## 2 Background Study
### Rainfall Rate Missing Data
#### 2.8.2 Temporal Missing: Generation by Synthetic Means  
The synthetic way to generate the rainfall rate of a locality needs local statistical distribution of the point rainfall intensity that feeds to the model, known as the exponential cell (EXCELL) model [67]. This model offers two types of rain cell structure, called the monoaxial model and the biaxial model. In a monoaxial model, the cell is presumed to be circular, and in a biaxial model, the model is assumed to be elliptical, and the monoaxial or biaxial property is determined by the orientation of central moments of inertia. In the EXCELL model, a typical rain cell can be defined by the following parameters: cell area (A), cumulative rain (Q), average rainfall rate (\(\overline{R}\)), root mean square (RMS) of rainfall rate (\(R_{rms}^{2}\)), rain cell gravity center (\(x_{0},y_{0}\)), and the central moments of inertia (\(I_{x},I_{y}\)). These parameters can be estimated from the local peak rainfall rate (\(R_{M}\)) and rain cell radius (\(\rho\)) in the (\(x,y\)) plane through which the rainfall rate decreases at the rate of \(e^{-1}\) (note the exponential form). It is notable that the simple rain attenuation model (SAM) [32] and the LU model [68] inherently include distance as a negative exponential factor to determine the spatial rainfall rate. This hypothesis induces error if the rain cell area is smaller than 5 km\({}^{2}\) and the rainfall rate is smaller than 5 mm/h (\(R_{1}\)). To describe the rain cell model for the EXCELL model, the parameters that must be considered are as per Table 6:  
Figure 1: Rain cell size for different rainfall rates: (**a**) \(R<31.6\) mm/h, (**b**) \(R\geq 31.6\) mm/h [43].  
A variant of the EXCELL model is the HYCELL model [69], where the rainfall rate is considered a result of an exponentially decreasing rainfall rate (\(e^{-1}\)), normally distributed (Gaussian process) rainfall rate, and as a finite linear combination of these two types of rainfall distribution. These two approaches of determining the rainfall rate are applicable separately depending on a specified rainfall rate (\(R_{2}\)) that acts as boundary value defining the application of the Gaussian process and the exponential rate to be used. Thus, if the rainfall rate satisfies \(R_{1}\leq R<R_{2}\), the rainfall rate distribution is generated through the exponential parameter, and if it is greater than \(R_{2}\) the Gaussian process parameter is applied. The EXCELL and HYCELL models contribute to developing a single rain cell that may not cover a long link. Consequently, an extended version of the EXCELL model covering a long link was proposed in the literature, which is known as the MultiEXCELL model (a kind of propagation-oriented precipitation model based on a cellular rain formulation) [70]. In the MultiEXCELL model, instead of the biaxial elliptical approach, the monoaxial approach was considered. The minimum rain cell area was set 4 km\({}^{2}\) and the other distinct parameters were similar to the EXCELL model (\(R_{M},\rho_{0}\)) and additional number \(k=1,\) 2 or 3. Compared with the EXCELL model, the MultiEXCELL model is additionally able to compute the synthetic rain cells' probability of occurrence and to consider the convective and stratiform rain dominance. Additionally it considers the distribution along the radii conditioned to \(R_{M}\) as \(p(\rho_{0}|R_{M})\).