# Investigation into the Effect of Atmospheric
## 2 Physics of Atmospheric Refraction in GNSS
If the refractive index of the particle is only slightly different than that of the surrounding medium [18]:  
\[\delta_{g}=\int\big{(}n_{g}-1\big{)}ds=\int\bigg{(}-\frac{a_{1}}{f^{2}}-\frac{a _{2}}{f^{3}}\bigg{)}ds \tag{2}\]0  
The unity inside the absolute value symbol in Equation (11) is the refractive index of the medium relative to itself, because \(n\) is the refractive index of the particle relative to that of the medium. If the condition expressed by Equation (11) is fulfilled, then the particle interacts very weakly with the incident light [18]; this may not be applicable to other electromagnetic frequencies. Nephelometry is used to measure the aerosol scattering coefficient, and it is suggested that PM\({}_{2.5}\) has the real part of refractive index in the range of 1.3-1.8 and the imaginary part of refractive index in the range of 0.000-0.200 in the visible light spectrum [19]; the wavelength in the visible light spectrum is shorter than the diameter of particulate matter and it is much shorter than the GNSS signal wavelengths. We know that a large variety of chemical compounds are involved in the formation of aerosols, and at the same time it is extremely difficult to determine the chemical composition of an atmospheric aerosol. A lack of accurate knowledge of this chemical composition makes it difficult to infer the property that strongly depends upon it, namely, the refractive index (both real and imaginary) [20]; in the literature a value of 1.55 has often been assigned for atmospheric aerosols [21]. The imaginary refractive index of atmospheric particulate matter in the 0.3-1.7 \(\upmu\)m spectral region shows seasonal and geographic variations [22]; note that this spectrum is much shorter than the GNSS spectrum and the imaginary refraction (absorption) is not a concern in this paper. In practice, most refractivity measurement methods such as a specular reflection technique of normal incidence at the surface of disks made out of aerosol and [PERSON]'s refracometer could cause chemical and/or physical changes and thus result in an error in the estimation of the in situ refractive index [21]. There is no physical model of the effect of particulate matter PM\({}_{2.5}\)/PM\({}_{10}\) on GNSS signals in the literature. An approximate generic model of non-gaseous atmospheric constituents (\(<\)1 mm), based on the Clausius-Mossotti equation for refractivity, is given in [10] as:  
\[\delta_{g}=\int\big{(}n_{g}-1\big{)}ds=\int\bigg{(}-\frac{a_{1}}{f^{2}}-\frac{a _{2}}{f^{3}}\bigg{)}ds \tag{2}\]1  
where \(M\) denotes the mass content of the particles per unit of air volume, \(\rho\) denotes the density of the particles, \(\varepsilon\) denotes the permittivity of the particles, and \(M/\rho\) is the mass fraction of the suspended particles. As stated above, due to the complex composition of particulate matter and the difficulties of measuring the refractive index of the particulate matter in the atmosphere, it is difficult to assess the impact of particulate matter (PM\({}_{2.5}\)/PM\({}_{10}\)) and its concentration on the tropospheric refractive index and on the GNSS signal propagation. This paper uses GPS raw carrier-to-noise ratio (CNR) data in consecutive sidereal days to analyze the impact of particulate matter on GNSS signal propagation, and validate the approximate generic model Equation (12).