*4.3. Sensitive Analysis of the Formula of KE-R Relationships*

Many previous studies [22,52,63] have shown that parameters of the KE-*R* relationship are highly sensitive to DSD, and for parameters closely related to DSD characteristics, such as μ, the change in μ represents the difference in rainfall characteristics, which can affect the value of the parameters. Equation (18) indicates the KE-*R* relationships under the impact of parameter *x*. The parameter *A* and m are used to simplify the KE-*R* formula (expressed as Equation (18)) and are defined as:

$$A = \frac{\eta \Gamma(\mathbf{x})}{4.1^{\mathbf{x}}} \tag{19}$$

$$\mathbf{m} = 0.21 \mathbf{x} \tag{20}$$

in which *A* and m are functions of parameter *x*, thus, *A* and m are affected by raindrop microphysical characteristics and the environmental conditions. The parameter *x* (known as a linear variation with the parameter μ according to the Equation (13)) varies differently due to the different regions, different time periods. Therefore, sensitivity analysis is conducted on the parameters *A* and m with the change of parameter *x*, in order to indicate the applicability of the KE-*R* relationships formula (derived as Equation (18)) under different rainfall conditions. S1 and S2 were defined as the parameters to analyze the sensitivity of *A* and m with the change of *x*, and the influence of change on parameter *x* on the KE-*R* relationships is further shown. The definition of sensitive parameters S1 and S2 are derived in Equations (21) and (22):

$$\mathbf{S}\_1 = \lim\_{\Delta \mathbf{x} \to 0} (\frac{\Delta A/A}{\Delta \mathbf{x}/\mathbf{x}}) = \frac{\mathbf{d}A}{\mathbf{d}\mathbf{x}} \cdot \frac{\mathbf{x}}{A} \tag{21}$$

$$\text{S}\_2 = \lim\_{\Delta x \to 0} (\frac{\Delta \text{m}/\text{m}}{\Delta x/\text{x}}) = \frac{\text{dm}}{\text{dx}} \cdot \frac{\text{x}}{\text{m}} = 1 \tag{22}$$

in which S1 (S2) is the ratio of the change rate of parameter *A* (parameter m) to the change rate of parameter *x*. It is obvious from Equation (20) that the parameter m is proportional to parameter *x*, thus, S2 is a constant, as shown in Equation (22). The sensitive parameter S1 is further calculated of stratiform and convective rain obtained via disdrometer in different seasons, and the results are summarized in Table 4.

**Table 4.** Sensitive analysis results for parameter *A* with *x* of different precipitation types in different seasons and the total year.


Table 4 shows the results of the sensitivity analysis for *A* with *x* of different precipitation types in different seasons. The sensitivity of parameter *A* varies with different precipitation types and seasons. In the whole-year scale, the S1 is 4.29 of stratiform rain and 3.88 of convective rain, which indicates the parameter *A* of stratiform rain is more sensitive to the change of parameter *x* in the total year. The S1 of stratiform rain in spring and summer is also more than that of convective rain in corresponding seasons. However, the S1 of stratiform rain (3.83) is less than that of convective rain (3.97) in autumn, and the S1 reaches its lowest in autumn among three seasons. The parameter *x* is affected by changes in environmental factors and rainfall types [64], and in autumn, the sensitivity of parameter *A* with the change of *x* is lower than in spring or summer. This explains the obvious decrease of parameter *A* in autumn compared with the other two seasons calculated in Section 3.4. For convective rain in different seasons, the S1 from small to large is summer (3.79) < spring (3.88) < autumn (3.97).

According to Teng [65], the values of μ for raindrop spectra are between −1 and 4, and different μ values correspond to different rainfall characteristics. The larger the μ, the more likely it is to cause convective rain. In order to analyze the KE-*R* relationship when the rainfall shape parameter μ takes different values, the KE-*R* relationship curve of different values of μ was therefore made based on Equation (18), as shown in Figure 12a–f. The rule of this study is that both parameter *A* and m increase with the increase of μ. Figure 12g further shows the relationship among KE-*R-*μ. At a certain *R*, KE increases as μ increases. At the same rainfall intensity, the rainfall kinetic energy is also related to the rainfall type: the more the rainfall type is inclined to convective rain, the greater the rainfall kinetic energy will be, which can be explained by the Equation (18) and corresponds to the conclusions in other studies [66,67]. KE-*R* relationship changes with the change of the parameter *x*, according to Equation (18). From Equation (13), the parameter *x* can be calculated with a linear function from parameter μ. As discussed in Section 2.2, the parameter μ is correlated with the rainfall types; thus, the KE can be interpreted by the rainfall types according to Equation (18). In addition, different fitted formulas obtained in previous studies can be approximated with Equation (18) by changing the parameter μ (see Table A1), and the specific results can be approximated by Figure A1. This shows that the derived theoretical formula (Equation (18)) in this study is universal in various regions; however, whether the formula can be directly used to further analyze the KE-*R* relationship in other semi-arid areas should be further discussed in the future.

**Figure 12.** The *R*-KE relationships under different values of shape factors μ. The curves are derived based on Equation (18). (**a**) KE-*R* relationship for μ = −1; (**b**) KE-*R* relationship for μ = 0; (**c**) KE-*R* relationship for μ = 1; (**d**) KE-*R* relationship for μ = 2; (**e**) KE-*R* relationship for μ = 3; (**f**) KE-*R* relationship for μ = 4; (g) KE-*R*-μ relationship.

#### **5. Conclusions**

Characteristics of raindrop size distributions (DSDs) are important for improving the accuracy of radar reflectivity-rainfall intensity (*Z-R*) relationships in remote sensing (QPE) and the estimation of soil erosivity. In this study, an OTT Parsivel-2 Disdrometer is used to measured raindrop spectra from 10 August 2018 to 10 August 2019 in Yulin Ecohydrological Station, Shaanxi Province, China. The precipitation events obtained are classified as stratiform and convective rain based on the rainfall intensity classifying processes. The conclusions are summarized as follows.

(1) The characteristics of microphysical variables (the mass median diameter *Dm* and the raindrop size distribution *Nw*) were analyzed. The average *Dm* of different precipitation types in different seasons shows that for stratiform rain, rainfall intensity *R* is affected more by the average raindrop diameter *Dm*; for convective rain, *R* is affected more by DSD. The yearly average *Dm* and log10*Nw* are 1.41 and 3.91 mm, respectively. The average *Dm* of stratiform (convective) rain from small to large is 1.38 mm (1.46 mm) in summer, 1.46 mm (1.52 mm) in spring and 1.51 mm (1.57 mm) in autumn. This reflects the semi-arid climate rainfall characteristics in Yulin Station.

(2) The variances of rainfall microphysical characteristics in different precipitation types and seasons are related. The distribution of rainfall terminal velocity-diameter (*v-D*) spectra of spring, summer and autumn is concentrated near the theoretical curve derived by Beard [45]. The base-10 logarithm of *Nw* is used to fit the relationship curves log10*Nw* = c*R*d, in which c and d are parameters fitted by measured data. The difference in parameter d is small among different seasons (0.07–0.08 for stratiform rain and 0.02–0.04 for convective rain), and the d of stratiform rain is larger than that of convective rain.

(3) The *Z-R* relationships of different rainfall events in spring, summer and autumn in this semi-arid area are derived in this study. The parameter a is larger in stratiform rain than in convective rain, while the parameter b is larger in convective rain, showing the impact of different rainfall types on a and b. The results show that the estimation of different seasons should be treated, respectively.

(4) The theoretical formula of KE-*R* relationships for stratiform precipitation in semi-arid areas is derived (KE = <sup>η</sup>Γ(*x*) 4.1*<sup>x</sup>* ·*R*0.21*x*, where the parameter <sup>η</sup> is constant and *<sup>x</sup>* = <sup>μ</sup> + 6.01), which indicates the characteristics of precipitation and environmental conditions represented by parameter μ. This formula gives a general expression of the KE-*R* relationships and is simple to use because the parameters are all derived from the parameter μ. The sensitivity analysis results show that the parameter *A* for stratiform rain is more sensitive to the change of different precipitation types and environmental conditions in a total year. The closer the precipitation types are to convective rain, the larger the KE is at the same level of *R*. By changing the parameter μ, different empirical formulas obtained in previous studies can be approximated with the derived theoretical formula (Equation (18)).

In summary, the DSD characteristics of Yulin Station were obtained and the results can help to understand the microphysical characteristics of precipitation and have a strong impact on the mechanism of soil erosivity in the semi-arid area. Additionally, the formula of KE-*R* relationships provides a convenient way to fit with different rainfall events in semi-arid areas by adjusting its parameters. But the results are not conclusive because of the limited sample records of different rainfall types. In this study, data in winter is not deeply investigated, e.g., the solid precipitation processes should be further considered and analyzed. Moreover, the impact of environmental conditions on the parameter *A* and m is still not well understood. In the future, the influence of environmental factors on the parameters in *Z-R* relationships should be further discussed.

**Author Contributions:** Conceptualization, Z.X.; Data curation, Z.X. and H.L.; Formal analysis, Z.X.; Funding acquisition, H.Y.; Methodology, Z.X. and H.Y.; Project administration, H.Y. and H.L.; Resources, H.L. and Q.H.; Supervision, H.Y.; Writing—original draft, Z.X.; Writing—review & editing, H.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially supported by funding from the National Natural Science Foundation of China (Grant Nos. 51622903, 51979140 and 51809147), the National Program for Support of Top-notch Young Professionals, and the Program from the State Key Laboratory of Hydro-Science and Engineering of China (Grant No. 2017-KY-01). APC was funding by the National Natural Science Foundation of China (Grant No. 51622903).

**Conflicts of Interest:** The authors declare no conflict of interest.
