## 2 Uncertainty Quantification Methods in Radar-Based Precipitation Estimation
### How to Quantify Uncertainty
#### 2.2.2 Uncertainty Delta Method  
The Delta method analyzes the variability of each stage using the second-moment \"variance\" based on a Taylor series expansion. Assuming that the random variable is \(X\) and a specific value of \(x\) is \(a\), the Taylor series expansion is as follows (ignoring high-order terms):  
\[f(x)\approx f(a)+f^{\prime}(a)(x-a) \tag{4}\]  
where specific value \(a\) is the average \(\mu_{X}\) of \(x\) and \(y\) is assumed to be \(f(x)\). Then, application of variance to both sides of Equation (4) produces the following:  
\[\text{Var}(Y)=\text{Var}(f(X))\approx[f^{\prime}(\mu_{X})]^{2}\text{Var}(X) \tag{5}\]  
Using Equation (5), this study improved the Delta method to UDM for quantifying uncertainty. The basic concept of the UDM in quantitative precipitation estimation based on hydrological and weather radar is shown in Figure 1(a).  
In Figure 1(a), the final uncertainty (or total uncertainty) \(U_{T}\) can be calculated as variances that occur in the expansion process of Equation (5), which can be expressed as follows:  
\[\begin{array}{ll}U_{T}=\text{Var}(P)=\text{Var}(l(Q))&=(l(Q)-l\big{(}\mu_{ Q}\big{)}^{2}\approx(l^{\prime}\big{(}\mu_{Q}\big{)}(Q-\mu_{Q})\big{)}^{2}=l^{ \prime}(\hat{Q})^{2}\text{Var}(Q)\\ &\approx(l^{\prime}(\mu_{P}))^{2}(h^{\prime}\big{(}\mu_{Q}\big{)}^{2}(g^{ \prime}(\mu_{S}))^{2}\text{Var}(R)\end{array} \tag{6}\]  
where \(R\), \(S\), \(Q\), and \(P\) represent each stage in the radar-based precipitation estimation process and \(l(\cdot)\), \(h(\cdot)\), and \(g(\cdot)\) are the probability functions of the expectations of the variables at each stage. Then, the uncertainty at each stage can be expressed as \(l^{\prime}(\mu_{P})^{2}=\text{Var}(P)/\text{Var}(Q)\) by transforming Equation (5). The same expression can be applied to the other stages, which can be summarized as follows:  
\[U_{T}=\text{Var}(P)\approx\left(\frac{\text{Var}(P)}{\text{Var}(Q)}\right) \times\left(\frac{\text{Var}(Q)}{\text{Var}(S)}\right)\times\left(\frac{ \text{Var}(S)}{\text{Var}(R)}\right)\times\text{Var}(R) \tag{7}\]  
Equation (7) can be expressed as (total uncertainty) = (uncertainty of post-processing) \(\times\) (uncertainty of QPE (quantitative precipitation estimation) model) \(\times\) (uncertainty of signal processing) \(\times\) (uncertainty of radar measurement). Using this equation, the total uncertainty can be quantified using the variances of each stage. The uncertainty at each stage can be quantified using the variances through Equations (6) and (7), and its process is shown in Figure 1(b). By taking the logarithm of both sides of Equation (7), the uncertainty can be quantified. Therefore, the UDM can evaluate the uncertainty at each stage and the total uncertainty.  
In comparison with the MEM, which quantifies uncertainty by applying the maximum and minimum values from all radar-based precipitation estimates to Equation (3), the UDM quantifies uncertainty by applying the variances from all precipitation estimates to Equation (7). These two methods are similar in that they use the statistical properties of the range of spread of all precipitation estimates.