Gaussian Random Process and Its Application for Detecting the Ionospheric Disturbances Using GPS

Transcription

1 Journal of Global Posonng Sysms (005) Vol. 4, No. 1-: Gaussan Random Procss and Is Applcaon for Dcng h Ionosphrc Dsurbancs Usng GPS H.. Zhang 1,, J. Wang 3, W. Y. Zhu 1, C. Huang 1 (1) Shangha Asronomcal Obsrvaory, Chns Acadmy of Scncs, 80# Nandan R.d, Shangha, Chna () Gradua School of Chns Acadmy of Scncs, Bjng , Chna (3) School of Survyng and Spaal Informaon Sysms, Unvrsy of Nw Souh Wals, Sydny, Ausrala -mal: hpzhang@shao.ac.cn Tl: ; Fax: Rcvd: 10 Dcmbr 004 / Accpd: 13 July 005 Absrac. Usually, onosphrc Toal Elcron Conn (TEC) varaon wh m can b vwd as a saonary random procss undr u condons. Howvr, suddn vns of h Sun and h Earh such as solar flar and suddn commncmn of gomagnc sorms may nduc h dsurbancs of h onosphr, so ha h saonary random procss s brokn; h sascal modl paramrs chang much. Basd on hs fac, hr w mak us of h m srs of TEC and h auocovaranc funcon of h saonary procss o consruc ndpndn dncal dsrbuon Gauss sampl so ha h χ s can b usd o dc h abnormy hddn n h sunc. In addon, GPS daa from svral IGS ss n Chna durng h svr solar flar occurrd on 14h July, 000 ar usd o vrfy h mhod. Th rsuls ndca ha h dsurbancs causd by h solar flar can b ffcvly dcd. Ky words: GPS, Toal Elcron Conn, Gaussan Random Procss, Ionosphrc dsurbancs 1 Inroducon Ovr h pos wo dcads, GPS has bn wdly usd n sudyng of h phnomna of Suddn Incras of Toal Elcron Conn (SITEC) causd by h solar flar (Wan. al., 000; Zhang al., 000 and 00; Edward al., 000), monorng h larg, mddl and small scal of h Travllng Ionosphr Dsurbanc (TID) (Zhang al., 00; Ho., al., 1996; Sao al., 1998), vrfyng h hory of Chapman Ionzaon, analysng h ffcs of magnc sorm on h onosphr(sao al., 1998; Ho., al., 1998), monorng h onosphrc rrgulars (P al., 1997). Du o h hgh spaal-mporal rsoluon of Toal Elcron Conn (TEC) daa provdd by h globally or rgonally covrd GPS connuously oprang rfrnc saons, GPS s hlpng us furhr h undrsandng of h characrscs, h prncpls and h law of h onosphrc acvs a h global, rgonal or local scal, whch graly promo h dvlopmn of h hgh-ar amosphr scnc and spac wahr suds. Howvr, mos of hs rsarchs ar accomplshd by mans of posprocssng, whl som spcal srvc, such as spac wahr prdcon, wrlss communcaon and hgh prcson GPS godc survyng, nds o dc and dal wh h dsurbng of h onosphr so ha h ffcs of h onosphr on hm can b bs conrolld. Thrfor, s ncssary ha h hory and mhods of dcng h onosphrc dsurbancs usng GPS should b sudd comprhnsvly and sysmacally. As o hs subjc, Yuan al. (001) offrd h random onosphrc dsurbanc dcng hory and schm for praccal opraon. Th prlmnary sng rsuls basd on such a hory usng h auo-covaranc smaon of varabl sampls (ACEVS) provdd by Yuan ( al. 001) suggs ha h onosphrc anomaly can b bs dcd. Hr w wll consruc h Indpndn Idncal Gauss Dsrbuon (IIDN(0,1)) sampls basd on h rlad characrscs of h saonary random procss of h varaon of h onosphr, hn h hypohss sng of h ch-suar s nvolvd n analyzng h m srs of TEC so ha h anomaly can b chckd ou. On h ohr hand, o valda such a mhod, h ACEVS mhod s nroducd and h rsuls producd usng h wo mhods ar compard.

2 Zhang al.: Gaussan Random Procss and s applcaon n GPS/TEC 77 Consrucng IIDN(0,1) sampl wh saonary random srs Assum a ralzaon of h rgodc Gaussan saonary random procss { x } wh zro xpcaon valu x = x + ( = 1,,..., N) (1) whr s rgodc Gaussan wh nos wh zro xpcaon valu (ndpndn of x); N s h numbr of sampls. For smplcy, h sochasc modl and ohr rlvan proprs of { x } and { } ar wrn as E( ) = E( x) = 0 COV ( x, x r) = E( xx r) = γ ( r) COV (, r) = E( r) = γ ( r) COV (, x) = E( x) = 0 COV ( x, x ) = γ ( r) + γ ( r) () r whr COV s h covaranc; γ and γ ar h auocovaranc funcon of { x } and { }, rspcvly. Snc s rgodc Gaussan wh nos wh zro xpcaon valu, hn h γ owns h followng characrscs γ ( y) = 0, y > 0 γ ( y) = D, y = 0 (3) Thrfor h auo-covaranc funcon of sampl srs { x } can b xprssd as: COV ( x, x ) = E[( x + )( x + )] r r = γ (r) + γ (r) = γ (r) r > 0 COV(x, x ) = + D r = 0 (4) whr D s h varanc of h rgodc Gaussan wh nos. Snc srs { x } s an rgodc Gaussan random procss, hn Y = ( x1, x, x3,..., x N ) can b vwd as N-vara random vcor accordng o h proprs of rgodc random procss (Lu, 000). Hr E( Y ) = 0. Whn varanc D of s known, covaranc marx of N-vara random vcor Y can Σ YY b drmnd by usng h auovaranc funcon of srs { x }. Thn vcor Y follows h N-vara normal dsrbuon wh zro xpcaon and covaranc marx Σ. Tha s: Y N(0, Σ ) YY YY whr covaranc marx Σ YY s non-ngav dfn. If d Σ YY > 0, hn random vcor 1/ Z = Σ (Y 0) (5) s N-vara random vcor wh E( Z ) = 0, Σ ZZ = I N, whr I N s n-vara un marx. Thn random vcor Z follows IIDN(0,1). Howvr, h ransfrrng procss dscrbd abov nds o us h auocovaranc funcon of h saonary random procss { x }, whch can no b accuraly known n pracc. Thn h smas of auocovaranc γ () r can b drvd usng sampls wh h followng formula (Pr al., 1991): N r 1 = 1 r γ (r) = N x x, 0 < r N 1 (6) Hr Euaon () s no h unbasd smas of γ () r, 1/ bu n h condon ha Y = Σ Z, { Z} IID(0, σ ), whn N, h asympocal dsrbuon γ () r s γ (r). Thn h sma γ ( r), r = 0,1,..., N 1 owns h auocovaranc marx n = γ (1) γ ( n 1) γ (1) γ ( n ) γ ( n 1) γ ( n ) Whch s non-ngav dfn whn n 1. As dscussd abov, h dscr srs of Gussan saonary random procss can b ransfrrd o a mulvara random vcor, hn h analyss of h sascal characrsc paramrs of saonary random procss can b subsud by h corrspondng analyss of a mulvara random vcor. Thrfor, whn happns o b anomaly a h h obsrvaon of { x }, h corrspondng h of { Z } should b anomaly. Thn h sascal ools can b usd o analyz. (7) 3 Drmnng saonary Ionosphrc TEC srs wh GPS Th GPS TEC obsrvaons nclud h drmnsc I par (such as a rnd and a prod) and sochasc ( δ I ) par du o h onosphr acvy. Usually, for shor m scals, for drmnsc ffcs, only h rnd varaon

3 78 Journal of Global Posonng Sysms can b consdrd, and I can b wrn as a polynomal m I = a = 0 Th sochasc ffcs δ I may b consdrd as a Gaussan random procss wh zro xpcaon valu. Whn random dsurbancs of h onosphr happn, hr ffcs on TEC wll usually dsroy h sady sa of δ I. Thrfor, s possbl o s h chang of sa of δ I usng sascal ools. Assum ha I s h onosphr TEC obsrvaon a an arbrary poch and ε s s Gaussan wh nos { E ( ε ) = 0}, ndpndn of δ I [.. E( δ I ε + ) = 0. Furhr, { δ I + ε} s a Gaussan random saonary procss wh zro xpcaon valu. Thus h onosphrc TEC obsrvaon modl can b xprssd as m I = I + δ I + ε = a + δi + =0 Dfn h dffrnc opraor as I = I I + 1 k k k 1 I ( I) ( 1) CkI+ k 1 = 0 ε (8) = = (9) whr C k s h combnaon opraor. To rduc h rnd rm I, a = m + 1 -ordr dffrnc opraon can b usd for E. (8) I = δ I + ε = ( δi + ε ) E( I ) = E( δ I + ε ) = E( δi ) + E( ε ) = 0 (10) Smlarly = + I+ h δ I+ h ε+ h E( I + h) = 0 (11) From h abov, can b sn ha I s a lnar combnaon of δi+ + ε+,( = 0,1,,..., ), whl { δ I+ + ε+ } s a Gaussan random varabl wh zro xpcaon valu. So accordng o h nvaranc propry of lnar ransformaons of Gaussan dsrbuons, I s a Gaussan random varabl wh zro xpcaon valu as wll. Thn can b provd ha { I } s saonary and obvously s an rgodc procss as wll [Yuan, al., 001]. I Bcaus { } s an rgodc Gaussan procss wh zro xpcaon valu, and f x = I +, h and x = I + h,, hn, undr normal obsrvaon condons, h srs { x = I + h, } may b consdrd as h approxma srs of { x = I + h, } and can b ransformd o IIDN(0,1) sampls accordng o h mhod dscussd n Sc.. In h m srs of TEC obsrvaons of GPS, h chang of h sascal proprs of h random onosphrc TEC { x I + h, = } from a saus of sably o on of dsurbanc can b dsngushd by h chang of s ransformd IIDN(0,1) sampls. So s possbl o s by usng h GPS m srs. 4 Applcaon and analyss 4.1 Schm for dcng h Anomaly Accordng o wha hav bn dscussd abov, w can consruc h schm for sng h anomaly usng h ransformd IIDN(0,1) sampls. Hr w brfly dscrb h schm as followng: 1) G h dffrncd TEC srs from GPS obsrvaons so ha s saonary. Usually, scond-ordr dffrncng s nough. ) Calcula h smas of h auo-covaranc of h dffrncd saonary TEC srs usng formula (6), consruc h auo-covaranc marx wh formula (7). Hr w s 40-ordr marx so ha h fxd-lngh sampls can b sldng along h m srs wh m. 3) Accordng o h fxd-lngh sampls wndows (40 sampls usd hr) sldng wh m, consruc h IIDN(0,1) sampls { Z, = 1,,..., N } usng formula (5). Thn χ hypohss sng can b usd o dc h anomaly. 4) If χ ( N) = Z1 + Z ZN χ α ( N), hn h sa of onosphr s sabl, ohrws an anomaly happns. To valda h schm abov, hr w also apply h ACEVS mhod o h sampl xampl. Th cor prncpl of ACEVS s o consruc an asympocally ndpndn normal Gaussan sunc { ( ) (0,1)} N ρ r N M so ha χ hypohss sng can b

4 Zhang al.: Gaussan Random Procss and s applcaon n GPS/TEC 79 usd. Hr M s h mnmum sampls usd o g γ () r usng formula (6), ρ () r = γ N+ 1, N r () / N( N + 1) N+ 1, N N+ 1 N γ () r = γ () r γ () r. (1) To smplfy h ACEVS mhod n applcaon, w brfly dscrb h schm for sng an anomaly usng ACEVS mhod as followng: 1) G h dffrncd TEC srs from GPS obsrvaons so ha s saonary. Usually, scond-ordr dffrncng s nough. ) Accordng o h N sampls w oband (hr w s r=10, M=100, N>100), calcula γ N () r and consruc sunc γ ( r ), = 1,,..., N M. 3) G h srs Fg. 1 provds h frs-ordr dffrncd TEC curvs drvd from h obsrvaons of GPS salls PRN9 and PRN1 obsrvd a IGS ss WUHN and BJFS rspcvly on 14 h, July, 000. Obvously, solar flar causd h phnomna of SITEC durng h prod of UTC10:0010:30. Th amplud of such suddn ncras was up o 0.76TECU/mn and lasd almos half an hour. Such onosphrc anomaly could b sn n h TEC obsrvaons of ohr salls obsrvd a ohr GPS rackng ss, whch mans ha h solar flar affcd h onosphr n h global scal. γ, + 1() r = γ+ 1() r γ(), r = 1,,..., N M + 1, hn consruc nw sunc ρ ( r) = γ, + 1( r) /, = 1,,..., N M + 1. N( N + 1) Ths sunc s an asympocally ndpndn normal Gaussan sunc. 4) Usng h fxd-lngh sldng sampl wndow (hr w s as 40, so N-M mus b mor han 40), consrucs k ρ = 1 h sascal uany χ ( k) = ( r). Thn slds h wndow wh m and s h saus of onosphr, f χ ( k) χ α ( k), h onosphr s n good condon. 4. Exampl and analyss Th ncras and dcras of h onosphrc TEC wh m ar h man phnomna ha rflc h saus of h onosphr. Usually, solar flar and h rrgular acvs of h amosphr may caus h phnomna of SITEC so ha h u saus of h onosphr s brokn. Hr w wll analyz h anomaly of h onosphr causd by h srong solar flar happnd on 14 h, July, 000. Th wo sng schms dscrbd abov ar usd n h followng scon. Fgur 1. Th frs-ordr dffrncd TEC srs obsrvd by PRN9 (up panl) and PRN1(down panl) a WUHN and BJFS rspcvly on 14 h, July, 000 In Fg. 1 w can s ha h rnd of h frs-ordr dffrncd TEC curvs s obvous, whch ndcas ha h srs ar no an saonary procss. So h scondordr dffrncd TEC srs of PRN9 obsrvd a WUHN s s drvd, whch s shown n Fg. (a). Fg. (b) provds h valu of h auo-corrlaon funcon of R(10) varyng wh m (h anomaly sampls ar kckd ou), whch s a consan and ndcas ha s ndpndn of m. Hr R(10) s as a xampl, ohr valus, whch s smlar o ha of Fg. (b), of h auocorrlaon funcon varyng wh m a dffrn

5 80 Journal of Global Posonng Sysms nrvals hav bn drvd and no shown hr. So h scond-ordr dffrncd TEC srs s saonary. achv h sam purpos of monorng and dcng h anomaly of h onosphr. Fgur (a). Th scond-ordr dffrncd TEC srs obsrvd by PRN9 a WUHN Fgur 3. Th rsul of χ sng wh h ransformd IIDN(0,1) Sampls Fgur (b). Th Auo-corrlaon Funcon of R(10) (h nrval s10 pochs) of h scond-ordrd dffrncd TEC srs Fg. 3 s wha s oband by h χ hypohss s usng h IIDN(0,1) sampls ransformd from h scond-ordr dffrncd TEC srs wh h approxma auocorrlaon marx drvd from such a random procss, whl Fg. 4 s h rsul of h χ hypohss s usng ACEVS mhod wh h sam ralzaon of such a random procss. In Fg. 3 and Fg. 4, w can s ha h wo fgurs ar n h smlar shap, wha s mor, h paks appars and corrsponds o h anomaly prod rflcd n Fg. 1. In addon, h valu of χ s mor han h hrshold valu 0.7 bfor h pak valu appar, whch ndca ha h onosphr bgun o b unsabl bfor h solar flar brak ou, whl h onosphr rcovrd u soon afr h solar flar. Ths rflcs h characrsc of h vns of ha solar flar on h onosphr. Th smlary of such curvs n Fg. 3 and Fg.4 show ha h wo schms dscrbd abov can Fgur 4. Th rsul of 4. Summary χ hypohss s usng h ACEVS mhod Th brakag of h u onosphr corrsponds o h chang of h sascal paramrs of h TEC m srs, whch can b usd o monor h acvs of h onosphr so ha h dsurbanc of h onosphr can b dcd. Monorng and dcng h onosphrc dsurbanc s mporan for h rsarch and prdcon of h spac wahr, as wll as GPS survyng, sall navgaon, sall communcaon and so on. So h hory and mhods of ral-m monorng of h onosphr wh GPS can lad us o know h saus of h onosphr accuraly and n ral m, so ha w can ak good sps o avod gra loss. Thrfor, h mhods usd o dc h dsurbanc of h onosphr

6 Zhang al.: Gaussan Random Procss and s applcaon n GPS/TEC 81 wh h saonary random procss n hs papr provd a good alrnav choc. Rfrncs Wan WX, Yuan H, al (001) Th Suddn Incras of h Toal Elcron Conn causd by h gra Solar Flar occurrd on 7h, July, 000. Scnc In Chna (A), 31(supp.), Zhang DH, Xao Z (00) Corrlav onosphrc dsurbancs n h sunl hmsphr durng h flar on July 14, 000, Chns Scnc Bulln, 47(1), Zhang DH, Xao Z (000) Th sudy of onosphrc TEC durng h flar on Nov., 1998 by mans of 4 GPS rcvrs ovr Chna, Aca Scnarum Nauralum, Unvrsas Pknnss, 36(3), Edward L. Aframovch, Eugn A. Kosogorov, Ludmla A. Lonovch (000) Th us of nrnaonal GPS nwork as h global dcor (GLOBDET) smulanously obsrvng suddn onosphrc dsurbancs, Earh Plans Spac, 5, Zhang DH, Igarash K, Xao Z, Ma GY (00), Th Obsrvaon of Larg Scal Travllng Ionosphrc Dsurbancs Basd on GPS Nwork, Chns Journal of Gophyscs, 45(4), Ho CM, Mannucc A, Lndwsr U, P X, Tsuruan B (1996) Global onosphr prurbaons monord by h worldwd GPS nwork, Gophyscal Rsarch Lrs, 3( ), Sao A, Fukao S (1998) Hgh rsoluon mappng of TEC prurbaons wh h GSI GPS nwork ovr Japan, Gophyscal Rsarch Lrs, 5(16), C.M. Ho, Mannucc A, Lndwsr U, P X, Tsuruan B, Sparks L, Ijma B, Wlson B, Harrs I, and Rys M (1998) Global onosphrc TEC varaon durng January 10, 1997 Sorm, Gophyscal Rsarch Lrs, 5(14), P X, Mannucc A, Lndwsr U, Ho C (1997) Monorng of global onosphrc rrgulars usng h worldwd GPS nwork, Gophyscal Rsarch Lrs, 4(18), Yuan Y, Ou J (001) Auo-covaranc smaon of varabl sampls (ACEVS) and s applcaon for monorng random onosphrc dsurbancs usng GPS, Journal of Godsy, 75: Lu CH (000) Sochasc Procss, Huazhong Unvrsy of Scnc and Tchnology Prss Pr J. Brockwll, Rchard A. Davs (1991) Tm Srs: Thory and Mhods, Sprngr-Vrlag Prss, Nw York Davs K, Harmann GK (1999) Sudyng h onosphr wh h Global Posonng Sysm, Rado Scnc, 3(4),