# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Propagation Path (Atmosphere)
#### 3.2.3 Liquid Water Delay  
In addition to dry neutral atmosphere and water vapor, there are also solid and liquid particles suspended within the atmosphere, such as ice crystals and liquid water droplets, which are the components of clouds [120]. When interacting with a radar wave, liquid water forms a secondary wave front owing to the dielectric medium, and subsequently the undisturbed and secondary wave fronts interfere with each other, resulting in a phase shift [28; 76]. Consequently, the refractivity induced by the dielectric medium is related to the liquid water content \(W\) (g/m\({}^{3}\)) as well as the thickness of the cloud layer \(L\) (km) regardless of the shape of cloud droplets. The zenith liquid water delay (\(\delta_{liq}^{z}\)) in mm is given in Equation (13) [28]. \[\delta_{liq}^{z}=1.4\times W\times L \tag{13}\]  
The variables \(W\) and \(L\) fluctuate according to different cloud types (see [120]: 131-133 for cloud classification). Since the values for \(W\) and \(L\) vary from different literature [28; 76; 121; 122], we categorize two groups for simplification: nonprecipitating clouds and precipitating clouds (including drizzle and rain). Liquid water content for nonprecipitating clouds ranges between 0.1 and 1 g/m\({}^{3}\), and for rain clouds, it can exceed 2 g/m\({}^{3}\)[123]. To define the range of liquid water content for precipitating clouds, we set a general range of 0.5-3 g/m\({}^{3}\) after referencing different literature [28; 122; 123]. As for the cloud layer, the vertical extent of the cloud could be up to 12 km for the most severe and extreme weather conditions such as thunderstorms [120]. The other values specified for the cloud layer column referenced [76]. Table 4 arranges the overall information and calculates the zenith liquid delay for each group. The zenith liquid water delay (\(\delta_{liq}^{z}\)) is calculated based on the given range of liquid water content and cloud layer. Non-precipitating clouds only induce up to 3 mm of liquid water delay. For situations such as lighter precipitation (drizzle or light rain), say \(W\) = 0.5-2 and \(L\) = 0.5-2, the resulting liquid delay is within 6 mm, which could produce a noticeable 0.2 phase cycle for C-band sensors and an unnoticeable 0.05 phase cycle for L-band sensors. More drastic situations are clouds of vertical developments where the extents extend to several kilometers in \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Group** & **Cloud Types** & \begin{tabular}{l} **Liquid Water** \\ **Content (g/m\({}^{3}\))** \\ \end{tabular} & \
\begin{tabular}{l} **Cloud Layer (km)** \\ **Delay (mm)** \\ \end{tabular} \\ \hline Non-precipitating clouds &
\begin{tabular}{l} Cirrus, altostratus, \\ altocumulus, stratus, \\ cumulus, stratocumulus \\ Nimobstratus, cumulus \\ congestus, cumulonimbus, \\ altostratus, stratus \\ \end{tabular} & 0.1–1 & 0.5–2 & 0.07–2.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation for zenith liquid delay for two groups: nonprecipitating clouds and precipitating clouds (reference the parameters from [28; 120; 123]. height, such as cumulus congestus and cumulonimbus. These clouds can produce heavy precipitation and are sometimes accompanied by lightning and thunder [120]. In such scenarios, say \(W>2\) and \(L>8\), liquid water could result in more than 20 mm of signal delay, corresponding to 0.7 phase shift for C-band sensors and 0.16 phase shift for L-band sensors. Although radar remote sensing is known for \"seeing through\" clouds and being operable during all weather conditions, and this is always considered one of its best merits over optical sensors, this does not mean that the effects of clouds and rain are negligible in all situations for radar measurements. In the past, phase delay contributed from liquid water has rarely been considered mainly due to two reasons. First, it is assumed that condensed water such as clouds and precipitation do not exist in the atmosphere under the clear gas hypothesis [28].