Study on the Uncertainty of Critical Rainfall for Flash Floods in Small Watersheds Based on the Random Rainfall Pattern

Wenlin Yuan Zhengzhou University;Key Laboratory of Lower Yellow River Channel and Estuary Regulation,M.W.R. Lu Lu Zhengzhou University Hanzhen Song (  songhanzhen1994@163.com ) Yellow River Engineering Consulting Co Ltd Xiang Zhang Key Laboratory of Lower Yellow River Channel and Estuary Regulation, M.W.R.; Yellow River Institute of Hydraulic Research, YRCC Linjuan Xu Key Laboratory of Lower Yellow River Channel and Estuary Regulation,M.W.R.; Yellow River Institute of Hydraulic Research,YRCC Chengguo Su Zhengzhou University Meiqi Liu Zhengzhou University Denghua Yan Zhengzhou University;China Institute of Water Resources and Hydropower Research Zening Wu Zhengzhou University

. Therefore, the accurate and effective determination of critical rainfall 38 has become an important issue in early warning for flash floods (Norbiato et al., 2009). In the early 39 stages, a data-driven method is often used to calculate the CR. However, this method is based purely 40 on statistical data, which ignores the physical mechanism of flash flood, so the CR is less consistent 41 with the real-life natural disaster scenario (Liu, 2019). Hydrology and hydraulics methods can fully 42 consider the hydrological characteristics of a watershed, which is embraced in current flash flood 4 simulation results of hydrological models. Many studies have investigated the influence of rainfall 66 spatial distribution on hydrological models (Douinot et al., 2016;Carreau and Bouvier, 2016; 67 Zoccatelli et al., 2010). However, the spatial size of a small watershed within a mountainous region 68 is small, hence the rainfall temporal distribution is mainly considered in this study. 69 The distribution of rainfall over time is defined as the rainfall pattern. The rainfall pattern does conditions, and found that the rainfall pattern has a significant influence on the flood peak discharge. believe that rainfall events have high self-similarity in statistical law, hence, a method of proportional 83 parameter scaling to design rainfall patterns was proposed, but the uncertainty of some characteristics 84 in rainfall time distribution was ignored. To describe the randomness of urban rainfall temporal 85 distribution, Thorndahl and Willems (2008) adopted Gaussian distribution to express the (2) Based on the RRP, the HEC-HMS model is utilized in the simulation of the rainfall-runoff 110 process for a small watershed, and the trial algorithm is used to calculate the CR threshold space.

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Moreover, the effect of the peak position coefficient and peak ratio on the CR is quantitatively 112 analyzed.

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(3) Based on the similarities in quantity and tendency, and the comprehensive similarity between 114 the RRP and current rainfall, the RRP that is best suited to the current rainfall conditions is identified.

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Then an early warning model with dynamic correction based on RRP identification is proposed to 116 carry out effective early warning for flash floods.

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In the following section, Section 2 presents the methodology and the model, including the 118 structure of the RRP, the calculation process of the rainfall-runoff process, detailed steps of the trial 119 algorithm and the establishment of the early warning model. Section 3 describes an overview of the 120 study area and the source of the data used. The results and discussion are displayed in Section 4.

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Finally, conclusions are drawn in Section 5. In this section, a complete flash flood early warning system including rainfall pattern module, 124 rainfall-runoff module, CR module and early warning module is introduced. Rainfall patterns 125 considering the randomness of rainfall are designed in the rainfall pattern module as input conditions 126 for the hydrological model, and then the rainfall-runoff process is simulated by the rainfall-runoff 127 module. Based on the hydrological model after debugging parameters, the CR module is used to 128 calculate CR threshold space. Finally, a warning signal is issued through the early warning module.

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The general schematic diagram of this study is displayed in Fig. 1. Atlas is the most widely used to calculate CR (Lin et al., 2005). However, for engineering safety, the 136 peak rainfall position is usually designed to be at the back of the TRP, which is greatly different from 137 actual rainfall. In addition, the randomness of rainfall is ignored, therefore, the RRP is proposed for 138 rainfall pattern design. This method not only follows the distribution law of regional rainfall, but also 139 takes into account the uncertainty of rainfall. There are many flood simulation methods, such as the   (1) Gather rainfall data, including information for rainfall which causes flash flood, and then 163 establish a database.

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(2) Determine the resolution of rainfall data including the total rainfall duration T and temporal 165 resolution t . T is equal to the duration with the largest proportion of the duration in the total 166 duration of the rainfall events. Based on T and t , the total number of time periods I can be 167 obtained, which is equal to the ratio of T to t .

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(3) Extract all rainfall events with a total duration of T from the database, and use j to denote 169 the number of these rainfall events. Then, a rainfall information matrix P can be obtained, as shown  (2) Where ji b denotes the ratio of rainfall in time period i to the total rainfall in rainfall event j . j

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(1) Calculation for net rainfall 216 The initial constant rate method is the most widely used in calculating net rainfall. This method 217 includes two parts of rainfall loss calculation, namely the initial loss and later loss, as shown in Eq.
Where et P represents the net rainfall at time t . a P is the antecedent soil moisture condition 221 (ASMC), which is an indicator of soil moisture. t P denotes the cumulative rainfall from t to tt  . 222 i P denotes the cumulative rainfall, and fc means the maximum infiltration capacity of the soil.

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(2) Calculation for direct runoff 224 In this study, the Soil Conservation Service Unit Hydrograph method is chosen to calculate direct   .
Where I denotes the channel inflow, Q denotes the channel outflow, W denotes the storage taken for the flood wave to pass along the section of river channel.

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Based on Eqs. (13)-(14), the expression of the flow equation can be acquired, as shown in Eqs.
Where 1 I and 2 I mean the inflow of the upper section at the beginning and finish time,

Obtaining Critical Rainfall 274
The trial algorithm is adopted in the current study for calculating the CR of the flash flood 275 corresponding to the RRP set. A flowchart of the trial algorithm is displayed in Fig.3. The detailed 276 steps are shown below. (1) Assume an initial rainfall H which is the total rainfall of the rainfall event.

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(2) To obtain the rainfall process, the time distribution of H is obtained using the RRP 281 designed in section 2.1.

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Due to the randomness of rainfall patterns, taking the uncertainty fully into consideration when 299 determining the CR may make the early warning process cumbersome. To simplify the early warning 300 process and improve the early warning efficiency while considering the uncertainty of the rainfall 301 pattern and issuing a warning signal which is suitable for current rainfall conditions, this study 302 proposes an early warning model with dynamic correction based on RRP identification. The detailed 303 early warning process is as follows.

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(1) Determine the range of control parameters according to the statistics of the measured rainfall 305 data and generate multiple RRP sets by using the method in Section 2.1.

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(2) Calculate the CR for each RRP by using the methods described in Sections 2.2 and 2.3. Then, 307 the rainfall of each time period in the RRP can be allocated according to the CR.
(3) Set the rainfall identification time period (ITP), which indicates the time period that needs to be compared in the early warning process up to the current period. As the rainfall process continues, 310 the ITP will increase dynamically.

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(4) Calculate the indices used to describe rainfall similarity corresponding to the ITP, including Where X and Y mean the rainfall process in RRPs and the current rainfall process, respectively. warning signal will be issued and step (7) should be followed.   with rainfall duration of 1h to 6h, occupied the largest proportion of the total rainfall events. In 357 addition, Short-duration heavy rainfall is the main causal factor of flash floods. Therefore, after 358 comprehensively considering the characteristics of rainfall and the warning time period, the total 359 duration was set as 6h and the temporal resolution was set as 1h. The GIS data contains DEM data (Fig.4), soil maps (Fig.5a) and land use (Fig.5b), which were homogenous, with surrounding forests on the hills and cultivated land along the river (Fig. 5b).  Table 2. which indicates that goodness of fit of the Gauss function is the best. Hence, the Gauss function was 389 adopted as the distribution function for each i B , and the fitting result of each i B is shown in Fig.6.   Table 3. 406 Table 3 Flood simulation results for the Xinxian watershed The results show that each relative deviation between the simulated value and the observed value 408 is less than 15%, the time deviations of peak flow occurrence are less than 1 h, and all the NSEs are 409 greater than 0.8, which indicate that the simulated results of this model are reliable and reasonable.

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Therefore, the HEC-HMS hydrological model has a good application in this small watershed, and the 411 calibrated HEC-HMS hydrological model can be used to simulate the rainfall-runoff process.

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The peak ratio and peak position coefficient form the basis of the CPR and CPPC, respectively. enough RRPs are generated in each RRP set, there must be a rainfall whose rainfall pattern is the 443 highly similar to the RRP.

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Combined with the disaster flow, the CRs corresponding to the RRPs in Fig.8 were calculated 445 by the method in Section 2.3. In addition, taking into account the moisture of the soil, three ASMCs 446 were set to 0.2Wm (dry), 0.5Wm (general) and 0.8Wm (wet), respectively, as shown in Fig.9.  distributions. In cases where the CPPC is small (i.e., the proportion of rainfall in the initial stage of 462 the rainfall event is large), the peak rainfall is small after rainfall loss such as infiltration, filling, and     Table 6 501 and Table 7. The early warning information is shown in Fig.10.  As can be seen from Table 6, the CSI of RT35 is the smallest value out of the five RRPs, which 507 indicates that when the rainfall lasts for 2h, RT35 is most similar to the current rainfall. Then, the 2h 508 CR was calculated according to RT35, and the 2h accumulated rainfall of the current rainfall was 509 calculated, as shown in Fig.10. It can be seen that the 2h accumulated rainfall did not exceed the 2h 510 CR line, hence, there was no need to issue an early warning signal. As the current rainfall continues, 511 so does the ITP. When the current rainfall lasted for 3h (i.e., ITP=3h), the CSI of RT362 was the 512 smallest, indicating that RT362 was the most similar to the current rainfall. Combined with Fig.10, 513 the 3h accumulated rainfall of the current rainfall was compared with the 3h CR, and it was found 514 that at 9:10 a.m., the 3h accumulated rainfall exceeded the 3h CR, and the early warning signal was 515 immediately issued. Rainfall 20160701 from the measured data did cause a flash flood at 10:00 a.m., 516 and based on the early warning model, the early warning signal was issued at 9:10 a.m., increasing 517 the forecast lead time and leaving enough time for the transfer of people. results and analysis, the conclusions of this study can be summarized as follows.

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(1) The generation method of the RRP based on parameter control is a practical and simple 527 method, and it presents a novel idea and an approach for investigating the uncertainty relationship 528 between CR and rainfall pattern. The uncertainty of CR caused by the uncertainty of the rainfall 529 pattern can be effectively controlled by solving the critical rainfall threshold space under the RRP set.

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(2) The peak flow deviations are less than 15%, the time deviations of the peak flow occurrences 531 are less than 1 h, and all the NSEs are greater than 0.8, which proves that the application of the HEC-532 HMS model in the rainfall-runoff process simulation of a small watershed is reasonable and reliable.

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(3) Rainfall pattern has a greater effect on critical rainfall than the ASMC. The disaster scenario 534 in (0.5, bmax) is changeable, difficult to control, and has a high frequency of occurrence, so this rainfall belongs to the dangerous rainfall pattern.

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(4) The early warning model with dynamic correction based on RRP identification is effective rainfall was issued 50 minutes in advance, which avoided loss of life and property.

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It was assumed that the time periods are independent from each other in the random rainfall 542 pattern, and all of them are single-peak rainfall patterns. Thus, it does not have the capacity to include 543 each possible rainfall pattern of the region. Furthermore, the disaster mechanism of flash flood is 544 complex, and there are many influencing factors. The coupling effect of multiple factors on the 545 uncertainty response mechanism between the rainfall pattern and critical rainfall remains to be studied 546 further.

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All authors have seen and agreed with the contents of the manuscript and are looking forward to 549 publishing this paper on "Water Resources Management" journal. We certify that the submission is

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The authors declare that they have no known competing financial interests or personal 565 relationships that could have appeared to influence the work reported in this paper.

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The code that supports the findings of this study is available from the corresponding author upon 568 reasonable request.