A Comprehensive Analysis of Mapping Functions Used in ModelingTropospheric Propagation Delay in Space Geodetic Data1

V.B. Mendes and R.B. LangleyGeodetic Research Laboratory

Department of Geodesy and Geomatics EngineeringUniversity of New BrunswickFredericton, N.B. E3B 5A3

e-mail: lang@unb.ca

1Paper presented at KIS94, International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Banff,Canada, August 30 – September 2, 1994.

BIOGRAPHIESABSTRACT

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. In 1990, he completed the SpecializationCourse in Hydrography at the Hydrographic Institute,Lisbon. In 1991, he enrolled as a Ph. D. student at theUniversity of New Brunswick. He is involved in theapplication of the Global Positioning System to themonitoring of crustal deformation.

The principal limiting error source in the GlobalPositioning System and other modern space geodesytechniques, such as very long baseline interferometry, isthe mismodeling of the delay experienced by radio wavesin propagating through the electrically neutralatmosphere, usually referred to as the tropospheric delay.This propagation delay is generally split into twocomponents, called hydrostatic (or dry) and wet, each ofwhich can be described as a product of the delay at thezenith and a mapping function, which models theelevation dependence of the propagation delay.

Richard Langley is a professor in the Department ofGeodesy and Geomatics Engineering at the University ofNew Brunswick, where he has been teaching since 1981.He has a B.Sc. in applied physics from the University ofWaterloo and a Ph.D. in experimental space science fromYork University, Toronto. After obtaining his Ph.D., Dr.Langley spent two years with the Department of Earthand Planetary Sciences of the Massachusetts Institute ofTechnology where he carried out research involvinglunar laser ranging and very long baselineinterferometry.

In the last couple of decades, a number of mappingfunctions have been developed for use in the analysis ofspace geodetic data. Using ray tracing through anextensive radiosonde data set covering different climaticregions as "ground truth", an assessment of accuracy ofmost of these mapping functions, including thosedeveloped by Saastamoinen, Lanyi, Davis (CfA-2.2),Santerre, Ifadis, Baby, Herring (MTT), and Niell (NMF),has been performed. The ray tracing was performed fordifferent elevation angles, starting at 3°.

Virtually all of the tested mapping functions provide sub-centimeter accuracy for elevation angles above 15°.However, stochastic techniques currently used to modelthe tropospheric zenith delay require low elevationobservations in order to reduce the correlation betweenthe estimates of the zenith tropospheric delay and thestation height. Based on this analysis, and for elevationangles below 10°, only a select few of the mappingfunctions were found to adequately meet therequirements imposed by the space geodetic techniques.

Dr. Langley has worked extensively with the GlobalPositioning System. He is a co-author of the best-sellingGuide to GPS Positioning published by Canadian GPSAssociates and is a columnist for GPS World magazine.He has helped develop and present a number of seminarcourses on GPS for both Canadian GPS Associates andthe American-based Navtech Seminars Inc. Dr. Langleyhas consulted extensively in the field of GPS with privatecompanies and government agencies both in Canada andabroad.

INTRODUCTION refractivity constants are the ones provided by Smith andWeintraub [1953] and Thayer [1974] (see Table 1).

The electromagnetic signals used by modern spacegeodetic techniques propagate through part of the earth'satmosphere – specifically through an ionized layer (theionosphere) and a layer that is electrically neutral,composed primarily of the troposphere and stratosphere,referred to as the neutral atmosphere. Unlike the ionizedpart of the atmosphere, the neutral atmosphere isessentially a non-dispersive medium at radio frequencies(except for the anomalous dispersion of the water vaporand oxygen spectral lines), i.e., the effects on phase andgroup delay are equivalent and the availability of morethan one transmitted frequency is of no advantage inremoving the tropospheric effect. Since the troposphereaccounts for most of the neutral atmosphere mass andcontains practically all the water vapor, the termtropospheric delay is often used to designate the globaleffect of the neutral atmosphere. The effect of the neutralatmosphere is a major residual error source in modernspace geodesy techniques, such as the Global PositioningSystem (GPS), very long baseline interferometry (VLBI),Doppler Orbitography and Radiopositioning Integratedby Satellite (DORIS), satellite altimetry (as featured onthe TOPEX/Poseidon and SEASAT satellites), andsatellite laser ranging (SLR).

Smith and Weintraub Thayer[1953] [1974]

K1 77.61±0.01 77.60±0.014K2 72±9 64.8±0.08K3 (3.75±0.03)⋅105 (3.776±0.004)⋅105

Table 1 - Experimentally determined values for therefractivity constants (K1 and K2 are in K⋅mbar-1, K3 isin K2⋅mbar-1).

The tropospheric delay contribution dtrop to a radiosignal propagating from a satellite to the earth's surfaceis given in first approximation by [Langley, 1992]:

d n r r dr dtropr

r

geos

a= −z +( ) csc ( )1 θ , (3)

where dgeo is the geometric delay that accounts for thedifference between the refracted and rectilinear ray paths(ray bending), given by

The neutral atmosphere affects the propagation ofmicrowaves, causing a propagation delay and, to a lesserextent, bending of the ray path. These effects depend onthe real-valued refractive index, n, along the signal raypath, more conveniently expressed by another quantity,the refractivity, N:

d r dr r drgeor

r

r

r

s

a

s

a= z − z

L

NMM

O

QPP

csc ( ) csc ( )θ ε ,

and where r is the geocentric radius, θ is the refracted(apparent) satellite elevation angle, ε is the non-refracted(geometric or true) satellite elevation angle, rs is thegeocentric radius of the earth's surface, and ra is thegeocentric radius of the top of the neutral atmosphere.

N n= − ⋅1 106a f , (1)

which can be expressed, in general, as [Thayer, 1974]

This equation is valid for a spherically symmetricatmosphere, for which n varies simply as a function ofthe geocentric radius. The first integral in equation (3)represents the difference between the electromagneticand geometric lengths of the refracted transmission path.For a signal coming from the zenith direction, thegeometric delay is zero; hence, for a sphericallysymmetric atmosphere equation (3) becomes at thezenith:

N KT

Z Ke

TK

e

TZd

d w= FHG

IKJ

+ FHG

IKJ

+ FHG

IKJ

L

NM

O

QP− −

11

2 3 21P , (2)

where Pd is the partial pressure of the dry gases in theatmosphere, e is the partial pressure of the water vapor, Tis the absolute temperature, Zd is the compressibilityfactor for dry air, Zw is the compressibility factor forwater vapor, and Ki are constants empiricallydetermined.

d n r dr N drtropz

r

r

r

r

s

a

s

a= −z = z

−( ) 1 10 6 . (4)The compressibility factors are corrections to account forthe departure of the air behavior from that of an ideal gasand depend on the partial pressure due to dry gases andtemperature [Owens, 1967]. The most often used sets of

The delay just defined is the tropospheric zenith delay.The integration of the total refractivity expressed by

2

equation (2) depends on the mixing ratio of moist air. Asshown in Davis et al. [1985], it is possible to derive anew refractivity definition for which the first term isdependent on the total mass density ρ only. Followingthis formalism, the total refractivity can be expressed asthe sum of a hydrostatic component (versus drycomponent using the formalism expressed in equation(2)) and a wet component:

The mapping functions were tested for the standardformulations specified by the authors. Site-optimizedmapping functions, such as those derived by Ifadis[1986], and additional functions by Chao [Estefan,1994], for example, are not included in our analysis. Thelatter represent a significant improvement over theoriginal function tested here, but remain unpublished.

The original model developed by Hopfield [1969] and amodel developed by Rahnemoon [1988] are not includedin this analysis. The first is numerically stable only ifcomputed in quadruple precision; using ray tracingthrough standard profiles, we found [Mendes andLangley, 1994] that its accuracy is very similar to thatprovided by the Yionoulis algorithm, as expected. Thesecond is a numerical-integration-based model, for whichan explicit product of a zenith delay and a mappingfunction can be formed only artificially.

N K R Ke

TK

e

TZd w= + F

HIK + F

HGIKJ

L

NM

O

QP −

1 2 3 21ρ ' , (5)

where Rd is the specific gas constant for dry air, and K2'

= (17±10) K⋅mbar-1 [Davis et al., 1985].

The zenith delay can be related to the delay that thesignal would experience at other elevation angles throughthe use of mapping functions. If the mapping functionsare determined separately for the hydrostatic and the wetcomponent, the tropospheric delay can be expressed as:

CODE REFERENCEBB Baby et al. [1988]

d d m d mtrop hz

h wz

w= ⋅ + ⋅ε εa f a f (6)BL Black [1978]BE Black and Eisner [1984]

where dhz is the zenith delay due to dry gases, dw

z is thezenith delay due to water vapor, mh is the hydrostaticcomponent mapping function, mw is the wet componentmapping function and ε, as above, is the non-refractedelevation angle at the ground station (some mappingfunctions use the refracted angle, θ).

CH Chao [1972]DA Davis et al. [1985]GG Goad and Goodman [1974]HE Herring [1992]HM Moffett [1973]IF Ifadis [1986]LA Lanyi [1984]

A recent advance in estimation techniques (especially inVLBI) is the inclusion of observations at low elevationangles, which reduce the correlation between theestimates of the zenith atmospheric delay and the stationheights [e.g. Rogers et al., 1993]. Lichten [1990] alsofound that GPS baseline repeatability improves when lowelevation GPS data is included in the estimation process.There is some irony here: errors in the mappingfunctions, which increase at low elevation angles, caninduce systematic errors in the estimation of thetropospheric delay, which will introduce errors in thevertical component of the estimated positions; on theother hand, the inclusion of low elevation observationswill likely improve the precision of the baseline vectorestimates [e.g. Davis et al., 1991; MacMillan and Ma,1994]. This constant need for better mapping functionsin the analysis of space geodetic data was the motivationfor the assessment of fifteen mapping functions throughcomparison with ray tracing through radiosonde data.The mapping functions assessed in this study and thecorresponding codes used in the tables of results arelisted in Table 2.

MM Marini and Murray [1973]NI Niell [1993a,1993b,1994]SA Saastamoinen [1973]ST Santerre [1987]YI Yionoulis [1970]

Table 2 - List of the mapping functions. (Note: BL, BE,GG, HM, ST, and YI are based on the Hopfield [1969]model; CH, DA, HE, IF, MM, and NI are based on theMarini [1972] continued fraction form.)

Typographical errors in some publications (e.g. Ifadis[1986], Lanyi [1984], Baby et al. [1988]) were corrected[Ifadis, 1994; Cazenave, 1994]. Whenever required, weused nominal values for the temperature lapse rate (6.5K/km) and tropopause height (11 231 m – a valuesuggested in Davis et al. [1985]). For the case of theBaby et al. (hereafter simply called Baby) andSaastamoinen functions, the apparent elevation anglerather than the geometric elevation angle was used, asrequired by the functions.

3

RAY TRACING mapping-function errors only, a different approach thanthat used by Janes et al. [1991].

To compute the tropospheric delay components to beused as benchmark values, radiosonde data from ninestations (see Table 3) representing different climaticregions was used. The data pertains to the year 1992,each station having typically two balloon launches perday, at 11 h and 23 h UT (U.S. controlled stations) and 0h and 12 h UT (Canadian stations), and consist of heightprofiles of pressure, temperature and relative humidity.Radiosonde profiles with obvious errors (e.g. no surfacedata recorded) were discarded.

HYD (m) WET (m)STATION mean rms mean rmsSan Juan 2.316 0.005 0.264 0.045

Guam 2.281 0.009 0.274 0.062Nashville 2.270 0.012 0.151 0.081Denver 1.908 0.012 0.073 0.040Oakland 2.313 0.010 0.115 0.035St. John's 2.268 0.026 0.092 0.056

Whitehorse 2.108 0.021 0.060 0.031

For each ray trace, the hydrostatic and the wetcomponents of the tropospheric delay were computedseparately using the refractivity constants determined byThayer [1974] and the hydrostatic/wet formalismexpressed in Davis et al. [1985]. The geometric delay(ray bending term) was added to the hydrostaticcomponent. The mean and the root-mean-square (rms)about the mean of the NP values of the hydrostatic andwet components of the zenith delay computed from theray traces for each station are shown in Table 4.

Kotzebue 2.299 0.025 0.056 0.042Alert 2.282 0.022 0.032 0.023

Table 4 - Statistical summary of the ray traces for thehydrostatic (HYD) and wet components of the zenithdelay for the different radiosonde stations.

For the total delay, virtually all of the tested mappingfunctions have discrepancies with respect to the ray-tracing results of less than 5 mm for elevation anglesabove 30°, but only the Baby, Herring, Ifadis, Lanyi, andNiell mapping functions showed submillimeter accuracy.The lack of correction values for elevation angles at 30°and above in the Saastamoinen mapping functionexplains a comparatively better performance of thefunction at 15° for most of the sites.

STATION ϕϕ (°°N) λλ (°°W) H (m) NPSan Juan 18.43 66.10 3 675

Guam 13.55 215.17 111 736Nashville 36.12 86.68 180 745Denver 39.75 108.53 1611 753

For elevation angles above 10°, the performance of themapping functions can be classified into three majorgroups. The least satisfactory performance is shared bythe Hopfield-based functions and the Marini-Murraymapping function. For this first group, the great part ofthe discrepancies is due to the disregard of the geometricdelay inherent in the definition of the tropospheric delay.The mapping function developed by Santerre [1987]represents a substantial improvement over the rest of thegroup. The best performances are achieved by theremaining functions based on the Marini [1972]continued fraction form, and by the Baby, Lanyi, andSaastamoinen functions. Both the Lanyi and Davismapping function differences with respect to ray tracingindicate some seasonal and/or latitude dependence,which might be caused by the use of nominal values forthe tropopause height and temperature lapse rate. Asthose parameters are generally not known exactly, theprocedure of using nominal values is likely the one thatwill be used in the implementations of these mappingfunctions in software for most space geodetic dataanalyses.

Oakland 37.73 122.20 6 740St. John's 47.62 52.75 140 713

Whitehorse 60.72 135.07 704 719Kotzebue 66.87 162.63 5 687

Alert 82.50 62.33 66 720

Table 3 - Approximate locations of the radiosonde sitesand the corresponding number of profiles (NP) used. H isthe height of the station above the geoid.

RESULTS AND CONCLUSIONS

The results of our assessment are summarized in Tables 5to 9. In Tables 5 to 8, the mean values of the differencesbetween the delays computed using the mappingfunctions and the ray-trace results corresponding toelevation angles of 30°, 15°, 10° and 3° (and separatelyfor the hydrostatic, wet, and total delay) are listed; therms differences for the total delay only, and for the sameelevation angles, are listed in Table 9.

Although some of the mapping functions were designedto model observations above a certain elevation angle,results for lower elevation angles are presented forillustrative purposes. It should be pointed out that thevalues tabulated represent the differences due to

Among the group of models performing the best, theprecision of the Niell, Herring, and Ifadis mapping

4

functions stands out even at high elevation angles and itis quite remarkable at very low elevation angles (lessthan about 10°), showing biases with respect to the"benchmark" ray-trace values which are about one to twoorders of magnitude smaller than those obtained usingother functions. The Niell and Ifadis functions havesmaller biases than Herring's at low elevation angles, butIfadis' functions show less scatter than Niell's. The worstperformance of the Niell mapping functions, both interms of bias and long-term scatter, happens for highlatitudes. The phenomena of temperature inversionsaffect some of the models and is responsible for the largeshort-term scatter shown by those mapping functionsusing temperature as a parameter. In these cases, thesurface temperature value driving the models is notrepresentative of the conditions aloft, resulting in abiased determination of the delay (see Figure 1).Although some of the functions were designed to be usedfor elevation angles above 10°, only the Baby andSaastamoinen functions break down very rapidly belowthis limit, as illustrated in Table 8.

waves from surface meteorological measurements."Radio Science, Vol. 23, No. 6, pp. 1023-1038.

Black, H.D. (1978). "An easily implemented algorithmfor the tropospheric range correction." Journal ofGeophysical Research, Vol. 38, No. B4, pp. 1825-1828.

Black, H.D and A. Eisner (1984). "Correcting satelliteDoppler data for tropospheric effects." Journal ofGeophysical Research, Vol. 89, No. D2, pp. 2616-2626.

Cazenave, A. (1994). Personal communication. Groupede Recherche de Géodésie Spatiale, Toulouse,France.

Chao, C.C. (1972). "A model for tropospheric calibrationfrom daily surface and radiosonde balloonmeasurements." JPL Technical Memorandum 391-350, Jet Propulsion Laboratory, Pasadena, CA.

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.

Based on our analysis, we conclude that a large numberof mapping functions can provide satisfactory resultswhen used for elevation angles above 15°. For high-precision applications, it is recommended that themapping functions derived by Lanyi, Herring, Ifadis orNiell be used. The last three functions are more accuratethan the first at lower elevation angles and are moreeffective in terms of ease of implementation andcomputation speed. Neverthless, and according to ouranalysis, errors can still be induced by these fourmapping functions at low elevation angles (of the orderof tens of centimeters at 3°).

Davis, J.L., T.A. Herring, and I.I. Shapiro (1991)."Effects of atmospheric modeling errors ondeterminations of baseline vectors from very longbaseline interferometry." Journal of GeophysicalResearch, Vol. 96, No. B1, pp. 643-650.

Estefan, J. (1994). Personal communication. JetPropulsion Laboratory, Pasadena, CA.

Goad, C.C. and L. Goodman (1974). "A modifiedHopfield tropospheric refraction correction model."Paper presented at the AGU Annual Fall Meeting,San Francisco, CA, Dec. 12-17 (abstract: EOS, Vol.55, p. 1106, 1974).

Herring, T.A. (1992). "Modeling atmospheric delays inthe analysis of space geodetic data." Proceedings ofthe Symposium on Refraction of TransatmosphericSignals in Geodesy, Eds. J.C. De Munck andT.A.Th. Spoelstra, Netherlands GeodeticCommission, Publications on Geodesy, No. 36, pp.157-164.

ACKNOWLEDGMENTS

We would like to express our appreciation to Arthur Niellfor providing the source code for the NMF (Niell)mapping functions and the ray-trace software (developedby James Davis, Thomas Herring, and Arthur Niell) andto Thomas Herring for the source code of the MTT(Herring) mapping functions. We also thank AnneCazenave for helping to clarify the formulation in theBaby mapping function.

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

Ifadis, I.I. (1986). "The atmospheric delay of radiowaves: modeling the elevation dependence on aglobal scale." Technical Report 38L, ChalmersUniversity of Technology, Göteborg, Sweden.

The support of the Programa Ciência/Junta Nacional deInvestigação Científica e Tecnológica, Portugal, and theUniversity of New Brunswick's University Research Fundis gratefully acknowledged. Ifadis, I.I. (1994). Personal communication. Aristotle

University of Thessaloniki, Department of CivilEngineering, Division of Geotechnical Engineering,Section of Geodesy, Thessaloniki, Greece.

REFERENCES

Janes, H.W., R.B. Langley, and S.P. Newby (1991)."Analysis of tropospheric delay prediction models:

Baby, H.B., P. Golé, and J. Lavergnat (1988). "A modelfor the tropospheric excess path length of radio

5

comparisons with ray-tracing and implications forGPS relative positioning." Bulletin Géodésique, Vol.65, pp. 151-161.

Owens, J.S. (1967). "Optical refractive index of air:dependence on pressure, temperature, andcomposition." Applied Optics, Vol. 6, No. 1, pp. 51-59.Lanyi, G. (1984). "Tropospheric delay effects in radio

interferometry." Telecommunications and DataAcquisition Progress, JPL Technical Report 42-78,Jet Propulsion Laboratory, Pasadena, CA, pp. 152-159.

Rahnemoon, M. (1988). Ein neues Korrekturmodell fürMikrowellen - Entfernungsmessungen zu Satelliten.Deutsche Geodätische Kommission, Reihe C, HeftNr. 335, München, Germany.

Langley, R.B. (1992). "The effect of the ionosphere andtroposphere on satellite positioning systems." Paperpresented at the Symposium on Refraction ofTransatmospheric Signals in Geodesy, The Hauge,The Netherlands, May 19-22.

Rogers, A.E.E., R.J. Cappallo, B.E. Corey, H.F.Hinteregger, A.E. Niell, R.B. Phillips, D.L. Smythe,A.R. Whitney, T.A. Herring, J.M. Bosworth, T.A.Clark, C. Ma, J.W. Ryan, J.L. Davis, I.I. Shapiro, G.Elgered, K. Jaldehag, J.M. Johansson, B.O.Rönnäng, W.E. Carter, J.R. Ray, D.S. Robertson,T.M. Eubanks, K.A. Kingham, R.C. Walker, W.E.Himwich, C.E. Kuehn, D.S. MacMillan, R.I. Potash,D.B. Shaffer, N.R. Vandenberg, J.C. Webber, R.L.Allshouse, B.R. Schupler, and D. Gordon (1993)."Improvements in the accuracy of geodetic VLBI."In Contributions of Space Geodesy to Geodynamics:Technology, Eds. D.E. Smith and D.L. Turcotte,Geodynamics Series, Vol. 25, American GeophysicalUnion, Washington, D.C. , pp. 47-62.

Lichten, S.M. (1990). "Precise estimation of troposphericpath delays with GPS techniques."Telecommunications and Data Acquisition ProgressReport, JPL Technical Report 42-100, Jet PropulsionLaboratory, Pasadena, California.

Marini, J.W. (1972). "Correction of satellite trackingdata for an arbitrary tropospheric profile." RadioScience, Vol. 7, No. 2, pp. 223-231.

Marini, J.W. and C.W. Murray (1973). "Correction oflaser range tracking data for atmospheric refractionat elevations above 10 degrees." Goddard SpaceFlight Center Report X-591-73-351, NASA GSFC,Greenbelt, MD.

Saastamoinen, J. (1973). "Contributions to the theory ofatmospheric refraction." In three parts. BulletinGéodésique, No. 105, pp. 279-298; No. 106, pp.383-397; No. 107, pp. 13-34.MacMillan, D.S. and C. Ma (1994). "Evaluation of very

long baseline interferometry atmospheric modelingimprovements." Journal of Geophysical Research,Vol. 99, No. B1, pp. 637-651.

Santerre, R. (1987). "Modification to the Goad &Goodman tropospheric refraction model."Unpublished internal report of the Department ofSurveying Engineering, University of NewBrunswick, Fredericton, N.B., Canada.

Mendes, V.B. and R.B. Langley (1994). "Modeling thetropospheric delay from meteorological surfacemeasurements: comparison of models." Presented atthe American Geophysical Union Spring Meeting,MD, 23-27 May 1994 (abstract: EOS, Vol. 75, No.16, Spring Meeting Supplement, p. 105, 1994).

Smith, E.K. and S. Weintraub (1953). "The constants inthe equation of atmospheric refractive index at radiofrequencies." Proceedings of the Institute of RadioEngineers, Vol. 41, No. 8, pp. 1035-1037.

Moffett, J.B. (1973). "Program requirements for two-minute integrated Doppler satellite navigationsolution." Technical Memorandum TG 819-1,Applied Physics Laboratory, The Johns HopkinsUniversity, Laurel, MD.

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

Yionoulis, S.M. (1970). "Algorithm to computetropospheric refraction effects on rangemeasurements." Journal of Geophysical Research,Vol. 75, No. 36, pp. 7636-7637.

Niell, A.E. (1993a). "A new approach for the hydrostaticmapping function." Proceedings of the InternationalWorkshop for Reference Frame Establishment andTechnical Development in Space Geodesy,Communications Research Laboratory, Koganei,Tokyo, Japan, 18-21 January 1993, pp. 61-68.

Niell, A. E. (1993b). "Improved global atmosphericmapping functions for VLBI and GPS." URSI/IAUSymposium on VLBI Technology - Progress andFuture Observational Possibilities, Kyoto, Japan, 6-10 September 1993 (abstract).

Niell, A.E. (1994). Personal communication. HaystackObservatory, Westford, MA.

6

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PP

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TIO

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T S

A N

I M

M L

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HM

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GG

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BE

BL

BB

ST

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2.7

2.2

-0.5

T1.9

1.2

-0.0

--

-0.2

1.7

0.0

1.8

-1.1

-4.4

-1.8

-H

1.1

0.1

-0.0

--

0.0

0.1

0.0

0.1

-0.3

0.1

-0.1

-W

WH

ITE

HO

RS

E

3.0

1.3

-2.9

0.0

1.5

0.2

-0.2

1.8

0.0

1.9

-1.4

-4.3

2.2

1.9

-0.8

T2.5

1.7

--0

.1-

--0

.22.0

0.1

2.4

-0.8

-4.7

-2.4

-H

1.1

0.1

-0.0

--

0.0

0.1

0.0

0.1

-0.2

0.1

-0.1

-W

KO

TZ

EB

UE

3.6

1.8

-3.6

-0.1

1.4

0.6

-0.2

2.1

0.1

2.5

-1.0

-4.6

2.5

2.5

-0.5

T2.8

2.0

--0

.3-

--0

.21.6

0.2

2.7

-0.5

-5.0

-2.7

-H

0.6

0.0

-0.0

--

0.0

0.0

0.0

0.0

-0.1

0.0

-0.1

-W

ALE

RT

3.4

2.0

-3.9

-0.3

1.1

0.9

-0.2

1.6

0.2

2.7

-0.6

-5.0

2.2

2.8

-0.5

T

Tab

le 5

- M

appi

ng fu

nctio

n m

ean

erro

r at

30

degr

ees

elev

atio

n an

gle.

The

val

ues

repr

esen

t the

mea

n di

ffere

nces

bet

wee

n th

e de

lay

com

pute

d us

ing

the

diffe

rent

map

ping

func

tions

and

the

ray-

trac

e re

sults

, for

the

hydr

osta

tic (

H),

wet

(W

) an

d to

tal (

T)

dela

y, in

mill

imet

res.

The

BB

, BE

, LA

, MM

, and

SA

func

tions

do

not h

ave

sepa

rate

hyd

rost

atic

and

wet

com

pone

nts.

MA

PP

ING

FU

NC

TIO

N Y

I S

T S

A N

I M

M L

A IF

HM

HE

GG

DA

CH

BE

BL

BB

ST

AT

ION

13.6

7.8

--0

.1-

--1

.631.3

-0.1

13.3

0.9

-5.2

-13.6

-H

4.2

3.5

-0.2

--

0.2

3.2

0.0

3.8

-8.8

3.4

-3.5

-W

SA

N J

UA

N

17.8

11.4

-1.7

0.1

26.4

-2.3

-1.4

34.5

-0.1

17.1

-7.9

-1.8

22.6

17.1

7.0

T13.0

7.4

-0.2

--

-1.5

31.1

0.0

12.6

0.7

-4.9

-12.9

-H

4.4

3.7

-0.3

--

0.4

3.5

0.2

4.0

-9.1

3.6

-3.7

-W

GU

AM

17.4

11.1

-1.5

0.5

26.8

-2.4

-1.1

34.6

0.2

16.6

-8.4

-1.3

22.4

16.6

8.2

T15.0

9.4

--0

.6-

--1

.128.0

0.2

14.8

2.4

-7.8

-14.9

-H

2.4

1.8

-0.1

--

0.2

1.8

0.1

1.9

-4.9

1.9

-2.0

-W

NA

SH

VIL

LE

17.4

11.2

0.4

-0.5

20.7

0.0

-0.9

29.8

0.3

16.7

-2.5

-5.9

21.8

16.9

-1.8

T10.5

6.4

--0

.2-

--1

.319.8

0.2

10.4

0.9

-10.3

-10.4

-H

0.9

0.7

--0

.2-

-0.0

0.7

0.1

0.7

-2.5

0.7

-0.7

-W

DE

NV

ER

11.4

7.1

2.9

-0.4

15.1

-0.9

-1.3

20.5

0.3

11.1

-1.6

-9.6

15.5

11.1

-6.4

T15.5

9.7

--0

.1-

--0

.929.1

0.2

15.4

2.5

-7.4

-15.4

-H

1.7

1.7

-0.0

--

0.0

1.3

0.0

1.8

-3.8

1.4

-1.4

-W

OA

KLA

ND

17.2

11.4

0.1

-0.1

19.7

0.1

-0.9

30.4

0.2

17.2

-1.3

-6.0

23.6

16.8

0.6

T17.6

11.8

--1

.1-

--0

.825.6

0.8

17.5

4.8

-10.2

-17.5

-H

1.5

0.8

-0.0

--

0.1

1.1

0.1

0.9

-2.8

1.2

-1.2

-W

ST

. JO

HN

'S

19.1

12.6

-1.8

-1.1

15.8

2.8

-0.7

26.7

0.9

18.4

2.0

-9.0

20.7

18.7

-3.7

T14.7

9.8

-0.1

--

-1.8

21.1

0.1

14.8

3.3

-12.2

-14.6

-H

0.9

0.6

--0

.1-

-0.0

0.6

0.1

0.6

-1.9

0.7

-0.7

-W

WH

ITE

HO

RS

E

15.6

10.4

-2.3

0.0

12.2

1.5

-1.8

21.7

0.2

15.4

1.4

-11.5

17.0

15.3

-6.4

T19.9

13.8

--0

.6-

--1

.323.5

0.9

20.0

6.6

-12.8

-19.7

-H

1.1

0.6

-0.0

--

0.0

0.7

0.1

0.6

-1.7

0.7

-0.7

-W

KO

TZ

EB

UE

21.0

14.4

-5.1

-0.7

11.9

4.7

-1.3

24.2

1.0

20.6

4.9

-12.1

19.3

20.4

-4.2

T22.4

16.1

--2

.2-

--1

.420.6

1.8

22.5

9.1

-15.4

-22.2

-H

1.6

0.1

-0.0

--

0.1

0.5

0.1

0.1

-0.8

0.5

-0.5

-W

ALE

RT

24.0

16.2

-8.1

-2.2

8.8

7.3

-1.3

21.2

1.9

22.6

8.3

-14.9

17.1

22.7

-3.5

T

Tab

le 6

- M

appi

ng fu

nctio

n m

ean

erro

r at

15

degr

ees

elev

atio

n an

gle;

oth

erw

ise,

as

for

Tab

le 5

.

MA

PP

ING

FU

NC

TIO

N Y

I S

T S

A N

I M

M L

A IF

HM

HE

GG

DA

CH

BE

BL

BB

ST

AT

ION

44.6

23.6

-0.2

--

-3.3

96.3

0.5

41.0

7.1

10.9

-41.6

-H

13.9

11.4

-0.7

--

0.8

10.7

0.0

12.4

-27.7

11.7

-11.6

-W

SA

N J

UA

N

58.5

35.0

-9.1

0.9

81.6

-7.5

-2.5

107.0

0.5

53.4

-20.6

22.6

62.4

53.2

27.2

T42.5

22.1

-1.3

--

-3.0

95.7

0.7

38.9

6.2

11.6

-39.6

-H

14.9

12.2

-1.0

--

1.4

11.5

0.6

13.2

-28.6

12.5

-12.4

-W

GU

AM

57.4

34.3

-8.3

2.3

83.0

-8.1

-1.6

107.2

1.3

52.1

-22.4

24.1

61.7

52.0

30.7

T48.9

28.3

--1

.4-

--1

.685.8

1.6

45.7

11.3

2.1

-45.6

-H

8.0

5.8

-0.5

--

0.6

6.1

0.4

6.1

-15.3

6.7

-6.6

-W

NA

SH

VIL

LE

56.9

34.1

-1.9

-0.9

63.4

-0.4

-1.0

91.9

2.0

51.8

-4.0

8.8

60.4

52.2

1.6

T34.6

19.3

--0

.1-

--2

.760.2

1.3

31.8

6.0

-10.1

-31.7

-H

3.1

2.2

--0

.5-

-0.2

2.2

0.5

2.2

-7.8

2.5

-2.4

-W

DE

NV

ER

37.7

21.5

6.6

-0.6

46.2

-3.2

-2.5

62.4

1.8

34.0

-1.8

-7.6

42.2

34.1

-12.6

T50.5

29.4

-0.4

--

-1.1

89.4

1.7

47.4

11.7

4.1

-47.2

-H

5.8

5.7

-0.1

--

0.1

4.4

-0.1

5.9

-11.8

4.8

-4.7

-W

OA

KLA

ND

56.3

35.1

-3.0

0.5

60.1

-0.1

-1.0

93.8

1.6

53.3

-0.1

8.9

66.1

51.9

9.2

T57.2

35.9

--2

.7-

--1

.078.2

3.2

54.1

18.6

-5.4

-53.5

-H

4.9

2.8

-0.1

--

0.3

3.7

0.5

3.0

-8.8

4.1

-4.0

-W

ST

. JO

HN

'S

62.1

38.7

-8.7

-2.6

47.9

8.2

-0.7

81.9

3.7

57.1

9.8

-1.3

57.2

57.5

-4.1

T47.9

29.4

-0.8

--

-4.1

64.0

1.0

45.4

13.5

-13.7

-44.4

-H

2.9

1.8

--0

.4-

-0.1

2.1

0.3

1.9

-5.9

2.3

-2.3

-W

WH

ITE

HO

RS

E

50.8

31.2

-9.6

0.4

36.4

4.0

-4.0

66.1

1.3

47.3

7.6

-11.4

46.0

46.7

-12.5

T64.4

42.1

--1

.5-

--2

.571.4

3.9

61.7

24.1

-13.4

-60.4

-H

3.0

1.9

--0

.1-

-0.1

2.3

0.2

1.9

-5.2

2.5

-2.4

-W

KO

TZ

EB

UE

67.4

44.0

-19.6

-1.6

35.3

14.0

-2.4

73.7

4.1

63.6

18.9

-10.9

52.7

62.8

-5.8

T72.4

49.3

--6

.3-

--2

.762.5

6.7

69.7

31.8

-21.7

-68.1

-H

1.9

0.4

-0.1

--

0.2

1.5

0.4

0.4

-2.5

1.6

-1.6

-W

ALE

RT

74.3

49.7

-29.2

-6.2

25.3

22.2

-2.5

64.0

7.1

70.1

29.3

-20.1

45.9

69.7

-4.5

T

Tab

le 7

- M

appi

ng fu

nctio

n m

ean

erro

r at

10

degr

ees

elev

atio

n an

gle;

oth

erw

ise,

as

for

Tab

le 5

.

MA

PP

ING

FU

NC

TIO

N Y

I S

T S

A N

I M

M L

A IF

HM

HE

GG

DA

CH

BE

BL

BB

ST

AT

ION

571

285

-4

--

-15

279

5562

623

792

-554

-H

300

244

-16

--

26

192

7269

-405

281

-288

-W

SA

N J

UA

N

871

529

-6834

20

178

293

11

471

12

831

218

1073

-537

842

13014

T541

265

-19

--

-13

286

6531

609

792

-525

-H

321

262

-22

--

41

209

21

289

-413

301

-308

-W

GU

AM

862

527

-6712

41

222

274

28

495

27

820

196

1093

-534

833

12485

T635

349

--1

3-

-13

147

30

632

622

649

-616

-H

171

122

-13

--

18

109

12

131

-226

160

-163

-W

NA

SH

VIL

LE

806

471

-6911

0-8

6413

31

256

42

763

396

809

-456

779

18835

T439

230

-9

--

-8-5

129

435

457

372

-422

-H

67

47

--1

0-

-5

37

11

49

-120

62

-63

-W

DE

NV

ER

506

277

-5791

-1-1

40

279

-3-1

440

484

337

434

-473

485

21534

T661

368

-18

--

25

184

37

659

643

696

-641

-H

123

124

-1

--

576

-1130

-179

115

-117

-W

OA

KLA

ND

784

492

-7117

19

-178

449

30

260

36

789

464

811

-368

758

18221

T748

447

--3

6-

-21

26

55

748

669

528

-726

-H

104

65

-3

--

866

10

68

-132

97

-98

-W

ST

. JO

HN

'S

852

512

-7197

-33

-353

536

29

92

65

816

537

625

-470

824

19630

T612

351

-13

--

-35

-113

23

617

551

355

-592

-H

60

39

--9

--

135

540

-92

55

-56

-W

WH

ITE

HO

RS

E

672

390

-6828

4-4

46

422

-34

-78

28

657

459

410

-534

648

20623

T853

528

--2

2-

-0

-100

75

859

709

410

-830

-H

62

39

--3

--

239

340

-80

58

-58

-W

KO

TZ

EB

UE

915

567

-7535

-25

-580

629

2-6

178

899

629

468

-536

888

18983

T979

635

--9

4-

-5

-227

131

986

777

278

-956

-H

39

10

-1

--

326

710

-39

36

-36

-W

ALE

RT

1018

645

-7628

-93

-723

744

8-2

01

138

996

738

314

-612

992

17470

T

Tab

le 8

- M

appi

ng fu

nctio

n m

ean

erro

r at

3 d

egre

es e

leva

tion

angl

e; o

ther

wis

e, a

s fo

r T

able

5.

MA

PP

ING

FU

NC

TIO

N Y

I S

T S

A N

I M

M L

A IF

HM

HE

GG

DA

CH

BE

BL

BB

εεS

TA

TIO

N1.8

0.1

0.1

0.1

0.2

0.1

0.1

0.1

0.1

0.1

0.2

0.1

0.1

0.1

0.3

30

1.1

0.8

1.0

0.5

1.7

0.9

0.6

1.0

0.6

0.8

1.7

1.1

0.7

1.1

2.7

15

SA

N J

UA

N3.7

2.4

3.3

1.7

5.6

2.8

1.9

3.4

2.0

2.5

5.3

3.6

2.5

3.4

7.9

10

73

38

72

34

104

44

35

64

36

41

63

77

59

72

1111

31.2

0.1

0.1

0.1

0.2

0.1

0.1

0.1

0.1

0.1

0.3

0.2

0.1

0.2

0.3

30

1.6

0.8

1.1

0.7

2.1

0.8

0.7

1.3

0.8

0.9

1.8

1.4

0.9

1.4

2.7

15

GU

AM

5.2

2.6

3.4

2.4

6.9

2.5

2.3

4.3

2.4

2.7

5.5

4.7

3.2

4.7

8.1

10

109

43

89

48

132

38

49

86

49

47

67

105

78

106

958

31.4

0.2

0.2

0.1

0.5

0.3

0.1

0.3

0.2

0.3

0.6

0.3

0.1

0.3

0.6

30

2.3

2.0

1.6

1.1

4.2

2.7

1.0

2.9

1.3

2.1

4.5

2.9

1.1

2.3

5.0

15

NA

SH

VIL

LE7.2

6.3

5.4

3.7

13.3

8.5

3.3

9.2

4.1

6.7

14.0

9.5

3.5

7.0

14.5

10

112

88

194

63

220

135

56

148

67

97

164

173

85

110

3591

30.7

0.3

0.3

0.1

0.4

0.3

0.1

0.3

0.1

0.3

0.5

0.2

0.2

0.3

0.6

30

2.4

2.5

2.4

0.9

3.0

2.7

0.9

2.2

1.3

2.6

3.4

2.2

1.3

2.4

4.6

15

DE

NV

ER

7.7

7.8

7.7

2.8

9.7

8.2

2.8

7.0

4.0

8.3

10.4

7.1

4.0

7.5

13.2

10

107

112

162

50

162

111

48

113

61

122

115

125

64

106

4358

30.7

0.2

0.2

0.1

0.3

0.2

0.2

0.3

0.2

0.2

0.3

0.3

0.2

0.2

0.5

30

2.1

2.0

2.1

1.1

2.6

2.0

1.6

2.2

1.7

2.1

2.4

2.2

1.9

2.1

4.3

15

OA

KLA

ND

6.7

6.4

6.7

3.5

8.4

6.4

5.1

7.1

5.4

6.6

7.5

7.1

6.1

6.6

12.6

10

106

99

129

60

134

97

81

111

84

103

100

117

98

105

2409

31.0

0.4

0.3

0.2

0.5

0.2

0.1

0.4

0.1

0.4

0.4

0.4

0.2

0.2

0.4

30

2.1

2.9

2.7

1.4

4.5

1.8

1.1

3.2

1.2

3.0

2.9

3.1

1.8

1.7

3.6

15

ST

. JO

HN

'S5.4

8.2

8.5

4.5

14.4

5.8

3.6

10.1

3.8

8.4

9.1

10.1

5.6

5.2

10.5

10

92

83

229

76

236

92

61

159

64

88

104

177

73

90

1991

30.5

0.3

0.3

0.2

0.4

0.4

0.1

0.3

0.2

0.3

0.5

0.3

0.2

0.4

0.3

30

3.3

2.7

2.7

1.6

3.2

3.4

1.1

2.4

1.5

2.9

3.8

2.3

1.5

3.1

2.7

15

WH

ITE

HO

RS

E10.0

8.5

8.7

4.9

10.4

10.2

3.5

7.6

4.7

9.2

11.9

7.5

4.7

9.6

7.8

10

146

118

167

73

168

144

59

118

74

135

135

128

57

145

2562

30.6

0.3

0.4

0.2

0.5

0.4

0.2

0.4

0.2

0.4

0.6

0.4

0.2

0.4

0.3

30

3.8

2.9

3.1

1.7

4.2

3.7

1.3

3.3

1.5

3.2

4.6

3.3

2.1

3.4

2.6

15

KO

TZ

EB

UE

10.9

9.0

9.9

5.3

13.4

11.5

4.0

10.3

4.9

10.1

14.4

10.4

6.3

10.7

7.6

10

159

121

209

79

212

172

65

161

77

146

164

174

74

159

1709

30.3

0.3

0.5

0.4

0.5

0.4

0.2

0.4

0.2

0.3

0.5

0.4

0.4

0.4

0.2

30

4.0

2.4

4.5

3.4

4.3

3.1

1.3

3.7

1.3

2.7

3.7

3.7

3.0

3.1

1.9

15

ALE

RT

9.9

7.5

14.3

10.6

13.4

9.9

4.0

11.5

4.1

8.7

11.6

11.5

9.3

9.7

5.4

10

161

114

188

148

197

155

56

164

65

142

138

171

114

163

2418

3

Tab

le 9

- R

oot-

mea

n-sq

uare

(rm

s) s

catte

rs a

bout

the

mea

n of

the

diffe

renc

es b

etw

een

the

dela

ys c

ompu

ted

usin

g th

e m

appi

ng

func

tions

and

the

ray

trac

e re

sults

for

vario

us e

leva

tion

angl

es, i

n m

illim

etre

s.N

ote:

Onl

y th

e rm

s fo

r th

e to

tal t

ropo

sphe

ric d

elay

is s

how

n.

050

100

150

200

250

300

350

-20020

Differences (mm)

Her

ring

050

100

150

200

250

300

350

-20020

Differences (mm)

Ifadi

s

050

100

150

200

250

300

350

-20020

Differences (mm)

Lany

i

050

100

150

200

250

300

350

-20020

Day

of Y

ear

1992

Differences (mm)

Nie

ll

Figu

re 1

- D

iffe

renc

es b

etw

een

the

dela

y co

mpu

ted

usin

g th

e H

E,

IF,

LA

, an

d N

I m

appi

ng f

unct

ions

and

the

ray

-tra

ce r

esul

ts,

for

the

tota

lde

lay

at 1

0° e

leva

tion

ang

le,

for

Oak

land

. T

his

stat

ion

is a

ffec

ted

by f

requ

ent

tem

pera

ture

inv

ersi

ons,

a p

heno

men

on w

hich

deg

rade

s th

epe

rfor

man

ce o

f th

e m

appi

ng f

unct

ions

usi

ng t

he s

urfa

ce t

empe

ratu

re a

s an

inp

ut p

aram

eter

, suc

h as

the

Lan

yi,

Her

ring

, an

d If

adis

map

ping

func

tion

s. T

he l

atte

r is

the

lea

st a

ffec

ted

of t

hese

thr

ee.

The

ind

epen

denc

e of

the

Nie

ll f

unct

ions

fro

m t

he m

eteo

rolo

gica

l pa

ram

eter

s is

refl

ecte

d in

a s

mal

ler

rms

scat

ter,

for

this

sta

tion

.