## 2 Uncertainty Quantification Methods in Radar-Based Precipitation Estimation
### How to Quantify Uncertainty
#### 2.2.1 Maximum Entropy Method  
In discussing information theory, [PERSON] (1954) first introduced the concept of entropy, which has been used widely as a method with which to measure expected values of a given average information content to generate the unpredictability or uncertainty of information. If the probability of occurrence of the given information is large, the amount of information and its uncertainty are small, and vice versa. If the information is a random variable \(X\) with probability \(p\), and the average amount of information is defined as \(I(X)\), the basic concept of [PERSON]'s entropy can be described as follows [PERSON] (1954):  
\[H(x)=\sum\
olimits_{i=1}^{n}p_{X}(x_{i})I(X)=\mathrm{E}[I(X)] \tag{1}\]  
where \(H\) is entropy of \(X\), \(x\) is the value of \(X\), \(p_{X}(\cdot)\) is the probability density function of \(X\), and \(n\) is the number of \(x\). The principle of maximum entropy (ME), based on [PERSON]'s entropy theory [PERSON] (1954), was first expounded by [PERSON] (1954). It represents the current state of a given information content using the most suitable probability distribution with the largest entropy. When a certain set of information content is given, based on the information content, ME theory can provide the probability density function that maximizes the entropy. ME has the moment constraint (\(\sum_{i=1}^{n}p_{X}(x_{i})f_{n}(x_{i})=y_{n}\)) and generalization constraint (\(\sum_{i=1}^{n}p_{X}(x_{i})=1\)) [PERSON] (1954). The key concepts of ME and entropy \(H(x)\) are as follows:  
\[\max_{p}H(x)=-\sum\
olimits_{i=1}^{n}p_{X}(x_{i})\ln p_{X}(x_{i})\ (p_{X}(x_{i})\geq 0) \tag{2}\]  
\[H(x)=-\int_{a}^{b}f_{X}(x)\ln f_{X}(x)dx=-\int_{a}^{b}\frac{1}{b-a}\ln\frac{1} {b-a}dx=-\ln(b-a) \tag{3}\]  
where \(f_{n}(\cdot)\) is the moment constraint and \(n\) is the number of samples. If the maximum value \(b\) and the minimum value \(a\) are given, the distribution maximizing the entropy is a uniform distribution (\(f(x)=1/(b-a)\)) subject to the normalization constraint \(\int_{a}^{b}f_{X}(x)dx=1\) and the occurrence probabilities of all events are the same on the interval (a, b) [PERSON] (1954). This means the probability of each value of precipitation estimate is equally distributed.  
The basic concept of uncertainty propagation in the radar-based precipitation estimation process can be described easily when the above ME is utilized as follows.  
First: Radar-based precipitation estimates for all combinations of methods are generated, for example: four methods in the quality control stage, three methods in the precipitation estimation stage, and three methods in the post-processing; all combinations = 24 precipitation estimates.  
Figure 1: Definition of uncertainty quantification and propagation in the radar-based precipitation estimation process (**a**) with increasing uncertainty, (**b**) with decreasing uncertainty.  
Second: For a certain method \(k\) among all \(K\) methods of a certain stage \(l\) that is to be demonstrated, the maximum and minimum values of radar-based precipitation estimates where method \(k\) is involved, and the corresponding ME \(H_{k,l}\) are calculated, for example: for QC1 method of the quality control stage, 9 precipitation estimates that passes through QC1 method were estimated. For 9 results, the maximum and minimum values were taken to calculate ME.  
Third: This process is iterated for all \(K\) methods in a given stage \(l\), for example: this process was repeated for three other methods. Each of the four methods had the ME value.  
Fourth: The calculated ME in stage \(l\) are combined to quantify the averaged uncertainty \(\overline{H_{k}}\) for stage \(l\). To combine the ME, we may use either the maximum or average operation, for example: in the quality control stage, four ME values were calculated from four methods. The ME in the quality control stage was calculated by averaging four MEs or selecting the maximum value of four MEs.