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Page 1: Tropospheric Zenith Delay Prediction Accuracy for Airborne ...gauss.gge.unb.ca/papers.pdf/ion54am.virgilio.pdf · Tropospheric Zenith Delay Prediction Accuracy for Airborne GPS High-Precision

Tropospheric Zenith Delay PredictionAccuracy for Airborne GPSHigh-Precision Positioning

V. B. Mendes1,2 and R. B. Langley21Faculty of Sciences of the University of Lisbon, R. Ernesto de Vasconcelos, Bloco C1, Piso 3, 1700 Lisboa, Portugal;e-mail: [email protected]. 2Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering,University of New Brunswick, P.O. Box 4400, Fredericton, N.B., E3B 5A3 Canada; e-mail: [email protected].

BIOGRAPHY

Virgílio Mendes received his Diploma in GeographicEngineering from the Faculty of Sciences of theUniversity of Lisbon, Portugal, in 1987. Since then, hehas been working as a teaching assistant at thisuniversity. His principal areas of research are themodeling of tropospheric delay in space techniques andthe application of the Global Positioning System to themonitoring of crustal deformation.

Richard Langley is a professor in the Department ofGeodesy and Geomatics Engineering at the Universityof New Brunswick, where he has been teaching andconducting research since 1981. He has a B.Sc. inapplied physics from the University of Waterloo and aPh.D. in experimental space science from YorkUniversity, Toronto. Dr. Langley is a co-author of theGuide to GPS Positioning and is a contributing editorand columnist for GPS World magazine.

ABSTRACT

When traversing the earth’s neutral atmosphere, GPSradio signals are affected significantly by thevariability of its refractive index, which causesprimarily a delay, usually referred to in the literature asthe tropospheric delay. An inaccurate modeling of thisdelay results in degradation of position estimates,affecting mainly the height component.

The tropospheric delay is commonly divided into twocomponents, “hydrostatic” and “wet”, each oneconsisting of the product of the delay at the zenith anda mapping function that projects the zenith delay ontothe desired line-of-sight. A few high-accuracymapping functions, parameterized by either specificmeteorological parameters or other site-dependentparameters, have been developed in recent years. Asregards the prediction of the zenith delay, the problemis far more complicated essentially due to the highspatial and temporal variability of the wet component.

We have determined mean bias and r.m.s. scatter for agreat number of zenith delay prediction modelsdeveloped in the last few decades, including the models

generally used in airborne navigation, by comparingthe models against ray-tracing using a one-year data setof radiosonde profiles from 50 stations distributedworldwide. We have concluded that the hydrostaticzenith delay can be predicted with submillimetreaccuracy, provided accurate measurements of stationpressure are available. As regards the wet zenith delay,the models differ significantly in accuracy but showvery similar r.m.s. scatter. Our analyses show that thewet zenith delay can typically be predicted with aprecision of ~3 cm (at the one-sigma level), usingmeteorological data. The prediction of the total delayby models typically used in airborne navigationindicates a much poorer accuracy, leading to predictionbias ranging from ~6 cm up to more than 20 cm. Ingeneral, all the models tested perform significantlybetter at mid-latitudes than at low latitudes.

INTRODUCTION

As radio signals used by radiometric space techniquestraverse the earth’s neutral atmosphere, they experiencea decrease in their speed of propagation and, as aconsequence of Fermat’s principle of least time, adeviation of their path from a straight line. This globaleffect is generally known as tropospheric propagationdelay − as the troposphere is responsible for most ofthis effect − and is induced by the variable refractiveindex of the neutral atmosphere.

A mismodeling of the tropospheric delay results in adegradation of the estimates of the height componentof position and may therefore constitute a limitation inhigh-accuracy airborne GPS positioning. Errors due toincorrect tropospheric delay modeling may alsointroduce errors in carrier phase ambiguity resolution[Mendes et al., 1995]. The modeling of troposphericdelay for precise airborne relative positioning is furthercomplicated due to the large height differencegenerally existing between the ground-based receiverand the airborne remote receiver. Shi and Cannon[1995] found a significant correlation between theresidual effect of tropospheric delay on the estimatedheight component and the height difference betweenthe reference and remote receivers.

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For modeling purposes, the tropospheric delay can beexplicitly written as the contribution of a hydrostatic(mostly dry) and a wet component, each one consistingof the product of the delay experienced in the zenithdirection and a mapping function that models theelevation angle dependence of the tropospheric delay:

)(md)(mdd wzwh

zhtrop ε⋅+ε⋅= , (1)

where dtrop is the tropospheric propagation delay at agiven elevation angle ε, z

hd and zwd are, respectively,

the hydrostatic and wet zenith delays, and )(mh ε and)(mw ε are the hydrostatic and wet mapping functions,

respectively.

The goal of this paper is to assess models for zenithdelay prediction models. For airborne positioning, themapping functions developed by Niell [1996] areparticularly interesting, due to their high-accuracy andindependence from meteorological parameters. A largenumber of mapping functions have been intercomparedby Mendes and Langley [1994].

ZENITH DELAY

The delay experienced by a radio signal in the zenithdirection is defined as

∫−=ar

sr

6ztrop dz N10d , (2)

where N is the refractivity, rs is the geocentric radius ofthe receiver antenna, ra is the geocentric radius of thetop of the neutral atmosphere, and dz has length units.

The refractivity of a parcel of air is expressed as[Thayer, 1974]:

ZT

eK

T

eKZ

TPKN 1

w2321

dd

1−−

+

+

= , (3)

where Ki are constants empirically determined inlaboratory (see Bevis et al. [1994] for a discussion onrefractivity constants), Pd is the partial pressure due todry gases, e is the water vapor pressure, T is thetemperature, Zd and Zw are the compressibility factorsfor dry air and water vapor, respectively [Owens,1967].

The first term on the right-hand side of Equation (3)does not depend on the water vapor content of theatmosphere and is therefore known as the drycomponent of the refractivity; the second termrepresents the wet component of the refractivity.

If we assume that air behaves as an ideal gas, then Pd =P − e and Zd = Zw = 1; hence Equation (3) is commonlyrewritten as

( )23121

T

eK

T

eKK

T

PKN +−+= . (4)

An alternate separation of the refractivity componentswas derived by Davis et al. [1985]. They rewriteEquation (3) as:

1w23

1w2 d1 Z

T

eKZ

Te

KRKN −−

+

+ρ= ' , (5)

where

−=

w

d122 R

RKKK ' , (6)

Rd is the specific gas constant for dry air, Rw is thespecific gas constant for water vapor, and ρ is thedensity of moist air.

The first term of the right-hand side of Equation (5) isno longer purely “dry”, as there is a contribution of thewater vapor hidden in the total density, and is known inthe literature as the “hydrostatic” component ofrefraction. On the other hand, the “wet” component isalso different from the wet component of the “dry/wet”formalism, as a consequence of the differentpartitioning of dry gases and water vapor contributions;it should probably be referred to as the “non-hydrostatic” component, but the term “wet” is stillcommonly used. In this paper, the designation “wet”will also be used and therefore no distinction betweenthe wet and non-hydrostatic components will be made,but it is important to note that these are actually two(slightly) different quantities.

If we consider the refractivity to be composed of ahydrostatic and a wet component, the zenith delay cantherefore be split into two components, denominatedthe hydrostatic zenith delay and the wet zenith delayand consequently Equation (2) can be written as:

∫∫ −− +=ar

sr

w6

ar

sr

h6z

trop dzN10dzN10d , (7)

or symbolically,

zw

zh

ztrop ddd += , (8)

where zhd represents the hydrostatic zenith delay and

zwd represents the wet zenith delay. With regards to

the hydrostatic component, we have specifically:

∫ ρ= −ar

sr

d16z

h dzRK10d . (9)

Using the hydrostatic equation, we can derive thefollowing relation:

m

sd1

6zh g

PRK10d −= , (10)

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where gm is the acceleration due to gravity at the centerof mass of the vertical column of air, and Ps is thepressure at the antenna location.

For the wet component, we write:

dzZT

eK

T

eK10d

1-w

ar

sr232

6zw ∫

+

= − ' . (11)

The direct solution of this integral requires a watervapor profile, which is generally not known, andsurface conditions are not universally stronglycorrelated with the conditions aloft. Nevertheless, alarge variety of models attempting to tackle thisproblem have been developed and a new empiricalmodel will be presented here.

A NEW MODEL

Before reviewing the existing zenith delay models wehave assessed in our study (see Appendix I), wepresent a new empirical model for the wet zenith delay.

Based on ray-tracing results obtained using data from50 radiosonde stations distributed worldwide (seeAppendix II for locations) for all of 1992, we havefound that there is a correlation between the wet zenithdelay and surface water vapor pressure (a correlationthat varies from weak to strong, depending on thelocation), which can be expressed by the followingmodel (labeled UNB98ZW, for discussion purposes):

e 00943.00122.0dzw += (12)

where zwd is given in metres and e in hectopascals.

The coefficients for this model were obtained from aleast-squares fit with 10,822 data points (sampled fromthe total number of traces). The uncertainties for theintercept and slope of the straight line are 4.1×10-4 mand 2.9×10-5 m hPa-1, respectively. The r.m.s. of the fitwas 0.024 m.

For airborne GPS navigation, and for many otherapplications, accurate meteorological data may bedifficult to acquire and therefore the use of wet zenithdelay prediction models using meteorological data asan input may be inconvenient or even impossible. Inthe case of our model, the use of the water vaporpressure as the only input would nevertheless requiremeasurements, e.g., of temperature and relativehumidity (as water vapor pressure is typically notmeasured directly), at each given height. There arehowever several sources of mean values of water vaporpressure, such as the International Organization forStandardization Reference Atmospheres for AerospaceUse (hereafter ISO) [ISO, 1982; 1983], that can beused to by-pass that difficulty. Values are available forthe months of January and July, and for altitudesranging from 0 m to 10,000 m (every 1,000 m) and forthe following latitudes in the northern hemisphere: 10°,30°, 50°, and 70°.

Using this information, we have developed a modelthat allows the determination of water vapor pressure ata given latitude, height, and day-of-year. We fitted amodel to the tabulated values of water vapor pressureof the ISO atmospheres, and subsequently, we carryout an interpolation to a given day-of-year using ascheme that follows the one adopted by Niell [1996] inthe development of his mapping functions (for detailssee Mendes [1998]).

OTHER MODELS

Our model has been incorporated into an analysis of 12different wet zenith delay models. We have alsoassessed hydrostatic and total zenith delay models, allof which are briefly described in this section.

If we follow a theoretical approach, the modeling ofthe hydrostatic zenith delay is straightforward, andmodels can only differ due to the choice of therefractivity constant (the effect of choosing differentrefractivity constants is not significant however) and onthe modeling of the height and latitude dependence ofthe acceleration of gravity. Saastamoinen [1973] andBaby et al. [1988] have developed models based onthis approach (see Equation (10)). Hopfield [1969]showed that the dry zenith delay could also be obtainedusing a quartic model for the dry refractivity profileand developed the following model:

5

HHN10d s

ed

ds6z

d−= − , (13)

where Nds is the surface dry refractivity, Hs is thestation height, and e

dH is the so-called equivalent

height, which can be determined as a function ofsurface temperature [Hopfield, 1972].

For the wet zenith delay we have chosen a much largerselection of models. This choice embraces modelsbased either on theoretical assumptions concerningwater vapor height profiles or on empirical models. Ingeneral, these models are highly dependent on thewater vapor pressure at the antenna location. AppendixI lists the models selected for this intercomparisonstudy and shows the input parameters used by eachone. The models have been used in their standardformulation and driven by the surface meteorologicaldata provided by the radiosonde soundings. In order toprovide reliable information on temperature lapse rateneeded for some models, monthly-averaged valueswere computed for all stations.

The Hopfield model is formulated on the assumptionthat the wet refractivity can also be described by thequartic profile (for convenience, as physically this isnot justified). The wet equivalent height was set to afixed value [Hopfield, 1972].

Saastamoinen [1973] and Askne and Nordius [1987]assumed that the water vapor pressure decreases withheight according to a power law. In its full expression,

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the Saastamoinen model can be tailored to a particularlocation (see Janes et al. [1991]); the model analyzedhere corresponds to the standard formulation derivedfor mid-latitude conditions. For the model developedby Askne and Nordius, the global value of the watervapor lapse parameter (λ) given by Smith [1966] wasused (this parameter can be tuned for a particularregion). Collins et al. [1996] used this model as thebasis of the UNB3w wet zenith delay model, wherevalues for the input meteorological parameters areobtained by interpolation through a set of averagevalues for different latitudes (a lookup table), compiledfrom different sources. In its full implementation, thehydrostatic component of UNB3 is modeled by theSaastamoinen model, using a lookup table of averagevalues of surface pressure, and the elevationdependence of the delay is modeled with the mappingfunctions developed by Niell [1996].

Callahan [1973] assumes an exponential height profilefor the water vapor pressure (empirically determined).In its simplest form, the model is only a function of thesurface water vapor pressure and temperature. Chao[1973] based his model on an adiabatic atmospheremodel, and derived an expression that is a function ofsurface water vapor pressure and temperature, andtemperature lapse rate.

Berman [1976] and Baby et al. [1988] have derivedmodels (codes B70w and BB1w, in Appendix I) thatbasically share the same assumptions: the relativehumidity is constant with altitude (and equal to itssurface value) up to a certain height and thetemperature decreases with height at a constant rate.The models differ in the computation of the watervapor pressure (Baby et al. proposed two models forthis computation, and the simplest one was used inBB1w). They also proposed empirical models − B74w(Berman 74), BTMw (Berman TMOD), and BB2w.The Berman empirical models are based on theexistence of a strong linear relation between the ratiosof the wet and hydrostatic delays and correspondingrefractivities. The empirical model by Baby et al. is afunction of surface relative humidity and temperature,and a set of empirical coefficients (ν and γ), which canbe adjusted for different climatic regions (the globalvalues were used in our analysis).

Ifadis [1986] found a weak correlation between the wetzenith delay and all “standard” surface meteorologicalparameters (temperature, pressure, and water vaporpressure) and, as with UNB98ZW (98aw), is based onray tracing through radiosonde data (but in our modelonly the water vapor pressure dependence isconsidered).

Finally, we have also included in our analysis a versionof UNB98ZW driven by our model for determinationof water vapor pressure (98bw). This version andUNB3 are therefore the only models analyzed whoseuse is totally independent of concurrently-measuredmeteorological data.

The total delay at the zenith is formally obtained byadding the contributions of the hydrostatic and wetzenith delay models. However, some models do nothave an explicit separation of the hydrostatic and thewet components. Such is the case of most of themodels specifically developed for airborne positioning.In general, these models are also independent of directmeasurements of meteorological data. The modelsincorporated in GPS receivers have generally a simplemathematical structure that allow fast computations bythe task-limited microprocessor. Other models aremore sophisticated and constitute, in general, analyticalapproximations to refractivity profile models.

Another feature of these models is that they alsoincorporate a mapping function. The zenith delaycomponent is therefore obtained by taking the value ofsuch “hybrid models” in the direction of the zenith.

We have selected one model known to be used in GPSreceivers, three models used in “precise” airbornenavigation, and the Saastamoinen model (acombination of the hydrostatic and wet zenith delaymodels, to be used as reference).

The model we have selected as representative ofmodels used in GPS receivers is documented in theliterature (e.g. Braasch [1990]; Lewandowski et al.[1992]), but the original source of the model isunknown to us. Lewandowski et al. [1992] state that itis “implemented in receivers manufactured by StanfordTelecommunication Inc.”. We will label it as STI. Theonly input for this model is the user’s height.

Another commonly-used model is the one developedby Altshuler and Kalaghan [1974]. This model is afunction of the user’s latitude, height, and surfacerefractivity, which is in turn estimated as a function ofthe user’s latitude, height, and season.

The zenith delay model adopted for the NATOstandard troposphere model [NATO, 1993] is based onthe CRPL Reference Atmosphere - 1958 [Bean andThayer, 1959] and is a function of the height above sealevel and the mean sea-level refractivity (a fixed valueof 324.8 N-units was used).

The Federal Aviation Administration (FAA) initiallyproposed a model for the Wide Area AugmentationSystem (WAAS) that is derived from the Altshuler andKalaghan model [DeCleene, 1995]. The input for thismodel is the user’s height above sea level, stationlatitude, and day-of-year.

MODEL ASSESSMENT

As benchmark values for model assessment we haveused ray tracing through the radiosonde data used forthe UNB98ZW model development, but the totalnumber of traces (32,467) was considered. Theradiosonde stations have heights above sea levelranging from 3 m to 2234 m and have typically tworadiosonde launches per day (at 0h and 12h UTC).

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Radiosonde soundings consist of height profiles ofpressure, temperature, and relative humidity or dew-point.

The performance of each prediction model wasassessed in terms of its bias, corresponding to the meanof the differences between the model values and raytracing, and root-mean-square (r.m.s.) scatter about thismean value (standard deviation). To assess the globalperformance (total number of traces) we have adopteda representation by box-and-whisker plots, where thefollowing statistical quantities are represented: medianand mean (thinner and thicker lines inside the boxes,respectively), 25th and 75th percentiles (vertical boxlimits), 10th and 90th percentiles (whiskers), and 5th and95th percentiles (open circles). We have assessed theperformance of the hydrostatic delay, wet delay, andtotal delay models separately.

HYDROSTATIC ZENITH DELAY

The results of our global assessment are presented inFigure 1 (for codes used in the figures see Appendix I).

The outstanding performance of the Saastamoinenmodel is very clear, with submillimetre bias andsubmillimetre r.m.s. scatter with respect to ourbenchmark values. The other two models show biasesof 3-4 mm and r.m.s. scatter of 2-3 mm.

The performance of each model at the differentradiosonde stations is shown in Figure 2.

The bias of each model is clearly dependent on thelatitude of the station, even though it may be

considered insignificant for the Saastamoinen model.The latitude-dependent bias seen for the Hopfieldmodel is probably explained by an inadequate value ofthe equivalent dry height (an error of 100 m in thisvalue induces a bias of about 5 mm in delay), but othereffects may also be responsible for these biases. Ther.m.s. scatter (1-2 mm) is within the values quoted byHopfield [1972] and again is likely connected tovariations in dry equivalent height, judging by thesmall r.m.s. scatter attained at stations with a verysteady surface temperature.

The Baby et al. model performs similarly to theHopfield model in terms of mean bias, but has an r.m.s.scatter which is comparable to that obtained for theSaastamoinen model. The source of the bias in this

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Figure 2 − Bias (top plot) and r.m.s. scatter (bottom plot) for the hydrostatic zenith delay models, for 50 radiosondestations.

BB

h

HO

h

SA

h

Mod

el m

inu

s Tra

ce (m

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-0.010

-0.005

0.000

0.005

0.010

Figure 1 − Box-and-whisker plot for the differencesbetween the hydrostatic zenith delay models and raytracing.

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model is associated with their gravity modelingapproach. The model used by Baby et al. to estimategm is a function of different quantities, such as surfacetemperature and temperature lapse rate. The values forthe temperature lapse rate used in our analysis aremonthly-averaged values determined at each station,which minimizes the influence of errors due to lapserate determination.

The Hopfield and the Baby et al. models tend to over-predict the zenith delay, except for the equatorialregion. The Saastamoinen performance is extremelygood for all the analyzed radiosonde stations and it istherefore expected that this model will provide veryaccurate predictions of the hydrostatic zenith delaygiven accurate station pressures.

WET ZENITH DELAY

For the group of models driven by meteorological data,and despite the heterogeneity of models, the globalanalysis allows us to infer that the models showdifferent biases but very similar r.m.s. scatter, exceptfor Berman 74, Berman TMOD and Callahan, whichachieve the worst performance of this major group (seeFigure 3).

The best overall performance is achieved by our model(UNB98ZW). This is partially due to the fact that wehave used part of the radiosonde data to derive thecoefficients for the model, resulting therefore in anessentially zero mean bias. The r.m.s. scatter (slightlyless than 3 cm) is also the lowest, by a very smallmargin.

The two models requiring no external meteorologicaldata (98bw and UNB3w) also perform similarly interms of r.m.s. scatter, but 98bw has a much lowermean bias. As compared to the best models usingmeteorological data, 98bw and UNB3w see the r.m.s.scatter increased by a factor of ~1.6.

None of the models has a clear advantage at allradiosonde stations used in our intercomparison, asillustrated in Figure 4. The histograms in this figurerepresent the percentage of best performancesaccomplished by the different models on a station-by-station basis, in terms of bias, r.m.s. scatter, and totalerror (resulting from the combination of the bias andr.m.s. scatter). In terms of bias, we can conclude thatthere is a good distribution of best rankings, with someadvantage for the group ANw, BB2w, SAw, and 98aw.BB1w achieves the best performance as regards ther.m.s. scatter, but the differences in r.m.s. scatterbetween this model and most of the best models is atthe sub-millimetre level. When we examine the totalerror, the advantage of UNB98ZW is to be noted. Themodels developed by Askne and Nordius and Baby etal. have identical performance according to thiscriterion.

Of special interest for airborne applications is theperformance of the two models using no externalmeteorological data. These perform very differentlyaccording to the latitude of the station. In general, theUNB3w model performs better for low-latitudestations, whereas 98bw performs better for middle andhigh latitudes, as illustrated in Figure 5. The 98bwmodel performs better than UNB3w for about 2/3 ofthe total number of stations, for any of the rankingcriteria. The poorer performance of the 98bw model atlow-latitudes can be explained by inadequate values ofwater vapor pressure given in the ISO standardatmospheres. For the common latitude of 30° N, thewater vapor pressure of the ISO Reference Atmosphereis more than 20% lower than the value we havecomputed from the U.S. Standard AtmosphereSupplements, 1966 (USSA) [ESSA/NASA/USAF,1966]. It is therefore possible that a combination of theinformation provided by both standard atmospherescould lead to an improvement of our model for watervapor pressure determination.

AN

w

BB

1w

BB

2w

B70w

B74w

BTM

w

CA

w

CH

w

HO

w

IFw

SA

w

UN

B3w

98aw

98bw

Mod

el m

inu

s T

race

(m

)

-0.12

-0.06

0.00

0.06

0.12

Figure 3 − Box-and-whisker plot for the differencesbetween the wet zenith delay models and ray tracing.

AN

w

BB

1w

BB

2w

B70w

B74w

BTM

w

CA

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IFw

SA

w

UN

B3w

98aw

98bw

%

0

5

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35

40

bias r.m.s. scatter total

Figure 4 − Histogram for the best performances, inpercentage.

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TOTAL ZENITH DELAY

For the total zenith delay models, the box-and-whiskerplot for the total number of differences with respect toray tracing shown in Figure 6 indicates that the modelscommonly used in navigation applications performpoorly both in an absolute sense (with respect to ourbenchmark values) and in a relative sense, whencompared with the Saastamoinen model.

Among the models used in navigation applications, theNATO and initially-proposed WAAS models performthe best, the latter showing a larger bias but a lowerr.m.s. scatter. The large r.m.s. scatter of the NATOmodel is likely associated with the use of a fixed meanvalue for surface refractivity. In the case of the

Altshuler and Kalahan model, the use of the refractivityas computed from meteorological data improved themean bias, but did not improve the r.m.s. scatter, ingeneral.

All the navigation models experience large biases atlow-latitude stations, as shown in Figure 7 (theAltshuler and Kalahan model is not shown in theseplots, for the sake of clarity of the figure). In the caseof the STI model, these biases reach more than 20 cm,but the model performs acceptably at high latitudes,with small mean offsets. The r.m.s. scatter for thismodel is the same as that obtained for the NATOmodel, as the only variable in these models is theheight of the station. We conclude that the NATOmodel performs reasonably well at mid-latitudes butdegrades towards high and low latitudes. The r.m.s.scatter for this model is larger than that obtained for theWAAS model, which also has a better performance athigh latitudes.

CONCLUSIONS

Using ray tracing through a large number of radiosondeprofiles, for the year 1992, we have evaluated theperformance of different models for troposphericzenith delay prediction. We have concluded that thehydrostatic zenith delay can be determined frommeasurements of station pressure with sub-millimetreaccuracy, using the Saastamoinen model. The modelsused for wet zenith delay prediction have seriouslimitations, due to the high temporal and spatialvariability of the wet delay. All models have r.m.s.scatter at the few centimetre-level, even if driven by

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station meteorological data. We have developed asimple model that shows an overall best performance,for different climatic conditions. The model degradessomewhat if driven by average values from standardatmospheres (rather than actual station water vapormeasurements), both in bias and r.m.s. scatter. For thisvariant of our model, the r.m.s. scatter increases by afactor of ~1.6 (up to ~5 cm), but the bias is onlyunsatisfactory for low latitudes. It has however theadvantage of being totally independent of directmeteorological measurements, which is likely thesituation encountered in most airborne positioning.The models frequently used in navigation applicationspresented severe limitations. Even though a clearimprovement in r.m.s. scatter cannot be expected formodels not using meteorological data, a bettercalibration for reduction of the bias has to beconsidered in order to meet the requirements of high-precision airborne positioning.

ACKNOWLEDGMENTS

We would like to express our appreciation to ArthurNiell, for providing us with some of the radiosondedata we used and the ray-trace software (developed byJames Davis, Thomas Herring, and Arthur Niell), andto Gunnar Elgered and the Swedish Meteorological andHydrological Institute, for providing us radiosonde datafor some Scandinavian stations.

The support of the “Laboratório de Tectonofísica eTectónica Experimental” is gratefully acknowledged.

REFERENCES

Altshuler, E.E. and P.M. Kalaghan (1974).“Tropospheric range error corrections for theNAVSTAR system.” AFCRL-TR-74-0198, Air ForceCambridge Research Laboratories, Bedford, Mass.

Askne, J. and H. Nordius (1987). “Estimation oftropospheric delay for microwaves from surfaceweather data.” Radio Science, Vol. 22, No. 3, pp.379-386.

Baby, H.B., P. Golé, and J. Lavergnat (1988). “Amodel for the tropospheric excess path length of radiowaves from surface meteorological measurements.”Radio Science, Vol. 23, No. 6, pp. 1023-1038.

Bean, B.R. and G.D. Thayer (1959). “Models of theatmospheric radio refractive index.” Proceedings ofInstitute of Radio Engineers, Vol. 47, pp. 740-755.

Berman, A.L. (1976). “The prediction of zenith rangerefraction from surface measurements ofmeteorological parameters.” JPL Technical Report32-1602, Jet Propulsion Laboratory, Pasadena, Calif.

Bevis, M., S. Businger, S. Chiswell, T.A. Herring, R.A.Anthes, C. Rocken, and R.H. Ware (1994). “GPSmeteorology: Mapping zenith wet delays ontoprecipitable water.” Journal of Applied Meteorology,Vol. 33, No. 3, pp. 379-386.

Braasch, M.S. (1990). “A signal model for GPS.”Navigation, Journal of The Institute of Navigation(U.S.), Vol. 37, No. 4, pp. 363-377.

BE

LH

OB

QU

IB

LO

FO

RB

LB

KP

PG

UA

JSJ

ME

XL

IHM

ZT

EY

WP

BI

CR

PM

AF

TU

SJA

NN

KX

AB

QT

AT

GS

OB

NA

OA

KW

AL

DE

NM

AD

BR

IS

LC

OV

NC

HH

MF

RA

LB

YS

AY

YT

GG

WM

UN

INL

YZ

TW

SE

YQ

DL

AN

YS

MY

XY

SU

NY

VN

FA

IL

UL

OT

ZY

LT

R.m

.s. sc

atte

r (m

)

0.00

0.05

0.10

BE

LH

OB

QU

IB

LO

FO

RB

LB

KP

PG

UA

JSJ

ME

XL

IHM

ZT

EY

WP

BI

CR

PM

AF

TU

SJA

NN

KX

AB

QT

AT

GS

OB

NA

OA

KW

AL

DE

NM

AD

BR

IS

LC

OV

NC

HH

MF

RA

LB

YS

AY

YT

GG

WM

UN

INL

YZ

TW

SE

YQ

DL

AN

YS

MY

XY

SU

NY

VN

FA

IL

UL

OT

ZY

LT

Bia

s (m

)

-0.30

-0.15

0.00

0.15

0.30NATOSTIWAASSA

NATOSTIWAASSA

Figure 7 − Bias (top plot) and r.m.s. scatter (bottom plot) for total zenith delay models, for 50 radiosonde stations.

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Callahan, P.S. (1973). “Prediction of tropospheric wet-component range error from surface measurements.”JPL Technical Report 32-1526, Vol. XVIII, pp. 41-46, Jet Propulsion Laboratory, Pasadena, Calif.

Chao, C.C. (1973). “A new method to predict wetzenith range correction from surface measurements.”JPL Technical Report 32-1526, Vol. XIV, pp. 33-41,Jet Propulsion Laboratory, Pasadena, Calif.

Collins, J.P., R.B. Langley, and L. LaMance (1996).“Limiting factors in tropospheric propagation delayerror modelling for GPS airborne navigation.”Proceedings of The Institute of Navigation 52nd

Annual Meeting, Cambridge, Mass., 19-21 June, pp.519-528.

Davis, J.L., T.A. Herring, I.I. Shapiro, A.E.E. Rogers,and G. Elgered (1985). “Geodesy by radiointerferometry: Effects of atmospheric modelingerrors on estimates of baseline length.” RadioScience, Vol. 20, No. 6, pp. 1593-1607.

DeCleene, B. (1995). Personal communication, FederalAviation Administration, May.

ESSA/NASA/USAF (Environmental Science ServicesAdministration, National Aeronautics and SpaceAdministration, and United States Air Force) (1966).U.S. Standard Atmosphere Supplements, 1966. U.S.Government Printing Office, Washington, D.C.

Hopfield, H.S. (1969). “Two-quartic troposphericrefractivity profile for correcting satellite data.”Journal of Geophysical Research, Vol. 74, No. 18,pp. 4487-4499.

Hopfield, H.S. (1972). “Tropospheric refraction effectson satellite range measurements.” APL TechnicalDigest, Vol. 11, No. 4, pp. 11-19.

Ifadis, I. (1986). “The atmospheric delay of radiowaves: Modeling the elevation dependence on aglobal scale.” Technical Report No. 38L, School ofElectrical and Computer Engineering, ChalmersUniversity of Technology, Göteborg, Sweden.

International Organization for Standardization (ISO)(1982). Reference Atmospheres for Aerospace Use.ISO International Standard 5878-1982, TechnicalCommittee ISO/TC 20, International Organization forStandardization, Geneva, Switzerland.

International Organization for Standardization (ISO)(1983). Reference Atmospheres for Aerospace Use,Addendum 2: Air Humidity in the NorthernHemisphere. ISO International Standard 5878-1982/Addendum 2-1983, Technical CommitteeISO/TC 20, International Organization forStandardization, Geneva, Switzerland.

Janes, H.W., R.B. Langley, and S.P. Newby (1991).“Analysis of tropospheric delay prediction models:Comparisons with ray tracing and implications forGPS relative positioning.” Bulletin Géodésique, Vol.65, No. 3, pp. 151-161.

Lewandowski, W., G. Petit, and C. Thomas (1992).“GPS standardization for the needs of time transfer.”Proceedings of the 6th European Frequency andTime Forum, Noordwijk, The Netherlands, 17-19March, ESA SP-340, pp. 243-248.

Mendes, V.B. (1998). “Modeling the neutral-atmosphere propagation delay in radiometric spacetechniques.” Ph.D. dissertation (in preparation),Department of Geodesy and Geomatics Engineering,University of New Brunswick, Fredericton, NewBrunswick.

Mendes, V.B. and R.B. Langley (1994). “Acomprehensive analysis of mapping functions used inmodeling tropospheric propagation delay in spacegeodetic data.” Proceedings of KIS94, InternationalSymposium on Kinematic Systems in Geodesy,Geomatics and Navigation, Banff, Canada, August30 - September 2, The University of Calgary,Calgary, Canada, pp. 87-98.

Mendes, V.B., P. Collins, and R.B. Langley (1995).“The effect of tropospheric propagation delay errorsin airborne GPS precision positioning.” Proceedingsof ION GPS-95, the 8th International TechnicalMeeting of the Satellite Division of The Institute ofNavigation, Palm Springs, Calif., 12-15 September,pp. 1681-1689.

NATO (North Atlantic Treaty Organization) (1993).Standardization Agreement (STANAG) Doc. 4294EL (Edition 1), Appendix 6 to Annex A, pp. A-6-34− A-6-37. North Atlantic Treaty Organization,Brussels.

Niell, A.E. (1996). “Global mapping functions for theatmosphere delay at radio wavelengths.” Journal ofGeophysical Research, Vol. 101, No. B2, pp. 3227-3246.

Owens, J.C. (1967). “Optical refractive index of air:Dependence on pressure, temperature andcomposition.” Applied Optics, Vol. 6, No. 1, pp. 51-59.

Saastamoinen, J. (1973). “Contributions to the theoryof atmospheric refraction.” In three parts. BulletinGéodésique, No. 105, pp. 279-298; No. 106, pp. 383-397; No. 107, pp. 13-34.

Shi, J. and M.E. Cannon (1995). “Critical error effectsand analysis in carrier phase-based airborne GPSpositioning over large areas.” Bulletin Géodésique,Vol. 69, No. 4, pp. 261-273.

Smith, W.L. (1966). “Note on the relationship betweentotal precipitable water and surface dew point.”Journal of Applied Meteorology, Vol. 5, pp. 726-727.

Thayer, G.D. (1974). “An improved equation for theradio refractive index of air.” Radio Science, Vol. 9,No. 10, pp. 803-807.

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Appendix I − Codes, references, and input parameters for the analyzed zenith delay models (e − water vapor pressure;T − temperature; P − total pressure; α − temperature lapse rate; doy − day-of-year; ϕ − station latitude; H − heightabove sea level; g − surface gravity; U − relative humidity; λ, ν, and γ − empirical coefficients).

Code Reference e T P α doy ϕ H other

BBh Baby et al. [1988] 9 9 9 9 9 g

HOh Hopfield [1972] 9 9

SAh Saastamoinen [1973] 9 9 9

ANw Askne and Nordius [1987] 9 9 9 9 9 λ

BB1w Baby et al. [1988] 9 9 9 U

BB2w Baby et al. [1988] 9 9 U, ν, γ

B70w Berman [1976] 9 9 9

B74w Berman [1976] 9 9

BTMw Berman [1976] 9 9

CAw Callahan [1973] 9 9

CHw Chao [1973] 9 9 9

HOw Hopfield [1972] 9 9

IFw Ifadis [1986] 9 9 9

SAw Saastamoinen [1973] 9 9

UNB3w Collins et al. [1996] 9 9 9

98aw This paper 9

98bw This paper 9 9 9

AL Altshuler and Kalaghan [1974] 9 9 9

NATO NATO [1993] 9

SA Saastamoinen [1973] 9 9 9 9 9

STI Lewandowski et al. [1992] 9

WAAS DeCleene [1995] 9 9 9

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Appendix II − Approximate location of radiosonde stations.

STATION CODE ϕ ( °N) λ ( °E) H(m)Bellingshausen, BAT BEL -62.20 -58.93 46Hobart, Tasmania, Australia HOB -42.83 147.50 28Quintero, Chile QUI -32.78 -71.52 8Bloemfontein, South Africa BLO -29.10 26.30 1359Fortaleza, Brazil FOR -3.72 -38.55 19

Balboa, Panama BLB 8.98 -79.60 66

Trinidad, Trinidad and Tobago KPP 10.58 -61.35 12Guam, Mariana Islands, USA GUA 13.55 144.83 111San Juan, Puerto Rico JSJ 18.43 -66.00 3Mexico City, Mexico MEX 19.43 -99.07 2234Lihue, HI, USA LIH 21.98 -159.35 36Mazatlan Sinaloa, Mexico MZT 23.18 -106.42 4Key West, FL, USA EYW 24.55 -81.75 3West Palm Beach, FL, USA PBI 26.68 -80.12 7Corpus Christi, TX, USA CRP 29.77 -97.50 14Midland, TX, USA MAF 31.93 -102.20 873Tuscon, AZ, USA TUS 32.12 -110.93 788Jackson, MS, USA JAN 32.32 -90.07 91Miramar, CA, USA NKX 32.87 -117.15 147Albuquerque, NM, USA ABQ 35.05 -106.62 1619Tateno, Japan TAT 36.05 140.13 27Greensboro, NC GSO 36.08 -79.95 277Nashville, TN, USA BNA 36.25 -86.57 180Oakland, CA, USA OAK 37.75 -122.22 6Wallops Island, VA, USA WAL 37.93 -75.48 13Denver, CO, USA DEN 39.77 -104.88 1611Madrid, Spain MAD 40.50 -3.58 633Brindisi, Italy BRI 40.65 17.95 15Salt Lake City, UT, USA SLC 40.77 -111.97 1288Omaha, NE, USA OVN 41.37 -96.02 400Chatham, MA, USA CHH 41.67 -69.97 16Medford, OR, USA MFR 42.37 -122.87 397Albany, NY, USA ALB 42.75 -73.80 85Sable Island, NS, Canada YSA 43.93 -60.02 4St.John’s, NF, Canada YYT 47.67 -52.75 140Glasgow, MT, USA GGW 48.22 -106.62 696Munique, Germany MUN 48.25 11.58 484International Falls, MN, USA INL 48.57 -93.38 359Port Hardy, BC, Canada YZT 50.68 -127.37 17Edmonton, AB, Canada WSE 53.55 -114.10 766The Pas, MB, Canada YQD 53.97 -101.10 273Landvetter, Sweden LAN 57.67 12.30 155Ft. Smith, NWT, Canada YSM 60.03 -111.95 203Whitehorse, YK, Canada YXY 60.72 -135.07 704Sundsvall, Sweden SUN 62.53 17.45 6Iqaluit, NWT, Canada YVN 63.75 -68.55 21Fairbanks, AK, USA FAI 64.82 -147.87 135Lulea, Sweden LUL 65.55 22.13 34Kotzebue, AK, USA OTZ 66.87 -162.63 5Alert, NWT, Canada YLT 82.50 -62.33 66