Triangles Technique for Time and Location Finding of the Lightning Discharge in Spherical Model of the Earth

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1 Joual of Geocece ad Evomet Potecto Publhed Ole Apl 06 ScRe Tagle Techque fo Tme ad Locato Fdg of the Lghtg Dchage Sphecal Model of the Eath Aatoly Lozb Yuy Shpad Alexade Ich Scetfc Space Sytem Laboatoy Ittute of Space Techque ad Techologe Almaty Kazakhta Receved 5 Febuay 06; accepted 5 Apl 06; publhed 8 Apl 06 Copyght 06 by autho ad Scetfc Reeach Publhg Ic Th wok lceed ude the Ceatve Commo Attbuto Iteatoal Lcee (CC BY Abtact The phecal model of tme ad locato calculato of the lghtg dchage gve The calculato ae made by mea of ado gal detecto by eo of the dtbuted etwok The full oluto of a poblem of lghtg dchage cloud-goud type locato fo thee eo gve Baed o th tak the lghtg locato method fo a etwok of eo wa developed By mea of computatoal expemet the aaly of accuacy of the model depedg o ado gal detecto accuacy at obevg tato wa doe Keywod Lghtg Tme of Aval Techque (TOA Atmophec Sphecal Tgoomety Itoducto Let P( ϕ θ be dffeet pot o the Eath uface wth logtude ϕ ad lattude θ I thee pot we have eo ecevg a ado gal fom lghtg dchage Let thee wa a lghtg dchage the tme momet t ome pot P( ϕ θ of the Eath uface Rado gal fom th dchage wee detected by the eo the t tme pot Aumg that the ado gal (low fequecy eache each eo o the hotet way alog a Eath uface we wll eceve ytem of thee equato fo lghtg dchage tme ad locato detemato PP c ( t t ( whee PP legth of a malle ach of a bg ccle of the teetal phee coectg pot c the peed of ado wave P ad How to cte th pape: Lozb A Shpad Y ad Ich A (06 Tagle Techque fo Tme ad Locato Fdg of the Lghtg Dchage Sphecal Model of the Eath Joual of Geocece ad Evomet Potecto P

2 A Lozb et al The ytem of thee Equato ( qute defed a t ha thee ukow: coodate ϕ θ ad tme of dchage t I a tak we beleve that the lghtg dchage ca occu ay place o Eath uface that : π π π < ϕ π θ Becaue of phycal ee tmepot of dchage t mut happeed befoe all tmepot of t detecto o all eo that : t m t ( O the othe had the way paed by a ado gal to each eo ca t be moe tha legth of a em-ccle of a bg ccle of the Globe Fom th follow that max t π c t ( R e whee R e Eath adu Baed o ( ad ( we coclude that tmepot of the lghtg dchage mut be the teval Sytem ( Featue c R e max π m t t t (4 Let code ome featue of the ytem ( Ug the geocetc coodate ytem whch coected wth Eath we wll ete gle vecto { x y z} ad { x y z} hodogaph of the pot P( ϕ θ ad P( ϕ θ Coodate of the vecto ad coected wth phecal coodate of the pot P ad P by the equato: x coϕ co θ π π y ϕ co θ π < ϕ π θ z θ x coϕ co θ π π y ϕ co θ π < ϕ π θ z θ Applyg a cala poduct of vecto we ca how dtace fom eo to a lghtg a PP R acco ( e I geocetc coodate ytem the cala poduct dcloe by the equato: ad phecal coodate ytem: xx + yy + zz coθ coθ co ϕ ϕ + θ θ (5 Let put (5 to ( ad dvde equato o R e I eult lead the ytem ( to acco C t t (6 R whee CR c Re t whch ft compoet defe locato of the lghtg dchage ad the ecod the momet coepodg to t the oluto of ytem (6 f t atfe to each equato of th ytem We wll code that tmepot t ae umbeed accodg to ceae of the value The pa { } 6

3 A Lozb et al Othewe we wll chage umbeg of pot P accodg to (7 Let code a dffeece of two equato of ytem (6 t t t (7 ( ( R( acco acco C t t > (8 Fo a phecal tagle wth vetexe P P ad P whch de located o the ame hemphee the tatemet mla to tagle o the plae fa the abolute value of a dffeece of two de alway le tha thd de Thu Fom th equato ad (8 t follow that ( ( ( acco acco acco C t t acco > (9 R The Iequato (9 epeet a eceay codto fo olvablty of the ytem of Equato ( Thu f the dffeece of the lghtg tmepot meaued by each couple of eo doe ot meet a equemet (9 to defe locato ad tme of a lghtg dchage baed o thee eo t mpoble The caue of daagemet of a Iequato (9 ca be a eo of detfcato of lghtg dchage at eo o be a coequece of eo of meaug equpmet Futhe we beleve that the equemet (9 met Geometcally each of the Equato (8 defe a et of pot P o the gle phee Dffeece of dtace of thee pot o a phee uface up to two fxed pot P ad P the cotat ad equal CR( t t Pot P P ad P ae adal poecto of pot P P ad P to the gle phee wth avg of the phecal coodate By aalogy wth the plae th et of pot P called a a hypebole Pot P ad P ae focue of a hypebole The focal legth equal acco ( c (0 em-tavee ax a CR ( t t > ( Futhe we aume that equato (9 doe ot ext ad the tct equato we have a < c > ( Theeby we exclude a cae whe the hypebole degeeate a ach teval Ulke a flat cae hypebole o the phee the lmted cloed cuve ad moeove t cocde wth a phecal ellpe Really ug the detcal equato: let u fd acco x π acco x x ( ( ˆ acco π acco ( whee ˆ - a ut vecto oppote vecto Plug ( to (8 we get a equato: acco + acco ˆ π a Th equato defe a geometcal et of pot P The um of dtace of thee pot o the phee up to two fxed pot P ( ϕ θ ad Pˆ ( ˆ ˆ ϕ θ P ( ϕ θ equal to cotat π a that coepod to deπ fto of a ellpe o the phee Focal legth of a ellpe equally c c ad mao em-ax π a a We wll otce that all pot of th ellpe ae located o a plae lmted hemphee ad th plae pag though the cete of Eath pepedcula to a vecto + ˆ 7

4 A Lozb et al The vualzato of the Equato (8 gve by fucto gaph: ( acco ( acco ( f ϕ θ whch epeeted Fgue a cyldcal ectagula poecto Hypebole (ellpe ae le of level of th fucto Focue of all hypebole ae located pot P ad P whch ae ot maked o gaph but ymmetzed o the equato wth epect to zeo meda ad the eal axe ae equal to value of fucto f ϕ θ o the epectve level le Paametezato of a Sphecal Hypebole Let u fd the paametcal equato fo a hypebole (8 Let u add γ half-plae whch a pa out fom vecto ad ca otate aoud th vecto We wll otce that at ay locato γ half-plae ha oe ad oly oe geeal pot wth a hypebole Th fact the ba fo defto of the paametcal equato of a hypebole Pevouly we wll ceate othogoal coodate ytem wth epect of whch thee wll be a γ half-plae otato To defe β plae whch vecto ad belogg to Let u cotuct a gle omal vecto to the plae β whee c (4 c defed by Equato (0 ad add vecto beleve that (5 vecto choe that ode ae fom the ght tple of gle vecto vecto a omal to β plae ad vecto ae belogg to th plae Double vecto poduct expadg we fd ( ( co c (6 c c c Thu the vecto a lea combato of vecto ad complaaty Add vecto ad that atual becaue of vecto γ coψ + ψ 0 ψ < π (7 Fgue Gaph of the fucto f ( ϕ θ Agle value ae degee 8

5 A Lozb et al ad dect γ half-plae alog th vecto At chage ψ agle wth 0 to π the vecto γ togethe wth half-plae γ wll make a complete evoluto aoud the ax pag though a vecto Rotato of a vecto wth ceae of the ψ wll happe couteclockwe f to look fom the ed of a vecto to the plae of vecto ad (potve otato utay vecto ytem Thu the ψ agle the value whch defg each pot of a hypebole Futhe we wll fd elatohp betwee vecto ad ψ The vecto wth half-plae γ theefoe t equal to a lea combato of vecto ad γ co λ + λ 0 < λ < π (8 γ whee λ - a fucto of the ψ The geometcal ee the λ t agle betwee ad Pluggg fom (7 to (8 we wll get vecto eoluto by utay vecto ad γ vecto co λ + λcoψ + λψ (9 Fo the fdg depedece λ fom ψ we wll plug (9 to ytem of Equato (8 Fom the Equato (4-(8 takg to accout degato (0-( ad featue of the vecto multplcato we wll fd the dot poduct: 0 Afte pluggg (9 to (8 we wll get a equato whch afte accoe veo chaged to ( c co co с co с с с с acco co с co λ с coψ λ a + λ co c co λ c coψ λ co a + λ (0 Becaue of ( the co a co c > 0 the fom (0 we get fom th we have a c coψ ctg λ co a co c a c coψ λ acctg 0 ψ < π co a co c The Equato ( gve u ukow elatohp betwee λ ad ψ The we wll ty to fd λ mag chage If the ψ 0 f the ψ π λ max λ m ( c a ( a co + c acctg acctg c a co a co c c a ( c + a a co c acctg acctg π + co a co c c + a Theefoe λ value ae atfy of the equato ( c a ( 0 < c a λ π c + a < π ( By ug e ad coe of the λ agle though cot we wll get: 9

6 A Lozb et al a c coψ co λ + co a co c λ ( co a co c ( a c coψ ( co a co c + ( a c coψ ( Thu the equed paametcal equato fo a hypebole (8 et by Equato (9 whch λ value defed by equato ( o fom the Equato ( 4 The Sytem ( Solvg By excludg t equato fom d ad d (6 we wll get equvalet ytem: ( CR( t t ( ( CR( t t ( ( C ( t t acco acco acco acco acco R Two lat equato decbe two hypebole whch have the geeal focu et by vecto Theeof the paametcal Equato (9 of thee hypebole ca be educed to oe paamete Fo th pupoe t eceay to combe otato of half-plae γ ad γ whch poceed fom the ame ax et by a vecto ad defe the poto of the cuet pot o the ft ad ecod hypebole A a eult two paametcal equato of hypebole depedg o oe geeal value we have A pot of teecto of hypebole wll defe the poto of a lghtg dchage We wll gve the coepodg fomula Paametcal equato of the d equato the ytem (4: co λ + λ coψ + λ ψ (5 whee co c a c coψ ctg λ co a co c c c (4 c acco ( a C ( t t R Paametcal equato of the d equato the ytem (4: co λ + λ coψ + λ ψ (6 whee co c a c coψ ctg λ co a co c c c c acco ( a C ( t t R At the ame value ψ ad ψ the agle ψ 0 betwee half-plae γ ad γ equal to agle betwee vecto ad e ψ 0 acco ( (7 Let the vecto ae ght o the cala tple poduct potve x y z x y z > 0 (8 x y z The at the half-plae γ ad γ couplg the ϕ agle wth ay value wll be exceed ϕ o the cotat ϕ 0 o the Equato (9 wll be ght 0

7 A Lozb et al ϕ ϕ ϕ0 (9 If the cala tple poduct of the vecto и egatve fo educto to a Iequato (8 t eough to chage umbeg of vecto ad ytem (4 By pluggg (9 (6 we wll eceve the paametcal equato of two hypebole whch deped o oe paamete ϕ Vecto ad a pot of teecto of hypebole mut be cocde I ode that ad wee equal the equalty of agle λ ad λ eceay ad uffcet A thee agle ae defed de 0 π teval o whch the cot bach alo defed the cot of thee agle ae equal ctg λ ctg λ By pluggg cot value we wll get equato elatve to ϕ ( ψ ψ a co c coψ a c co a co c co a co c 0 By detcal tafomato the Equato (0 wll be a whee Beleve that we get equato: Aψ + Bcoψ C c ψ 0 A co a co c c coψ 0 c B co a co c co a co c a a C co a co c co a co c A coς A + B B ς A + B ( ς C whee A + B Depedg o value thee cae ae poble: If < the hypebole ae coed two pot (0 ψ + ( k ψ ς + ac + kπ k 0 If ± the hypebole have oe geeal pot π ψ ς ± If > the hypebole have o geeal pot ad the tak of lghtg poto defto ha o oluto Afte ψ fdg the λ calculated ad the vecto defg the poto of a lghtg dchage The vecto a pot of a lghtg dchage cocde wth a vecto To lghtg dchage thee ca coepod oly oe pot of teecto of hypebole whch we wll call actual The ecod pot a coequece of cog of two cloed covex cuve leag at each othe Th

8 A Lozb et al pot we wll call phatom Tmepot of both lghtg dchage (actual ad phatom ae defe by fomula ( whch develop fom the ft equato of ytem (8: t t acco ( CR Thu f the lghtg detecto etwok cot oly of thee eo the wth codto ( keepg we have actual poto of a lghtg ad a a ule thee a phatom pot Fo ytem of the Equato (4 both of t oluto ae equal Theefoe to allocate a lghtg actual pot addtoal fomato eceay Fo example fo th pupoe t poble to ue a vecto of ducto of a magetc compoet of the accepted ado gal of a lghtg whch coodate ca be eceved by mea of the bdectoal magetc atea [] [] Howeve a appea fom the aaly of the Equato ( ome tuato depedg o mutual poto of eo ad lghtg dchage poto of the actual ad phatom gal ca be vey cloe that doe a poblem wth eal gal epaato 5 Example I a demotato example thee meauemet pot whch ae codtoally placed ea the cte of Almaty Taldykoga ad Kaphagay (Republc of Kazakhta wee elected The coepodg tmepot ae calculated the aumpto that the lghtg dchage occued ea Ataa cty Zeo wa take fo tal coutg of tme Gve data fo calculato ae peeted Table I tet calculato of dchage tme the tak oluto the followg value wee take: Speed of ado wave km/ Aveage Eath adu 6708 km A a eult of the tak oluto two pot of the lghtg dchage wth vaou tmepot of a dchage ae defed Reult of calculato ae gve Table Paamete of the actual ad phatom pot of the lghtg dchage ae mutually eveble That f accodg to a phatom pot povded Table to defe tme of egtato of the lghtg dchage ad to make calculato of a lghtg dchage the we wll eceve the ame eult gve Table Lghtg dchage locato ad tmepot a phatom pot ae dplaced wth epect to actual 6 Tagle Techque Applcato fo Tmepot ad Locato of the Lghtg Dchage fo Set of Seo Let code a et of N eo located adomly o the Eath uface the pot P( ϕ θ N N > Let pot P ( ϕ θ thee wa a lghtg dchage the gal fom whch eache each tato momet t We wll put complace to each pot P( ϕ θ a ut vecto of t hodogaph { x y z} ad pot I th cae the ytem ( wll expad to N of lghtg dchage P ( ϕ θ -ut vecto { x y z } Table Seo locato ad tmepot of the lghtg dchage detecto Seo aea Logtude of the eo degee Lattude of the eo degee Lghtg dchage detecto tmepot Almaty Taldykoga Kaphagay Table Calculato eult Soluto # Logtude degee Lattude degee Tmepot of lghtg dchage (Ataa ego (Phatom pot

9 A Lozb et al equato ( R( acco C t t N ( We wll make ubytem fom the equato of ytem ( cludg thee equato to each ubytem We wll call uch ubytem a tad I total t poble to make M tad fom N equato whee N N M ( ( N Let each tad of the Equato ( have two oluto We wll cotuct the et of oluto { k tk} k M of all tad ad calculate fuctoal value o each oluto If the fuctoal F( t ( ϕθ acco ( ( 6 N R (4 F t C t t ϕθ at ay oluto { k tk} wll become zeo (t poble wth gve accuacy ε > 0 the th pa wll be a ytem ( oluto If F( ϕθ t > 0 at ay oluto { k tk} the fo appoxmate oluto of the ytem ( we chooe oluto whee fuctoal F( ϕθ t ha lowe value Wthout log a geealty we wll uppoe that { t } { t} Fo a aemet of the eceved oluto of ytem ( we wll allocate the ubet cludg the M actual oluto fom a et of all oluto of tad Fo th pupoe we ue the atual aumpto that o the actual oluto the fuctoal F( ϕθ t have lee value tha o phatom oluto ( cae of thee eo thee value cocde ad both ae equal to zeo If both oluto of ome tad ae cloe to each othe the ae cae th aumpto ca be volated We wll umbe th ubet value a k M We wll deteme a athmetc aveage value o a ubet of actual oluto M M t M t M Fo the accuacy akg of the appoxmato { t } dtace betwee ad at tme-dffeece δ Re (5 at the poto of the lghtg dchage we take a acco ( t t t (6 δ (7 The gade of dpeo of teecto pot of hypebole chaactezed by a aveage quae devato fom the poto of the oluto whch calculated by fomula ( acco M Re M (8 Fo the aaly of the gve techque a umbe of umecal expemet wa executed I all expemet the ame goup of 6 eo wth coodate ae gve Table wa ued Numecal expemet ae gve fo thee pot of a lghtg placed vaou ego of Kazakhta emoved fom each othe the Noth Wet ad Eat Fo each lghtg coodate ae et ad codtoally exact tme of detecto wth p accuacy calculated The zeo momet of each lghtg dchage each expemet takg Lghtg paamete ae gve Table 4 I addto fo the ubequet aaly the dtace fom tato to a pot of the lghtg pecfed Table 4 The poblem of each expemet coted lghtg poto ad tal momet calculato wth tagle techque depedg o the accuacy of lghtg detecto o tato Tme of lghtg detecto wa et by oudg of exact value of the momet of detecto gve Table 4 by tep-by-tep appoxmato to 0 00 ad μ Reult of calculato ae gve Table 5-7

10 A Lozb et al Table Seo locato Stato # Logtude degee Lattude degee Rego of the Stato Almaty Taldykoga Kaphagay Taaz Balkhah Shu Table 4 Lghtg dchage paamete Stato # Lghtg coodate: -Logtude 7 -Lattude 5 -Ataa cty Lghtg detecto tmepot Dtace fom tato to lghtg km Lghtg coodate: -Logtude 5 -Lattude 44 -Aktau cty Lghtg detecto tmepot Dtace fom tato to lghtg km Lghtg detecto tmepot Lghtg coodate: -Logtude 85 -Lattude 47 -Zaa cty Dtace fom tato to lghtg km Table 5 The eult of expemet fo lghtg Ataa cty ego (5 N 7 E Lghtg tmepot meauemet accuacy Lghtg logtude degee Lghtg lattude degee Tme of lghtg Accuacy δ km The mea tme of lghtg Mea quae devato km Devato fom the exact poto km μ Table 6 The eult of expemet fo lghtg Aktau cty ego (44 N 5 E Lghtg tmepot meauemet accuacy Lghtg logtude degee Lghtg lattude degee Tme of lghtg Accuacy δ km The mea tme of lghtg Mea quae devato km Devato fom the exact poto km μ

11 A Lozb et al Table 7 The eult of expemet fo lghtg Zaa cty ego (47 N 85 E Lghtg tmepotmeaumet accuacy Lghtg logtude degee Lghtg lattude degee Tme of lghtg Accuacy δ km The mea tme of lghtg Meaquaedevato km Devato fom the exact poto km μ Cocluo Compag chage of eult of lghtg dchage tme ad locato calculato o ato of ceae of a eo of lghtg detecto tme t poble to make a cocluo that the peeted techque of calculato effectve ad elable f the detecto accuacy doe ot exceed 00 aoecod Sce the accuacy of mcoecod age of hypebole teecto pot dpeo fo dffeet tad whch chaactezed by ceae dtace betwee them ad accdet of the mutual poto codeably ceae Ackowledgemet The wok uppoted by the Gat 000/GF4 of Mty of Educato ad Scece of the Republc of Kazakhta Refeece [] Kohak WJ ad Solakewcz RJ (00 TOA Lghtg Locato Reteval o Sphecal ad Oblate Spheodal Eath Geomete Joual of Atmophec ad Oceac Techology [] Kute FA Bulatov AA ad Shlyugaev YuV (04 The Developmet of the Lghtg Detecto Netwok Baed o BoltekStomTacke Hadwae XV Iteatoal Cofeece o Atmophec Electcty Noma 5-0 Jue