A three-dimensional time-dependent algorithm for ionospheric imaging using GPS

Transcription

1 ANNALS OF GEOPHYSICS, VOL. 46, N. 4, August 2003 A three-dmensonal tme-dependent algorthm for onospherc magng usng GPS Cathryn N. Mtchell and Paul S.J. Spencer Department of Electronc and Electrcal Engneerng, Unversty of Bath, U.K. Abstract Global Postonng System (GPS) satellte recevers provde a world-wde network of phase and group delay measurements. The combnaton of two-frequency measurements can be used to derve the ntegral of the electron concentraton along each satellte-to-recever path, a parameter known as the Total Electron Content (TEC). At ths stage these slant TEC data are dffcult to nterpret as they orgnate from a combnaton of a temporally changng onosphere and spatally changng observaton geometry. In ths paper TEC data are nverted to evaluate the underlyng dstrbuton and tme evoluton of electron concentraton. Accordngly, a new three-dmensonal, tme-dependent algorthm s presented here for magng onospherc electron concentraton usng GPS sgnals. The nverson results n a three-dmensonal move rather than a statc mage of the electron-concentraton dstrbuton. The technque s demonstrated usng smulated ground-based GPS data from actual measurement geometry over Europe. Key words onosphere magng GPS nverson tomography 1. Introducton The onosphere results from the nteracton of solar emssons wth the Earth s atmosphere. Its evoluton s nfluenced by changes n solar emssons, atmospherc dynamcs and the nterplanetary- and geo-magnetc felds. Ionospherc morphology results from the nteracton of these drvers and san nterestng topc for scentfc study. Specal nstrumentaton devoted to such studes has been developed and used extensvely for many years. The onosphere affects electro-magnetc waves by refracton and the magntude of ths Malng address: Dr. Cathryn N. Mtchell, Department of Electronc and Electrcal Engneerng, Unversty of Bath, Claverton Down, Bath, BA2 7AY, U.K.; e-mal: C.N.Mtchell@bath.ac.uk s frequency dependent. The result of ths s that sgnals are bent and delayed n a manner that may be ether advantageous or dsadvantageous to rado systems. For example, HF communcaton beyond the horzon s possble only because oblque sgnals can be refracted and returned to the ground.in ths case the user can determne the path of a sgnal by ray-tracng through a map of the electron concentraton. L-band sgnals used n satellte navgaton systems based on the tme-of-flght of the sgnal are put n error by the unknown onospherc delay. For the user needng to compensate for ths error, the requrement s to be able to know the delay ntroduced by the Total Electron Content (TEC), along specfc satellte-to-recever ray paths. Agan, ths requres accurate knowledge of the dstrbuton of electron concentraton. Recently, dual-frequency satellte sgnals have provded a new source of onospherc nformaton. The Global Postonng System (GPS) satelltes are montored by a network of dual-frequency recevers, recordng the phase and tme delay of each sgnal. These measure- 687

2 Cathryn N. Mtchell and Paul S.J. Spencer ments provde a valuable source of nformaton about the onosphere n the form of ray-path ntegratons of electron concentraton. These measurements are rather uneven n dstrbuton and coverage but have some smlartes to polarorbtng low-earth orbt satellte TECs derved from the Navy Ionospherc Montorng System (NIMS) satelltes. In contrast to GPS TECs, NIMS satellte TECs map easly nto a plane and thus can be nverted nto mages of electron concentraton usng tomographc methods. Tomography s the nverson of networks of lne-ntegral measurements to yeld the twodmensonal spatal dstrbuton of the ntegrated parameter. Ths s very mportant for onospherc data combnaton snce many measurements of the lne ntegral of the electron concentraton (.e. the TEC) are easly obtaned, but the parameter of nterest s often the electron concentraton tself. In the case of conventonal onospherc tomography, TEC measurements are made from dual-frequency trans-onospherc sgnals along many ntersectng raypaths between a LEO satellte and a chan of ground-based recevers. These TEC data are subsequently analysed and nverted usng a sutable mathematcal algorthm to produce two-dmensonal mages of electron concentraton. A revew of the expermental results from onospherc tomography has been publshed by Bernhart et al. (1998). The applcaton of the tomographc technque to onospherc magng was frst proposed n 1986 by Austen et al. LEO satellte-to-earth measurement geometry only allows observatons over a lmted number of vewng angles and Yeh and Raymund (1991) nvestgated some of the theoretcal lmtatons ths would mpose on onospherc tomography. In fact the orentatons of the satellte-to-recever ray paths are based n a vertcal sense wth no ray paths runnng horzontally through the onosphere. Consequently the vertcal electronconcentraton gradents are poorly defned by the TEC measurements alone. Ths geometrcal constrant means that the reconstructon of the tomographc mages s not straghtforward and the reconstructon cannot be mplemented drectly usng conventonal nverson algorthms alone. There have been several solutons to the problem and the one descrbed by Fremouw et al. (1992) has been wdely used. Ths algorthm, adapted from another geophyscal applcaton, uses a set of vertcal orthonormal functons, created from onospherc models, to mage the vertcal profle and a power law spectrum to select the horzontal structures from a Fourer bass. The algorthm presented n ths paper s an extenson nto three-dmensons of that presented by Fremouw et al. (1992) for two-dmensonal tomographc magng. In addton, a tme-dependence has been ncorporated nto the nverson, allowng for changes n electron concentraton throughout the magng perod. Ths allows the algorthm to nvert mult-drectonal slant TEC data derved from GPS sgnals over a perod of typcally one hour nto moves of electron concentraton. 2. Theory In ths secton the theory relatng drectly to the nverson problem s outlned. A detaled descrpton of the propagaton of electromagnetc waves through an onsed medum and the effects of the onosphere on rado systems can be found n Daves (1990). Total electron content s defned as the lne ntegral of the electron concentraton along a path from a satellte, S, to a recever, R. The TEC (b s ) may be expressed as R bs N r, θφ, ds = ( ) S (2.1) n whch N s the electron concentraton, r the radal dstance from the centre of the Earth, q s the lattude, f the longtude and s the dstance along the satellte-to-recever ray path. TEC can be measured usng Faraday rotaton or dfferental Doppler technques. In the latter case, dualfrequency rado sgnals that propagate through the onosphere are subject to a dfferental phase change due to the dspersve nature of the plasma. As a frst order approxmaton the change n the dfferental phase shft s drectly proportonal to the change n Total Electron Content (TEC) between the transmtter and recever. In addton to the phase technque, GPS sgnals also contan nformaton about the sgnal delays. The measurement of dfferental delay s subject 688

3 A three-dmensonal tme-dependent algorthm for onospherc magng usng GPS to systematc errors known as nterfrequency bases assocated wth the satellte and recever hardware and to random and systematc errors due to multpath. Nevertheless, t s stll useful to utlse dfferental delay measurements for TEC calbraton as has been done by others, for example see Craolo and Spalla (1997). The frst stage of the nverson s to set up a three-dmensonal grd of voxels (.e. volume pxels), each bounded n lattude, longtude and alttude, and to compute the length of each element of a satellte-to-recever sgnal propagaton path though each ntersected voxel. For TEC measurements and j voxels the unknown electron concentraton s defned to be constant wthn each voxel and contaned n the column vector x of j components. The problem may now be expressed as Ax = b (2.2) where A s an j matrx of the path lengths wthn each voxel and b are the observed TECs. Ths cannot be solved drectly as the matrx A s rectangular, hghly sngular and ncorporates no pror nformaton as to the lkely soluton. To overcome ths dffculty a mappng matrx, X, s used to transform the problem to one for whch the unknowns are n coeffcents of orthonormal bass functons, the combnaton of whch wll gve the fnal mage of electron concentraton. The choce of orthonormal bass functons s crtcal n the determnaton of the fnal soluton to ths underdetermned nverson problem. Here the bass functons (X) were generated usng a sphercal harmonc expanson to represent the horzontal varaton and Emprcal Orthonormal Functons (EOFs) for the radal varaton n electron concentraton. The sphercal harmoncs provde a flexble bass to determne the horzontal dstrbuton of onsaton whch should be well defned by the measurements. The EOFs form a constrant to the vertcal profle, only allowng a certan range of possble solutons. Ths s now expressed mathematcally as AXW = b (2.3) where the matrx X contans the bass functons, such that AX defnes a bass set of lne ntegratons of electron concentraton. Snce the raypath ntegrals derved from dfferental phase along the same satellte-to-recever paths are subject to an unknown cycle offset adjacent rows of the matrces AX and b can be dfferenced n a manner to negate the effect of ths on the soluton. The n unknowns, W, now represent the relatve contrbuton of the bass functons where W=(AX) -1 b. (2.4) Here (AX) -1 s a generalzed nverse matrx such that W s the maxmum lkelhood soluton. In ths case Sngular Value Decomposton (SVD) has been chosen to solve the nverson. Applyng sngular value decomposton to the matrx AX returns two orthogonal matrces U and V and a dagonal matrx of sngular values, w, thus ( ) 1 ( AX) = V dag ( 1 / w) U T. (2.5) The soluton to the nverse problem s then gven by ( ( )) T W = V dag( 1 / w) U b (2.6) where the recprocal of the terms n w that are suffcently small, typcally 10 7 of the domnant sngular weght, are zeroed to account for degeneracy n the AX matrx. A descrpton of the use of SVD can be found n Golub and Van Loan (1989). Fnally, the electron denstes wthn each voxel, j, are recovered usng x = XW. (2.7) GPS satelltes have an orbtal perod of just under 12 h and are nclned at 55 to the equatoral plane. From a fxed locaton on the Earth they appear to move slowly across the sky and for hgh-elevaton passes take several hours from rse to set tme. A seres of TEC measurements from any sngle GPS-to-ground lnk consst of both spatal and temporal changes n the onosphere. The tme-ndependent algorthm descrbed s based on the assumpton of statonarty of the medum and for the onosphere ths generally requres a short measurement tme of a few mnutes. Ths short measurement perod would result n lmtng the data quantty and spatal 689

4 Cathryn N. Mtchell and Paul S.J. Spencer coverage when usng GPS. Extendng ths algorthm nto a tme-dependent nverson allows a great ncrease n the quantty and angular coverage of measurements that can be used n each nverson. In addton, snce there s no mplct statonarty on the onospherc electron concentraton dstrbuton t allows movng features to be maged n a consstent manner rather than beng averaged out. The algorthm can be extended nto a tme-dependent nverson by ncorporatng a pror nformaton about the evoluton of the electron concentraton durng a specfed perod of tme. Assumng that the change of the electron concentraton wthn a voxel wth tme s lnear, whch makes sense f short enough tme perods are dealt wth, then t s possble to wrte the same system of equatons to solve for the change n the relatve contrbutons of each bass functon. A relatvely short perod s chosen for the tme-dependent nverson, for example one hour, and data collected at typcally 30 s ntervals are consdered. The change n the ray path geometry, defned n the D matrx, multpled by the unknown change n electron concentraton (y) s equal to the change n TEC (c) Dy=c. (2.8) Thus the matrx, D, s formed from the dfference n ray-path geometry at successve tme ntervals, y s the unknown change n electron concentraton between each tme nterval and c are the measured changes n TEC. The mappng matrx, X, s agan used to transform the problem to one for whch the unknowns are the lnear changes n coeffcents (G) of a set of n approprately selected ortho-normal bass functons D(XG)=c. (2.9) The matrces can be re-wrtten such that the ray path geometry s multpled drectly by the mappng matrx to create the bass set DX(G)=c (2.10) and the change n the unknown contrbutons of each of these lne ntegratons of electron concentraton s solved for G=(DX) -1 c (2.11) The tme-dependent soluton to the nverse problem s then gven by y=xg (2.12) and ths can be combned wth the statc soluton of (2.2)-(2.7). Crtcal factors n the soluton are the choce of the vertcal profle set to form the bass functons and the total tme over whch lnear changes can be assumed. Lnear changes are generally enforced over tme perods of onehour. 3. Results In ths secton smulatons of two- and threedmensonal statc and tme evolvng nversons are demonstrated. For the ntal smulaton a twodmensonal lattude versus alttude model of the electron concentraton was created. The model was not ntended to represent any partcular onosphere but to contan features that mght be dffcult to mage usng a statc soluton. Ths s to demonstrate that a tme-evolvng onosphere can be maged wth the algorthm presented here. The vertcal profle of the onsaton n the model was formed from Chapman functons (Chapman, 1931). In the ntal model the onosphere had a constant peak heght at 285 km and scale heght of 110 km, wth a peak electron concentraton of m -3. In the fnal model the peak heght to the north of the mage was ncreased to 370 km, the scale heght to 140 km and the peak electron concentraton was ncreased to m -3. A large depleton n onsaton was supermposed onto the mages to represent the nght-tme md-lattude structure known as the man trough. The centre lattude of the trough was 48º n the frst frame and 45ºN n the fnal frame. Thus the changes to the onosphere ncluded a 3º southward movement of the trough, an ncrease n electron concentraton and an ncrease n the peak and scale heght to the north of the trough. The 690

5 A three-dmensonal tme-dependent algorthm for onospherc magng usng GPS a a b b c Fg. 1a-c. Two-dmensonal maps of electron concentraton. a) Intal frame of the model onosphere. b) Mddle frame of the model onosphere. c) Fnal frame of the model onosphere. The contours are shown every electrons m - 3. c Fg. 2a-c. Two-dmensonal maps of electron concentraton resultng from the tme-dependent nverson. a) Intal frame of the nverson. b) Mddle frame of the nverson. c) Fnal frame of the nverson. The contours are shown every electrons m

6 Cathryn N. Mtchell and Paul S.J. Spencer Fg. 3. Electron concentraton determned usng the tme-ndependent (statc) nverson. The contours are shown every electrons m -3. modelled electron concentraton vared lnearly between the extremes of the two mages shown n fg. 1a,c. Sx ground-based recevers, evenly spaced every 5 between 35 and 60 N, were used n the smulaton. Two counter-orbtng NIMS type satelltes at 1200 km alttude were modelled. NIMS were chosen snce ther polarorbts are applcable to two-dmensonal magng. A mnmum of two satelltes s requred to separate temporal and spatal changes n ths type of magng. TECs were smulated between each satellte and recever for 1 steps n satellte lattude. Thus a sngle set of twelve (two satelltes and sx recevers) smulated TEC measurements were found by ntegraton through a sngle frame of the onosphere. Subsequent sets of TECs were found by ntegraton through subsequent frames of the onosphere accordng to changes descrbed above, relatng to the tme-evoluton. The tme-dependent nverson results are shown n fg. 2a-c. The three mages are taken from the frst, mddle and fnal frames of the nverson. Comparng fg. 2a wth fg. 1a t can be seen that the mage represents the peak heght, scale heght and electron densty qute well. The lattudnal locaton of the trough and ts depth are n agreement wth the model. Comparng fg. 2b wth fg. 1b t can be seen that the electron concentraton s slghtly low to the south of the trough and hgh to the north. The contour to the far north of the mage s closed, ndcatng that the nverson has produced a slghtly low peak densty to the northern edge of the mage. Nevertheless, the scale heghts are well reproduced by the nverson and the step n peak heght across the trough s clearly seen. The lattudnal locaton of the trough has also been reconstructed well. The fnal mage (fg. 2c compared to fg. 1c) shows that the features are well replcated but the electron concentraton s slghtly hgher than that of the model to the north of the trough. There s a slght gradent n the peak heght n the reconstructed mage that s not n the model. Nevertheless the overall features and the changes n the onosphere have been very well accommodated n ths tmedependent nverson. In order to make a comparson, the nverson was also performed usng the tme-ndependent algorthm. The result s shown n fg. 3. Surprsngly, the resultng mage has a lower peak concentraton than that found n the mddle frame of the model. Some artfcal structurng has appeared n the magng, caused by the nconsstences between the measurements. Ths shows that t s not possble to rely on the statc magng f extreme densty gradents and movng features could be present n the onosphere. The tme-dependent algorthm n threedmensons s now demonstrated usng a onehour smulaton. The onosphere over Europe was modelled usng the IRI model (Bltza, 1990) at 2 mn ntervals from 06 to 07 UT on 20 Aprl The IRI model was chosen because of ts ablty to represent the onosphere globally n three-dmensons wthout any dscontnutes, thus allowng smooth ntegraton wth whch to smulate TEC. Actual GPS satellte orbts and 15 recever locatons from the Internatonal GPS Servce network were used. The slant TECs were smulated by ntegraton along the satellterecever path every two mnutes throughout the hour through hgh-resoluton grds of IRI electron denstes. The hgh-resoluton grd, ensurng no dscontnutes n the smulated TECs, was 1º lattude by 2º longtude and 25 km alttude extendng from 35ºN to 65ºN n lattude, 10º W to 40º E n longtude and 100 km to 1200 km n alttude. The nverson was based on the same grd. Fgure 4a,b shows the vertcal TEC from the IRI model at 06 UT (fg. 4a) and 07 UT (fg. 4b). A smooth gradent n TEC s seen rsng 692

7 A three-dmensonal tme-dependent algorthm for onospherc magng usng GPS a b Fg. 4a,b. Vertcal TEC from the IRI-95 model for (a) 06 UT and (b) 07 UT on 20 Aprl Contours are shown every 2 TECU. a b Fg. 5a,b. Vertcal TEC resultng from ntegraton through the tme-dependent nverson of model phase data from ground-based recevers and GPS satelltes for (a) 06 UT and (b) 07 UT. Contours are shown every 2 TECU. n magntude toward the south-east. Over ths one-hour perod the model shows an overall ncrease n TEC of around 15%. The lower two panels of the fgure show the vertcal TEC obtaned by ntegraton through the frst and last frames of the three-dmensonal tme-dependent reconstructon. Comparng fg. 5a wth fg. 4a, t can be seem that the nverson has produced a very accurate representaton of the zonal and merdonal gradents n TEC. The reconstructon shows a slght underestmaton of TEC towards the south of the mage and overestmaton n 693

8 Cathryn N. Mtchell and Paul S.J. Spencer the north. There s also some evdence of an artfcal change n the zonal TEC gradent to the north of the mage, resultng from the use of sphercal harmoncs and a lmted number of dscrete voxels n the mage reconstructon. The last frame of the nverson (fg. 5b) shows the vertcal TEC at 07 UT. Agan the mage shows an accurate representaton of the zonal and merdonal gradents and a slght underestmaton of TEC. Ths underestmaton may be the result of the lmted number of EOFs used n the nverson. Twenty-four smulatons were run to test the nverson method over an entre day. One three-dmensonal tme-dependent nverson was done for each one-hour perod of the 20 Aprl Each reconstructon was evaluated by comparson of vertcal TEC between the reconstructon and the IRI model at the start of the hour, to determne the mean and the mean absolute error n both TEC unts and as percentages of the model TEC. The mean errors n TEC were found usng 1 ( bnv (3.1) br) where br are the 750 () vertcal TECs found by ntegraton through the IRI model represented on the 30 lattude 12 longtude grd used for the nverson and bnv are the correspondng values found by ntegraton through the electron concentratons n the nverson. The mean absolute errors n TEC were found usng the followng equaton: 1 bnv. (3.2) br The mean errors n TEC as percentages were found usng 100 ( ) bnv br br (3.3) and the mean absolute error n TEC, as a percentage, has been found usng 100 bnv br br. (3.4) Fg. 6. Errors found between the nverson and the model over twenty-four, one-hour runs. The errors were calculated at the start of each hour usng eqs. (3.1)-(3.4). Fgure 6 shows the errors,.e. the dfferences between the vertcal TEC n the mage and the vertcal TEC n the model, over the twenty-four hour perod. It can be seen from the graphs that there s a slght underestmaton of vertcal TEC of the order of 1%. Slght varatons occur for each reconstructon, but the accuracy s no worse than 1 TECU. The absolute values ndcate that the TEC may be underestmated n some areas of the mage and over n others, snce at 4 UT, for example, the error s 2.5% n absolute but only 1.5% n mean error. It s mportant to note that n ths test a general set of bass functons were generated from sphercal harmoncs and orthonormal vertcal functons. These bass functons happen to ft the vertcal profles of certan mages better than others and the choce of them remans a crtcal factor n the accuracy of the mages. 4. Conclusons A new algorthm has been presented whch could use carrer-phase observatons to mage the electron concentraton n the onosphere. 694

9 A three-dmensonal tme-dependent algorthm for onospherc magng usng GPS The algorthm can be appled to two- or threedmensonal spatal magng ether wth or wthout a tme evolvng soluton. The advantage of usng the tme-evolvng soluton has been demonstrated by means of a two-dmensonal tomographc smulaton. It was shown to be approprate to use the tme-dependent approach to mage an onosphere wth a movng trough and ncreasng electron concentraton. In partcular, the tme-ndependent algorthm can prevent magng artefacts caused by movng structures and changng layer heghts. Watermann et al. (2002) have already recognsed ths lmtaton n onospherc magng and suggest that onospherc magng n the polar cap can be reled upon when the magnetometers suggest a slower plasma convecton. The tme-dependent algorthm shown here could be useful for magng the polar onosphere, provded that suffcent satellte data are avalable to separate the spatal and temporal varatons. It has already been establshed that GPS data can be used n onospherc magng by usng a short measurement perod and combnng wth other measurements or models. Bust et al. (2001) have shown such results from ther combned onospherc campagn n the Carbbean regon. The work presented here opens up the possblty to produce mages of the onosphere usng GPS data on ts own. The key to ths s the ncorporaton of the tme-dependence n the magng process and ths has been demonstrated wth an IRI-based smulaton wth realstc satellte and recever geometry taken from an actual case study. The work demonstrates the potental of the technque to use GPS data for applcatons requrng onospherc mappng and for geophyscal research. In the work presented here only the dfferental phase technque s utlsed to provde relatve TEC from ground-based recevers. A future development of the work wll nvolve the ncorporaton of many dfferent data sources nto the mage. It s already feasble to extend the nput matrx to contan any approxmately lnear onospherc measurement such as nformaton derved from onograms. Hajj et al. (1994) suggested usng the satellte-to-satellte transmsson of GPS to LEO satellte measurements n a tomographc framework to provde the so-called «mssng horzontal rays» and mprove the vertcal resoluton. In addton, LEOs provde measurements over the oceans and nto remote polar caps, thus enablng the onosphere to be studed contnuously n both space and tme. Thus ths technque to create global mage of the onosphere wll soon be used to solve practcal rado propagaton problems and to study the Earth s onosphere on a global scale. Acknowledgements The authors would lke to thank the UK EPSRC for supportng ths research and the Internatonal GPS Servce for nformaton used n the smulatons. They also are very grateful for the use of the Internatonal Reference Ionosphere. REFERENCES AUSTEN, J.R., S.J. FRANKE and C.H. LIU (1988): Ionospherc magng usng computersed tomography, Rado Sc., 23, BERNHARDT, P.A., R.P. MCCOY, K.F. DYMOND, J.M. PICONE, R.R. MEIER, F. KAMALABADI, D.M. COTTON, S. CHARKRABARTI, T.A. COOK, J.S. VICKERS, A.W. STEPHAN, L. KERSLEY, S.E. PRYSE, I.K. WALKER, C.N. MITCHELL, P.R. STRAUS, H. NA, C. BISWAS, G.S. BUST, G.R. KRONSCHNABL and T.D. RAYMUND (1998): Two dmensonal mappng of the plasma densty n the upper atmosphere wth Computerzed Ionospherc Tomography (CIT), Phys. Plasmas, 5, BILITZA, D. (Edtor) (1990): Internatonal Reference Ionosphere, Natonal Space Scence Data Centre/ World Data Centre for Rockets and Satelltes, Lanham, MD, Report n. NSSDC/WDC-A-R&S, BUST, G.S., D.S. COCO and T.L. GAUSSIRAN (2001): Computerzed onospherc tomography analyss of the combned onospherc campagn, Rado Sc., 36 (6), CHAPMAN, M.A. (1931): The absorpton and dssocaton onsaton effect of monochromatc radaton n a rotatng Earth, Proc. Phys. Soc., 43 (1), 26. CIRAOLO, L. and P. SPALLA (1997): Comparson of onospherc total electron content from the NNSS and GPS, Rado Sc., 32 (3), DAVIES, K. (1990): Ionospherc Rado (Peter Peregrnus Ltd., London, U.K.), pp FREMOUW, E.J., J.A. SECAN and B.M. HOWE (1992): Applcaton of stochastc nverse theory to onospherc tomography, Rado Sc., 17, GOLUB, G.H. and C.F. VAN LOAN (1989): Matrx Computatons (Johns Hopkns Unversty Press, Baltmore), pp

10 Cathryn N. Mtchell and Paul S.J. Spencer HAJJ, G.A., R. IBAÑEZ-MEIER, E.R. KURSINSKI and L.J. ROMANS (1994): Imagng the onosphere wth the Global Postonng System, Int. J. Imag. Syst. Technol., 5, WATERMANN, J., G.S. BUST, J.P. THAYER, T. NEUBERT and C. COKER (2002): Mappng plasma structures n the hgh-lattude onosphere usng beacon satellte, ncoherent scatter radar and ground-based magnetometer observatons, Ann. Geophyscs, 45 (1), YEH, K.C. and T.D. RAYMUND (1991): Lmtatons of onospherc magng by tomography, Rado Sc., 26, (receved February 4, 2003; accepted July 28, 2003) 696