## 2 Satellite Systems and Errors in Position Determination
### Errors in Position Determination
In the case of satellite systems, two atmosphere layers have a significant impact on their operation: the troposphere and the ionosphere. The troposphere is the layer of the Earth's atmosphere closest to the Earth, occupying the smallest area in relation to the other layers, but having the largest mass. It makes up 75% of the total mass of the Earth's atmosphere and is a place where most weather phenomena occur. The troposphere consists of a mixture of gases, mainly azide and oxygen, as well as industrial gases such as carbon dioxide, sulfur, etc. The troposphere is characterized by a continuous decrease in temperature and pressure with increasing altitude. The average decrease in temperature is 0.6 \({}^{\circ}\)C per 100 m, while the average pressure above sea level is 1013 hPa, and about 300 hPa at an altitude of 10 km. Tropospheric error, influencing satellite systems operation, is caused by the refraction of the electromagnetic wave. The dry part of the troposphere, characterized by atmospheric pressure, is responsible for 90% of the positioning error, while 10% is the wet part, characterized by water vapor. The parameter describing the tropospheric delay of the satellite signal related to atmospheric pressure is zenith hydrostatic delay (ZHD). ZHD depends mainly on the refraction associated with the hydrostatic part of the atmosphere, i.e., with the dry part of the atmosphere, and can be calculated based on deterministic models--the Hopfield model for example [55]:  
\[ZHD_{Hop}=2*10^{7}*N_{hyd}*H_{d} \tag{1}\]  
where  
\[N_{hyd}=k_{1}*\frac{P}{T_{k}} \tag{2}\]  
\[H_{d}=40136+148.72*T_{c} \tag{3}\]  
\(k_{1}\)--refraction index, always greater than 1;  
\(P\)--atmospheric pressure;  
\(T_{K}\)--temperature expressed in Kelvins;  
\(T_{C}\)--temperature expressed in degrees Celsius. The error due to the wet part of the atmosphere is usually neglected. In order to eliminate the tropospheric refraction error, corrections are applied to the measurement in a form of special mathematical models, such as the Saastamoinena model for example [55]:  
\[ZWD_{Saas}=0.00227768*\left(\frac{1255}{T_{K}}+0.05\right)*e \tag{4}\]  
where  
\(ZWD\)--zenith wet delay;  
\(T_{K}\)--temperature expressed in degrees Celsius;  
\(e\)--water vapor pressure at the height of the reference station. The ionosphere is an ionized layer of the atmosphere, starting at 85 km above the Earth's surface, reaching an altitude of 2000 km in the exosphere. The essence of the ionosphere is that it contains plasma, which is formed by the ionization of gas particles in the atmosphere by cosmic rays and ultraviolet solar radiation. The ionosphere, through its structure, affects the refraction, reflection, absorption, and polarization of radio waves. These phenomena cause interference in radio communication. That is why positioning errors caused by the ionospheric layer of the Earth's atmosphere consist of electromagnetic waves' propagation speed changes when passing through this layer. Wave speed changes arise from ionized gases influence, as a result of solar activity and cosmic radiation, and are dependent on the frequency of the electromagnetic wave. This phenomenon is called dispersion. The position is determined based on the propagation time of the signal. Once the speed changes, the time and position are also different. The magnitude of the ionospheric effect is proportional to the number of free total electron content (TEC) electrons that fit in an unit cuboid from the receiver to the satellite. TEC is a function of many variable factors: time of day, solar activity, geographic location, and height of zenith satellites. To eliminate the ionospheric error, dual-frequency receivers are used. In single-frequency receivers, the approximate correction is calculated and sent in the navigation message. To eliminate the positioning error caused by ionosphere when single-frequency GPS receivers are used, the Klobuchar model is used [56].