) an (2) use in the fiel alulations. Therefore the appliation of the metho evelope here to the simultaneous multiple station fiel measurements require a orretion of the input fiel ata for the propagation an hannel orientation effets. Appenix Algorithm A: Survey of the Geneti Optimization methos an be ivie into two main groups, namely, graient searh methos an global searh methos. A graient searh metho is base on a eterministi algorithm, an it onverges rapily to a loal minimum epening on the hoie of an initial guess. Global searh methos inlue simulate annealing [e.g., Garnero et al., 1991], neural network methos [e.g., Lu an Berryman, 1990] an geneti algorithms [e.g., Golberg, 1989], an they o not have the problems at the loal minima an o not epen on the initial hoies. In the present paper we use the geneti algorithm ue to its relatively fast onvergene an sim- pliity of implementation. Here we esribe the geneti algorithm that minimizes the objetive funtional given by (5). Genes are the basi builing bloks in a geneti algorithm. A gene is a binary enoing of a parameter, an a hromosome is an array of genes (here the array orrespons to the set of parameters rd, v, an u for the DU moel, an v for the TL moel). Eah hromosome is assoiate with a value of the objetive funtional. After ranking, seletion, rossover, an mutation in eah iteration, the global minimum is obtaine after a number of iterations [Golberg, 1989; Haupt, 1995; Weile an Mihielssen, 1997]. The geneti algorithm starts from an initial population whih is generate ranomly in the searh spae. We hoose the size of the initial population Pini, larger than the size of the working population P (an even in- There are many ways to arry out the seletion (the proeure to hoose parents). In the present paper we hoose the proportional (roulette-wheel) seletion sheme (see equation (A2) below an Golberg [1989] an Weile an Mihielssen [1997]). To perform the proportional seletion, the minimization problem shoul be reformulate as a maximization problem, whih is ahieve in the present paper by the following transfor- where h, x(t), is the maximum height of the return stroke at time t, ontributing to the fiel but not so far teger). After the initial population is generate, the away as to justify negleting the inution omponent orresponing values of the objetive funtional are al- in the magneti fiel (see the first term of equation (2)). ulate. The best (with the smallest F values) P/2 Note that this onition will also apply if the observahromosomes are kept (the others are isare) an tion time T, x is relatively small ompare to -. Let form pairs of parents to proue the new P/2 offspring. us all this onition "the intermeiate fiel onition" mation: F' exp(-af), (AI) where a is a saling fator. The parents are hosen with the following probability proportional to their values of the objetive funtional F', pi- /2 ' n--1 (A2) where fi is the value of the objetive funtional F' for the ith hromosome. The hosen pairs form the new offspring by a rossover operator. There are also many ifferent ways to perform the rossover. In the present paper a one-point rossover is use [e.g., Golberg, 1989; Haupt, 1995; Weile an Mihielssen, 1997], an the rossover point is hosen ranomly for every pair of parents. Mutation is then applie for the offspring an a small perentage of the genes may be hange to the opposite values (1 to 0, or 0 to 1). This mutation operation allows the algorithm to investigate new regions in the searh spae whih have not been overe by the initial population an also to esape from loal minima if all (or almost all) hromosomes in the population are trappe in a loal minimum. This proess is repeate many times until the final solution is foun or a stop onition is satisfie. Appenix B' Derivation of an Integral-Differential Equation Relating the Remote Magneti Fiel an Channel-Base Current Using the DU Moel an Both Components for the Magneti Fiel Assume that the magneti fiel is measure suffiiently far away to satisfy the onition nmx(t) <<1, (B1)

11 POPOV ET AL.' RECONSTRUCTION OF LIGHTNING CURRENTS 24,479 an the fiel itself as an "intermeiate fiel". Beause onition (B1) an be also written as /i}, one an write (2) as I o 0 vt Oi 0 vt, 27r ' i(z' t)z' o (z' t) z' + 2vr Ot ' (B2) istribution (4) for the DU moel as a superposition of two omponents where i( '. t) - i ( '. t) - i (z'. Z! ix (z', t) - i(0, t + --), (B3) (B4) t Z t Z t t z i2(z', t) -- i(o, )exp( ). (B5) 73 u T D We erive the mathematial relation between eah of the fiel omponents B (, t) an B2(, t) an the orresponing urrent omponents i (z', t) an i2(z', t) separately. Consier the omponent B orresponing to the urrent omponent i (a, t + -) 0 Taking into onsieration that oi(o, + - ) ot one an write (B6) as 0 0 vt t + -- zt u )z' Z t /zo fo t Oi(O, t + T) z'. 2 2 i(o, + 2vr Ot /zo oi(o, t + v) OZ t vt Z t ' i(o t + --)z' 2w 2 ' u lzou (B6) +5- [i(0, kt) - i(0, t)], (B7) where k Using the Leibnitz formula, one an evaluate the time erivative of (B7), x(, t + ) t o [ i(o t)- i(o t)] 2w 2,, / ou + 2 & [i(o, t)- i(o, t)]. (n8) Now onsier the seon omponent B2 of the magneti fiel. The relation between i2(z', t) an assoiate magneti fiel B2(z', t) is (,t+-) o vt l o i 2 ( z', t) z' 2w 2 lzo Oi2(z, t) z. + 27r Ot One an fin the erivative of i2(z', t) in (BS) 0i2 ( z', t) z' z' 0 t t at - i(o, ) u exp( T D ) 1 4 [ 7- D., t + Substituting (B9)into (B2), one obtains (,t+-) 27r 2 27r -D i2(z', t)z'. (B10) Evaluating the time erivative of (B10), one obtains (, t t + - ) o( - 27r 2 v-d a)[ (o ' t) TD However, from (B10) it follows that.0 ( -o- ) 2 vr 2 -D vt i2(z, )z. Jo i (z', t) z' - (, t + -). Substituting (B12)into (Bll), one obtains (Bll) (B12) B2(, t + ) V/Zo( -D -- ) i(o kt) I B2(, or t u (,t+-) t + ) 27r 2 -o ' -o + - (, t t + ) :.o( o- ) (o. 2 7r 2 ß (B13) To obtain a single expression, multiply (B8) by -D an a (B7), then subtrat (B13) to obtain a yo (a, yo (, + _)+ t l-to 2 i vt i(o, Z t +,) 2 r t + --)z + o [i(o t)- i(o, t)] 2w ' + o [ i(o t)- i(o, 2w 2 ' + o [i(o, t)- i(o, t)] 2w t v o (ro - ) - 2w 2 i(0, kt). (B14)

12 24,480 POPOV ET AL.: RECONSTRUCTION OF LIGHTNING CURRENTS Making the substitution t - t +- u for the integral in (B14), one obtains the final relation between the hannel-base urrent an the magneti fiel / o t + -) z! q-q- t t + Aknowlegments. The partial support of the Sweish Researh Counil for Engineering Sienes an the Sweish Institute is gratefully aknowlege. The simultaneous triggere lightning urrent an eletri file in Figure 2 are from the experiment of J. Willet an C. Leteinturier arrie out at the NASA Kenney Spae Center in The authors are also thankful to V. Rakov for his omments on this researh. Referenes Brue, C. E. R., an R. H. Gole, The lightning isharge, J. Inst. Eletr. Eng. (Lonon), 88, , Cooray, V., A moel for the subsequent return strokes, J. Eletrostat., 30, , Cooray, V., an C. Gomes, Estimation of peak return stroke urrents, urrent erivatives an return stroke spee from measure fiels, J. Eletrostat., 43, , Cooray, V., an S. 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13 POPOV ET AL.: RECONSTRUCTION OF LIGHTNING CURRENTS 24,481 Willett, J. C., J. C. Bailey, V. P. Ione, A. Eybert-Berar, an L. Barret, Submiroseon interomparison of raiation fiels an urrents in triggere lightning return strokes base on the transmission-line moel, J. Geophys. Res., 9 1, 13,275-13,286, S. He, Centre for Optial an Eletromagneti Researh, State Key Laboratory of Moern Optial Instrumentation, Zhejiang University, Yu-Quan, Hangzhou, People's Republi of China. ( sailing@tet.kth.se) M. Popov, Division of Eletromagneti Theory, Royal Institute of Tehnology, S Stokholm, Sween. (e-maih h.se) R. Thottappillil, Institute of High Voltage Researh, Uppsala University, S Uppsala, Sween. (Reeive February 26, 1999; revise June 29, 1999; aepte April 26, 2000.)