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# Galileo Tropospheric Correction Model Fundamentals
Title Galileo Tropospheric Correction Model
Author(s) A. Martellucci, ESA
Level Intermediate
Year of Publication 2012

For GNSS working in L-band the main effects are due to Earth’s atmosphere and the characteristics of the local environment of the receiver. On this sense, Earth’s atmosphere can be classified on troposphere, whose main effect is a group delay on the navigation signal due to water vapour and the gas components of the dry air, this delay is non-dispersive (independent of frequency); and the ionosphere, the ionised part of the atmosphere, that induces a dispersive group delay that is several of orders of magnitude larger than the one from the troposphere. Others ionospheric effects such as scintillations and refraction may be present.

## Tropospheric Effects

Due to the refractive index N of the earth’s neutral atmosphere (N > 1) GNSS microwave signals suffer from tropospheric propagation delays. The total tropospheric delay in direction of a particular satellite - slant path delay (SPD) - can be divided into a hydrostatic and a wet component. Using mapping functions, these two delays are projected into zenith direction and viceversa.

$\displaystyle{ SPD \left( \epsilon \right)= m_h \left( \epsilon \right) \cdot ZHD + m_W \left( \epsilon \right) \cdot ZWD \qquad \mbox{(1)} }$

• SPD$\displaystyle{ \left( \epsilon \right) = }$ slant path delay [m] along elevation angle $\displaystyle{ \epsilon }$
• ZHD = zenith hydrostatic delay [m]
• ZWD = zenith wet delay [m]
• $\displaystyle{ m_h \left( \epsilon \right), m_W \left( \epsilon \right) }$ are the mapping functions of the hydrostatic and wet path delay along elevation angle $\displaystyle{ \varepsilon }$

For the mapping functions, $\displaystyle{ m_h \left(\epsilon \right), m_W \left(\epsilon \right) }$ the model of Niell is currently used.

The ZHD and ZWD are related to the air refractivity N, which in turn is determined by values of air total pressure, P [hPa], temperature, T [K], and water vapour pressure, e [hPa].

$\displaystyle{ N = K_1 \left(P-e \right)+ K_2\frac{e}{T}+ K_3\frac{e}{T^2} \equiv k_1\left(P-e \right)Z_d^{-1} + \left(k_2\frac{e}{T} + k_3\frac{e}{T^2}\right)Z_v^{-1} \qquad \mbox{(2)} }$

$\displaystyle{ Z_d }$ and $\displaystyle{ Z_v }$ are the compressibility factors of dry air and water vapour. Capital K is used for the formulation that does not include compressibility. In the following tables some experimental values of air refractivity coefficients available from different authors are given. The uncertainty of air refractivity parameters, with particular regard to $\displaystyle{ K_3 }$, produces an rms fluctuation of the ZTD of about 6 mm on a global scale.

Author(s) $\displaystyle{ K_1 }$
$\displaystyle{ [1/hPa] }$
$\displaystyle{ K_2 }$
$\displaystyle{ [K/hPa] }$
$\displaystyle{ K_3 * 10^{-5} }$
$\displaystyle{ [K^2/hPa] }$
Smith & Wintraub 77.607
± 0.013
71.6
± 8.5
3.747
± 0.013
Thayer 77.64
± 0.014
64.79
± 0.08
3.776
±0.004
Hasegawa & Strokesb 77.6
±0.032
69.4
± 0.146
3.701
± 0.003
Liebe 77.676
± 0.023
71.631 3.74656
Bevis
weighted
77.6 69.4 3.701
Bevis
unweighted
77.6
± 0.05
70.4
±2.2
3.739
± 0.012
Rueger
best avail. 
77.695
± 0.013
71.97
± 10.5
3.75406
± 0.03
Rueger
best average. 
77.689
±0.0094
71.295
± 1.3
3.75463
± 0.0076

The zenith hydrostatic delay ZHD can be modelled using total pressure at the antenna site. The model of Saastamoinen  is a rather accurate hydrostatic model:

$\displaystyle{ ZHD = \dfrac{0.0022767 \cdot p}{1-0.00266 \cdot cos 2 \varphi - 0.00028 \cdot h} \qquad \mbox{(3)} }$

• $\displaystyle{ \varphi }$: Ellipsoidal latitude
• $\displaystyle{ h }$: Surface height above the ellipsoid in [km]
• $\displaystyle{ p }$: Total Surface pressure in [hPa]

Therefore the ZWD is given by the following integral along the atmospheric profile:

$\displaystyle{ ZWD = 10^{-3} \cdot \int\limits_{h_0}^{\infty}\left( K_2'\frac{e}{T} +K_3\frac{e}{T^2}\right)\,dh \qquad \mbox{(4)} }$

$\displaystyle{ K_2' =K_2 - K_1\frac{M_w}{M_d}= 22.1 [K/hPa] }$
$\displaystyle{ M_d = 28.9644 }$ , $\displaystyle{ M_w = 18.0152 }$ are the Molar weights of dry and wet air [g/mol]

The zenith wet delay can also be estimated, with a reduced accuracy, from the vapour pressure at the height of the receiver, $\displaystyle{ e_0 }$, and the local climatological parameters,$\displaystyle{ \lambda }$ and $\displaystyle{ T_m }$ related to the vertical profiles of air total pressure, temperature and water vapour pressure.

$\displaystyle{ ZWD\left(e_0,\lambda,T_M\right) = 10^{-6} \cdot \frac{R_d}{g_m} \cdot \left(K_2' + \frac{K_3}{T_M}\right)\cdot \frac{e_0}{\lambda +1} \approx 10^{-6} \cdot \frac{R_d}{g_m}\frac{K_3}{T_M} \frac{e_0}{\lambda +1} \qquad \mbox{(5)} }$

This equation is assumed to be valid under the following condition $\displaystyle{ e = e_0 \left(p/p_0\right)^{\lambda+1} }$

$\displaystyle{ R_d = 287.054 [J/(Kg K)] }$
$\displaystyle{ T_M = \int\limits_{h_0}^{\infty}\frac{e}{T}\,dh / \int\limits_{h_0}^{\infty}\frac{e}{T^2}\, dh = }$ Effective Mean temperature of the water vapour column above the receiver [K]
$\displaystyle{ g_m=g\left(\varphi,h_0\right) = 9.784 \cdot \left(1-0.00266 \cdot cos\left(2\cdot\varphi\right)-0.00028\cdot h_0\right) = }$ Gravity acceleration at the mass centre of air $\displaystyle{ [m/s^2] }$

In the framework of the Galileo project ESA used the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis product ERA15 covering the period from 1979 to 1993 to derive the input parameters and the coefficients for the calculation of ZHD ZWD. ERA15 global fields are characterized by a horizontal resolution of 1.5 x 1.5deg, a vertical resolution of 31 levels and a temporal resolution of 6 hours (00, 06, 12, 18 UTC). The 31 levels are defined in terms of pressure (hPa), with a higher resolution in the planetary boundary layer, where the levels follow the earth surface. Values of temperature (K) and specific humidity (kg/kg), associated with these pressure levels, are also provided with the dataset. The analysis produced among the others, monthly statistics (unconditioned and conditioned to the hour of the day) of “surface data”, $\displaystyle{ P\, , e\, , T_m\, , \lambda }$, with a world-wide coverage.

Air total pressure, P, is assumed to be affected by a seasonal variation:

$\displaystyle{ X_i\left(D_y\right)=a1_i-a2_i cos \left [ 2\pi \dfrac{\left(D_y-a3_i\right)}{365.25}\right ] \qquad \mbox{(6)} }$
$\displaystyle{ a1_i = }$ average value of the parameter
$\displaystyle{ a2_i = }$ seasonal fluctuation of the parameter
$\displaystyle{ a3_i = }$ day of the minimum value of the parameter
$\displaystyle{ D_y = }$ day of the year [1 ..365.25]

In addition $\displaystyle{ T_m\, , e\, , \lambda }$ are assumed to be affected by both seasonal and diurnal fluctuations:

$\displaystyle{ X_i\left(D_y,H_d\right)=a1_i - a2_i cos \left [ 2\pi \dfrac{\left(D_y-a3_i\right)}{365.25}\right ] - b2_i \left(D_y\right)cos \left [ 2\pi \dfrac{\left(H_d – b3_i \left(D_y\right)\right)}{24}\right ] \qquad \mbox{(7)} }$
$\displaystyle{ H_d = }$ hour of the day [0..24)
$\displaystyle{ b2_i = }$ amplitude of daily fluctuation.
$\displaystyle{ b3_i = }$ hour of the day at which the minimum value occur.

The model has been validated using and independent ECMWF global product covering the period from March to October 2002 with a spatial resolution of 1 x 1 deg. The global map of the model rms error is given in Figure 1. The overall rms errors of ZTD, ZHD and ZWD are 4.5, 1.6 and 4.4 cm with biases of 9, 0.6 and 8 mm, respectively. Those errors can be reduced by using as input actual meteorological ground measurements and as an additional step actual values of $\displaystyle{ \lambda }$

The input parameters of the model can be scaled from surface to height h by assuming:

$\displaystyle{ T_m\left(h\right) = T_{ms} - \alpha_m \cdot \left( h - h_s\right) \qquad \mbox{(8)} }$
$\displaystyle{ a_m }$ lapse rate of the mean temperature of water vapour, [K/km], also derived from ERA15.

As an example the error of the model at 5000 m, in the upper part of the troposphere, the model error rms reduces to 2.5, 2.2 and 1 cm for ZTD, ZHD and the ZWD.