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\write 0 {9.  TROPOSPHERIC MODEL \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\large CHAPTER 9} {\bf TROPOSPHERIC MODEL}
\bigskip

\write 0 {\indent Optical Techniques \noexpand\dotfill\the
\pageno\noexpand\break}
\noindent{\bf Optical Techniques}

\nobreak

     The formulation of Marini and Murray (1973) is commonly used
in laser ranging.  The formula has been tested by comparison with
ray-tracing radiosonde profiles.

     The correction to a one-way range is
$$\Delta R={f(\lambda)\over f(\phi,H)}\cdot{A+B\over\sin E+{B/(A+B)
\over\sin E+0.01}},\eqno(1)$$
where
$$A=0.002357P_0+0.000141e_0,\eqno(2)$$
$$B=(1.084\times 10^{-8})P_0T_0K+(4.734\times 10^{-8}){P_0^2\over T_0}
{2\over(3-1/K)},\eqno(3)$$
$$K=1.163-0.00968\cos 2\phi-0.00104T_0+0.00001435P_0,\eqno(4)$$
where
\item{$\Delta R$} = range correction (meters),
\vskip-1pc
\item{$E$} = true elevation of satellite,
\vskip-1pc
\item{$P_0$} = atmospheric pressure at the laser site (in $10^{-1}$ 
kPa, equivalent to millibars),
\vskip-1pc
\item{$T_0$} = atmospheric temperature at the laser site (degrees  
              Kelvin),
\vskip-1pc
\item{$e_0$} = water vapor pressure at the laser site ($10^{-1}$ kPa,   
               equivalent to millibars),
\vskip-1pc
\item{$f(\lambda)$} = laser frequency parameter ( $\lambda$ =
wavelength in micrometers), and
\vskip-1pc
\item{$f(\phi,H)$} = laser site function.

\noindent Additional definitions of these parameters are available.
The water vapor pressure, $e_0$, can be calculated from a relative
humidity measurement, $R_h$(\%) by
$$e_0={R_h\over 100}\times 6.11\times 10^{7.5(T_0-273.15)\over
237.3+(T_0-273.15)}.$$
The laser frequency parameter, $f(\lambda)$, is
$$f(\lambda)\equiv 0.9650+{0.0164\over\lambda^2}+{0.000228\over
\lambda^4}.$$
$f(\lambda) = 1$ for a ruby laser, [{\it i.e.} $f$(0.6943) =
1], while $f(\lambda_G) = 1.02579$ and $f(\lambda_{IR})
= 0.97966$ for green and infrared YAG lasers.

\noindent The laser site function is
$$f(\phi,H) = 1 - 0.0026 \cos 2\phi - 0.00031 H,$$
where $\phi$ is the latitude and $H$ is the geodetic height (km).


\write 0 {\indent Radio Techniques \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\bf Radio Techniques}

\nobreak

     The differences between mathematical tropospheric models are
often less than the errors which would be introduced by the
character and distribution of the wet component and by the
departures of the refractivity from azimuthal symmetry.  For this
reason it is customary in the analysis of geodetic data to
estimate the zenith atmospheric delay and to model only the
mapping function, which is the ratio of delay at a given
elevation angle to the zenith delay.  The mapping function may be
for the hydrostatic, wet, or total troposphere delay. 
Accordingly, the IERS conventional model applies primarily to
the mapping functions.  For the most accurate {\it a priori}
hydrostatic delay, desirable when the accuracy of the estimate of
the zenith wet delay is important, the formula of Saastamoinen
(1972) as given by Davis {\it et al.} 1985) should be used.

     Comparisons of many mapping functions with the ray tracing of
a global distribution of radiosonde data have been made by Janes
{\it et al.} (1991) and by Mendes and Langley (1994).  For observations
below 10$^\circ$ elevation, which may be included in geodetic 
programs in order to increase the precision of the vertical
component of site position, the mapping functions of Lanyi
(1984), Ifadis (1986), Herring (1992, designated MTT) and Niell
(1996, designated NMF) are the most accurate.  Only the last
three were developed for observations below an elevation of 6$^\circ$,
with MTT and NMF being valid to 3$^\circ$ and Ifadis to 2$^\circ$.

     Each of these mapping functions consists of a component for
the water vapor and a component for either the total atmosphere
(Lanyi) or the hydrostatic contribution to the total delay
(Ifadis, MTT, and NMF).  In all cases the wet mapping should be
used as the function partial derivative for estimating the
residual atmosphere zenith delay.

     The parameters of the atmosphere that are readily accessible
at the time of the observation are the surface temperature,
pressure, and relative humidity.  The mapping functions of Lanyi,
Ifadis, and Herring were developed to make use of this
information.  Lanyi additionally allows for parameterization in
terms of the height of a surface isothermal layer, the lapse rate
from the top of this layer to the tropopause, and the height of
the tropopause.  Including the surface meteorology without these
data results in larger discrepancies from radiosonde data than
the Ifadis and Herring models.

     The hydrostatic mapping function of Niell differs from the
other three by being independent of surface meteorology.  It
relies instead on the greater contribution by the conditions in
the atmosphere above approximately 1 km, which are strongly
seasonal dependent.  The RMS variation is comparable to those
using Ifadis and MTT, and all three are less than that from the
Lanyi model when only surface data are available.  Thus NMF
offers comparable precision and accuracy to Ifadis and MTT, when
they are provided with accurate surface meteorology data, but
with no dependence on external measurements.

     Thus, if information is available on the vertical
temperature distribution in the atmosphere, Lanyi is preferred. 
Otherwise one of the other three mapping functions should be
used.

\noindent{\bf References}

\nobreak

\item{}\kern-2pc
Davis, J. L., Herring, T. A., Shapiro, I. I., Rogers, A. E. E., and
     Elgered, G., 1985, ``Geodesy by Radio Interferometry: Effects
     of Atmospheric Modelling Errors on Estimates of Baseline
     Length," {\it Radio Science}, {\bf 20}, No. 6, pp. 1593--1607.

\item{}\kern-2pc
Herring, T. A., 1992, ``Modeling Atmospheric Delays in the Analysis
     of Space Geodetic Data," {\it Proceedings of Refraction of
     Transatmospheric Signals in Geodesy}, Netherlands Geodetic
     Commission Series, {\bf 36}, The Hague, Netherlands, pp. 157--164.

\item{}\kern-2pc
Ifadis, I. I., 1986, ``The Atmospheric Delay of Radio Waves: Modeling
     the Elevation Dependance on a Global Scale," {\it Technical
     Report No. 38L}, Chalmers U. of Technology, G\"oteburg, Sweden.

\item{}\kern-2pc
Janes, H. W., Langley, R. B., and Newby, S. P., 1991, ``Analytical
     Tropospheric Delay Prediction Models:  Comparison with Ray-
     tracing and Implications for GPS Relative Positioning," {\it
     Bull. G\'eod}, {\bf 65}, pp 151--161.

\item{}\kern-2pc
Lanyi, G., 1984, ``Tropospheric Delay Affecting Radio Interferometry,"
{\it TDA Progress Report}, pp. 152--159; see also {\it Observation
     Model and Parameter Partials for the JPL VLBI Parameter
     Estimation Software 'MASTERFIT'-1987}, 1987, JPL Publication
     83-39, Rev. 3.

\item{}\kern-2pc
Marini, J. W. and Murray, C. W., 1973, ``Correction of Laser
     Range Tracking Data for Atmospheric Refraction at Elevations
     Above 10 Degrees," NASA GSFC X-591-73-351.

\item{}\kern-2pc
Mendes, V. B. and Langley, R. B., 1994, ``A Comprehensive Analysis
     of Mapping Functions Used in Modeling Tropospheric Propagation
     Delay in Space Geodetic Data," International Symposium on 
     Kinematic Systems in Geodesy, Geomatics and Navigation, Banff
     Canada.

\item{}\kern-2pc
Niell, A. E., 1996, ``Global Mapping Functions for the Atmospheric
     Delay of Radio Wavelengths," {\it J. Geophys. Res.}, {\bf 101},
     pp. 3227--3246.

\item{}\kern-2pc
Saastamoinen, J., 1972, ``Atmospheric Correction for the Troposphere
     and Stratosphere in Radio Ranging of Satellites,"
     {\it Geophysical Monograph 15}, Henriksen (ed), pp. 247--251.

\vfil\eject
