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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.
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5
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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.
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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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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.
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Niell, A.E. (1994). Personal communication. HaystackObservatory, Westford, MA.
6
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BB
ST
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1.8
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-1.5
-4.2
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
.