Atmospheric Sensing Using GNSS RO - GNSS Remote Sensing: Theory, Methods and Applications - page 124

6.1.2 Dry Atmosphere Retrieval (Density, Pressure

and Temperature)

In the regions where the water vapor is negligible, Eq. ( 6.1 ) reduces to N D c 1 P / T .
As the saturation vapor pressure decreases rapidly with decreasing temperature
according to Clausius-Clapeyron equation, the water vapor pressure P w can be
neglected in the upper troposphere where temperature is low (e.g., T < 250 K)
(Kursinski et al. 2000a ;Hajjetal. 2002 ). Combining the Eq. ( 6.1 ) with the
ideal gas law (i.e., P D ( R / m ) T , where m is the mean molecular mass of
air), the atmospheric density ( r )(inkgm 3 ) can be expressed as a function of
refractivity by

N.r/m d
c 1 R

P.r/m d
T.r/R ;

.r/ D D (6.2)

where R D 8.314 (J K 1 mol 1 ) is the universal gas constant. The m d is the mean
molecular mass of dry air, which can be deemed as a constant as the atmosphere is
well-mixed below the homopause ( 100 km altitude) (Kursinski et al. 1997 ). With
a further assumption of hydrostatic equilibrium:

@P .r /
@r

D .r/g; (6.3)

where g is the gravitational acceleration, the atmospheric pressure ( P ) can be derived
from the density by integrating the hydrostatic equation given the density profile
obtained from the refractivity in Eq. ( 6.2 ).

Z r top Z r top

P.r/ D P r top C

c 1 N r top T r top C

1 m d

c 1 R

.r/g.r/dr D N.r/g.r/dr

r r

(6.4)

Note that an upper boundary condition of pressure, P ( r top ) is required to obtain
the pressure profile at any given altitude. The simplest option is to use zero pressure
at the top of the atmosphere (e.g., P D 0 when r !1 ). However, the vertical extent
of the density profile is constrained by the RO refractivity retrieval. In practice
the temperature from weather model analysis or climatology at a high altitude
(e.g., in the upper stratosphere) is generally used. Fractional pressure errors due
to such initial temperature guess decrease rapidly as the integration moves deeper
into the atmosphere (Kursinski et al. 1997 ). Given the pressure and refractivity (or
density) profiles, accurate temperature profiles can be easily derived based on Eq.
( 6.2 ) throughout the stratosphere down to mid-troposphere and below depending on
latitudes, where the water vapor is negligible, e.g.,

T.r/ D c 1 P.r/

N.r/ :

(6.5)