# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Propagation Path (Atmosphere)  
When microwave signals propagate through the atmosphere, they enter the exosphere, thermosphere (contains the ionosphere in its lowest portion), mesosphere, stratosphere, and then the troposphere, respectively. In this process, a couple of propagation impairments occur due to different physical reasons, including phase delay or advance, signal attenuation, signal depolarization, tropospheric refractive fading, and ionospheric signal scintillations [76]. These propagation impairments influence InSAR measurements in different ways with varying magnitudes. Nevertheless, because phase delay or advance is the primary cause of systematic error in InSAR-derived measurements, this section focuses on phase delay or advance. Since the refractivity index of the mediums in the ionosphere and troposphere differs slightly from that of a vacuum, wave velocity decreases or increases, resulting in phase advances in the ionosphere and phase delays in the troposphere [77]. The errors caused by this effect are called clock timing errors [30]. Phase advance or delay can be characterized by the dimensionless refractivity index, \(N\)[28; 78], and is composed of four different elements: hydrostatic component, wet component, ionospheric component, and liquid component (Equation (5)) [79; 80; 81]. \[N=\left(k_{1}\frac{P}{T}\right)_{hydr}+\left(k_{2}^{\prime}\frac{e}{T}+k_{3} \frac{e}{T^{2}}\right)_{wet}-\left(4.03\times 10^{7}\frac{n_{e}}{f^{2}} \right)_{iono}+\left(1.4W\right)_{liquid} \tag{5}\]  
In Equation (5), \(k_{1}\), \(k_{2}^{\prime}\), and \(k_{3}\) are empirical constants [80]; \(P\) is total atmospheric pressure in hPa; \(T\) is temperature in Kelvin; \(e\) is the partial pressure of water vapor in hPa; \(k_{1}\) is 77.6 K hPa\({}^{-1}\); \(k_{2}^{\prime}\) is 23.3 K hPa\({}^{-1}\); \(k_{3}\) is 3.75 \(\times\) 10\({}^{5}\) K\({}^{2}\) hPa\({}^{-1}\); \(n_{e}\) is the electron number density per cubic meter;\(f\) is radar frequency in GHz; and \(W\) is the liquid water content in g/m\({}^{3}\). The signal delay induced by the hydrostatic and wet components is known as tropospheric delay, the signal delay caused by the liquid component is known as liquid water delay, and the signal advance generated by the ionospheric component is known as an ionospheric advance. tropospheric delays and ionospheric advances both cause phase patterns at various scales ranging from short wavelength to long wavelength artifacts. Both error sources are more substantial than the liquid water delay in most cases [28]. As liquid water delay accounts for relatively much smaller delays, it is usually ignored during InSAR processing and analysis. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Satellite Operation** & **Launch Year** & **Range Uncertainties** & **Azimuth Uncertainties** \\ \hline ERS & 1991 & 1.5 & \textless{}1.5 \\ ENVISAT & 2002 & 0.5 & \textless{}1.5 \\ TerraSAR-X & 2007 & 0.2 & \textless{}0.5 \\ Sentinel-1 & 2014 & 0.2 & \textless{}0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Range and azimuth uncertainties of the velocity gradients caused by orbital errors for each satellite in mm yr\({}^{-1}\) 100 km\({}^{-1}\) (modified from [75]). It should be noted that the term \"advance\" is used in this article to distinguish it from the \"delay\" that occurs within the troposphere. Since the phase velocity in the ionosphere is greater than in a vacuum, there is a phase advance along the propagation route rather than the delay seen in the troposphere [82; 83]. In other words, the increased total electron content along the propagation path in the ionosphere will result in a decreased observed range, so the phase is \"advanced\" [28]. Due to the fact that the phase is advanced in the ionosphere but delayed in the troposphere, the sign of the ionospheric component in Equation (5) is negative, whereas the other components (all of which occur within the troposphere) are positive.