Lightning return stroke current reconstruction or vertical and variable channel shape

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1 nenaona Confeene on Lgnng oeon (CL) Sanga Cna Lgnng eun soe uen eonsuon o vea and vaabe anne sape Ande Cean Adan oos Dan D. Mu Son Spnean Levene Cumb Depamen of ea ngneeng Depamen of Maemas Tena Unves of Cu-Napoa Cu-Napoa omana Ande.Cean@em.uu.o Amedeo Andeo Depamen of ea ngneeng Unves of Napes Fedeo Napes a Absa fs a new maemaa appoa s pesened o evauae e ee and magne fed of e gnng va engneeng mode w vaabe sape eun soe anne; ne an nvese poedue s eposed fo e eonsuon of bo spaa and empoa wavefoms of e gnng eun soe uen ougou a numea fed sness poedue based on eguaaon of -posed pobems. Te appoa uses as npu daa e aquson of me doman eodngs of ee ando magne fed geneaed b e gnng uen a vaous oaons on e gound and ansfoms ese sgnas no amons b Foue deomposon. Ts ombnaon beween e poposed sovng poedues and amon feng eds numea esus a ae n good ageemen w e esng funons. Kewods-gnng eun soe uen vaabe anne posed nvese pobem amon eonsuon fed sness.. NSGTS AND CONCTS GADNG LGTNNG Lgnng eun soe modeng s of nees fo a vaous ange of easons as pa of evauaons no e pss of gnng as an nsumen b w eun soe uens a gound an be denfed fom neab o fa awa measued eeomagne feds and ene b w uens of ndvdua gnng ses o sasa dsbuons of e soe uens an be evauaed [] [] [] [] [5] [6]. n s pape s pesened e maemaa modeng of a de and en nvese emoe sensng poedue n ode o denf and eonsu e spaa and me doman wavefom of e gnng eun soe uen. s based on e aquson of e ee ando magne fed geneaed b e dsage anne a vaous oaons on e gound and a vaous fequenes. Ts ma be a meansm fo auang eas feds ne o be used n oupng auaons su as o deemne e gnng-ndued voages appeang on powe gds o eeommunaon nes wen gnng ous nea ose nes. Afe e denfaon of a maemaa mode of e eun soe uen beomes possbe o evauae e ee and magne fed vaues n an nees aea and subsequen usng e ansmsson ne meod o esmae an ndued poena n neab powe gds. Fo s sud ee mus be used Sommefed negas [7] [8]. Some auos onsde a a onvenen wave souon fo bo uen dsbuon aong e gnng anne and assoaed eeomagne feds an be aeved on wen usng eeomagne modes. On e oe and s we nown a engneeng modes fo e eun soe uen and feds onss n bo spaa and empoa vaaon as sepaaed vaabes. Te eoea esmaon of eun soe uens fom emoe eeomagne feds depends on e adoped eun soe mode. pessons eang adaed feds and eun soe anne base uens ave been deved fo vaous engneeng eun soe modes. ngneeng eun-soe modes ave been evewed n man papes [] [9]. Te pobem of deemnaon of e eun soe uen fom emoe measued ee ando magne feds onsdeab faaes e oeon of daa on e gnng eun soe uen wou avng o nsumen owes o gge e gnng afa and wou e neen eave neffen of ese meods. Ts s espea ue now beause of e wdespead use of gnng oaon ssems LLS [].. XSTNG ALTNATV CONSTUCTON TCNQUS Sevea auos ave suded e ab of e engneeng modes o ped e eeomagne fed adaed b eun soes; een n [9] ee ae menoned wo pma appoaes of evauaon: Te fs appoa nvoves usng a pa anne-base uen wavefom and a pa eun soe popagaon speed as mode npus and en ompang e mode-peded eeomagne feds w pa obseved feds; Te seond appoa nvoves usng e anne-base uen wavefom and e popagaon speed measued fo e same ndvdua even and ompang ompued feds w measued feds fo a same gnng. Te seond appoa s abe o povde a moe defnve answe egadng mode vad bu s feasbe on n e ase of ggeed-gnng eun soes o wen naua gnng ses o a owes wee anne-base uen an

2 be measued. Wen ng o eonsu e eun soe uen f e measued and e auaed fed do no agee e anne uen spaa and empoa paamees ae anged. Te poedue s epeaed un e mang beomes sasfao good. Some pevous eseaes a wee denfed [9] [] [] [] popose dffeen vesons of a and eo poedues fo e eonsuon of e eun uen usng e measued fed vaues dung gnng ouene. Anoe aenave s o de sove e nega equaons w e ep of e ooaon meod usng Cebasev o Gegenbaue base funons [] [] [5]. Ou poposed poedue mpes e foowng: app Foue sees o e me doman sgna of e ee o magne fed and ea some N omponens fom - ampudes and pases; f avaabe app Foue sees o e me doman sgna of e anne base uen and ea some N omponens fom - ampudes and pases; a s momen a oeaon an be pefomed beween e fequen doman of e eemagne eoded feds and of e eun soe uen wavefom; o ea uen amon w oespond an eemagne fed amon ned b e fs nd Fedom nega equaon; s appoa as a psa meanng aso and ees on e supeposon meod; pass en fom e anaa nega equaon o a nea ssem of equaons oug numea mesng of e spaa vaabe - anne eg on one sde and ange of oona sensos on e oe sde; s numea ssem of equaons as a sevee -posed souon a fa epessed b s ondon numbe. Te modua agoms on e pesen appoa an be suessfu apped fo gnng eun soe uen eonsuon ne o be used n powe engneeng eeomagne ompab pobems n e esea of e adaed gnng eeomagne puse and s oupng w e oveead nes and oe mea suues.. MATMATCAL AOAC OF T TUN STOK Tee s a wde ange of ee o magne fed sness appaons a ave o be modeed w Fedom nega equaons of e fs nd as -posed nvese pobems. s e ase of: magne esonane magng os fo unfom feds; poson denfaon of sps fom e gavaona magneaon; undewae deemnaon of ooson fo oean pafoms [6]; gnng eun soe uen denfaon fom fed measuemens [] e. Fo e engneeng modes e ause and effe eaons beween e spaa-empoa epesson of e asendan eade and e ee and magne geneaed fed ma be modeed oug Fedom nega equaons of e fs nd. Te modeng ma aso assume a e eun soe uen dsbuon an be summed up b e ndvdua onbuons of e mpuse uens w popagae upwad w dffeen speeds. f some adops as modeng poess e fa a e so as a fa sape of omogeneous maea and w pefe onduv en e enes of e Fedom nega equaons of e fs nd ae epessed b aona funons weged b deang eponenas. f s admed a ose o ea poess a of nonomogeneous and fne essv so e epesson of e enes depends on Besse funons [7]. ee ae some of e poess a ae used n e modeaon of e gnng: e gnng anne s epesened as beng D vaabe aong w uen popagaes as a movng fon; e so s omogeneous as a fa sape and pefe onduv. Te ne paagaps am o evea e maemaa epessons of e fed omponens n nda oodnaes bo fo ee and magne feds as Fedom nega equaons of e fs nd. Above an be seen a gven appomae geome mode of e gnng eonsuon w an be geneaed o an spaa uve as gnng anne: Fgue. Geomea paamees used n auang eun soe feds. Wee: - anne eg; - eun soe uen spaa and me dependene; = - s e oodnae of pon n spae wee e poenas ae ompued evauaon oaon of sensos; - s e oodnae of a pon of e soue of e uen. Sang fom e Mawe equaons n a nea omogeneous and soop medum we ave e souon as magne veo poena and e equaon o n e fed and poena: A V A A dv () A d () Tese ae nomogeneous souons epessed b e magne veo and ee poenas dependen one o ea oe. We aso assume a e uen dsbuon and e age dsbuon ae eo a =.

3 n e ase of e gnng eun soe e soue s movng aong a uve sang on ea and gong up e C be s genea uve w beng e eg paamee: ] [; ý () And e adus fom e vaabe anne o e obsevaon pon o be: () Afe a sees of negaons and subsuons and e use mages meod o add e onbuon of e ea we oban e epesson of e ee fed n espe w s vaabe spaa uve of e gnng anne as equaon o desbe e penomena: adaon nduon sa (5) Wee: d d sa 5 d nduon d adaon (6) Ae e sa e nduve and e adaon omponens of e ee fed. Ts epesson fo e ee fed was dedued n [] fo e ase wee e uve C s a segmen of a ne usng Foue ansfom; us we now ave a genea equaon o use. Sma onsdeaons and maemaa seps wee used o dedue e magne fed fomua: d (7) Up o now aoug e eaons ae vad fo an vaabe spaa anne uve so as ese eaons o be numea ompuabe e us nodue some ssues egadng e eun soe uen. Genea epesson of e uen: d d u (8) W e agumens. Te me dea d s e duaon fo e uen o ave fom e gound (=) o e pon of e anne oespondng o and an be ompued w e ne nega: d v ds (9) Wee v s e popagaon speed of e uen a funon of eg. Ne f we onsde a e gnng anne s omposed of n nea segmens obaned b onng e pons n... e fs pon s e ogn and denoe b aso e angen veo of oespondng o e segmen. Fgue. Lgnng anne numea dseaon Denoe b v e speed of e uen oespondng o ea segmen. Consde e me fo gng o ave fom o. Tus we an ompue: v v... () avng ese onsdeaons ea of e fed epessons an now be numea evauaed ve eas as foows: sa n... () Wee s e nega oespondng o e segmen.... n a segmen as e equaon: () We aso oban: () Usng eaons () and () we an use e fna ompuabe nega (): d d 5 Tese maemaa epessons of e fed omponens n

4 nda oodnaes bo fo ee and magne feds as Fedom nega equaons of e fs nd ae used as e engneeng modes. We found a a mss nepeaon pesss egadng e fequen doman of ese ada aa and poa fed equaons. Te do no epesen e Foue ansfom of e me doman fed epessons bu foma wen w ompe numbes eaons. Tus () onves n () and e same w e oe omponens: G (5) d ep n s ase e ene funon G nopoaes e sa nduon and adaon onbuons as n oespondene w e eaons fom (). Te eun soe uen (SC) as a funon (6) and (7) sows dependene o an na pea vaue a e anne base a spaa aenuaon aong e anne and o e popagaon speed of e uen bo fo me and fequen dependene: u v v (6) ep v (7) Wee: fequen onveed anne base uen (CBC); () - spaa aenuaon of e eun soe uen. V. GULAZATON OF LL-OSD QUATONS Tang aoun a ese desbed nvese Fedom posed nega equaons onss n ompuaon e ause fom e effe s epeed a sma nose n e g-and sde measued fed omponens ae e o geneae numea SCs g onamnaed b undesed gfequen osaons [] [8]. Tus f one b sandad numea poedues evauaes e souon s as ee mao nonvenen aaess: mpeson nsab o sma npu fed modfaons and psa nonssene. Te -posed eeomagne nvese pobems -posed ae ve we deaed n e eaue espea fo Fedom nega equaons [] [9]. Fo s eason n ou sud ee as been adoped e onep of Wong eguaaon Agom (WA) [] as a funona mue of ee faos: a eguaaon agom a paamee oe meod and e mpemenaon of ese meods. Fo effen we ae no aoun an avaabe maemaa suue n e pobem (sngua smme spase). B usng e ondon numbe n e na evauaon we an sow a ea onneon beween e souon nsab and e ondon numbe as eaed o an peubaon a ma ou n e measued fed o n e pobem suue e ene ma. Tus e nose as on e effe veo as: X X u u X [%] K A K A u[%] (8) X u u - peubed effe veo; X - e esued souon (aenuaon funon) as eaed o e peubed effe; K A - ondon numbe; Mnmng a Tonov funona [] [8] epessed w e ep of veo noms (8) s nong bu a onsan meod w ms e unonoed gow of e souon: ag mn A X u C X (9) f Tonov Wee: A - ma ssem; u - fed veo; eguaaon paamee; A X=u e ssem of equaons ognaed fom e nega equaon (). Te em C X onsss n a pena apped o e souon n ode no o aow s nsab. Aso e opeao C ma embed geomea and psa onsans fo e souon. Ts eguaaon poedue and s devaons ma be egaded as a pena meod []. Tunaed sngua vaue σ deomposon (TSVD) apped as a eguaaon meod w e maon of ean ems a ene n e sum as eaed o a sngua vaue saed as esod s nepeed as beng a poeon meod; an evauaon of a veo b summng up of oe veos wou undesed omponens. X f U T u V () Wee: f - fe faos; U V - sngua maes; - unaon oeffen w as as eguaaon paamee. Tus e eguaaon effes as a pena meod o as a poeon one. We assfed ese eguaaon poedues as foowng: T - pena meod based on Tnov eo efeed b eaon (9); DVST onoff - poeon meod unaed sngua vaue deomposon w onoff fe faos f =σ f o f >; DVSTA - poeon meod damped unaed deomposon of e sngua vaues based on eaon () w f =σ (σ +α) f o >; oe sandad meods: GCS - onugae gaden meod; TA - ageba eonsuon enque []; GCV - geneased ossed vadaon meod epessed as e mnmum of GCV()= C X fo e oe of e eguaaon paamee; LC - L sape uve funon a dependen vaaon beween e eo and souon noms as nodued n (9) o w e one epesens e opmum eguaaon paamee []. Fo ea of e above eguaaon meods e ogna onbuon of e auos s eaed o e defnon and evauaon of e feng faos. Te esod fom w e feng sas s b sef a eguaaon paamee n eaon w e deomposon of e me doman sgna. Bo WA pena and poeon meods onss na n a amon anass fo e noms of e sngua veos V fom e deomposon and afewads n a feng of ose sngua veos a ave a owe nom an an mposed m f e ma be affeed b e ampfaon due o e sngua vaues σ n e souon eonsuon. Afe ompung e souon b an of ese meods an eo evauaon s pefomed usng e eaons: es [%] souon ()

5 mas a [%] () effe mas Some eo auses a appea fo ee o magne fed measuemen: LLS deves; uen efeons n nsumened owes [5]. n e evauaons s aso esed e sensv of e souon wen nose ous n e fed veo. V. NUMCAL SULTS AND DSCUSSONS No avng an avaabe daa abou e eg dependen aenuaon funon () numea e veo X s an be evauaed fo ea fequen fom e Foue speum b sovng e above nega equaon () fo a vea anne and e oe eaed fed negas. Ten e numea souons an be ompaed w e poposed es funons MTL and MTLL as on ese ave non una spaa aenuaon dependene. Tee wee used sevea npu daa egadng e oaon of e fed sensos (ange of 5 o 5 m) eg of e measuemen sensos ( o 5 m) eg of e uen anne ( o 7.5 m) and e sampng fequenes of e measued feds ( o 5 amons eaed o a mamum duaon of - se). A of ese numea ases apped o e Fedom nega eq. modes vea eea fed () and eaed oona ee and amu magne fed sengs ead o -posed and ve sevee -ondoned na ssems of equaons and equed eguaaon. Le us onsde e esus fo a 7.5 m anne eg and an na mposed CBC as w ndaed epesson and paamees gven n []. Fgue. nvese eonsuon of eun uen spaa dsbuon w poeon and pena meods avng e eonsued MTL mode spaa aenuaon funon w e Tonov usng e L uve eon and 75 `amons` n no pefe fng e CBC (oespondene w e Foue speum of e fed) an be evdened e eun soe uen a dffeen egs as n fg. 5. Te moe `amons` evauaed e owe w be e fuuaon n e souon. We assume a e mode evdened dsonnu n e SC a dffeen egs ma be due o dspeson of e uen usng e suppo of [6]. Fgue 5. eun soe uen denfaon fo dffeen aenuaon funons Ten was pefomed e ombnaon of e fed equaons fo e senao of bo ee and magne measued fed omponens w on one senso and apped eguaaon poedues. A sampe of e souon eos eds e opmum appoa fo e oaon of e senso a 5 m and MTL: Fgue. Vea eea fed omponen sampe a 5 m fom e se n fg. an be seen e esu of evauang de e ee fed seng fo e TL MTLL and MTL modes [] [5] [] bo n me doman and e fequen speum fo MTL mode a a dsane of 5 m fom e se oaon fo a duaon of 5 μs. Ts s aouned fo e senao w on one ee fed senso fo e emoe sensng poedue o denf e spaa dsbuon of e uen aong e anne eg. Usng ese eea fed vaues w a 5% added nose as fg. sows we deemned e aenuaon funon b e WA. n fg. s epesened a sampe esu fo e denfaon of e MTL (Tes ) mode: Fgue 6. Sensve anass of e eguaaon apped on e ombnaon of e nega modes fo a snge oaon ee and magne fed sensos n e eonsuon of e MTLL (Tes ) mode we aso aeved easonabe pefomane as eaed o oe epoed

6 esus [9] - []. egadng e epemena aspes of e pesen sud s wo menonng a we used smuaed vaues as a esng appoa. s ou nenon o ande aso naua o ggeed gnng eodngs povded b LNT Geman. Moe daa ave o be evauaed n ode o adequae vadae e modes and o mpove em n ode o epodue as ose as possbe epemena vaues. We fnd ou a wou eguaaon on fo ge fequenes s epeed o ave an mpovemen n e sab of e souons f usng snge fequen eodngs bu mupe fed senso oaons. Usng e pesened assumpons we epoed e souon beavou fo ea of e poposed esng ondons. Wen ang no aoun wa s e bes fequen speum fo w o eonsu e spaa aenuaon funon soud be noed a n e ange M (w added DC omponen) e eos ea mnmum vaues. V. CONCLUDNG MAKS Te pesen wo fouses on e sness of gnng eun soe uens fom emoe measued geneaed feds. Afe e denfaon of a maemaa mode of e eun soe uen beomes possbe o evauae e ee and magne fed vaues n an nees aea. As e pobem poves o be sevee -posed we poposed a WA as a goup of eguaaon poedues a based on e amon feng of e sngua veos. Te effeveness of e agoms as been poved espea fo onoff DVST and DVSTA; aso fo Tonov eguaaon n e ombnaon of wo pe of fed measuemens ada aa. n ode o vef e obusness of e nvese poedue we added nose o e fee em of e ssem.e. o e fed measuemens. Te vea gnng anne s no moe aepabe as one aes no aoun a ea gnng s aaeed b ouos and banng n ode o be abe o usf e fne suue of e feds adaed b gnng dsages wose me-doman beavo ebs a nos sape w spea [6] [7]. Tese feaues ae nvesgaed and epoed o mpove e eun soe uen eonsuons. Te auo s onbuons eae o e an ogna maemaa appoa of vaabe annes noduon and vadaon of e Foue fequen deomposon of e fed me doman sgnas and numea eguaaon n s gnng eun soe uen pobem eonsuon. FNCS [] C. Gomes V. Cooa Coneps of gnng eun soe modes Tans. eomagne Compab No [] V. A. aov M. A. Uman evew and evauaon of gnng eun soe modes nudng some aspes of e appaon Tans. eomagne Compab No [] A. Andeo F. Defno and L. Veono An denfaon poedue fo gnng eun soes Jouna of eosas Vo. 5 No. 5 pp. 6-. [] A. Andeo F. Defno. Gdno and L. Veono A fed based nvese agom fo e denfaon of dffeen eg gnng eun soes COML Vo. No. pp [5] A. Andeo U. De Mans L. Veono An nvese poedue fo e eun soe denfaon Tansaons on eomagne Compab No. pp [6] F. ad C. A Nu M. ano C. Mae nfuene of a oss gound on gnng-ndued voages on oveead nes Tans. eomagne Compab No pp [7]. Degauque and J. amen (edos) eomagne Compab Ofod Unves ess 99. [8] C. A. Nu Lgnng-ndued voages on dsbuon ssems: nfuene of gound essv and ssem opoog Jouna Lgnng esea Vo. 7 pp [9] V. A. aov Lgnng eun soe speed Jouna of Lgnng esea Vo. 7 pp [] F. Defno e a Lgnng eun soe uen denfaon va fed measuemens ea ngneeng (Av fu eoen) Vo. 8 No. pp. 5. [] M.A. Uman and D.K. MLan Lgnng eun soe uen fom magne and adaon fed measuemens Jouna of Geopsa esea Vo pp [] D. avaneo eomagne adaon fom gnng eun soes o a suues. D. Tess No. 7 FL Lausanne 7. [] A. Andeo D. Asane S. Fao and L. Veono An mpoved poedue fo e eun soe uen denfaon Tansaons on Magnes Vo. No. 5 5 pp [] A. Andeo S. Fao L. Veono Some negas nvovng ede s gnng eun soe uen epesson ea ngneeng No [5] A. Andeo U. De Mans C. eaa V. A. aov and L. Veono Lgnng eeomagne feds and ndued voages: nfuene of anne ouos [6] O. Cadebe e a ow o we pose a magneaon denfaon pobem Tans. Magn. Vo. 9 No. pp [7] J. L. Bemude Lgnng uens and eeomagne feds assoaed w eun soes o eevaed se obes. D. Tess No. 7 FL Lausanne. [8] V. Cooa G. Cooa Te eeomagne feds of an aeeang age appaons n gnng eun soe modes nenaona Confeene on Lgnng oeon CL Caga a. [9]. C. ansen eguaaon oos A Maab paage fo anass and souon of dsee posed pobems Denma 8 p []. C. ansen Dsee nvese pobems nsg and agoms SAM U.S.A. [] A. Cean L. Cumb D.D. Mu On some eun soe gnng denfaon poedues b nvese fomuaon and eguaaon nenaona Confeene on Lgnng oeon CL Caga Sep.. [] J.C.We D.M. Le Vne and V.. done Lgnng eun soe uen wavefoms aof fom measued fed ange uen and anne geome Jouna of Geopsa esea Vo. 8 pp [] J. D. Jason Cassa eodnams Jon We & Sons n. New Yo 999 d edon. [] D. M. Le Vne and. Menegn A souon fo e eeomagne feds ose o a gnng dsage 8 nenaona Aeospae and Gound Confeene on Lgnng and Sa e (COLS) Na. neagen Cood. Goup Na. Amos. e. aads o. ogam Fo Wo Teas U.S.A. June 98. [5] M. J. Mase and M. A. Uman Tansen ee and magne feds assoaed w esabsng a fne eeosa dpoe Ame. J. s [6] M. ubnsen and M. A. Uman Meods fo auang e eeomagne feds fom a nown soue dsbuon: Appaon o Lgnng Tansaons on eomagne Compab [7] A. Cean V. Topa D.D. Mu A. Andeo Lgnng nvese eonsuon b emoe sensng and numea-fed sness Tansaons on Magnes Vo. 9 No. 5 pp. -5.