# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Propagation Path (Atmosphere)
#### 3.2.2 Tropospheric Delay  
When radar waves travel through the troposphere, they are refracted and scattered by molecules as well as solid and liquid particles floating in the atmosphere. The refractivity index, \(N\), deviates from unity (i.e., 1) owing to the polarizability of the molecules and the particles in the air [76]. Accordingly, the three terms that impact the refractivity inside the troposphere, including the hydrostatic term, moist term, and liquid term, may be generated based on the molecules and their polarizability [76; 81; 101]:  
* Hydrostatic term: The dry constituents (primarily non-polar nitrogen and oxygen molecules) have an induced dipole moment when interacting with radar microwaves. This means the molecules are polarized, and during the precise moment of polarization, the center of charges is displaced towards the direction of the electric field. Since sometimes the dry constituents include non-polar water vapor, the \"hydrostatic delay\" is a more precise term than \"dry delay\", which is misleading [102]. * Wet term: The wet constituents (mainly polar water vapor molecules) have an induced dipole moment when interacting with radar microwaves. * Liquid water term: the water molecules (polar liquid water molecules) have a permanent dipole moment when interacting with radar microwaves. The three terms characterize the delays happening in the troposphere. Tropospheric delays are the addition of delays caused by the hydrostatic term and the wet term. The delay induced by the liquid water term causes liquid water delay (explain in Section 3.2.3). Unlike free electrons in the ionosphere, tropospheric media are nondispersive. Therefore, tropospheric delays are independent of the carrier frequency. The tropospheric phase can be written as Equation (8), where \(z_{1}\) is ground surface height, and \(z\) is the height of the tropopause (which varies spatially). As a reference, the tropopause in the tropics is about 17 km, at middle latitudes about 11 km, and in polar regions is about 9 km [84]. Although most tropospheric delays happen within the lower troposphere [102], the integral water vapor along the propagation path within the whole troposphere should be obtained to accurately calculate the tropospheric phase. Since the refractivity index \(N\) is a very small number, the value is scaled by a factor of \(10^{-6}\) by definition [77]. Also, as the microwave propagates through the atmosphere along the slant range, the calculation of the delay should be multiplied by the reciprocal of the cosine of incidence angle. \[\phi_{\it tropo}=-\frac{4\pi}{\lambda}\times\frac{10^{-6}}{\cos\theta}\int_{z _{1}}^{z}\Biggl{[}\biggl{(}k_{1}\frac{P}{T}\biggr{)}_{\it hydr}+\biggl{(}k_{ 2}\frac{e}{T}+k_{3}\frac{e}{T^{2}}\biggr{)}_{\it wet}\Biggr{]}dz \tag{8}\]  
Since microwave wavelengths are inherently sensitive to phase changes, the same amount of signal delay will produce larger phase shifts in short-wavelength sensors than in long-wavelength sensors. For instance, a phase variation caused by spatiotemporal variations of 40 mm/km can result in 2.6 phase cycles for X-band wavelength, 1.4 phase cycles for C-band wavelength, 0.67 phase cycles for S-band wavelength, and 0.3 phase cycle for L-band wavelength. Therefore, phase variations are inversely proportional to microwave wavelength. This calculation is achieved with Equation (9), by which we could convert the number of spatiotemporal variations in mm to the number of phase cycles. Figure 4 depicts the number of phase cycles produced when the spatiotemporal variation ranges from 0 to 120 mm/km. When the spatiotemporal variations are 120 mm/km, it causes eight times more phase shifts in the X-band than in the L-band.