Using a Numerical Weather Model to Improve Geodesy

0.89 MB·English

by  A. Niell

Using a Numerical Weather Model to Improve Geodesy
Arthur Niell
4
0 MIT Haystack Observatory,Westford, MA, USA 01886 ([email protected])
0
2
Leonid Petrov
n
NVI, Inc./NASA Goddard Space Flight Center, Greenbelt, MD, USA ([email protected])
a
J

3
2

]
h Abstract. The use of a Numerical Weather imum elevation above which data should be in-
p Model(NWM) to providein situatmospherein- cluded.
- formation for mapping functions of atmosphere The total atmosphere path delay is expressed
o
e delay has been evaluated using Very Long Base- as the product of the zenith path delay and the
g lineInterferometry(VLBI)dataspanningeleven mapping function, m(e), defined as the ratio of
s. years. Parameters required by the IMF map- the excess path delay at geometric elevation, e,
c ping functions (Niell 2000, 2001) have been cal- i.e., the vacuum elevation, to the path delay in
i
s culated from the NWM of the National Centers the zenith direction. The hydrostatic and wet
y forEnvironmentalPrediction(NCEP)andincor- mapping functions are calculated separately be-
h
poratedinthe CALC/SOLVEVLBI dataanaly- causetheverticaldistributionsofrefractivitydif-
p
sisprogram. ComparedwiththeuseoftheNMF fer(MacMillanandMa,1994;Niell,1996). Many
[
mappingfunctions(Niell1996)theapplicationof geodetic analyses for GPS and VLBI currently
1
IMF for global solutions demonstrates that the use very simple models for mapping functions
v
hydrostatic mapping function, IMFh, provides that are based on site location and time of year
8
1 both significant improvement in baseline length (Niell, 1996;hereafterreferredto asNMF). This
1 repeatabilityandnoticeablereductionintheam- climatological approach is used due to the dif-
1 plitude of the residual harmonic site position ficulty of obtaining in situ meteorological infor-
0
variations at semidiurnal to long-period bands. mationalongthepathtraversedbytheincoming
4
For baseline length repeatability the reduction radio waves.
0
/ in the observed mean square deviations achieves The next best thing would be mapping func-
s
c 80% of the maximum that is expected for the tionsbasedonthe verticaldistributionofrefrac-
i change from NMF to IMF. On the other hand, tivity above the site, assuming azimuthal sym-
s
y the use of the wet mapping function, IMFw, as metry. While measurement of such a profile, for
h implemented using the NCEP data, results in a example by launching radiosondes several times
p slight degradation of baseline length repeatabil- per day, has not been deemed feasible, some
:
v ity, probablydue to the largegridspacingofthe amount of information on the state of the atmo-
i NWM that is used. spherecanbeobtainedfromthemeteorologicnu-
X
merical weather models (NWM), which provide
r
a 1 Introduction estimatesoftheverticaldistributionoftempera-
ture, pressure, and water vapor content at spec-
The accuracy of the estimate of the local verti- ified horizontal grid points over the entire globe
calcoordinatefromGPSandVLBI observations everysixhours. Niell(2000,2001)demonstrated
increases as the minimum elevation cutoff of in- howthis informationcanbe usedto improvethe
cludeddataisreduced,duetoimproveddecorre- mapping functions.
lationofthe estimatesoftropospherepathdelay In this note we show that these mapping
and the site vertical coordinate. From the other functions (IMF), using data from NCEP on a
side, the errors in mapping the zenith tropo- 2.5 by 2.5 grid interpolated to the site loca-
◦ ◦
sphere delay increase rapidly with decreasing el- tion, provide the expected improvementin base-
evationangleandcauseadegradationofthepre- line length repeatability, and that, furthermore,
cisionoftheestimateofsitepositioncoordinates. they provide a noticeable reduction in the am-
Thus,itisimportanttoassesstheoptimummin- plitudes ofresidual harmonic site position varia-
1
tions. The goal of mapping function research is to
While the wet mapping function (IMFw) has find the functional form and dependence on ex-
thepotentialtobe animprovementoverNMFw, ternal information of the parameters a, b, c ...
asdemonstratedusingradiosondeprofiles(Niell, that best matches the actual values of the map-
2000), this is not achieved in the current im- ping function for real atmospheres at arbitrary
plementation using the NCEP NWM in the locations and times.
VLBI analysis package CALC/SOLVE (Ma et In order to provide a useful tool for estimat-
al., 1990). ing or calculating the atmosphere path delay for
In section2 we give a briefdescriptionof IMF geodetic observations, the parameters a, b, and
and in section 3 outline the implementation in c should be calculable in terms of available in-
CALC/SOLVE. The VLBI analysis is described formation. The following procedure was used to
insection4andsomeofthegeodeticimplications determine the relation of the parameters to ex-
are presented in sections 5. Section 6 contains a ternal data.
summaryoftheresultsandsuggestionsforfuture
1. For radiosondeprofilesfroma largenumber
developments.
of sites and spanning at least one year, cal-
2 The Isobaric Mapping Functions (IMF) culate, by raytracing, the hydrostatic and
wet mapping functions at nine elevations
Currently the closest approximation to a ”true” from 90 degrees down to 3 degrees.
mapping function is obtained by using ra-
diosonde data to define the verticalprofile of ra- 2. Calculate by least-squares fit the continued
dio refractivity and, assuming spherical symme- fractionparametersa,b,andcforeachpro-
try about the radiosonde launch site, integrat- file for both hydrostatic and wet mapping
ingalongthepathdeterminedbytherefractivity functions.
to determine the delay through the atmosphere
3. Assumethattheparametersarelinearfunc-
both in the zenith direction and at the elevation
tions of some other parameters, qi, that are
for which the mapping function is to be calcu-
available to characterize the mapping func-
lated. Thedelaysduetothehydrostaticandwet
tion, such as geographic location, time of
components of the atmosphere (Niell 1996) are
year,oracombinationofmeteorologicalpa-
retained independently in the raytrace integra-
rameters. Expandeachoftheparametersa,
tion, and the mapping functions are calculated
b, and c in terms of the qi.
separately.
Marini (1972) showed that, for a spherically 4. Use linear regression of all a values on pa-
symmetric but vertically arbitrary profile of at- rameters qi for all sites and profiles to de-
mosphericrefractivity,themappingfunctioncan termine the regression coefficients that de-
be approximated by a continued fraction of the scribe a in terms of the qi. Repeat for b
form and c; do for both the hydrostatic and wet
parameters.
a
1+
b To determine the parameter dependence, the
1+ hydrostaticandwetmappingfunctionswerecal-
1+c
m(e)= (1) culated from radiosonde profiles for twenty-six
a
sites for a one year period (1992) with usually
sine+
b two profiles per day, at 00 and 12 UT. Based
sine+
sine+c ontheexpectationthatthehydrostaticmapping
function would be strongly correlated with the
Thenumeratorisincludedtonormalizethefrac- thickness of the atmosphere, a possible empiri-
tion at an elevation angle of 90 , and the num- cal relation to the geopotential heights of con-
◦
ber of terms (only three are shown) should be stant pressure levels (isobaric surfaces) for each
determined by the desired accuracy of the fit. of the parameters ah, bh, and ch was investi-
Niell (1996) found that three terms is sufficient gated. The 200 hPa level gave the best agree-
tokeeptheerrortolessthan1mmforelevations ment (Niell, 2000). The relation was found to
down to 3 if the delays are calculated at eight require a dependence on the cosine of twice the
◦
elevations in addition to the zenith direction. latitude. Thecoefficientsrelatingtheparameters
2
ah, bh, and ch to the geopotential height and tection (NCEP)thatare output everysix hours.
the latitude were obtained by linear regression We used the Reanalysis model (Kalnay et al.,
onthe200hPaheightsfromtheradiosondedata 1996) in our study. It appears on their Web site
and cos(2*latitude). Since the 200 hPa isobar with a time lag of 3 to 7 days. All Reanalysis
has little sensitivity to the topography below, a model products have been analyzed with a con-
correctionforheightofasiteabovesealevelwas sistent, though not necessarily the most recent,
calculatedbylinearregressionofthe site heights model. The current model was adopted in 1996.
and the residual mapping function error using The files that are downloadedcontaingeopoten-
the derived coefficients. This was found to be tial height and temperature at eighteenpressure
consistentwiththatusedforNMF,andtheNMF levels, specific humidity at eight pressure levels,
height correction (Niell 1996) was adopted. and the surface height for each grid point. The
Althoughtheconceptofatmosphericthickness values of geopotential height at 200 hPa are ex-
failedtoproduceareasonablerelationforthewet tracted,andthevaluesofρarecalculatedateach
mapping function, we found that there exists a gridpoint. FortheVLBIapplicationthe param-
linearregressionbetweentheparametersaw,bw, eters are subsequently interpolated to the hori-
and cw and a ”wet pseudo-mapping function”. zontal position of each VLBI site at each NCEP
Thewetpseudo-mappingfunction,ρ,isgivenby epoch (0, 6, 12, 18 UT). During interpolation,
r(3◦.3) the unit vector of the normal to the surface of
ρ= (2)
geopotentialheightat 200 hPais computed. We
r(90 )
◦
assumethatthehydrostaticatmospherepathde-
where 1+ h lay is azimuthally symmetric withrespect to the
ev(h) R normaltothegeopotentialheight,ratherthanlo-
k3 2 ⊕ dh
T (h) 2 cal zenith. Thus, we imply that the atmosphere
Z h
1+ −cos2ǫ is tilted with respect a local observer. Typical
s R
r(ǫ)= (cid:18) ⊕(cid:19) values of the tilt are 0.1 to 1.0 mrad.
ev(h)
k3 2 dh
T (h)
Z 4 VLBI analysis and results
(3)
ρistheratiooftheintegralofthewetrefractivity
The troposphere path delay at each station is
along an elevation of 3.3 to the integral in the
◦ modeledasalinearsplinewithtypicaltimespan
vertical direction, and r(ǫ) is the integral of wet
of20minutes,andparametersofthelinearspline
refractivity along elevation angle ǫ. Here ev is are estimated together with other parameters of
specific humidity, T is temperature, h is height,
interest in a single least-squares solution. The
and the integration is done through the whole
hydrostatic mapping function is used for com-
atmosphere. r(ǫ) is a pseudo-mapping function,
puting the apriori hydrostatic path delay (the
becausebendingoftheraypathduetothespatial
aprioriwetpathdelayissettozero)andthewet
change in the index of refraction is not included
mappingfunctionisusedasthepartialderivative
in the calculation. This parameter was inves-
for estimation of the wet zenith path delay. If
tigated in order to be able to avoid the large
the hydrostaticmapping function is wrong,then
amount of computing required for an accurate
the apriori total path delay in the direction of
ray-trace.
the observed source will be biased. Since the
These regression coefficients allows us to
hydrostatic mapping function and the wet map-
compute mapping functions using geopotential
ping function are slightly different at low eleva-
height and the ratio of the wet refractivity inte-
tions(thedifferenceis4–7%atelevation5 ),the
◦
grals obtained from a numerical weather model
estimation procedure will not remove this bias
(NWM) for the case when no radiosonde atmo-
completely. Thus, systematic errors in the hy-
sphere profile near an observing station is avail-
drostatic mapping function result in systematic
able,asisusuallythecaseduringVLBIandGPS
errors in estimates of site position. Due to the
observations.
errorsinthewetmappingfunction,whichisused
as the partial derivative, the contribution of the
3 Implementation at GSFC for VLBI
wet troposphere path delay to the observablesis
The NWM data used are the gridded data from not completely removed by estimating the wet
theU.S.NationalCenterforEnvironmentalPro- path delay in the zenith direction.
3
Figures 1 and 2 show the differences between first degree with time span 20 minutes, clocks,
the hydrostatic NMF and IMF mapping func- Earth orientation parameters, source positions,
tions at two sites. If the IMF is closer to the and other nuisance parameters. The hydrostatic
true mapping function, we could expect that us- zenith path delay was computed using the for-
ing IMF would reduce systematic errors in GPS mula of Saastamoinen (1972), and IMFh was
and VLBI results. used to transform the apriori zenith hydrostatic
path delay to the path delay in the source di-
rection. However, in the right hand-side of the
equationsofconditions,insteadofthedifferences
betweenobservedandmodeleddelayweinserted
the differences between hydrostatic path delays
calculated using NMFh and IMFh but with the
same apriori zenith hydrostatic path delay com-
putedfromtheSaastamoinenformula. AllVLBI
observations for the period of 1993–2003 were
used. Weobtainedfromthissolutiontheseriesof
baseline length estimates which have the mean-
ingofchangesinbaselinelengthsdueto changes
in the hydrostatic mapping function, and we
Figure1: NMFh(thickline)andIMFh(dots)at
computedtheweightedmeanvalueofthe length
elevation angle 5 at station Gilcreek (Alaska).
◦ change for each baseline. Since replacement of
themappingfunctionaffectsmainlytheestimate
of vertical site coordinate, the changes in base-
line length, δl, will depend on baseline length as
L
∆l = (∆h1+∆h2), where L is the baseline
2R
length an⊕d R is the radius of the Earth. Fig-
⊕
ure 3 shows the mean changes in baseline length
as a function of the total baseline length for all
baselines in the solution.
Figure2: NMFh(thick line)andIMFh(dots)at
elevation angle 5 at station Fortleza (Atlantic
◦
coast, Brazil).
To evaluate quantitatively what kind of im-
provement in geodetic results is provided by the
use of IMF instead of NMF, we performed a se-
ries of VLBI solutions and investigated changes
in baseline length repeatability and changes in Baselinelengthinkm
the estimated amplitudes of the harmonic site
Figure 3: Changes in baseline lengths due to re-
position variations.
placement of NMFh with IMFh (in mm) as a
4.1 Baseline length repeatability test function of baseline length.
First, we need to evaluate how a replacement of
the NMFh with IMFh affects baseline lengths. We then made two other solutions. In solu-
Forthis purposewemade a specialsolution. We tionAweusedNMFhandinsolutionBweused
used the standard parameterization in the esti- IMFh. Parameterizationwasexactlythesameas
mation model: we solved for site positions, tro- in the previous special solution. For each base-
posphere path delay modeled by spline of the line we got the series of length estimates and
4
evaluated the formal uncertainties of these esti- that NMFh is much closer to the (unknown to
mates. A linear model was fitted to each series, us) ”true” mapping function than IMFh is, the
and the wrms deviation from the linear model use of IMFh will result in degradation of base-
was computed for each baseline. We call this line length repeatability by the amount σ2 , and
m
wrms baseline length repeatability. We formed then R ≈ 0. If IMFh is closer to the true map-
the differences of the squares of repeatabilities ping function than NMFh by an amount equal
in the sense solution A (IMFh) minus solution to the difference (NMFh - IMFh), then the dif-
B (NMFh). The differences, smoothed with a ference in baseline length repeatability is equal
Gaussian filter, are presented as a function of to σ2 , and R=1.
m
baseline length in figure 4. We havecomputedthe coefficientofreduction
ofvarianceforeachbaselineobserved16ormore
times and have calculated the weighted mean
value of R over all baselines. We found that
R = 0.81±0.02. This means that 80% of the
power in the change of baseline lengths due to
the changein mapping function is presentin the
data, and the baseline length repeatability has
improvedby approximately80%of the expected
changeunder theassumptionthatthe new map-
ping function, IMFh, is closer to the true map-
pingfunctionthantheoldNMFhbytheamount
of the differences between these two functions.
Baselinelengthinkm This result is very encouraging. It demon-
strates that using only one value, the geopoten-
Figure 4: Differences in squares of baseline
tial height of the 200 hPa surface, we can re-
lengths in mm2 using IMFh versus NMFh for
construct a mapping function (an infinite set of
computing apriorihydrostatic path delay, in the
values) with a surprisingly good precision: not
sense IMFh-NMFh.
worse than 80%.
We see that the differences are negative. This 4.2 Changes in harmonic site position varia-
means that using IMFh instead of NMFh re- tions
ducedthebaselinelengthrepeatabilityand,thus,
As we see in figures 1 and 2, NMFh does not
improvedthequalityofthegeodeticresults. But
model seasonal changes perfectly, and it com-
we would like to have a quantitative measure of
pletely ignores harmonic variations of the map-
how replacement of NMFh by IMFh brings us
ping function at frequencies others than annual.
closer to the true mapping function. We in-
Thus, we can expect that these errors will cause
troduce a coefficient of reduction of variance R
harmonicvariationsintheestimatesofsite posi-
(see Petrov and Boy (2003) for more details) as
tions. As wesawin the previoussection,the use
a quantitative measure of changes in baseline
of IMFh improves baseline length repeatability.
length repeatability:
Therefore we can expect that observed residual
∆σ2+σ2 harmonicsitepositionvariationswillbe reduced
m
R= 2 (4) when IMFh is used instead of NMFh.
2σ
m
In order to evaluate this effect quantitatively,
where ∆σ2 is the difference in the squares of wemadetwosolutions. Inbothsolutionsweesti-
baselinelengthrepeatabilityofthe solutionwith matedtropospherepathdelay,clockparameters,
NMFh minus the solutionwith IMFh, andσ2 is Earth orientation parameters, station position
m
the square of differences in baseline lengths, de- andvelocities,sourcecoordinates,andharmonic
rived from the special solution, which has the sitepositionvariationsat32frequenciesforeach
meaning of the square of expected changes in station. NMFh/NMFw were usedinthe firstso-
baseline lengths. This coefficient runs from 0 to lution, and IMFh/NMFw were used in the sec-
1. In the case that the change of mapping func- ond solution. We included all observations from
tiondid not resultina change ofbaseline length 1980 through 2003 for 40 VLBI stations with
repeatability, the coefficient is 0.5. In the case a long history of observations. Our theoretical
5
model included site displacements due to solid cataloguefromthesolutionthatusedIMFhwith
Earth tides, ocean loading, atmosphere loading, respect to the catalogue from the solution that
andhydrologyloading. Thus,weobtainedresid- usedNMFh. Thistransformationincludestrans-
ual harmonic site position variations. lation, rotation, and the scale factor. We found
For each frequency we computed the ratio of thatforthesitepositioncataloguefromsolutions
the weightedsumofsquaresofthe observedam- with IMFh the scale is greater by 0.21 ± 0.05
plitudes of harmonic site position variations to ppb with respect to the solutions using NMFh.
2
its mathematical expectation, χ /f. In the ab- This changes in the scale factor corresponds to
sence of a signal χ2/f should be less than 1.2. an approximately 1 mm increase of vertical co-
Detailedanalysisofthetechniqueforsolutionsof ordinates of site positions.
this type canbe foundin PetrovandMa(2003).
We noticed differences in these statistics at 4.4 Wet IMF
six frequencies. They are presented in table 1.
WemadethebaselinelengthtestforusingIMFw
We see that the power of the residual signal is
in place of NMFw, but instead of improvement
reduced at all frequencies when IMFh is used
in the baseline length repeatability, we found
instead of NMFh. To our surprise the power
a slight degradation. It should be noted that
was reduced not only at annual (Sa), diurnal
the expected improvement, 1.0 mm at baseline
(S1)andsemi-diurnal(S2)frequencies,wherewe
lengths10000km,isverysmall,afactorof5less
expected improvement, but also at semi-annual
thanthe improvementdue to replacementofthe
(Ssa), sidereal (K1) and semi-sidereal (K2) fre-
hydrostaticmappingfunction. Atthe sametime
quencies. Changesofthepowerofresidualsignal
comparisonofIMFw andNMFw withrespectto
at the K1 frequency may have a significant con-
the mapping function derived using radiosonde
sequence: it means that using IMFh instead of
dataatthesitesofradiosondelaunchesindicates
NMFh may result in a non-negligible change in
that IMFw better approximates the radiosonde
the estimate of the precessionrate, since preces-
mapping function than NMFw, in contradiction
sioncorrespondstotheharmonicvariationinthe
to the results of the VLBI data analysis. We
Earth rotation at the sidereal frequency.
think that this contradiction is related to the
fact that the relatively large grid of the NCEP
Table1: χ2/f statisticsofresidualharmonicsite Reanalysis NWM, 200x270km at mid-latitudes,
position variations in two solutions which used is not adequate to represent the wet refractivity
different mapping functions. field.
Tide χ2/f 5 Geodetic implications
NMFh, NMFw IMFh, NMFw
If there were no model errors, it would be ad-
vert horiz vert horiz
vantageous to make observations to as low an
K2 1.80 2.01 1.61 2.05 elevationaspermittedbytheequipmentandthe
S2 2.55 1.73 2.15 1.76 physical surrounding, for both GPS and VLBI.
K1 2.13 3.45 1.89 3.34 Including the widest range of elevations reduces
S1 3.80 2.32 3.58 2.22 the uncertainty in the height estimate and im-
Ssa 2.42 1.12 1.85 1.09 provesseparationofthe estimates ofthe vertical
Sa 6.34 2.56 5.77 2.43 component of position from the estimates of the
zenith path delays and the clock offsets. How-
ever, any errorin the azimuthally symmetric at-
4.3 Changes in the scale factor of the TRF
mospheremodelwillpropagateintoerrorsinthe
Systematic changes in the estimates of the ver- estimated vertical coordinate, and these errors
tical component of site positions may result in a willincreasewithdecreasingminimumelevation.
change of the scale factor of the output site po- Thus, there is a trade-off between atmosphere
sition catalogue. In order to evaluate quantita- uncertainty and the accuracy of determination
tivelythiseffect,wehaveanalyzedthecatalogue of the vertical, and it is possible to choose the
of site positions obtained in the solutions de- optimum minimum elevation if the error model
scribedintheprevioussection. Weestimatedthe is sufficiently well known.
7-parameter transformation of the site position The formal uncertainty in the vertical, taking
6
into account atmosphere stochastic variability, becalculatedastheproductofthedelayerrorat
assuming the clock is modeled as white noise, the lowestobservedelevation and the sensitivity
and adding a fixed amount of delay noise, is of the height estimate to delay change at that
calculated in the precise point positioning mode elevation.
(PPP)oftheGPSanalysissoftwareGipsy(Zum- MacMillan and Ma (1994) evaluated the sen-
bergeetal.,1997),althoughothereffects,suchas sitivity of the vertical error to the error in at-
satellite orbit error, are not included explicitly. mosphere path delay at the minimum elevation.
For twenty-four hour sessions the uncertainties They found that an error of 1 mm in atmo-
for minimum elevations of 15◦ down to 5◦ de- sphere path delay at the minimum elevation an-
crease from 4.0 mm to 2.0 mm (the data were gle causes 0.22 mm error in the vertical coordi-
taken with an Ashtech Z-12 with good sky visi- nate ifthe minimum elevationis 5 , and0.4 mm
◦
bility). iftheminimumelevationis10 . Wemadeasim-
◦
The effect of the atmosphere model error has ilar test for a later set of VLBI data and found
beencalculatedbycomparingtheIMFandNMF asensitivityof0.3–0.35foraminimumelevation
mapping functions obtained from the NWM at of 5 . A simulation of GPS observations using a
◦
twenty-six radiosonde launch sites for the year simple model of estimating vertical, zenith tro-
1992, with the mapping functions calculated by pospheredelay,andclockforasingleepochgave
raytracing along the path of the radio waves at sensitivity of 0.4 to 0.8 over the minimum eleva-
various elevations, assuming a vertical distribu- tionrange5 –15 . ForGPStheactualsensitivity
◦ ◦
tion of refractivity given by the radiosonde tem- will depend on what satellites are visible at the
perature, pressure, and humidity profiles. Using time of observation, but for illustration of the
thesurfacepressureforthezenithhydrostaticde- technique we will use the values from our GPS
lay,andthezenithwetdelayfromtheradiosonde simulation.
data, the rms delay errors for an elevation of 5◦ To show the effect of the mapping function
are shown in figure 5 for the hydrostatic com- errors on height at mid-latitude the rms delay
ponent of IMF and for the hydrostatic and wet differences were found between both the NMF
components of NMF. The errors for IMFw are and IMF mapping functions and mapping func-
very close to those of NMFw. tionsobtainedbyraytracingtheradiosondedata
(as described in section 2) for the site ALB (Al-
25 radiosonde sites (1992) bany,stateNewYork,USA)atlatitude42 . The
◦
m) NMFh differences were converted to height uncertain-
m40 IMFh tiesusingthesensitivitiesdescribedintheprevi-
ev ( NMFw ous paragraph(figure 6). We calculated inflated
el
m formal uncertainties of the height estimates by
30
mu adding in quadrature the point-positioning un-
ni certainties and the uncertainty due to errors in
mi
° 20 themappingfunction. Thesetotalerrorsarealso
at 5 shown in figure 6. From this study we conclude
y that the use of NMFh significantly increases the
a10
el total height uncertainty when data are included
d
S below about 10 for a twenty-four hour session.
M ◦
R 0 AtthesametimetheuseofIMFhdoesnotresult
−50 0 50
latitude in an increase of total height uncertainties until
data below about 7 are included.
◦
Figure5: Rmsdelayerrorformappingfunctions This type of evaluation can be important for
at radiosonde sites for 5 elevation. decidinghowmuchdatatoinclude inasolution.
◦
For example, the lower elevation data might be
How dothese delay errorsaffectthe heightes- desirable for improving estimates of atmosphere
timates? Forerrorsthatincreasewithdecreasing gradients, but only if the uncertainty of vertical
elevation, such as those that are due to the at- estimates is not degraded.
mosphere,thelowestelevationobservationshave An important point to notice from figure 6 is
the greatest effect on the height uncertainty. As thatforaminimumelevationof15 ,whichisthe
◦
a result, the effect onthe heightuncertainty can standardformanyGPSanalyses,theerrorinthe
7
ALB 7 Acknowledgment
10
pt posn
We used NCEP Reanalysis data provided by
NMF total
IMF total the NOAA-CIRES Climate Diagnostics Center,
m) 8
m Boulder, Colorado, USA, from their Web site at
y ( http://www.cdc.noaa.gov/
nt 6
ai
rt References
e
c
n 4
u Kalnay E. et al., (1996). The NCEP/NCAR 40-
ht
heig 2 ← NMFh height error Yteeoarrol.ReSaonca.,ly7s7i,spP.4ro37je–c4t7,1B.ullet. Amer. Me-
IMFh height error → Ma, C., J.M. Sauber, L.J. Bell, T.A. Clark,
0 D. Gordon, W.E. Himwich, and J.W. Ryan,
0 5 10 15 20
minimum elevation (°) (1990). Measurement of Horizontal Motions
in Alaska Using Very Long Baseline Interfer-
Figure 6: Height uncertainty for different mini- ometry,J.Geophys. Res.,95(B13),p.21,991–
mumelevationsforprecisepoint-positionandfor 22,011.
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tionofverylongbaselineinterferometryatmo-
spheric modeling improvements, J. Geophys.
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than 2 mm and is therefore not significant for Marini, J. W., (1972). Correction of satellite
daily solutions. However, in computing weekly tracking data for an arbitrary tropospheric
site position averages, the atmosphere modeling profile, Radio Science, 7, p. 223–231.
error may contribute more than expected since Niell,A.E.,(1996). Globalmappingfunctionsfor
weather systems often persist over many days, the atmospheredelayatradiowavelengths,J.
and the actual uncertainty of the average will Geophys. Res., 101, No. B2, p. 3227–3246.
decreasemoreslowlythan 1 ,becausetheerrors Niell,A.E.,(2000). Improvedatmosphericmap-
√n
are correlatedover longer than one day. ping functions for VLBI and GPS, Earth,
Planets, and Space, 52, p. 699–702.
Niell, A. E., (2001). Preliminary evaluation of
6 Summary, conclusions, future work
atmospheric mapping functions based on nu-
Atmosphere delay mapping functions, desig- merical weather models, Phys. Chem. Earth,
nated IMF, based on in situ meteorological in- 26, p. 475–480.
formation from the NCEP numerical weather Petrov, L., and C. Ma, (2003). Study of har-
model,havebeenimplementedintheVLBIanal- monic site position variations determined by
ysis package CALC/SOLVE. When applied to VLBI, J. Geophys. Res., vol. 108, No. B4,
globalVLBIsolutionsthatincludealldataabove p. 2190, doi: 10.1029/2002JB001801.
a minimum elevation of 5 , their use, compared Petrov L., and J.-P. Boy, (2003). Study of the
◦
to the model NMF, improves baseline length re- atmospheric pressure loading signal in VLBI
peatability and reduces the power in residual observations, submitted to J. Geophys. Res..
harmonic site position variations. Saastamoinen,J.(1972). Atmosphericcorrection
IfusedforGPS,IMFwillallowlowerelevation for the troposphere and stratosphere in radio
data (down to approximately 7 ) to be included ranging of satellites. In The Use of Artifi-
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cialSatellitesforGeodesy, Geophys. Monogr.,
inthesolutionswhilemaintainingtheformalun-
certainty in the vertical that is obtained for 15 AGU, 15, p. 247–251.
◦
Zumberge, J. F., M. B. Heflin, D. C. Jeffer-
minimum elevation, and will reduce amplitudes
son, M. M. Watkins, and F. H. Webb, (1997).
of observed residual harmonic site position vari-
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analysis of GPS data from large networks. J.
With improvement in modeling of annual and
Geophys. Res., 102, p. 5005–5017.
semi-annual atmosphere errors, it is time to be-
gin modeling systematic temperature-dependent
antenna height errors for GPS and VLBI.
8