THE ESTIMATION OF THE PROPAGATION DELAY THROUGH THE
TROPOSPHERE FROM MICROWAVE RADIOMETER DATA
J. M. Moran and B. R. Rosen
Harvard-Smithsonian Center for Astrophysics
ABSTRACT
Uncertainty in the estimate of the microwave propagation delay through the troposphere is a
principal limiting factor to the accuracy of the technique of very long baseline interferometry
(VLBI). This uncertainty is due primarily to tropospheric water vapor, the total amount and verti-
cal distribution of which is variable. Because water vapor both delays and attenuates microwave
signals, the propagation delay, or wet path length, can be estimated from the microwave brightness
temperature near the 22.235 GHz transition of water vapor.
We analyzed the data from a total of 240 radiosonde launches taken simultaneously in 1974 at
Chatham, Massachusetts; Albany, New York; and Portland, Maine. Estimates of brightness tempera-
ture at 19 and 22 GHz and wet path length were made from these data. The wet path length in the
zenith direction could be estimated from the surface water vapor density to an accuracy of 5 cm for
the summer data and 2 cm for winter data. Using the brightness temperatures, the wet path could
be estimated to an accuracy of 0.3 cm.
Two dual-frequency radiometers constructed by the National Radio Astronomy Observatory
(NRAO) were refurbished in order to test these techniques. These radiometers were capable of
measuring the difference in the brightness temperature at 30” elevation angle and at the zenith to an
accuracy of about 1°K. In August 1975,45 radiosondes were launched from Haystack Observatory
over an 1 l-day period. Brightness temperature measurements were made simultaneously at 19 and
22GHz with the NRA0 radiometers.
The rms error for the estimation of wet path length from
surface meteorological parameters was 3.2 cm, and from the radiometer brightness temperatures,
1Scm.
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RADIO INTERFEROMETRY
INTRODUCTION
The neutral atmosphere retards and attenuates propagating electromagnetic waves. The retardation
limits the accuracy to which very long baseline interferometry can be used to measure radio source
positions and baseline vectors between the antennas. Atmospheric water vapor can contribute up to
about 40cm of excess propagation path length in the zenith direction at microwave frequencies.
The exact amount cannot be predicted accurately from ground level meteorological variables since
the water vapor is not well mixed in the atmosphere. However, since the index of refraction and ab-
sorption coefficient are functions of the water vapor density, the brightness temperature due to the
self-emission, a weighted integral of the absorption coefficient, is related to the path length, the
integral of the index of refraction. Hence, ground-based radiometric measurements can be used to
estimate the excess phase path. Early evaluations of the effectiveness of this technique were made
by Waters (1967) and Shaper, Staelin, and Waters (1970).
The vertical profiles of temperature and water vapor density are routinely obtained from radio-
sondes launched by the National Weather Service (NWS). From these data, the index of refraction
can be calculated and the path delay estimated. The microwave absorption coefficient can also be
calculated and the brightness temperature estimated from the equation of radiative transfer. We
completed a theoretical study based on radiosondes launched at three locations in New England in
order to determine how well the path delay could be estimated from: (1) surface meteorological
data, (2) radiosonde data from a remote station, and (3) microwave radiometry. We also conducted
an 1 l-day experiment during which we compared actual radiometry data from two microwave radi-
ometers with values of brightness temperatures and path length derived from radiosonde data. Our
study was focused on establishing how well various techniques work. We used the method of linear
regression analysis and examined the residuals. Substantial progress has been made recently by Wu
(1979) and Clafhn, Wu, and Resch (1978) on establishing useful a priori prediction algorithms. A
full report of our work is available in Moran and Penfield (1976).
BACKGROUND PHYSICS
The excess propagation path length is given by
L = 1o-6 N(h)dh
(1)
where N(h) is the refractivity of the air as a function of height. The refractivity of moist air is de-
scribed by the Smith-Weintraub equation (Bean and Dutton, 1966),
N -
77.6
T
p
where
WAVES OF THE FUTURE AND OTHER EMISSIONS
365
T = temperature (“K)
P = total pressure (mb)
e = partial pressure of water vapor (mb).
The first term in equation (2) arises from the displacement polarizations of all the air constituents
including about a l-percent contribution from water vapor at the surface. It is called the “dry
term.”
The second term in equation (2) is due to the dipole moment of the water vapor molecule
and is called the “wet term.”
The refractivity of air is essentially independent of frequency from 0 to 30 GHz. The dispersive
component of the index of refraction associated with the 22.235 GHz transition of water vapor has
a refractivity which is less than 0.02 (Liebe, 1969).
Using the ideal gas law, the dry and wet components of the refractivity, No and N,, can be written
as
ND
_ 77.6p -
T
2.70 x lo4 PD
(3)
NV = 3.73 x lo5 $ = 1720 $,
where pu and pv are the densities of dry air and water vapor in grams per cubic meter. At the sur-
face, No typically varies between 250 and 300; while in New England, N, varies from between
about 10 and 100.
The dry gas obeys the equation of hydrostatic equilibrium which leads immediately to the result
that the excess dry path is
LD
= lo6
s
ND dh = AP,
(5)
where P, is the total pressure at the surface, A = 77.6
R
- = 0.2276 cm mb-1 , R is the universal
gm
gas constant, m is the molecular weight of dry air, and g is the surface gravity constant. Hence, the
value of Lu at sea level at the standard pressure of 1013 mb is 231 cm at 45” latitude. Lu can
therefore be estimated to an accuracy of less than 1 cm provided the pressure is measured to an ac-
curacy of a few millibars since departures from hydrostatic equilibrium are small (Hopfield, 1971).
The partial pressure and density of water vapor are related by the ideal gas law, so that
217e
Pv
= 7f- gmS3 .
(6)
lllll
lIlllllIlllll
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RADIO INTERFEROMETRY
The wet path length is therefore
& = 1720 Iom g dh.
(7)
The assumption that pv is an exponential function with a scale height of 2.2 km and T is constant at
290°K leads to the approximate formulas
and
L,
= 1.3pv
(8)
LV
= l.Oe.
(9)
More exact analysis for prediction of path length from surface meteorological data are given by
Saastamoinen (1973).
The brightness temperature at the surface of the earth, derived from the equation of radiative trans-
fer, can be written as
TB(v) = T,emTV +
Jrn
T (I!) QI (v, !2) e-7’ (VY n)d!?,
0
where
and
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1 _-
TV -
a(v, I?)d!J
(11)
a(v, IZ)dQ
(12)
and where Q is the distance along the ray path from the observer, CY (v, !2) is the absorption coeffi-
cient, T, is the brightness temperature of any extraterrestial radiation source, rv is the total atmo-
spheric opacity along the ray path, and rv is the opacity between the point of emission and the
observer.
The principal contribution to the microwave absorption coefficient are water vapor and oxygen.
The opacity due to oxygen at 22 GHz is about 0.013 nepers; therefore, the brightness temperature
contribution is only about 4°K. This contribution changes very little with time. The absorption
coefficient for oxygen, taken from Meeks and Lilley (1960) has been included in our calculations.
The absorption coefficient due to the water vapor for the 6,, -5,, transition having a rest fre-
quency, ve, of 22.23508 GHz is given by Staelin (1966) as
WAVES OF THE FUTURE AND OTHER EMISSIONS
V2 PP”
CX(V,Q) = 3.24
x 1&4&64/T -
T3.125
1 + 0.0147 y
>
1 (v-v,)2 +A,2
' (v+v,)2 +A9 1
+ 2.55x 1O-8 pvv2
-$$ cm-l
where
I
Au = 2.58 x 10-s
P”T
p
>
(T/3 l&625
and v is the frequency in GHz.
The brightness temperature at center line, for the case where 7v Q 1, is therefore
TB a
PV
p~.875
c&44/T dh
whereas
Lv a
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(13)
(14)
(15)
(16)
To a first approximation, we find that T, (H,O, v = 22 GHz) (“K) - 2.1 Lv (cm). P decreases by
10 percent per kilometer and T by 2 percent per kilometer so that a given amount of water vapor
contributes more heavily at higher altitudes to the brightness temperature than to the wet path
length although saturation tends to reduce the discrepancy. Frequencies can be chosen to maximize
the correlation between path length and brightness temperature (Wu, 1979).
RADIOSONDE STUDIES
Radiosondes are launched routinely at 11 and 23 hours UT from Portland, Maine; Chatham, Massa-
chusetts; and Albany, New York by the National Weather Service. We obtained data for 50 occa-
sions during which simultaneous launches were made at the three stations in July and August 1974
and for 30 occasions in January and February 1974. The data pf the significant reporting points
were used in our analysis. The geopotential height was calculated from the pressure and tempera-
ture (Hess, 1959), and the water vapor density was calculated from the dew point depression and
temperature. The absorption coefficient was calculated as a function of height from pv, P, and T.
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