A technical description of atmospheric sounding by GPS occultation

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1 Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) A technicl description of tmospheric sounding by GPS occulttion G.A. Hjj, E.R. Kursinsi, L.J. Romns, W.I. Bertiger, S.S. Leroy Jet Propulsion Lbortory, Cliforni Institute of Technology, Mil Stop , 4800 O Grove Dr., Psden, CA 91109, USA Received 10 April 2001; ccepted 12 November 2001 Abstrct In recent yers, the globl positioning system (GPS) hs been exploited vi rdio occulttion techniques to obtin proles of refrctivity, temperture, pressure nd wter vpor in the neutrl tmosphere nd electron density in the ionosphere. The GPS=MET experiment, which plced GPS receiver in low-erth orbit, provided welth of dt which ws used to test this concept nd the ccurcy of the retrievls. Severl investigtions hve lredy demonstrted tht the retrievl ccurcies obtined with GPS=MET is lredy comprble, if not better, thn the more trditionl tmospheric sensing techniques (e.g., rdiosondes). Even though the concept of tmospheric proling vi rdio occulttion is quite simple one, cre must be ten to seprte the numerous fctors tht cn ect the occulted signl. These include the motion of the stellites, cloc drifts, reltivistic eects, the seprtion of the ionosphere nd the neutrl tmosphere, nd the contribution of the upper tmosphere where sensitivity of the GPS signl is we. In ddition, cre must be ten to use proper boundry conditions, use proper smoothing intervls nd interpoltion schemes to void retrieving rticil tmospheric structures, nd most importntly detect nd correct phse mesurement errors introduced by shrp refrctivity grdients in the tmosphere. This wor describes in some detil the severl steps involved in processing such dt. In prticulr, it describes system tht ws developed t the Jet Propulsion Lbortory nd used to process the GPS=MET dt. Severl exmples of retrieved refrctivity, temperture nd wter vpor proles re shown nd compred to nlyses from the Europen Center for Medium-rnge Wether Forecst (ECMWF). Sttisticl comprisons of GPS=MET nd ECMWF tempertures for dt collected during June 21 July 4, 1995, indicte tht dierences re of order 1 2 K t northern ltitudes where the ECMWF nlyses re most ccurte. c 2002 Elsevier Science Ltd. All rights reserved. Keywords: GPS rdio occulttion; Remote sensing; Refrctivity; Temperture; Wter vpor; Pressure; Electron density 1. Introduction The rdio occulttion technique hs three decdes of history s prt of NASA s plnetry explortion missions (e.g., Fjeldbo nd Eshlemn, 1969; Fjeldbo et l., 1971; Tyler, 1987; Lindl et l., 1990; Lindl, 1992). Applying the technique to the Erth s tmosphere using the globl positioning system (GPS) signl ws conceived decde go (e.g., Yunc et l., 1988; Gurvich nd Krsil niov, 1990) Corresponding uthor. Tel.: ; fx: E-mil ddress: george.hjj@jpl.ns.gov (G.A. Hjj). nd demonstrted for the rst time with the GPS=MET experiment in 1995 (Wre et l., 1996). Since then severl missions hve own with GPS occulttion receivers including Oersted (see, e.g., Escudero et l., 2001), SUNSAT (Mostert nd Koeemoer, 1997), CHAMP (see, e.g., Wicert et l., 2001), nd the Argentinin SAC-C (lunched in 2000). The promises of the technique hve generted much interest from severl disciplines including meteorology, climtology nd ionospheric physics. The technique relies on very ccurte mesurements of the GPS dul-frequency phse delys collected from receiver in low-erth orbit (LEO) trcing GPS stellite setting or rising behind the Erth s tmosphere. The extr phse dely /02/$ - see front mtter c 2002 Elsevier Science Ltd. All rights reserved. PII: S (01)

2 452 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) induced by the tmosphere cn be converted to tmospheric bending which cn then be interpreted in terms of refrction due to tmospheric refrctivity chnges t dierent heights. Assuming sphericl symmetry in the loclity of the occulting tmosphere, the index of refrction cn therefore be determined from the height of the LEO down to the Erth s surfce. Index of refrction cn then be converted into electron density bove 60 m, neutrl tmospheric density, pressure nd temperture between 60 m nd the middle troposphere, nd, with independent nowledge of temperture, into wter vpor density in the middle nd lower troposphere. Numerous rticles nd reports hve been written describing the technique, its resolution nd ccurcy, nd its relevnce to climte, wether nd ionospheric reserch. On the theoreticl front, severl ppers hve ddressed the expected resolution nd ccurcy of the technique (e.g., Gorbunov nd Soolovsiy, 1993; Hjj et l., 1994; Melbourne et l., 1994; Kursinsi et l., 1995; Hoeg et l., 1995; Gorbunov, 1996; Gorbunov et l., 1996,b; Kursinsi et l., 1997; Kryel nd Hinson, 1997; Mortensen nd Hoeg, 1998; Ahmd nd Tyler, 1998; Ahmd nd Tyler, 1999; Kursinsi et l., 2000; Hely, 2001). On the observtionl front, GPS=MET dt hve been used to derive temperture, wter vpor, geopotentil heights of constnt pressure levels nd ionospheric electron density proles. GPS=MET-derived temperture proles gree with those from rdiosondes nd nlyses from the Europen Center for Medium-rnge Wether Forecst (ECMWF) to better thn 1:5 K between 5 nd 30 m ltitudes (Hjj et l., 1995; Kursinsi et l., 1996; Wre et l., 1996; Rocen et l., 1997; Steiner et l., 1999; Gorbunov nd Kornblueh, 2001). Similr results hve lso been shown from CHAMP (Wicert et l., 2001). GPS=MET-derived geopotentil heights of constnt pressure levels gree with those of the ECMWF to 20 gpm (Leroy, 1997). GPS=MET-derived specic humidities gree with those of the ECMWF to 0:1 g=g in the men (Kursinsi nd Hjj, 2001). In the ionosphere, GPS=MET dt were nlyzed to derive electron density proles in the E nd F regions (Hjj nd Romns, 1998; Schreiner et l., 1999) nd pe electron densities were shown to gree with those of digisondes t the 20% level (1 ). In ddition, tomogrphic inversions of GPS=MET nd Oersted dt were performed to obtin 2-dimensionl (2-D) nd 3-D imges of electron density (Leitinger et l., 1997; Rius et l., 1998; Hjj et l., 2000; Escudero et l., 2001). Severl vritions nd renements on the technique of processing GPS rdio occulttions hve been considered in recent yers. Those include: (1) inversion of rdio occulttions using mplitude dt (Soolovsiy, 2000), (2) improved upper strtospheric retrievls (Hely, 2001,b), (3) improved lower tropospheric trcing nd retrievls (Soolovsiy, 2001,b), (4) use of rdiohologrphic methods for better hndling of tmospheric multipth (Hoce et l., 1999; Gorbunov et l., 2000), use of vritionl nd non-liner optimiztion pproches for seprtion of hydrosttic nd moist terms in refrctivity (Hely nd Eyre, 2000; Plmer et l., 2000). Other studies hve considered the use of GPS occulttion products in pplictions such s climte chnge detection (Yun et l., 1993; Leroy nd North, 2000), numericl wether predictions (Eyre, 1994; Zou et l., 1995; Kuo et l., 1998; Zou et l., 1999; Anthes et l., 2000; Kuo et l., 2000; Zou et l., 2000), nd grvity wves morphology (Tsud et l., 2000; Steiner nd Kirchengst, 2000; Hoce nd Tsud, 2001). The purpose of this pper is to describe in detil system developed t the Jet Propulsion Lbortory (JPL) for processing GPS rdio occulttion dt to obtin proles of refrctivity, pressure, temperture in the lower neutrl tmosphere (below 50 m ltitude), wter vpor in the middle nd lower troposphere, nd electron density in the ionosphere. While the technique is conceptully rther simple, there re severl issues to be crefully considered in optimlly nlyzing the dt. These issues include the proper clibrtion of phse delys mesured from GPS in order to isolte the tmospheric dely, the detection nd correction of mesurement errors such s dt outges nd cycle slips, the proper smoothing of the dt nd the ssocited mesurement resolution, nd the evlution of errors ssocited with the estimted tmospheric dely, Doppler shift nd bending. The processing steps re illustrted with specic occulttion. Other exmples of retrieved refrctivity, temperture nd wter vpor re discussed. Sttisticl comprisons of GPS=MET nd ECMWF tempertures for dt collected during June 21 July 5, 1995, re lso shown. The pper is structured s follows. In Section 2 we describe the bsics of the GPS signl nd how it is modeled. Section 3 describes the process of extrcting the tmospheric dely during n occulttion nd mens of detecting nd correcting mesurement errors. Section 4 describes the inversion process which includes deriving the tmospheric induced Doppler shift nd bending, removing the ionospheric eects, the Abel inversion, nd then the derivtion of the geophysicl prmeters from refrctivity. In Section 5 we present other exmples of GPS=MET retrievls nd comprisons to the ECMWF nlyses. A conclusion is given in Section GPS signl structure nd observbles The GPS constelltion currently consists of 29 stellites t 26; 500 m rdius, 12 h period, orbiting in six dierent plnes inclined t 55. Ech GPS stellite brodcsts two signls t L-bnd (f 1 = 1575:42 MHz nd f 2 = 1227:60 MHz). The L1 nd L2 signls received from ech GPS stellite cn be written s (Spiler, 1980) S L1(t)= 2C C=A D(t)X (t) sin(2f 1t + 1) + 2C P1D(t)P(t) cos (2f 1t + 1); (1) S L2(t)= 2C P2D(t)P(t) cos(2f 2t + 2); (2)

3 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) with C C=A, C P1 the received power of the in-phse nd qudrture components of the L1 signl, respectively; C P2 the received power of L2; D(t) n mplitude modultion for L1 nd L2 contining nvigtion dt; X (t) pseudorndom sequence of ±1 nown s cler cquisition or C=A code modulting the in-phse component of L1 t rte of 1:023 Mhz; P(t) pseudorndom sequence of ±1 nown s P-code modulting the qudrture component of L1 nd L2 t rte of 10:23 Mhz. A properly equipped receiver will detect mplitude, pseudornge 1 nd phse mesurements for ech of the C=A, L1 P-code (P1) nd L2 P-code (P2) signls. The C=A nd P1 mesurements essentilly contin identicl informtion, however C=A is preferred over P1 becuse its power is stronger by 3 db nd is not encrypted. Therefore, the bsic observbles used during n occulttion experiment re the C=A phse nd the P2 phse mesurements between the low-erth orbit (LEO) stellite nd the occulting GPS stellite. These phse mesurements cn be modeled (in dimension of distnce) s L ij c f ij = ij + ij + Ci + C j + ; f 2 (3) ij = ij + d TECij ; (3b) with ij the recorded phse in cycles for the signl propgted from trnsmitter i to receiver j; c the speed of light in vcuum; = 1 or 2 for L1 nd L2, respectively; ij the rnge corresponding to the trvel light time (in vcuum) between the trnsmitter nd the receiver; ij the extr dely due to neutrl tmosphere nd ionosphere; C i ;C j time dependent trnsmitter nd receiver cloc errors, respectively; mesurement noise which contins the receiver s therml noise nd locl multipth. ij the extr dely due to the neutrl tmosphere; d constnt; the integrted electron density long the rypth; TEC ij 1 Pseudornge is n bsolute mesurement of group dely between the time signl is trnsmitted nd received. It is the sum of the ctul rnge between the trnsmitter nd the receiver, tmospheric nd ionospheric delys nd trnsmitter nd receiver clocs osets. Fig. 1. GPS occulttion geometry dening the tngent point, the symptote miss distnce,, nd depicting how the L1 nd L2 signls trvel slightly dierent pths due to the dispersive ionosphere. Also shown re the other non-occulting GPS trnsmitter nd ground receiver used for clibrtion. Eq. (3) ignores the following terms: (1) A bis corresponding to lrge integer number of cycles which is constnt over connected rc (i.e., GPS stellite trced continuously during the occulttion). It is the time derivtive of the phse tht is of interest to us during n occulttion, therefore ll dditive constnts cn be ignored. (2) The wind-up term tht ccounts for the reltive orienttion of the trnsmitting nd the receiving ntenns. Becuse the geometry nd the reltive orienttion of the trnsmitting nd receiving ntenns re well nown, this term is modeled nd removed (Wu et l., 1993). (3) Trnsmitting nd receiving ntenns phse center vritions (which cn be clibrted if necessry). Eq. (3b) ignores higher order ionospheric terms (order 1=f 3 or higher) which results from the expnsion of the Appleton Hrtree formul (see, e.g., Bssiri nd Hjj, 1993). This term is normlly smll, but it becomes dominnt error term t high ltitudes ( m) during solr-mximum dy-time conditions (Kursinsi et l., 1997). Subscripts on ny term in Eqs. (3 nd b) implies tht it depends on the frequency. The neutrl tmosphere is non-dispersive t rdio frequencies; however, since the electromgnetic signl hs to trvel through the dispersive ionosphere before nd fter it reches the lower neutrl tmosphere, the L1 nd L2 signls received t given time sense slightly dierent prts of the neutrl tmosphere (s depicted by the solid, L1, nd dotted, L2, occulted signls of Fig. 1). This seprtion of L1 nd L2 signls is the reson to hve subscripts on the terms ij nd TECij in Eq. (3b).

4 454 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) Isolting tmospheric dely 3.1. The clibrtion process nd TEC ij Isolting the extr delys induced by the Erth medi, ij in Eq. (3b), is the rst necessry step towrd reconstructing proles of refrctivity. This is ccomplished by computing or modeling the cloc terms on the right side of Eq. (3), procedure tht we refer to here s the clibrtion process. Depending on the stbility of the trnsmitter s nd receiver s clocs, we my or my not need to solve for the cloc terms in Eq. (3). (For the eects of cloc instbility on tmospheric retrievls, see Kursinsi et l., 1997.) For the se of generlity, we will here ssume tht both the trnsmitter s nd the receiver s clocs re suciently unstble nd require clibrtion. 2 In order to be ble to solve for both the trnsmitter s nd receiver s clocs, the following geometry is required (see Fig. 1): An occulting receiver (LEO(4)) must view simultneously n occulting trnsmitter (GPS(2)) nd non-occulting trnsmitter (GPS(3)). A second non-occulting receiver (GS(1)) must simultneously view both the GPS(2) nd GPS(3). In order to understnd how the clibrtion of the vrious clocs is performed we describe in some detil the modeling of the time dely. We distinguish between three dierent types of time: () cloc time, t, which is the time recorded by the trnsmitter s or receiver s cloc (we cll time tg) nd contins time vrying oset; (b) proper time, t, which is the time recorded by perfect cloc in frme moving with the trnsmitter or receiver; nd (c) coordinte time, t, which is the time recorded by perfect cloc in given coordinte system. A GPS receiver mesures the rnge between the trnsmitter nd the receiver by essentilly dierencing the trnsmitter s time tg ssocited with given sequence of code from the receiver s time tg t the time tht sequence is received. Up to constnt bis, phse mesurements cn be thought of in the sme mnner. Therefore, we cn write L 21 = c( t 1 t 2 )=c{( t 1 t 1 )+(t 1 t 1 ) +(t 1 t 2 )+(t 2 t 2 )+(t 2 t 2 )}; (4) where 1 nd 2 corresponds to the id s of the receiver nd the trnsmitter, respectively, s shown in Fig. 1. The dierence between the received nd trnsmit time is modeled s the sum of ve terms on the right-hnd side of Eq. (4). In their respective order, these terms correspond to the following: (1) receiver s cloc error; (2) proper time coordinte time t receiver (due to specil nd generl reltivistic eects); 2 This is usully true of the LEO cloc nd sometime of the GPS cloc. (3) light trvel time; (4) coordinte time proper time t trnsmitter; (5) trnsmitter s cloc error. Similrly for the other lins of Fig. 1 we cn write L 31 = c( t 1 t 3 )=c{( t 1 t 1 )+(t 1 t 1 ) +(t 1 t 3 )+(t 3 t 3 )+(t 3 t 3 )}; L 24 = c( t 4 t 2 )=c{( t 4 t 4 )+(t 4 t 4 ) +(t 4 t 2 )+(t 2 t 2 )+(t 2 t 2 )}; L 34 = c( t 4 t 3 )=c{( t 4 t 4 )+(t 4 t 4 ) +(t 4 t 3 )+(t 3 t 3 )+(t 3 t 3 )}: (4b) (4c) (4d) Under norml opertion, for given receiver time tg, the receiver will record the time dely from ll trced stellites. Therefore, in writing Eqs. (4), L 21 nd L 31 hve the sme received time, nd similr for L 24 nd L 34 ; but L 21 nd L 24 hve dierent trnsmit time ( t 2 nd t 2, respectively), nd similrly for L 31 nd L 34, in order to ccount for the dierence in the trvel light time. In n occulttion geometry, the only term tht is of interest to us is the light trvel time ssocited with the L 24 lin, which includes the dely induced on the lin by the tmosphere. In order to obtin t 4 t 2 we either solve for or compute ll the other terms s follows: (1) Bsed on nowledge of the positions of both trnsmitters nd receivers (which re obtined from solutions of the orbit using ground networ of GPS sttions nd ll other GPS stellites), we solve for the light time ssocited with lins L 21, L 31 nd L 34 nd for the time dierences between coordinte nd proper times for ll lins. This is done by ccounting for specil nd generl reltivistic eects in the mnner described by (Wu et l., 1990) nd (Sovers nd Border, 1990). Eqs. (5) (7) summrize their results which is pplied to lin L 21 s n exmple: The dierence between proper time nd coordinte time for receiver xed on the ground is given by t 1 t 1 = [(TAI UTC)+(TDT TAI)]; (5) where TAI-UTC is n integer number of lep seconds which chnges pproximtely once yer nd TDT -TAI is dened to be 32:184 s. The trvel tight time is given by t 1 t 2 = r12 c +2GM Erth r1 + r2 + r12 ln ; (6) c 3 r 1 + r 2 + r 12 where r 1; r 2; r 12 re the position of the phse center of the receiver, trnsmitter nd their dierence, respectively, in n erth-centered inertil frme; t 2 in Eq. (6) is solved for itertively given nowledge of the trnsmitter s orbit. For trnsmitter or receiver in spce, the proper nd coordinte

5 times re relted by dt 2 dt G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) = GM Erth ṙ2 2 + const; (7) c 2 r 2 2c2 where the constnt is chosen such tht proper nd coordinte times gree on the surfce of the erth t the equtor. Eq. (7) is integrted by stndrd techniques to give the dierence between proper nd coordinte times (Wu et l., 1990). (2) Given the estimtes of (t 1 t 1 ); (t 1 t 2 ); (t 2 t 2 ) from the previous step, we solve for the drift of cloc (2), (t 2 t 2 ), in Eq. (4) reltive to cloc (1). Similrly, we solve for t 3 t 3 in Eq. (4b). (3) Assuming tht clocs (2) nd (3) re smooth between smples, we interpolte the corrections obtined in the previous step over the dierentil light time, therefore we cn solve for t 2 t 2 nd t 3 t 3. (4) We solve for t 4 t 4 in Eq. (4d). (5) We solve for t 4 t 2 in Eq. (4c) which is the desired term nd corresponds to the rst two terms on the right-hnd side of Eq. (3). Note tht in steps (2) (5) ll cloc solutions re reltive to cloc (1), which mes the nl solution of step (5) independent of tht cloc. The only requirement is tht clocs be smooth enough between smples for proper interpoltion. For the GPS=MET experiment, the GPS clocs were solved for every 1 s bsed on 1 s ground mesurements nd interpolted with cubic spline to the 50 Hz rte of the receiver. This 1 Hz rte ws necessry to solve for the GPS cloc vritions which were mostly due to selective vilbility (SA). (SA is the dithering of the GPS cloc by the Deprtment of Defense in order to reduce the ccurcy for non-uthorized users.) After the termintion of SA in My of 2000, GPS cloc solutions obtined every 30 s or even 5 min cn be suciently smooth for occulttion processing s demonstrted on CHAMP occulttions (Wicert et l., 2001b). The clibrtion steps described bove re performed with GIPSY=OASIS, the softwre developed t JPL for precise positioning nd orbit determintion pplictions using GPS. An exmple of the tmospheric dely (i.e., ij in Eq. (3b)) is illustrted in Fig. 2 for n occulttion obtined from the GPS=MET experiment. The occulttion tngent point re locted ner 16 S nd 171 E geodetic ltitude nd longitude, respectively. The leveling of the dely t the beginning of the occulttion is n indiction of where the ionospheric dely is dominnt. The tmospheric dely of 2:4 m ner the surfce is lrger thn verge; s the wter vpor retrievl will lter show, this cn be ttributed to lrge moisture concentrtion. Another, more strightforwrd but less ccurte, technique (now s double dierencing ) is to form the liner combintion: L 24 L 34 (L 21 L 31 ) (8) Fig. 2. Extr phse dely induced by the ionosphere nd the neutrl tmosphere on n occulted signl t 50 Hz rte. The occulttion event is between GPS=MET nd GPS stellite No. 31. Time 0 corresponds to : UT. The ttening of the curves t the beginning of the occulttion is due to the dominnt ionospheric dely t these heights. which cuses number of terms in Eqs. (4) to cncel out. However, only if we ignore cloc drifts over the dierentil light time would ll the cloc error terms cncel out completely. For receiver in LEO, the dierentil light time rnges between :03 s. Prior to the demise of SA, typicl GPS cloc drift over short time scle (seconds minutes) ws of order 6 cm=s (Wu et l., 1990b). Therefore, ignoring the dierentil light time would introduce n error of mm of phse mesurement. The time vrition of this error, which is the relevnt number for occulttion mesurements, depends on the ctul spectrum of SA. Fig. 3 shows n exmple of this error s function of time during period when AS ws turned on. Even though this error is smll (0.1 0:3 mm=s in generl), it is not insignicnt for retrievls t high ltitudes ( 45 m). On the other hnd, fter SA ws terminted, error introduced by the double dierencing scheme becomes insignicnt t ll relevnt ltitudes Detecting nd xing bres In the lower troposphere severl eects cuse signl trcing to be dicult. These eects include the following: (1) the ttenution of the signl due to defocusing (s discussed lter in Section 4 nd shown in Fig. 7). (2) The signicnt ccelertion of the tmospheric phse dely. (3) The scintilltion of the signl s mplitude nd phse due to dirction nd multipth propgtion cused by shrp verticl refrctivity grdients. While the rst two eects cn be esily compensted for by hving dequte receiver s ntenn gin nd proper trcing strtegies, the third eect cn be

6 456 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) Fig. 3. The error introduced by stright double dierencing over the spn of n occulttion. This is estimted by ssuming dierentil trvel light time of 0:01 s nd bsed on the behvior of the GPS clocs with SA on. This error is eliminted by properly solving for the clocs ccounting for the trvel light time. chllenging nd requires creful exmintion of the signl s power spectrum to identify the dierent tones corresponding to dierent modes of propgtion. The exmintion of the signl s spectrum will te us outside the scope of this pper (however, see Hoce et l., 1999 nd Gorbunov et l., 2000). Here, it suces to sy tht the GPS receiver used for the GPS=MET experiment ws not optimized to hndle the complicted dynmics of the signl t the lower troposphere nd it only trced, when possible, the totl sum of the multiple signls received t the LEO. During some occulttions the eects mentioned bove cused the receiver (which trced t 50 Hz) to () slip n integer number of hlf-cycles t certin time updtes, s shown in Fig. 4, (b) slip hlf-cycle t ech time updte for certin number of consecutive time updtes cusing Doppler shift bis, s shown in Fig. 4b, or (c) loose loc completely. Figs. 4 nd b show exmples of these eects. In order to detect hlf-cycle slips, which re bout 10 cm, in delys tht re of order 1 m, we exmine the unsmoothed Doppler shift obtined by dierentiting consecutive phse mesurements divided by the time between mesurements, t (e.g., 0:02 s for GPS=MET). In this procedure hlf-cycle corresponds to 1=(2t) Hz (or 25 Hz for GPS=MET). Fig. 4 shows n exmple of the tmospheric Doppler shift obtined in this mnner nd detrended by subtrcting second order polynomil t. In tht exmple we observe two hlf-cycle slips t 71 nd 77 s. Such hlf-cycle slips re esily xble. Additionl informtion on the qulity of the dt is obtined from the voltge signl-to-noise rtio for ech mesurement s recorded by the receiver. By denition, the phse therml noise of mesurement with voltge signl-to-noise rtio of SNR v is given by =(2SNR v), Fig. 4. (Top-) The tmospheric Doppler shift on the occulted signl corresponding to the occulttion of Fig. 2 fter subtrcting second order polynomil t. The Figure shows two outlirs t 71 nd 77 s which correspond to hlf-cycle slips in the receiver. The error br re estimted from the signl-to-noise rtio informtion recorded in the receiver. The solid line is third order polynomil t to the dt. (Bottom-b) The tmospheric Doppler shift on the occulted signl corresponding to n occulttion between GPS=MET-GPS23 strting t UT. Close exmintion of the Doppler shift shows: (1) one hlf-cycle slip t 60 s, (2) hlf-cycle slip t ech of the 50 Hz point fter 68 s resulting in the Doppler shift bis of +25 Hz, (3) chnge in the slope of the Doppler shift t bout 73 s, where the voltge signl-to-noise rtio indictes tht the receiver is out-of-loc beyond tht point. where is the operting wvelength. Assuming therml noise to be independent between mesurements, the noise on the Doppler shift computed in the mnner described bove is the root-squre-sum of the noise of the two phse mesurements used for ech point. The error brs indicted in Fig. 4 re obtined in this mnner nd they help identify

7 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) Fig. 6. Residul tmospheric Doppler shift (i.e., mesured predicted) s function of the occulttion tngent height for 85 GPS=MET occulttions collected on July 1, The tngent height is determined bsed on stright line connecting the trnsmitter nd the receiver. Fig. 5. (Bottom) L1 Bending s function of ( 0 ) for 99 GPS=MET occulttions collected on July 2, is the seprtion ngle between the trnsmitter nd the receiver in coordinte tht represents the locl curvture of the Erth ner the occulttion tngent point (see Fig. 8). 0 is when the tngent point height is t 30 m ltitude. (Top) Residul bending fter subtrcting 14th order polynomil t to L1 vs. ( 0 ) in the bottom gure. when the devition from the men Doppler is due to eects other thn therml noise. Other possible type of bises in the Doppler shift cn been seen in Fig. 4b, where, strting t 68 s pst the beginning of the occulttion, the Doppler shift is bised by +25 Hz reltive to the time before 68 s. This is due to hlf-cycle slip t ech mesurement fter 68 s. In the exmple of Fig. 4b, the low voltge signl-to-noise rtio fter 73 s indictes tht the receiver is no longer in loc. A systemtic method of detecting cycle slips nd bises is to subtrct predicted vlue of the tmospheric Doppler shift from the mesured one nd exmine the residuls. Such prediction cn be obtined by rytrcing through n tmospheric model nd by nowledge of the geometry of the stellites. However, here we describe simpler nd more direct method of predicting the tmospheric Doppler shift in mnner which is independent of n tmospheric model. By plotting the L1 bending ngle ( L1) estimted from the L1 phse mesurements s function of the seprtion ngle between the trnsmitter nd the receiver ( in Fig. 8) for lrge number of occulttions, we nd tht they fol- low very closely the sme functionl form. (Both L1 nd re computed in coordinte frme tht represents the locl curvture of the Erth ner the occulttion tngent point s explined in more detil in Section 4.2.) The bottom pnel of Fig. 5 shows L1 vs. ( 0) for 99 GPS=MET occulttions collected on July 2, 1995, where 0 is the seprtion ngle between the two stellites when the tngent point is t 30 m. The top pnel of Fig. 5 shows the residul bending fter subtrcting functionl form, F( 0), which is polynomil t of the 14th order to L1 vs. ( 0)inthe bottom pnel. It should be cler from Fig. 5 tht, except for severl outlirs which re due to improper trcing in the receiver, for ny given ( 0); L1 cn be predicted to within ±0:03. Once F( 0) is determined, then, by process which is the inverse of deriving the bending from the Doppler mesurements (described in Section 4.2), the tmospheric Doppler shift cn be predicted with n uncertinty of 10 Hz s seen in Fig. 6. In Fig. 6, the residul tmospheric Doppler (i.e., mesured predicted bsed on F( 0)) for 85 GPS=MET occulttions collected on July 1, 1995 re shown s function of the tngent point height which is computed bsed on stright line connecting the trnsmitter nd the receiver. Once the mesured Doppler shift wonders suciently fr ( 10 Hz) from the prediction, dt pst tht point for tht occulttion cn be eliminted. A similr procedure cn be done for L2 dt. This provides systemtic nd robust mnner by which bd episodes of n occulttion cn be detected nd discrded.

8 458 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) Inversion process 4.1. Deriving tmospheric Doppler The result of the clibrtion process is to extrct the sum of the neutrl tmospheric nd ionospheric delys for both the L1 nd L2 occulted lins. The tmospheric dely is normlly obtined t high rte (50 Hz for the GPS=MET experiment) for the purpose of extrcting multipth signls in the lower troposphere, or for smpling the dirction pttern induced by the tmosphere for high resolution retrievls optics (e.g., Kryel nd Hinson, 1997). Our pproch is to smooth the high rte dt over period corresponding to the time it tes ech signl to cross Fresnel dimeter. The Fresnel dimeter of the occulted signl is determined bsed on L1 nd L2 mplitude mesurements vi the following set of reltions: DtD r F 0 = ; (9) D t + D r [ M = 1 d ] 1 D td r ; (9b) dh D t + D r M = I ( ) 2 SNRv = ; (9c) I 0 SRNv 0 F = F 0 M; V = V 0M; T = 2F V = with 2F0 V 0 M ; (9d) (9e) (9f) F 0;F the rst Fresnel zone rdius in free spce nd in Erth tmosphere, respectively; rdio lin wvelength; D t;d r distnce from tngent point to trnsmitter nd receiver, respectively; M tmospheric scle fctor; bending due to Erth tmosphere; d=dh the derivtive of with respect to the tngent point height; I 0;I the signl intensity in free spce nd in Erth tmosphere; SNR 0 v; SNR v the instrumentl voltge signl-to-noise rtio recorded by the receiver in free spce nd in Erth tmosphere respectively; V 0;V verticl velocity of the rypth tngent point in free spce nd in Erth tmosphere, respectively; T time to cross Fresnel dimeter in the presence of Erth medi. Knowledge of the stellites ephemerides llows us to determine F 0 (by use of Eq. (9)) nd V 0. On the other hnd, during n occulttion, SNR v is recorded s function of time, while SNR 0 v correspond to the SNR v t the beginning of n occulttion. Bsed on smoothed version of SNR v,we compute M ccording to Eq. (9c) from which we cn compute F; V nd T vi Eqs. (9d f ). The high-rte L1 nd L2 phse points re then smoothed over time intervl T with second-order polynomil t, nd the Doppler shift is derived by ting the time derivtive of the t t the middle of tht intervl. This smoothing gurntees tht dirction ptterns cused by sub-fresnel tmospheric structure do not pper in the retrievl. However, this procedure does not eliminte mbiguities or errors in determining the Doppler when multiple tones re generted in the tmosphere with tngent point distnces lrger thn the Fresnel zone. This tmospheric multipth sitution is triggered by shrp chnges in refrctivity such s round the mrine boundry lyer cusing the occulted signl to trvel two or more seprte pths connecting the trnsmitter nd the receiver. Identifying the dierent tones in multipth sitution is reserch topic nd is not prt of our routine processing t this point. In the cse of multipth, the smoothing procedure described bove will eectively nd the verge Doppler shift of the dierent tones. For the occulttion corresponding to Fig. 2, Fig. 7 shows the rw nd smoothed L1 SNR v, the corresponding intensity, Fresnel dimeter (which roughly corresponds to the verticl resolution of the mesurement), the velocity of the tngent point, nd the time it tes the signl to cross Fresnel dimeter. The high concentrtion of wter vpor cuses the Fresnel dimeter to decrese ( 100 m), the ry tngent point to slow down (the minimum tngent point velocity in Fig. 7b is 10 m=s) t the lowest prt of the tmosphere before the signl disppers. The corresponding Doppler shift for this occulttion is shown in Fig. 7c. In order to ssocite forml error to the derived Doppler, we consider the following: When verging over N number of phse mesurements, the noise in deriving the slope (which correspond to Doppler shift) for liner or second order t of the phse is given by (see e.g., Bevington, 1969, p. 115) Doppler 2 = 2 ; (10) t 2 i where is the stndrd devition of phse mesurement (in cycles) ssumed to be uncorrelted in time; nd t i is the time of mesurement i (for ech t the reference time is set to the middle of the intervl). Concentrting on rndom phse error cused by therml noise in the receiver, is equl to 1=2SNR cycles. For lrge N, with the time between dt points (e.g., 20 ms for GPS=MET), we cn pproximte the sum in the denomintor of Eq. (10) by 2 N 3 =12 nd the error ssocited with the Doppler estimte is then given by Doppler = ; (11) N 3=2

9 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) We te dvntge of this rpid decrese in the Doppler noise with N in order to minimize the noise in the estimted Doppler t high ltitudes where the tmosphere is tenuous. For instnce, if we desire to rech specic ccurcy Desired, then we need to verge over N points, where N is given by ( ) 2=3 N : (12) Desired In prctice, when deciding on the pproprite number of phse mesurement points to verge, we choose the mximum of N (of Eq. (12)) or T= (where T is given by Eq. (9f)). Bsed on Eq. (11), for n SNR of 300 nd 1 s verging, Doppler =0:0018 Hz. This would be the Doppler error due to therml noise if only single frequency nd single lin is used. In relity, two frequencies nd four lins re used, ech contributing its own noise with the L2 being dominnt due to its lower SNR, ming the therml Doppler noise n order of mgnitude bigger. Furthermore, other, usully non-rndom, sources of error contribute to the Doppler estimte, including orbitl velocity errors ( 0:01 0:1 mm=s), multipth in the surrounding of one of the two trnsmitters or two receivers involved in the occulttion, nd residul ionosphere Deriving tmospheric bending The tmospheric Doppler shift cn be used to derive the tmospheric bending,, s function of the symptotic miss distnce, (Fig. 1). The tmospheric Doppler shift, f, cn be relted to the direction of the trnsmitted nd received signls by the expression d dt = f =[vt ˆ t v r ˆ r (v t v r) ˆ] (13) with Fig. 7. (Top-) High rte (thin line) nd smoothed (thic line) L1 voltge signl-to-noise rtio for the sme occulttion event s in Fig. 2. (Middle-b) The Corresponding Fresnel dimeter (which sets roughly the verticl resolution), the verticl velocity of the tngent point nd the corresponding verging time of the signl, derived bsed on Eqs. (9 f ). A running time verge with time steps of 0:33 s is used to derive the Doppler shift nd bending. (Bottom-c) The bsolute vlue of tmospheric Doppler shift of the occulttion. For descending occulttions, the tmospheric dely increses with time cusing the signl to be blue shifted, therefore the tmosphere introduces negtive Doppler shift. The reverse is true of scending occulttions. where = 12. (The generl form of Eq. (11) is vlid for polynomil t to the dt of ny order but the vlue of chnges with the order of the polynomil t; e.g., = 76 for third order polynomil.) v t nd v r the trnsmitter nd receiver s velocity, respectively, ˆ t; ˆ r the unit vectors in the direction of the trnsmitted nd received signl, respectively (Fig. 8), ˆ the unit vector in the direction from the trnsmitter to the receiver. Eq. (13) is derived by dierencing the Doppler shift observed in the presence of the tmosphere nd the Doppler shift tht would be observed for the sme trnsmitter receiver geometry in the bsence of the tmosphere. The rst-order reltivistic contributions to the Doppler re identicl in the two situtions nd cncel out. Note tht this equlity is true in generl regrdless of the shpe of the tmosphere; however, Eq. (13) hs n innite number of solutions since ˆ t nd ˆ r re both unnown, corresponding to four independent prmeters. Therefore, certin ssumptions hve to be mde on the shpe of the tmosphere in order to derive the tmospheric bending from Eq. (13).

10 460 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) the Erth s center of symmetry hs been implemented erly on in the GPS occulttion processing system t JPL (e.g., Kursinsi et l., 1993), full chrcteriztion of the error introduced by ignoring such correction for GPS occulttions ws rst discussed by Syndegrd (1998). With the sphericl symmetry ssumption, Eq. (13) reduces to the eqution in two unnowns d =(vtcos(t t) vr cos(r r)) dt (v t cos t v r cos r) (14) (the ngles re dened in Fig. 8). In ddition, the formul of Bouguer (equivlent to Snell s lw in sphericlly symmetric medium) sttes tht (e.g., Born nd Wolf, 1980, p. 123) Fig. 8. Occulttion geometry dening the ngles used in Eqs. (14 nd b). = r tn t sin( t + t)=r rn r sin( r + r) with (14b) To very good pproximtion, the neutrl tmosphere is sphericlly symmetric. In order to ccount for the ellipticity of the Erth, the center of symmetry is ten to correspond to the circle in the occulttion plne which best ts the geoid ner the tngent point. The eect of oblteness on rdio occulttion observtions ws relized soon fter the rst such observtions of Jupiter s tmosphere were interpreted ignoring oblteness (Kliore et l., 1974, 1975) nd yielded tempertures in the deep troposphere severl hundred Kelvin hotter thn other observtions nd theory (Eshlemn, 1975). Subsequently the verge center of curvture s dened by the plnet s shpe in the vicinity of the tngent point of the occulted ry pths brought the results in line with other observtions nd theory (Kliore et l., 1976). Becuse of the Erth s oblteness, with n equtoril rdius roughly 20 m lrger thn its polr rdius, the coordintes used to invert Doppler to bending is djusted to represent the locl curvture of the Erth ner the occulttion tngent point (Fig. 8). This is done s follows: (1) Since the shpe of the geoid vries by less thn 100 m with respect to n ellipsoid, n ellipsoid cn be used to represent the geoid shpe. (2) We dene n occulttion plne which contins the trnsmitter, the receiver nd the norml to the geoid ner the tngent point. (Since n occulttion my not te plce entirely in one plne, this plne is determined by the lowest lin of the occulttion.) (3) The intersection of the occulttion plne nd the ellipsoid denes n ellipse. The center of symmetry is ten to be the center of circle in the occulttion plne which is tngent to the ellipse t the ry pth tngent point with rdius equl to the ellipse rdius of curvture t the sme tngent point. This center is then xed for the entire occulttion, nd cn be s fr s 40 m from the rel center of the Erth. Once center of symmetry is dened, ll vribles (such s nd ) re dened with respect to tht center. Even though this correction to r t; r r the vectors from the center of curvture to trnsmitter nd receiver, respectively; r t = r t ; nd r r = r r ; n t nd n r the indices of refrction t the trnsmitter nd receiver, respectively. At the heights of the trnsmitter ( 20; 000 m) nd receiver (low-erth orbit), the indices of refrction re ssumed equl to 1. This pproximtion cn be shown to introduce very smll error in the estimted bending in the ionosphere (Hjj nd Romns, 1998), nd it cncels completely fter doing the ionospheric clibrtion to estimte the neutrl tmospheric bending. The ngles t nd r re determined by simultneously solving Eqs. (14) nd (14b) (esily ccomplished using Newton s method nd rst guess of t =0 nd r =0). The totl bending is = t + r, nd the corresponding is obtined from Eq. (14b). Fig. 9 shows n exmple of the bending derived for L1 nd L2. The error ssigned to the bending is obtined vi the simple scling = Doppler ; (15) V 0 where V 0 is the sme s in Eq. (9e), nd is given in units of rdins Ionospheric clibrtion Becuse of the dispersive nture of the ionosphere, the L1 nd L2 lins trvel long slightly dierent pths nd hve slightly dierent bending (s depicted in Fig. 1). The seprtion of the two signls t ionospheric heights ner the tngent point vries between 100 m nd 5 m, depending on the tngent height of the occulted signl, the solr conditions, locl time nd the loction of the occulttion. (For this eect nd other ionospheric eects on GPS occulttions

11 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) Asymptote miss distnce, m L1 L2 Ionospheric free Atmospheric bending, deg. Fig. 9. Estimted totl tmospheric bending for L1 nd L2, nd ionospheric clibrted bending, s function of, the symptote miss distnce, for the occulttion of Fig. 2. At this stge of the processing, the height of the tngent point is not yet nown, but cn be pproximted s - rdius of curvture nd is shown on the right scle. see, e.g., Hjj nd Romns, 1998.) In the most generl sitution, n ionospheric correction is needed in order to estimte the neutrl tmospheric bending. In our pproch we follow procedure rst suggested by Vorob ev nd Krsil niov (1994). Let 1( 1) nd 2( 2) be the bending s function of the symptote miss distnce for the L1 nd L2 signls, respectively. The liner combintion f1 2 neut( o)= (f1 2 f2 2 ) 1(o) f2 2 (f1 2 2(o); (16) f2 2 ) where 1 nd 2 re interpolted to the sme vlue of o, removes the rst-order ionospheric bending (which is proportionl to f 2 ). The two coecients on the right-hnd side of Eq. (16) hve the numericl vlues to 2:5457 ::: nd 1:5457 :::, respectively. The interpoltion scheme we use is piecewise cubic, within ech intervl t i t t i+1 between dt points. The interpolting cubic polynomil f(t) is determined by four conditions xing the t nd its derivtive t the endpoints, in terms of the four dt vlues f i 1;f i;f i+1;f i+2. Speciclly, the conditions re: f(t j)=f j nd f (t j)= (f j+1 f j 1)=(t j+1 t j 1) for j = i; i + 1. This smooth-cubic interpoltion scheme voids introducing shrp vritions between the points when the dt is noisy (in contrst to trditionl cubic splines), nd is used throughout the reminder of the inversion process (including the Abel trnsform nd the hydrosttic integrls). Bsed on Eq. (16), the noise ssocited with the neutrl tmospheric bending is given by neut =(2:54) (1:54) : (17) Approximte TP Height, m L2 phse mesurements re usully noisier thn L1 phse mesurements for severl resons. First, the L2, which hs lower frequency thn L1, is more inuenced by ionospheric scintilltion nd dely. Second, the C=A code is trnsmitted with 3 db more power tht the P1 code, which in turn is 1 3 db stronger thn the P2. The reltive strength of C=A reltive to P1 nd P2 lso depends on whether or not the Deprtment of Defense (DoD) nti-spoong (AS) is ctivted, nd on the type of the receiver nd trcing strtegy. In generl, L1 phse derived bsed on C=A is more ccurte thn L2. However, even under conditions where both L1 nd L2 noise re comprble, Eq. (17) implies tht the noise introduced by the ionospheric-free liner combintion is bout fctor of 3 lrger thn individul signl noise. We overcome this limittion by redening the bending in Eq. (16) s neut( o)= 1( o)+1:54( 1( o) 2( o)); (18) where 1 nd 2 re the L1 nd L2 bending smoothed over longer intervls thn discussed in Section 4.2. The longer smoothing intervl is normlly of order 2 s (100 points of 50 Hz dt), which, reltive to Fresnel dimeter smoothing ( 26 points of 50 Hz dt), should correspond to noise bout fctor of 7 smller, ccording to Eq. (11). The clibrtion should not be continued bove certin height, when the neutrl tmosphere signture on the occulted signl is comprble to residul ionospheric eects or the receiver s therml noise. This height is determined by computing moving 2 bsed on deprtures from n exponentil t to ionospheric-free bending, nd discontinuing the clibrtion fter 2 exceeds specied vlue. This tends to occur t height of order m, depending on the ionospheric conditions. Deeper in the tmosphere, due to defocusing eects nd the weening of the signl, the L2 signl is not used when the SNR v drops below certin limit. In tht cse, n extrpoltion of the ionospheric correction (i.e., the term 1( o) 2( o) in Eq. (18) is used from higher ltitudes to correct for the ionosphere. 3 The ionospheric-free bending for our exmple occulttion is shown in Fig. 9. For ionospheric retrievls, the bending from one frequency is used bove 60 m The Abel inversion In sphericlly symmetric tmosphere, from the formul of Bouguer (Born nd Wolf, 1980, p. 123), the signl s bending cn be relted to the index of refrction vi the integrl ()= 2 1 dln(n) d : (19) 2 2 d 3 Extrpoltion of the ionospheric correction in this mnner ws used to generte erly results in Kursinsi et l. (1996), nd lso independently suggested nd used by Rocen et l. (1997).

12 462 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) This integrl eqution cn then be inverted by using n Abel integrl trnsform given by (see, e.g., Tricomi, 1985, p. 39) ln(n()) = 1 ( ) 2 2 d : (20) The upper limit of the integrl in Eq. (19) requires nowledge of the bending s function of up to the top of the tmosphere. In prctice, however, the estimted bending is resonbly ccurte only up to certin upper height, u, s described in Section 4.3. Therefore, in crrying out the integrl of Eq. (20) we follow the procedure introduced by Soolovsiy nd Hunt (1996) where we use lest-squre estimtor of bsed on mesured bending, m, nd estimted bending (from n priori model), e, weighted by their corresponding uncertinties, m(= 2 1 +(1:54) (1:54) bsed on Eq. (18)) nd e. Therefore, ( m() ()=A() m() + e() ) ; 2 e() 2 ( 1 A()= m() + 1 ) 1 : 2 e() 2 (21) Since the ccurcy of the climtologicl model is not well nown, we choose n dhoc vlue of e=0:05 e bove certin height, u(=50 m in our exmple), nd e below u. Since refrctivity decys exponentilly with height, the dependence of the retrieved prole on the climtology used is very smll, two scle heights below u. After substituting Eq. (21) in Eq. (20), we obtin ln(n()) = 1 u m( ) 2 2 d + 1 m( ) 2 2 d + 1 A( ) u m( 2 ) A( ) e( ) u e( 2 ) 2 2 d : (22) Furthermore, in order to void numericlly integrting over we singulrity t the lower boundry of the rst integrl on the rightside of (22), we rewrite it s u m( ) 2 2 d = [ ( int )ln( int + 2 int 2 ) int ()ln() ln( 2 + ] 2 2 ) d( ) d d + u m( ) int 2 2 d (23) where int is n intermedite vlue between nd u nd is normlly chosen to be slightly lrger thn. The terms in brcets on the right-hnd side of Eq. (23) re the result of integrtion by prts. Eq. (22) yield the index of refrction, n, s function of, t the tngent point (TP). The TP rdius is obtined from r ==n. The rdius in turn is converted into height bove n ellipsoidl t to the men se-level geoid. Fig. 10 shows the retrieved refrctivity s function of height for our occulttion exmple. We cn convert the independent coordinte from height to geopotentil height. First we convert ll of the tngent point loctions into displcements from the center of the Erth. We compute the sum of the grvittionl nd centrifugl potentil energies per unit mss t ech position. The grvittionl potentil energy per unit mss is computed using version of the JGM-3 grvity model (Tpley et l., 1996). This version is 64th degree sphericl hrmonic expnsion which cn reproduce the men se-level geoid with n ccurcy of tens of centimeters. The potentil of the men se-level geoid is subtrcted from the sum of the grvittionl nd centrifugl potentils, nd the dierence is then divided by stndrd vlue of grvittionl ccelertion to give the geopotentil height. The geopotentil height cn then substitute for height s the independent coordinte for ll derived proles (Leroy, 1997). For ionospheric retrievls, the retrievl is performed from 60 m up to the height of the LEO stellite. Above the LEO stellite, the bending is extrpolted with n exponentil t (Hjj nd Romns, 1998) Deriving geophysicl prmeters from refrctivity The refrctivity, N, is relted to geophysicl quntities vi N =(n 1) 10 6 P = 1 T + P w 2 T 40:3 n e f 2 ( ) 1 + O + ww f 3 w + i W i ; (24) with 1 =77:6 K=mbr; 2 =3: K 2 =mbr; P totl pressure; T temperture; P w wter vpor prtil pressure; n e electron density (m 3 ); f operting frequency (Hz); W w nd W i re liquid wter nd ice content, respectively, in grms per cubic meter s nd i re 1.4 nd 0.6 (cubic meter=grms), respectively. For relistic suspensions of wter or ice, the lst two terms of Eq. (24) re smll in comprison with other terms nd will be neglected here (for discussion of the eects of these terms on GPS occulttions see Kursinsi, 1997 nd Solheim et l., 1999). When the signl is pssing through the ionosphere (tngent point height 60 m), the rst two terms

13 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) Height, m (b) Height, m () Height, m (c) Refrctivity Pressure, mbr Pressure or Refrctivity GPS/MET ECMWF Temperture, K GPS/MET ECMWF Specific Humidity, g/g Fig. 10. () GPS=MET derived refrctivity nd pressure. The hydrosttic integrl is strted t 50 m ltitude. (b) GPS=MET nd ECMWF nlysis tempertures (left) nd their dierences (right). The ECMWF highest level is t 23 m. The GPS=MET temperture is retrieved from 50 m down to 8 m where T = 250 K. The greement is better thn 2 K everywhere except ner the tropopuse, where we scribe the discrepncy to the model due to insucient resolution to cpture the double tropopuse detected from GPS=MET. (c) Specic humidity from GPS=MET nd ECMWF nlysis. In deriving the GPS=MET specic humidity, the ECMWF nlysis temperture ws used. on the right-hnd side of Eq. (24) cn be ignored, s well s higher order ionospheric terms in the ionosphere. Therefore, mesurement of n directly corresponds to electron density in the ionosphere. In the neutrl tmosphere (tngent point height 60 m), the ionospheric clibrtion process described in Section 4.3 bove eectively removes the rst order ionospheric term (1=f 2 ) in Eq. (24). (Higher order contributions constitute the mjor source of error during dy-time solr mximum t high ltitudes; see Kursinsi et l., 1997, for n estimte of these errors nd Bssiri nd Hjj (1994), for possible mens of correcting them.) In order to solve for P; T, nd=or P w given N we use the dditionl constrints of hydrosttic equilibrium nd the idel gs lw: dp = g; (25) dh = d + w = m dp TR + (mw m d)p w ; (26) TR with h height; g grvittionl ccelertion; ; d, w totl, dry ir nd wter vpor densities respectively; m d ;m w men moleculr mss of dry ir (28:97 g=mole) nd wter vpor (18:0 g=mole), respectively; R universl gs constnt. Combining Eqs. (25) nd (26), nd using Eq. (24) (ignoring the ionospheric terms) to substitute for P=T, we obtin dp dh = gm d N + 2gm d P w 1R 1R T + g(m d m w) P w 2 R T : (27) Given N, we hve system of two equtions (24 nd 27) nd three unnowns (T; P, nd P w). Since sturtion vpor pressure decreses rpidly with decresing temperture, s dictted by the Clusius Clpeyron eqution, P w cn be ignored bove the tropospheric height corresponding to T = 250 K; therefore, given N, both T nd P cn be solved for in the upper troposphere nd the strtosphere from Eqs. (24) nd (27) nd boundry condition (usully ten to be temperture boundry condition t 50 m). The solution to P nd T s function of height for our occulttion exmple re shown in Figs. 10 nd 10b, respectively. For comprison, T from the ECMWF nerest 6 hourly nlysis nd interpolted to the loction of the occulttion, is lso shown in Fig. 10b. Both the GPS=MET nd ECMWF nlysis temperture gree to better thn 2 K everywhere except ner the tropopuse, where the nlysis misses the double tropopuse detected by the GPS=MET retrievl. When P w is signicnt, such s in the middle nd lower troposphere, it is necessry to hve n independent nowledge of one of the three prmeters (T; P; P w) in order to

14 464 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) solve for the other two. Given tht temperture is generlly better nown nd less vrible thn wter vpor, it is more ecient to ssume nowledge of T, nd then solve for P nd P w. The exct reltion between errors in T; P nd P w is function of ltitude nd height nd is described in detil by Kursinsi et l. (1995). Assuming nowledge of T (h) nd pressure t some height for boundry condition, then Eqs. (24) nd (27) re solved itertively s follow: (1) Assume P w(h) = 0 for rst guess, (2) Integrte Eq. (27) to obtin P(h), (3) Use P(h) nd T (h) in Eq. (24) to updte P w(h), (4) repet steps (2) nd (3) until convergence. Given P nd P w, specic humidity, q (dened s the rtio of wter vpor density to the moist ir density), is given by [ ( ) ] 1 md P q = 1 +1 mw P w m w P w m d P : (28) The solution of specic humidity for our exmple, using T from the ECMWF nlysis, is shown in Fig. 10c. For comprisons, specic humidity from the ECMWF nlysis is lso shown on Fig. 10c. The retrieved specic humidity t the surfce is close to the mximum tht is normlly observed on Erth. This explins why the dely nd bending observtions re quite lrger thn verge ner the surfce. It is notble tht in the occulttion exmple shown the signl ws trced virtully down to the surfce, in spite the very lrge humidity there. The time during which this occulttion ws ten (June 23, 1995 which is during the second prime GPS=MET period), corresponded to time where specil trcing softwre ws operting which hd improved the trcing in the lower troposphere substntilly over the other two prime time periods (for more detil on this see discussion by Kursinsi nd Hjj, 2001). 5. Exmples of GPS=MET retrievls nd comprisons to ECMWF Here, we briey present other exmples of GPS=MET proles obtined with the system described bove, nd compre them to corresponding proles obtined from the nerest 6 h ECMWF nlysis interpolted to the loction of the occulttion. Fig. 11 shows the results from four dierent GPS=MET occulttion (the three plots ligned horizontlly correspond to the sme occulttion). The dte, time, occulting GPS stellite, ltitude nd longitude of the occulttion is indicted t the top of ech plot. For ech occulttion we plot the frctionl dierence between the GPS=MET nd ECMWF refrctivity (plots on left), the GPS=MET (solid line) nd ECMWF (dshed line) tempertures (middle plots), the GPS=MET (solid line) nd ECMWF (dshed line) wter vpor pressures (plots on right). These occulttions re representtive of other occulttions obtined during the period of June 21 July 4, 1995, nd re chosen to represents dierent tmospheric conditions (i.e., dierent ltitudes nd moisture content). GPS=MET nd ECMWF frctionl refrctivity plots indicte tht refrctivity gree to better thn 1% in dry regions. Lrge refrctivity dierences (up to 6%) pper t lower ltitude nd cn be ttributed to dierences in moisture. Theoreticl estimte indictes tht refrctivity derived from GPS occulttions is ccurte to bout 1% ner the surfce nd improves t higher ltitude (up to 40 m) (Kursinsi et l., 1997). Sttisticl comprison of GPS=MET temperture proles to EMCWF nlysis for 579 occulttions collected during the period June 21 July 4, 1995 re shown in Fig. 12. These sttistics re divided bsed on the loction of the occulttions into three regions corresponding to northern (ltitude 30 ), tropicl ( 30 ltitude 30 ) nd southern (ltitude 30 ) regions. Dierences between GPS=MET nd ECMWF is 0:5 K in the men nd 1:5 K in stndrd devition in the northern region. The greement is worse in the southern region. Tht this degrdtion is due to the ECMWF nlyses cn be concluded bsed on (1) the fct tht GPS=MET retrievl is independent of the region in the globe, (2) the ECMWF nlysis is less ccurte in the southern hemisphere thn in the northern hemisphere due to lc of dt. The wvy structure in the tropics round nd bove the tropopuse cn be due to grvity wves detected by the GPS=MET but smoothed out by the nlysis. More temperture retrievls obtined from GPS=MET using the described occulttion retrievl system s well s sttisticl comprisons to tmospheric models re discussed by Hjj et l. (1995) nd Kursinsi et l. (1996). Also retrievls of geopotentil height s function of pressure from GPS=MET nd comprison to ECMWF nlysis re discussed by Leroy (1997). A detiled wter vpor nlysis from GPS=MET is given by (Kursinsi nd Hjj, 2001). 6. Summry=conclusion We described system developed t JPL to process GPS occulttion dt for retrieving refrctivity, temperture, Fig. 11. Retrievls of refrctivity, temperture, nd wter vpor pressure (ligned horizontlly for the sme occulttion) re shown for four dierent GPS=MET occulttions. The dte, time, occulting GPS, ltitude nd longitude of ech occulttion re indicted on the top of ech plot. Retrieved prmeters re compred to vlues derived from the nerest 6 h ECMWF nlysis interpolted to the loction of the occulttion. Frctionl refrctivity dierences between GPS=MET nd the nlyses re shown on the left. GPS=MET tempertures nd wter vpor pressures re indicted in solid lines, those of the nlysis re indicted in dshed lines. GPS=MET tempertures re shown only bove the middle troposphere t height where T 250 K.

15 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002)

16 466 G.A. Hjj et l. / Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) Fig. 12. Sttisticl comprisons of tempertures derived from GPS=MET nd the ECMWF nlysis for the period June 21 July 4, The comprisons is done for three regions: northern mid nd high-ltitude ( 30N), tropicl (30S 30N), nd mid nd high-southern ltitudes ( 30S). The number of occulttions included in the sttistics re indicted on the top of ech pnel. pressure nd geopotentil height in the neutrl tmosphere nd ionospheric free electron density in the ionosphere. Although the concept of rdio occulttion is simple one, cre must be ten t dierent steps in the processing of the dt in order to obtin ccurte retrievls. The system described cn be divided into two mjor prts. First, the clibrtion of the signl, which implies isolting the tmospheric dely induced on the occulted signl from ll other eects such s geometricl motion of the stellites, clocs nd ground troposphere. Second, the inversion of the tmospheric dely to obtin physicl prmeters such s refrctivity nd other derived products. Our system relies on the most bsic pproch of using the Abel inversion to obtin refrctivity in the tmosphere. More dvnced pproches for inverting GPS occulttion dt include dt ssimiltion into wether models which cn be done t dierent levels, including tmospheric phse dely s function of time, Doppler shift s function of time, bending ngle s function of symptotic miss distnce, nd refrctivity s function of geopotentil height. All of these pproches, however, would still require the rst stge of processing, nmely, the clibrtion stge nd some ionospheric clibrtion. Even though the scope of this wor ws not to demonstrte how lower tropospheric sensing cn be done with GPS on routine bsis, our exmple demonstrtes tht it is t lest fesible to trc the GPS signl down to the surfce under very humid conditions. In future wor, we will discuss how lower tropospheric sensing cn be obtined t virtully ll times with GPS occulttion. Acnowledgements This wor ws done t the Jet Propulsion Lbortory of the Cliforni Institute of Technology, with funding from the Ntionl Aeronutics nd Spce Agency, Integrted Progrm Oce, nd JPL s Director Reserch Discretionry Funds. References Ahmd, B., Tyler, G.L., The two-dimensionl resolution ernel ssocited with retrievl of ionospheric nd tmospheric refrctivity proles by Abelin inversion of dio occulttion phse dt. Rdio Science 33 (1), Ahmd, B., Tyler, G.L., Systemtic errors in tmospheric proles obtined from Abelin inversion of rdio occulttion dt: eects of lrge-scle horizontl grdients. Journl of Geophysicl Reserch 104 (D4), Anthes, R.A., Rocen, C., Kuo, Y., Applictions of COSMIC to Meteorology nd Climte. Terrestril Atmospheric nd Ocenic Science 11 (1), Bssiri, S., Hjj, G.A., Higher-order ionospheric eects on the Globl Positioning System observbles nd mens of modeling them. Mnuscript Geodetic 18, Bevington, P.R., Dt Reduction nd Error Anlysis for the Physicl Sciences. McGrw-Hill boo compny, New Yor. Born, M., Wolf, E., Principles of Optics, 6th Edition. Pergmon, Trrytown, New Yor. Escudero, A., Schlesier, A.C., Rius, A., Flores, A., Rube, F., Lrsen, G.B., Syndergrd, S., Hoeg, P., Ionospheric

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