Estmatng the Resdual Tropospherc Delay for Arborne Dfferental GPS Postonng J. Paul Collns and Rchard B. Langley Geodetc Research Laboratory, Department of Geodesy and Geomatcs Engneerng, Unversty of New Brunswc, Fredercton, N.B., Canada. BIOGRAPHIES Paul Collns graduated from the Unversty of East London n 1993 wth a B.Sc. (Hons) degree n Surveyng and Mappng Scences. He s currently enrolled n the M.Sc.E. degree program n the Department of Geodesy and Geomatcs Engneerng at the Unversty of New Brunswc (UNB), where he s nvestgatng the effects of the troposphere on nematc GPS postonng. Rchard Langley s a professor n the Department of Geodesy and Geomatcs Engneerng at UNB, where he has been teachng snce 1981. He has a B.Sc. n appled physcs from the Unversty of Waterloo and a Ph.D. n expermental space scence from Yor Unversty, Toronto. Prof. Langley has been actve n the development of GPS error models snce the early 1980s and s a contrbutng edtor and columnst for GPS World magane. ABSTRACT In post-processng dual frequency GPS carrer phase data, the resdual tropospherc delay can easly be the largest remanng error source. Ths error can contrbute a bas n heght of several centmetres even f smultaneously recorded meteorologcal data are used. Ths shortcomng s prmarly due to the poor representaton of the water vapour profle n the tropospherc delay models. In addton, a lac of realtme meteorologcal data would force the scalng of ether surface values or standard atmosphere values, nether of whch are lely to accurately represent the ambent atmosphere. To obtan the hghest precson n nematc GPS some advantage may be obtaned by estmatng the resdual tropospherc delay along wth the poston of the movng platform. The smple tests reported n ths paper removed bases of upto ten centmetres n heght when estmatng the resdual tropospherc delay from GPS data recorded at an arcraft n flght. However, mportant lmtatons exst n the geometry of the satellte coverage whch must be consdered before the full relablty of the technque can be assessed. INTRODUCTION Ths paper descrbes an nvestgaton nto the estmaton of the resdual tropospherc delay from GPS sgnals. Ths parameter s the remanng part of the tropospherc delay not predcted by emprcal models. In post-processed dual frequency carrer phase data, t can easly be the largest remanng error source. Unle most applcatons of ths technque, where the recever s statc, we have used data recorded at an arcraft n flght. Ths dea was motvated by the fact that hghly accurate arcraft postons are requred for gravmetrc, altmetrc and photogrammetrc surveyng purposes. Increasngly, GPS s beng used to provde the decmetre-level accuracy requred for some of these technques. Ths level of precson can be acheved usng carrer phase observables, but we wll show that unmodelled tropospherc effects could potentally contrbute a bas of a smlar magntude. The Tropospherc Delay An electromagnetc sgnal propagatng through the neutral atmosphere s affected by the consttuent gases. The fact that the refractve ndex s slghtly greater than unty gves rse to a decrease n the sgnal s velocty. Ths ncreases the tme taen for the sgnal to reach a GPS recever s antenna, ncreasng the equvalent path length (both effects are often referred to as the delay ). Refracton also bends the raypath and thereby lengthens t, further ncreasng the delay. Because the bul of the delay occurs wthn the troposphere, the whole delay s often referred to solely as the tropospherc delay. Poster presentaton at ION GPS 97, 10th Internatonal Techncal Meetng of the Satellte Dvson of The Insttute of Navgaton, Kansas Cty, Mo., September 16-19, 1997.
the By assumng that the neutral atmosphere s both horontally stratfed and amuthally symmetrc, the tropospherc delay can be modelled n two parts: the delay experenced n the enth drecton and the scalng of that delay to the delay experenced at the enth angle of the raypath. The functons that undertae the scalng are usually termed mappng functons, although oblquty factor s sometmes used. Ths leads to the common formulaton of enth delays and mappng functons seen n the space geodetc lterature. The typcal formulaton of the tropospherc delay s descrbed as: ( ) ( ) ( ) ( ) T = t hyd m hyd + t wet m wet, ( 1 ) where at the antenna of recever, the delay on the sgnal from satellte s a functon of the delays n the enth drecton caused by the atmospherc gases n hydrostatc equlbrum and by those gases not n hydrostatc equlbrum (prmarly water vapour), t ( hyd) and t ( wet) respectvely; and ther mappng functons, m ( hyd) and m ( wet) respectvely. The mappng functons are usually descrbed as functons of the satellte elevaton angle ± complement of the enth angle. For smplcty, we wll consder equaton (1) as: where m ( ) ( ) T = t total m combned, ( 2 ) ( combned) = t ( hyd) m ( hyd) + t ( wet) m ( wet) t ( hyd) + t ( wet). ( 3 ) When processng GPS observatons, a value for the tropospherc delay s predcted usng emprcal models whch n general must be provded wth values of the ambent temperature, pressure and relatve humdty. Unfortunately, even wth accurate values, these models rarely predct the true delay wth a hgh degree of accuracy. In theory, the hydrostatc component of the delay can be predcted n the enth to the mllmetre level, however the hghly varable nature of atmospherc water vapour means that the accuracy of the nonhydrostatc delay s at the centmetre, or even decmetre level. In addton, when recordng GPS data at an arcraft, t s often the case that no meteorologcal nformaton s recorded at the same tme. When processng ths data, assumed meteorologcal values must be used, and n addton to the poorly modelled wet component, there could also be a bas contrbuted by the hydrostatc component. The recovery of these errors s the am of the technques presented here. The GPS Observables Because of ts hgh precson and low nose characterstcs we used the GPS carrer phase observatons. In general, the pseudorange measurements are too nosy to allow for the accurate estmaton of the tropospherc delay. To remove the bas of the satellte and recever clocs we have used the double-dfference observable. Ignorng multpath and nose, we have l Φ l l l l = ρ + T I + λ N, ( 4 ) where ρ represents the dfferental geometrc range between satelltes and l and statons and ; T s the dfferental delay caused by the troposphere; I s the dfferental delay caused by the onosphere; λ s the carrer frequency wavelength; and N s the dfferental nteger cycle ambguty. The dfferental onospherc delay can be removed from equaton (4) by usng dual frequency data wth the standard nter-frequency combnaton. The doubledfference nteger ambguty term must also be resolved by some sutable method. It s mportant that the ambgutes then reman fxed for the soluton to be consstent. For ths reason, t s mportant that the carrer phase data be free from cycle-slps and data gaps. THE RESIDUAL TROPOSPHERIC DELAY There are two ways of estmatng the resdual tropospherc delay: ether as a scale factor, s; or as a resdual enth component, r : ( ) T = 1 + s t m, ( 5 ) ( ) T = t + r m. ( 6 ) where we have dropped the enth delay and mappng functon parenthetcal labels for clarty. By restrctng the resdual error to the enth delay, we are assumng that there are no errors n the mappng functon. Ths s obvously untrue, however recent mappng functons such as those of Nell [1996] have been shown to be very accurate and any remanng error wll lely come from unmodelled atmospherc gradents and amuthal asymmetry. In theory these gradents can also be modelled, however, our data may not have the senstvty to detect them. 2
Modellng Consderatons The dfferental tropospherc delay s gven by: l l l T = T T T + T, ( 7 ) whch, because the enth delay at a partcular staton wll be the same for satelltes l and, can be wrtten as: l l l ( ) ( ) T = t m m t m m. ( 8 ) Estmatng a staton dependent scale factor, s, gves: l l l ( ) ( ) ( ) ( ) T = 1 + s t m m 1 + s t m m, ( 9 ) wth partal dervatves for a least-squares adustment: T s l t ( m l m = ) and t ( m m l ) T s l =, whch are the between-satellte sngle dfference tropospherc delays. Estmatng a resdual enth delay, r gves: ( )( ) ( )( ) T = t + r m m t + r m m, ( 10 ) l l l wth partal dervatves: T r l l = m m and T r l = m m, whch are the dfferental mappng functons. Condton of the Normal Equatons Prevous studes of estmatng the tropospherc delay from GPS data (e.g. Van Hove et al. [1993]) have hghlghted the problem of usng dfferenced data over short baselnes. For ths stuaton, there exsts a strong mathematcal correlaton between the partal dervatves of the tropospherc delay at the two statons. For baselne lengths up to several hundred lometres the elevaton angles to a partcular satellte wll be smlar and hence so wll the partal dervatves (but wth opposng sgns). Even f the meteorologcal condtons are drastcally dfferent at the ends of such a baselne, t s dffcult for a least-squares model to separate the two contrbutons. The usual technque to overcome ths problem s nown as leverng [Rocen et al., 1995] and wors by smply l fxng the tropospherc delay at the reference staton and estmatng the relatve delay at the secondary staton. Our use of real-tme meteorologcal data at the reference staton wll help mnmse the error n the estmated resdual delay. Whle some error wll be present, t wll be constant for all solutons computed wth dfferent tropospherc delay models at the arcraft. Ths problem of estmatng the tropospherc delay over short baselnes s compounded by the fact that the determnaton of the heght component of poston s senstve to the exstence of any unmodelled tropospherc enth delay and vce-versa. To explan ths we can consder the observaton equaton for a sngle staton/satellte measurement contrbutng to the doubledfference observable. Ignorng the onospherc and nteger ambguty terms (whch we can remove), we have: Φ ( ) ( ) = ρ + T = E E + N N ( ) 2 2 1 2 2 + + H H t m ( 11 ) where the recever and satellte coordnates are defned n the local geodetc coordnate system (eastng, northng and heght). Lnearsng around approxmate values and subtractng from the observaton gves: ρ = Φ T = E cos( e) sn( a) + N cos( e) cos( a) + H sn( e) + t csc( e), ρ 0 0 ( 12 ) where the ero superscrpts ndcate predcted values; represents the small correctons to the a-pror estmates; and a and e represent the amuth and elevaton angle of the satellte. The mappng functon s approxmated wth the cosecant of the elevaton angle. If we consder that the correctons to the horontal poston components E and N are largely decoupled from the heght and tropospherc delay components, then we can see how the presence of heght and enth delay errors affect the retreval of each other. Fgure 1 shows the effect on the modelled range dfference of a enth delay error of 2 mm, a heght error of 200 mm and ther combned effect ( ρ) from the enth down to an elevaton angle of fve degrees. Addtonally, we show what happens when we try to recover the heght and enth delay components assumng (ncorrectly) that the contrbuton of the other component s ero. We are able to accurately recover the heght component down to an elevaton angle of 3
approxmately thrty degrees, beyond whch the accuracy ncreasngly degrades. At the same tme we are only able to recover well the enth delay error at the low elevaton angles. At very hgh elevaton angles an error n the tropospherc enth delay s almost ndstngushable from the unmodelled heght component. Range / Heght / Zenth delay (m) 0.5 0.4 0.3 0.2 0.1 0.0 H.sn(e), H = 0.2 m t.csc(e), t = 2 m ρ = H.sn(e) + t.csc(e) ρ/sn(e) = H ρ/csc(e) = t 0 10 20 30 40 50 60 70 80 90 Elevaton Angle (e, deg) Fgure 1. Smulaton of the mpact and recovery of heght and enth delay components on a GPS range. What ths means s that an unmodelled tropospherc enth delay error causes an error n heght determnaton, whch ncreases wth the ncluson of lower elevaton data. Ths s a well nown fact n GPS, but mportantly for us we can see that attemptng to solve for the enth delay s hndered wthout low elevaton angle data. These results wll be modfed f tght constrants are placed on the staton heght components n the least-squares adustment. By closely constranng the heght to ts nown value, the least-squares model s better able to estmate the tropospherc enth delay. IMPLEMENTATION AND DATA PROCESSING A least-squares postonng model usng doubledfferenced, dual-frequency, GPS carrer phase observatons s mplemented n the KARS processng software [Mader, 1996]. The code has been modfed at UNB to allow for the estmaton of the tropospherc delay as ether a scale factor or a enth delay resdual at ether the secondary rovng recever or at both the rover and the reference recever. To test the estmaton of the resdual delay, some of the flght data from the St. John s, Newfoundland based Frle 95 campagn was used (see Collns and Langley [1997] for more detals). Meteorologcal data consstng of pressure, temperature and relatve humdty were smultaneously recorded along wth dual frequency GPS data at an arcraft and at a ground reference staton. Both data types are avalable at a two second samplng nterval. It has been possble to use the carrer phase data down to an elevaton angle of fve degrees on one of the data sets. Of the data recorded on the other days of ths experment, there are too many cycle slps and data gaps for the software to adequately process the data. Fgure 2 shows the flght path over whch the data was recorded on March 3rd 1995. The maxmum dstance reached from the reference staton was 210 lometres. As Fgure 2 ndcates, data collecton was halted n md-flght. Ths was due to a lac of memory n the arcraft recever. Heght (m) 5000 4000 3000 2000 1000 0-52 Longtude (deg) Fgure 2. Flght path of arcraft for data used n ths study. -51 46 48 47 49 Lattude (deg) A set of fxed carrer phase nteger ambgutes for all satelltes on both frequences was derved. Ths was done by processng the flght data at varous elevaton cut-off angles whle resolvng the ambgutes on-the-fly. Comparng the ambgutes from these solutons wth ambgutes computed for the short statc perod on the ground before the flght, has enabled stable sets of ntegers to be selected. Whle confdent that these are the correct values, wthout actually estmatng these values n flght (n a smlar manner to Sonntag et al. [1995]), we can only confrm ths by examnng the resduals of the poston solutons to see f they dverge over tme. Of the remanng error sources, the prmary one s the satellte poston error. To mnmse t as much as possble, Internatonal GPS Servce for Geodynamcs (IGS) precse orbts were used. Ths leaves multpath and nose as remanng unmodelled errors whch should be of the order of centmetres or less for the carrer phase observable. 4
RESULTS One tropospherc enth delay and mappng functon combnaton was adopted for the reference staton for all the solutons. These were the Saastamonen [1973] enth delays usng the smultaneously recorded meteorologcal data and the mappng functons of Nell [1996], whch only requre poston and day-of-year nformaton. Ths combnaton was also used at the arcraft and provded wth the smultaneously recorded meteorologcal data. Ths model s denoted as SAANf ( f for full-met. nput). The model denoted as SAANx ( x for extrapolated) used the reference staton meteorologcal data scaled to the heght of the arcraft usng standard atmospherc scalng equatons (see Collns and Langley [1997]); and the SAANh model ( h for hydrostatc only) whch predcted only the hydrostatc delay at the arcraft from the real-tme data whle the wet delay was set to ero. Three other models were also tested for modellng the delay at the arcraft: UNB4, whch supples meteorologcal data based on the 1966 U.S. Standard Atmosphere Supplements to the Saastamonen and Nell algorthms; the ntally proposed WAAS model; and the NATO recommended model (see Collns and Langley [1997] for more detals). One soluton was computed for each of these models estmatng the three-dmensonal Cartesan postons of the arcraft along wth the resdual tropospherc delay as a scale factor at each epoch. No flterng was appled and no a-pror constrants were placed on any of the parameters. Each epoch provded an ndependent soluton. Epoch resduals rms (m) 0.20 0.16 0.12 0.08 0.04 SAANf SAANx UNB4 NATO Tme (mn) Fgure 3. Root-mean-square double-dfference carrer phase resduals wthout resdual tropospherc delay estmaton. Epoch resduals rms (m) 0.05 0.04 0.03 0.02 0.01 SAANf SAANx UNB4 NATO Tme (mn) Fgure 4. Root-mean-square double-dfference carrer phase resduals wth resdual tropospherc delay estmaton. 5
Adustment Resduals Consderng frst of all the double-dfferenced carrer phase resduals after the least-squares adustment, comparson of Fgure 3 and Fgure 4 shows the general mprovement ganed by estmatng a resdual tropospherc delay parameter. Only four of the models are plotted for clarty. Fgure 3 represents the root-meansquare (rms) of the resduals computed wthout estmatng the resdual tropospherc delay. Almost every plot has a dstnct trace ndcatng the mpact of each tropospherc delay model. The excepton s for the traces of the SAANf and UNB4 solutons, whch closely follow each other. In Fgure 4 however, all the traces have merged to become almost ndstngushable, ndcatng that estmaton of the resdual tropospherc delay has largely removed the mpact of the choce of a partcular model on the soluton (note also the change of scale). Epoch resduals rms (m) 0.05 0.04 0.03 0.02 0.01 SAANf - no estmaton SAANf - scale factor Tme (mn) Fgure 5. Root-mean-square double-dfference carrer phase resduals wth and wthout resdual tropospherc delay estmaton SAANf model used wth arcraft data. Epoch resduals rms (m) 0.40 0.32 0.24 0.16 0.08 WAAS - no estmaton WAAS - scale factor Tme (mn) Fgure 6. Root-mean-square double-dfference carrer phase resduals wth and wthout resdual tropospherc delay estmaton WAAS model used wth arcraft data. Examnng two models more closely, Fgure 5 shows the rms resdual for each epoch for the SAANf model. In general, only a small mprovement has been made n estmatng a resdual correcton. Ths s to be expected because the tropospherc delay predcton of ths model s farly good due to the use of meteorologcal measurements recorded at the arcraft. A better ndcaton of the mprovement possble wth estmatng the resdual delay s ganed from examnng the mpact of the ntally-proposed WAAS model on the soluton. Ths model was left out of Fgure 3 and Fgure 4 because of ts large mpact. Ths can be seen n Fgure 6 where large umps correspond to the rsng and settng of satelltes at 5 degrees elevaton angle. Ths model uses only a modfed cosecant of the elevaton angle mappng functon and consequently large errors would be expected 6
wth these satelltes. As ths plot shows, some, but not all, of the error has been absorbed by estmatng the resdual tropospherc enth delay. In addton, the spes vsble n ths plot at approxmately 12 and 40 mnutes nto the flght are an artfact of a dscontnuty n the formulaton of ths model. Ths occurs whenever the arcraft crosses the 1500m heght level (see Collns and Langley [1997] for further dscusson of ths feature). 0.10 0.05 Zenth resdual (m) -0.05-0.10-0.15-0.20-0.25 UNB4 WAAS NATO Tme (mn) Fgure 7. Resdual tropospherc delay estmates for models wth no real-tme meteorologcal data. 0.10 Zenth resdual (m) 0.05-0.05-0.10 SAANf SAANh SAANx -0.15 Tme (mn) Fgure 8. Resdual tropospherc delay estmates for models wth real-tme meteorologcal data. Resdual Delay Estmates Turnng to the actual resdual delays estmated, Fgure 7 shows the values for the models wthout real-tme meteorologcal nput and Fgure 8 for those wth realtme nput. Both plots can be consdered n two halves before and after the 45 mnute epoch. Consderaton of Fgure 9 shows that before ths pont n the flght there are no satelltes at low elevaton angles (< 10 degrees). As ndcated prevously, ths lmts the potental for adequately estmatng the tropospherc delay. As an example, gven that the SAANh model predcts only the hydrostatc delay we would expect postve enth resduals to represent the remanng wet delay. Ths s generally true n the second half of Fgure 8, but the wde varaton n the frst half, coupled wth the large negatve values could mean that the resdual estmates for ths tme span are unrelable. At the same tme however, t s nterestng to note from Fgure 7 that the resdual estmates for the NATO model soluton have an almost constant bas component over the 7
whole tme span. Ths trend s what we would expect gven that the NATO model s formulated wth a constant value of surface refractvty whch, to a frst order consderaton, s based from the real surface refractvty experenced over the flght path. Both of the other models used n the solutons shown n Fgure 7 change ther atmospherc parameter nputs prmarly as a functon of lattude, hence there s no constant bas n ther solutons. Consderng the results for the other models, Fgure 7 shows that UNB4 gves the smallest resdual tropospherc delay for the models wthout real-tme meteorologcal nput; and that the ntally-proposed WAAS model s 0.04 0.02 greatly nfluenced by low elevaton satelltes n the soluton. Gven the correct enth delay, the ntallyproposed WAAS model wll over-predct the delay at low elevaton angles. Ths can be seen at ~45 mnutes when a new satellte appears. The resdual estmate umps to a large negatve value to try to account for the overpredcton. The resdual then ncreases toward ero as the satellte rses n the sy. Fgure 8 shows that the full-meteorologcal model, SAANf, has the smallest resdual delay, whch s what we would hope for. The extrapolaton of surface values ntroduces some bas (model SAANx), but the largest resdual delay s observed when estmatng the whole wet enth delay (model SAANh). 22 20 18 Scale factor -0.02-0.04-0.06-0.08 Scale factor estmate Scale factor uncertanty Lowest elevaton angle 16 14 12 10 8 Elevaton angle (deg) -0.10 6-0.12 4 Tme (mn) Fgure 9. Comparson of SAANf resdual tropospherc delay estmates and ther uncertanty wth the lowest elevaton angle used n the soluton. Poston Dfferences Turnng to the mpact of the resdual tropospherc delay estmaton on the poston determnaton, we can frst of all consder the resdual delay that remans when usng real-tme meteorologcal data. Wthout resdual delay estmaton, we would consder the SAANf soluton to be the best obtanable because of ts realstcally-modelled enth delays and mappng functons drven by real-tme meteorologcal data. By estmatng the resdual delay we would hope to model any devatons from the average atmospherc structure mpled by these models. Fgure 10 shows the dfference n the poston components for solutons computed usng the SAANf model wth and wthout resdual tropospherc delay estmaton. The dfference n the heght component s consderable: of the two sets of statstcs for ths data, even when consderng only the good estmates after the 45 mnute epoch, there s a mean bas n heght of ~5 cm wth an rms of ~9 cm. In addton, usng the soluton wth the SAANf model and resdual tropospherc delay estmaton as a benchmar, we can compare the mpact of estmaton wth other models and confrm that estmaton of the resdual delay helps to remove the mpact of less accurate models. Fgure 11 shows the poston dfferences of the soluton computed usng UNB4 at the arcraft and Fgure 12 shows the nfluence of usng the NATO model at the arcraft. The bases n these two plots are predomnantly a functon of the lowest elevaton angle (cf. Fgure 9). 8
Poston dfference (m) 0.4 0.3 0.2 0.1 0.0-0.1-0.2-0.3-0.4 Lattude Longtude Heght Heght Dfference Statstcs All data: mean = 0.078 m, rms = 0.115 m After t=45: mean = 0.045 m, rms = 0.085 m Tme (mn) Fgure 10. Dfference n poston solutons wth and wthout resdual delay estmaton from predctons wth real-tme meteorologcal data (SAANf model). Poston dfference (cm) 1.0 Lattude Longtude 0.5 Heght 0.0-0.5-1.0 Heght dfference rms = 0.4 cm Tme (mn) Fgure 11. Poston dfferences between the UNB4 soluton and the SAANf soluton. (Resdual delay estmated n both solutons.) Poston dfference (cm) 10.0 5.0 0.0-5.0-10.0 Lattude Longtude Heght Heght dfference rms = 2.6 cm Tme (mn) Fgure 12. Poston dfferences between the NATO soluton and the SAANf soluton. (Resdual delay estmated n both solutons.) 9
Where there are no low elevaton satelltes to magnfy the errors, all three solutons compare at the mllmetre level n all three poston components. Even so, the mpact of usng the UNB4 model nstead of real-tme meteorologcal data s stll only of the order of 1 cm n heght at maxmum. The mpact of the NATO model s one order of magntude larger. CONCLUSIONS We have attempted to show n ths paper the effect of mplementng resdual tropospherc delay estmaton from GPS data recorded at an arcraft n flght. The am was to remove any unmodelled effects of the troposphere that cannot be predcted by emprcal models, even when usng meteorologcal measurements of the ambent atmosphere. Problems were encountered wth the general stablty of the least-squares soluton when attemptng to estmate the resdual delay at the reference staton as well as at the arcraft. The smlarty of the normal equaton coeffcents prevents the separaton of the contrbuton of the atmosphere at the two statons n the doubledfference observable. By estmatng only the resdual delay at the arcraft we are assumng that there s no resdual effect at the reference staton, or that ths effect s absorbed by the estmate for the arcraft. Estmatng the resdual delay appeared to almost wholly remove the mpact of a partcular tropospherc delay model. However, the accuracy of the mappng functon and the mpact of the satellte geometry are mportant. It appears crucal that there exsts data at low elevaton angles (less than 10 degrees) for the tropospherc resdual estmate to be meanngful. Wth ths condton, plus an accurate mappng functon, estmatng the resdual delay not only removes the mpact of one partcular tropospherc delay model but also the bases from usng non-real-tme meteorologcal parameter values at the arcraft. Therefore, f the hghest possble precson s requred for arcraft postonng the estmaton of a resdual delay should be consdered, otherwse bases of the order of ten centmetres may be present n the soluton. The wor reported n ths paper has been only a prelmnary study and further nvestgatons are requred to study the condton of the least-squares normal equatons and the overall relablty of the technque. New nvestgatons could nclude the mpact of antenna phase centre correctons, as the data s partcularly senstve to these at low elevaton angles. Addtonally, the mplementaton of a Kalman or other type of constranng least-squares flter could sgnfcantly enhance the technque by provdng some a-pror constrants to the estmates. ACKNOWLEDGMENTS The support of Nav Canada (formerly Transport Canada Avaton), the Natural Scences and Engneerng Research Councl of Canada, the Atmospherc Envronment Servce and the U.S. Federal Avaton Admnstraton are gratefully acnowledged. REFERENCES Collns, J.P. and R.B. Langley (1997). A Tropospherc Delay Model for the User of the Wde Area Augmentaton System. Fnal contract report prepared for Nav Canada, Department of Geodesy and Geomatcs Engneerng Techncal Report No. 187, Unversty of New Brunswc, Fredercton, N.B., Canada. Mader, G.L. (1996). Knematc and rapd statc (KARS) GPS postonng: Technques and recent experences. IAG Symposa No. 115, Eds. G. Beutler, G.W. Hen, W.G. Melbourne and G. Seeber. IUGG/IAG, Boulder, Colo., 3-4 July. Sprnger- Verlag, Berln, pp. 170-174. Nell, A.E. (1996). Global mappng functons for the atmosphere delay at rado wavelengths. Journal of Geophyscal Research, Vol. 101, No. B2, pp 3227-3246. Rocen, C., T. Van Hove, J. Johnson, F. Solhem and R. Ware (1995). GPS/STORM GPS sensng of atmospherc water vapour for meteorology. J. Atmos. Oceanc Technology, Vol. 12, pp. 468-478. Saastamonen, J. (1973). Contrbutons to the theory of atmospherc refracton. In three parts. Bulletn Géodésque, No. 105, pp. 279-298; No. 106, pp. 383-397; No. 107, pp. 13-34. Sonntag, J.G., C.F. Martn and W.B. Krabll (1995). Ambguty resoluton over long baselnes for arborne dfferental GPS postonng. Proceedngs of ION GPS-95, Palm Sprngs, Calf., September 12-15, pp. 1117-1125. Van Hove, T.M., C. Alber and J.M. Johnson (1993). Atmospherc water vapour as nose and sgnal for Global Postonng System applcatons. Proceedngs of ION GPS-93, Salt Lae Cty, Utah, September 22-24, pp. 797-804. 10