# Investigation into the Effect of Atmospheric
## 2 Physics of Atmospheric Refraction in GNSS  
According to the laws of reflection and refraction, when a plane wave falls on to a boundary between two homogeneous media of different optical properties, it is split into two waves--a transmitted wave proceeding into the second medium, and a reflected wave propagated back into the first medium [11]. In GNSS, it is well known that refraction occurs when GNSS signals propagate in the ionosphere and troposphere, and reflection occurs when there is multipath [12; 13]. The ionospheric effects on the phase (\(\delta_{p}\)) and code (\(\delta_{g}\)) signal transmission path (\(s\)) can be represented as [6]:  
\[\delta_{p}=\int\big{(}n_{p}-1\big{)}ds=\int\bigg{(}\frac{a_{1}}{f^{2}}+\frac{a _{2}}{f^{3}}\bigg{)}ds \tag{1}\]  
where \(n_{p}\) denotes the refractive index of ionosphere on phase, coefficients \(a_{1}\) and \(a_{2}\) depend on the electronic density \(N_{e}\), and \(f\) is the frequency:  
\[\delta_{g}=\int\big{(}n_{g}-1\big{)}ds=\int\bigg{(}-\frac{a_{1}}{f^{2}}-\frac{a _{2}}{f^{3}}\bigg{)}ds \tag{2}\]  
where \(n_{g}\) denotes the refractive index of ionosphere on group. Omitting the second term on the right-hand side of Equations (1) and (2), we get:  
\[\delta_{p}=-\delta_{g}=\int\bigg{(}\frac{a_{1}}{f^{2}}\bigg{)}ds \tag{3}\]  
The ionospheric effects on the phase and code measurements have the opposite signs and have approximately the same magnitude. The coefficient \(a_{1}\) is \(-40.3N_{e}\), where \(N_{e}\) is the electronic density. The total electronic content (TEC) in the zenith direction can be defined as:  
\[TEC=\int N_{e}ds \tag{4}\]  
which can be computed from special models; see [6] for the details. Particulate matters are trapped in the planetary boundary layer that is usually below 2 km measured from sea level [14] while the ionosphere is in the region of 50-1500 km measured from sea level [15]. Therefore, no physical interaction between particulate matters in the lower atmosphere with ions/electrons in the ionosphere is possible. Tropospheric effect depends on the temperature, pressure, humidity and altitude of the antenna location, it causes the same delays in different GNSS frequencies and measurement types (i.e., pseudorange and carrier phase). Similar to the ionospheric path delay, the tropospheric path delay can be written as [6]:  
\[\delta=\int(n-1)ds \tag{5}\]  
where \(n\) is the refractive index of the troposphere, the integration is taken along the signal path. Scaling of the refractive index anomaly (\(n-1\)) is usually made by:  
\[N=10^{6}(n-1) \tag{6}\]  
where \(N\) is called tropospheric refractivity. \(N\) can be divided into the wet (about 10%) and dry (about 90%) components:  
\[N=N_{w}+N_{d} \tag{7}\]  
where indices \(w\) and \(d\) denote the wet and dry components, which are caused by the water vapour and dry atmosphere, respectively. Equation (5) becomes:  
\[\delta=\delta_{w}+\delta_{d}=10^{-6}\int Nds \tag{8}\]where the wet component of the tropospheric path delay (\(\delta_{w}\)):  
\[\delta_{w}=10^{-6}\int N_{w}ds \tag{9}\]  
and, the dry component of the tropospheric path delay (\(\delta_{d}\)):  
\[\delta_{d}=10^{-6}\int N_{d}ds \tag{10}\]  
Particulate matter (PM\({}_{2.5}\) and PM\({}_{10}\)) may float in the air/gas or dissolve in water and become an aerosol [16], suspended in the troposphere [17]. The presence of particulate matter may change the refractivity of the troposphere. When an electromagnetic wave propagates through the \"layer\" of particulate matter, the possible physical interactions are reflection, refraction, absorption and scattering. GNSS frequency range (about 1.2-1.6 GHz) can be considered as very short radio waves or very long microwaves in the electromagnetic spectrum. We know that we can still receive GNSS signals when there is PM\({}_{2.5}\)/PM\({}_{10}\) particulate matter in the atmosphere, therefore, the most likely and important physical interaction is refraction. However, it could be that the PM\({}_{2.5}\)/PM\({}_{10}\) particulate matter has no physical effect on GNSS signals.