RadioScience,Volume 20, Number 6, Pages1593-1607,November-December1985
Geodesyby radio interferometry:
Effectsof atmosphericmodelingerrors
on estimatesof baselinelength
J. L. Davis,
• T. A. Herrinch
2 I. I. Shapiro,
2 A. E. E. Rollers,
3 andG. El•lered
4
(ReceivedMarch 14, 1985;revisedJune27, 1985;acceptedJune27, 1985.)
Analysisof verylongbaselineinterferometry
data indicatesthat systematic
errorsin prior estimates
of baselinelength,of order 5 cm for • 8000-kmbaselines,
weredue primarilyto mismodeling
of the
electricalpath length of the troposphereand mesosphere
("atmosphericdelay"). Here we discuss
observationalevidencefor the existenceof sucherrorsin the previouslyusedmodelsfor the atmospher-
ic delayand developa new"mapping"functionfor the elevationangledependence
of thisdelay.The
delaypredicted
by thisnewmappingfunctiondiffersfromray traceresultsby lessthan • 5 ram,at all
elevationsdown to 5ø elevation, and introduceserrors into the estimatesof baselinelength of •< 1 cm,
for the multistationintercontinentalexperimentanalyzedhere.
1.
in which refractivity profiles, averaged over all seasonsand various sites,were used.This mapping function therefore contains no parametrization based on
INTRODUCTION
A signalfrom a distant radio sourcereceivedby an
antenna located on the surface of the earth will have
surface weather conditions at the site. However, for
been refractedby the terrestrialatmosphere.The correspondingdelay introduced by the atmospheredependson the refractiveindex along the actual path
traveled by the receivedsignal. For an atmosphere
which is azimuthally symmetric about the receiving
antenna, this delay dependsonly on the vertical profile of the atmosphereand the elevation angle of the
the refractivity profiles used, the mapping function
describesthe elevation angle behavior of the atmospheric delay to better than 1% for elevation angles
above 1ø. On the other hand, the Marini mapping
function [Marini, 1972; J. W. Marini, unpublished
manuscript, 1974] contains terms which depend on
surfacemeteorological conditions, but it is based on
approximations that degrade its accuracy below
about 10ø. Mapping functions other than these two
have appeared in the literature [e.g., Hopfield, 1969;
Saastamoinen, 1972; Black, 1978; Black and Eisner,
1984], but these other mapping functions have not
undergonetestingwith VLBI data.
In the following, we review the manner in which
the atmosphericdelay is modeled, and the effectsof
these models on estimates of baseline length made
from radio interferometric data. We presentevidence
of systematicerrors in estimatesof baseline length
radio
source. The function
which
describes the de-
pendenceupon elevation angle of the atmospheric
delay has becomeknown as the mapping function.
This mapping function is used, along with some
model for the zenith delay, to account for the atmospheric delay in models for the interferometricobservables.Historically, analysesof very long baseline
interferometry(VLBI) data for geodesy[e.g., Robertson, 1975] have made use of the "Chao" mapping
function and, more recently [Clark et al., 1985], of
the "Marini" mapping function. The Chao mapping
function [Chao, 1972] is basedon ray tracing studies made from VLBI data, and we demonstrate that
these errors are caused by the mismodeling of the
XDepartmentof Earth, Atmospheric,and Planetary Sciences, atmosphere. Finally, we describe the development
and testingof a new mapping function.
MassachusettsInstitute of Technology,Cambridge.
2Harvard-SmithsonianCenter for Astrophysics,Cambridge,
2.
Massachusetts.
3HaystackObservatory,
Westford,Massachusetts.
•Onsala SpaceObservatory,ChalmersUniversityof Technology, Sweden.
Copyright 1985by the AmericanGeophysicalUnion.
MODELING
The models
THE
ATMOSPHERIC
for the interferometric
DELAY
observables
of
group and phasedelay, and of phasedelay rate, must
account for the atmosphericdelay:
•a=•atm
dS
n(s)
- fvac
dS
Paper number 5S0522.
0048-6604/85/005S-0522508.00
1593
(1)
1594
DAVIS
ET AL.: ATMOSPHERIC
DELAY
where the first integral is evaluatedalong the path of
a hypothetical ray originating from the direction of
the radio sourceand passingthroughthe atmosphere
to a receivingantenna,and n(s)is the index of refraction at the point s along the path; the secondintegral
is evaluated along the path the ray would take were
the atmospherereplacedby vacuum. For simplicity,
we have chosenunits in which the speedof light is
unity. (Delay will thereforebe expressedin units of
equivalentlength.)The difference,r?-r?,
to the surface of the antenna and be
reflectedback along the axis. For a prime focus antenna, this paraxial ray would continue to travel
until it enters the antenna
VLB
INTERFEROMETRY
minus the erroneouslyadded path from the vertexto
the intersection.These paths should contribute less
than • 1 mm amplitude of diurnal variation (due to
diurnal variationsof temperatureand humidity) and
lessthan 0.01 mm variation with antenna pointing
angle.
Evaluation of the second integral on the righthand side of (1) requires only knowledge of the
source and antenna coordinates. However, evalu-
for the ation of the first integralrequires,as well, knowledge
two antennas i and j, of an interferometergives the
contribution of the atmosphereto the model of the
interferometricdelay. (Using the term "tropospheric
delay" here would be inaccurate,sinceabout 25% of
the atmosphericdelay occursabove the troposphere.)
We can find the point at which the integration in
both parts of (1) is terminated at the earth by visualizing the path of the (hypothetical)paraxial ray. This
ray would strike the vertex of the paraboloid of the
antenna normal
IN
feed at the focus. For a
of the index of refractionin the neighborhoodof the
correct ray path, which is necessaryin order to
obtain the path itself via Fermat's principle [Born
and Wolf, 1970]. Sincein practiceit is not possibleto
obtain this knowledge,one usually relies on models
of the structureof the atmosphere.For example,one
often assumes that the index of refraction of the at-
mosphereis constantfrom the surfaceof the earth up
to an altitude H; for altitudes above H, the index of
refractionis assumedto be unity, and the bendingof
the ray at the atmosphere/vacuum
boundaryis ignored. Then for a plane parallel model of the earth
and the atmosphere,(1) reducesto
Cassegrainfocusantenna,the ray would be reflected
za=csc
e•oHdz
(n
o--1)
once more at the subreflector and then enter the feed
located at or above the vertex along the axis. The
path(s) after the initial reflectioncan be ignored in
evaluating(1), becausethe delay is very nearly constant. The daily variation is usuallylessthan 0.5 mm
per 10 m of travel. (The largest diurnal variation in
this delay, calculated from data taken by meteorological sensorslocated at the sites,was recorded for
the Westfordantennasite; the value was0.8 mm per
10 m of travel and was associatedwith a rapid decreasein the humidity.)A constantdelay of any type
at one of the sites is indistinguishablefrom a constant clock offsetor instrumentaldelayfor that site.
For most antennas,the vertex of the primary reflector moves when pointing is changed;the size of
the movement is usually a few meters. This movement can usuallybe ignored,with consequentnegligible error, and a fixed referencepoint used in the
evaluation of (1). For example, the intersectionof
axesof rotation of the Haystackantenna(one of the
antennasused in VLBI experiments;see below) is
located 4.3 m from the vertex along the axis of the
parabola in a directionoppositeto that of the prime
focus. In this case, if we use the axis intersection as
the fixed referencepoint for the evaluationof (1), the
errorsintroducedwill be equal to the neglecteddelay
from vertex to the subreflector and back to the sec-
ondary focus(a total distancetraveled of 25.2 m),
(2)
wheree is the elevationangleof the radio sourceand
no is the index of refraction at the surface of the
earth.
It is possibleto write (1) in a form which is motivated by the simpleform of (2). Quite generally,we
can write
•a = m(e,P)
dz In(z)-
1]
(3)
The function m(e,P), which is defined by (1) and
(3), dependson the elevationangle e as well as on the
parameter vector P, which is a parametrizedrepresentation of the behavior of the index of refraction in
the atmosphere.The number of elements(parameters) in P dependson the assumptionsmade about
"regular" atmosphericstructure.For example,if one
assumesthat no discernible atmospheric structure
exists, then P will be an infinite-dimensional vector
containingthe index of refractionat all points. Since,
as previouslydiscussed,
the refractionat all points is
not known, this assumptionwould void (3) of any
possible advantages. Instead, one usually makes
some assumptionsand approximations concerning
the structureof the atmosphereand its effectson the
ray path. A simple set of assumptionsand approxi-
DAVIS
ET AL.: ATMOSPHERIC
DELAY
mations is, for example, that which led to the cosecant law. In this case,the dependenceof the function
m on parametersP other than elevation is completely
absent. (See below for a discussionof the assumptions usedto developthe new mapping function.)
The integral in (3) is the atmospheric delay for a
radio source at zenith. This integral will be denoted
z•, the "zenith delay." Its dependenceon atmospheric conditions directly above the antenna is discussed
in the Appendix A. The function re(s, P) is known
also as the "mapping function." For simplicity, the
dependenceon the parametersP will be suppressed,
and we will write simply re(s).
Often separatemapping functionsare used for the
"wet" and "dry" componentsof the delay:
•. = •,•r%(s)+ r•wrnw(S)
(4)
where the subscriptd on the zenith delaysand mapping functionsrefersto "dry" and w to "wet." Such a
form is usedwhen, for instance,water vapor radiometer (WVR) data are used to estimate the "wet" component directly [Resch,1984]. Then (4) is replacedby
IN
VLB
INTERFEROMETRY
1595
more completediscussionof the group delay model.)
The feature of importancein (6) is the dependenceof
the group delay on the elevation of the source at
each site. Any elevation-dependenterror (such as a
mapping function error) in the group delay model
which
correlates
with the sine of the elevation
will
corrupt the least squares adjustment of the radial
component of the site position. An error Ar introduced into the estimate of the radial component of
the positionof either site introducesa corresponding
error Ab in the estimateof the length of the baseline
between
the sites:
b
Ab • • Ar
2re
(7)
where b is the baselinelength and re is the radius of
the earth. The above equation is accurate to order
(Ar)2/b.
Mapping function errors introduce systematic
errors into the estimatesof other parametersas well.
In fact, all estimated parameterswill be systematically affected,althoughthe magnitudewith which the
12
a -- '17•
ma(e)+ Zwv.
(5)
mapping function error manifests itself in the estiwhere rwwt is obtained from the WVR data. The user mate of a particular parameter dependson the funcof such formulas, however, must be extremely careful tional dependenceof the group delay on that paramto understand exactly what is meant by the terms eter. Thus one can expect systematicerrors in esti"dry" and "wet," becausethe path the radio signal mates of source position, earth orientation, nutation,
travels through the atmosphereis dependent on the and any and all other estimated parameters; howcontributions
to the index of refraction from all atever, for illustration, this paper will confine itself to
mospheric constituents. Furthermore, the so-called studyingonly errors in baselinelength estimates.
Do we have evidenceof mapping function errors,
"dry" zenith delay also contains contributions from
and, if so, how large are they? A useful method
water vapor (seeAppendix A).
which can be used to indicate the presenceand size
of elevation-dependent systematic errors, such as
3. SYSTEMATIC
ERRORS
IN ESTIMATES
OF
mapping function errors, is the "elevation angle
BASELINE
LENGTH
cutoff test." In this test, all baselines are estimated
The manner in which estimatesof baseline length
are affectedby errors in the mapping function used
to model the atmosphericdelay can be understood
by first examiningthe approximate expressionfor the
above, the discussion will be confined for illustration
to the effectson the estimatesof baselinelength.)The
"geometric"term'•geom
of thegroupdelaymodel
•geom
-- -- bø • -- -- (r2 sine2- r• sine•)
simultaneously using all the data available. (Of
course, other parameters are estimated along with
baselines,but here and in the following, as stated
(6)
where b is the baselinevector (directed from site 1 to
site 2), g is a unit vector in the direction of the source,
ri is the distance from the center of the earth to the
ith site, and ei is the elevation of the sourceat the ith
site (i = 1, 2), and where the total group delay (of
which'rgeo
mis but oneterm)is definedas the time of
arrival of the signalat site 2 minus the time of arrival
of the signal at site 1. (See Robertson [1975] for a
baselines are then reestimated with the data limited
to observations
above some minimum
elevation
angle. More estimates can be made with different
elevation angle minima. If there are no elevationdependent systematicerrors, the mean of the differencesbetweenthe correspondingbaseline-lengthestimates should tend toward zero. Significant biasesindicate mapping function errors. Figure 1 contains the
results from such a test. Plotted
are the differences in
baseline-lengthestimates for 5ø and 15ø minimum
1596
DAVIS
1
ET AL.' ATMOSPHERIC
2
3
DELAY
4
5
IN
VLB
INTERFEROMETRY
6
7
8
9
10
Baseline length (1000 km)
Fig. 1. Differencein baselinelengthestimatesfor the 15ø-5ø elevationanglecutoff test of the Marini mapping
function.The error barsare the statisticalstandarddeviationsof the differences
(seeAppendixB). The straightline
represents
the effectsof a changein the local verticalcomponentof sitepositionof 3 cm at eachof the sites.
elevation angles.These differencesare plotted against
the length of the baseline.The error bars shown are
the standard deviations of the differences, obtained
for this site for some periods of the campaign.The
phasedelay rate data were not included.
From Figure 1 it can be seenthat the differencesin
from the statistical standard deviations
of the indithe estimatesof baselinelength seem to be nearly
vidual estimates.(It can be quite easily shown that proportional to baselinelength. Recalling(7), we can
the variancesof the differencesof the baseline-length interpret these differencesas due to corresponding
estimates are the differences of the corresponding differencesin the estimatesof radial positionsof the
variances resulting from the two least squaressolu- individual sites,if theselatter differencesare nearly
tions; seeAppendix B.) The group delay data usedto equal. For reference,Figure 1 containsa line repregeneratethesedifferencesare the entire yield of VLBI
senting the effect of a 3-cm radial differenceat each
group delay data from the project MERIT short site. (The senseof the radial differenceis, from (7),
campaign of Septemberand October 1980 [Robert- such that the estimatesof the radial positionsfrom
son and Carter, 1982], with the exception noted the 0ø cutoff solution were greater than those from
below. These data were processedas describedby the 15ø cutoff solution.)It can be seenthat this nearly
Clark et al. [1985]. The atmosphericdelay was mod- represents the actual situation. We thus conclude
eled by usingthe Marini formula, which requiressur- that the differencesevident in Figure 1 are due to
face weather data. The group delay data involving mapping function errors, on the assumption that
the site at Chilbolton, United Kingdom, were de- there do not exist any other elevation angleleted, since surface weather data were not available dependenterrors of this magnitude.
DAVIS
ET AL.: ATMOSPHERIC
DELAY
IN VLB INTERFEROMETRY
1597
that height. The accelerationdue to gravity was assumedto be constantwith height. This simple set of
Marini [1972] showedthat the continuedfraction
assumptionsconcerningthe structure of the atmoform of the mapping function
sphereallowed us to examinethe dependenceof the
1
mapping function on variations about the nominal
me) =
a
values of the following parameters:surfacepressure,
surfacerelative humidity, surfacepartial pressureof
sine+
b
sin • +
water vapor, temperature of the tropopause, and
sin • +
(8) height of the tropopause.However, the samplingof
sin • +...
parameter space was not done in a systematic
where a, b, c,... are constants, can be used to ap- manner due to the large number of ray trace analyses
proximatethe elevationangledependence
of the at- which this would entail. For example, if just three
mosphericdelay. Only two terms are used in the valuesfor eachparameterwere used,there would be
Marini mappingfunction(J. W. Marini, unpublished 35= 243 differentcombinationsof parameters.Inmanuscript, 1974). Chao [1972] uses two terms as stead, 57 analyseswere performed,and there are rewell, excepthe replacesthe secondsin e with tan e, sultinggapsin the samplingof the parameterspace.
For each set of atmosphericconditions, then, we
therebyensuringthat m(90ø)= 1. We have attempted
to developa mappingfunctionfor the "dry" or "hy- determined the ray trace values for the mapping
drostatic"componentof the atmosphere(seeAppen- function, in stepsof 1ø for elevationsfrom 5ø to 90ø.
dix A) basedon the Chao model,but with improved We then used least squaresto estimate a, b, and c.
accuracyat low elevationangles.In order to achieve However, c could be fixed at some nominal value and
subcentimeteraccuracy at 5ø elevation, we have not appreciably degrade the solution; the nominal
"continued the fraction" by adding one more term value ultimately decidedupon (see below) for c was
taken to be the approximateaverageof the valuesfor
but keepingthe tangent'
the first severalray traces performed.The mapping
m(•) =
function form given in (9) was, for each set of atmosphericconditions,able to model the elevation angle
sin • +
dependenceof the delay to within 3 mm for all elevation anglesdown to 5ø, and with an rms deviation of
4.
THE
NEW
MAPPING
FUNCTION
tan e + •
sin e + c
(9)
less than 1.5 mm.
The advantageof using this form is its simplicity,
both in calculatingthe mappingfunctionitselfand in
calculatingpartial derivativesof the mapping function with respectto the parametersto be estimated.
The disadvantageof this form is that for higherelevation angles(20ø-60ø),tan e does not approach
sin e quicklyenough.As a result,one can expect1- to
2-mm errors in representingthe atmosphericdelay
with (9) for theseelevations.
In order to determine the mapping function parametersa, b, and c, we performedray trace analyses
The ray trace analysesthus provided a set of estimates of each of the mapping function parameters,a
and b, coveringa variety of atmosphericconditions.
We then representeda and b each as a linear function of the various atmosphericparametersthat were
varied and usedleast squaresto determinethe coefficients. Such a linear model fits the mapping function parameter a within 0.2% (correspondingto • 5
mm at 5ø elevation) and the parameter b to within
0.5% (• 2 mm at 5ø elevation)in all cases;the rms fit
for a is 0.08% (• 2 mm at 5ø elevation)and for b is
0.15% (•0.6 mm at 5ø elevation). In particular, we
for various values of a limited number of atmospher-
have
ic conditions.The ray trace algorithm we used was
based on a sphericallysymmetric,layered atmo- a =0.001185 [1 + 0.6071x 10-'•(Po- looo)
sphere.The temperatureprofilewas taken to havea
-0.1471 x 10-3eo
linear dependencewith height up to the tropopause,
+0.3072 x 10-2(To- 20)
above which the temperaturewas assumedconstant.
The total pressurewasassumedto resultfrom hydro+0.1965 x 10-•(fi + 6.5)
staticequilibrium,and the relativehumiditywas as-0.5645 x 10-2(ht- 11.231)]
sumed to be constant up to 11 km and zero above
(10)
1598
DAVIS
ET AL.:
ATMOSPHERIC
DELAY
b = 0.00114411
+ 0.1164x 10-'•(Po- 1000)
IN
VLB
INTERFEROMETRY
TABLE 1. Sensitivitiesof the Path Delay From Model CfA-2.2
to Changesin AtmosphericParametersfor ALz½
n = 240 cm and
+0.2795 x 10-3eo
for Different
+0.3109 x 10-2(To- 20)
P,
cm/mbar
+0.3038 x 10- •(• + 6.5)
-0.1217 x 10-X(ht- 11.231)]
(11)
c = -0.0090
(12)
15ø
10ø
-9.1
-2.8
5ø
x 10 -4
x 10 -3
-0.017
T,
cm/øC
Elevations
e,
cm/mbar
e
/•,
cm/(K/km)
h,,
cm/km
-0.046
-0.14
0.002
0.007
-0.29
-0.88
0.082
0.25
-0.75
0.053
-4.4
1.1
See text for explanation of model CfA-2.2. P, pressure-T, tem-
where Po is the total surfacepressurein millibars, eo perature; e, partial pressureof water vapor; ]•, temperaturelapse
is the partial pressureof water vapor at the surfacein rate; ht,height of tropopause.
millibars, To is the surface temperature in degrees
Celsius,/• is the tropospherictemperature lapse rate water vapor producesa change of approximately 5
in K km-x, and ht is the heightof the tropopause
in mm in the predicteddelay at 5ø elevation.(It is fortukilometers.This version of the "dry" mapping func- nate, in fact, that the mapping function is not very
tion has been dubbed CfA-2.2. The sensitivities of the
sensitiveto the amount of water vapor in the atmoCfA-2.2 mapping function to changesin these atmo- sphere,sincethis quantity is spatially highly variable
spheric parameters are summarized in Table 1. For and not well predicted by surface measurements.)
example, a 10-mbar change in the partial pressureof Figures 2 and 3 contain the differencesbetweenray
40
I
I
I
i
i
i
I
I
I
i
30
•
2o
Marini
10
o
CfA-2.2
•
0
.,•
-10
Chao
5
10
15
20
Elevation angle (deg)
Fig. 2. Differencesfrom ray tracingof the new mappingfunction,the Chao mappingfunction,and the Marini
mappingfunctionfor Po = 850 mbar and TO= 15øC.The partial pressureof water vapor,temperaturelapserate,
and tropopauseheight are all at their nominal valuesof 0 mbar, -6.5 K/km, and 11.231km, respectively.
The
correspondingvalue of the zenithdelay is 1.935m of equivalentlength(6.5 ns).
DAVIS
ET AL.' ATMOSPHERIC
DELAY
IN VLB INTERFEROMETRY
1599
40
30
20
Chao
10
Marini
0
C[A-2.2
-10
-20
5
15
lO
20
Elevation angle (deg)
Fig. 3.
Same as Figure 2, except for Po = 1000 mbar and To = -30øC. The correspondingvalue of the zenith
delay is 2.277 m.
tracing and CfA-2.2 for the different atmospheric
conditions indicated. These conditions, which represent the nominal conditions of humidity, lapse rate,
and tropopause height, were chosen because they
representlocationsin the pressure-temperature
plane
near which no ray trace analyses were performed.
Their agreementwith ray tracing is thereforean indication
of the robustness of the method
used in the
plane of temperatureand pressure.Also shown are
the differencesfrom ray tracing for the Chao and
Marini
models. These models are the most common-
ly used mapping functions in VLBI data analysis
[Fanselow,1983; Clark et al., 1985].
,
The accuracy of the CfA-2.2 mapping function
model seemshigher near latitudesof 45øN, for which
the nominal values of tropopause height and lapse
rate usedin CfA-2.2 are representative.For example,
for conditions representativeof a latitude of 30øN
(h, = 16 km, /• =-4.7
K/km to -5.9 K/km), the
difference between CfA-2.2 and ray trace values
reaches• 4 cm at 5ø elevation.Relatively large differ-
enceshave also been noted for higher latitudes in the
extremeof winter: For a latitude of 60øN (h, - 8 km,
/• = -3.9 K/km), the differencesfrom the ray trace
values reach •2.5 cm at 5ø. These (comparatively)
large differencesfrom the ray trace values seemto be
due to the simultaneousdeparturesfrom the nominal
valuesof lapse rate and tropopauseheight. Although
our choiceof -6.5 K/km is the standard one for the
lapse rate in the troposphere(U.S. Standard Atmosphere,1976),it seemsto be somewhatlarge (in magnitude) when one considerscompilationsof temperature profiles found, for example, in the work of
Smith et al. [1963]. However, even with a better
choice of nominal value, a site-dependentmodel of
sometype will have to be developed:lapse rate and
tropopauseheight do not truly vary independently,
since the temperature of the tropopausevaries less
than the surface temperature. Thus those climates
with a very low tropopause height (high latitudes)
can be expected to have correspondinglysmall (in
magnitude) lapse rates. Those climates with a high
1600
DAVIS
ET AL.: ATMOSPHERIC
DELAY
IN VLB INTERFEROMETRY
tropopause height (equatorial latitudes) will have TABLE 2. Values for Tropopause Height ht and Temperature
Lapse Rate fl Used in Elevation Angle Cutoff Test of CfA-2.2
correspondinglylarge (in magnitude)lapserates.
Mapping Function
The lack of dependenceof the new mapping function on azimuth resultsdirectly from the assumption
Site
Geographic
North
ht,*
•,
Name
Location
Latitude
km
K/km
of azimuthal symmetry. Gardner [1977] expressed
the index of refraction in (1) in cylindrical coordiOnsala
south Sweden
57 ø
10.5
- 5.7
nates (with the z coordinate aligned along the local Effelsberg
West Germany
51ø
9.6
- 5.7
vertical) and expandedit in powersof horizontal dis- Haystack
east Massachusetts
43ø
13.6
- 5.6
tance from
the z axis. He showed that the zeroth-
order term representsthe sphericallysymmetricterm.
Thus our new mapping function represents this
zeroth-order
term. The first-order
term in Gardner's
expansion arises from horizontal refractivity gradients; Gardner showed that this term can be as large
as 5 cm at 10ø elevation angle. However, this term
has never been included in our VLBI data analysis
because of the lack of a network of meteorological
sensorsin the near vicinity of our sitesfrom which to
determine the refractivity gradients. In principle,
though, there is no reason that this gradient term
could not be introduced into our atmospheric
models; its utility would depend on (1) a dense
enough network of meteorologicalsensorsbeing in
place around each site, (2) models for the gradient
being developedthat depend on the meteorological
conditionsat the site only (suchas wind direction
and speed),and possiblyon climateand/or season,or
(3) the ability to estimate accurately the gradient
term being demonstratedfor data from a network of
distant (> 100 km) meteorological sensorssuch as
exist at airports and other weatherstations.
5.
PROCESSING
VLBI
DATA
WITH
CfA-2.2
We have performed the elevation-angle-cutofftest
on the CfA-2.2 mapping function. For this test, the
Saastamoinen formula for the zenith delay [Saastamoinen, 1972] was used to be consistent with the
zenith delay valuesusedfor the Marini formula. The
"wet" part of the delay (see Appendix A) was
mapped by using (9)-(12) as well, even though this
use introduces a small error which is, from (4), the
"wet" delay multiplied by the difference between
CfA-2.2 and the "true" wet mapping function. The
values listed in Table 2 were used for tropopause
height and lapse rate. These values are based on
tables of mean temperature profiles near the 80th
meridian west [Smith et al., 1963]. No attempt was
made to obtain the exact profiles of temperaturethat
prevailed at the sites,since for this elevation angle
cutoff test we were attempting only to remove the
grosseffectsof differencesfrom the nominal valuesof
Owens Valley
south California
37ø
12.8
-5.6
Fort
southwest Texas
31 ø
13.4
-6.3
Davis
*Height of tropopausegiven as height above station for direct
usein CfA-2.2 mappingfunctionformula; seetext.
/• and hr. The procedure used was first to obtain
estimatesof/• and ht at the latitudesof 30ø, 40ø, and
50ø by fitting a linear function of height to the values
given in the tables; possiblevariations of these parameterswith longitude were ignored. The three estimates for each of the parameters/• and ht were then
expressedvia least squares as second-order polynomials in latitude. For each North American site,
the latitude
of the antenna
was then substituted
to
determine/• and ht. Each European site was treated
as though it were 5ø south of its true position to
account approximatelyfor the warmer climate at European longitudes in the choice of /• and hr. (The
value of 5ø was based upon visual inspection of
world maps of tropopause height found by Bean et
at. l-1966].)
The results of this elevation angle cutoff test are
shown in Figure 4. Any systematictrend that may be
present in this figure is clearly much smaller than
that seenin Figure 1. Table 3 allows us to compare
the results from the two tests more quantitatively.
The second column contains least squares estimates
of the differencesin the radial site positions which,
from (7), would yield the baseline-lengthdifferences
evident in Figure 1. The third column contains the
same information, except for Figure 4. The fourth
column
contains
the differences
of the second two
columns. The numbers in this fourth column, then,
represent the changesin the inferred differencesof
the radial position we obtained in performing the
15ø-5ø elevation angle cutoff tests.It can be seenthat
for the sites at Haystack, Onsala, and Effelsberg,
thesechangeswere • 4 cm, over 10 timesthe changes
at Ft. Davis and Owens Valley. This differencecan
be explained by the entries in the fifth column. This
column
contains
15 ø elevation
the fraction
of data obtained
at these sites. That
below
these last two sites
had no data from these lower elevation angles
DAVIS
ET AL.'
ATMOSPHERIC
DELAY
IN
VLB
INTERFEROMETRY
1601
-1
-2
-3
-4
-5
-6
-7
0
1
2
3
4
5
6
7
[5
9
10
Baseline length (1000 km)
Fig. 4.
Differencein baselinelengthestimatesfor the 15ø-5ø elevationanglecutoff test of the CfA-2.2 mapping
function.The error barsare the statisticalstandarddeviationsof the differences
(seeAppendixB).
implies that there should be little differencebetween
the resultsfor either of thesesitesfrom usingdifferent
mapping functions.
Further testing of the CfA-2.2 mapping function is
underway.Single-baseline
experimentsare now being
carried out in which a large fraction of the observations from one site are obtainedfor elevationangles
below 5ø elevation, and for a very large fraction
below 10ø elevation, while observations from the
TABLE 3. Comparisonof ElevationAngle Cutoff Tests
Fraction
Site
Name
Onsala
Ar, cm
(Marini)
- 3.7 + 1.0
0.9
Effelsberg
-4.9 ___1.3
Haystack
- 3.9 _+0.8
Owens Valley -1.0 _+0.3
-0.7
0.1
-0.8
-4.2
-4.0
-0.2
7.7%
3.1%
0%
Fort Davis
- 1.1
-0.3
0%
- 1.4 _+0.5
of
Ar, cm Difference, Data Below 15ø
(CfA-2.2)
cm
Elevation
-4.6
other site remain at relatively high (> 10ø) elevation
angles. This procedure should enable us to isolate
mapping function errors for the site at which the low
elevation observations are taken, since we would be
relatively insensitiveto mapping function errors for
the other site.
10.4%
Plans are also being made to optimize the coefficientsin (10)-(12) for site location, and to develop
seasonalatmosphericstructure parameters. For this
purpose, radiosondedata obtained from the National
The entries in the columnsheaded by Ar are the changesin the Climatic Data Center for U.S. sites, and from various
estimatesof the local vertical positionsof the sitescorresponding European centers, will be used. Simultaneously,an
to the baselinelength differencesfrom each of the elevation angle
effort will be made to attempt to increasethe accucutoff tests,shown in Figure 1 (Marini) and Figure 4 (CfA-2.2).
The
column
labeled
"difference"
is the difference
between
the
changesin the radial estimates.The uncertaintiesfor the valuesof
Ar (CfA-2.2) are the sameasfor the valuesfor Ar (Marini).
racy of the mapping function at all elevation angles
(but with emphasisat extendingthe mapping function for use at elevation anglesbelow 5ø) and to in-
1602
DAVIS
ET AL.'
ATMOSPHERIC
DELAY
vestigatepossiblemeans for modeling of horizontal
gradients.
6.
SUMMARY
Errors in modeling the elevation angle dependence
of the atmosphericdelay can causesystematicerrors
in the estimatedradial positionsof the antenna sites.
These radial errors will "map" into the estimatesof
baseline length by an amount approximately proportional to the baselinelength. An elevation angle
cutoff test performedwith the Marini mapping function indicated
that the errors in the estimates of base-
line lengthintroducedby this mappingfunctionwere
of the order of --,5 cm for a baselinelength of • 8000
km and that theseerrors display a systematicdependence on baseline length indicative of a mapping
function error. A new mapping function has been
developed which is based on ray-tracing through
model atmospheres.Repetition of the elevation angle
cutoff test with this new mapping function yields apparent errors in baseline-lengthestimatesof •< 1 cm,
with the differencesshowinglittle or no dependence
on baselinelength.
IN
VLB
INTERFEROMETRY
Z•-• and Z• • are the respective
inversecompressibilities,with the subscriptshaving the samemeaning
as for the pressures.The symbol e is usually usedin
place of pw.Thayer's values for the constantsk•, k2,
and k3 are summarized in Table A1. The uncertainties of thesevalueslimit the accuracywith which
the refractivitycan be calculatedto about 0.02%.
The first term in (A1) representsthe effect of the
induced dipole moment ("displacement polarization") of the dry constituents.The secondterm represents the same effect for water vapor, whereas the
third term representsthe dipole orientation effectsof
the permanent dipole moment of the water molecule.
None of the primary constituentsof dry air (shownin
Table A2) possesses
a permanentdipole moment.
The values for k2 and k3 listed in Table A1 have
been disputed by Hill et al. [1982]. They point out
that Thayer'sextrapolationof the value of k2 from its
value at optical wavelengthsignores the effect of the
rotational
and vibrational
resonances in the infrared
[Van Vleck, 1965]. Hill et al. calculate a theoretical
value for k2 and k3 and find kz = 98 + 1 K/mbar
and k3 = (3.583+ 0.003)x 105 K•/mbar. However,
theseresultsare so greatly in disagreementwith published
values of k2 and k3, which have been obtained
APPENDIX
A: ZENITH
DELAY
FORMULAS
by measurementsin the microwave region [BouThe purposeof this appendix is to derive an accu- douris, 1963; Birnbaum and Chatterjee, 1952], that
rate expressionfor the zenith delay from the wet and Hill recommendsusing the measuredvalues instead
dry refractivityformulas.We pay particular attention of either his or Thayer's. Birnbaum and Chatterjee
to the treatment of the wet/dry mixing ratio. We also find k2 = 71.4 + 5.8 K/mbar and k3 = (3.747
obtain an estimatefor the accuracyof the hydrostatic +0.029) x 105K•/mbar, while Boudouris finds
(i.e., "dry") delay formula, and derive an expression k: = 72 __+
11 K mbar and k3 = (3.75__+
0.03)x 105
we keepThayer'svalues
for the "wet" zenith delay which is consistentwith K'/mbar. As a compromise,
the "dry" zenith delay formula. This "wet" delay for k• and k3 (which differ from the experimental
formula makes use of the most recent expressionfor values by lessthan the uncertaintiesof the latter), but
the wet refractivity and can be used to establishthe choose the (rounded) experimental uncertainties,as
relationship between the observablesof instruments shown in Table A1.
which measurethe radiative emissionof atmospheric
The grouping together of all the dry constituents
water vapor (e.g., water vapor radiometers)and the into one refractivity term is possiblebecausethe relative mixing ratios of thesegassesremain nearly conline-of-sightdelay due to water vapor.
Derivationof the zenithdelayfrom the
stant
in
time
and
over
the
surface
of the
earth
[Glueckauf,1951]. The eight main constituentsof the
refractive index
dry atmosphere are listed in Table A2, along with
The three-term formula for the total refractivity of their molar weight and fractional volume, and a stanmoist air, as given by Thayer [ 1974], is
dard deviation representingthe variability of that
constituentin the atmosphere.Using thesenumbers,
N= k•-• Z•-• + k2-• Z•,• + k3•¾Z•• (A1)we find the mean molar weight Md of dry air to be
Md = 28.9644 + 0.0014 kg/kmol, where the standard
Here T is the temperature,Pdis the partial pres- deviation is an upper bound on the variability of Ma
sure of the "dry" constituents ("dry" is defined basedon the valuesin Table A1 and on the assumpbelow), Pw the partial pressureof water vapor, and tion that theseconstituentsvary independently.
DAVIS
ET AL.'
ATMOSPHERIC
TABLE A1.
DELAY
IN
VLB
INTERFEROMETRY
1603
Constants Used in the Appendix
Constant
Value*
Uncertainty*
Uncertainty Used
(See Text)
Units
kx
k2
k3
77.604
64.79
377600
0.014
0.08
400
0.014
10
3000
K mbar- •
K mbar- 1
K 2 mbar- •
Derived
Constant
Equivalent
Units
k2'
k2-k• M._•.•
= 17+_10
Ma
Kmbar
-x
k3'
k3 + k,_'Tm= (3.82+_0.04)x 105
K'- mbar-•
Here Ma = 28.9644+_0.0014kg kmol-•, molarmassof dry air' M,• = 18.0152kg kmol-•, molarmass
of H20' r m= 260 _+20 K, "meantemperature"(seetext).
*From Thayer [1974].
The inverse compressibilitiesin (A1) represent the
nonideal behavior of their respective atmospheric
constituents.This behavior is describedby the equation of state for the ith constituent Pi = Zip•R•T,
where Pi is the partial pressure,Z• is the compressibility, p• is the massdensity,and Ri is the specificgas
constant for that constituent(Ri = R/M•, where R is
the universal gas constant and Mi is the molar mass),
and T is the absolute temperature. For an ideal gas
Z = 1; Z differs from unity by a few parts per thousand for the atmosphere.The expressionsfor the in-
and
Z•, 1= 1 + 1650(pw/T3X1 - 0.01317t
+ 1.75 x 10-'•t2 + 1.44x 10-6t3)
(A3)
where t is the temperature in degreesCelsius,Pdand
Pware in millibars, and T is in Kelvins. Owens found
that (^2) and (^3) model the compressibility to
within a few parts per million.
The total zenith delay Lz is
Lz -- 10- 6
versecompressibility
Z•-• for dry air and Z• • for
©dz N(z)
(A4)
water vapor were determined by Owens [1967] by
least squaresfitting to thermodynamic data. These Integration of the refractivity in the form given in
expressionsare
(^1) requires knowledge of the profiles of both the
wet and dry constituents,the mixing ratio of which is
Z•-•= 1 + pd[57.97x 10-8(1 + 0.52/T)
highly variable. However, it is possible to create a
-9.4611 x 10-4t/T 2]
(A2) term nearly independentof this mixing ratio. We can
rewrite the first two terms in (A1) by using the equaTABLE A2.
Primary Constituentsof Dry Air and Their
Variability
tion of state as
Pd
Molar
Constituent
Weight,*
kg/kmol
Volume,J'
(Unitless)
28.0134
31.9988
Ar
39.948
CO 2
44.00995
0.000314
0.000010
Ne
20.183
0.00001818
0.0000004
4.0026
Kr
Xe
83.30
•
131.30
• 0.00934
0.00004
0.00002
0.00001
0.00000524
0.(X)(X)(X)04
0.00000114
0.0000001
0.000000087
0.000000001
*U.S. StandardAtmosphere(1976).
t'Gleuckauf[1951].
+
, Pw
N2
O •_
He
0.78084
0.209476
Pw
z; ' + -7' ' =
Fractional
= k•Rap
+ k2-• ZT•
•
(A5)
where the total massdensityp - Pa + P,• is indicated
by the absenceof Subscripts,
and the new constantk•
is given by
Ra
M,•
k•=k2- k,• = k2- k,Ma
(A6)
If a value for the molar weight of water M• =
18.0152 kg/kmol is used ICRC Press, 1974], and if
1604
DAVIS
ET
AL.'
ATMOSPHERIC
DELAY
IN
VLB
INTERFEROMETRY
independent errors in kx, k2, and Md are assumed, above the geoid. Saastamoinenclaims that this exthen we find k• = (17 ___
10) K mbar-x. By using pressionis accurateto within 0.4 km for all latitudes
(A5), we find the expressionfor the total refractivity and for all seasons.Substituting for He into (All)
to be
yields
gm= 9.784m s-2(1 -0.00266 cos2it- 0.00028H)
N= k,R.,p
+ k2-• Z•*+ k3'• Z,7,,* (A7)
+ 0.001rn s-2 = gO,,[f(j.,
H) + 0.0001]
(A13)
It is important to note that the first term in (A7) is
dependent only on the total density and not on the
wet/dry mixing ratio. This term can be integrated by
applying the condition that hydrostatic equilibrium
wheregO,,
= 9.784m s-2. Combiningall theconstants
is satisfied:
L, - [(0.0022768
+ 0.0000005)
rn mbar-•] •
in (A9), along with their uncertainties(assumeduncorrelated),gives
Po
f(,•, H)
(A14)
dP
dz
- -p(z)g(z)
(A8)
where g(z) is the accelerationdue to gravity at the
vertical coordinate z, P(z) is the total pressure,and,
as above, p(z) is the total massdensity.Denoting the
result of the integration of the first term in (A7) as L•,
we find that
L, = [10-6k,Rdg,••]Po
(A9)
where Po is the total atmospheric pressure at the
intersection
of rotation
axes of the radio
antenna
(not the surfacepressure,sincethe antenna is located
some height above the ground; see text), and where
gmis givenby
©Clz
•(z)•(z)
o•zP(Z)
(AlO)
By expandingg(z) to first order in z, it can be seen
that (A10) very nearly representsthe accelerationdue
to gravity at the center of mass of the vertical
column. The value of gmat this point is [Saastamoinen, 1972]
g.•= 9.8062m s-2(1 - 0.00265cos2it - 0.00031He)
(All)
where ,• is the geodetic site latitude and Hc is the
height in kilometersof the center of massof the vertical column of air. The quantity He and thereforegm
is dependentupon the atmospherictotal densityprofile, but Saastamoinen[19723 used his simple model
atmosphereand "average" conditionsto generatethe
expression
H• = 0.9H + 7.3 km
(A12)
where H is the height in kilometers of the station
where a value of R = 8314.34 q-0.35 J kmol -x K -x
has been used for the universal gas constant ICRC
Press,1974]. The uncertainty of the constantin (A14)
takes into account the uncertainty of kx, the uncertainty in g,•, the uncertainty in R, and the variability
of the dry mean molar mass.It does not include the
effect due to nonequilibrium conditions.It is, in fact,
difficult to assessthis effect without actually integrating vertical profiles of vertical wind acceleration
(which, in general, are not available); no attempt to
assessthis effect will be made here. Fleagle and Businger [1980] state that only under extreme weather
conditions (thunderstorm or heavy turbulence) do
thesevertical accelerationsreach 1% of gravity, correspondingto an error in Lx of about 20 mm/1000
mbar. Exactly where the true uncertainty lies between thesevaluesof 0.5 and 20 mm/1000 mbar must
be left to future investigation.
Because the uncertainty associated with L• in
(A14) is so small, and becausevariability is associated with water vapor, Lx is usually (and inaccurately) termed the "dry delay." Something like the
"hydrostatic delay" would be more descriptive,for in
principle the uncertainty of the dry density at any
point is no lessthan the uncertainty of the wet density, whereasthe total densityis very predictable.
The remaining two terms in the expressionfor the
refractivity are wet terms
Nw
=
+k3P]z
T2
j w
-'
The partial pressureof water is not by itself in equilibrium, and water vapor can remain relatively unmixed, making the wet delay very unpredictable.
Water vapor radiometers(WVR's) will, we hope, obviate this problem. However, there are large amounts
of VLBI
and other data for which no WVR
calibra-
tion is available, and more such data are being con-
DAVIS
ET
AL.'
ATMOSPHERIC
DELAY
IN
VLB
INTERFEROMETRY
1605
tinually generated. Thus there is still a need for whencethe (nearly)constantk• is givenby
models of the zenith wet delay. No attempt will be
k• = k3 + k• Tm
(A19)
made to develop one here. All suchmodelsin current
use [e.g., Chao, 1972; Berman 1976; Saastamoinen,
Most investigatorschoosea constantvalue for Tm
1972] use obsolete values for the refractivity con- for all sitesand seasons.For example,for Tm= 260
stants ke and k3. However, these old values induce + 20 K, we find, assumingindependenterrors in k•
errors on the submillimeter level, much less than the
inherent error in the prediction of the wet delay. On
the other hand, these models also tend to be based
on empirical models for the wet atmosphere,
averagedover location and season.However, we believe that site and seasondependenceof the atmospheric profile could cause seasonal and site-
dependentbiasesin these wet models of up to 1020%, based on a comparison of expressionsfor
"average" profilesreportedthroughoutthe literature.
Water vapor radiometers
A water vapor radiometeris a multichannelradiometer which uses the sky brightnesstemperature
near the 22-GHz rotational absorptionline of atmosphericwater vapor to obtain an estimateof the integral of the wet refractivity in (A15). The WVR's now
coming into use should have their "retrieval coefficients" [see Resch, 1984] "optimized" for site and
seasonaldependenceof the atmosphericprofiles.For
this optimization, one uses radiosondeestimatesof
the wet delay ALw to determine the retrieval coefficientsax and a2 definedin the equation
and k3, k• = (3.82+ 0.04)x 105 K 2 mbar-x. This
approach is adequate, since the k• Tm term is only
about 1% of k3, and basedon seasonaltemperature
profiles,seasonalvariationsin k• Tmare one order of
magnitude smaller, or <0.2 mm for a zenith delay.
However, it is fairly common in the literature to use
an incorrect value for k•. This usagearisesfrom implicitly assumingthat Md = M• in (A6), which actually changesthe sign of k•. The value then found for
k• is approximately 0.373, or about •2.5% smaller
than the 0.382 number derived here. This (incorrect)
value results in an underestimateby • 5 mm for a
zenith wet path delay of • 20 cm.
APPENDIX
OF
B: COVARIANCE
DIFFERENCED
PARAMETERS
In this appendix we derive the expressionfor the
covariancematrix for the differenceof two (different)
least squaresestimatesof the sameparameter vector.
We assume that one of the estimates
is based on a
subset of the data used to make the estimate of the
other. We begin by writing the linearized equation
relating the observationsto the parameters.
Yl = AlX -{-t•1
ALw= alf(WVR) + a2g(Po, To)
(A16)
wheref(WVR) is somefunctionof the WVR observables, and g(Po, To) is some function of the surface
temperature and pressure [Resch, 1983]. Both
f(WVR) and g(Po, To)are determinedby theory.By
"radiosonde estimates of the wet delay" we mean
that ALw is determinedby numericalintegration of
the wet refractivity given in (A15) using radiosonde
profilesof p• and T. In practice,most investigators
usea one-termexpressionfor the wet delay:
(B1)
where x is a vector of parametersto be estimated,
is an unknown, Gaussian, zero-mean random vector
whosecovariancematrix is Gy.•, and whosemean
squareis to be minimized,and where y• is a vector of
observations.(The subscripty was chosen for the
covarianccmatrix of • to emphasizethat it representsthe experimentalerrors of the observationsy•.)
The least squaresestimate:• of x basedon y• is
•1 -- [A1
TGy,1
-1 All -1ATt%'.-1
I •'•y,1
Yl
(B2)
The covariancematrix G•,• of the parameterestimate
• •'•
AL,•
= 10-6k•dzT•
]•1 is
(A•7)
where k• is the k3 in (A15) modified for the effect of
k•. This modification is made possibleby using the
mean value theorem to introduce a "mean temperature"
via
-•=
•
•dz
p•Tr,,fdZT'P•
Gx,1= [AT
-1
1Gy,]A1]
(B3)
Let us now consider the least squaresestimate of
•t given a set of observationsYt which are composed
of the previous observationsy x as well as a distinct
set of observations
(A18)
yt =
(B4)
Y2
1606
DAVIS
ET AL.' ATMOSPHERIC
DELAY
We will assumethat y• and Y2 are uncorrelated,so
that
IN
VLB INTERFEROMETRY
In terms of this paper, y x would be composedof
the VLBI
observations
from elevations
above the ele-
vation angle cutoff, while Y2 would be composed of
Gy•=E[y•y•r]=[G•,•
0]
(B5)
,
Gy,2
vector ix is the least squaresestimate of x resulting
observations
•
from below this cutoff in elevation.
The
where E[ ] indicatesexpectation,and Gy,2is the from the observationsy x, while it resultsfrom using
covariance matrix of e2, the observational errors as- all the data (both yx and y2). From (B14) it can be
sociatedwith Y2. The least squaresestimatei t based seen that the covariance matrix of the difference between ix and it is the difference of their respective
on y, is thereforegiven by
covariance
•t = [A,TGy,,
- 1At]- •ArG
t y,t
-•
matrices.
Yt
_[Ar1Gy,1A
-• 1+ A2
r Gy,2
-• A2]-• (A1
r Gy,
- •Y1+ Ara-•
Y2)
-2
__Gx,t G•..•
T -1 Y2
-, 1 q_Gx,tA2Gy,2
(B6)
Acknowledgments.This work was supported by Air Force
GeophysicsLaboratory contract F19628-83-K-0031, NASA contract NAS5-27571, and NSF grants EAR-83-02221 and EAR-8306380.
where
REFERENCES
At =
(B7)
A2
and
Bean, B. R., B. A. Cahoon, C. A. Samson, and G. D. Thayer, A
World Atlas of AtmosphericRadio Refractivity,ESSA Monogr.,
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r - • A2]- •
Gx,t= [A•G•,•A•+ A2Gy,2
(B8) Berman, A. L., The prediction of zenith refraction from surface
The differencebetweenthe parameter estimatesi•
and it will be denoted Ai. The covariance matrix
Gaxof Ai is
measurementsof meteorologicalparameters,Rep. JPL TR 321602, Calif. Inst. of Technol. Jet Propul. Lab., Pasadena,Calif.,
1976.
Birnbaum, G., and S. K. Chatterjee, The dielectric constant of
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G•x = E[Ai Air]
- E[((Gx,t G•.,• - l)i• + Gxt
• Y2)x transpose]
-, Ar2Gy,2
(B9)
Black, H. D., An easily implemented algorithm for the troposphericrange correction,J. Geophys.Res.,83, 1825-1828, 1978.
Black, H. D., and A. Eisner,CorrectingsatelliteDoppler data for
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New York, 1970.
Boudouris, G., On the index of refraction of air, the absorption
and dispersionof centimeterwavesby gasses,J. Res. Natl. Bur.
where I is the identity matrix.
Since yx and Y2 are uncorrelated,we have
E[ixy2r] = E[y2ixr] = 0
Stand., 67D, 631-684, 1963.
(B10)
and therefore
G,•x= (Gx,tG•,x
x- l)Gx,x(Gff,
xXGx,tI)
+ Gx.,A
%.z
Chao, C. C., A model for troposphericcalibration from daily surface and radiosondeballoon measurements,Tech. Memo Calif.
Inst. Technol.,let Propul.Lab., 391-350, 17 pp., 1972.
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AzGx.,
(B
CRC Press,Handbookof Chemistryand Physics,55th ed., pp. B-1
and F-223, Boca Raton, Fla., 1974.
where we have used the fact that a covariance
matrix
is symmetric.Algebraicmanipulation of (B 11) yields
Gax= Gx,,G•Gx,, + Gx,x-- 2Gx,,
+ Gx,Ar
-• A2Gxt
, 2Gy,2
,
(B12)
From (B3) and (B8) it can be seenthat
A• Gy•A2= C-' - C•
•,t
,
1977.
(B13)
Substitution of (B13) into (B12) and cancellation
yield
Gax= G•,x - Gx,,
Fanselow,J. L., Observationmodel and parameter partials for the
JPL VLBL parameter estimation software "MASTERFITV1.0," JPL Publ.,83-39, 54 pp., 1983.
Fleagle, R. G., and J. A. Businger,An Introduction to Atmospheric
Physics,pp. 15-16, Academic,Orlando, Fla., 1980.
Gardner, C. S., Correction of laser tracking data for the effectsof
horizontal refractivity gradients, Appl. Opt., 16, 2427-2432,
(B14)
Glueckauf,E., The compositionof the atmosphere,in Compendium
of Meteorology, edited by T. F. Malone, pp. 3-10, American
MeteorologicalSociety,Boston,Mass., 1951.
Hill, R. J., R. S. Lawrence, and J. T. Priestly, Theoretical and
calculational aspectsof the radio refractive index of water
vapor, Radio $ci., 17, 1251-1257, 1982.
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ET
AL.:
ATMOSPHERIC
DELAY
IN
VLB
INTERFEROMETRY
1607
stratospherein radio ranging of satellites,in The Use of Artificial Satellites for Geodesy, Geophys. Monogr. Ser., vol. 15,
1969.
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