## 5 Discussion  
In Section 4, Examples 1-4 are the simulations under the maximum rain rate of 10 mm/h with different shapes of the rain-cloud cells, that is the rectangular rain-cloud cell, the trapezoidal rain-cloud cell, the triangular rain-cloud cell and the rectangular bimodal rain-cloud cell. It is obvious that the simulated result using this method is in good agreement with the given distribution. Besides different shapes of the rain-cloud cells, this method can also be applied to different rain rates. Examples 1, 5 and 6 are the simulations under the maximum rain rates of 10 mm/h, 30 mm/h and 50 mm/h with a triangular rain-cloud cell. The typical value 30 mm/h is chosen because the melting layer, the region where the precipitation changes from snow to rain, exists at lower rain rates, approximately below 30 mm/h [30]. Furthermore, the result has shown that the simulation result using this method is also in good agreement with the given one. However, simulation errors exist. Large errors occur at the edges of the precipitation area, which are also where the biggest errors of the MOS method and the MOSVI method occur. For some demanding applications, more studies should be carried out to detect the edges of the precipitation area more accurately. Besides, fluctuations exist at the junctions of intervals introduced in Section 3 and Appendix B in detail. This is mainly because the unavoidable non-linearity of the function \(p(x)\). Since the precipitation rate I(x) is derived from the function \(p(x)\), the simulated precipitation rate I(x) is nonlinear as well. Thus, the fluctuations may occur at the junctions of the adjacent intervals. Furthermore, the fluctuations may affect the retrieval that follows. It seems that the method introduced in this paper does not decrease the errors greatly, but taking a deep insight into the process of MOS method, great improvements can be sensed. The MOS method will determine the shape of the rain-cloud cell according to some of the features of the NRCS values. Furthermore, the final retrieval of the horizontal variation will be determined by using a certain equation, such as Equations (6) or (7) in this paper. This seems quite simple but the equation of the horizontal variation differs from shape to shape. To cover more shapes, large number of equations for the horizontal variation should be prepared in advance, which is a large workload at the early stages. Furthermore, for those rain-cloud cells which are not previously prepared, their shapes will be approximated to the closest one. It is clear that not all shapes can be covered previously, this method will add some errors when retrieving the uncovered shapes. The method introduced in this paper, however, can obtain the horizontal variation \(I(x)\) within the range of interest analytically, which will increase the accuracy of the retrieval of the horizontal variation. Another merit of this method is that it uses less empirical coefficients. Thus, this method can reduce the workload of the early stage and the simulation process. What is more, since each empirical coefficient is calculated under a certain rain rate, slant angle of the radar or other conditions, the fewer empirical coefficients the method uses, the morescenarios in which the method can be easily applied. For the MOS and MOSVI method, the simulated widths of rectangular distributions, triangular distributions and trapezoidal distributions are calculated with empirical coefficients:  
\[\hat{w}=0.97(\hat{x}_{min}-\hat{x}_{L}), \tag{26}\]  
\[\hat{w}=1.61(\hat{x}_{min}-\hat{x}_{L})^{0.93}, \tag{27}\]  
\[\hat{w}=\Big{[}0.97(\hat{x}_{min}-\hat{x}_{L})+1.61(\hat{x}_{min}-\hat{x}_{L})^ {0.93}\Big{]}/2, \tag{28}\]  
where \(\hat{x}_{min}\) is the simulated result of the \(x\)-coordinate of the point whose NRCS reaches the minimum and \(\hat{x}_{L}\) is the simulated result of \(x\)-coordinate of the left edge of the precipitation area. Furthermore, for the MOS method, the identification of the shape of the rain-cloud cell is also quite complex.