*2.3. Methodology*

### 2.3.1. RTI Method

Since 2005, the RTI model has been widely applied in flash flood warnings, which e ffectively reduced the casualties caused by flash floods. Considering that flash floods are mainly caused by hourly peak rainfall, this method is used to predict flash flood by multiplying the e ffective cumulative rainfall (Rt) and rainfall intensity (I). Among them, the e ffective accumulated rainfall mainly uses the accumulated rainfall in the previous 7 days before the flash flood occurs. Based on historical rainfall, this method obtains the flash flood probability under di fferent rainfall conditions by calculating the RTI, and then divides the critical rainfall map into three regions for early warning (low possible, medium, and high) [13]. Meanwhile, it also defines how to segmen<sup>t</sup> the rainfall events, i.e., the start time is defined as the rainfall per hour exceeding 4 mm, and the end time is the rainfall per hour falling below 4 mm for six consecutive hours. A single day refers to the period from yesterday at 08:00 a.m. to today at 08:00 a.m. Rainfall data were selected according to the flash flood disaster events. If the flash flood disaster occurs in an area without a monitoring station, calculations are performed using data from the nearest rainfall stations (within a radius of 50 mm). The RTI equations are as follows:

$$RTI\_t = I \ast R\_t \tag{1}$$

$$R\_t = \sum\_{i=0}^{n} \alpha^i R\_i \tag{2}$$

where *RTIt* is the RTI calculated at time t; *I* is the rainfall intensity; *Rt* is the e ffective cumulative rainfall; *i* means the antecedent day number from one to *n*; α is the rainfall attenuation coe fficient, mainly taken from the measured value of 0.78 by Cui Peng in Jiangjiagou, Yunnan Province [26]; α*i* is the reduction factor of the previous *i* day; and *Ri* is the 24-h cumulative rainfall of the previous *i*-day, where the initial cumulative rainfall *R*0 is 50 mm, which is mainly obtained through actual statistical analysis.

Since the previous rainfall was calculated using "t" days of accumulated rainfall, the false alarms rate is higher in certain rainfall patterns (e.g., long-term duration and low rainfall intensity). Besides, this method does not consider the e ffects of intermittent rainfall and rainfall segmentation, etc.; all of the above results in low accuracy. Therefore, based on the reduction period and the reduction coe fficient being unchanged, Chen et al. (2018) proposed an improved RTI method and provides a detailed flowchart describing the method [16]. The specific equation of this method is

$$R\_t = I\_t + R\_{t-1} \* (\alpha)^{\frac{1}{24}} \tag{3}$$

where *It* is the current rainfall intensity at time *t* (mm/h); and *Rt*−<sup>1</sup> is the e ffective cumulative rainfall one hour earlier. In Formula (2), each operation needs to calculate *Ri* separately, multiply it by α*i*, and then accumulate; however, in Formula (3), it has only one rainfall intensity and one accumulated rainfall, which greatly reduces the calculation amount and contributes to the future subsequent large-scale grid operations.

## 2.3.2. Evaluation Metrics

Six indicators were employed to evaluate the accuracy of the satellite precipitation products. Table 2 shows the formulas and optimal values of these indexes. CC (correlation coe fficient) represents the correlation between the satellite precipitation data and site precipitation data, the greater the better; BIAS (Relative Bias) and RMSE (Root Mean Square Error) are quantitative indicators, representing the deviation degree between the satellite precipitation data and ground station precipitation data. The method also includes three classification indicators: Probability Of Detection (POD), False Alarm Ratio (FAR), and Critical Success Index (CSI), which can comprehensively reflect the ability of the precipitation products to estimate the probability of precipitation event occurrence. Among them, POD is the correct forecast rate, while FAR is the wrong forecast rate. The critical success index (CSI) is the function of POD and FAR that provides a more balanced estimate of the satellite products [27].


**Table 2.** List of the formulas and optimal values of the indexes.

Variables: *n*, total number of flash floods; *Di*, the *i*-th of the evaluated; *Mi*, the *i*-th of the reference data; *D*, mean of *Di*; *M*, mean of *Mi*; *O*, number of hits; *U*, number of misses; *P*, number of false positive.

#### 2.3.3. Systematic or Random Error

Random error is an avoidable error caused by measurement or calculation. Systematic error is the unavoidable error caused by the experimental instrument or accuracy. Systematic and random errors are determined by objective and subjective factors, respectively, both of which can be reduced but not eliminated [28]. The spatiotemporal variability of precipitation, measurement error, and the uncertainty of sampling affects the satellite precipitation data accuracy, where the uncertainties include systematic and random errors (hereafter Syst and Rand), mainly from (1) sensor observation; (2) the algorithm used to estimate rainfall; and (3) sampling error. Reference [29] developed a method for estimating Syst and Rand for satellite precipitation products. The system of mean square error (MSE) and the formula for random error are

$$\text{MSE} = MSE\_{\text{S}} + MSE\_{R} \tag{4}$$

$$\frac{\sum\_{n}(\mathbf{x} - \mathbf{y})^2}{n} = \frac{\sum\_{n}(\mathbf{x} - \mathbf{y})^2}{n} + \frac{\sum\_{n}(\mathbf{x} - \hat{\mathbf{x}})^2}{n} \tag{5}$$

where *x* is the satellite precipitation; *y* is the CMA precipitation; and *n* is the time steps number (here days); the formula for calculating *x*ˆ is as follows:

$$
\pounds = ay + b
\tag{6}
$$

where *a* and *b* are the slope and intercept parameters that need to be calibrated, respectively. The calculation formula is

$$\text{Syst} = \frac{\sum\_{\text{ll}} (\pounds - y)^2}{n} / \frac{\sum\_{\text{ll}} (\pounds - y)^2}{n} \tag{7}$$

$$Rand = \frac{\sum\_{n} (\mathbf{x} - \hat{\mathbf{x}})^2}{n} / \frac{\sum\_{n} (\mathbf{x} - \mathbf{y})^2}{n} \tag{8}$$

#### **3. Results and Discussion**

#### *3.1. Spatial Distribution of Precipitation*

Figure 2 shows the spatial distributions of the daily average precipitation in Yunnan Province captured by IMERG-E, IMERG-F, and CMA from March 2015 to December 2018. Using CMA data as the reference calibration data, the rainfall distribution in Yunnan Province is shown in Figure 2a. In Yunnan Province, the precipitation increases from northeast to southwest, with the largest precipitation occurring in the western border region. Referring to Figure 1, Yunnan's terrain is complex and changeable, with a high northwest and low southeast, descending stepwise from north to south. Moreover, flash floods are mainly concentrated in the southwest region and densely distributed in parts of the northeast, reflecting that the flash flood disaster is mainly a ffected by multiple factors with heavy rainfall as the main trigger. Meanwhile, there is a clear precipitation zone in the western region, with the smallest precipitation in the northwest and more precipitation in the central-western region. Figure 2b,c presents the distribution of the estimated precipitation in Yunnan Province using IMERG-E and IMERG-F, respectively. The overall trend is consistent with the precipitation distribution measured by CMA; that is, the estimated precipitation is relatively large in the southwest area, while low in the northeast relatively. Among them, the estimated rainfall of IMERG-E is less in the northeast region and has a larger coverage area, which is lower than that of CMA. The estimated maximum rainfall area is consistent with the CMA, but its coverage area is much smaller than the CMA. Therefore, IMERG-E underestimates the precipitation, especially in the southwest region. IMERG-F is the opposite of IMERG-E; its estimated rainfall covers a larger area, and the minimum and maximum rainfall are higher than the CMA. IMERG-F overestimates precipitation, especially in the central and western regions.

**Figure 2.** Spatial distributions as per the CMA (**a**), IMERG-E (**b**), and IMERG-F (**c**).

#### *3.2. Evaluation of IMERG-E and IMERG-F*

Figure 3 presents the spatial patterns of six indicators obtained from IMERG-E and the CMA using hourly data over the Yunnan Province. Generally, CC was relatively low, changing from 0.2 to 0.5, especially over the northwest regions (CC < 0.2) with the highest altitudes. As for the RMSE and BIAS, their spatial patterns are similar, with an increasing trend for the RMSE and BIAS and a decreasing trend for relative error from northwest to southeast. Many previous studies have confirmed that satellite precipitation estimates are usually low with large errors in mountainous areas. For example, compared with the highest regions in the Northwest, IMERG-E's BIAS is underestimated by about 50% [30]. As for satellite-based precipitation metrics, there is also a significant trend; that is, POD and CSI are increasing, while FAR is decreasing, which is in harmony with the overall trend of the IMERG-E data. Table 3 shows the evaluation metrics for IMERG calculated with the mean value at hourly and daily timescales; the evaluation index calculated by the mean is larger on a daily scale than that on an hourly scale. Given the relatively low accuracy of IMERG-E, the di fferences between the IMERG-E and CMA were relatively obvious.

**Figure 3.** The spatial patterns of six metrics (CC (**a**), RMSE (**b**), BIAS (**c**), POD (**d**), FAR (**e**), and CSI (**f**)) generated from IMERG-E and CMA using hourly data in Yunnan Province.


**Table 3.** Calculation results for evaluating IMERG.

The spatial patterns of the six metrics derived from IMERG-F and CMA (Figure 4) exhibited similar trends as those from IMERG-E and CMA. Table 2 revealed the four indicators CC, BIAS, POD, and CSI of NRL IMERG-E products are lower than these for PRL IMERG-F, regardless of daily and hourly data, while the two indicators RMSE and FAR are higher than that for IMERG-F. Therefore, IMERG-F indicated a better performance in this region. For instance, the POD and CSI of IMERG-F and CMA were also overall higher than those of IMERG-F and CMA, especially for CSI. Besides, in both Figures 3 and 4, the POD and CSI in these two figures show significant differences inside and outside of the Yunnan Province, which is mainly due to the complex and changeable terrain that induces large systematic errors in satellite precipitation. Besides, there is no actual rainfall measurement site outside the Yunnan border, and the rainfall distribution obtained only by interpolation appears discontinuous.

The following is a further analysis of the error source (i.e., systematic error or random error) of EMERG-E. The spatial patterns of systematic and random errors in IMERG-E at the 1 and 24 h temporal scales in Yunnan Province, respectively, demonstrated almost the same spatial patterns (Figure 5). Overall, the system error is mainly distributed in the northwest region of Yunnan Province with a high altitude (about 80%), and the coverage area is relatively small. Meanwhile, except for the northwestern part of Yunnan Province, the systematic error at 1 h is significantly higher than that at 24 h, especially the error at 24 h mostly disappeared. Random error mainly occurs in the southern region of Yunnan Province with relatively high rainfall (about 80%), but a relatively low in the northwest region. Besides, the random error is higher at 24 h than at 1 h. Therefore, for the IMERG-E data, the estimated error in the high-altitude and low-precipitation areas of Yunnan Province is mainly determined by the systematic error; the southern area of Yunnan Province with a low elevation and high precipitation is dominated by random errors.

**Figure 4.** The spatial pattern of six indicators generated from hourly data of IMERG-F and CMA in Yunnan, where the indicators indicated by (**a**), (**b**), (**c**), (**d**), (**e**), and (**f**) are CC, RMSE, BIAS, POD, FAR, CSI.

**Figure 5.** The spatial patterns of the system and random errors for IMERG-E at the 1 h and 24 h temporal scales in Yunnan Province, respectively, where (**a**) and (**b**) are the systematic errors of 1 h and 24 h, respectively; (**c**) and (**d**) are random error of 1 h and 24 h.

As for IMERG-F, systematic errors accounted for less than 20% of the total errors across the whole study area at both the 1 and 24 h temporal scales. Compared with IMERG-E, IMERG-F has been significantly improved because the systematic errors were effectively reduced, especially in mountainous areas with the complex terrain. Figure 6 shows that the IMERG-F data error (≥80%) is primarily a random error, independent of the underlying surface features. Besides, Figures 6 and 7 behave differently inside and outside the Yunnan border. This phenomenon is mainly due to the complex and changeable terrain of Yunnan Province, which further triggers the discontinuous changes in satellite precipitation errors. It is a discontinuous problem in the original satellite's precipitation error, rather than using different spatiotemporal resolution rainfall. Therefore, the overall IMERG-F data is better than the IMERG-E data.

**Figure 6.** The spatial patterns of the system and random errors in IMERG-F at the 1 and 24 h temporal scales, respectively; where (**a**) and (**b**) are the systematic errors of 1 h and 24 h, respectively; (**c**) and (**d**) are random error of 1 h and 24 h.

#### *3.3. Applicability Analysis of IMERG in Flash Flood Warning*

Based on historical flash flood disaster events, combined with three types of rainfall products (IMERG-E and -F, and CMA), the multi-period rainfall (1 h, 3 h, 6 h, and 24 h) is obtained. The resultant effective accumulated multi-period rainfall is calculated by the improved RTI method. Meanwhile, combined with the flash floods' actual frequency and the rainstorm statistical parameter atlas of China, the multi-period critical rainfall (1 h, 3 h, 6 h, and 24 h) (hereafter, CR1, CR3, CR6, and CR24) is obtained; the G (x) early warning model is constructed for effective cumulative rainfall (Rt) and corresponding period critical rainfall (CRt). Since this model does not take into account the potential multiple influencing factors, such as slope, vegetation, and human activity, when issuing the flash flood warning, we should make a comprehensive analysis regarding the flash flood risk distribution to determine whether a flash flood event has been captured. Considering the flash flood risk map obtained by Ma et al. (2019), the obtained results are shown in Figures 7 and 8 [31].

**Figure 7.** The performance of the flash floods captured by the three products (IMERG-F, EMERGE, and CMA) for different times (1 h, 3 h, 6 h, and 24 h). Note: Red indicates that the warning issued has captured the flash flood event; blue indicates that the issued warning has not captured the flash flood event, where (**a**), (**b**), (**c**), and (**d**) are the distribution of IMERG-E products in 1 h, 3 h, 6 h and 24 h to capture flash flood disasters; (**e**), (**f**), (**g**), and (**h**) are the distribution of IMERG-F products in 1 h, 3 h, 6 h and 24 h to capture flash flood disasters; (**i**), (**j**), (**k**), and (**l**) are the distribution of CMA products in 1 h, 3 h, 6 h and 24 h to capture flash flood disasters.

**Figure 8.** Percentage of flash floods caught by CMA, IMERG-E, and IMERG-F. Note: CRt represents the critical rainfall at time t.

Obviously, for the same product, the captured flash flood events has decreased over time, where the hit rate of both CR6 and CR24 is less than 60%. Among them, the flash floods events captured by these three rainfall products are concentrated in western Yunnan, while there are fewer flash flood events in the relatively flat southwest and central Yunnan. In general, if a flash flood event cannot be

captured in a short period, it will not be anyhow captured. Moreover, the e ffect of IMERG-E products on capturing disasters is significantly lower than that of IMERG-F, and the capture rate in each period is less than 50%. However, the capture accuracy of IMERG-F at CR1 and CR3 is almost comparable to the CMA—only a 1% di fference—and the capture rate of CR1 is nearly 80%, with an extremely high accuracy. Nonetheless, with the time marching, the capture e ffect decreased significantly. Meanwhile, if the flash flood events cannot be captured by IMERG-F and CMA, the same phenomenon occurs in IMERG-F; but, as time goes on, there is an out-of-sync phenomenon in the catching flash flood events by IMERG-F and CMA. Besides, the CMA data is a fusion of measured data and CMORPH, leading to possible delays in data acquisition. Therefore, the hit rate is decreasing with an increasing temporal resolution or increasing the averaging time of the satellite images.
