Study on urban rainstorm pattern of short-duration double-peak

At present, researches on urban short-duration rainstorm patterns mainly focus on single-peak rainstorm patterns, and rarely involve double-peak rainstorm patterns, or convert double-peak patterns into single-peak patterns directly, even ignore the impact of double-peak patterns, which directly affects the urban flood planning and early warning and rescue. To scientifically and rationally deduce the urban short-duration double-peak rain pattern, this paper proposes a new function fitting rain pattern method by constructing double-peak virtual rain peak rainfall and virtual rain peak coefficient, based on the idea of convert double-peak to single-peak, then revert to the double-peak, directly deducing the double-peak rain pattern. The results show that (1) The rain pattern derived by the function fitting rain pattern method (FFRPM) can effectively improve the accuracy of the double-peak rain pattern and is also more practical; (2) The fitting degree of function fitting rainfall pattern and actual rain pattern is more than 90%, accounting for 80%, the fitting degree of main and secondary peak rainfall is more than 90%, with an average of about 95%; the accuracy of the main and secondary peak positions is also relatively high; (3) Compared with the P&C rain pattern method, whether the overall accuracy or local peak rainfall, the FFRPM has the higher accuracy, especially more accurate on rain peak rainfall.


Introduction
Urban waterlogging caused by the large-scale short-duration rainstorm can pose a serious threat to the safety of people's lives and property. According to China's 2020 statistics, there are 178.96 billion RMB (Jun et al. 2020) loss in property caused by urban rainstorm and 1 3 flood disasters. Usually, the short-duration rainstorm triggers urban waterlogging more easily. Short-duration rainstorm refers to a rainstorm within 120 min (Chunwei et al. 2015), and its rain pattern shows the temporal distribution process of rainfall, namely the rainstorm intensity changes with time (Khaled et al. 2015). Actually, the proportion of short-duration single-peak rainstorm is far higher than that of double-peak rainstorm, and it is with sudden rise and fall pattern, and more prone to cause flood disaster. Therefore, the characteristics of single-peak rainstorm, especially its rain pattern, are often the focus in the current study of urban storm flood, and currently many abundant research achievements are available and also are used to guide urban flood control and waterlogging planning and practice work (Meimei and Zening 2021). For example, by studying the relationship between rainfall intensity and duration, the Chicago rain pattern was analyzed and then proposed (Onof and Arnbjerg-Nielsen 2009), and the rainfall data of rainfall stations in central and eastern Illinois were analyzed and summarized, the time is divided into 4 types of rainfall (Huff 1967). And in 1975, the P&C method places the rain peak at the most possible position and takes the average ratio of the rain peak of each rainfall event to its total rain amount as the rain peak proportion, and the positions and proportions of rainfall in other periods are also determined by the same method, so the rainfall type is obtained through a sequential average method (Pilgrim and Cordery 1975). Proposed a single-peak triangle Shape rain pattern, used to calculate runoff in drainage area of small watershed (Chao et al. 2018).
However, affected by climate changes and urbanization development, some scholars found that the double-peak rainstorm occurred more frequently and even occupies about 20.5% of the short-duration rainstorms in China. So, the urban waterlogging caused by double-peak rainstorm cannot be ignored anymore, and it must be paid more attention (Arnell er al. 1984). Nevertheless, it is rare in the study on double-peak rainstorm now, and the existing research is simply to convert double-peak rainstorm into single-peak (Yuan-Yuan et al. 2020). This converting double-peak into single-peak method (CDPISPM) is simply and convenient with a certain precision (Wenjie et al. 2018). However, the double-peak rainfall and its location in a double-peak rainstorm play a crucial role in urban rainstorm warning and emergency response, while they are often fuzzed by the CDPISPM. So, the CDPISPM cannot truly reflect the characteristics of the actual double-peak rainstorm, which will reduce the accuracy of rainstorm forecasting, and also affect the urban drainage, waterlogging prevention and emergency early warning (Shao et al. 2020).
This study proposes a new FFRPM to fit the actual double-peak rain pattern method (DPRPM).
The paper aims to that (1) analyzing the characteristics of short-duration double-peak rainstorm in China firstly; (2) by constructing double-peak combined virtual rain peak coefficient (DPCVRPC), the double-peak rainstorm is converted into single-peak rainstorm firstly, and then they are reverted again, thus the FFRPM is derived; (3) by fitting the actual urban doublepeak rainstorm, the FFRPM is verified, and also compared with the P&C RPM. Through the above research, it will provide reference for the city's waterlogging risk analysis and rain pattern design and provide a scientific basis for flood control and drainage.

Data sources
Zhengzhou city lies in the North-Central part of Henan Province of China and also in the middle and lower reaches of the Yellow River. It belongs to the North temperate continental monsoon climate with four distinct seasons (Chi et al. 2015). The annual average of rainfall is about 542.2 mm with the variously seasonality in Zhengzhou City. The rainfall from June to September in wet season accounts for 65.7% of the whole year, from March to May in spring accounts for 19.5%, from October to November in autumn accounts for 10.3%, and from December in winter to February relatively less, only accounts for 4.5%. The annual average rainfall is more concentrated rainfall in July, and its value can attain to 151.5 mm which easier to cause serious urban waterlogging. However, in December, the rainfall is only 7.8 mm, and the monthly maximum rainfall is 19.4 times the minimum rainfall (Hongbo er al. 2007).
The study area is the five districts within Zhengzhou City, and date uses the 10-min rainstorm except from the ground hydrological station from 2009 to 2018, including Jinshui District, Huiji District, Erqi District, Zhongyuan District and Guancheng Hui Nationality District. There are 14 rainfall observation stations (as shown in Fig. 1 With the statistics of rainfall stations in Zhengzhou city, it is found that for the short-duration double-peak rainfall event, the number of rainfall range is 15-70 mm; the maximum rainfall intensity in 1 h is generally at 15-25 min or 35-45 min, and the main peak location easily occurs in 15-25 min; the maximum rainfall intensity in 1.5 h is generally at 15-35 min or 45-65 min; and at 30-50 min or 60-80 min for the maximum rainfall intensity in 2 h. While for the minimum rainfall intensity, it often appears in the first or the last period.

Short-duration rainstorm classification
According to the actual investigation and statistical data analysis, the rainstorm causing the flood disaster in Zhengzhou city is mainly the short-duration rainstorm within 2 h (Xuefeng and John 2019). Short-duration rainstorm refers to a rainstorm lasting no more than 120 min, and usually 30, 60, 90 and 120 min are used as the rainfall statistics periods. According to the 10-min observed rainstorm data from 2009 to 2018, the rainstorm events are classified by no rainfall or rainfall less than 0.1 mm within 2 h, thus a series of continuous rainfall data are classified into several independent rainstorm events (WenLin and Lei 2019). The classification of rainstorms events is shown in

Selection of double-peak rainstorm samples
According to the statistics of the short-duration double-peak rainstorm in Zhengzhou from 2009 to 2018, it is found that the double-peak rainstorm event in 0.5 h occurs 2 times, double-peak rainstorm events in 1 h happen 21 times, rainstorm events in 1.5 h appear 19 times, and rainstorm events in 2 h are 10 times. So, the double-peak rainstorms in Zhengzhou City are mainly concentrated within 1 h and 1.5 h. The double-peak rainfall event in 1.5 h lasts longer than 1 h, and its rainfall process is more complicated, so the rainstorm event in 1.5 h is selected to analyze (for the rainstorm event in 1 h, its analysis results are given in Sect. 3.4). Figure 2 show the main peak rain (MPR) and secondary peak rain (SPR) of double-peak rainstorm within 1.5 h as well as their locations.
From the data in Fig. 2, the characteristic values of double-peak rainstorms in 1.5 h in Zhengzhou city are calculated: the mean value of the MPR is 7.82 mm, the mean value of the SPR is 5.22 mm, the difference between MPR and SPR is 2.6 mm, and the proportion of the main peak in the front and the behind is 57.9% and 42.1%, respectively.
Based on the amount of statistical analysis of double-peak rainstorms (Diomede et al. 2008), it is known that in the double-peak rainstorm event, the value of SPR cannot be too small, otherwise the impact of SPR on the whole rain pattern will be lower that will result in unevidently double-peak rain. Similarly, the ratio of SPR to MPR as well as the difference between SPR and trough should reach a certain value so as to reflect the double-peak characteristics.
The selection standard (Qihui et al. 2018) of double-peak rainstorm is as follows: ① SPR ≥ 2.5 mm; ② The secondary peak accounts for at least 0.25 of the main peak amount. ③ The difference between the secondary peak and peak valley is more than 1.5 mm. Therefore, the double-peak rainstorm samples within 1.5 h can be obtaind by the following two steps: (1) classfing the short-duration rainstorm according to Table 1 to obtain the rainstorm data within 1.5 h firstly, and finding out all double-peak rainstroms, then arranging them in order of total rainfall from large to small; (2) with the selection standard through ①, ②, ③, the double-peak rainstorm samples are selected to derive the double-peak rainstorm pattern.

Double-peak virtual rain peak coefficient (DPVRPC)
For a single-peak rainstorm, its RPC (Dan et al. 2015) is calculated as follows: where i is the i-th rainstorm, i = 1, 2, … n, r i is the single-peak RPC, t i is the peak occuring time, and T is the total duration of the whole rainstorm. For double-peak rainstorm, its double rain peaks is divided into the MRP and the SRP, so if the formula (1) is used, there will be two rain peak coefficients. However, these two rain peaks can be converted into one peak with the formula (1) by the CDPISPM. The idea of the CDPISPM is to assume a virtual rain peak of single-peak rainstorm which is located between the MRP and the SRP of double-peak rainstorm, and also just at the position of the combined CRPC, thus the position of this virtual rain peak is called as the virtual rain peak cofficient of double-peak rainstorm.
Based on the above assumption, the double-peak virtrual rain peak cofficient (DPVRPC) is calculeted as: where r i is the DPVRPC, t i and t j are the occuring time of the MRP and SRP in doublepeak rainstorm, h i and h j are the MPR and SPR, respectively, and T is the total duration of rainstorm.

DPCVRPC
The DPCVRPC of double-peak rainstrom within 1.5 h is (Guoping et al. 1998) expressed as follows: where r j represents the DPCVRPC, n represents the total number of double-peak rainstorm within 1.5 h, and r i represents the DPVRPC of each double-peak rainstorm.
In the formula, n represents the number of time periods, and x k is the position corresponding to the comprehensive virtual rain peak coefficient (CVRPC).

Calculation of double-peak virtual peak rain (DPVPR)
For a double-peak rainstorm with the FMPR and behind BSPR, suppose the MPR is h i , the SPR is h j , and the DPVPR is H i satifing the following requirements (1)-(4): (1) After the conversion of double-peak to single-peak, the virtual rainfall peak value meets the following requirements: where y i and y j , respectively, represent the coefficients of MRP and SRP.
(2) The virtual rain peak rainfall should also satisfy the following requirements: where y i and y j satisfies 0 < a ≤ y i ≤ 1, 0 < b ≤ y j ≤ 1, a, b ∈ (0, 1).
If y i = y j = 1, the MRP and SRP are both at the position of the DPVRPC, thus the double-peak rainstorm pattern becomes the single-peak one.
(3) The DPVRPC is related to the position of MRP and SRP. If the position of the MRP x i is closer to the DPVRPC, the coefficient y i in formula (5) is more larger, and the relationship between Xi and Yi is linear. Similarly, if the position x j of the SRP is closer to the DPVRPC, the coefficient y j in formula (5) will be larger, and the relations between x j and y j are linear as well. (4) Set any initial value in the range of a and b, and then carry out linear fitting for the DPVPR and the total rainfall. If the best fitting is found and the DPVPR also satisfies the formula (6), the corresponding values of a and b at this time are the required values. (2) For a double-peak rainstorm with the front sencondary peak rain (FSPR) and BMPR, y i and y j represent the coefficients of SRP and MRP in formula (5), respectively, and x i and x j represent the positions of SRP and MRP, respectively.
Based on the above assumptions, it can be derived: (5) is calculated as: where n represents the number of time periods (for example, if the rain data extracted every 10 min in the rainstrom lasting 1.5 h, n is 9). The relations between y i , y j and x i , x j in the formula (7) and (8) are shown in Fig. 3. It can be seen from Fig. 3 that if the MRP is in front, y i ∈ [a, 1] and y j ∈ [b, 1]; conversely, if the MRP is in behind, y i ∈ [b, 1] and y j ∈ [a, 1], so the DPVPR can be derived as follows: In formulas (9) and (10), H i is the DPVPR, the rests are the same meaning as before. x i x j Fig. 3 The relations between y i , y j and x i , x j 2.6 Calculation of double-peak rainstorm pattern (CDPRPP)

P&C rain pattern method
① Divide each rainstorm into N periods, and then sort and number them. The larger rainfall corresponds to the smaller numbers, thus, each rainstorm event has N serial numbers; ② Take the average value of the serial number for each period, and sort them from small to large, then sort their correspoinding rainstorm intensity from large to small; ➂ Calculate the ratio of rainfall to total rainfall in each period, and then average them; ➃ The maximum possible sequence of peak rainfall location is determined by step ②, and determining the distribution rainfall proportion of each period is also known by step 3 (Xing et al. 2013). Thus, the rainfall process curve is achieved.

Function fitting rain pattern method
① The peak coefficient and location of the main peak and secondary peak of double-peak rainstorm were statistically analyzed, Calculate the mode and average of the main and secondary peak positions, and select the position with the best fit as the position of the main and secondary peaks in the rain pattern of the double-peak rainstorm design; ② The virtual peak coefficients of double-peak rainstorm pattern in all measured events are calculated by formula (2), and the comprehensive virtual peak coefficient of double-peak rainstorm pattern is obtained by formula (3). ➂ Take any initial value of coefficient a and b in (0, 1) through step (4) in 2.5, and substitute it into formula (9) and formula (10). The least square method is used to perform linear fitting for the peak rainfall and total rainfall of the double-peak virtual rain in all events, fields, when the fitting is optimal, the values of a and b are the desired values. ➃ Substituting the values of a and b into Eqs. (9) and (10), the calculation formula of virtual peak rainfall of double-peak rainstorm can be obtained, from which the virtual peak rainfall of double-peak rainstorm can be calculated. ➄ Perform linear fitting on the sum of virtual rain peak rainfall and main peak rainfall, secondary peak rainfall, main and secondary peak rainfall (MASPRA) of all the actual measured double-peak rainstorms to achieve the best, and substitute the fitting formula to obtain the fitted MASPRA. ➅ After subtracting the main and sub-peak rainfall from the total rainfall, the rainfall in the rest of the period is calculated by the P&C RPM.
It can be seen that the design rain pattern of double-peak rainstorm can be obtained by the steps ① determining the location of the main and sub-peak rainstorm, ②-⑤ determining the MASPRA and ⑥ determining the time-history distribution of the remaining rainfall.

Selection of rainfall envents
(1) Arrange the rainfall of 19 double-peak rainstorms in 1.5 h from large to small, numbered 1-19, respectively. (2) Calculate the corresponding frequency of each rainfall through hydrological empirical frequency formula (Chun and Dongming 2010) Eq. (11), find the frequency corresponding to each rainfall. Then use the formula of frequency and period (Mei et al. 2021) Eq. (12) to select the rainfall in the return period of 1a, 3a, 5a and 10a.
In formulas (11) and (12), P is the frequency, m is the rainfall number of the events, n is the total rainfall events, and T is the return period.
(3) Obtain the rainfall of 15 mm, 28.7 mm, 38 mm, 46 mm corresponding to the return period of 1a, 3a, 5a and 10a. Choose four rainstorms with similar or the same total rainfall to replace 1a, 3a, 5a, 10a, and select the total rainstorm results to be 15.5 mm, 28 mm, 38 mm, 46 mm.

Determination of main and secondary peak locations
The positions of the main peak and the secondary peak during the 1.5 h double-peak rainstorm are shown in Fig. 4. In Fig. 4, in the first 11 double-peak rainstorms, the relative position of the main and secondary peaks is that the main peak is in the front, and the secondary peak is in the back, while in the second 12-19 double-peak rainstorms, the relative position of the main and secondary peaks is that the main peak is in the back, and the secondary peak is in the front. By calculating the data in Fig. 4 that when the main peak is in the front, the average positions of the main peak and the secondary peak are 2.6 and 6.6, respectively, and the positions where the main peak and the secondary peak appear most are 2 and 6; when the main peak is in the back, the average positions of the main peak and the secondary peak are 2.1 and 5.4, respectively, and the positions where the main peak and the secondary peak appear most are 6 and 2. Taking into account the CVRPC and the fitting degree of rain pattern, the positions of the main and secondary peak of the double-peak are 2 and 6 when the main peak is in the front and the secondary peak is in the back, and 5 and 2 when the main peak is in the back and the secondary peak is in the front.

Double-peak CVRPC
According to Eqs. (2) and (3), the virtual rain peak coefficients and comprehensive virtual rain peak coefficients of all 1.5 h measured double-peak rainstorms are obtained. In order to facilitate comparison, the rain peak coefficients of all 1.5 h measured single-peak rainstorms are also calculated. The rain peak coefficients of 1.5 h measured single-peak and double-peak rainstorms are shown in Fig. 5 and Table 2 below.
As shown in Table 2, the RPC of 1.5 h single-peak rainstorm is very similar to that of double-peak rainstorm. The proportion of RPC ≤ 0.5 is relatively high, and the peak rainfall is mainly in front, and the difference between the two comprehensive rain peak coefficient (CRPC) is not significant.

Calculation of double-peak virtual peak rainfall
(1) Take any value in (0,1) for a and b, and then put them into formula (9) and (10) to calculate the virtual peak rainfall, and then make linear fitting between the virtual peak rainfall and the total rainfall to make the best fitting. At this time, a = 0.5 and b = 0.5, as shown in Fig. 6. (2) Substitute a = 0.5 and b = 0.5 into Eqs. (9) and (10), and r j = 0.39 into Eq. (4) to get the rainfall and the position of virtual rain peak.
The main and secondary peak coefficient formula: Therefore: When the main peak is in front, the peak rainfall of the virtual rain peak is: When the main peak is behind, the peak rainfall of the virtual rain peak is: According to formulas (18) and (19), the virtual peak rainfall of the double-peak rainstorm can be calculated in 1.5 h for 19 envents, and the peak rainfall of the singlepeak rainstorm in 1.5 h for 21 envents can be compared. The results are shown in Fig. 7 and Table 3.
It can be seen from Fig. 7 and Table 3 that the peak rainfall of the single-peak and double-peak rainstorms during the 1.5 h field measurement are in good agreement, whether it is the average value, median or sample standard deviation, the peak rainfall of double-peak and single-peak. The result of peak rainfall is very close. 3.5, 9], y j = −0.0909x j + 1.3182 Virtual peak rainfall (mm) Total rainfall(mm) Fig. 6 Relationship between total rainfall and virtual peak rainfall

Calculation of MASPRA
The best linear fit is performed on the sum of the virtual peak rainfall and the secondary peak rainfall, the main peak rainfall, the MASPRA of all the measured double-peak rainstorms, and the results are shown in Fig. 8a-c. The rain peak-correlation coefficient is the worst at 0.7662, so the MASPRA can be obtained according to the relationship between total rainfall-virtual rain peak, main peak virtual rain peak, major and secondary peaks and virtual rain peak.

Rainfall distribution in other periods
After subtracting the MASPRA from the total rainfall, the rainfall during the rest of the period is calculated by the P&C RPM (Jingming 2017), and finally, the distribution ratios of the main peak before and after the main peak of the double-peak rainstorm can be obtained, respectively, such as Tables 4 and 5.

Fitting rain pattern results
Selecting four double-peak rainstorms with the return period of 1a, 3a, 5a, and 10a as examples, the compared results of fitting rain pattern and actual rain pattern are shown in Fig. 9. Rainfall(mm)

Rainfall events
Single-peak Double-peak Fig. 7 Comparison of 1.5 h measured Single-peak and Double-peak rainfall

P&C rain pattern method
According to the obtained 1.5 h double-peak rainstorm data, and apply with the P&C method, the rainfall proportion results of each period within 1.5 h double-peak rainstorm can be obtained, as shown in Table 6. Thus, the rain pattern time distribution of the four double-peak rainstorms with the different return periods of 1a, 3a, 5a and 10a can be achieved. Figure 10 is the comparsion of the P&C rain pattern and actual rain pattern with the different return periods below. Total rainfall(mm) Fig. 8 The relationship between Main peak or secondary peak or total peak with virtual rain peak

Overall rain pattern fitting analysis
Using the fuzzy recognition method (Srivastava et al. 2019), the fitting degree of the rainstorm patterns and the corresponding measured rainstorm pattern can be calculated as fourmula (17).
where e is the overall rain pattern fitting degree, x i is the distribution ratio for the i-th period of the measured rainstorm, y i is the distribution ratio for the i-th period of the corresponding P&C rain pattern, i = 1, 2, …, n. Fitting menthod Actual value Fig. 9 Comparison of the fitted rain pattern and the actual rain pattern with the different return period According to formula (17) and P&C method, for the return periods of 1a, 3a, 5a and 10a, their overall rain pattern fitting degrees are 0.934, 0.961, 0.940 and 0.923, respectively. Larger value of fitting degreee means the P&C RPM is more avaiable, so from Table  @@10, it can be seen that the P&C RPM is relatively good for the overall rain pattern fit for the double-peak rainstorm with the return period of 1a, 3a, 5a and 10a.

Main and secondary peak analysis
The main and secondary peak fitting formula (18) is used to calculate and analyze the main and secondary peak errors of the four rainstorm types, and the formula is as follows: In formula (18), x 1 and x 2 , respectively, represent the proportion of the main peak rainfall, and the secondary peak rainfall calculated by the P&C rain pattern to the total rainfall, and y 1 and y 2 , respectively, represent the proportion of the main peak rainfall and the secondary peak rainfall of the measured rainfall in the total rainfall. The larger the value of e , the higher the degree of fitting, so as to verify the accuracy and rationality of the fitting of P&C rain patterns on the local main and secondary peaks.
According to formula (18) and P&C method, for the return periods of 1a, 3a, 5a and 10a, their main and secondary peaks fitting degree are 0.926, 0.942, 0.736 and 0.804, respectively.
It can be seen that for the double-peak rainstorm with the return period of 1a and 3a, the P&C RPM fits better in the main and secondary peaks, but for the double-peak rainstorm with the return period of 5a and 10a, the P&C RPM is in the main. There is a big gap between the secondary peak fit degree and the true value, and the calculated main peak is only about half of the actual value; and the P&C RPM cannot distinguish between the main peak before and the main peak after.

Analysis of overall rainfall
Using formula (19) to calculate and analyze the overall rainfall difference of the four rainstorm patterns, the results are as follows in Table 7, and the formula is as follows: In formula (19), m is the total rainfall difference, n is the number of periods, h i is the rainfall obtained in each period of the P&C method, and H i is the actual measuredrainfall in each period.

Analysis of the results of FFRPM
Through the comparison and analysis with the measured rainstorm data, the calculation results of the fitting degree of the overall rain pattern, the fitting degree of the main and secondary peaks, and the difference of total rainfall can be obtained as shown in Table 8 For the return periods of 1a, 3a, 5a and 10a, according to formula (17) and fitting method, their overall rain pattern fitting degrees are 0.936, 0.968, 0.838 and 0.962, respectively. While their main and secondary peaks fitting degrees are 0.939, 0.987, 0.916 and 0.962 with formula (18) and fitting method, respectively.

Comparison of results between FFRPM and P&C RPM
From Table 8, it can be seen that for the double-peak rainstorms with the return period of 1a and 3a, the P&C RPM has a better fit between the main and secondary peaks, but for the double-peak rainstorm with return period of 5a and 10a, the P&C RPM has a large gap between the main and secondary peaks and the true value, and the calculated main peak is only about half of the actual value; And the P&C RPM cannot distinguish between the main peak in the front and the main peak in the back. It can be seen from Table 8 that the P&C RPM only has a small total rainfall difference in 3a, the total rainfall difference between 1 and 10a is relatively large, and the 5a total rainfall deviation is the largest, with a deviation ratio of 42.6%.
From Table 8, it can be seen that the fitting rain pattern method is superior to P&C method in three events of the overall rain pattern. Except for the rainstorm with a recurrence period of 5a, the fitness of the main and secondary peaks are higher than that of P&C method. Because FFRPM divides double-peak rainstorm into two types, i.e., the front of main peak and the back of main peak, the calculation results are more realistic than P&C method which does not distinguish the position of main and secondary peak. On the total rainfall difference, the difference between the total rainfall of 3a and 10a is smaller, the difference between the total rainfall of 1a is larger, and the difference between the total rainfall of 5a is the largest, and the comparison results are similar to P&C method. It can be seen that the FFRPM improves the fitness of main and secondary peaks greatly and is more accurate in the similarity of the overall rain pattern, which is more in line with the actual situation.

Deducing results of all double-peak rainstorms with short-duration in Zhengzhou City
The short-duration double-peak rainstorms were mainly concentrated within 1 h and 1.5 h, there were a total of 20 double-peak rainstorms within 1 h, accounting for 23%, and 19 double-peak rainstorms within 1.5 h, accounting for 47.5%. In order to more clearly verify the practicability of the FFRPM proposed in this paper, P&C method and FFRPM are, respectively, used to calculate and compare all 1 h and 1.5 h double-peak rainstorms in Zhengzhou City the calculation steps are the same as above.
(1) The fitting degree of rain pattern and the main and secondary peak fitting degree of double-peak rainstorm 1 h 20 rainstorms were analyzed, and the results are as follows: Fig. 11 a and b. (2) Analyze the fit degree of the double-peak rainstorm pattern and the fit degree of the main and secondary peaks during 1.5h19 in Zhengzhou City, and the results are shown in Fig. 12 a and b (Table 9). It can be seen from Fig. 11a and b, the overall rain pattern fitting degree of 20 doublepeak rainfall within 1 h shows that the fitting degree of 14 of 20 double-peak rainfall events is greater than that of P&C, and only 6 events are less than that of P&C; In the fitting degree of local main and secondary peak rain patterns, the fitting method was larger than P&C method in 18 of 20 events, and only 2 events were smaller than that of P&C method; It can be seen from Fig. 12a and b, for 1.5 h 19 double-peak rainfall overall rain pattern fit, 11 of 19 overall rain pattern fits are greater than the P&C method, and 8 are less than the P&C method; For the fitting degree of local primary and secondary peak rain patterns, the fitting method was larger than P&C method in 18 of 19 events, and only 1 event was smaller than P&C method; On the whole, whether it is 1 h or 1.5 h double peaks, the fitting method is better than the P&C method in the degree of fit between the primary and secondary peaks; It can be seen from Table 10 that the fitting degree of overall rainfall pattern, main and secondary peak rainfall pattern in 1H is greater than that of P% C method, the fitting degree of main and secondary peak rain patterns is improved more, and the main and secondary peak rain pattern fitting degree of 1.5 h is slightly lower than that of the P&C method, but the main and secondary peak rain pattern fitting degree are also greatly  improved. The fitting method is superior to the P & C method in 1 and 1.5 h for the overall rain pattern fitting degree.

Discussion
The fitting rain pattern method proposed in this paper is based on the characteristics of short-duration double-peak rainstorm patterns and the characteristics of P&C rain pattern. The advantage of this method is that the rainfall of the main peak and the secondary peak in the double-peak can be calculated very accurately, at the same time, the rest of the rainfall also maintains a higher accuracy, so that the overall rain pattern can maintain a higher similarity.
In order to better explain the applicability of FFRPM, this paper takes P&C rain pattern method as an example to compare and analyze the differences between them., and the rain pattern of rainfall is always unchanged, which is quite inconsistent with the actual short-duration rainstorm in the city (Shufang et al. 2018). Compared with P&C rain pattern, the precision of fitting rain pattern, main and secondary peak rainfall and location are improved, especially the precision of MASPRA is improved more. Urban flood disasters are often caused by the large and concentrated quantity of main and secondary peaks. This method can effectively estimate the peak quantity, which is of great significance to urban flood control and waterlogging prevention.
For rainfall with a return period of 3a, it can be seen that in the FFRPM, the MASPRA differs greatly from the measured rainfall. The main reason is that the locations of the main and secondary peaks are not accurate enough. The measured locations are 3 and 7, while the deduced locations are 2 and 6, there is a big difference between them. At present, no matter the research of double-peak rain pattern or single-peak rain pattern, it is impossible to accurately determine the location of the rain peak of each rainstorm.
For the rainfall with a return period of 1a, it can be seen that in the measured rainfall, there are three identical rainfall locations 1, 4 and 5 (all peak rainfall is 6 mm), and there should be a higher peak between 4 and 5. However, due to the low precision of data, the main peak location can only be set as 4.5, and 6 mm for the peak rainfall. This leads to certain errors in the location of the rain peak and the rainfall of rain peak, and the final result is not accurate enough. This study is based on the 10-min rainstorm extract data of Zhengzhou City. If there are 5-min or 1-min rainstorm extract data, it is believed that the simulation result will be more accurate.

Conclusions
(1) Based on the idea of CDPISPM and then reverting to double-peak, the proposed FFRPM can be used to calculate the DPRPM with short-duration in Zhengzhou City.
(2) The short-duration double-peak rainstorms mainly concentrated within 1 h and 1.5 h.
The comprehensive peak coefficient of double-peak rainstorms is 0.3918 by using the FFRPM, indicating that the rainfall of double-peak rainstorms mainly concentrated in the first half, which is consistent with the performance of single-peak.
(3) Compared with the traditional P&C method, the fitting degree of the overall rain pattern of the fitting method is 0.75-1, and the fitting degree of the main and secondary peak rainfall is 0.9-1, and the location accuracy of the MASPRA is also higher.
(4) Using the proposed fitting rain pattern method, the most of main and secondary rainfall peaks is fitted well. Combined with the P&C method, it makes the designed rain pattern to reveal the actual rain process more appropriately. Moreover, if the interval of rainstorm periods can be shorten, the location of the rain peak maybe more accurately.